5553_cross___takabeya___rigideces-1540809676.xlsx
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EJEMPLO Analizar la estructura mostrada en la figura 4 Tn/m 6 Tn 3
6
10 6 Tn/m 4 Tn 2
5
1
4
Datos de los elementos de la estructura L b Barra (m)
1-2 2-3 4-5 5-6 7- 8 2-5 3-6 5-8
(m)
3.2 2.7 3.2 2.7 3.2 4 4 3.5
0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
h
I
(m)
(m4)
0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.4
0.000675 0.000675 0.000675 0.000675 0.000675 0.0016 0.0016 0.0016
Comparacion de momentos finales y desplazamientos (Solucion) Takabeya Rigidez Mij Mji Mij Mji Barra (Tn-m)
Tn
8
7
1-2 2-3 4-5 5-6 7- 8 2-5 3-6 5-8
(Tn-m)
(Tn-m)
(Tn-m)
4.4855066 2.0451152 4.4855055 2.045111 0.1962165 0.993532 0.196214 0.9935343 6.4059877 5.8860774 6.40599 5.8860801 7.0791765 7.9310749 7.0791783 7.9310735 6.7010704 6.4762428 6.7010711 6.4762424 -2.2413143 -14.599513 -2.241325 -14.599517 -0.9935338 -7.9310667 -0.9935343 -7.9310735 1.6342597 -6.4762382 1.634259 -6.4762424
E
EI/L
(Tn/m2)
(Tn-m)
2000000 421.875 2000000 500 2000000 421.875 2000000 500 2000000 421.875 2000000 800 2000000 800 2000000 914.28571
k
Mij
Mji
(Tn-m)
(Tn-m)
1.00 0 0 1.19 0 0 1.00 0 0 1.19 0 0 1.00 0 0 1.90 8 -8 1.90 5.3333333 -5.3333333 2.17 4.375 -4.375
Cross Mij
Mji
(Tn-m)
(Tn-m)
4.4850218 2.0455162 0.1959796 0.9937935 6.4056023 5.8865489 7.0788986 7.9313283 6.7007297 6.4765812 -2.2414958 -14.599749 -0.9937935 -7.9313283 1.6343019 -6.4765812
θ1(rad)= θ2(rad)= θ3(rad)= θ4(rad)= θ5(rad)= θ6(rad)= θ7(rad)= θ8(rad)= Δ1(m)= Δ2(m)=
Takabeya 0.00E+00 -2.89E-03 -2.10E-03 0.00E+00 -6.16E-04 2.36E-04 0.00E+00 -2.66E-04 8.76E-03 7.27E-03
Rigidez 0.00E+00 -2.89E-03 -2.09E-03 0.00E+00 -6.16E-04 2.36E-04 0.00E+00 -2.66E-04 8.76E-03 7.27E-03
Cross
8.76E-03 7.27E-03
EJEMPLO Resolucion de la estructura mediante el metodo de TAKABEYA
3 -0.1923 0.9935
-0.75
0.1962 -0.1452 2 -0.1225 2.0451
-0.5
4.4855 -0.5 1
-0.3077 -0.9935 5.3333 0 -0.8654 -1.3905 -1.7379 -1.7817 -1.7758 -1.7704 1.19 -1.7684 -1.7678 -1.7677 -1.7677 -1.7677 -1.7677 0 -0.2323 -2.2413 8 0 -0.98 -2.0713 -2.3352 -2.4022 -2.4265 -2.4357 1.00 -2.4389 -2.44 -2.4403 -2.4404 -2.4404 -2.4404 0
-7.9311 -0.3077 1.90 -5.3333 6 3.4172 -0.1923 5.7475 7.9311 6.5457 6.7533 6.8003 6.8105 6.813 -0.75 6.8137 6.8139 6.814 6.814 6.814 7.0792 -0.0948 -14.6 -0.1517 1.90 -8 5 5.3333 -0.08 6.5592 5.8861 6.8334 6.8992 6.9175 6.9232 6.9251 -0.5 6.9256 6.9258 6.9259 6.9259 6.9259 6.406 -0.5 4
0 0.8654 0.7031 0.3884 0.2546 0.2134 0.2025 1.19 0.1998 0.1991 0.199 0.1989 0.1989 0.1989 0 -0.1734 1.6343 4.375 0 0.2901 -0.3483 -0.4866 -0.5189 -0.5219 -0.5209 1.00 -0.5202 -0.52 -0.5199 -0.5199 -0.5199 -0.5199 0
