calculo e detalhamento de pilares de concreto armado
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7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 127
UNIVERSIDADE DO ESTREMO SUL CATARINENSE ndash UNESC
CURSO DE ENGENHARIA CIVIL
FERNANDO EBERHARDT DE OLIVEIRA
CALCULO E DETALHAMENTO DE PILARES
CRICIUacuteMA DEZEMBRO DE 2012
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 227
FERNANDO EBERHARDT DE OLIVEIRA
CALCULO E DETALHAMENTO DE PILARES
Trabalho apresentado como requisito parcialpara aprovaccedilatildeo na disciplina de ConcretoArmado II da 8ordm fase do curso de EngenhariaCivil da Universidade do Extremo SulCatarinense ndash UNESC
Professora Daiane dos Santos da Silva
CRICIUacuteMA DEZEMBRO DE 2012
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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1 DADOS DO TRABALHO
Deve-se dimensionar e detalhar a armadura transversal e longitudinal dos
pilares P3 P5 e P6 da planta de formas da paacutegina 4
Os dados satildeo os seguintes
bull Concreto C25
bull Accedilo CA50
bull concreto = 25 KNmsup3
bull Cobrimento miacutenimo 25cm
bull drsquo = 40 cm
bull Laje maciccedila espessura 15 cm
bull Diacircmetro do estribo 50mm
bull αb = 10
bull lef = 280 cm
Abaixo tabela com os dados das barras de accedilos
Oslash (mm) As (cmsup2) Peso (Kgfm)
50 020 016
63 032 025
80 050 040
100 080 063
125 125 100
160 200 160
200 315 247
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 427
2 CARGAS NOS PILARES
21 Descarregamento das lajes
Cargas atuantes nas lajes
Enchimento 70 kgfmsup2Revestimento 50 kgfmsup2Carga Acidental 650 kgfmsup2Peso Proacuteprio (015m 2500 kgfmsup3) 375 kgfmsup2Total 1145 kgfmsup2Total 1145 kNmsup2
Planta baixa do pavimento
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 527
Descarregamento das lajes nas vigas
22 Caacutelculo das vigas
V101
Tramo 01
983153 = 983233983154983141983137 C983137983154983143983137 = 1439 983149983218 1145 983147N983149983218L 983140983137 983158983145983143983137 8081 983149
983153 983101 203983097 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 070 983149 983160 25 983147N = 21 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 627
Tramo 02
983153 = 786 983149983218 1145 983147N983149983218
556 983149
983153 983101 1617 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 18 983147N983149
Cargas totais na viga
Reaccediloes nos apoios
V102
Tramo 01
983153 = 1439 983149983218 1145 983147N983149983218 + 1566 983149983218 1145 983147N983149983218
7916 983149 7916 983149
983153 983101 4347 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 015 983149 983160 070 983149 983160 25 983147N = 263 983147N983149
Tramo 02
983153 = 786 983149983218 1145 983147N983149983218 + 792 983149983218 1145 983147N983149983218
5554 983149 5554 983149
983153 983101 3253 983147983118983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 727
983120983141983155983151 983152983154983283983152983154983145983151 015 983149 983160 070 983149 983160 25 983147N = 263 983147N983149
Cargas totais na viga
Reaccediloes nos apoios
V105
Tramo 01
983153 = 659 983149983218 1145 983147N983149983218 + 659 983149983218 1145 983147N9831499832185095 983149 5095 983149
983153 983101 298309762 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 180 983147N983149
Tramo 02
983153 = 941 983149983218 1145 983147N983149983218 + 935983149983218 1145 983147N9831499832186095 983149 6095 983149
983153 983101 3524 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 180 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 827
Cargas totais na viga
Reaccediloes nos apoios
V106
Tramo 01
983153 = 935 983149983218 1145 983147N983149983218
6095 983149
983153 983101 1756 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 050 983149 983160 25 983147N = 150 983147N983149
Tramo 02
983153 = 659 983149983218 1145 983147N9831499832185095 983149
983153 983101 149830961 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 050 983149 983160 25 983147N = 150 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 927
Cargas totais na viga
Reaccediloes nos apoios
23 Cargas nos pilares
P3
Reaccedilatildeo da Viga 101 2536 kNReaccedilatildeo da Viga 106 2736 kNPeso Proacuteprio do Pilar(020 m 025 m 280 m 25 kNmsup3) 350 kNTotal 5622 kN
P5
Reaccedilatildeo da Viga 102 36225 kN
Reaccedilatildeo da Viga 105 24341 kNPeso Proacuteprio do Pilar(025 m 025 m 280 m 25 kNmsup3) 438 kNTotal 61004 kN
P6
Reaccedilatildeo da Viga 102 4937 kNReaccedilatildeo da Viga 106 12569 kNPeso Proacuteprio do Pilar(020 m 040 m 280 m 25 kNmsup3) 560 kNTotal 18066 kN
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1027
3 CALCULO DO PILAR DE EXTREMIDADE ldquoP6rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 18066
983118983140 983101 2529830972 983147983118
Vatildeo efetivo da viga
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 le05 9831561 = 05 25 = 125 983139983149
9831371 983101 125 98313998314905 983144 = 05 70 = 350 983139983149
9831372 le05 9831562 = 05 40 = 200 983139983149
9831372 983101 200 98313998314905 983144 = 05 70 = 350 983139983149
983148983141983142 = 5229 + 125 + 20
983148983141983142 983101 5554 983139983149
Momentos de ligaccedilatildeo viga-pilar
983122983155983157983152 = 983122983145983150983142 =I
=(983138 983144983219) 12
=(20 40983219) 12
= 7619830970 983139983149983219983148983141 983148983141 2 280 2
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1127
983122983158983145983143983137 =I983158983145983143983137
=(15 70983219) 12
= 7719830977 983139983149983219983148983141983142 5554
Momento de engastamento
M983141983150983143 =983120 L983218
=3516 5554983218
= 98309703983096 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 9038 3 76190
3 76190 + 4 77197 + 3 76190
983117983155983157983152 983101 983117983145983150983142 983101 269830977 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 2697 + 2697
983117983158983145983143983137 983101 539830974 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147M983140983156983151983152983151 = 14 10 2697
983117983140983156983151983152983151 983101 3776 983147983118983149 983117983140983138983137983155983141 983101 9830853776 983147983118983149
Excentricidade inicial no topo e na base
983141983145 = M983140 N983140
983141983145 = 3776 983147N983139983149 25292 983147N
983141983145 983101 149830974 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1227
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 25292 (0015 + 003 04)
9831171983140983149983277983150 983101 69830963 983147983118983149
Iacutendice de esbeltes
λ = (346 983148983141) 983144
λ = (346 280) 40
λ 983101 2422
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 40)
10λ1 = 25
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λ = 2422 eacute menor que λ1 = 35 natildeo se considera efeitos de segunda ordem
assim adota-se M1d = 3776 kNm = 3776 kNcm
Coeficientes adimensionais
η =N983140
=25292
= 018983138 983144 983142983139983140 20 40 (2514)
983140=
4= 01
983144 40
μ = η 983141983156983151983156983137983148 = 018 1494 = 007983144 40
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1327
Utilizando o aacutebaco A-25 de VENTURINI (1987) obtecircm-se ω = 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 40 (2514) 005
= 164 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 40 = 64 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 40 = 32 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
983119B983123 A983140983151983156983137983154 A983155 = 32 983139983149983218
Detalhamento
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1427
Portanto o pilar ldquoP6rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
4 CALCULO DO PILAR DE CANTO ldquoP3rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147N983140 = 14 10 5622
983118983140 983101 798309671 983147983118
Vatildeo efetivo da viga
bull Eixo ldquoxrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1527
9831371 le05 9831561 = 05 30 = 150 983139983149
9831371 983101 150 98313998314905 983144 = 05 60 = 300 983139983149
9831372 le
05 9831562 = 05 25 = 125 983139983149
9831372 983101 125 98313998314905 983144 = 05 60 = 300 983139983149
983148983141983142 = 5289 + 125 + 15
983148983141983142 983101 5564 983139983149
bull Eixo ldquoyrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 = 9831372 le05 983156 = 05 20 = 100 983139983149
9831371 983101 9831372 983101 100 98313998314905 983144 = 05 50 = 250 983139983149
983148983141983142 = 4895 + 100 + 100
983148983141983142 983101 509830975 983139983149
Momentos de ligaccedilatildeo viga-pilar
bull
Eixo ldquoxrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(20 25983219) 12
= 1983096601 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 60983219) 12
= 398309698309621 983139983149983219983148983141983142 5564
bull Eixo ldquoyrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(25 20983219) 12
= 1198309705 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 50983219) 12
= 24534 983139983149983219983148983141983142 5095
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1627
Momento de engastamento
bull Eixo ldquoxrdquo
M983141983150983143 =983120 L983218
=1797 5564983218
= 4636 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 4636 3 18601
3 18601 + 4 38821 + 3 18601
983117983155983157983152 983101 983117983145983150983142 983101 9830976983097 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 969 + 969
983117983158983145983143983137 983101 19830973983096 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 969
983117983140983156983151983152983151 983101 1357 983147983118983149 983117983140983138983137983155983141 983101 9830851357 983147983118983149
bull Eixo ldquoyrdquo
M983141983150983143 =983120 L983218
=1631 5095983218
= 352983096 98314798311898314912 12
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1727
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 3528 3 11905
3 11905 + 4 24534 + 3 11905
983117983155983157983152 983101 983117983145983150983142 983101 743 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 743 + 743
983117983158983145983143983137 983101 149830966 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 743
983117983140983156983151983152983151 983101 1040 983147983118983149 983117983140983138983137983155983141 983101 9830851040 983147983118983149
Excentricidade inicial no topo e na base
Eixo ldquoxrdquo
983141983145 = M983140 N983140
983141983145 = 1357 983147N983139983149 7871 983147N
983141983145 983101 1724 983139983149
Eixo ldquoyrdquo
983141983145 = M983140 N983140
983141983145 = 1040 983147N983139983149 7871 983147N
983141983145 983101 1321 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1827
Momento miacutenimo
Eixo ldquoxrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 025)
9831171983140983149983277983150 983101 177 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1357 983147N983149
Eixo ldquoyrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 020)
9831171983140983149983277983150 983101 165 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1040 983147N983149
Iacutendice de esbeltes
Eixo ldquoxrdquo
λ = (346 983148983141) 983144
λ = (346 280) 25
λ 983101 398309675
Eixo ldquoyrdquo
λ = (346 983148983141) 983144
λ = (346 280) 20
λ 983101 498309644
Iacutendice de esbeltes limite
Eixo ldquoxrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1724 25)
10λ1 = 3362
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Eixo ldquoyrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1321 20)
10λ1 = 3326
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
Eixo ldquoxrdquo
η =N983140
=7871
= 009983138 983144 983142983139983140 20 25 (2514)
1
=
0005
le
0005
983154 983144 (η + 05) 983144
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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FERNANDO EBERHARDT DE OLIVEIRA
CALCULO E DETALHAMENTO DE PILARES
Trabalho apresentado como requisito parcialpara aprovaccedilatildeo na disciplina de ConcretoArmado II da 8ordm fase do curso de EngenhariaCivil da Universidade do Extremo SulCatarinense ndash UNESC
Professora Daiane dos Santos da Silva
CRICIUacuteMA DEZEMBRO DE 2012
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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1 DADOS DO TRABALHO
Deve-se dimensionar e detalhar a armadura transversal e longitudinal dos
pilares P3 P5 e P6 da planta de formas da paacutegina 4
Os dados satildeo os seguintes
bull Concreto C25
bull Accedilo CA50
bull concreto = 25 KNmsup3
bull Cobrimento miacutenimo 25cm
bull drsquo = 40 cm
bull Laje maciccedila espessura 15 cm
bull Diacircmetro do estribo 50mm
bull αb = 10
bull lef = 280 cm
Abaixo tabela com os dados das barras de accedilos
Oslash (mm) As (cmsup2) Peso (Kgfm)
50 020 016
63 032 025
80 050 040
100 080 063
125 125 100
160 200 160
200 315 247
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2 CARGAS NOS PILARES
21 Descarregamento das lajes
Cargas atuantes nas lajes
Enchimento 70 kgfmsup2Revestimento 50 kgfmsup2Carga Acidental 650 kgfmsup2Peso Proacuteprio (015m 2500 kgfmsup3) 375 kgfmsup2Total 1145 kgfmsup2Total 1145 kNmsup2
Planta baixa do pavimento
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Descarregamento das lajes nas vigas
22 Caacutelculo das vigas
V101
Tramo 01
983153 = 983233983154983141983137 C983137983154983143983137 = 1439 983149983218 1145 983147N983149983218L 983140983137 983158983145983143983137 8081 983149
983153 983101 203983097 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 070 983149 983160 25 983147N = 21 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Tramo 02
983153 = 786 983149983218 1145 983147N983149983218
556 983149
983153 983101 1617 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 18 983147N983149
Cargas totais na viga
Reaccediloes nos apoios
V102
Tramo 01
983153 = 1439 983149983218 1145 983147N983149983218 + 1566 983149983218 1145 983147N983149983218
7916 983149 7916 983149
983153 983101 4347 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 015 983149 983160 070 983149 983160 25 983147N = 263 983147N983149
Tramo 02
983153 = 786 983149983218 1145 983147N983149983218 + 792 983149983218 1145 983147N983149983218
5554 983149 5554 983149
983153 983101 3253 983147983118983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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983120983141983155983151 983152983154983283983152983154983145983151 015 983149 983160 070 983149 983160 25 983147N = 263 983147N983149
Cargas totais na viga
Reaccediloes nos apoios
V105
Tramo 01
983153 = 659 983149983218 1145 983147N983149983218 + 659 983149983218 1145 983147N9831499832185095 983149 5095 983149
983153 983101 298309762 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 180 983147N983149
Tramo 02
983153 = 941 983149983218 1145 983147N983149983218 + 935983149983218 1145 983147N9831499832186095 983149 6095 983149
983153 983101 3524 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 180 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Cargas totais na viga
Reaccediloes nos apoios
V106
Tramo 01
983153 = 935 983149983218 1145 983147N983149983218
6095 983149
983153 983101 1756 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 050 983149 983160 25 983147N = 150 983147N983149
Tramo 02
983153 = 659 983149983218 1145 983147N9831499832185095 983149
983153 983101 149830961 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 050 983149 983160 25 983147N = 150 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Cargas totais na viga
Reaccediloes nos apoios
23 Cargas nos pilares
P3
Reaccedilatildeo da Viga 101 2536 kNReaccedilatildeo da Viga 106 2736 kNPeso Proacuteprio do Pilar(020 m 025 m 280 m 25 kNmsup3) 350 kNTotal 5622 kN
P5
Reaccedilatildeo da Viga 102 36225 kN
Reaccedilatildeo da Viga 105 24341 kNPeso Proacuteprio do Pilar(025 m 025 m 280 m 25 kNmsup3) 438 kNTotal 61004 kN
P6
Reaccedilatildeo da Viga 102 4937 kNReaccedilatildeo da Viga 106 12569 kNPeso Proacuteprio do Pilar(020 m 040 m 280 m 25 kNmsup3) 560 kNTotal 18066 kN
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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3 CALCULO DO PILAR DE EXTREMIDADE ldquoP6rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 18066
983118983140 983101 2529830972 983147983118
Vatildeo efetivo da viga
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 le05 9831561 = 05 25 = 125 983139983149
9831371 983101 125 98313998314905 983144 = 05 70 = 350 983139983149
9831372 le05 9831562 = 05 40 = 200 983139983149
9831372 983101 200 98313998314905 983144 = 05 70 = 350 983139983149
983148983141983142 = 5229 + 125 + 20
983148983141983142 983101 5554 983139983149
Momentos de ligaccedilatildeo viga-pilar
983122983155983157983152 = 983122983145983150983142 =I
=(983138 983144983219) 12
=(20 40983219) 12
= 7619830970 983139983149983219983148983141 983148983141 2 280 2
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983122983158983145983143983137 =I983158983145983143983137
=(15 70983219) 12
= 7719830977 983139983149983219983148983141983142 5554
Momento de engastamento
M983141983150983143 =983120 L983218
=3516 5554983218
= 98309703983096 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 9038 3 76190
3 76190 + 4 77197 + 3 76190
