Analisis Matricial Excel Ejercicio Resuelto
October 11, 2022 | Author: Anonymous | Category: N/A
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Ensamble de la matriz de rigidez global k221 +k115 +k113
k125
k123
0
k215
k222 +k225 +k114
0
k124
k213
0
k223 +k116
k126
0
k214
k216
k224 +k226
K
Encontramos el vector fuerza (global)
" #
r1z
5
r1 m1!
0 0
r2z
5
r2
0
m2!
#
0
r3z
%&5
r3
0
m3!
0
r4z
%&5
r4
0
m4!
0
$on
Ensamblamos la matriz de des'lazamien des'lazamiento to 1 #
1 u #
2 #
u1z
u1z
u1
u1
*1!
*1!
u2z
u2z
u2
u2
1
*2!
1
u3z
u3z
u3
u3
*3!
*3!
u4z
u4z
u4
u4
*4!
*4!
1
3 #
4 #
#
*2!
'ara el elemento 12 * # 0&00 1& 1&5%0%6326,
k221
0&5625
0&0000
1&1250
0&0000
5&0000
0&0000
k222
1&1250
0&0000
3&0000
k000
0
0
0
ki.
0
0
0
k000
0
0
0
cos * # 0&00 sen * # 1&00
-i.
/ # 5&00 # 0&5625 # 3&00 # 1&125 E # 1&13
// E EE7$8 3 4 * # 0&00 1&5%0%633
// E EE -113
cos * # 0&00 999
-i. -114
0&,,,
0&0000
1&3333
* # 0&00
0&0000
6&666%
0&0000
cos * # 1&00
1&3333
0&0000
2&666%
999
/ # 6&666%
/ # 2&5000
# 0&,,,
# 0&0234
# 2&666%
# 0&5000
# 1&3333
# 0&03,
E # 1&3333
E # 0&03,
* # 0&00 1&5%0%633
-123
cos * # 0&00 999
-i. -124
0&,,,
0&0000
1&3333
* # 0&00
0&0000
6&666%
0&0000
cos * # 1&00
1&3333
0&0000
1&3333
999
/ # 6&666%
/ # 2&5000
# 0&,,,
# 0&0234
# 1&3333
# 0&2500
# 1&3333
# 0&03,
E # 1&3333
E # 0&03,
* # 0&00 1&5%0%633
-213
cos * # 0&00 999
-i. -214
0&,,,
0&0000
1&3333
* # 0&00
0&0000
6&666%
0&0000
cos * # 1&00
1&3333
0&0000
1&3333
999
/ # 6&666%
/ # 2&5000
# 0&,,,
# 0&0234
# 1&3333 # 1&3333
# 0&2500 # 0&03,
E # 1&3333
E # 0&03,
* # 0&00 1&5%0%633
-223
cos * # 0&00 999
-i. -224
0&,,,
0&0000
1&3333
* # 0&00
0&0000
6&666%
0&0000
cos * # 1&00
1&3333
0&0000
2&666%
999
/ # 6&666%
/ # 2&5000
# 0&,,,
# 0&0234
# 2&666%
# 0&5000
# 1&3333
# 0&03,
E # 1&3333
E # 0&03,
Ensa Ensamb mble le de de la matr matriz iz de de rigi rigid d k221 +k115 +k113 K
k215 k213 0
7$8 5 6
3&514 0
-115 -i. -116
2&5000
0&0000
0&0000
0&0000
0&0000
0&0234
0&03,
0&20,3
0&0000
0&03,
0&5000
2&5000 0&0000 k
0&0000 0&,,, 0&0000 1&3333 0&0000
0
-125 -i. -126
2&5000
0&0000
0&0000
0&0000
0&0000
0&0234
0&03,
0&0000
0&0000
0&03,
0&2500 8btenci cio on de de llo os de des'lazamie
[]= 〖 [] 〗 ^(
0
-215 -i. -216
2&5000
0&0000
0&0000
0&0000
0&0234
0&03,
0&0000
0&03,
0&2500 ;u< #
0
-225 -i. -226
2&5000
0&0000
0&0000
0&0000
0&0234
0&03,
0&0000
0&03,
0&5000
;u< #
% alculo de defo
=ue =ue lasf lasfue uerz rzas as e
'ara el elemento 6
3
4
2
3
4 5
1
2
2
1
2
1
a
