eng 1204 – anÁlise de estruturas ii – 2º semestre – 2017...

ENG 1204 ndash ANAacuteLISE DE ESTRUTURAS II ndash 2ordm Semestre ndash 2017

Segunda Prova ndash 1ordf Parte ndash 30102017 ndash Duraccedilatildeo 145 hs ndash Sem Consulta

1ordf Questatildeo (55 pontos) Empregando-se o Meacutetodo dos Deslocamentos obter o diagrama de momentos fletores para o quadro abaixo (barras inextensiacuteveis) Todas as barras tecircm a mesma ineacutercia agrave flexatildeo EI = 24x104 kNm2 com exceccedilatildeo da barra horizontal superior que eacute infinitamente riacutegida agrave flexatildeo

Soluccedilatildeo de um sistema de 2 equaccedilotildees a 2 incoacutegnitas

1

1

2

2

0

0

bf deD

ad bce a b D

f c d Dce af

Dad bc

minus=

minus

+ = rArr

minus =

minus


Segunda Prova ndash Parte 2 ndash 01112017 ndash Duraccedilatildeo 115 hs ndash Sem Consulta

2ordf Questatildeo (35 pontos) Empregando-se o Meacutetodo das Forccedilas obter os diagramas de momentos fletores e momentos torccedilores para a grelha abaixo Todas as barras tecircm a relaccedilatildeo indicada entre a rigidez agrave torccedilatildeo GJt e a rigidez agrave flexatildeo EI

2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


1

1

2

2

0

0

bf deX

ad bce a b X

f c d Xce af

Xad bc

minus=

minus

+ = rArr

minus =

minus

1ordf Questatildeo

Equaccedilotildees de equiliacutebrio

=++

=++

0

0

22212120

21211110

DKDK

DKDK

β

β

1

2

9 53 18 0

675 18 11120 0

DEI

D

minus + + rArr + sdot sdot = + + +

1

2

67534

82846

DEI

DEI

= +

rArr = minus

Momentos Fletores Finais

22110 DMDMMM sdot+sdot+=

Caso (1) ndash Deslocabilidade D1 isolada no SH Caso (2) ndash Deslocabilidade D2 isolada no SH

Sistema Hipergeomeacutetrico

Caso (0) ndash Solicitaccedilatildeo externa isolada no SH

[kNm]

M

β20 = +675 kN

β10 = ndash9 kNm

ndash18 +18

0

0

[kNm]

M0

0

0 0

0

0 0

+27 ndash27

ΣFy = 0 rArr β20 ndash 6sdot18 + 225 + 180 = 0 uArr

β10 = ndash27 + 18 = ndash9 uArr

ΣFy = 0 rArr K21 = ndash2EI(86) + 6EI62

K11 = + 4EI8 + 3EI6 + 4EI6

M1

D1 = 1

+4EI6

K21 = +EI8 K11 = +5EI3

0

0 0 +2EI6

2EI6

0 +3EI6

6EI62

x D1

0

0

+2EI8

ndash2EI8

+4EI8

2EI(86)

2 1

SH

M2

D2 = 1 +6EI62

K12 = +EI8 12EI63

ΣFy = 0 rArr K22 = +2EI(156) +3EI63 + 12EI63

K22 = +11EI120

0

0 0

0

0

+6EI62

6EI62

+2EI15 ndash(3EI10)θ2

ndash3EI62

K12 = ndash2EI(86) + 6EI62

3EI63

x D2 3EI62

2 16θ =

ndash(4EI8)θ2

ndash(2EI8)θ2

2EI(156)

0

0 0

ndash1273

ndash1336

0

+859

+68

+683

ndash751 +960

+414

2ordf Questatildeo ndash 1ordf opccedilatildeo para Sistema Principal

2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m

Sistema Principal e Hiperestaacutetico (g = 1)

X1

Caso (0) ndash Solicitaccedilatildeo externa isolada no Sistema Principal

Diagrama de momentos fletores M0

Diagrama de momentos torccedilores T0

Caso (1) ndash Hiperestaacutetico X1 isolado no Sistema Principal



Equaccedilatildeo de compatibilidade

011110 =+ Xδδ

[ ]

