ejercicio por metodo matricial de la rigidez
TRANSCRIPT
Se tiene una estructura se pide resolver por el metodo matricial de la riguidez directa
SOLUCION
x y1 0 02 1500 2598.083 3000 0
CONU 4 4500 2598.085 6000 06 7500 2598.087 9000 0
x y1 0 02 d1 d23 d3 d4
GLNU 4 d5 d65 d7 06 d8 d97 d10 d11
1. Origen de coordenadas de los nudos
2. Grados de libertad de los nudos
15 KN
30 KN
20 KN
1 1
1 3
2
8 6
1
4
7
NI NF1 1 32 3 53 5 74 2 45 4 6
NINF 6 1 27 2 38 3 49 4 5
10 5 611 6 7
NI NF L
1 1 3 3000 0 3000.00
2 3 5 3000 0 3000.00
3 5 7 3000 0 3000.00
4 2 4 3000 0 3000.00
5 4 6 3000 0 3000.00
6 1 2 1500 2598.08 3000.00
7 2 3 1500 -2598.08 3000.00
8 3 4 1500 2598.08 3000.00
9 4 5 1500 -2598.08 3000.00
10 5 6 1500 2598.08 3000.00
11 6 7 1500 -2598.08 3000.00
Barra 1d3 d4
53.33 0.00 -53.33 0.00K1 = 0.00 0.00 0.00 0.00
-53.33 0.00 53.33 0.000.00 0.00 0.00 0.00
Barra 3d7 d0 d10 d11
3. Nudo inicial y final por cada barra
4. Propiedades de la barra
X2-X1 Y2-Y1
5. Riguidez de cada barra o elemento
53.33 0.00 -53.33 0.00K3 = 0.00 0.00 0.00 0.00
-53.33 0.00 53.33 0.000.00 0.00 0.00 0.00
Barra 5d5 d6 d8 d9
53.33 0.00 -53.33 0.00K5 = 0.00 0.00 0.00 0.00
-53.33 0.00 53.33 0.000.00 0.00 0.00 0.00
Barra 7d1 d2 d3 d4
13.33 -23.20 -13.33 23.20K7 = -23.20 40.37 23.20 -40.37
-13.33 23.20 13.33 -23.2023.20 -40.37 -23.20 40.37
Barra 9d5 d6 d7 d0
13.33 -23.20 -13.33 23.20K9 = -23.20 40.37 23.20 -40.37
-13.33 23.20 13.33 -23.2023.20 -40.37 -23.20 40.37
Barra 11d8 d9 d10 d11
13.33 -23.20 -13.33 23.20K11 = -23.20 40.37 23.20 -40.37
-13.33 23.20 13.33 -23.2023.20 -40.37 -23.20 40.37
d1 d2 d3 d4 d51 79.99 0.00 -13.33 23.20 -53.332 0.00 80.74 23.20 -40.37 0.003 -13.33 23.20 133.32 0.00 -13.334 23.20 -40.37 0.00 80.74 -23.205 -53.33 0.00 -13.33 -23.20 133.326 0.00 0.00 -23.20 -40.37 0.007 0.00 0.00 -53.33 0.00 -13.338 0.00 0.00 0.00 0.00 -53.339 0.00 0.00 0.00 0.00 0.00
6. Ensamblaje de la matriz general
10 0.00 0.00 0.00 0.00 0.0011 0.00 0.00 0.00 0.00 0.00
15.00 80 0 -13.333333-30.00 0 80.736 23.2-14.14 -13.333333 23.2 133.32-14.14 23.0933333 -40.368 00.00 -53.333333 0 -13.333333330.00 = 0 0 -23.20.00 0 0 -53.333333330.00 0 0 0
-30.00 0 0 0-20.12 0 0 0-40.25 0 0 0
0.0351938432 -0.009440199 0.0140864889 -0.010784715-0.009440199 0.0240024382 -0.008090623 0.01858299640.0140864889 -0.008090623 0.018768338 -0.005394844-0.010784715 0.0185829964 -0.005394844 0.03406550760.0258211926 -0.00405227 0.0140882503 -7.84382E-006