;u<
$
a>ora calculamos
;k?<
z global con las res'ec:vas submatrices
k125 k1
k123
0
k222 +k225 +k114
0
k124
0
k223 +k116
k126
k214
k216
k224 +k226
0&0000
0&20,3
2&5000
0&0000
0&0000
0&,,,
11&601
0&03,
0&0000
0&0234
0&03,
0&0000
0&03,
6&166%
0&0000
0&03,
0&2500
1&3333
0&0000
0&0000
3&514
0&0000
0&20,3
0&0000
0&0234
0&03,
0&0000
11&601
0&03,
0&0000
0&03,
0&2500
0&20,3
0&03,
6&166%
0&0000
0&0000
1&3333
0&0000
0&0000
0&0000
3&3,,
6&666%
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
1&3333
0&0000
0&0000
0&0000
1&3333
0&0000
0&0000
0&,,,
0&0000
1&3333
2&5000
0&0000
0&0000
0&0000
6&666%
0&0000
0&0000
0&0000
0&0000
1&3333
0&0000
1&3333
0&0000
tos giros globales
1) ∗[] u1z
2&06331%
0&02544%16 0&55163123,1
1&,1221330,
u1
0&02544%16
0&144340, 0&014%%235,
0&02544%16
*1!
0&5 0&551 5163 6312 123, 3,1 1
0& 0&01 014% 4%% %23 235, 5, 0& 0&36 36,% ,%3 344 442, 2,
0& 0&53 53%0 %056 5643 4334 34
u2z
1&,1221330,
0&02544%16 0&53%0564334
2&06331%
u2
0&02544%16
0&000550652
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0&02544%16
0&53%0564334
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0&1%555033
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*2!
#
u3z u3
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0&0,156%3, 1&1612%%2% 0&131,0,%3 0&01,340103,
3&1%06561 0&0366,020%6
*3!
0&2 0&220 206 641 412 24 4
0& 0&02 0201 0112 1225 2515 15 0& 0&0, 0,2 20 06 66 6
0& 0&30 3012 1215 152 205 05
u4z
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0&0,156%3, 1&134662511
3&1104,6013
u4
0&0366,020%6
*4!
0&3 0&301 0121 2152 520 05 5
u1z
6%&55,0
u1
1&5253
*1!
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u2z
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u2
1&5253
*2!
#
22&66%
u3z
13,&30%2
u3
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0&0006,112%
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0&0366,020%6
0& 0&02 0201 0112 1225 2515 15 0& 0&1% 1%1% 1%45 4504 041 1
0& 0&22 2206 064 41 124 24
*3!
1,&,%50
u4z
13,&30%2
u4
2&0421
*4!
1,&,%50
maciones en coordenadas locales de elementos mecanicos
local
global
n los elementos elementos en coordenadas coordenadas locales 'ueden 'ueden obtenerse obtenerse a 'ar:r de la ecuacion
1 datods / # 20&00 @ @ # 3&00 @
5&0000 E@
0&5625 E@
1&1250 E@
4&00 m 2 3&0000 E@
1&5000 E@
1 b
uaz
0
0
ua
0
0
a
*a!
0
1
1
u1z
6%&55,0
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u1
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*1!