2.17
Barra 1-2 2-3 4-5 5-6 7- 8 2-5 3-6 5-8
-6.4762 -0.3421 -4.375
Mij
Mji
(Tn-m) (Tn-m)
4.4855 2.0451 0.1962 0.9935 6.406 5.8861 7.0792 7.9311 6.7011 6.4762 -2.2413 -14.6 -0.9935 -7.9311 1.6343 -6.4762
8 -0.1579 6.4762
-0.5
6.7011 -0.5 7
0 0.6907 -0.0321 -0.1783 -0.2106 -0.2199 -0.2232 1.00 -0.2243 -0.2247 -0.2248 -0.2248 -0.2248 -0.2248 0
φ1= φ2= φ3= φ4= φ5= φ6= φ7= φ8= δ1= δ2=
0 -2.4404 -1.7677 0 -0.5199 0.1989 0 -0.2248 6.9259 6.814
θ1= θ2= θ3= θ4= θ5= θ6= θ7= θ8= Δ1= Δ2=
0 rad -0.0028923 rad -0.002095 rad 0 rad -0.0006162 rad 0.0002357 rad 0 rad -0.0002665 rad 0.0087557 m 0.0072683 m
SISTEMA GLOBAL DE COORDENADAS 2 4 6
SISTEMA LOCAL DE COORDENADAS 14 4 13
1 3
5
3 2
7
12 11
1 Matriz de rigidez del sistema local Barra 1-2 EI/L= 421.88 k= 1687.5 843.75 843.75 1687.5 Barra 11-12 k=
EI/L= 3200 1600
800 1600 3200
Barra 3-4 k=
Barra 13-14 k=
Matriz de transformacion "A" D1=1 D2=1 D3=1 d1 0.3125 0 d2 0.3125 0 d3 -0.3704 0.3704 d4 -0.3704 0.3704 d5 0.3125 0 d6 0.3125 0 A= d7 -0.3704 0.3704 d8 -0.3704 0.3704 d9 0.3125 0 d10 0.3125 0 d11 0 0 d12 0 0 d13 0 0 d14 0 0 d15 0 0 d16 0 0 Momentos finales en ejes locales "mij" m 1-2= 4.4855 m 3-4= 0.1962 2.0451 0.9935 m 9-10= 6.7011 6.4762
m 11-12= -10.241 -6.5995
EI/L= 2000 1000
500 1000 2000
Barra 5-6
EI/L= 3200 1600
800 1600 3200
Barra 15-16 k=
k=
EI/L= 1687.5 843.75 EI/L= 3657.1 1828.6
Ensamblaje de la matriz de rigid D4=1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0
D5=1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0
D6=1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0
D7=1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1
m 5-6= 6.406 5.8861
m 7-8= 7.0792 7.9311
m 13-14= -6.3269 -2.5977
m 15-16= -2.7407 -2.1012
K=
3129.2 -1646 -320.1 -1111 -320.1 -1111 791.02
Vector de desplazamientos "D" 0.0088 0.016 -0.0029 D= -0.0021 -0.0006 0.0002 0
CAL DE COORDENADAS 8
7 6
Mij
Barra
10
5
421.88 843.75 1687.5
(Tn-m) (Tn-m)
1-2 2-3 4-5 5-6 7- 8 2-5 3-6 5-8
16 15
Mji
4.4855 2.0451 0.1962 0.9935 6.406 5.8861 7.0792 7.9311 6.7011 6.4762 -2.2413 -14.6 -0.9935 -7.9311 1.6343 -6.4762
9
Barra 7-8 k=
EI/L= 2000 1000
500 1000 2000
Barra 9-10
EI/L= 421.88 1687.5 843.75 843.75 1687.5
k=
914.29 1828.6 3657.1
mblaje de la matriz de rigidez "K"
Matriz de rigidez lateral "Kl"
-1646 -320.1 -1111 -320.1 -1111 791.02 1646.1 1111.1 1111.1 1111.1 1111.1 0 1111.1 6887.5 1000 1600 0 0 1111.1 1000 5200 0 1600 0 1111.1 1600 0 10545 1000 1828.6 1111.1 0 1600 1000 5200 0 0 0 0 1828.6 0 5344.6
or de desplazamientos "D"
500.08 -415.08 -415.08 552.11 Kl=
Desplazamientos en ejes locales "d" d 1-2= 0.0027 d 3-4= 0 0 0.0006 d 9-10= 0.0027 0.0025