983117983155983157983152 983101 983117983145983150983142 983101 269830977 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 2697 + 2697
983117983158983145983143983137 983101 539830974 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147M983140983156983151983152983151 = 14 10 2697
983117983140983156983151983152983151 983101 3776 983147983118983149 983117983140983138983137983155983141 983101 9830853776 983147983118983149
Excentricidade inicial no topo e na base
983141983145 = M983140 N983140
983141983145 = 3776 983147N983139983149 25292 983147N
983141983145 983101 149830974 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1227
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 25292 (0015 + 003 04)
9831171983140983149983277983150 983101 69830963 983147983118983149
Iacutendice de esbeltes
λ = (346 983148983141) 983144
λ = (346 280) 40
λ 983101 2422
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 40)
10λ1 = 25
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λ = 2422 eacute menor que λ1 = 35 natildeo se considera efeitos de segunda ordem
assim adota-se M1d = 3776 kNm = 3776 kNcm
Coeficientes adimensionais
η =N983140
=25292
= 018983138 983144 983142983139983140 20 40 (2514)
983140=
4= 01
983144 40
μ = η 983141983156983151983156983137983148 = 018 1494 = 007983144 40
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1327
Utilizando o aacutebaco A-25 de VENTURINI (1987) obtecircm-se ω = 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 40 (2514) 005
= 164 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 40 = 64 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 40 = 32 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
983119B983123 A983140983151983156983137983154 A983155 = 32 983139983149983218
Detalhamento
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1427
Portanto o pilar ldquoP6rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
4 CALCULO DO PILAR DE CANTO ldquoP3rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147N983140 = 14 10 5622
983118983140 983101 798309671 983147983118
Vatildeo efetivo da viga
bull Eixo ldquoxrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1527
9831371 le05 9831561 = 05 30 = 150 983139983149
9831371 983101 150 98313998314905 983144 = 05 60 = 300 983139983149
9831372 le
05 9831562 = 05 25 = 125 983139983149
9831372 983101 125 98313998314905 983144 = 05 60 = 300 983139983149
983148983141983142 = 5289 + 125 + 15
983148983141983142 983101 5564 983139983149
bull Eixo ldquoyrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 = 9831372 le05 983156 = 05 20 = 100 983139983149
9831371 983101 9831372 983101 100 98313998314905 983144 = 05 50 = 250 983139983149
983148983141983142 = 4895 + 100 + 100
983148983141983142 983101 509830975 983139983149
Momentos de ligaccedilatildeo viga-pilar
bull
Eixo ldquoxrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(20 25983219) 12
= 1983096601 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 60983219) 12
= 398309698309621 983139983149983219983148983141983142 5564
bull Eixo ldquoyrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(25 20983219) 12
= 1198309705 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 50983219) 12
= 24534 983139983149983219983148983141983142 5095
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1627
Momento de engastamento
bull Eixo ldquoxrdquo
M983141983150983143 =983120 L983218
=1797 5564983218
= 4636 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 4636 3 18601
3 18601 + 4 38821 + 3 18601
983117983155983157983152 983101 983117983145983150983142 983101 9830976983097 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 969 + 969
983117983158983145983143983137 983101 19830973983096 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 969
983117983140983156983151983152983151 983101 1357 983147983118983149 983117983140983138983137983155983141 983101 9830851357 983147983118983149
bull Eixo ldquoyrdquo
M983141983150983143 =983120 L983218
=1631 5095983218
= 352983096 98314798311898314912 12
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1727
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 3528 3 11905
3 11905 + 4 24534 + 3 11905
983117983155983157983152 983101 983117983145983150983142 983101 743 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 743 + 743
983117983158983145983143983137 983101 149830966 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 743
983117983140983156983151983152983151 983101 1040 983147983118983149 983117983140983138983137983155983141 983101 9830851040 983147983118983149
Excentricidade inicial no topo e na base
Eixo ldquoxrdquo
983141983145 = M983140 N983140
983141983145 = 1357 983147N983139983149 7871 983147N
983141983145 983101 1724 983139983149
Eixo ldquoyrdquo
983141983145 = M983140 N983140
983141983145 = 1040 983147N983139983149 7871 983147N
983141983145 983101 1321 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1827
Momento miacutenimo
Eixo ldquoxrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 025)
9831171983140983149983277983150 983101 177 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1357 983147N983149
Eixo ldquoyrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 020)
9831171983140983149983277983150 983101 165 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1040 983147N983149
Iacutendice de esbeltes
Eixo ldquoxrdquo
λ = (346 983148983141) 983144
λ = (346 280) 25
λ 983101 398309675
Eixo ldquoyrdquo
λ = (346 983148983141) 983144
λ = (346 280) 20
λ 983101 498309644
Iacutendice de esbeltes limite
Eixo ldquoxrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1724 25)
10λ1 = 3362
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Eixo ldquoyrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1321 20)
10λ1 = 3326
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
Eixo ldquoxrdquo
η =N983140
=7871
= 009983138 983144 983142983139983140 20 25 (2514)
1
=
0005
le
0005
983154 983144 (η + 05) 983144
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2027
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2127
19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2227
Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2327
Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 327
1 DADOS DO TRABALHO
Deve-se dimensionar e detalhar a armadura transversal e longitudinal dos
pilares P3 P5 e P6 da planta de formas da paacutegina 4
Os dados satildeo os seguintes
bull Concreto C25
bull Accedilo CA50
bull concreto = 25 KNmsup3
bull Cobrimento miacutenimo 25cm
bull drsquo = 40 cm
bull Laje maciccedila espessura 15 cm
bull Diacircmetro do estribo 50mm
bull αb = 10
bull lef = 280 cm
Abaixo tabela com os dados das barras de accedilos
Oslash (mm) As (cmsup2) Peso (Kgfm)
50 020 016
63 032 025
80 050 040
100 080 063
125 125 100
160 200 160
200 315 247
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 427
2 CARGAS NOS PILARES
21 Descarregamento das lajes
Cargas atuantes nas lajes
Enchimento 70 kgfmsup2Revestimento 50 kgfmsup2Carga Acidental 650 kgfmsup2Peso Proacuteprio (015m 2500 kgfmsup3) 375 kgfmsup2Total 1145 kgfmsup2Total 1145 kNmsup2
Planta baixa do pavimento
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 527
Descarregamento das lajes nas vigas
22 Caacutelculo das vigas
V101
Tramo 01
983153 = 983233983154983141983137 C983137983154983143983137 = 1439 983149983218 1145 983147N983149983218L 983140983137 983158983145983143983137 8081 983149
983153 983101 203983097 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 070 983149 983160 25 983147N = 21 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 627
Tramo 02
983153 = 786 983149983218 1145 983147N983149983218
556 983149
983153 983101 1617 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 18 983147N983149
Cargas totais na viga
Reaccediloes nos apoios
V102
Tramo 01
983153 = 1439 983149983218 1145 983147N983149983218 + 1566 983149983218 1145 983147N983149983218
7916 983149 7916 983149
983153 983101 4347 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 015 983149 983160 070 983149 983160 25 983147N = 263 983147N983149
Tramo 02
983153 = 786 983149983218 1145 983147N983149983218 + 792 983149983218 1145 983147N983149983218
5554 983149 5554 983149
983153 983101 3253 983147983118983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 727
983120983141983155983151 983152983154983283983152983154983145983151 015 983149 983160 070 983149 983160 25 983147N = 263 983147N983149
Cargas totais na viga
Reaccediloes nos apoios
V105
Tramo 01
983153 = 659 983149983218 1145 983147N983149983218 + 659 983149983218 1145 983147N9831499832185095 983149 5095 983149
983153 983101 298309762 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 180 983147N983149
Tramo 02
983153 = 941 983149983218 1145 983147N983149983218 + 935983149983218 1145 983147N9831499832186095 983149 6095 983149
983153 983101 3524 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 180 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 827
Cargas totais na viga
Reaccediloes nos apoios
V106
Tramo 01
983153 = 935 983149983218 1145 983147N983149983218
6095 983149
983153 983101 1756 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 050 983149 983160 25 983147N = 150 983147N983149
Tramo 02
983153 = 659 983149983218 1145 983147N9831499832185095 983149
983153 983101 149830961 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 050 983149 983160 25 983147N = 150 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 927
Cargas totais na viga
Reaccediloes nos apoios
23 Cargas nos pilares
P3
Reaccedilatildeo da Viga 101 2536 kNReaccedilatildeo da Viga 106 2736 kNPeso Proacuteprio do Pilar(020 m 025 m 280 m 25 kNmsup3) 350 kNTotal 5622 kN
P5
Reaccedilatildeo da Viga 102 36225 kN
Reaccedilatildeo da Viga 105 24341 kNPeso Proacuteprio do Pilar(025 m 025 m 280 m 25 kNmsup3) 438 kNTotal 61004 kN
P6
Reaccedilatildeo da Viga 102 4937 kNReaccedilatildeo da Viga 106 12569 kNPeso Proacuteprio do Pilar(020 m 040 m 280 m 25 kNmsup3) 560 kNTotal 18066 kN
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1027
3 CALCULO DO PILAR DE EXTREMIDADE ldquoP6rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 18066
983118983140 983101 2529830972 983147983118
Vatildeo efetivo da viga
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 le05 9831561 = 05 25 = 125 983139983149
9831371 983101 125 98313998314905 983144 = 05 70 = 350 983139983149
9831372 le05 9831562 = 05 40 = 200 983139983149
9831372 983101 200 98313998314905 983144 = 05 70 = 350 983139983149
983148983141983142 = 5229 + 125 + 20
983148983141983142 983101 5554 983139983149
Momentos de ligaccedilatildeo viga-pilar
983122983155983157983152 = 983122983145983150983142 =I
=(983138 983144983219) 12
=(20 40983219) 12
= 7619830970 983139983149983219983148983141 983148983141 2 280 2
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1127
983122983158983145983143983137 =I983158983145983143983137
=(15 70983219) 12
= 7719830977 983139983149983219983148983141983142 5554
Momento de engastamento
M983141983150983143 =983120 L983218
=3516 5554983218
= 98309703983096 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 9038 3 76190
3 76190 + 4 77197 + 3 76190
983117983155983157983152 983101 983117983145983150983142 983101 269830977 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 2697 + 2697
983117983158983145983143983137 983101 539830974 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147M983140983156983151983152983151 = 14 10 2697
983117983140983156983151983152983151 983101 3776 983147983118983149 983117983140983138983137983155983141 983101 9830853776 983147983118983149
Excentricidade inicial no topo e na base
983141983145 = M983140 N983140
983141983145 = 3776 983147N983139983149 25292 983147N
983141983145 983101 149830974 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1227
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 25292 (0015 + 003 04)
9831171983140983149983277983150 983101 69830963 983147983118983149
Iacutendice de esbeltes
λ = (346 983148983141) 983144
λ = (346 280) 40
λ 983101 2422
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 40)
10λ1 = 25
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λ = 2422 eacute menor que λ1 = 35 natildeo se considera efeitos de segunda ordem
assim adota-se M1d = 3776 kNm = 3776 kNcm
Coeficientes adimensionais
η =N983140
=25292
= 018983138 983144 983142983139983140 20 40 (2514)
983140=
4= 01
983144 40
μ = η 983141983156983151983156983137983148 = 018 1494 = 007983144 40
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1327
Utilizando o aacutebaco A-25 de VENTURINI (1987) obtecircm-se ω = 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 40 (2514) 005
= 164 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 40 = 64 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 40 = 32 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
983119B983123 A983140983151983156983137983154 A983155 = 32 983139983149983218
Detalhamento
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1427
Portanto o pilar ldquoP6rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
4 CALCULO DO PILAR DE CANTO ldquoP3rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147N983140 = 14 10 5622
983118983140 983101 798309671 983147983118
Vatildeo efetivo da viga
bull Eixo ldquoxrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1527
9831371 le05 9831561 = 05 30 = 150 983139983149
9831371 983101 150 98313998314905 983144 = 05 60 = 300 983139983149
9831372 le
05 9831562 = 05 25 = 125 983139983149
9831372 983101 125 98313998314905 983144 = 05 60 = 300 983139983149
983148983141983142 = 5289 + 125 + 15
983148983141983142 983101 5564 983139983149
bull Eixo ldquoyrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 = 9831372 le05 983156 = 05 20 = 100 983139983149
9831371 983101 9831372 983101 100 98313998314905 983144 = 05 50 = 250 983139983149
983148983141983142 = 4895 + 100 + 100
983148983141983142 983101 509830975 983139983149
Momentos de ligaccedilatildeo viga-pilar
bull
Eixo ldquoxrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(20 25983219) 12
= 1983096601 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 60983219) 12
= 398309698309621 983139983149983219983148983141983142 5564
bull Eixo ldquoyrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(25 20983219) 12
= 1198309705 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 50983219) 12
= 24534 983139983149983219983148983141983142 5095
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1627
Momento de engastamento
bull Eixo ldquoxrdquo
M983141983150983143 =983120 L983218
=1797 5564983218
= 4636 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 4636 3 18601
3 18601 + 4 38821 + 3 18601
983117983155983157983152 983101 983117983145983150983142 983101 9830976983097 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 969 + 969
983117983158983145983143983137 983101 19830973983096 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 969
983117983140983156983151983152983151 983101 1357 983147983118983149 983117983140983138983137983155983141 983101 9830851357 983147983118983149
bull Eixo ldquoyrdquo
M983141983150983143 =983120 L983218
=1631 5095983218
= 352983096 98314798311898314912 12
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1727
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 3528 3 11905
3 11905 + 4 24534 + 3 11905
983117983155983157983152 983101 983117983145983150983142 983101 743 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 743 + 743
983117983158983145983143983137 983101 149830966 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 743
983117983140983156983151983152983151 983101 1040 983147983118983149 983117983140983138983137983155983141 983101 9830851040 983147983118983149
Excentricidade inicial no topo e na base
Eixo ldquoxrdquo
983141983145 = M983140 N983140
983141983145 = 1357 983147N983139983149 7871 983147N
983141983145 983101 1724 983139983149
Eixo ldquoyrdquo
983141983145 = M983140 N983140
983141983145 = 1040 983147N983139983149 7871 983147N
983141983145 983101 1321 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1827
Momento miacutenimo
Eixo ldquoxrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 025)
9831171983140983149983277983150 983101 177 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1357 983147N983149
Eixo ldquoyrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 020)
9831171983140983149983277983150 983101 165 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1040 983147N983149
Iacutendice de esbeltes
Eixo ldquoxrdquo
λ = (346 983148983141) 983144
λ = (346 280) 25
λ 983101 398309675
Eixo ldquoyrdquo
λ = (346 983148983141) 983144
λ = (346 280) 20
λ 983101 498309644
Iacutendice de esbeltes limite
Eixo ldquoxrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1724 25)
10λ1 = 3362
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Eixo ldquoyrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1321 20)
10λ1 = 3326
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
Eixo ldquoxrdquo
η =N983140
=7871
= 009983138 983144 983142983139983140 20 25 (2514)
1
=
0005
le
0005
983154 983144 (η + 05) 983144
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 427
2 CARGAS NOS PILARES
21 Descarregamento das lajes
Cargas atuantes nas lajes
