Porticos Ejercicios Resueltos

(1)

UNIVERSIDAD NACIONAL DE SAN ANTONIO ABAD

DEL CUSCO

FACULTAD DE INGENIERIA CIVIL

ASIGNATURA:

ASIGNATURA: ANALISIS ANALISIS MAMATRICIATRICIAL DE EL DE ESRUCTURASRUCTURASS ESTUDIANTE:

ESTUDIANTE:

TORRES APAZA DIEGO ARMANDO

TORRES APAZA DIEGO ARMANDO 111845111845

1.

1. Resolver por el método de rigidez, para el siguiente pórtico que se muestra en la figura:

Resolver por el método de rigidez, para el siguiente pórtico que se muestra en la figura:

Considere I=500in^, !="0in^#, $=#%&"0^'()si

EJERCICIOS RESUELTOS

(2)

(3)

A

AE

E//L

L

0

0 --A

AE

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L

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1

12 2E

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6 6E

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2

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-6

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Solución

1.

1.1.

1. Ma

Matr

triz d

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e rigi

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das lo

s loca

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les pa

s para l

ra la ba

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rra 1

a 1

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ks

sii)

)

A

A((iin

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2)

)

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L((iin

n))

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(4)

2 2#

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1

10

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120%&3333

3

3 12&$%6%0$

12&$%6%0$

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1$10&!166

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3 N

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4 6 5 1 2 3 4 6 5 1 2 3 120%&3 120%&3 3 3 0 0 00 --120%&3 120%&3 3 3 0 0 00 44 11 00 00 00 00 00 0 0 12&$%6 12&$%6 % % 1$10&! 1$10&! 2 2 00 --12&$%6 12&$%6 % % 1$10&! 1$10&! 2 2 66 00 11 00 00 00 00

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00 1$10&! 1$10&! 2 2 22!!116666' ' 00 --1$10&! 1$10&! 2 2 12120%0%3333 55

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00 00 11 00 00 00 --120%&3 120%&3 3 3 0 0 00 120%&3 120%&3 3 3 0 0 00 11 00 00 00 11 00 00 0 0 --12&$%6 12&$%6 % % --1$10&! 1$10&! 2 2 00 12&$%6 12&$%6 % % --1$10&! 1$10&! 2 2 22 00 00 00 00 11 00 0 0 1$10&! 1$10&! 2 2 112200%%333 3 00 --1$10&! 1$10&! 2 2 2!2!16166'6' 33 00 00 00 00 00 11 4 6 5 1 2 3 4 6 5 1 2 3 1 0 0 0 0 0 1 0 0 0 0 0 1208 1208 . .3 3 0 0 00 -1 122008 8 0 0 00 44 0 1 0 0 0 0 0 1 0 0 0 0 00 12.58 12.58 7 7 1510 1510 .4 .4 00 -12.59 12.59 1510 1510 .4 .4 66

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0 0 1510. 1510. 4 4 2416 2416 67 67 00 -1510 1510 1208 1208 33 33 55 0 0 0 1 0 0 0 0 0 1 0 0 -1 122008 8 0 0 00 1208 1208 . .3 3 0 0 00 11 0 0 0 0 1 0 0 0 0 0 1 0 00 -12.59 12.59 -1 155110 0 00 12.5 12.5 87 87 -1510 1510 22 0 0 0 0 0 1 0 0 0 0 0 1 00 1510. 1510. 4 4 1208 1208 33 33 00 -1510 1510 2416 2416 67 67 33 1.2.

1.2.