-0.00674265 0.0100670435 -0.002700078 0.01858231790.0187855079 -0.01078952 0.0187775812 -0.010787327
0.025826035 -0.00405303 0.0140908924 -7.84529E-006-0.00404628 -0.003871546 0.0026934993 -0.006195105
0.0187878563 -0.010790869 0.0187799287 -0.010788676-0.00809121 -0.007743868 0.0053883478 -0.012390984
d1 0.0351938432 -0.009440199d2 -0.009440199 0.0240024382d3 0.0140864889 -0.008090623d4 -0.010784715 0.0185829964d5 0.0258211926 -0.00405227d6 = -0.00674265 0.0100670435d7 0.0187855079 -0.01078952
7.Matriz de rigudez generica de la estructura en coordenadas globales
8. Matriz inversa de la matriz de riguidez generica de la estructura en coordenadas globales
9. Operación de la Matriz de desplazamiento
�^(−1)=
d8 0.025826035 -0.00405303d9 -0.00404628 -0.003871546
d10 0.0187878563 -0.010790869d11 -0.00809121 -0.007743868
Resolucion
d1d2d3d4d5d6 =d7d8d9
d10d11
53.33
Barra 1
F1= 53.33(0.41)F1= -21.90582947
Barra 2 d3
F2= 53.33(-1.04)F1= -55.4632
Barra 3 d7
F3= 53.33(-0.82)
10. Calculo de los esfuerzos en cada barra
�_1= _1 _1) _1 ((� � /� �_1)
� = ) (_� (�_� �_� /�_� )�_�
�_2= _2 _2) _2 ((� � /� �_3)
�_3= _3 _3) _3 ((� � /� �_3)
) (�_� �_� /�_� =
F3= -43.7306
Barra 4 d1
F4= 53.33(0.14)F4= 7.4662
Barra 5 d5
F5= 53.33(1.2)F5= 64.00
Barra 6
F6= 53.33(0.093)F6= 4.97
Barra 7
F7= 53.33(-0.75)F7= -40.00
Barra 8
F8= 53.33(1.048)F8= 55.89
Barra 9
�_4= _4 _4) _4 ((� � /� �_4)
�_5= _5 _5) _5 ((� � /� �_5)
�_6= _6 _6) _6 ((� � /� �_6)
�_7= _7 _7) _7 ((� � /� �_7)
�_8= _8 _8) _8 ((� � /� �_8)
�_9= _9 _9) _9 ((� � /� �_7)
F9= 53.33(-1.33)F9= -71.02
Barra 10
F10= 53.33(-1.51)F10= -80.53
Barra 11
F11= 53.33(-1.33)F11= 45.86
barra 1 COMPRECION
barra 2 COMPRECION
barra 3 COMPRECION
barra 4 TRACCION
barra 5 TRACCION
barra 6 TRACCION
barra 7 COMPRECION
barra 8 TRACCION
barra 9 COMPRECION
barra 10 COMPRECION
barra 11 TRACCION
�_10= _10 _10) _10 ((� � /� �_10)
�_11= _11 _11) _11 ((� � /� �_11)
Se tiene una estructura se pide resolver por el metodo matricial de la riguidez directa
30 KN
45 KN
2
1
5 7
4 6
5
9
2
10
3
11
1
C S C2 S2 CS A1 0 1 0 0 8001 0 1 0 0 8001 0 1 0 0 8001 0 1 0 0 8001 0 1 0 0 800
0.5 0.87 0.25 0.76 0.44 8000.5 -0.87 0.25 0.76 -0.44 8000.5 0.87 0.25 0.76 0.44 8000.5 -0.87 0.25 0.76 -0.44 8000.5 0.87 0.25 0.76 0.44 8000.5 -0.87 0.25 0.76 -0.44 800
Barra 2d3 d4 d7 d0
53.33 0.00 -53.33 0.00K2 = 0.00 0.00 0.00 0.00
d3 -53.33 0.00 53.33 0.00d4 0.00 0.00 0.00 0.00