Enchimento 70 kgfmsup2Revestimento 50 kgfmsup2Carga Acidental 650 kgfmsup2Peso Proacuteprio (015m 2500 kgfmsup3) 375 kgfmsup2Total 1145 kgfmsup2Total 1145 kNmsup2
Planta baixa do pavimento
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Descarregamento das lajes nas vigas
22 Caacutelculo das vigas
V101
Tramo 01
983153 = 983233983154983141983137 C983137983154983143983137 = 1439 983149983218 1145 983147N983149983218L 983140983137 983158983145983143983137 8081 983149
983153 983101 203983097 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 070 983149 983160 25 983147N = 21 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Tramo 02
983153 = 786 983149983218 1145 983147N983149983218
556 983149
983153 983101 1617 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 18 983147N983149
Cargas totais na viga
Reaccediloes nos apoios
V102
Tramo 01
983153 = 1439 983149983218 1145 983147N983149983218 + 1566 983149983218 1145 983147N983149983218
7916 983149 7916 983149
983153 983101 4347 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 015 983149 983160 070 983149 983160 25 983147N = 263 983147N983149
Tramo 02
983153 = 786 983149983218 1145 983147N983149983218 + 792 983149983218 1145 983147N983149983218
5554 983149 5554 983149
983153 983101 3253 983147983118983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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983120983141983155983151 983152983154983283983152983154983145983151 015 983149 983160 070 983149 983160 25 983147N = 263 983147N983149
Cargas totais na viga
Reaccediloes nos apoios
V105
Tramo 01
983153 = 659 983149983218 1145 983147N983149983218 + 659 983149983218 1145 983147N9831499832185095 983149 5095 983149
983153 983101 298309762 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 180 983147N983149
Tramo 02
983153 = 941 983149983218 1145 983147N983149983218 + 935983149983218 1145 983147N9831499832186095 983149 6095 983149
983153 983101 3524 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 180 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Cargas totais na viga
Reaccediloes nos apoios
V106
Tramo 01
983153 = 935 983149983218 1145 983147N983149983218
6095 983149
983153 983101 1756 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 050 983149 983160 25 983147N = 150 983147N983149
Tramo 02
983153 = 659 983149983218 1145 983147N9831499832185095 983149
983153 983101 149830961 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 050 983149 983160 25 983147N = 150 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Cargas totais na viga
Reaccediloes nos apoios
23 Cargas nos pilares
P3
Reaccedilatildeo da Viga 101 2536 kNReaccedilatildeo da Viga 106 2736 kNPeso Proacuteprio do Pilar(020 m 025 m 280 m 25 kNmsup3) 350 kNTotal 5622 kN
P5
Reaccedilatildeo da Viga 102 36225 kN
Reaccedilatildeo da Viga 105 24341 kNPeso Proacuteprio do Pilar(025 m 025 m 280 m 25 kNmsup3) 438 kNTotal 61004 kN
P6
Reaccedilatildeo da Viga 102 4937 kNReaccedilatildeo da Viga 106 12569 kNPeso Proacuteprio do Pilar(020 m 040 m 280 m 25 kNmsup3) 560 kNTotal 18066 kN
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1027
3 CALCULO DO PILAR DE EXTREMIDADE ldquoP6rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 18066
983118983140 983101 2529830972 983147983118
Vatildeo efetivo da viga
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 le05 9831561 = 05 25 = 125 983139983149
9831371 983101 125 98313998314905 983144 = 05 70 = 350 983139983149
9831372 le05 9831562 = 05 40 = 200 983139983149
9831372 983101 200 98313998314905 983144 = 05 70 = 350 983139983149
983148983141983142 = 5229 + 125 + 20
983148983141983142 983101 5554 983139983149
Momentos de ligaccedilatildeo viga-pilar
983122983155983157983152 = 983122983145983150983142 =I
=(983138 983144983219) 12
=(20 40983219) 12
= 7619830970 983139983149983219983148983141 983148983141 2 280 2
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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983122983158983145983143983137 =I983158983145983143983137
=(15 70983219) 12
= 7719830977 983139983149983219983148983141983142 5554
Momento de engastamento
M983141983150983143 =983120 L983218
=3516 5554983218
= 98309703983096 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 9038 3 76190
3 76190 + 4 77197 + 3 76190
983117983155983157983152 983101 983117983145983150983142 983101 269830977 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 2697 + 2697
983117983158983145983143983137 983101 539830974 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147M983140983156983151983152983151 = 14 10 2697
983117983140983156983151983152983151 983101 3776 983147983118983149 983117983140983138983137983155983141 983101 9830853776 983147983118983149
Excentricidade inicial no topo e na base
983141983145 = M983140 N983140
983141983145 = 3776 983147N983139983149 25292 983147N
983141983145 983101 149830974 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1227
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 25292 (0015 + 003 04)
9831171983140983149983277983150 983101 69830963 983147983118983149
Iacutendice de esbeltes
λ = (346 983148983141) 983144
λ = (346 280) 40
λ 983101 2422
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 40)
10λ1 = 25
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λ = 2422 eacute menor que λ1 = 35 natildeo se considera efeitos de segunda ordem
assim adota-se M1d = 3776 kNm = 3776 kNcm
Coeficientes adimensionais
η =N983140
=25292
= 018983138 983144 983142983139983140 20 40 (2514)
983140=
4= 01
983144 40
μ = η 983141983156983151983156983137983148 = 018 1494 = 007983144 40
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1327
Utilizando o aacutebaco A-25 de VENTURINI (1987) obtecircm-se ω = 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 40 (2514) 005
= 164 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 40 = 64 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 40 = 32 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
983119B983123 A983140983151983156983137983154 A983155 = 32 983139983149983218
Detalhamento
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1427
Portanto o pilar ldquoP6rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
4 CALCULO DO PILAR DE CANTO ldquoP3rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147N983140 = 14 10 5622
983118983140 983101 798309671 983147983118
Vatildeo efetivo da viga
bull Eixo ldquoxrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1527
9831371 le05 9831561 = 05 30 = 150 983139983149
9831371 983101 150 98313998314905 983144 = 05 60 = 300 983139983149
9831372 le
05 9831562 = 05 25 = 125 983139983149
9831372 983101 125 98313998314905 983144 = 05 60 = 300 983139983149
983148983141983142 = 5289 + 125 + 15
983148983141983142 983101 5564 983139983149
bull Eixo ldquoyrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 = 9831372 le05 983156 = 05 20 = 100 983139983149
9831371 983101 9831372 983101 100 98313998314905 983144 = 05 50 = 250 983139983149
983148983141983142 = 4895 + 100 + 100
983148983141983142 983101 509830975 983139983149
Momentos de ligaccedilatildeo viga-pilar
bull
Eixo ldquoxrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(20 25983219) 12
= 1983096601 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 60983219) 12
= 398309698309621 983139983149983219983148983141983142 5564
bull Eixo ldquoyrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(25 20983219) 12
= 1198309705 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 50983219) 12
= 24534 983139983149983219983148983141983142 5095
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1627
Momento de engastamento
bull Eixo ldquoxrdquo
M983141983150983143 =983120 L983218
=1797 5564983218
= 4636 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 4636 3 18601
3 18601 + 4 38821 + 3 18601
983117983155983157983152 983101 983117983145983150983142 983101 9830976983097 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 969 + 969
983117983158983145983143983137 983101 19830973983096 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 969
983117983140983156983151983152983151 983101 1357 983147983118983149 983117983140983138983137983155983141 983101 9830851357 983147983118983149
bull Eixo ldquoyrdquo
M983141983150983143 =983120 L983218
=1631 5095983218
= 352983096 98314798311898314912 12
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1727
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 3528 3 11905
3 11905 + 4 24534 + 3 11905
983117983155983157983152 983101 983117983145983150983142 983101 743 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 743 + 743
983117983158983145983143983137 983101 149830966 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 743
983117983140983156983151983152983151 983101 1040 983147983118983149 983117983140983138983137983155983141 983101 9830851040 983147983118983149
Excentricidade inicial no topo e na base
Eixo ldquoxrdquo
983141983145 = M983140 N983140
983141983145 = 1357 983147N983139983149 7871 983147N
983141983145 983101 1724 983139983149
Eixo ldquoyrdquo
983141983145 = M983140 N983140
983141983145 = 1040 983147N983139983149 7871 983147N
983141983145 983101 1321 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1827
Momento miacutenimo
Eixo ldquoxrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 025)
9831171983140983149983277983150 983101 177 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1357 983147N983149
Eixo ldquoyrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 020)
9831171983140983149983277983150 983101 165 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1040 983147N983149
Iacutendice de esbeltes
Eixo ldquoxrdquo
λ = (346 983148983141) 983144
λ = (346 280) 25
λ 983101 398309675
Eixo ldquoyrdquo
λ = (346 983148983141) 983144
λ = (346 280) 20
λ 983101 498309644
Iacutendice de esbeltes limite
Eixo ldquoxrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1724 25)
10λ1 = 3362
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Eixo ldquoyrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1321 20)
10λ1 = 3326
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
Eixo ldquoxrdquo
η =N983140
=7871
= 009983138 983144 983142983139983140 20 25 (2514)
1
=
0005
le
0005
983154 983144 (η + 05) 983144
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2027
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2127
19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2227
Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2327
Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 527
Descarregamento das lajes nas vigas
22 Caacutelculo das vigas
V101
Tramo 01
983153 = 983233983154983141983137 C983137983154983143983137 = 1439 983149983218 1145 983147N983149983218L 983140983137 983158983145983143983137 8081 983149
983153 983101 203983097 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 070 983149 983160 25 983147N = 21 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 627
Tramo 02
983153 = 786 983149983218 1145 983147N983149983218
556 983149
983153 983101 1617 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 18 983147N983149
Cargas totais na viga
Reaccediloes nos apoios
V102
Tramo 01
983153 = 1439 983149983218 1145 983147N983149983218 + 1566 983149983218 1145 983147N983149983218
7916 983149 7916 983149
983153 983101 4347 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 015 983149 983160 070 983149 983160 25 983147N = 263 983147N983149
Tramo 02
983153 = 786 983149983218 1145 983147N983149983218 + 792 983149983218 1145 983147N983149983218
5554 983149 5554 983149
983153 983101 3253 983147983118983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 727
983120983141983155983151 983152983154983283983152983154983145983151 015 983149 983160 070 983149 983160 25 983147N = 263 983147N983149
Cargas totais na viga
Reaccediloes nos apoios
V105
Tramo 01
983153 = 659 983149983218 1145 983147N983149983218 + 659 983149983218 1145 983147N9831499832185095 983149 5095 983149
983153 983101 298309762 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 180 983147N983149
Tramo 02
983153 = 941 983149983218 1145 983147N983149983218 + 935983149983218 1145 983147N9831499832186095 983149 6095 983149
983153 983101 3524 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 180 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 827
Cargas totais na viga
Reaccediloes nos apoios
V106
Tramo 01
983153 = 935 983149983218 1145 983147N983149983218
6095 983149
983153 983101 1756 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 050 983149 983160 25 983147N = 150 983147N983149
Tramo 02
983153 = 659 983149983218 1145 983147N9831499832185095 983149
983153 983101 149830961 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 050 983149 983160 25 983147N = 150 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 927
Cargas totais na viga
Reaccediloes nos apoios
23 Cargas nos pilares
P3
Reaccedilatildeo da Viga 101 2536 kNReaccedilatildeo da Viga 106 2736 kNPeso Proacuteprio do Pilar(020 m 025 m 280 m 25 kNmsup3) 350 kNTotal 5622 kN
P5
Reaccedilatildeo da Viga 102 36225 kN
Reaccedilatildeo da Viga 105 24341 kNPeso Proacuteprio do Pilar(025 m 025 m 280 m 25 kNmsup3) 438 kNTotal 61004 kN
P6
Reaccedilatildeo da Viga 102 4937 kNReaccedilatildeo da Viga 106 12569 kNPeso Proacuteprio do Pilar(020 m 040 m 280 m 25 kNmsup3) 560 kNTotal 18066 kN
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1027
3 CALCULO DO PILAR DE EXTREMIDADE ldquoP6rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 18066
983118983140 983101 2529830972 983147983118
Vatildeo efetivo da viga
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 le05 9831561 = 05 25 = 125 983139983149
9831371 983101 125 98313998314905 983144 = 05 70 = 350 983139983149
9831372 le05 9831562 = 05 40 = 200 983139983149
9831372 983101 200 98313998314905 983144 = 05 70 = 350 983139983149
983148983141983142 = 5229 + 125 + 20
983148983141983142 983101 5554 983139983149
Momentos de ligaccedilatildeo viga-pilar
983122983155983157983152 = 983122983145983150983142 =I
=(983138 983144983219) 12
=(20 40983219) 12
= 7619830970 983139983149983219983148983141 983148983141 2 280 2
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1127
983122983158983145983143983137 =I983158983145983143983137
=(15 70983219) 12
= 7719830977 983139983149983219983148983141983142 5554
Momento de engastamento
M983141983150983143 =983120 L983218
=3516 5554983218
= 98309703983096 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 9038 3 76190
3 76190 + 4 77197 + 3 76190
983117983155983157983152 983101 983117983145983150983142 983101 269830977 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 2697 + 2697
983117983158983145983143983137 983101 539830974 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147M983140983156983151983152983151 = 14 10 2697
983117983140983156983151983152983151 983101 3776 983147983118983149 983117983140983138983137983155983141 983101 9830853776 983147983118983149
Excentricidade inicial no topo e na base
983141983145 = M983140 N983140
983141983145 = 3776 983147N983139983149 25292 983147N
983141983145 983101 149830974 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1227
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 25292 (0015 + 003 04)
9831171983140983149983277983150 983101 69830963 983147983118983149
Iacutendice de esbeltes
λ = (346 983148983141) 983144
λ = (346 280) 40
λ 983101 2422
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 40)
10λ1 = 25
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λ = 2422 eacute menor que λ1 = 35 natildeo se considera efeitos de segunda ordem
assim adota-se M1d = 3776 kNm = 3776 kNcm
Coeficientes adimensionais
η =N983140
=25292
= 018983138 983144 983142983139983140 20 40 (2514)
983140=
4= 01
983144 40
μ = η 983141983156983151983156983137983148 = 018 1494 = 007983144 40
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1327
Utilizando o aacutebaco A-25 de VENTURINI (1987) obtecircm-se ω = 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 40 (2514) 005
= 164 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 40 = 64 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 40 = 32 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
983119B983123 A983140983151983156983137983154 A983155 = 32 983139983149983218
Detalhamento
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1427
Portanto o pilar ldquoP6rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
4 CALCULO DO PILAR DE CANTO ldquoP3rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147N983140 = 14 10 5622
983118983140 983101 798309671 983147983118
Vatildeo efetivo da viga
bull Eixo ldquoxrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1527
9831371 le05 9831561 = 05 30 = 150 983139983149
9831371 983101 150 98313998314905 983144 = 05 60 = 300 983139983149
9831372 le
05 9831562 = 05 25 = 125 983139983149
9831372 983101 125 98313998314905 983144 = 05 60 = 300 983139983149
983148983141983142 = 5289 + 125 + 15
983148983141983142 983101 5564 983139983149
bull Eixo ldquoyrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 = 9831372 le05 983156 = 05 20 = 100 983139983149
9831371 983101 9831372 983101 100 98313998314905 983144 = 05 50 = 250 983139983149
983148983141983142 = 4895 + 100 + 100
983148983141983142 983101 509830975 983139983149
Momentos de ligaccedilatildeo viga-pilar
bull