_{MATRIZ DE RIGIDEZ EN !!RDENADAS "!A"ES #ARA "A $ARRA %}

1 2 3 7 8 9 1 2 3 7 8 9 120%&3 120%&3 3 3 0 0 00 --120%&3 120%&3 3 3 0 0 00 11 00 --11 00 00 00 00

λ

λλ

(5)

K

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0 1$10&! 2 2!166' 0 -1$10&! 2 120%33 3

"

0 0 1 0 0 0 -120%&3 3 0 0 120%&3 3 0 0 7 0 0 0 0 -1 0 0 -12&$%6 % -1$10&! 2 0 12&$%6 % -1$10&! 2 8 0 0 0 1 0 0 0 1$10&! 2 120%33 0 -1$10&! 2 2!166' 9 0 0 0 0 0 1 1 2 3 7 8 9 0 1 0 0 0 0 12.58 7 0 1510 .4 -12.5 9 0 1510 .4 1 -1 0 0 0 0 0 0 1208 .3 0 0 -1208 0 2

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_1510. 4 0 2416 67 -1510 0 1208 33 3 0 0 0 0 1 0 -12.59 0 -1510 12.5 87 0 -1510 7 0 0 0 -1 0 0 0 -1208 0 0 1208 .3 0 8 0 0 0 0 0 1 1510. 4 0 1208 33 -1510 0 2416 67 9 1.3.

_{MATRIZ DE RIGIDEZ EN !!RDENADAS G"!$A"ES}

1 2 3 4 5 6 7 8 9 1220&#20 1! 0 1$10&!16 6' -120%&333 3 0 0 -12&$%6%0 6 0 1$10&!16 6' 1 0 1220&#20 1! -1$10&!16 ' 0 -1$10&!16 ' -12&$%6%0 6 0 -120%&333 3 0 2 1$10&!16 6' -1$10&!16 ' !%3333&3 33 0 120%33&3 33 1$10&!16 6' -1$10&!16 ' 0 120%33&3 33 3 K*+ -120%&333 3 0 0 120%&333 33 0 0 0 0 0 4 0 -1$10&!16 ' 120%33&3 33 0 2!1666&6 6' 1$10&!16 6' 0 0 0 5 0 -12&$%6%0 6 1$10&!16 6' 0 1$10&!16 6' 12&$%6%0 $6 0 0 0 6 -12&$%6%0 6 0 -1$10&!16 ' 0 0 0 12&$%6%0 $6 0 -1$10&!16 ' 7 0 -120%&333 3 0 0 0 0 0 120%&333 33 0 8 1$10&!16 0 120%33&3 0 0 0 - 0 2!1666&6 9

(6)

'

1.&.

A"'"! DE "!S DES#"AZAMIENT!S DES!N!ID!S

$ 1220&#20 1! 0 1$10&!16 6' -120%&333 3 0 ,1 0 0 1220&#20 1! -1$10&!16 ' 0 -1$10&!16 ' ,2 0

=

1$10&!16 6' -1$10&!16 ' !%3333&3 33 0 120%33&3 33 ,3 0 -120%&333 3 0 0 120%&333 33 0 ,! 0 0 -1$10&!16 ' 120%33&3 33 0 2!1666&6 6' ,$ ,1 0&6#$'$3#3 ,2 -0&001$$0'1 ,3  -0&002!%'6 ,! 0&6#$'$3#3 ,$  0&00123!11

1.(.

A"'"! DE "AS REAI!NES DES!N!IDAS

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A"'"! DE *'ERZAS #ARA "A $ARRA 1

(7)

q1 ! 0 6 -1&%'3'%0 1 $

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$&6%!3E-1! F1 0 F2 1&%'3'%0 0# F3 -!!#&'0'2 2

1.+.

A"'"! DE *'ERZAS #ARA "A $ARRA %

q2  ! " 1 120%&3 3 0 0 -120%&3 3 0 0 0 -1 0 0 0 0 0&6#$'$ ! 2 0 12&$%6 % 1$10&! 2 0 -12&$%6 % 1$10&! 2 1 0 0 0 0 0 -0&001$$ 3  0 1$10&! 2 2!166 ' 0 -1$10&! 2 120%33 0 0 1 0 0 0 -0&002!# F' -120%&3 3 0 0 120%&3 3 0 0 0 0 0 0 -1 0 0 F% 0 -12&$%6 % -1$10&! 0 12&$%6 % -1$10&! 2 0 0 0 1 0 0 0 F# 0 1$10&! 2 120%3 3 0 -1$10&! 2 2!166' 0 0 0 0 0 1 0 q2 1 1&%'3' % 2 $ 3