Barra 4d1 d2 d5 d6
d7 53.33 0.00 -53.33 0.00d0 K4 = 0.00 0.00 0.00 0.00d10 -53.33 0.00 53.33 0.00d11 0.00 0.00 0.00 0.00
Barra 6d0 d0 d1 d2
d5 13.33 23.20 -13.33 -23.20d6 K6 = 23.20 40.37 -23.20 -40.37d8 -13.33 -23.20 13.33 23.20d9 -23.20 -40.37 23.20 40.37
Barra 8 d3 d4 d5 d6
d1 13.33 23.20 -13.33 -23.20d2 K8 = 23.20 40.37 -23.20 -40.37d3 -13.33 -23.20 13.33 23.20d4 -23.20 -40.37 23.20 40.37
Barra 10d7 d0 d8 d9
d5 13.33 23.20 -13.33 -23.20d6 K10 = 23.20 40.37 -23.20 -40.37d7 -13.33 -23.20 13.33 23.20d0 -23.20 -40.37 23.20 40.37
d8d9d10d11
d6 d7 d8 d9 d10 d110.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.00
-23.20 -53.33 0.00 0.00 0.00 0.00-40.37 0.00 0.00 0.00 0.00 0.000.00 -13.33 -53.33 0.00 0.00 0.00
80.74 23.20 0.00 0.00 0.00 0.0023.20 133.32 -13.33 -23.20 -53.33 0.000.00 -13.33 79.99 0.00 -13.33 23.200.00 -23.20 0.00 80.74 23.20 -40.37
0.00 -53.33 -13.33 23.20 66.66 -23.200.00 0.00 23.20 -40.37 -23.20 40.37
23.0933333 -53.333333 0 0 0 0-40.368 0 0 0 0 0
0 -13.33333333 -53.33333333 0 0 080.736 -23.2 -40.368 0 0 0-23.2 133.32 0 -13.33333333 -53.33333333 0
-40.368 0 80.736 23.2 0 00 -13.33333333 23.2 133.32 -13.33333333 -23.20 -53.33333333 0 -13.33333333 79.99 00 0 0 -23.2 0 80.7360 0 0 -53.33333333 -13.33333333 23.20 0 0 0 23.2 -40.368
0.025821193 -0.00674265 0.018785508 0.025826035 -0.00404628 0.018787856-0.00405227 0.010067043 -0.01078952 -0.00405303 -0.003871546 -0.0107908690.01408825 -0.002700078 0.018777581 0.014090892 0.002693499 0.018779929
-7.8438E-006 0.018582318 -0.010787327 -7.8453E-006 -0.006195105 -0.0107886760.035201473 -0.001354384 0.018787857 0.035208074 -0.009436907 0.018790206
-0.001354384 0.024000887 -0.010786484 -0.001354638 -0.005420601 -0.0107878330.018787857 -0.010786484 0.037544597 0.01879138 0.010777711 0.037549290.035208074 -0.001354638 0.01879138 0.053968193 -0.020216559 0.018793729
-0.009436907 -0.005420601 0.010777711 -0.020216559 0.042584895 0.0107790580.018790206 -0.010787833 0.03754929 0.018793729 0.010779058 0.056306329
-0.018872463 -0.010841978 0.021558119 -0.040431768 0.060398469 0.032338023
0.014086489 -0.010784715 0.025821193 -0.00674265 0.018785508 0.025826035-0.008090623 0.018582996 -0.00405227 0.010067043 -0.01078952 -0.004053030.018768338 -0.005394844 0.01408825 -0.002700078 0.018777581 0.014090892
-0.005394844 0.034065508 -7.8438E-006 0.018582318 -0.010787327 -7.8453E-0060.01408825 -7.8438E-006 0.035201473 -0.001354384 0.018787857 0.035208074