Eixo ldquoxrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(20 25983219) 12
= 1983096601 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 60983219) 12
= 398309698309621 983139983149983219983148983141983142 5564
bull Eixo ldquoyrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(25 20983219) 12
= 1198309705 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 50983219) 12
= 24534 983139983149983219983148983141983142 5095
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1627
Momento de engastamento
bull Eixo ldquoxrdquo
M983141983150983143 =983120 L983218
=1797 5564983218
= 4636 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 4636 3 18601
3 18601 + 4 38821 + 3 18601
983117983155983157983152 983101 983117983145983150983142 983101 9830976983097 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 969 + 969
983117983158983145983143983137 983101 19830973983096 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 969
983117983140983156983151983152983151 983101 1357 983147983118983149 983117983140983138983137983155983141 983101 9830851357 983147983118983149
bull Eixo ldquoyrdquo
M983141983150983143 =983120 L983218
=1631 5095983218
= 352983096 98314798311898314912 12
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1727
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 3528 3 11905
3 11905 + 4 24534 + 3 11905
983117983155983157983152 983101 983117983145983150983142 983101 743 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 743 + 743
983117983158983145983143983137 983101 149830966 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 743
983117983140983156983151983152983151 983101 1040 983147983118983149 983117983140983138983137983155983141 983101 9830851040 983147983118983149
Excentricidade inicial no topo e na base
Eixo ldquoxrdquo
983141983145 = M983140 N983140
983141983145 = 1357 983147N983139983149 7871 983147N
983141983145 983101 1724 983139983149
Eixo ldquoyrdquo
983141983145 = M983140 N983140
983141983145 = 1040 983147N983139983149 7871 983147N
983141983145 983101 1321 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1827
Momento miacutenimo
Eixo ldquoxrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 025)
9831171983140983149983277983150 983101 177 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1357 983147N983149
Eixo ldquoyrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 020)
9831171983140983149983277983150 983101 165 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1040 983147N983149
Iacutendice de esbeltes
Eixo ldquoxrdquo
λ = (346 983148983141) 983144
λ = (346 280) 25
λ 983101 398309675
Eixo ldquoyrdquo
λ = (346 983148983141) 983144
λ = (346 280) 20
λ 983101 498309644
Iacutendice de esbeltes limite
Eixo ldquoxrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1724 25)
10λ1 = 3362
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Eixo ldquoyrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1321 20)
10λ1 = 3326
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
Eixo ldquoxrdquo
η =N983140
=7871
= 009983138 983144 983142983139983140 20 25 (2514)
1
=
0005
le
0005
983154 983144 (η + 05) 983144
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2027
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2127
19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 627
Tramo 02
983153 = 786 983149983218 1145 983147N983149983218
556 983149
983153 983101 1617 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 18 983147N983149
Cargas totais na viga
Reaccediloes nos apoios
V102
Tramo 01
983153 = 1439 983149983218 1145 983147N983149983218 + 1566 983149983218 1145 983147N983149983218
7916 983149 7916 983149
983153 983101 4347 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 015 983149 983160 070 983149 983160 25 983147N = 263 983147N983149
Tramo 02
983153 = 786 983149983218 1145 983147N983149983218 + 792 983149983218 1145 983147N983149983218
5554 983149 5554 983149
983153 983101 3253 983147983118983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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983120983141983155983151 983152983154983283983152983154983145983151 015 983149 983160 070 983149 983160 25 983147N = 263 983147N983149
Cargas totais na viga
Reaccediloes nos apoios
V105
Tramo 01
983153 = 659 983149983218 1145 983147N983149983218 + 659 983149983218 1145 983147N9831499832185095 983149 5095 983149
983153 983101 298309762 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 180 983147N983149
Tramo 02
983153 = 941 983149983218 1145 983147N983149983218 + 935983149983218 1145 983147N9831499832186095 983149 6095 983149
983153 983101 3524 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 180 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Cargas totais na viga
Reaccediloes nos apoios
V106
Tramo 01
983153 = 935 983149983218 1145 983147N983149983218
6095 983149
983153 983101 1756 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 050 983149 983160 25 983147N = 150 983147N983149
Tramo 02
983153 = 659 983149983218 1145 983147N9831499832185095 983149
983153 983101 149830961 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 050 983149 983160 25 983147N = 150 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Cargas totais na viga
Reaccediloes nos apoios
23 Cargas nos pilares
P3
Reaccedilatildeo da Viga 101 2536 kNReaccedilatildeo da Viga 106 2736 kNPeso Proacuteprio do Pilar(020 m 025 m 280 m 25 kNmsup3) 350 kNTotal 5622 kN
P5
Reaccedilatildeo da Viga 102 36225 kN
Reaccedilatildeo da Viga 105 24341 kNPeso Proacuteprio do Pilar(025 m 025 m 280 m 25 kNmsup3) 438 kNTotal 61004 kN
P6
Reaccedilatildeo da Viga 102 4937 kNReaccedilatildeo da Viga 106 12569 kNPeso Proacuteprio do Pilar(020 m 040 m 280 m 25 kNmsup3) 560 kNTotal 18066 kN
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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3 CALCULO DO PILAR DE EXTREMIDADE ldquoP6rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 18066
983118983140 983101 2529830972 983147983118
Vatildeo efetivo da viga
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 le05 9831561 = 05 25 = 125 983139983149
9831371 983101 125 98313998314905 983144 = 05 70 = 350 983139983149
9831372 le05 9831562 = 05 40 = 200 983139983149
9831372 983101 200 98313998314905 983144 = 05 70 = 350 983139983149
983148983141983142 = 5229 + 125 + 20
983148983141983142 983101 5554 983139983149
Momentos de ligaccedilatildeo viga-pilar
983122983155983157983152 = 983122983145983150983142 =I
=(983138 983144983219) 12
=(20 40983219) 12
= 7619830970 983139983149983219983148983141 983148983141 2 280 2
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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983122983158983145983143983137 =I983158983145983143983137
=(15 70983219) 12
= 7719830977 983139983149983219983148983141983142 5554
Momento de engastamento
M983141983150983143 =983120 L983218
=3516 5554983218
= 98309703983096 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 9038 3 76190
3 76190 + 4 77197 + 3 76190
983117983155983157983152 983101 983117983145983150983142 983101 269830977 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 2697 + 2697
983117983158983145983143983137 983101 539830974 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147M983140983156983151983152983151 = 14 10 2697
983117983140983156983151983152983151 983101 3776 983147983118983149 983117983140983138983137983155983141 983101 9830853776 983147983118983149
Excentricidade inicial no topo e na base
983141983145 = M983140 N983140
983141983145 = 3776 983147N983139983149 25292 983147N
983141983145 983101 149830974 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 25292 (0015 + 003 04)
9831171983140983149983277983150 983101 69830963 983147983118983149
Iacutendice de esbeltes
λ = (346 983148983141) 983144
λ = (346 280) 40
λ 983101 2422
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 40)
10λ1 = 25
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λ = 2422 eacute menor que λ1 = 35 natildeo se considera efeitos de segunda ordem
assim adota-se M1d = 3776 kNm = 3776 kNcm
Coeficientes adimensionais
η =N983140
=25292
= 018983138 983144 983142983139983140 20 40 (2514)
983140=
4= 01
983144 40
μ = η 983141983156983151983156983137983148 = 018 1494 = 007983144 40
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Utilizando o aacutebaco A-25 de VENTURINI (1987) obtecircm-se ω = 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 40 (2514) 005
= 164 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 40 = 64 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 40 = 32 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
983119B983123 A983140983151983156983137983154 A983155 = 32 983139983149983218
Detalhamento
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1427
Portanto o pilar ldquoP6rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
4 CALCULO DO PILAR DE CANTO ldquoP3rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147N983140 = 14 10 5622
983118983140 983101 798309671 983147983118
Vatildeo efetivo da viga
bull Eixo ldquoxrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1527
9831371 le05 9831561 = 05 30 = 150 983139983149
9831371 983101 150 98313998314905 983144 = 05 60 = 300 983139983149
9831372 le
05 9831562 = 05 25 = 125 983139983149
9831372 983101 125 98313998314905 983144 = 05 60 = 300 983139983149
983148983141983142 = 5289 + 125 + 15
983148983141983142 983101 5564 983139983149
bull Eixo ldquoyrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 = 9831372 le05 983156 = 05 20 = 100 983139983149
9831371 983101 9831372 983101 100 98313998314905 983144 = 05 50 = 250 983139983149
983148983141983142 = 4895 + 100 + 100
983148983141983142 983101 509830975 983139983149
Momentos de ligaccedilatildeo viga-pilar
bull
Eixo ldquoxrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(20 25983219) 12
= 1983096601 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 60983219) 12
= 398309698309621 983139983149983219983148983141983142 5564
bull Eixo ldquoyrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(25 20983219) 12
= 1198309705 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 50983219) 12
= 24534 983139983149983219983148983141983142 5095
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1627
Momento de engastamento
bull Eixo ldquoxrdquo
M983141983150983143 =983120 L983218
=1797 5564983218
= 4636 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 4636 3 18601
3 18601 + 4 38821 + 3 18601
983117983155983157983152 983101 983117983145983150983142 983101 9830976983097 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 969 + 969
983117983158983145983143983137 983101 19830973983096 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 969
983117983140983156983151983152983151 983101 1357 983147983118983149 983117983140983138983137983155983141 983101 9830851357 983147983118983149
bull Eixo ldquoyrdquo
M983141983150983143 =983120 L983218
=1631 5095983218
= 352983096 98314798311898314912 12
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1727
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 3528 3 11905
3 11905 + 4 24534 + 3 11905
983117983155983157983152 983101 983117983145983150983142 983101 743 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 743 + 743
983117983158983145983143983137 983101 149830966 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 743
983117983140983156983151983152983151 983101 1040 983147983118983149 983117983140983138983137983155983141 983101 9830851040 983147983118983149
Excentricidade inicial no topo e na base
Eixo ldquoxrdquo
983141983145 = M983140 N983140
983141983145 = 1357 983147N983139983149 7871 983147N
983141983145 983101 1724 983139983149
Eixo ldquoyrdquo
983141983145 = M983140 N983140
983141983145 = 1040 983147N983139983149 7871 983147N
983141983145 983101 1321 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1827
Momento miacutenimo
Eixo ldquoxrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 025)
9831171983140983149983277983150 983101 177 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1357 983147N983149
Eixo ldquoyrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 020)
9831171983140983149983277983150 983101 165 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1040 983147N983149
Iacutendice de esbeltes
Eixo ldquoxrdquo
λ = (346 983148983141) 983144
λ = (346 280) 25
λ 983101 398309675
Eixo ldquoyrdquo
λ = (346 983148983141) 983144
λ = (346 280) 20
λ 983101 498309644
Iacutendice de esbeltes limite
Eixo ldquoxrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1724 25)
10λ1 = 3362
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Eixo ldquoyrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1321 20)
10λ1 = 3326
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
Eixo ldquoxrdquo
η =N983140
=7871
= 009983138 983144 983142983139983140 20 25 (2514)
1
=
0005
le
0005
983154 983144 (η + 05) 983144
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2027
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2127
19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2227
Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 727
983120983141983155983151 983152983154983283983152983154983145983151 015 983149 983160 070 983149 983160 25 983147N = 263 983147N983149
Cargas totais na viga
Reaccediloes nos apoios
V105
Tramo 01
983153 = 659 983149983218 1145 983147N983149983218 + 659 983149983218 1145 983147N9831499832185095 983149 5095 983149
983153 983101 298309762 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 180 983147N983149
Tramo 02
983153 = 941 983149983218 1145 983147N983149983218 + 935983149983218 1145 983147N9831499832186095 983149 6095 983149
983153 983101 3524 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 060 983149 983160 25 983147N = 180 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Cargas totais na viga
Reaccediloes nos apoios
V106
Tramo 01
983153 = 935 983149983218 1145 983147N983149983218
6095 983149
983153 983101 1756 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 050 983149 983160 25 983147N = 150 983147N983149
Tramo 02
983153 = 659 983149983218 1145 983147N9831499832185095 983149
983153 983101 149830961 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 050 983149 983160 25 983147N = 150 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Cargas totais na viga
Reaccediloes nos apoios
23 Cargas nos pilares
P3
Reaccedilatildeo da Viga 101 2536 kNReaccedilatildeo da Viga 106 2736 kNPeso Proacuteprio do Pilar(020 m 025 m 280 m 25 kNmsup3) 350 kNTotal 5622 kN
P5
Reaccedilatildeo da Viga 102 36225 kN
Reaccedilatildeo da Viga 105 24341 kNPeso Proacuteprio do Pilar(025 m 025 m 280 m 25 kNmsup3) 438 kNTotal 61004 kN
P6
Reaccedilatildeo da Viga 102 4937 kNReaccedilatildeo da Viga 106 12569 kNPeso Proacuteprio do Pilar(020 m 040 m 280 m 25 kNmsup3) 560 kNTotal 18066 kN
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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3 CALCULO DO PILAR DE EXTREMIDADE ldquoP6rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 18066
983118983140 983101 2529830972 983147983118
Vatildeo efetivo da viga
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 le05 9831561 = 05 25 = 125 983139983149
9831371 983101 125 98313998314905 983144 = 05 70 = 350 983139983149
9831372 le05 9831562 = 05 40 = 200 983139983149
9831372 983101 200 98313998314905 983144 = 05 70 = 350 983139983149
983148983141983142 = 5229 + 125 + 20
983148983141983142 983101 5554 983139983149
Momentos de ligaccedilatildeo viga-pilar
983122983155983157983152 = 983122983145983150983142 =I
=(983138 983144983219) 12
=(20 40983219) 12
= 7619830970 983139983149983219983148983141 983148983141 2 280 2
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1127
983122983158983145983143983137 =I983158983145983143983137
=(15 70983219) 12
= 7719830977 983139983149983219983148983141983142 5554
Momento de engastamento
M983141983150983143 =983120 L983218
=3516 5554983218
= 98309703983096 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 9038 3 76190
3 76190 + 4 77197 + 3 76190
983117983155983157983152 983101 983117983145983150983142 983101 269830977 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 2697 + 2697
983117983158983145983143983137 983101 539830974 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147M983140983156983151983152983151 = 14 10 2697
983117983140983156983151983152983151 983101 3776 983147983118983149 983117983140983138983137983155983141 983101 9830853776 983147983118983149
Excentricidade inicial no topo e na base
983141983145 = M983140 N983140
983141983145 = 3776 983147N983139983149 25292 983147N
983141983145 983101 149830974 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1227
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 25292 (0015 + 003 04)
9831171983140983149983277983150 983101 69830963 983147983118983149
Iacutendice de esbeltes
λ = (346 983148983141) 983144
λ = (346 280) 40
λ 983101 2422
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 40)
10λ1 = 25
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λ = 2422 eacute menor que λ1 = 35 natildeo se considera efeitos de segunda ordem
assim adota-se M1d = 3776 kNm = 3776 kNcm
Coeficientes adimensionais
η =N983140
=25292
= 018983138 983144 983142983139983140 20 40 (2514)
983140=
4= 01
983144 40