!!#&'0 ' F' -1&%'3% F% -$ F# '$0&2# 3 q1  ! " ! 120%&3 3 0 0 -120%&3 3 0 0 1 0 0 0 0 0 0&6#$'$ ! 6 0 12&$%6 % 1$10&! 2 0 -12&$%6 % 1$10&! 2 0 1 0 0 0 0 0 $  0 1$10&! 2 2!166' 0 -1$10&! 2 120%33 0 0 1 0 0 0 0&00123 ! F1 -120%&3 3 0 0 120%&3 3 0 0 0 0 0 1 0 0 0&6#$'$ ! F2 0 -12&$%6 % -1$10&! 2 0 12&$%6 % -1$10&! 2 0 0 0 0 1 0 -0&001$$ F3 0 1$10&! 2 120%33 0 -1$10&! 2 2!166' 0 0 0 0 0 1 -0&002!#

(8)

1.,.

GRA*IA DE "A S!"'I!N

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(9)

Solución

%.1.

Matriz de rigidez en coordenadas locales para la barra 1

E(N/#2) A(#2) I(#^4) L(#) 1#000000 0&1$ 0&00312$ 6 AE/L(N/#) 12EI/L^3(N/# ) 6EI/L^2(N) 4EI/L(N. #) 2EI/L(N. #)

!'$000 32#%&611111 #%#$&%33333 3#$%3&3333 3 1#'#1&6666 ' Nodos   1 0 3 (6-0)/6 (3-3)/6 2 6 3 1 0

λ

(10)

1

2

3

4

5

6

!'$00 0 0 0 -!'$00 0 0 0 1 0 0 0 0 0 0 32#%&6 1 #%#$&% 3 0 -32#%&6 #%#$&% 3 0 1 0 0 0 0

K4



0 #%#$&% 3 3#$%3& 3 0 -#%#$&% 1#'#1& '

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0 0 1 0 0 0 -!'$00 0 0 0 !'$00 0 0 0 0 0 0 1 0 0 0 -32#%&6 -#%#$&% 0 32#%&6 1 -#%#$&% 0 0 0 0 1 0 0 #%#$&% 3 1#'#1& ' 0 -#%#$&% 3#$%3& 3 0 0 0 0 0 1

1

2

3

4

5

6

1 0 0 0 0 0 475000 0 0 -475000 0 0

1

0 1 0 0 0 0 0 3298.61 11 9895.83 33 0 -3298.61 1 9895.83 33

2 "4t



0 0 1 0 0 0

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0 9895.83 33 39583.3 33 0 -9895.83 3 19791.6 67

3

0 0 0 1 0 0 -475000 0 0 475000 0 0

4

0 0 0 0 1 0 0 -3298.61 1 -9895.83 3 0 3298.61 11 -9895.83 3

5

0 0 0 0 0 1 0 9895.83 33 19791.6 67 0 -9895.83 3 39583.3 33

6 %.%.

Matriz de rigidez en coordenadas locales para la barra %

E(N/#2) A(#2) I(#^4) L(#) 1#000000 0&0# 0&0006'$ 3 AE/L(N/#) 12EI/L^3(N/# ) 6EI/L^2(N) 4EI/L(N. #) 2EI/L(N. #) $'0000 $'00 %$$0 1'100 %$$0 5d5s   3 0 0 (0-0)/3 (3-0)/3 1 0 3 0 1 $'000 0 0 0 -$'000 0 0 0 0 1 0 0 0 0

λ

(11)