22&66%
22&66%,65156
* # 0 0&0 &00 0
;u?<
0 1&52525,004
1&5% 1&5%0% 0%6 632 326, 6,
cos * # 0&00 sen * # 1&00 0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
1&0000
la matriz de rigidez de coordenadas locales
5&0000 E@
0
0
5&0000 E@
0
0
0
0&5625 E@
1&1250 E@
0
0&5625 E@
1&1250 E@
0
1&1250 E@
3&0000 E@
0
1&1250 E@
1&5000 E@
5&0000 E@
0
0
5&0000 E@
0
0
0
0&5625 E@
1&1250 E@
0
0&5625 E@
1&1250 E@
0
1&1250 E@
1&5000 E@
0
1&1250 E@
3&0000 E@
%&6264%002 a 12&5 ;"?<
a
42&000,,,6% a %&6264%002
1
12&5
1
%&101133
1
0&0000
1&3333
0&0000
0&0000
0&0000
6&666%
0&0000
0&0000
0&0000
0&0000
0&0000
1&3333
0&0000
0&0000
0&0000
0&0000
0&0000
0&,,,
0&0000
1&3333
0&0000
0&0000
0&0000
6&666%
0&0000
0&0000
0&0000
1&3333
0&0000
1&3333
0&0000
1&3333
2&5000
0&0000
0&0000
6&601
0&03,
0&0000
0&0234
0&03,
0&03,
3&166%
0&0000
0&03,
0&2500
0&0000
0&0000
3&3,,
0&0000
1&3333
0&0234
0&03,
0&0000
6&601
0&03,
0&03,
0&2500
1&3333
0&03,
3&166%
0&02 0&025 54 44% 4%16 16 0&53 0&53%0 %056 5643 4334 34
3& 3&1 110 104, 4,60 6013 13
0& 0&03 0366 66,0 ,020 20%6 %6 0 0&2 &220 206 641 412 24 4 3& 3&1% 1% 0 066 66
0&00 0&0005 0550 5065 652 2 0&01 0&014% 4%% %23 235, 5,
0& 0&0, 0,1 156 56%3 %3, ,
0& 0&1 13 31, 1,0, 0,%3 %3 0&0 0&020 2011 1122 2251 515 5 0& 0&0, 0,1 156 56%4 %4
0& 0&0 014% 14%%2 %235, 35,
0&1%55 1%555 503 033
1&16 161 12 2%%2 %%2%
0& 0&0 0254 544 4%16 %16 0&5 0&5516 516312 3123,1 3,1
3&1 &1% % 0 065 6561
0& 0&0 01,340 ,3401 103, 03,
0&0, 0,20 2066 66 1& 1&1 1346 346625 625
0& 0&0 036 366 6,02 ,020%6 0%6 0&301 301215 21520 205
3&110 104,6 4,6
0& 0&1 14 443 4340, 40,
0&014% 14%%23 %235 5,
0&0, 0,1 15 56%3 6%3,
0& 0&0 000 006 6,1 ,112% 12%
0&020 020112 1122515 2515 0& 0&0 0,1 ,156% 56%
0& 0&0 014% 14%%2 %235, 35,
0&36,% 6,%344 3442 2,
1&13 1346 466 6251 2511
0& 0&0 01,340 ,3401 103, 03,
0&1%1 1%1%45 %45041 041 1& 1&1 161 612%% 2%%
0&0, 0&0,1 156 56%3 %3, , 1&13 1&1346 4662 6251 511 1 0&00 0&0006 06,1 ,11 12% 2% 0&01 0&01,3 ,340 4010 103, 3,
%& %&1 13% 3%01 0125 2564 64 0& 0&11 1116 16,6 ,645 4561 61
0& 0&0 0201 201122 122515 515
0&1%1% %1%4504 45041 1
1&11 11, ,6 6514 51453
0&0, 0&0,1 156 56%3 %3, , 1&16 1&161 12% 2%%2 %2% %
6& 6& 5, 5,65 653 3
0& 0&11 1116 16,6 ,645 4561 61 1 1&1 &11 1,6 ,651 5145 453 3 6& 6& 5, 5,66 66 0& 0&34 34, ,,% ,%02 026, 6, 0&0 0&031 3166 665 53% 3%1 1 0& 0&11 1116 16,6 ,646 46 0& 0&0 031665 1665 3%1 3%1
0&643 643405 405%14 %14 1& 1&0 04,, 4,,55 55
0& 0&11 1116 16,6 ,645 4561 61 1 1&0 &04, 4,,5 ,55, 5,%5 %5% % %& %&1 13% 3%01 0126 26
0& 0&1 13 31,0 1,0,%3 ,%3
0&01,3 1,34010 40103 3,
0&11 1116 16, ,645 64561
0& 0&0 001 010 0122 122232 232
0&031 031665 6653%1 3%1 0& 0&1 1116 116,65 ,65
0& 0&0 0201 201122 122515 515
0&0,2 ,206 06 6
1&04 04,, ,,5 55,% 5,%5%
0& 0&0 031665 1665 3%1 3%1
0&13 13554 554%3 %3 1& 1&1 11, 1,651 651
'ara el elemento 2 6
4
3
datods / # 20&00 @
2
3 1
@ # 3& 3&00 00 @ 5& 5&00 0000 00 E@
4
4&00 m
5
2
2
2
1
3&0000 E@
2
1
1
a
b
uaz ua
;u<
a
*a!