μ = η 983141983156983151983156983137983148 = 018 1494 = 007983144 40
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1327
Utilizando o aacutebaco A-25 de VENTURINI (1987) obtecircm-se ω = 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 40 (2514) 005
= 164 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 40 = 64 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 40 = 32 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
983119B983123 A983140983151983156983137983154 A983155 = 32 983139983149983218
Detalhamento
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1427
Portanto o pilar ldquoP6rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
4 CALCULO DO PILAR DE CANTO ldquoP3rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147N983140 = 14 10 5622
983118983140 983101 798309671 983147983118
Vatildeo efetivo da viga
bull Eixo ldquoxrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1527
9831371 le05 9831561 = 05 30 = 150 983139983149
9831371 983101 150 98313998314905 983144 = 05 60 = 300 983139983149
9831372 le
05 9831562 = 05 25 = 125 983139983149
9831372 983101 125 98313998314905 983144 = 05 60 = 300 983139983149
983148983141983142 = 5289 + 125 + 15
983148983141983142 983101 5564 983139983149
bull Eixo ldquoyrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 = 9831372 le05 983156 = 05 20 = 100 983139983149
9831371 983101 9831372 983101 100 98313998314905 983144 = 05 50 = 250 983139983149
983148983141983142 = 4895 + 100 + 100
983148983141983142 983101 509830975 983139983149
Momentos de ligaccedilatildeo viga-pilar
bull
Eixo ldquoxrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(20 25983219) 12
= 1983096601 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 60983219) 12
= 398309698309621 983139983149983219983148983141983142 5564
bull Eixo ldquoyrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(25 20983219) 12
= 1198309705 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 50983219) 12
= 24534 983139983149983219983148983141983142 5095
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1627
Momento de engastamento
bull Eixo ldquoxrdquo
M983141983150983143 =983120 L983218
=1797 5564983218
= 4636 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 4636 3 18601
3 18601 + 4 38821 + 3 18601
983117983155983157983152 983101 983117983145983150983142 983101 9830976983097 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 969 + 969
983117983158983145983143983137 983101 19830973983096 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 969
983117983140983156983151983152983151 983101 1357 983147983118983149 983117983140983138983137983155983141 983101 9830851357 983147983118983149
bull Eixo ldquoyrdquo
M983141983150983143 =983120 L983218
=1631 5095983218
= 352983096 98314798311898314912 12
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1727
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 3528 3 11905
3 11905 + 4 24534 + 3 11905
983117983155983157983152 983101 983117983145983150983142 983101 743 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 743 + 743
983117983158983145983143983137 983101 149830966 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 743
983117983140983156983151983152983151 983101 1040 983147983118983149 983117983140983138983137983155983141 983101 9830851040 983147983118983149
Excentricidade inicial no topo e na base
Eixo ldquoxrdquo
983141983145 = M983140 N983140
983141983145 = 1357 983147N983139983149 7871 983147N
983141983145 983101 1724 983139983149
Eixo ldquoyrdquo
983141983145 = M983140 N983140
983141983145 = 1040 983147N983139983149 7871 983147N
983141983145 983101 1321 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1827
Momento miacutenimo
Eixo ldquoxrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 025)
9831171983140983149983277983150 983101 177 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1357 983147N983149
Eixo ldquoyrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 020)
9831171983140983149983277983150 983101 165 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1040 983147N983149
Iacutendice de esbeltes
Eixo ldquoxrdquo
λ = (346 983148983141) 983144
λ = (346 280) 25
λ 983101 398309675
Eixo ldquoyrdquo
λ = (346 983148983141) 983144
λ = (346 280) 20
λ 983101 498309644
Iacutendice de esbeltes limite
Eixo ldquoxrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1724 25)
10λ1 = 3362
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Eixo ldquoyrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1321 20)
10λ1 = 3326
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
Eixo ldquoxrdquo
η =N983140
=7871
= 009983138 983144 983142983139983140 20 25 (2514)
1
=
0005
le
0005
983154 983144 (η + 05) 983144
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2027
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2127
19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2227
Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2327
Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 827
Cargas totais na viga
Reaccediloes nos apoios
V106
Tramo 01
983153 = 935 983149983218 1145 983147N983149983218
6095 983149
983153 983101 1756 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 050 983149 983160 25 983147N = 150 983147N983149
Tramo 02
983153 = 659 983149983218 1145 983147N9831499832185095 983149
983153 983101 149830961 983147983118983149
983120983141983155983151 983152983154983283983152983154983145983151 012 983149 983160 050 983149 983160 25 983147N = 150 983147N983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 927
Cargas totais na viga
Reaccediloes nos apoios
23 Cargas nos pilares
P3
Reaccedilatildeo da Viga 101 2536 kNReaccedilatildeo da Viga 106 2736 kNPeso Proacuteprio do Pilar(020 m 025 m 280 m 25 kNmsup3) 350 kNTotal 5622 kN
P5
Reaccedilatildeo da Viga 102 36225 kN
Reaccedilatildeo da Viga 105 24341 kNPeso Proacuteprio do Pilar(025 m 025 m 280 m 25 kNmsup3) 438 kNTotal 61004 kN
P6
Reaccedilatildeo da Viga 102 4937 kNReaccedilatildeo da Viga 106 12569 kNPeso Proacuteprio do Pilar(020 m 040 m 280 m 25 kNmsup3) 560 kNTotal 18066 kN
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1027
3 CALCULO DO PILAR DE EXTREMIDADE ldquoP6rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 18066
983118983140 983101 2529830972 983147983118
Vatildeo efetivo da viga
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 le05 9831561 = 05 25 = 125 983139983149
9831371 983101 125 98313998314905 983144 = 05 70 = 350 983139983149
9831372 le05 9831562 = 05 40 = 200 983139983149
9831372 983101 200 98313998314905 983144 = 05 70 = 350 983139983149
983148983141983142 = 5229 + 125 + 20
983148983141983142 983101 5554 983139983149
Momentos de ligaccedilatildeo viga-pilar
983122983155983157983152 = 983122983145983150983142 =I
=(983138 983144983219) 12
=(20 40983219) 12
= 7619830970 983139983149983219983148983141 983148983141 2 280 2
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1127
983122983158983145983143983137 =I983158983145983143983137
=(15 70983219) 12
= 7719830977 983139983149983219983148983141983142 5554
Momento de engastamento
M983141983150983143 =983120 L983218
=3516 5554983218
= 98309703983096 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 9038 3 76190
3 76190 + 4 77197 + 3 76190
983117983155983157983152 983101 983117983145983150983142 983101 269830977 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 2697 + 2697
983117983158983145983143983137 983101 539830974 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147M983140983156983151983152983151 = 14 10 2697
983117983140983156983151983152983151 983101 3776 983147983118983149 983117983140983138983137983155983141 983101 9830853776 983147983118983149
Excentricidade inicial no topo e na base
983141983145 = M983140 N983140
983141983145 = 3776 983147N983139983149 25292 983147N
983141983145 983101 149830974 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1227
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 25292 (0015 + 003 04)
9831171983140983149983277983150 983101 69830963 983147983118983149
Iacutendice de esbeltes
λ = (346 983148983141) 983144
λ = (346 280) 40
λ 983101 2422
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 40)
10λ1 = 25
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λ = 2422 eacute menor que λ1 = 35 natildeo se considera efeitos de segunda ordem
assim adota-se M1d = 3776 kNm = 3776 kNcm
Coeficientes adimensionais
η =N983140
=25292
= 018983138 983144 983142983139983140 20 40 (2514)
983140=
4= 01
983144 40
μ = η 983141983156983151983156983137983148 = 018 1494 = 007983144 40
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1327
Utilizando o aacutebaco A-25 de VENTURINI (1987) obtecircm-se ω = 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 40 (2514) 005
= 164 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 40 = 64 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 40 = 32 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
983119B983123 A983140983151983156983137983154 A983155 = 32 983139983149983218
Detalhamento
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1427
Portanto o pilar ldquoP6rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
4 CALCULO DO PILAR DE CANTO ldquoP3rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147N983140 = 14 10 5622
983118983140 983101 798309671 983147983118
Vatildeo efetivo da viga
bull Eixo ldquoxrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1527
9831371 le05 9831561 = 05 30 = 150 983139983149
9831371 983101 150 98313998314905 983144 = 05 60 = 300 983139983149
9831372 le
05 9831562 = 05 25 = 125 983139983149
9831372 983101 125 98313998314905 983144 = 05 60 = 300 983139983149
983148983141983142 = 5289 + 125 + 15
983148983141983142 983101 5564 983139983149
bull Eixo ldquoyrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 = 9831372 le05 983156 = 05 20 = 100 983139983149
9831371 983101 9831372 983101 100 98313998314905 983144 = 05 50 = 250 983139983149
983148983141983142 = 4895 + 100 + 100
983148983141983142 983101 509830975 983139983149
Momentos de ligaccedilatildeo viga-pilar
bull
Eixo ldquoxrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(20 25983219) 12
= 1983096601 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 60983219) 12
= 398309698309621 983139983149983219983148983141983142 5564
bull Eixo ldquoyrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(25 20983219) 12
= 1198309705 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 50983219) 12
= 24534 983139983149983219983148983141983142 5095
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1627
Momento de engastamento
bull Eixo ldquoxrdquo
M983141983150983143 =983120 L983218
=1797 5564983218
= 4636 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 4636 3 18601
3 18601 + 4 38821 + 3 18601
983117983155983157983152 983101 983117983145983150983142 983101 9830976983097 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 969 + 969
983117983158983145983143983137 983101 19830973983096 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 969
983117983140983156983151983152983151 983101 1357 983147983118983149 983117983140983138983137983155983141 983101 9830851357 983147983118983149
bull Eixo ldquoyrdquo
M983141983150983143 =983120 L983218
=1631 5095983218
= 352983096 98314798311898314912 12
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1727
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 3528 3 11905
3 11905 + 4 24534 + 3 11905
983117983155983157983152 983101 983117983145983150983142 983101 743 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 743 + 743
983117983158983145983143983137 983101 149830966 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 743
983117983140983156983151983152983151 983101 1040 983147983118983149 983117983140983138983137983155983141 983101 9830851040 983147983118983149
Excentricidade inicial no topo e na base
Eixo ldquoxrdquo
983141983145 = M983140 N983140
983141983145 = 1357 983147N983139983149 7871 983147N
983141983145 983101 1724 983139983149
Eixo ldquoyrdquo
983141983145 = M983140 N983140
983141983145 = 1040 983147N983139983149 7871 983147N
983141983145 983101 1321 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1827
Momento miacutenimo
Eixo ldquoxrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 025)
9831171983140983149983277983150 983101 177 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1357 983147N983149
Eixo ldquoyrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 020)
9831171983140983149983277983150 983101 165 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1040 983147N983149
Iacutendice de esbeltes
Eixo ldquoxrdquo
λ = (346 983148983141) 983144
λ = (346 280) 25
λ 983101 398309675
Eixo ldquoyrdquo
λ = (346 983148983141) 983144
λ = (346 280) 20
λ 983101 498309644
Iacutendice de esbeltes limite
Eixo ldquoxrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1724 25)
10λ1 = 3362
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Eixo ldquoyrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1321 20)
10λ1 = 3326
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
Eixo ldquoxrdquo
η =N983140
=7871
= 009983138 983144 983142983139983140 20 25 (2514)
1
=
0005
le
0005
983154 983144 (η + 05) 983144
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2027
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2127
19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2227
Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2327
Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 927
Cargas totais na viga
Reaccediloes nos apoios
23 Cargas nos pilares
P3
Reaccedilatildeo da Viga 101 2536 kNReaccedilatildeo da Viga 106 2736 kNPeso Proacuteprio do Pilar(020 m 025 m 280 m 25 kNmsup3) 350 kNTotal 5622 kN
P5
Reaccedilatildeo da Viga 102 36225 kN
Reaccedilatildeo da Viga 105 24341 kNPeso Proacuteprio do Pilar(025 m 025 m 280 m 25 kNmsup3) 438 kNTotal 61004 kN
P6
Reaccedilatildeo da Viga 102 4937 kNReaccedilatildeo da Viga 106 12569 kNPeso Proacuteprio do Pilar(020 m 040 m 280 m 25 kNmsup3) 560 kNTotal 18066 kN
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1027
3 CALCULO DO PILAR DE EXTREMIDADE ldquoP6rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 18066
983118983140 983101 2529830972 983147983118
Vatildeo efetivo da viga
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 le05 9831561 = 05 25 = 125 983139983149
9831371 983101 125 98313998314905 983144 = 05 70 = 350 983139983149
9831372 le05 9831562 = 05 40 = 200 983139983149
9831372 983101 200 98313998314905 983144 = 05 70 = 350 983139983149
983148983141983142 = 5229 + 125 + 20
983148983141983142 983101 5554 983139983149
Momentos de ligaccedilatildeo viga-pilar
983122983155983157983152 = 983122983145983150983142 =I
=(983138 983144983219) 12
=(20 40983219) 12
= 7619830970 983139983149983219983148983141 983148983141 2 280 2
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1127
983122983158983145983143983137 =I983158983145983143983137
=(15 70983219) 12
= 7719830977 983139983149983219983148983141983142 5554
Momento de engastamento
M983141983150983143 =983120 L983218
=3516 5554983218
= 98309703983096 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 9038 3 76190
3 76190 + 4 77197 + 3 76190
983117983155983157983152 983101 983117983145983150983142 983101 269830977 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 2697 + 2697
983117983158983145983143983137 983101 539830974 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147M983140983156983151983152983151 = 14 10 2697
983117983140983156983151983152983151 983101 3776 983147983118983149 983117983140983138983137983155983141 983101 9830853776 983147983118983149
Excentricidade inicial no topo e na base
983141983145 = M983140 N983140
983141983145 = 3776 983147N983139983149 25292 983147N
983141983145 983101 149830974 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1227
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 25292 (0015 + 003 04)
9831171983140983149983277983150 983101 69830963 983147983118983149
Iacutendice de esbeltes
λ = (346 983148983141) 983144
λ = (346 280) 40
λ 983101 2422
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 40)
10λ1 = 25
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λ = 2422 eacute menor que λ1 = 35 natildeo se considera efeitos de segunda ordem
assim adota-se M1d = 3776 kNm = 3776 kNcm
Coeficientes adimensionais
η =N983140
=25292
= 018983138 983144 983142983139983140 20 40 (2514)
983140=
4= 01
983144 40
μ = η 983141983156983151983156983137983148 = 018 1494 = 007983144 40
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1327
Utilizando o aacutebaco A-25 de VENTURINI (1987) obtecircm-se ω = 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 40 (2514) 005
= 164 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 40 = 64 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 40 = 32 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
983119B983123 A983140983151983156983137983154 A983155 = 32 983139983149983218
Detalhamento
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1427
Portanto o pilar ldquoP6rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
4 CALCULO DO PILAR DE CANTO ldquoP3rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147N983140 = 14 10 5622
983118983140 983101 798309671 983147983118
Vatildeo efetivo da viga