K4



0 %$$0 1'100 0 -%$$0 %$$0

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0 0 1 0 0 0 -$'000 0 0 0 $'000 0 0 0 0 0 0 0 1 0 0 -$'00 -%$$0 0 $'00 -%$$0 0 0 0 -1 0 0 0 %$$0 %$$0 0 -%$$0 1'100 0 0 0 0 0 1 7 8 9 1 2 3 0 -1 0 0 0 0 5700 0 -8550 -5700 0 -8550 7 1 0 0 0 0 0 0 570000 0 0 -570000 0 8

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0 0 1 0 0 0

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4

-8550 0 17100 8550 0 8550 9 0 0 0 0 -1 0 -5700 0 8550 5700 0 8550 1 0 0 0 1 0 0 0 -570000 0 0 570000 0 2 0 0 0 0 0 1 -8550 0 8550 8550 0 17100 3

%.-.

Matriz de rigidez en coordenadas locales para la barra

-E(K/2) A(2) I(^!) L()

1#000000 0&0# 0&0006'$ 3

AE/L(K/) 12EI/L^3(K/) 6EI/L^2(K) !EI/L(K&) 2EI/L(K&) $'0000 $'00 %$$0 1'100 %$$0 5d5s   2 6 3 (6-6)/3 (0-3)/3 ! 6 0 0 -1 $'000 0 0 0 -$'000 0 0 0 0 -1 0 0 0 0 0 $'00 %$$0 0 -$'00 %$$0 1 0 0 0 0 0

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0 %$$0 1'100 0 -%$$0 %$$0

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0 0 1 0 0 0 -$'000 0 0 0 $'000 0 0 0 0 0 0 0 -1 0 0 -$'00 -%$$0 0 $'00 -%$$0 0 0 0 1 0 0 0 %$$0 %$$0 0 -%$$0 1'100 0 0 0 0 0 1

λ

(12)

4 5 6 10 11 12 0 1 0 0 0 0 5700 0 8550 -5700 0 8550 4 -1 0 0 0 0 0 0 570000 0 0 -570000 0 5

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0 0 1 0 0 0

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8550 0 17100 -8550 0 8550 6 0 0 0 0 1 0 -5700 0 -8550 5700 0 -8550 10 0 0 0 -1 0 0 0 -570000 0 0 570000 0 11 0 0 0 0 0 1 8550 0 8550 -8550 0 17100 12

2.4.

_{MATRIZ DE RIGIDEZ EN !!RDENADAS G"!$A"ES}

1

2

3

4

5

6

7

8

9

10

11

12

!%0'00 0 %$$0 -!'$000 0 0 -$'00 0 %$$0 0 0 0

1

0 $'32#% &6 #%#$&% 33 0 -32#%&6 1 #%#$&% 33 0 -$'0000 0 0 0 0

2

%$$0 #%#$&% 33 $66%3& 33 0 -#%#$&% 3 1#'#1& 6' -%$$0 0 %$$0 0 0 0

3

-!'$000 0 0 !%0'00 0 %$$0 0 0 0 -$'00 0 %$$0

4

0 -32#%&6 1 -#%#$&% 3 0 $'32#% &6 -#%#$&% 3 0 0 0 0 -$'0000 0

5 K*+

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0 #%#$&% 33 1#'#1& 6' %$$0 -#%#$&% 3 $66%3& 33 0 0 0 -%$$0 0 %$$0

6

-$'00 0 -%$$0 0 0 0 $'00 0 -%$$0 0 0 0

7

0 -$'0000 0 0 0 0 0 $'0000 0 0 0 0

8

%$$0 0 %$$0 0 0 0 -%$$0 0 1'100 0 0 0

9

0 0 0 -$'00 0 -%$$0 0 0 0 $'00 0 -%$$0

1

0 0 0 0 -$'0000 0 0 0 0 0 $'0000 0

1

0 0 0 %$$0 0 %$$0 0 0 0 -%$$0 0 1'100

1 %.(.