Ensamblaje del vector d
d 11-12= -0.0029 -0.0006
2629 -1231
-1231 1094
Q=
d 5-6= 0.0027 0.0021
d 7-8=
d 13-14= -0.0021 0.0002
d 15-16=
θ1= θ2= θ3= θ4= θ5= θ6= θ7= θ8= Δ1= Δ2=
0 rad -0.0028923 rad -0.002095 rad 0 rad -0.0006162 rad 0.0002357 rad 0 rad -0.0002665 rad 0.0087557 m 0.0072683 m
Ensamblaje del vector defuerzas "Q" 4.00 6.00 -8.00 -5.33 3.63 5.33 4.38
0.0021 0.0029 -0.0006 0
ṁ 11-12=
8 -8 ṁ 13-14= 5.3333 -5.3333 ṁ 15-16= 4.375 -4.375
Resolucion del estado restringido (Estado 0)
1.90 0.2904 0.5 0 -2.323 -1.0256 0.1381 -0.0922 0.056 -0.09 0.0325 -0.0178 0.0107 -0.0125 0.0065 -0.0034 0.002 -0.0022 0.0012 -0.0007 0.0004 0 0.0002 -3.320
0.3846 0.5 0 -2.0513 -1.1615 -0.1844 0.0691 1.19 -0.1801 0.028 -0.0357 0.0163 -0.0249 0.0053 -0.0068 0.0032 -0.0043 0.001 -0.0013 0.0006 -0.0008 0.0002 0 -3.528
0.6154 0.5 5.3333 -3.2821 1.641 -0.2951 0.3992 -0.2881 0.0648 -0.0571 0.0486 -0.0399 0.0124 -0.0109 0.008 -0.0069 0.0024 -0.0021 0.0015 -0.0013 0.0005 0 3.528
0.6154 0.5 -5.3333 3.2821 -1.641 0.7983 -0.1475 0.1295 -0.1441 0.0971 -0.0286 0.0248 -0.0199 0.0161 -0.0055 0.0047 -0.0035 0.003 -0.001 0.0009 -0.0006 0.0006 -2.968
0.1897 0.5 0 0.6876 1.0256 -0.1259 0.2495 -0.0275 0.0405 -0.0237 0.0303 -0.0123 0.0078 -0.0045 0.005 -0.0027 0.0015 -0.0009 0.0009 -0.0005 0.0003 0 1.851
0.3846 0.5 0 2.0513 0.3438 0.4989 -0.063 1.19 0.081 -0.0137 0.0607 -0.0118 0.0155 -0.0062 0.01 -0.0022 0.003 -0.0013 0.0018 0 0.0006 0 0.0003 2.968
0.3035 0.5 -8 1.1001 -1.8584 -0.2015 0.1105 -0.044 0.0448 -0.0378 0.026 -0.0197 0.0085 -0.0072 0.0052 -0.0043
0 0 0 0 0.2901 0 -0.0531 0 -0.0116 0 -0.01 0 -0.0052 0 -0.0019 0
0.16 0.5 0 0.5801 0 -0.1063 0 1.00 -0.0232 0 -0.02 0 -0.0104 0 -0.0038 0 -0.0023
1.90 0 0 0 0 -0.98 0 0.0583 0 0.0236 0 0.0137 0 0.0045 0 0.0027 0
0.245 0.5 0 -1.9601 0 0.1165 0 1.00 0.0473 0 0.0274 0 0.009 0 0.0055 0 0.0017
0.4646 0.5 8 -3.7169 0.55 0.221 -0.1008 0.0897 -0.022 0.052 -0.0189 0.0171 -0.0099 0.0104 -0.0036 0.0033
0.0009 0 0.0005 0 0.0002 0 -0.876
0 0.0011 0 0.0003 0 0.0002 -1.751
-0.0021 0.002 -0.0007 0.0006 0 0.0004 5.071
0.0016 -0.0014 0.001 -0.0008 0.0003 0 -8.877
-0.0011 0 0 0 0 0 0.207
0 -0.0007 0 0 0 0 0.413
Resolucion del estado con desplazamiento en el primer piso (Estado 1)
1.90 0.2904 0.5 -1111 92.95 213.68 -76.151 -74.684 22.454 29.348 -15.197 -9.3273 4.5719 5.0899 -2.9293 -1.56 0.8851 0.9484 -0.5602 -0.2878 0.1695 0.1799 -0.1069
0.3846 0.5 -1111 427.35 46.475 -149.37 -38.076 1.19 58.696 11.227 -18.655 -7.5986 10.18 2.286 -3.12 -1.4647 1.8968 0.4426 -0.5757 -0.2801 0.3599 0.0847 -0.109