1
u2z u2 *2!
* # 0& 0&00 00 1&5 1&5%0% %0%63 633 3 cos * # 0&00 sen * # 1&00
$
0&0000
1&0000
1&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
a>or a>ora a cal calcu cula lamo moss la la ma matr triz iz de rig igid ide ez de de coo coorrde
;u?<
0
5&0000 E@
0
0
0
0&5625 E@
0
1&1250 E@
1&52525,004
5&0000 E@
0
6%&55%52534
0
0&5625 E@
22&66%,65156
0
1&1250 E@
0
;k?<
;"?<
;"< 0&0366,02 0&3012153
5
0&0006,11 0&0201123
0
0&01,3401 0&1%1%4504
0
0&0366,02 0&220642
5
0&131,0 0&02011225
0
0&01,3401
0&0,20%
0&1116,65 1&04,,55 0&00101222 0&031665 0&0316654
0&13555
0 %&5 0 0
0&1116,65 1&11,651
%&5
0&34,,%03 0&0316654
0
0&0316654 0&643405%
0
'ara el elem
0&5625 E@ E@
6
4
3
2
1&1250 E@ E@
3 1
4 5
2 2
1
1&5000 E@
2
1
0
0
0
0
0
1
6%&55,0
E@
;u?<
0
;u<
1&52525,
1&5253
6%&55%53
22&66%
22&66%,65
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
0&0000
1&0000
nadas locales
a
$
a>ora calcula
0
5&0000 E@
0
0
0
1&1250 E@
0
0&5625 E@
1&1250 E@
0
3&0000 E@
0
1&1250 E@
1&5000 E@
0
5&0000 E@
0
0
1&52525,
1&1250 E@
0
0&5625 E@
1&1250 E@
6%&55%53
1&5000 E@
0
1&1250 E@
3&0000 E@
22&66%,65
%&6264% 12&5 42&000,, %&6264% 12&5 %&10113
;u?<
0
;k?<
nto 3 datods
4
3
/ # 20&00 @ @ # 2&00 @ 6&666% E@ E@
0&,,, E@ E@
1&3333 E@ E@
3&00 m
1
2
2 2&666% E@
1&3333 E@
1 b
u1z
6%&55,0
1&52525,
u1
1&5253
6%&55%53
a
*1!
22&66%
1
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1
22&66%,65 2&0420,2
u3
2&0421
13,&30%24
*3!
1,&,%50
1,&,%455
* # 0& 0&00 00 1&5%0 1&5%0%6 %633 33 cos * # 0&00 999
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
1&0000
mos la matriz de rigidez de coordenadas locales
6&666% E@
0
0
6&666% E@
0
0
1&52525,
0
0&,,, E@
1&3333 E@
0
0&,,, E@
1&3333 E@
6%&55%53
0
1&3333 E@
2&666% E@
0
1&3333 E@
1&3333 E@
6&666% E@
0
0
6&666% E@
0
0
2&0420,2
0
0&,,, E@
1&3333 E@
0
0&,,, E@
1&3333 E@
13,&30%24
0
1&3333 E@
1&3333 E@
0
1&3333 E@
2&666% E@
1,&,%455
3&4453541 %&4%,214 ;"?<
,&%1,136, 3&44535412 %&4%,21 13&%%53504
;u?<
22&66%,65
'ara el elemento 4 6
datods
3
@ # 2&00 @ 6&666% E@ E@
4
0&,,, E@ E@
1&3333 E@ E@
3&00 m
5
1
2
2
2
1
3
/ # 20&00 @
2
2&666% E@
2
1
1
a
b ;u<
1&3333 E@
u2z
6%&55,0
1&52525,
u2
1&5253
6%&55%53
a
*2!
22&66%
1
u4z
13,&30%2 E@
;u?<
1
22&66%,65 2&0420,
u4
2&0421
13,&30%24
*4!