A"'"! DE "!S DES#"AZAMIENT!S DES!N!ID!S

0 !%0'00 0 %$$0 -!'$000 0 0 ,1 -'$ 0 $'32#%& 6 #%#$&%3 3 0 -32#%&61 1 #%#$&%3 3 ,2 -'$

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%$$0 #%#$&%3 3 $66%3&3 3 0 -#%#$&%3 3 1#'#1&6 ' ,3 0 -!'$000 0 0 !%0'00 0 %$$0 ,! -32#%&61 -#%#$&%3 $'32#%& -#%#$&%3

(13)

'$ 0 #%#$&%3 3 1#'#1&6 ' %$$0 -#%#$&%3 3 $66%3&3 3 ,6 ,1 1&%22$E-0$  ,2 -0&000131 6  ,3

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-0&00203' 2 rad ,! -1&%23E-0$  ,$ -0&000131 6  ,6 0&00203' 2 rad

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A"'"! DE "AS REAI!NES DES!N!IDAS

.' -$'00 0 -%$$0 0 0 0 1&%22$E-0$ 1'&31! K .% 0 -$'0000 0 0 0 0 -0&000131$ % '$&000 K .#

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%$$0 0 %$$0 0 0 0 -0&00203'2

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-1'&262 K&  .10 0 0 0 -$'00 0 -%$$0 -1&%22$E-0$ -1'&31! K .11 0 0 0 0 -$'0000 0 -0&000131$ % '$&000 K .12 0 0 0 %$$0 0 %$$0 0&00203'2 1'&262 K& 

%.+.

A"'"! DE *'ERZAS #ARA "A $ARRA 1

1 K " , 1 !'$000 0 0 -!'$000 0 0 1 0 0 0 0 0 1&%22$E-0$ 1'&31 ! K 2 0 32#%&61 1 #%#$&%3 3 0 -32#%&61 1 #%#$&%3 3 0 1 0 0 0 0 -0&000131 6 0 K 3  0 #%#$&%3 3 3#$%3&3 3 0 -#%#$&%3 3 1#'#1&6 ' 0 0 1 0 0 0 -0&00203' 2



-!0&32 K&  F! -!'$000 0 0 !'$000 0 0 0 0 0 1 0 0 -1&%23E-0$ -1'&31 ! K F$ 0 -32#%&61 1 -#%#$&%3 3 0 32#%&61 1 -#%#$&%3 3 0 0 0 0 1 0 -0&000131 6 0 K F6 0 #%#$&%3 3 1#'#1&6 ' 0 -#%#$&%3 3 3#$%3&3 3 0 0 0 0 0 1 0&00203' 2 !0&32 K& 

(14)

2 K " , ' $'0000 0 0 -$'0000 0 0 0 1 0 0 0 0 0 '$ K % 0 $'00 %$$0 0 -$'00 %$$0 -1 0 0 0 0 0 0 -1'&31 ! K #  0 %$$0 1'100 0 -%$$0 %$$0 0 0 1 0 0 0 0



-1'&26 2 K&  F1 -$'0000 0 0 $'0000 0 0 0 0 0 0 1 0 1&%22$E-0$ -'$ K F2 0 -$'00 -%$$0 0 $'00 -%$$0 0 0 0 -1 0 0 -0&000131 6 1'&31 ! K F3 0 %$$0 %$$0 0 -%$$0 1'100 0 0 0 0 0 1 -0&00203' 2 -3!&6% K& 

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-3 K " , ! $'0000 0 0 -$'0000 0 0 0 -1 0 0 0 0 -1&%23E-0$ '$ K $ 0 $'00 %$$0 0 -$'00 %$$0 1 0 0 0 0 0 -0&000131 6 1'&31 ! K 6  0 %$$0 1'100 0 -%$$0 %$$0 0 0 1 0 0 0 0&00203' 2



3!&6% K&  F10 -$'0000 0 0 $'0000 0 0 0 0 0 0 -1 0 0 -'$ K F11 0 -$'00 -%$$0 0 $'00 -%$$0 0 0 0 1 0 0 0 -1'&31 ! K F12 0 %$$0 %$$0 0 -%$$0 1'100 0 0 0 0 0 1 0 1'&26 2 K& 

%.1/.