bull Eixo ldquoxrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1527
9831371 le05 9831561 = 05 30 = 150 983139983149
9831371 983101 150 98313998314905 983144 = 05 60 = 300 983139983149
9831372 le
05 9831562 = 05 25 = 125 983139983149
9831372 983101 125 98313998314905 983144 = 05 60 = 300 983139983149
983148983141983142 = 5289 + 125 + 15
983148983141983142 983101 5564 983139983149
bull Eixo ldquoyrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 = 9831372 le05 983156 = 05 20 = 100 983139983149
9831371 983101 9831372 983101 100 98313998314905 983144 = 05 50 = 250 983139983149
983148983141983142 = 4895 + 100 + 100
983148983141983142 983101 509830975 983139983149
Momentos de ligaccedilatildeo viga-pilar
bull
Eixo ldquoxrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(20 25983219) 12
= 1983096601 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 60983219) 12
= 398309698309621 983139983149983219983148983141983142 5564
bull Eixo ldquoyrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(25 20983219) 12
= 1198309705 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 50983219) 12
= 24534 983139983149983219983148983141983142 5095
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1627
Momento de engastamento
bull Eixo ldquoxrdquo
M983141983150983143 =983120 L983218
=1797 5564983218
= 4636 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 4636 3 18601
3 18601 + 4 38821 + 3 18601
983117983155983157983152 983101 983117983145983150983142 983101 9830976983097 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 969 + 969
983117983158983145983143983137 983101 19830973983096 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 969
983117983140983156983151983152983151 983101 1357 983147983118983149 983117983140983138983137983155983141 983101 9830851357 983147983118983149
bull Eixo ldquoyrdquo
M983141983150983143 =983120 L983218
=1631 5095983218
= 352983096 98314798311898314912 12
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1727
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 3528 3 11905
3 11905 + 4 24534 + 3 11905
983117983155983157983152 983101 983117983145983150983142 983101 743 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 743 + 743
983117983158983145983143983137 983101 149830966 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 743
983117983140983156983151983152983151 983101 1040 983147983118983149 983117983140983138983137983155983141 983101 9830851040 983147983118983149
Excentricidade inicial no topo e na base
Eixo ldquoxrdquo
983141983145 = M983140 N983140
983141983145 = 1357 983147N983139983149 7871 983147N
983141983145 983101 1724 983139983149
Eixo ldquoyrdquo
983141983145 = M983140 N983140
983141983145 = 1040 983147N983139983149 7871 983147N
983141983145 983101 1321 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1827
Momento miacutenimo
Eixo ldquoxrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 025)
9831171983140983149983277983150 983101 177 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1357 983147N983149
Eixo ldquoyrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 020)
9831171983140983149983277983150 983101 165 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1040 983147N983149
Iacutendice de esbeltes
Eixo ldquoxrdquo
λ = (346 983148983141) 983144
λ = (346 280) 25
λ 983101 398309675
Eixo ldquoyrdquo
λ = (346 983148983141) 983144
λ = (346 280) 20
λ 983101 498309644
Iacutendice de esbeltes limite
Eixo ldquoxrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1724 25)
10λ1 = 3362
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Eixo ldquoyrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1321 20)
10λ1 = 3326
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
Eixo ldquoxrdquo
η =N983140
=7871
= 009983138 983144 983142983139983140 20 25 (2514)
1
=
0005
le
0005
983154 983144 (η + 05) 983144
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2027
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2127
19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1027
3 CALCULO DO PILAR DE EXTREMIDADE ldquoP6rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 18066
983118983140 983101 2529830972 983147983118
Vatildeo efetivo da viga
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 le05 9831561 = 05 25 = 125 983139983149
9831371 983101 125 98313998314905 983144 = 05 70 = 350 983139983149
9831372 le05 9831562 = 05 40 = 200 983139983149
9831372 983101 200 98313998314905 983144 = 05 70 = 350 983139983149
983148983141983142 = 5229 + 125 + 20
983148983141983142 983101 5554 983139983149
Momentos de ligaccedilatildeo viga-pilar
983122983155983157983152 = 983122983145983150983142 =I
=(983138 983144983219) 12
=(20 40983219) 12
= 7619830970 983139983149983219983148983141 983148983141 2 280 2
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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983122983158983145983143983137 =I983158983145983143983137
=(15 70983219) 12
= 7719830977 983139983149983219983148983141983142 5554
Momento de engastamento
M983141983150983143 =983120 L983218
=3516 5554983218
= 98309703983096 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 9038 3 76190
3 76190 + 4 77197 + 3 76190
983117983155983157983152 983101 983117983145983150983142 983101 269830977 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 2697 + 2697
983117983158983145983143983137 983101 539830974 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147M983140983156983151983152983151 = 14 10 2697
983117983140983156983151983152983151 983101 3776 983147983118983149 983117983140983138983137983155983141 983101 9830853776 983147983118983149
Excentricidade inicial no topo e na base
983141983145 = M983140 N983140
983141983145 = 3776 983147N983139983149 25292 983147N
983141983145 983101 149830974 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 25292 (0015 + 003 04)
9831171983140983149983277983150 983101 69830963 983147983118983149
Iacutendice de esbeltes
λ = (346 983148983141) 983144
λ = (346 280) 40
λ 983101 2422
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 40)
10λ1 = 25
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λ = 2422 eacute menor que λ1 = 35 natildeo se considera efeitos de segunda ordem
assim adota-se M1d = 3776 kNm = 3776 kNcm
Coeficientes adimensionais
η =N983140
=25292
= 018983138 983144 983142983139983140 20 40 (2514)
983140=
4= 01
983144 40
μ = η 983141983156983151983156983137983148 = 018 1494 = 007983144 40
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Utilizando o aacutebaco A-25 de VENTURINI (1987) obtecircm-se ω = 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 40 (2514) 005
= 164 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 40 = 64 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 40 = 32 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
983119B983123 A983140983151983156983137983154 A983155 = 32 983139983149983218
Detalhamento
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1427
Portanto o pilar ldquoP6rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
4 CALCULO DO PILAR DE CANTO ldquoP3rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147N983140 = 14 10 5622
983118983140 983101 798309671 983147983118
Vatildeo efetivo da viga
bull Eixo ldquoxrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1527
9831371 le05 9831561 = 05 30 = 150 983139983149
9831371 983101 150 98313998314905 983144 = 05 60 = 300 983139983149
9831372 le
05 9831562 = 05 25 = 125 983139983149
9831372 983101 125 98313998314905 983144 = 05 60 = 300 983139983149
983148983141983142 = 5289 + 125 + 15
983148983141983142 983101 5564 983139983149
bull Eixo ldquoyrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 = 9831372 le05 983156 = 05 20 = 100 983139983149
9831371 983101 9831372 983101 100 98313998314905 983144 = 05 50 = 250 983139983149
983148983141983142 = 4895 + 100 + 100
983148983141983142 983101 509830975 983139983149
Momentos de ligaccedilatildeo viga-pilar
bull
Eixo ldquoxrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(20 25983219) 12
= 1983096601 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 60983219) 12
= 398309698309621 983139983149983219983148983141983142 5564
bull Eixo ldquoyrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(25 20983219) 12
= 1198309705 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 50983219) 12
= 24534 983139983149983219983148983141983142 5095
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1627
Momento de engastamento
bull Eixo ldquoxrdquo
M983141983150983143 =983120 L983218
=1797 5564983218
= 4636 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 4636 3 18601
3 18601 + 4 38821 + 3 18601
983117983155983157983152 983101 983117983145983150983142 983101 9830976983097 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 969 + 969
983117983158983145983143983137 983101 19830973983096 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 969
983117983140983156983151983152983151 983101 1357 983147983118983149 983117983140983138983137983155983141 983101 9830851357 983147983118983149
bull Eixo ldquoyrdquo
M983141983150983143 =983120 L983218
=1631 5095983218
= 352983096 98314798311898314912 12
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1727
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 3528 3 11905
3 11905 + 4 24534 + 3 11905
983117983155983157983152 983101 983117983145983150983142 983101 743 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 743 + 743
983117983158983145983143983137 983101 149830966 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 743
983117983140983156983151983152983151 983101 1040 983147983118983149 983117983140983138983137983155983141 983101 9830851040 983147983118983149
Excentricidade inicial no topo e na base
Eixo ldquoxrdquo
983141983145 = M983140 N983140
983141983145 = 1357 983147N983139983149 7871 983147N
983141983145 983101 1724 983139983149
Eixo ldquoyrdquo
983141983145 = M983140 N983140
983141983145 = 1040 983147N983139983149 7871 983147N
983141983145 983101 1321 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1827
Momento miacutenimo
Eixo ldquoxrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 025)
9831171983140983149983277983150 983101 177 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1357 983147N983149
Eixo ldquoyrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 020)
9831171983140983149983277983150 983101 165 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1040 983147N983149
Iacutendice de esbeltes
Eixo ldquoxrdquo
λ = (346 983148983141) 983144
λ = (346 280) 25
λ 983101 398309675
Eixo ldquoyrdquo
λ = (346 983148983141) 983144
λ = (346 280) 20
λ 983101 498309644
Iacutendice de esbeltes limite
Eixo ldquoxrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1724 25)
10λ1 = 3362
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Eixo ldquoyrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1321 20)
10λ1 = 3326
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
Eixo ldquoxrdquo
η =N983140
=7871
= 009983138 983144 983142983139983140 20 25 (2514)
1
=
0005
le
0005
983154 983144 (η + 05) 983144
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2027
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2127
19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2227
Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2327
Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1127
983122983158983145983143983137 =I983158983145983143983137
=(15 70983219) 12
= 7719830977 983139983149983219983148983141983142 5554
Momento de engastamento
M983141983150983143 =983120 L983218
=3516 5554983218
= 98309703983096 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 9038 3 76190
3 76190 + 4 77197 + 3 76190
983117983155983157983152 983101 983117983145983150983142 983101 269830977 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 2697 + 2697
983117983158983145983143983137 983101 539830974 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147M983140983156983151983152983151 = 14 10 2697
983117983140983156983151983152983151 983101 3776 983147983118983149 983117983140983138983137983155983141 983101 9830853776 983147983118983149
Excentricidade inicial no topo e na base
983141983145 = M983140 N983140
983141983145 = 3776 983147N983139983149 25292 983147N
983141983145 983101 149830974 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1227
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 25292 (0015 + 003 04)
9831171983140983149983277983150 983101 69830963 983147983118983149
Iacutendice de esbeltes
λ = (346 983148983141) 983144
λ = (346 280) 40
λ 983101 2422
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 40)
10λ1 = 25
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λ = 2422 eacute menor que λ1 = 35 natildeo se considera efeitos de segunda ordem
assim adota-se M1d = 3776 kNm = 3776 kNcm
Coeficientes adimensionais
η =N983140
=25292
= 018983138 983144 983142983139983140 20 40 (2514)
983140=
4= 01
983144 40
μ = η 983141983156983151983156983137983148 = 018 1494 = 007983144 40
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1327
Utilizando o aacutebaco A-25 de VENTURINI (1987) obtecircm-se ω = 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 40 (2514) 005
= 164 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 40 = 64 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 40 = 32 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
983119B983123 A983140983151983156983137983154 A983155 = 32 983139983149983218
Detalhamento
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1427
Portanto o pilar ldquoP6rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
4 CALCULO DO PILAR DE CANTO ldquoP3rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147N983140 = 14 10 5622
983118983140 983101 798309671 983147983118
Vatildeo efetivo da viga
bull Eixo ldquoxrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1527
9831371 le05 9831561 = 05 30 = 150 983139983149
9831371 983101 150 98313998314905 983144 = 05 60 = 300 983139983149
9831372 le
05 9831562 = 05 25 = 125 983139983149
9831372 983101 125 98313998314905 983144 = 05 60 = 300 983139983149
983148983141983142 = 5289 + 125 + 15
983148983141983142 983101 5564 983139983149
bull Eixo ldquoyrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 = 9831372 le05 983156 = 05 20 = 100 983139983149
9831371 983101 9831372 983101 100 98313998314905 983144 = 05 50 = 250 983139983149
983148983141983142 = 4895 + 100 + 100
983148983141983142 983101 509830975 983139983149
Momentos de ligaccedilatildeo viga-pilar
bull
Eixo ldquoxrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(20 25983219) 12
= 1983096601 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 60983219) 12
= 398309698309621 983139983149983219983148983141983142 5564
bull Eixo ldquoyrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(25 20983219) 12
= 1198309705 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 50983219) 12
= 24534 983139983149983219983148983141983142 5095
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1627
Momento de engastamento
bull Eixo ldquoxrdquo
M983141983150983143 =983120 L983218
=1797 5564983218
= 4636 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 4636 3 18601
3 18601 + 4 38821 + 3 18601
983117983155983157983152 983101 983117983145983150983142 983101 9830976983097 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 969 + 969
983117983158983145983143983137 983101 19830973983096 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 969
983117983140983156983151983152983151 983101 1357 983147983118983149 983117983140983138983137983155983141 983101 9830851357 983147983118983149
bull Eixo ldquoyrdquo
M983141983150983143 =983120 L983218
=1631 5095983218
= 352983096 98314798311898314912 12
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1727
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 3528 3 11905
3 11905 + 4 24534 + 3 11905
983117983155983157983152 983101 983117983145983150983142 983101 743 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 743 + 743
983117983158983145983143983137 983101 149830966 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 743
983117983140983156983151983152983151 983101 1040 983147983118983149 983117983140983138983137983155983141 983101 9830851040 983147983118983149
Excentricidade inicial no topo e na base
Eixo ldquoxrdquo
983141983145 = M983140 N983140
983141983145 = 1357 983147N983139983149 7871 983147N
983141983145 983101 1724 983139983149
Eixo ldquoyrdquo
983141983145 = M983140 N983140
983141983145 = 1040 983147N983139983149 7871 983147N
983141983145 983101 1321 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1827
Momento miacutenimo
Eixo ldquoxrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 025)
9831171983140983149983277983150 983101 177 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1357 983147N983149
Eixo ldquoyrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 020)
9831171983140983149983277983150 983101 165 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1040 983147N983149
Iacutendice de esbeltes
Eixo ldquoxrdquo
λ = (346 983148983141) 983144
λ = (346 280) 25
λ 983101 398309675
Eixo ldquoyrdquo
λ = (346 983148983141) 983144
λ = (346 280) 20
λ 983101 498309644
Iacutendice de esbeltes limite
Eixo ldquoxrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1724 25)