0.6154 0.5 0 683.76 341.88 -239 -114.53 93.914 37.275 -29.847 -18.869 16.288 5.826 -4.992 -3.467 3.0349 1.0542 -0.9211 -0.6555 0.5758 0.1986 -0.1743
0.6154 0.5 0 683.76 341.88 -229.07 -119.49 74.55 46.957 -37.738 -14.924 11.652 8.1439 -6.9339 -2.496 2.1084 1.5174 -1.3111 -0.4605 0.3971 0.2879 -0.2496
0.1897 0.5 -1111 60.712 213.68 -3.3009 -71.584 28.734 23.297 -8.0216 -11.793 6.2475 3.6412 -1.8603 -2.1669 1.2261 0.6589 -0.3696 -0.4097 0.2355 0.1241 -0.0712
0.3846 0.5 -1111 427.35 30.356 -143.17 -1.6505 1.19 46.594 14.367 -23.586 -4.0108 7.2825 3.1238 -4.3337 -0.9301 1.3177 0.6131 -0.8194 -0.1848 0.2482 0.1177 -0.156
-921.643
-771.359
771.359
758.580
-872.136
-758.580
0.3035 0.5 0 97.14 74.36 -5.2815 -60.921
0 0 791.02 0 25.613 0 -1.3926
0.16 0.5 791.02 51.226 0 -2.7851 0
1.90 0 0 791.02 0 39.213 0 -32.126
0.245 0.5 791.02 78.426 0 -64.252 0
0.4646 0.5 0 148.72 48.57 -121.84 -2.6407
0 9.4726 0 -6.4113 0 1.9288 0 -1.2358 0 0.3734 0 -0.2363 0 0.0715 0
1.00 18.945 0 -12.823 0 3.8576 0 -2.4716 0 0.7468 0 -0.4727 0 0.143 0 -0.0902
35.926 22.987 -24.316 -6.4173 7.3151 4.998 -4.687 -1.4882 1.4162 0.9809 -0.8964 -0.2957 0.2711 0.1884 -0.1711
45.975 17.963 -12.835 -12.158 9.9961 3.6575 -2.9764 -2.3435 1.9618 0.7081 -0.5914 -0.4482 0.3767 0.1356 -0.1139
0 12.122 0 -3.3841 0 2.6357 0 -0.7848 0 0.5173 0 -0.1559 0 0.0993 0
1.00 24.244 0 -6.7682 0 5.2714 0 -1.5696 0 1.0346 0 -0.3119 0 0.1987 0 -0.0601
802.07
813.02
108.62
154.61
826.29
861.50
Resolucion del estado con desplazamiento en el segundo piso (Estado 1)
1.90 0.2904 0.5 1111.1 -322.65 -213.68 111 96.77 -48.888 -37.138 21.391 16.51 -9.4418 -6.9238 4.0908 3.0628 -1.8065 -1.3138 0.7809 0.5805 -0.3449 -0.2503 0.149
0.3846 0.5 1111.1 -427.35 -161.32 193.54 55.502 1.19 -74.276 -24.444 33.02 10.696 -13.848 -4.7209 6.1257 2.0454 -2.6276 -0.9033 1.161 0.3905 -0.5007 -0.1725 0.2212
0.6154 0.5 0 -683.76 -341.88 309.66 137.62 -118.84 -61.407 52.832 25.308 -22.156 -11.206 9.8011 4.7863 -4.2041 -2.1152 1.8575 0.9113 -0.8011 -0.4026 0.3539
0.6154 0.5 0 -683.76 -341.88 275.23 154.83 -122.81 -59.421 50.616 26.416 -22.412 -11.078 9.5726 4.9005 -4.2305 -2.1021 1.8226 0.9288 -0.8051 -0.4005 0.3476
0.1897 0.5 1111.1 -210.74 -213.68 89.485 86.01 -45.66 -38.38 20.007 15.818 -8.9549 -7.0037 3.948 2.9914 -1.7194 -1.322 0.7591 0.5696 -0.3285 -0.2516 0.1451
0.3846 0.5 1111.1 -427.35 -105.37 172.02 44.742 1.19 -76.759 -22.83 31.635 10.003 -14.007 -4.4775 5.9829 1.974 -2.6441 -0.8597 1.1391 0.3796 -0.5032 -0.1643 0.2172
723.022
703.646
-703.646
-724.237
802.804
724.237
1.90 0 0 0 0 -136.12 0 46.83 0 -20.625 0 9.0245 0 -3.9833 0 1.7258 0 -0.7621 0 0.3295 0 -0.1455 0