1,&,%50
1,&,%455
* # 0& 0&00 00 1& 1&5%0 5%0%6 %633 33 cos * # 0&00 999
$
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
1&0000
a>ora calculamos la matriz de rigidez de coordenadas locales
;k?<
6&666% E@
0
0
6&666% E@
0
0
0
0&,,, E@
1&3333 E@
0
0&,,, E@
1&3333 E@
0
1&3333 E@
2&666% E@
0
1&3333 E@
1&3333 E@
6&666% E@
0
0
6&666% E@
0
0
0
0&,,, E@
1&3333 E@
0
0&,,, E@
1&3333 E@
0
1&3333 E@
1&3333 E@
0
1&3333 E@
2&666% E@
3&44535412 %&4%,214 ;"?<
,&%1,136, 3&4453541 %&4%,21 13&%%53504
'ara el elemento 5 6
datods / # 20&00 @
2
3
@ # 1&00 @ 2&5000 E E@@
4
0&03, E@ E@
,&00 m
5
2
2
1
0&0234 E@ E@
0&5000 E@
2
1
1
a
b ;u<
0&2500 E@
u1z
6%&55,0
u1
1&5253
a
*1!
22&66%
1
1
u2z
6%&55,0
E@
u2
1&5253
*2!
22&66%
* # 0&00
0
cos * # 1&00 999
$
1&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
a>ora calculamos la matriz de rigidez de coordenadas locales
;u?<
1&52525,
2&5000 E@
0
0
2&5000 E@
6%&55%53
0
0&0234 E@
0&03, E@
0
0
0&03, E@
0&5000 E@
0
2&0420,
2&5000 E@
0
0
2&5000 E@
13,&30%24
0
0&0234 E@
0&03, E@
0
1,&,%455
0
0&03, E@
0&2500 E@
0
22&66%,65
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0&0000 E@ 4&1%,% E@ ;"?<
16&%14 E@ 0&0000 E@ 4&1%,% E@ 16&%14 E@
'ara el elemento 5 6
4
3
datods / # 20&00 @
2
3 1
,&00 m
5
2
2
2
1 6%&55%525
@ # 1& 1&00 00 @ 2& 2&50 5000 00 E@
4
0&5000 E@
2
1
1
a
b
u3z
1&52525, ;u?<
22&66%,65
u3 ;u<
6%&55%525
a
*3!
1
u4z
1&52525,
u4
22&66%,65
*4! * # 0&00
0
cos * # 1&00 999 0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
$
a>ora a>or a calculamos calculamos la matriz matriz de de rigidez rigidez d
0
0
6%&55%525
2&5000 E@
0
0&0234 E@
0&03, E@
1&52525,
0
0&0234 E@
0&03 03, E@
0&2500 500 E@
0
0&03, E@
0
0
6%&55%525
2&5000 E@
0
0&0234 E@
0&03, E@
1&52525,
0
0&0234 E@
0&03, E@
0&5000 E@
22&66%,65
0
0&03, E@
;u?<
22&66%,65
;k?<
;"?<
0&0234 E@ E@
0&03, E@ E@
0&2500 E@ 13,&30%2
13,&30%241
2&0421
2&0420,2
1,&,%50
;u?<
1
13,&30%2 E@
1,&,%455 13,&30%241
2&0421
2&0420,
1,&,%50
1,&,%455
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
1&0000
0&0000
0&0000
0&0000
0&0000
1&0000
coordenadas locales
0
2&5000 E@
0
0
13,&30%241
0&03, E@
0
0&0234 E@
0&03, E@
2&0420,2
0&5000 E@
0
0&03, E@
0&2500 E@
0
2&5000 E@
0
0
13,&30%241
0&03, E@
0
0&0234 E@
0&03, E@
2&0420,
0&2500 E@
0
0&03, E@
0&5000 E@
1,&,%455
0&0000 E@ 3&4433 E@ 13&%%33 E@ 0&0000 E@ 3&4433 E@ 13&%%3323
;u?<
1,&,%455
2 nudos osea :ene 6 grados de libertad encontramos el vector fuerza
;"<
f1z
%5
f1
0
f1
m1!
#
0
f2
f2z
%5
f2
0
m2!
0
ton
encontramos los des'lazamientos des'lazamientos u1z u1 ;u<
1
*1!
2
u2z u2 *!