10

1 1 1 1 1 1 1 13 72 3 3 18 3 3 216 3 3 18 3 3 144 3 3 18 3 3 216 3

3 3 3 3 3 3 3

1 1566 1296 41583 72 3 3 216 3

t t

EI

GJ EI GJ EI

δ

= minus sdot sdot sdot + sdot sdot sdot + sdot sdot sdot minus sdot sdot sdot + sdot sdot sdot + sdot sdot sdot + sdot sdot sdot sdot +

minus sdot sdot + sdot sdot sdot = + + = +

[ ]11

1 1 1 36 54 1444 3 3 3 2 ( 3) ( 3) 3

3 t tEI GJ EI GJ EIδ

= sdot + sdot sdot sdot sdot + sdot minus sdot minus sdot sdot = + + = +

1

4158 1440X

EI EIrArr + + sdot = 1 28875 kNXthere4 = minus

Momentos fletores finais 110 XMMM sdot+= (kNm)

Momentos torccedilores finais 110 XTTT sdot+= (kNm)


2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


X1








011110 =+ Xδδ

10

1 3 1 3 1 3 1 3 1 3 1 3 1 3 172 3 18 3 216 3 18 3 144 3 18 3 216 3

3 2 3 2 3 2 3 2 3 2 3 2 3 2

3 3 1 783 648 207972 3 216 3

2 2 t t

EI

GJ EI GJ EI

δ

= + sdot sdot sdot minus sdot sdot sdot minus sdot sdot sdot + sdot sdot sdot minus sdot sdot sdot minus sdot sdot sdot minus sdot sdot sdot sdot +

+ sdot sdot minus sdot sdot sdot = minus minus = minus

11

1 3 3 1 3 3 1 9 27 364 3 2 3

3 2 2 2 2 2t tEI GJ EI GJ EIδ

= sdot + sdot sdot sdot sdot + sdot sdot sdot sdot = + + = +

1

2079 360X

EI EIrArr minus + sdot = 1 57750 kNXthere4 = +




2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


X1








011110 =+ Xδδ

[ ]

10

1 1 1 1 1 13 288 3 3 18 3 3 18 3 3 72 3 3 18 3

3 3 3 3 3

1 1026 2592 62103 288 3

t t

EI

GJ EI GJ EI

δ

= minus sdot sdot sdot + sdot sdot sdot minus sdot sdot sdot minus sdot sdot sdot + sdot sdot sdot sdot +

minus sdot sdot sdot = minus minus = minus

[ ]11

1 1 1 36 54 1444 3 3 3 2 ( 3) ( 3) 3



1

6210 1440X





Segunda Prova ndash Parte 2 ndash 01112017 ndash Duraccedilatildeo 115 hs ndash Sem Consulta

2ordf Questatildeo (35 pontos) Empregando-se o Meacutetodo das Forccedilas obter os diagramas de momentos fletores e momentos torccedilores para a grelha abaixo Todas as barras tecircm a relaccedilatildeo indicada entre a rigidez agrave torccedilatildeo GJt e a rigidez agrave flexatildeo EI

2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


1

1

2

2

0

0

bf deX

ad bce a b X

f c d Xce af

Xad bc

minus=

minus

+ = rArr

minus =

minus

1ordf Questatildeo


=++

=++

0

0

22212120

21211110

DKDK

DKDK

β

β

1

2

9 53 18 0

675 18 11120 0

DEI

D


1

2

67534

82846

DEI

DEI

= +

rArr = minus






[kNm]

M

β20 = +675 kN

β10 = ndash9 kNm

ndash18 +18

0

0

[kNm]

M0

0

0 0

0

0 0

+27 ndash27




K11 = + 4EI8 + 3EI6 + 4EI6

M1

D1 = 1

+4EI6

K21 = +EI8 K11 = +5EI3

0

0 0 +2EI6

2EI6

0 +3EI6

6EI62

x D1

0

0

+2EI8

ndash2EI8

+4EI8

2EI(86)

2 1

SH

M2

D2 = 1 +6EI62

K12 = +EI8 12EI63

ΣFy = 0 rArr K22 = +2EI(156) +3EI63 + 12EI63

K22 = +11EI120

0

0 0

0

0

+6EI62

6EI62


ndash3EI62

K12 = ndash2EI(86) + 6EI62

3EI63

x D2 3EI62

2 16θ =

ndash(4EI8)θ2

ndash(2EI8)θ2

2EI(156)

0

0 0

ndash1273

ndash1336

0

+859

+68

+683

ndash751 +960

+414


2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


X1








011110 =+ Xδδ

[ ]