0.245 0.5 0 -272.23 0 93.66 0 1.00 -41.249 0 18.049 0 -7.9665 0 3.4516 0 -1.5243 0 0.6589 0 -0.291 0 0.1257
0.4646 0.5 0 -516.23 -168.6 177.61 71.588 -78.221 -36.528 34.226 16.005 -15.107 -7.164 6.5453 3.1584 -2.8905 -1.3755 1.2495 0.6073 -0.5518 -0.2628 0.2384
0.3035 0.5 0 -337.19 -258.12 143.18 88.803 -73.056 -39.11 32.011 17.113 -14.328 -7.5534 6.3169 3.2727 -2.7511 -1.4452 1.2146 0.6247 -0.5257 -0.2759 0.2321
0 0 0 0 -88.908 0 37.751 0 -19.263 0 8.4403 0 -3.7779 0 1.6656 0 -0.7254 0 0.3202 0 -0.1386 0
0.16 0.5 0 -177.82 0 75.503 0 1.00 -38.526 0 16.881 0 -7.5557 0 3.3312 0 -1.4507 0 0.6405 0 -0.2772 0 0.1224
-103.72
-207.32
-515.70
-441.59
-64.64
-129.15
Calculo de desplazamientos totales Kl=
2629.2 -1231 -1231 1094
R=
Calculo de momentos finales en las barras Barra 1-2 2-3 4-5 5-6 7- 8 2-5 3-6 5-8
Mij
Mji
(Tn-m) (Tn-m) 4.49 2.05 0.20 0.99 6.41 5.89 7.08 7.93 6.70 6.48 -2.24 -14.60 -0.99 -7.93 1.63 -6.48
3.2945 6.7515
D=
0.0088 0.016
8.5362
-8.5362
R2= -6.7515
-2.5362
1.7848
2.5362
-1.7848 2.2846
-2.2846
3.8757
-3.8757
-0.8208
0.1936
0.5812
0.8208
-0.1936
-0.5812
R1 y R2 representa las reacciones en el estado restringido Barra 1-2 2-3 4-5 5-6 7- 8 2-5 3-6 5-8
2.17 0.3468 0.5 4.375 1.2572 1.4968 -0.2303 -0.2151 -0.0502 0.0394 -0.0433 0.0086 -0.0225 0.0074 -0.0082 0.0039 -0.0049
R1=
0.6843 0.5 -4.375 2.9937 0.6286 -0.4301 -0.1152 0.0788 -0.0251 0.0172 -0.0216 0.0148 -0.0113 0.0077 -0.0041 0.0028
0 0 0 0 0.6907 0 -0.0992 0 0.0182 0 0.004 0 0.0034 0 0.0018 0
0.3157 0.5 0 1.3813 0 -0.1985 0 1.00 0.0364 0 0.0079 0 0.0068 0 0.0036 0 0.0013
Mij
Mji
(Tn-m) (Tn-m)
-0.876 -3.320 0.207 1.851 0.620 5.071 3.528 6.613
-1.751 -3.528 0.413 2.968 1.240 -8.877 -2.968 -1.240
0.0014 -0.0016 0.0008 -0.001 0.0003 0 6.613
-0.0024 0.0017 -0.0008 0.0005 0 0.0003 -1.240
0.0006 0 0.0004 0 0.0001 0 0.620
627.04
0 0.0008 0 0.0002 0 0.0002 1.240
-627
R2=
-627
-604
627.04
603.97 -1132
1131.8
-1231
-2263
2263.2
504.72
527.43
366.01
-504.72
-527.43
-366
R1 y R2 representa las reacciones en el estado restringido Barra 1-2 2-3 4-5 5-6 7- 8 2-5 3-6 5-8
2.17 0.3468 0.5 0 111.02 -270.63 -6.036 -18.991
R1=
0.6843 0.5 0 -541.26 55.509 -37.982 -3.018
0 0 791.02 0 -124.88 0 -8.763
0.3157 0.5 791.02 -249.75 0 -17.526 0
Mij
Mji
(Tn-m) (Tn-m) 802.07
813.02
-921.64
-771.36
826.29
861.50
-872.14
-758.58
654.11
517.13
108.62
154.61
771.36
758.58
-143.96
-517.13
52.543 1.0325 -14.668 -8.9882 11.424 2.5092 -3.4017 -1.9543 2.2421 0.5819 -0.6759 -0.3835 0.4306 0.1156 -0.1302
2.0651 26.271 -17.976 -7.334 5.0184 5.712 -3.9085 -1.7008 1.1638 1.121 -0.7671 -0.338 0.2313 0.2153 -0.1473
0 0.4764 0 -4.1474 0 1.1578 0 -0.9018 0 0.2685 0 -0.177 0 0.0534 0