'ara los elementos 12 'ro'iedades geometricas de las columnas de seccion transversal 40
cuadrad area
40 1600
inercia
87A@$B
213333&333
2&% 100
2 3
%200 16,3000
entonces los coeCcentes de rigidez serianD
5&25 E
0&1301 E
1%&55,3 E
3160&43, E
15,0&246 E
submatricez de rigidez en coordenadas locales elementos 1 2 1
5&25 E
0
0
2%0
;k?11< 2 1
0
0&1301 E
1%&55,3 E
0
1%&55,3 E
999
5&25 E
0
0
0
0&1301 E
1%&55,3 E
0 1 1%&55,3 E
999
;k?12< 2 1
5&25 E
2
0 0
1
5&25 E
;k?21<
;k?22<
0 2
0
0
0&1301 E 1%&55,3 E 1%&55,3 E 999 0
0
0&1301 E 1%&55,3 E
0 1 1%&55,3 E
999
de la Cgura nos interesan k221 k222 'ara el elemento 12 * # 0&00 1&5%0%633
k221
0&1301
0&0000
1%&55,3
0&0000
5&25
0&0000
k222
1%&55,3
0&0000
3160&43,
k000
0
0
0
ki.
0
0
0
k000
0
0
0
cos * # 0&00 sen * # 1&00
-i.
/ # 5&25 # 0&1301 # 3160&4 # 1%&55,3 E # 1%&55,3
'ara los elementos 3
'ara los ele
'ro'iedades geometricas de las columnas de seccion transversal transversal
'ro'iedades
cuadrad
cuadrad
25
area
40 1000
inercia
87A@$B
3&5
133333&333
100 2 3
350
area 8E"@@E7$E
122500 42,%5000
entonces los coeCcentes de rigidez serianD ki
2&,5%1 E
0&03%3 E
6&5306 E
1523&,05 E
%61&04, E
submatricez de rigidez en coordenadas locales
ki
elementos 1 2 1
submatriz en 2&,5%1 E
0
0
;k?11< 2 1
0
0&03%3 E
6&5306 E
0
6&5306 E
999
= 〖 〖 1)
2&,5%1 E 2
0
0
0
0&03%3 E
6&5306 E
2
0
6&5306 E %61&04, E
1
2&,5%1 E
2
0 0
1
2&,5%1 E
0
0
0
0&03%3 E
6&5306 E
0
6&5306 E
999
;k?12<
;k?21<
;k?22< 2
0
'or lo tanto
0
0&03%3 E 6&5306 E 6&5306 E %61&04, E
FB/$@EF FB/ $@EF E7 88E7//F 8/EF // E EE7$8 3 * # 0&00
* # 0&00 0
-113
2&,5%1
0&0000
0&0000
0&0000
0&03%3
6&5306
-114
0&0000
6&5306
1523&,05
-123
2&,5%1
0&0000
0&0000
0&0000
0&03%3
6&5306
-124
0&0000
6&5306
%61&04,
-213
2&,5%1
0&0000
0&0000
0&0000
0&03%3
6&5306
-214
0&0000
6&5306
%61&04,
-223
2&,5%1
0&0000
0&0000
0&0000
0&03%3
6&5306
0&0000
6&5306
1523&,05
cos * # 1&00
-i.
sen * # 0&00 / # 2&,5%1 # 0&03%3 # 1523&,05 # 6&5306 E # 6&5306 * # 0&00
0
cos * # 1&00
-i.
sen * # 0&00 / # 2&,5%1 # 0&03%3 # %61&04, # 6&5306 E # 6&5306 * # 0&00
0
cos * # 1&00
-i.
sen * # 0&00 / # 2&,5%1 # 0&03%3 # %61&04, # 6&5306 E # 6&5306 * # 0&00
0
cos * # 1&00 sen * # 0&00 / # 2&,5%1 # 0&03%3
-i. -224
# 1523&,05 # 6&5306 E # 6&5306
entos 4 geometricas de las columnas de seccion transversal transversal 15
15 2
E E @A@E
t #
14
87A@$Bz
3&5
350
87A@$B
2&%
2%0
longit lon gitud ud @/ 442&040% 442&040%22 22
0&6560 E
0&6560 E
0&6560 E
0&6560 E 0&6560 E coordenadas globales
〗 ^(−
0&%%1 0&%%142, 42,5% 5% 0&65%0 0&65%0%4 %44 4 3% 3%&64 &64%62 %6206 06 cos * # 0&% 999
ecordando =ue la matriz encoordenadas globales
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