10λ1 = 3362
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Eixo ldquoyrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1321 20)
10λ1 = 3326
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
Eixo ldquoxrdquo
η =N983140
=7871
= 009983138 983144 983142983139983140 20 25 (2514)
1
=
0005
le
0005
983154 983144 (η + 05) 983144
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2027
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2127
19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2227
Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2327
Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1227
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 25292 (0015 + 003 04)
9831171983140983149983277983150 983101 69830963 983147983118983149
Iacutendice de esbeltes
λ = (346 983148983141) 983144
λ = (346 280) 40
λ 983101 2422
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 40)
10λ1 = 25
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λ = 2422 eacute menor que λ1 = 35 natildeo se considera efeitos de segunda ordem
assim adota-se M1d = 3776 kNm = 3776 kNcm
Coeficientes adimensionais
η =N983140
=25292
= 018983138 983144 983142983139983140 20 40 (2514)
983140=
4= 01
983144 40
μ = η 983141983156983151983156983137983148 = 018 1494 = 007983144 40
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1327
Utilizando o aacutebaco A-25 de VENTURINI (1987) obtecircm-se ω = 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 40 (2514) 005
= 164 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 40 = 64 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 40 = 32 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
983119B983123 A983140983151983156983137983154 A983155 = 32 983139983149983218
Detalhamento
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1427
Portanto o pilar ldquoP6rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
4 CALCULO DO PILAR DE CANTO ldquoP3rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147N983140 = 14 10 5622
983118983140 983101 798309671 983147983118
Vatildeo efetivo da viga
bull Eixo ldquoxrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1527
9831371 le05 9831561 = 05 30 = 150 983139983149
9831371 983101 150 98313998314905 983144 = 05 60 = 300 983139983149
9831372 le
05 9831562 = 05 25 = 125 983139983149
9831372 983101 125 98313998314905 983144 = 05 60 = 300 983139983149
983148983141983142 = 5289 + 125 + 15
983148983141983142 983101 5564 983139983149
bull Eixo ldquoyrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 = 9831372 le05 983156 = 05 20 = 100 983139983149
9831371 983101 9831372 983101 100 98313998314905 983144 = 05 50 = 250 983139983149
983148983141983142 = 4895 + 100 + 100
983148983141983142 983101 509830975 983139983149
Momentos de ligaccedilatildeo viga-pilar
bull
Eixo ldquoxrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(20 25983219) 12
= 1983096601 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 60983219) 12
= 398309698309621 983139983149983219983148983141983142 5564
bull Eixo ldquoyrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(25 20983219) 12
= 1198309705 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 50983219) 12
= 24534 983139983149983219983148983141983142 5095
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1627
Momento de engastamento
bull Eixo ldquoxrdquo
M983141983150983143 =983120 L983218
=1797 5564983218
= 4636 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 4636 3 18601
3 18601 + 4 38821 + 3 18601
983117983155983157983152 983101 983117983145983150983142 983101 9830976983097 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 969 + 969
983117983158983145983143983137 983101 19830973983096 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 969
983117983140983156983151983152983151 983101 1357 983147983118983149 983117983140983138983137983155983141 983101 9830851357 983147983118983149
bull Eixo ldquoyrdquo
M983141983150983143 =983120 L983218
=1631 5095983218
= 352983096 98314798311898314912 12
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1727
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 3528 3 11905
3 11905 + 4 24534 + 3 11905
983117983155983157983152 983101 983117983145983150983142 983101 743 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 743 + 743
983117983158983145983143983137 983101 149830966 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 743
983117983140983156983151983152983151 983101 1040 983147983118983149 983117983140983138983137983155983141 983101 9830851040 983147983118983149
Excentricidade inicial no topo e na base
Eixo ldquoxrdquo
983141983145 = M983140 N983140
983141983145 = 1357 983147N983139983149 7871 983147N
983141983145 983101 1724 983139983149
Eixo ldquoyrdquo
983141983145 = M983140 N983140
983141983145 = 1040 983147N983139983149 7871 983147N
983141983145 983101 1321 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1827
Momento miacutenimo
Eixo ldquoxrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 025)
9831171983140983149983277983150 983101 177 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1357 983147N983149
Eixo ldquoyrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 020)
9831171983140983149983277983150 983101 165 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1040 983147N983149
Iacutendice de esbeltes
Eixo ldquoxrdquo
λ = (346 983148983141) 983144
λ = (346 280) 25
λ 983101 398309675
Eixo ldquoyrdquo
λ = (346 983148983141) 983144
λ = (346 280) 20
λ 983101 498309644
Iacutendice de esbeltes limite
Eixo ldquoxrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1724 25)
10λ1 = 3362
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Eixo ldquoyrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1321 20)
10λ1 = 3326
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
Eixo ldquoxrdquo
η =N983140
=7871
= 009983138 983144 983142983139983140 20 25 (2514)
1
=
0005
le
0005
983154 983144 (η + 05) 983144
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2027
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2127
19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2227
Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2327
Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1327
Utilizando o aacutebaco A-25 de VENTURINI (1987) obtecircm-se ω = 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 40 (2514) 005
= 164 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 40 = 64 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 40 = 32 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
983119B983123 A983140983151983156983137983154 A983155 = 32 983139983149983218
Detalhamento
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1427
Portanto o pilar ldquoP6rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
4 CALCULO DO PILAR DE CANTO ldquoP3rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147N983140 = 14 10 5622
983118983140 983101 798309671 983147983118
Vatildeo efetivo da viga
bull Eixo ldquoxrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1527
9831371 le05 9831561 = 05 30 = 150 983139983149
9831371 983101 150 98313998314905 983144 = 05 60 = 300 983139983149
9831372 le
05 9831562 = 05 25 = 125 983139983149
9831372 983101 125 98313998314905 983144 = 05 60 = 300 983139983149
983148983141983142 = 5289 + 125 + 15
983148983141983142 983101 5564 983139983149
bull Eixo ldquoyrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 = 9831372 le05 983156 = 05 20 = 100 983139983149
9831371 983101 9831372 983101 100 98313998314905 983144 = 05 50 = 250 983139983149
983148983141983142 = 4895 + 100 + 100
983148983141983142 983101 509830975 983139983149
Momentos de ligaccedilatildeo viga-pilar
bull
Eixo ldquoxrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(20 25983219) 12
= 1983096601 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 60983219) 12
= 398309698309621 983139983149983219983148983141983142 5564
bull Eixo ldquoyrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(25 20983219) 12
= 1198309705 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 50983219) 12
= 24534 983139983149983219983148983141983142 5095
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1627
Momento de engastamento
bull Eixo ldquoxrdquo
M983141983150983143 =983120 L983218
=1797 5564983218
= 4636 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 4636 3 18601
3 18601 + 4 38821 + 3 18601
983117983155983157983152 983101 983117983145983150983142 983101 9830976983097 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 969 + 969
983117983158983145983143983137 983101 19830973983096 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 969
983117983140983156983151983152983151 983101 1357 983147983118983149 983117983140983138983137983155983141 983101 9830851357 983147983118983149
bull Eixo ldquoyrdquo
M983141983150983143 =983120 L983218
=1631 5095983218
= 352983096 98314798311898314912 12
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1727
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 3528 3 11905
3 11905 + 4 24534 + 3 11905
983117983155983157983152 983101 983117983145983150983142 983101 743 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 743 + 743
983117983158983145983143983137 983101 149830966 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 743
983117983140983156983151983152983151 983101 1040 983147983118983149 983117983140983138983137983155983141 983101 9830851040 983147983118983149
Excentricidade inicial no topo e na base
Eixo ldquoxrdquo
983141983145 = M983140 N983140
983141983145 = 1357 983147N983139983149 7871 983147N
983141983145 983101 1724 983139983149
Eixo ldquoyrdquo
983141983145 = M983140 N983140
983141983145 = 1040 983147N983139983149 7871 983147N
983141983145 983101 1321 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1827
Momento miacutenimo
Eixo ldquoxrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 025)
9831171983140983149983277983150 983101 177 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1357 983147N983149
Eixo ldquoyrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 020)
9831171983140983149983277983150 983101 165 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1040 983147N983149
Iacutendice de esbeltes
Eixo ldquoxrdquo
λ = (346 983148983141) 983144
λ = (346 280) 25
λ 983101 398309675
Eixo ldquoyrdquo
λ = (346 983148983141) 983144
λ = (346 280) 20
λ 983101 498309644
Iacutendice de esbeltes limite
Eixo ldquoxrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1724 25)
10λ1 = 3362
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Eixo ldquoyrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1321 20)
10λ1 = 3326
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
Eixo ldquoxrdquo
η =N983140
=7871
= 009983138 983144 983142983139983140 20 25 (2514)
1
=
0005
le
0005
983154 983144 (η + 05) 983144
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2027
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2127
19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2227
Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2327
Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1427
Portanto o pilar ldquoP6rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
4 CALCULO DO PILAR DE CANTO ldquoP3rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147N983140 = 14 10 5622
983118983140 983101 798309671 983147983118
Vatildeo efetivo da viga
bull Eixo ldquoxrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1527
9831371 le05 9831561 = 05 30 = 150 983139983149
9831371 983101 150 98313998314905 983144 = 05 60 = 300 983139983149
9831372 le
05 9831562 = 05 25 = 125 983139983149
9831372 983101 125 98313998314905 983144 = 05 60 = 300 983139983149
983148983141983142 = 5289 + 125 + 15
983148983141983142 983101 5564 983139983149
bull Eixo ldquoyrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 = 9831372 le05 983156 = 05 20 = 100 983139983149
9831371 983101 9831372 983101 100 98313998314905 983144 = 05 50 = 250 983139983149
983148983141983142 = 4895 + 100 + 100
983148983141983142 983101 509830975 983139983149
Momentos de ligaccedilatildeo viga-pilar
bull
Eixo ldquoxrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(20 25983219) 12
= 1983096601 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 60983219) 12
= 398309698309621 983139983149983219983148983141983142 5564
bull Eixo ldquoyrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(25 20983219) 12
= 1198309705 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 50983219) 12
= 24534 983139983149983219983148983141983142 5095
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1627
Momento de engastamento
bull Eixo ldquoxrdquo
M983141983150983143 =983120 L983218
=1797 5564983218
= 4636 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 4636 3 18601
3 18601 + 4 38821 + 3 18601
983117983155983157983152 983101 983117983145983150983142 983101 9830976983097 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 969 + 969
983117983158983145983143983137 983101 19830973983096 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 969
983117983140983156983151983152983151 983101 1357 983147983118983149 983117983140983138983137983155983141 983101 9830851357 983147983118983149
bull Eixo ldquoyrdquo
M983141983150983143 =983120 L983218
=1631 5095983218
= 352983096 98314798311898314912 12
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1727
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 3528 3 11905
3 11905 + 4 24534 + 3 11905
983117983155983157983152 983101 983117983145983150983142 983101 743 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 743 + 743
983117983158983145983143983137 983101 149830966 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 743
983117983140983156983151983152983151 983101 1040 983147983118983149 983117983140983138983137983155983141 983101 9830851040 983147983118983149
Excentricidade inicial no topo e na base
Eixo ldquoxrdquo
983141983145 = M983140 N983140
983141983145 = 1357 983147N983139983149 7871 983147N
983141983145 983101 1724 983139983149
Eixo ldquoyrdquo
983141983145 = M983140 N983140
983141983145 = 1040 983147N983139983149 7871 983147N
983141983145 983101 1321 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1827
Momento miacutenimo
Eixo ldquoxrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 025)
9831171983140983149983277983150 983101 177 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1357 983147N983149
Eixo ldquoyrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 020)
9831171983140983149983277983150 983101 165 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1040 983147N983149
Iacutendice de esbeltes
Eixo ldquoxrdquo
λ = (346 983148983141) 983144
λ = (346 280) 25
λ 983101 398309675
Eixo ldquoyrdquo
λ = (346 983148983141) 983144
λ = (346 280) 20
λ 983101 498309644
Iacutendice de esbeltes limite
Eixo ldquoxrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1724 25)
10λ1 = 3362
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Eixo ldquoyrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1321 20)
10λ1 = 3326
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
Eixo ldquoxrdquo
η =N983140
=7871
= 009983138 983144 983142983139983140 20 25 (2514)
1
=
0005
le
0005
983154 983144 (η + 05) 983144
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2027
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2127
19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2227
Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2327
Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1527
9831371 le05 9831561 = 05 30 = 150 983139983149
9831371 983101 150 98313998314905 983144 = 05 60 = 300 983139983149
9831372 le
05 9831562 = 05 25 = 125 983139983149
9831372 983101 125 98313998314905 983144 = 05 60 = 300 983139983149
983148983141983142 = 5289 + 125 + 15
983148983141983142 983101 5564 983139983149
bull Eixo ldquoyrdquo
983148983141983142 = 983148983151 + 9831371 + 9831372
9831371 = 9831372 le05 983156 = 05 20 = 100 983139983149
9831371 983101 9831372 983101 100 98313998314905 983144 = 05 50 = 250 983139983149
983148983141983142 = 4895 + 100 + 100
983148983141983142 983101 509830975 983139983149
Momentos de ligaccedilatildeo viga-pilar
bull
Eixo ldquoxrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(20 25983219) 12
= 1983096601 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 60983219) 12
= 398309698309621 983139983149983219983148983141983142 5564
bull Eixo ldquoyrdquo
983122983155983157983152 = 983122983145983150983142 =I
=(983138 H983219) 12
=(25 20983219) 12
= 1198309705 983139983149983219983148983141 983148983141 2 280 2
983122983158983145983143983137 =I983158983145983143983137
=(15 50983219) 12
= 24534 983139983149983219983148983141983142 5095
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1627
Momento de engastamento
bull Eixo ldquoxrdquo
M983141983150983143 =983120 L983218
=1797 5564983218
= 4636 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 4636 3 18601
3 18601 + 4 38821 + 3 18601
983117983155983157983152 983101 983117983145983150983142 983101 9830976983097 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 969 + 969
983117983158983145983143983137 983101 19830973983096 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 969