1.00 0.9529 0 -8.2948 0 2.3156 0 -1.8035 0 0.537 0 -0.354 0 0.1067 0 -0.068
-143.96
-517.13
654.11
517.13
-528.4
528.4
R2=
528.4
565.57
-528.4
-565.57 625.6
-625.6
1094
1251.7
-1252
R1=
-97.2
-60.557
20.717
97.2
60.557
-20.717
R1 y R2 representa las reacciones en el estado restringido Barra 1-2 2-3 4-5 5-6 7- 8 2-5
Mij
Mji
(Tn-m) (Tn-m) -103.72
-207.32
723.02
703.65
-64.64
-129.15
802.80
724.24
22.07
44.23
-515.70
-441.59
3-6 5-8
2.17 0.3468 0.5 0 -385.36 0 163.63 65.922 -83.493 -27.991 36.584 14.283 -16.375 -6.2582 7.2193 2.8012 -3.1441 -1.235 1.3881 0.5378 -0.6008 -0.2375 0.2653
0.6843 0.5 0 0 -192.68 131.84 81.815 -55.983 -41.747 28.566 18.292 -12.516 -8.1874 5.6023 3.6096 -2.4699 -1.572 1.0757 0.694 -0.4749 -0.3004 0.2055
0 0 0 0 0 0 30.418 0 -12.916 0 6.5905 0 -2.8877 0 1.2925 0 -0.5698 0 0.2482 0 -0.1096 0
0.3157 0.5 0 0 0 60.836 0 1.00 -25.832 0 13.181 0 -5.7754 0 2.5851 0 -1.1397 0 0.4963 0 -0.2191 0 0.0948
-232.07
-44.23
22.07
44.23
-703.65
-724.24
-232.07
-44.23
-3.2945
2629.2
-1231
6
10 6 Tn/m 4 Tn 2
5
1
4
Datos de los elementos de la estructura L b Barra (m)
1-2 2-3 4-5 5-6 7- 8 2-5 3-6 5-8
(m)
3.2 2.7 3.2 2.7 3.2 4 4 3.5
0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
h
I
(m)
(m4)
0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.4
0.000675 0.000675 0.000675 0.000675 0.000675 0.0016 0.0016 0.0016
Comparacion de momentos finales y desplazamientos (Solucion) Takabeya Rigidez Mij Mji Mij Mji Barra (Tn-m)
Tn
8
7
1-2 2-3 4-5 5-6 7- 8 2-5 3-6 5-8
(Tn-m)
(Tn-m)
(Tn-m)
4.4855066 2.0451152 4.4855055 2.045111 0.1962165 0.993532 0.196214 0.9935343 6.4059877 5.8860774 6.40599 5.8860801 7.0791765 7.9310749 7.0791783 7.9310735 6.7010704 6.4762428 6.7010711 6.4762424 -2.2413143 -14.599513 -2.241325 -14.599517 -0.9935338 -7.9310667 -0.9935343 -7.9310735 1.6342597 -6.4762382 1.634259 -6.4762424
E
EI/L
(Tn/m2)
(Tn-m)
2000000 421.875 2000000 500 2000000 421.875 2000000 500 2000000 421.875 2000000 800 2000000 800 2000000 914.28571
k
Mij
Mji
(Tn-m)
(Tn-m)
1.00 0 0 1.19 0 0 1.00 0 0 1.19 0 0 1.00 0 0 1.90 8 -8 1.90 5.3333333 -5.3333333 2.17 4.375 -4.375
Cross Mij
Mji
(Tn-m)
(Tn-m)
4.4850218 2.0455162 0.1959796 0.9937935 6.4056023 5.8865489 7.0788986 7.9313283 6.7007297 6.4765812 -2.2414958 -14.599749 -0.9937935 -7.9313283 1.6343019 -6.4765812
θ1(rad)= θ2(rad)= θ3(rad)= θ4(rad)= θ5(rad)= θ6(rad)= θ7(rad)= θ8(rad)= Δ1(m)= Δ2(m)=
Takabeya 0.00E+00 -2.89E-03 -2.10E-03 0.00E+00 -6.16E-04 2.36E-04 0.00E+00 -2.66E-04 8.76E-03 7.27E-03
Rigidez 0.00E+00 -2.89E-03 -2.09E-03 0.00E+00 -6.16E-04 2.36E-04 0.00E+00 -2.66E-04 8.76E-03 7.27E-03
Cross
8.76E-03 7.27E-03