GRA*IA DE "A S!"'I!N

(15)

(16)

(17)

(18)

(19)

-.1.

Matriz de rigidez en coordenadas locales para la barra 1

1

2

3

4

5

6

0&1666 ' 0 0 -0&166' 0 0 1 0 0 0 0 0 0 0&1111 1 0&3333 3 0 -0&1111 0&3333 3 0 1 0 0 0 0

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0 0&3333 3 1&3333 3 0 -0&3333 0&6666 '

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0 0 1 0 0 0 -0&166' 0 0 0&1666 ' 0 0 0 0 0 1 0 0 0 -0&1111 -0&3333 0 0&1111 1 -0&3333 0 0 0 0 1 0 0 0&3333 3 0&6666 ' 0 -0&3333 1&3333 3 0 0 0 0 0 1

(20)

1

2

3

4

5

6

1 0 0 0 0 0 0.1666 67 0 0 -0.1666 67 0 0

1

0 1 0 0 0 0 0 0.1111 11 0.3333 33 0 -0.1111 11 0.3333 33

2 "4t

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0 0 1 0 0 0

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0 0.3333 33 1.3333 33 0 -0.3333 33 0.6666 67

3

0 0 0 1 0 0 -0.1666 67 0 0 0.1666 67 0 0

4

0 0 0 0 1 0 0 -0.1111 11 -0.3333 33 0 0.1111 11 -0.3333 33

5

0 0 0 0 0 1 0 0.3333 33 0.6666 67 0 -0.3333 33 1.3333 33

6 -.%.

Matriz de rigidez en coordenadas locales para la barra %

7

8

9

1

2

3

0&3333 3 0 0 -0&3333 0 0 0 1 0 0 0 0 0 0&!!!! ! 0&6666 ' 0 -0&!!!! 0&6666 ' -1 0 0 0 0 0

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0 0&6666 ' 1&3333 3 0 -0&666' 0&6666 '

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0 0 1 0 0 0 -0&3333 0 0 0&3333 3 0 0 0 0 0 0 1 0 0 -0&!!!! -0&666' 0 0&!!!! ! -0&666' 0 0 0 -1 0 0 0 0&6666 ' 0&6666 ' 0 -0&666' 1&3333 3 0 0 0 0 0 1 7 8 9 1 2 3 0 -1 0 0 0 0 0.4444 44 0 -0.6666 67 -0.4444 44 0 -0.6666 67 7 1 0 0 0 0 0 0 0.3333 33 0 0 -0.3333 33 0 8

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Matriz de rigidez en coordenadas locales para la barra

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Matriz de rigidez en coordenadas locales para la barra (

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0&6 1 0 0&6 ' -0&2 0 0 -0&! 0 0&6 ' 0 0 0 0 0 0 0 0 0 0 0 0 0&! ! 0&3 3 0 -0&1 0&3 3 0 -0&3 0 0 0 0 0 0 0 0 0 0 0 0 0&6 ' 0&3 3 2&6 ' 0 -0&3 0&6 ' -0&' 0 0&6 ' 0 0 0 0 0 0 0 0 0 0 0 -0&2 0 0 0&6 1 0 0&6 ' 0 0 0 -0&! 0 0&6 ' 0 0 0 0 0 0 0 0 0 -0&1 -0&3 0 0&! ! -0&3 0 0 0 0 -0&3 0 0 0 0 0 0 0 0 0 K*+4  0 0&3 3 0&6 ' 0&6 ' -0&3 2&6 ' 0 0 0 -0&' 0 0&6 ' 0 0 0 0 0 0 0 0 -0&! 0 -0&' 0 0 0 0&% 0 -0&3 -0&2 0 0 0 0 0 -0&2 0

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