10

1 1 1 1 1 1 1 13 72 3 3 18 3 3 216 3 3 18 3 3 144 3 3 18 3 3 216 3

3 3 3 3 3 3 3

1 1566 1296 41583 72 3 3 216 3

t t

EI

GJ EI GJ EI

δ



[ ]11

1 1 1 36 54 1444 3 3 3 2 ( 3) ( 3) 3



1

4158 1440X





2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


X1








011110 =+ Xδδ

10

1 3 1 3 1 3 1 3 1 3 1 3 1 3 172 3 18 3 216 3 18 3 144 3 18 3 216 3

3 2 3 2 3 2 3 2 3 2 3 2 3 2

3 3 1 783 648 207972 3 216 3

2 2 t t

EI

GJ EI GJ EI

δ



11

1 3 3 1 3 3 1 9 27 364 3 2 3



1

2079 360X





2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


X1








011110 =+ Xδδ

[ ]

10

1 1 1 1 1 13 288 3 3 18 3 3 18 3 3 72 3 3 18 3

3 3 3 3 3

1 1026 2592 62103 288 3

t t

EI

GJ EI GJ EI

δ



[ ]11

1 1 1 36 54 1444 3 3 3 2 ( 3) ( 3) 3



1

6210 1440X




1ordf Questatildeo


=++

=++

0

0

22212120

21211110

DKDK

DKDK

β

β

1

2

9 53 18 0

675 18 11120 0

DEI

D


1

2

67534

82846

DEI

DEI

= +

rArr = minus






[kNm]

M

β20 = +675 kN

β10 = ndash9 kNm

ndash18 +18

0

0

[kNm]

M0

0

0 0

0

0 0

+27 ndash27




K11 = + 4EI8 + 3EI6 + 4EI6

M1

D1 = 1

+4EI6

K21 = +EI8 K11 = +5EI3

0

0 0 +2EI6

2EI6

0 +3EI6

6EI62

x D1

0

0

+2EI8

ndash2EI8

+4EI8

2EI(86)

2 1

SH

M2

D2 = 1 +6EI62

K12 = +EI8 12EI63

ΣFy = 0 rArr K22 = +2EI(156) +3EI63 + 12EI63

K22 = +11EI120

0

0 0

0

0

+6EI62

6EI62


ndash3EI62

K12 = ndash2EI(86) + 6EI62

3EI63

x D2 3EI62

2 16θ =

ndash(4EI8)θ2

ndash(2EI8)θ2

2EI(156)

0

0 0

ndash1273

ndash1336

0

+859

+68

+683

ndash751 +960

+414


2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


X1








011110 =+ Xδδ

[ ]

10

1 1 1 1 1 1 1 13 72 3 3 18 3 3 216 3 3 18 3 3 144 3 3 18 3 3 216 3

3 3 3 3 3 3 3

1 1566 1296 41583 72 3 3 216 3

t t

EI

GJ EI GJ EI

δ



[ ]11

1 1 1 36 54 1444 3 3 3 2 ( 3) ( 3) 3



1

4158 1440X





2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


X1








011110 =+ Xδδ

10

1 3 1 3 1 3 1 3 1 3 1 3 1 3 172 3 18 3 216 3 18 3 144 3 18 3 216 3

3 2 3 2 3 2 3 2 3 2 3 2 3 2

3 3 1 783 648 207972 3 216 3

2 2 t t

EI

GJ EI GJ EI

δ



11

1 3 3 1 3 3 1 9 27 364 3 2 3



1

2079 360X





2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


X1








011110 =+ Xδδ

[ ]

10

1 1 1 1 1 13 288 3 3 18 3 3 18 3 3 72 3 3 18 3

3 3 3 3 3

1 1026 2592 62103 288 3

t t

EI

GJ EI GJ EI

δ



[ ]11

1 1 1 36 54 1444 3 3 3 2 ( 3) ( 3) 3



1

6210 1440X





2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


X1








011110 =+ Xδδ

[ ]

10

1 1 1 1 1 1 1 13 72 3 3 18 3 3 216 3 3 18 3 3 144 3 3 18 3 3 216 3

3 3 3 3 3 3 3

1 1566 1296 41583 72 3 3 216 3

t t

EI

GJ EI GJ EI

δ



[ ]11

1 1 1 36 54 1444 3 3 3 2 ( 3) ( 3) 3



1

4158 1440X





2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


X1








011110 =+ Xδδ

10

1 3 1 3 1 3 1 3 1 3 1 3 1 3 172 3 18 3 216 3 18 3 144 3 18 3 216 3

3 2 3 2 3 2 3 2 3 2 3 2 3 2

3 3 1 783 648 207972 3 216 3

2 2 t t

EI

GJ EI GJ EI

δ



11

1 3 3 1 3 3 1 9 27 364 3 2 3



1

2079 360X





2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


X1








011110 =+ Xδδ

[ ]