EJEMPLO Resolucion de la estructura mediante el metodo de TAKABEYA
3 -0.1923 0.9935
-0.75
0.1962 -0.1452 2 -0.1225 2.0451
-0.5
4.4855 -0.5 1
-0.3077 -0.9935 5.3333 0 -0.8654 -1.3905 -1.7379 -1.7817 -1.7758 -1.7704 1.19 -1.7684 -1.7678 -1.7677 -1.7677 -1.7677 -1.7677 0 -0.2323 -2.2413 8 0 -0.98 -2.0713 -2.3352 -2.4022 -2.4265 -2.4357 1.00 -2.4389 -2.44 -2.4403 -2.4404 -2.4404 -2.4404 0
-7.9311 -0.3077 1.90 -5.3333 6 3.4172 -0.1923 5.7475 7.9311 6.5457 6.7533 6.8003 6.8105 6.813 -0.75 6.8137 6.8139 6.814 6.814 6.814 7.0792 -0.0948 -14.6 -0.1517 1.90 -8 5 5.3333 -0.08 6.5592 5.8861 6.8334 6.8992 6.9175 6.9232 6.9251 -0.5 6.9256 6.9258 6.9259 6.9259 6.9259 6.406 -0.5 4
0 0.8654 0.7031 0.3884 0.2546 0.2134 0.2025 1.19 0.1998 0.1991 0.199 0.1989 0.1989 0.1989 0 -0.1734 1.6343 4.375 0 0.2901 -0.3483 -0.4866 -0.5189 -0.5219 -0.5209 1.00 -0.5202 -0.52 -0.5199 -0.5199 -0.5199 -0.5199 0
2.17
Barra 1-2 2-3 4-5 5-6 7- 8 2-5 3-6 5-8
-6.4762 -0.3421 -4.375
Mij
Mji
(Tn-m) (Tn-m)
4.4855 2.0451 0.1962 0.9935 6.406 5.8861 7.0792 7.9311 6.7011 6.4762 -2.2413 -14.6 -0.9935 -7.9311 1.6343 -6.4762
8 -0.1579 6.4762
-0.5
6.7011 -0.5 7
0 0.6907 -0.0321 -0.1783 -0.2106 -0.2199 -0.2232 1.00 -0.2243 -0.2247 -0.2248 -0.2248 -0.2248 -0.2248 0
φ1= φ2= φ3= φ4= φ5= φ6= φ7= φ8= δ1= δ2=
0 -2.4404 -1.7677 0 -0.5199 0.1989 0 -0.2248 6.9259 6.814
θ1= θ2= θ3= θ4= θ5= θ6= θ7= θ8= Δ1= Δ2=
0 rad -0.0028923 rad -0.002095 rad 0 rad -0.0006162 rad 0.0002357 rad 0 rad -0.0002665 rad 0.0087557 m 0.0072683 m
SISTEMA GLOBAL DE COORDENADAS 2 4 6
SISTEMA LOCAL DE COORDENADAS 14 4 13
1 3
5
3 2
7
12 11
1 Matriz de rigidez del sistema local Barra 1-2 EI/L= 421.88 k= 1687.5 843.75 843.75 1687.5 Barra 11-12 k=
EI/L= 3200 1600
800 1600 3200
Barra 3-4 k=
Barra 13-14 k=
Matriz de transformacion "A" D1=1 D2=1 D3=1 d1 0.3125 0 d2 0.3125 0 d3 -0.3704 0.3704 d4 -0.3704 0.3704 d5 0.3125 0 d6 0.3125 0 A= d7 -0.3704 0.3704 d8 -0.3704 0.3704 d9 0.3125 0 d10 0.3125 0 d11 0 0 d12 0 0 d13 0 0 d14 0 0 d15 0 0 d16 0 0 Momentos finales en ejes locales "mij" m 1-2= 4.4855 m 3-4= 0.1962 2.0451 0.9935 m 9-10= 6.7011 6.4762
m 11-12= -10.241 -6.5995
EI/L= 2000 1000
500 1000 2000
Barra 5-6
EI/L= 3200 1600
800 1600 3200
Barra 15-16 k=
k=
EI/L= 1687.5 843.75 EI/L= 3657.1 1828.6
Ensamblaje de la matriz de rigid D4=1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0
D5=1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0
D6=1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0
D7=1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1
m 5-6= 6.406 5.8861
m 7-8= 7.0792 7.9311
m 13-14= -6.3269 -2.5977
m 15-16= -2.7407 -2.1012
K=
3129.2 -1646 -320.1 -1111 -320.1 -1111 791.02