983117983140983156983151983152983151 983101 1357 983147983118983149 983117983140983138983137983155983141 983101 9830851357 983147983118983149
bull Eixo ldquoyrdquo
M983141983150983143 =983120 L983218
=1631 5095983218
= 352983096 98314798311898314912 12
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1727
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 3528 3 11905
3 11905 + 4 24534 + 3 11905
983117983155983157983152 983101 983117983145983150983142 983101 743 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 743 + 743
983117983158983145983143983137 983101 149830966 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 743
983117983140983156983151983152983151 983101 1040 983147983118983149 983117983140983138983137983155983141 983101 9830851040 983147983118983149
Excentricidade inicial no topo e na base
Eixo ldquoxrdquo
983141983145 = M983140 N983140
983141983145 = 1357 983147N983139983149 7871 983147N
983141983145 983101 1724 983139983149
Eixo ldquoyrdquo
983141983145 = M983140 N983140
983141983145 = 1040 983147N983139983149 7871 983147N
983141983145 983101 1321 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1827
Momento miacutenimo
Eixo ldquoxrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 025)
9831171983140983149983277983150 983101 177 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1357 983147N983149
Eixo ldquoyrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 020)
9831171983140983149983277983150 983101 165 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1040 983147N983149
Iacutendice de esbeltes
Eixo ldquoxrdquo
λ = (346 983148983141) 983144
λ = (346 280) 25
λ 983101 398309675
Eixo ldquoyrdquo
λ = (346 983148983141) 983144
λ = (346 280) 20
λ 983101 498309644
Iacutendice de esbeltes limite
Eixo ldquoxrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1724 25)
10λ1 = 3362
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Eixo ldquoyrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1321 20)
10λ1 = 3326
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
Eixo ldquoxrdquo
η =N983140
=7871
= 009983138 983144 983142983139983140 20 25 (2514)
1
=
0005
le
0005
983154 983144 (η + 05) 983144
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2027
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2127
19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2227
Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2327
Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1627
Momento de engastamento
bull Eixo ldquoxrdquo
M983141983150983143 =983120 L983218
=1797 5564983218
= 4636 98314798311898314912 12
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 4636 3 18601
3 18601 + 4 38821 + 3 18601
983117983155983157983152 983101 983117983145983150983142 983101 9830976983097 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 969 + 969
983117983158983145983143983137 983101 19830973983096 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 969
983117983140983156983151983152983151 983101 1357 983147983118983149 983117983140983138983137983155983141 983101 9830851357 983147983118983149
bull Eixo ldquoyrdquo
M983141983150983143 =983120 L983218
=1631 5095983218
= 352983096 98314798311898314912 12
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1727
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 3528 3 11905
3 11905 + 4 24534 + 3 11905
983117983155983157983152 983101 983117983145983150983142 983101 743 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 743 + 743
983117983158983145983143983137 983101 149830966 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 743
983117983140983156983151983152983151 983101 1040 983147983118983149 983117983140983138983137983155983141 983101 9830851040 983147983118983149
Excentricidade inicial no topo e na base
Eixo ldquoxrdquo
983141983145 = M983140 N983140
983141983145 = 1357 983147N983139983149 7871 983147N
983141983145 983101 1724 983139983149
Eixo ldquoyrdquo
983141983145 = M983140 N983140
983141983145 = 1040 983147N983139983149 7871 983147N
983141983145 983101 1321 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1827
Momento miacutenimo
Eixo ldquoxrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 025)
9831171983140983149983277983150 983101 177 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1357 983147N983149
Eixo ldquoyrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 020)
9831171983140983149983277983150 983101 165 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1040 983147N983149
Iacutendice de esbeltes
Eixo ldquoxrdquo
λ = (346 983148983141) 983144
λ = (346 280) 25
λ 983101 398309675
Eixo ldquoyrdquo
λ = (346 983148983141) 983144
λ = (346 280) 20
λ 983101 498309644
Iacutendice de esbeltes limite
Eixo ldquoxrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1724 25)
10λ1 = 3362
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Eixo ldquoyrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1321 20)
10λ1 = 3326
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
Eixo ldquoxrdquo
η =N983140
=7871
= 009983138 983144 983142983139983140 20 25 (2514)
1
=
0005
le
0005
983154 983144 (η + 05) 983144
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2027
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2127
19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2227
Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2327
Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1727
M983155983157983152 = M983141983150983143 3 983122983155983157983152
3 983122983155983157983152 + 4 983122983158983145983143983137 + 3 983122983145983150983142
M983145983150983142 = M983141983150983143 3 983122983145983150983142
3 983122983145983150983142 + 4 983122983158983145983143983137 + 3 983122983155983157983152
M983155983157983152 = M983145983150983142
M983155983157983152 = M983145983150983142 = 3528 3 11905
3 11905 + 4 24534 + 3 11905
983117983155983157983152 983101 983117983145983150983142 983101 743 983147983118983149
M983158983145983143983137 = M983155983157983152 + M983145983150983142
M983158983145983143983137 = 743 + 743
983117983158983145983143983137 983101 149830966 983147983118983149
M983140983156983151983152983151 = 983085M983140 983138983137983155983141
M983140983156983151983152983151 = 983129983139 983129983150 M983147
M983140983156983151983152983151 = 14 10 743
983117983140983156983151983152983151 983101 1040 983147983118983149 983117983140983138983137983155983141 983101 9830851040 983147983118983149
Excentricidade inicial no topo e na base
Eixo ldquoxrdquo
983141983145 = M983140 N983140
983141983145 = 1357 983147N983139983149 7871 983147N
983141983145 983101 1724 983139983149
Eixo ldquoyrdquo
983141983145 = M983140 N983140
983141983145 = 1040 983147N983139983149 7871 983147N
983141983145 983101 1321 983139983149
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1827
Momento miacutenimo
Eixo ldquoxrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 025)
9831171983140983149983277983150 983101 177 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1357 983147N983149
Eixo ldquoyrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 020)
9831171983140983149983277983150 983101 165 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1040 983147N983149
Iacutendice de esbeltes
Eixo ldquoxrdquo
λ = (346 983148983141) 983144
λ = (346 280) 25
λ 983101 398309675
Eixo ldquoyrdquo
λ = (346 983148983141) 983144
λ = (346 280) 20
λ 983101 498309644
Iacutendice de esbeltes limite
Eixo ldquoxrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1724 25)
10λ1 = 3362
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Eixo ldquoyrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1321 20)
10λ1 = 3326
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
Eixo ldquoxrdquo
η =N983140
=7871
= 009983138 983144 983142983139983140 20 25 (2514)
1
=
0005
le
0005
983154 983144 (η + 05) 983144
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2027
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2127
19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2227
Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2327
Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1827
Momento miacutenimo
Eixo ldquoxrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 025)
9831171983140983149983277983150 983101 177 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1357 983147N983149
Eixo ldquoyrdquo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 7871 (0015 + 003 020)
9831171983140983149983277983150 983101 165 983147983118983149
A983140983151983156983137983140983151 M1983140983137 = 1040 983147N983149
Iacutendice de esbeltes
Eixo ldquoxrdquo
λ = (346 983148983141) 983144
λ = (346 280) 25
λ 983101 398309675
Eixo ldquoyrdquo
λ = (346 983148983141) 983144
λ = (346 280) 20
λ 983101 498309644
Iacutendice de esbeltes limite
Eixo ldquoxrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1724 25)
10λ1 = 3362
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Eixo ldquoyrdquo
λ1 =
25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (1321 20)
10λ1 = 3326
35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
Eixo ldquoxrdquo
η =N983140
=7871
= 009983138 983144 983142983139983140 20 25 (2514)
1
=
0005
le
0005
983154 983144 (η + 05) 983144
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2027
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
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19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2227
Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2327
Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 1927
1=
0005le
0005
983154 25 (009 + 05) 25
1= 0000339 ge 0002 rarr
1983101 00002983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 13570 + 7871 (280983218 10) 00002
983117983140983156983151983156 983101 14983096042 983147983118983139983149
Eixo ldquoyrdquo
η =
N983140
=
7871
= 009983138 983144 983142983139983140 20 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 20 (009 + 05) 20
1
= 0000424 ge 00025 rarr
1
983101 000025983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 10400 + 7871 (280983218 10) 000025
983117983140983156983151983156 983101 11983097427 983147983118983139983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
μ =M983140983156983151983156
=148042
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2027
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2127
19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2227
Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2327
Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2027
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 7871 991251 3875983218 25 7871 991251 19200 10 1357) M983140 983156983151983156 991251 3840
10 25 7871 1357 = 0
983117983140983156983151983156 983101 14798309654 983147983118983149
Eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
μ =M983140983156983151983156
=119427
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2127
19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2227
Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2327
Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2127
19200 M983140983156983151983156983218 + (3840 20 7871 991251 4844983218 20 7871 991251 19200 10 1040) M983140 983156983151983156 991251 3840
10 20 7871 1357 = 0
983117983140983156983151983156 983101 1198309721983097 983147983118983149
Coeficientes adimensionais
Eixo ldquoxrdquo
η = 009
Eixo ldquoyrdquo
Utilizando o aacutebaco A-67 de VENTURINI (1987) obtecircm-se ω = 02
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=20 25 (2514) 02
= 411 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 20 25 = 400 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 20 25 = 20 983139983149983218 rarr 983105983155983149983277983150 983100 983105983155 983119983147
μ =M983140983156983151983156
=147854
= 007983144 A983139 983142983139983140 25 20 25 (25 14)
983140=
4= 016 rarr 015
983144 25
μ =M983140983156983151983156
=119219
= 007983144 A983139 983142983139983140 20 20 25 (25 14)
983140=
4= 020
983144 20
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2227
Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2327
Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2227
Detalhamento
Observa-se que os momentos calculados pelo meacutetodo da curvatura satildeo
proacuteximos aos do meacutetodo da rigidez assim os coeficientes exatamente iguais
resultando aacutereas de accedilo iguais
As = 411 cmsup2 rarr 4Oslash125mm (As = 50 cmsup2)
Estribos
983256983156 ge
983256983148 4 = 125 4 = 313 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 150 98313998314912 983256983148 = 12 125 = 150 983139983149
20 983139983149
OBS Natildeo eacute necessaacuterio utilizar estribos suplementares
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2327
Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2327
Portanto o pilar ldquoP3rdquo teraacute 4Oslash125mm como armadura longitudinal e
1Oslash50mm a cada 150cm como armadura transversal
5 CALCULO DO PILAR INTERMEDIAacuteRIO ldquoP5rdquo
Carga de Projeto
N983140 = 983129983139 983129983150 N983147
N983140 = 14 10 61004
983118983140 983101 9830965406 983147983118
Momento miacutenimo
M1983140983149983277983150 = N983140 (0015 + 003 983144)
M1983140983149983277983150 = 85406 (0015 + 003 025)
9831171983140983149983277983150 983101 65336 983147983118983149
Iacutendice de esbeltes
λ983128 = λ983129 = (346 983148983141) 983144
λ983128 = λ983129 = (346 280) 25
λ983128 983101 λ983129 983101 398309675
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2427
Iacutendice de esbeltes limite
λ1 =25 + 125 (983141983145 983144)
α983138
λ1 =25 + 125 (0 25)
10λ1 = 250 35 le λ1 le 90 983152983151983154983156983137983150983156983151 λ1 983101 35
Como λx e λy eacute maior que λ1 = 35 se considera efeitos de segunda ordem para os
dois eixos
Momento de segunda ordem meacutetodo da curvatura
η =N983140
=85406
= 077983138 983144 983142983139983140 25 25 (2514)
1=
0005le
0005
983154 983144 (η + 05) 983144
1=
0005le
0005
983154 25 (077 + 05) 25
1= 000015748 ge 00002 rarr
1983101 00001574983096
983154 983154
M983140983156983151983156 = α983138 M1983140983137 + N983140 (983148983218 10) (1 983154)
M983140983156983151983156 = 10 65336 + 85406 (280983218 10) 000015748
983117983140983156983151983156 983101 17079830962 983147983118983139983149
Coeficientes adimensionais
η = 077
μ =M983140983156983151983156
=170782
= 006983144 A983139 983142983139983140 25 25 25 (25 14)
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2527
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-se ω = 009
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 009
= 231 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Momento de segunda ordem meacutetodo da rigidez ldquokrdquo
Eixo ldquoxrdquo igual a eixo ldquoyrdquo
19200 M983140983156983151983156983218 + (3840 983144 N983140 991251 λ983218 983144 N983140 991251 19200 α983138 M1983140983137) M983140983156983151983156 991251 3840 α983138 983144 N983140
M1983140983137 = 0
19200 M983140983156983151983156983218 + (3840 25 85406 991251 3875983218 25 85406 991251 19200 10 65336) M983140 983156983151983156 991251
3840 10 25 85406 65336 = 0
983117983140983156983151983156 983101 98309759830979830960 983147983118983149
Coeficientes adimensionais
η = 077
μ = M983140983156983151983156 = 95980 = 003983144 A983139 983142983139983140 25 25 25 (25 14)
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2627
Utilizando o aacutebaco A-12 de VENTURINI (1987) obtecircm-seω
= 005
Caacutelculo da aacuterea de accedilo
A983155 =A983139 983142983139983140 ω
=25 25 (2514) 005
= 12983096 983139983149983218983142983161983140 (50 115)
A983155983149983265983160 = 8 A983139 = 008 25 25 = 500 983139983149983218 rarr 983105983155983149983265983160 983102 983105983155 983119983147
A983155983149983277983150 = 04 A983139 = 0004 25 25 = 25 983139983149983218 rarr 983105983155983149983277983150 983102 983105983155 983118983267983151 983119983147
Obs Asmiacuten = 250 cmsup2 satildeo necessaacuterias 4 barras longitudinais com essa aacuterea cada
uma teria 0625 cmsup2 esta aacuterea unitaacuteria eacute menor que 080 cmsup2 (aacuterea de Oslash100mm)
que eacute a barra miacutenima portanto adotar As = 320 cmsup2 (4Oslash100mm)
Detalhamento
Seraacute detalhada a aacuterea de accedilo do meacutetodo da rigidez ldquokrdquo pois esta foi
menor que a outra sendo que as duas estatildeo de acordo com a norma NBR
61182003 seraacute utilizada a que representaraacute maior economia de accedilo
As = 320 cmsup2 rarr 4Oslash100mm
Estribos
983256983156 ge983256983148 4 = 10 4 = 25 983149983149
983256983156 983101 50 98314998314950 983149983149
E983155983152983137983271983137983149983141983150983156983151 le
15 983139983149 (983152983137983154983137 983138 lt 19983139983149)
983109983155983152983137983271983137983149983141983150983156983151 983101 120 98313998314912 983256983148 = 12 10 = 120 983139983149
20 983139983149
983140=
4= 016 rarr 015
983144 25
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
7232019 Calculo e Detalhamento de Pilares de Concreto Armado
httpslidepdfcomreaderfullcalculo-e-detalhamento-de-pilares-de-concreto-armado 2727
Portanto o pilar ldquoP5rdquo teraacute 4Oslash100mm como armadura longitudinal e
1Oslash50mm a cada 120cm como armadura transversal
top related