10

1 1 1 1 1 13 288 3 3 18 3 3 18 3 3 72 3 3 18 3

3 3 3 3 3

1 1026 2592 62103 288 3

t t

EI

GJ EI GJ EI

δ



[ ]11

1 1 1 36 54 1444 3 3 3 2 ( 3) ( 3) 3



1

6210 1440X








011110 =+ Xδδ

[ ]

10

1 1 1 1 1 1 1 13 72 3 3 18 3 3 216 3 3 18 3 3 144 3 3 18 3 3 216 3

3 3 3 3 3 3 3

1 1566 1296 41583 72 3 3 216 3

t t

EI

GJ EI GJ EI

δ



[ ]11

1 1 1 36 54 1444 3 3 3 2 ( 3) ( 3) 3



1

4158 1440X





2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


X1








011110 =+ Xδδ

10

1 3 1 3 1 3 1 3 1 3 1 3 1 3 172 3 18 3 216 3 18 3 144 3 18 3 216 3

3 2 3 2 3 2 3 2 3 2 3 2 3 2

3 3 1 783 648 207972 3 216 3

2 2 t t

EI

GJ EI GJ EI

δ



11

1 3 3 1 3 3 1 9 27 364 3 2 3



1

2079 360X





2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


X1








011110 =+ Xδδ

[ ]

10

1 1 1 1 1 13 288 3 3 18 3 3 18 3 3 72 3 3 18 3

3 3 3 3 3

1 1026 2592 62103 288 3

t t

EI

GJ EI GJ EI

δ



[ ]11

1 1 1 36 54 1444 3 3 3 2 ( 3) ( 3) 3



1

6210 1440X





2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


X1








011110 =+ Xδδ

10

1 3 1 3 1 3 1 3 1 3 1 3 1 3 172 3 18 3 216 3 18 3 144 3 18 3 216 3

3 2 3 2 3 2 3 2 3 2 3 2 3 2

3 3 1 783 648 207972 3 216 3

2 2 t t

EI

GJ EI GJ EI

δ



11

1 3 3 1 3 3 1 9 27 364 3 2 3



1

2079 360X





2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


X1








011110 =+ Xδδ

[ ]

10

1 1 1 1 1 13 288 3 3 18 3 3 18 3 3 72 3 3 18 3

3 3 3 3 3

1 1026 2592 62103 288 3

t t

EI

GJ EI GJ EI

δ



[ ]11

1 1 1 36 54 1444 3 3 3 2 ( 3) ( 3) 3



1

6210 1440X








011110 =+ Xδδ

10

1 3 1 3 1 3 1 3 1 3 1 3 1 3 172 3 18 3 216 3 18 3 144 3 18 3 216 3

3 2 3 2 3 2 3 2 3 2 3 2 3 2

3 3 1 783 648 207972 3 216 3

2 2 t t

EI

GJ EI GJ EI

δ



11

1 3 3 1 3 3 1 9 27 364 3 2 3



1

2079 360X





2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


X1








011110 =+ Xδδ

[ ]

10

1 1 1 1 1 13 288 3 3 18 3 3 18 3 3 72 3 3 18 3

3 3 3 3 3

1 1026 2592 62103 288 3

t t

EI

GJ EI GJ EI

δ



[ ]11

1 1 1 36 54 1444 3 3 3 2 ( 3) ( 3) 3



1

6210 1440X





2t

EIGJ =

16 kNm

16 kNm

16 kNm

3 m

3 m

3 m


X1








011110 =+ Xδδ

[ ]

10

1 1 1 1 1 13 288 3 3 18 3 3 18 3 3 72 3 3 18 3

3 3 3 3 3

1 1026 2592 62103 288 3

t t

EI

GJ EI GJ EI

δ



[ ]11

1 1 1 36 54 1444 3 3 3 2 ( 3) ( 3) 3



1

6210 1440X








011110 =+ Xδδ

[ ]

10

1 1 1 1 1 13 288 3 3 18 3 3 18 3 3 72 3 3 18 3

3 3 3 3 3

1 1026 2592 62103 288 3

t t

EI

GJ EI GJ EI

δ



[ ]11

1 1 1 36 54 1444 3 3 3 2 ( 3) ( 3) 3



1

6210 1440X




eng 1204 – anÁlise de estruturas ii – 2º semestre – 2017...

Documents