-0.002700078 0.018582318 -0.001354384 0.024000887 -0.010786484 -0.0013546380.018777581 -0.010787327 0.018787857 -0.010786484 0.037544597 0.01879138
.Matriz de rigudez generica de la estructura en coordenadas globales
. Matriz inversa de la matriz de riguidez generica de la estructura en coordenadas globales
0.014090892 -7.8453E-006 0.035208074 -0.001354638 0.01879138 0.0539681930.002693499 -0.006195105 -0.009436907 -0.005420601 0.010777711 -0.0202165590.018779929 -0.010788676 0.018790206 -0.010787833 0.03754929 0.0187937290.005388348 -0.012390984 -0.018872463 -0.010841978 0.021558119 -0.040431768
0.83-0.37-0.41-0.220.970.19-1.452.17-3.82-2.27-7.36
d3
-0.41
d7
-1.45+0.41 -1.04
d10
d10-d7 -2.27+1.45
δ1= d3 - 0 δ1=
δ2= d7-d3
δ2= δ2=
δ3= δ3=
-0.82
d5
d5-d1 0.97-0.83 0.14
d8
d8-d5 2.17-0.97 1.2
C (d1-0) + S (d2-0) 0.5(0.83) + 0.87(-0.37)0.0931
C (d3-d1) + S (d4-d2) 0.5(-0.41-0.83) - 0.87(-0.22+0.37)-0.75
C (d5-d3) + S (d6-d4) 0.5(0.97+0.41) + 0.87(0.14+0.22)1.048
C (d7-d5) + S (d6-0)
δ3=
δ4= δ4= δ4=
δ5= δ5= δ5=
δ6= δ6= δ6=
δ7= δ7= δ7=
δ8= δ8= δ8=
δ9=
0
0
d2
d1
d1
d2
d4
d3
d4
d6
d5
d6
0
0.5(-1.45-0.97) - 0.87(0.14)-1.3318
C (d8-d7) + S (d4-0) 0.5(2.17+1.45) + 0.87(-3.62)-1.51
C (d10-d8) + S (d11-d9) 0.5(-2.27-2.17) - 0.87(-7.36+3.82)0.86
δ9= δ9=
δ10= δ10= δ10=
δ11= δ11= δ11=
d7
0
d9
d8
d9
d11
3 5 7
2 4 6
8 6
1
4
7
85
9
2
10
3
11
�_1
�_2
�_5
�_6
�_8
�_9
�_10
�_11
�_7
�_0
�_3
�_4
53.3353.3353.3353.3353.3353.3353.3353.3353.3353.3353.33
d3d4d7d0
EA/L KN/mm
d1d2d5d6
d0d0d1d2
d3d4d5d6
d7d0d8d9
d1d2d3d4d5d6d7d8d9
d10d11
0 0 d10 0 d20 0 d30 0 d40 0 d50 0 X d6
-53.33333333 0 d7-13.33333333 23.2 d8
23.2 -40.368 d966.66 -23.2 d10-23.2 40.368 d11
-0.00809121-0.0077438680.005388348
-0.012390984-0.018872463-0.0108419780.021558119
-0.0404317680.0603984690.0323380230.126992284
-0.00404628 0.018787856 -0.00809121 15-0.003871546 -0.010790869 -0.00774387 -300.002693499 0.018779929 0.00538835 -14.14213525
-0.006195105 -0.010788676 -0.01239098 -14.14213525-0.009436907 0.018790206 -0.01887246 X 0-0.005420601 -0.010787833 -0.01084198 00.010777711 0.03754929 0.02155812 0
-0.020216559 0.018793729 -0.04043177 00.042584895 0.010779058 0.06039847 -300.010779058 0.056306329 0.03233802 -20.124611590.060398469 0.032338023 0.12699228 -40.24922319
d3
d5
d7
d8
d11
d10
DATOSL de todas las barras = 3 marea = 800 mm2
E= 200000 MpaE= 200