Vector de desplazamientos "D" 0.0088 0.016 -0.0029 D= -0.0021 -0.0006 0.0002 0
CAL DE COORDENADAS 8
7 6
Mij
Barra
10
5
421.88 843.75 1687.5
(Tn-m) (Tn-m)
1-2 2-3 4-5 5-6 7- 8 2-5 3-6 5-8
16 15
Mji
4.4855 2.0451 0.1962 0.9935 6.406 5.8861 7.0792 7.9311 6.7011 6.4762 -2.2413 -14.6 -0.9935 -7.9311 1.6343 -6.4762
9
Barra 7-8 k=
EI/L= 2000 1000
500 1000 2000
Barra 9-10
EI/L= 421.88 1687.5 843.75 843.75 1687.5
k=
914.29 1828.6 3657.1
mblaje de la matriz de rigidez "K"
Matriz de rigidez lateral "Kl"
-1646 -320.1 -1111 -320.1 -1111 791.02 1646.1 1111.1 1111.1 1111.1 1111.1 0 1111.1 6887.5 1000 1600 0 0 1111.1 1000 5200 0 1600 0 1111.1 1600 0 10545 1000 1828.6 1111.1 0 1600 1000 5200 0 0 0 0 1828.6 0 5344.6
or de desplazamientos "D"
500.08 -415.08 -415.08 552.11 Kl=
Desplazamientos en ejes locales "d" d 1-2= 0.0027 d 3-4= 0 0 0.0006 d 9-10= 0.0027 0.0025
Ensamblaje del vector d
d 11-12= -0.0029 -0.0006
2629 -1231
-1231 1094
Q=
d 5-6= 0.0027 0.0021
d 7-8=
d 13-14= -0.0021 0.0002
d 15-16=
θ1= θ2= θ3= θ4= θ5= θ6= θ7= θ8= Δ1= Δ2=
0 rad -0.0028923 rad -0.002095 rad 0 rad -0.0006162 rad 0.0002357 rad 0 rad -0.0002665 rad 0.0087557 m 0.0072683 m
Ensamblaje del vector defuerzas "Q" 4.00 6.00 -8.00 -5.33 3.63 5.33 4.38
0.0021 0.0029 -0.0006 0
ṁ 11-12=
8 -8 ṁ 13-14= 5.3333 -5.3333 ṁ 15-16= 4.375 -4.375
Resolucion del estado restringido (Estado 0)
1.90 0.2904 0.5 0 -2.323 -1.0256 0.1381 -0.0922 0.056 -0.09 0.0325 -0.0178 0.0107 -0.0125 0.0065 -0.0034 0.002 -0.0022 0.0012 -0.0007 0.0004 0 0.0002 -3.320
0.3846 0.5 0 -2.0513 -1.1615 -0.1844 0.0691 1.19 -0.1801 0.028 -0.0357 0.0163 -0.0249 0.0053 -0.0068 0.0032 -0.0043 0.001 -0.0013 0.0006 -0.0008 0.0002 0 -3.528
0.6154 0.5 5.3333 -3.2821 1.641 -0.2951 0.3992 -0.2881 0.0648 -0.0571 0.0486 -0.0399 0.0124 -0.0109 0.008 -0.0069 0.0024 -0.0021 0.0015 -0.0013 0.0005 0 3.528
0.6154 0.5 -5.3333 3.2821 -1.641 0.7983 -0.1475 0.1295 -0.1441 0.0971 -0.0286 0.0248 -0.0199 0.0161 -0.0055 0.0047 -0.0035 0.003 -0.001 0.0009 -0.0006 0.0006 -2.968
0.1897 0.5 0 0.6876 1.0256 -0.1259 0.2495 -0.0275 0.0405 -0.0237 0.0303 -0.0123 0.0078 -0.0045 0.005 -0.0027 0.0015 -0.0009 0.0009 -0.0005 0.0003 0 1.851
0.3846 0.5 0 2.0513 0.3438 0.4989 -0.063 1.19 0.081 -0.0137 0.0607 -0.0118 0.0155 -0.0062 0.01 -0.0022 0.003 -0.0013 0.0018 0 0.0006 0 0.0003 2.968
0.3035 0.5 -8 1.1001 -1.8584 -0.2015 0.1105 -0.044 0.0448 -0.0378 0.026 -0.0197 0.0085 -0.0072 0.0052 -0.0043
0 0 0 0 0.2901 0 -0.0531 0 -0.0116 0 -0.01 0 -0.0052 0 -0.0019 0
0.16 0.5 0 0.5801 0 -0.1063 0 1.00 -0.0232 0 -0.02 0 -0.0104 0 -0.0038 0 -0.0023
1.90 0 0 0 0 -0.98 0 0.0583 0 0.0236 0 0.0137 0 0.0045 0 0.0027 0
0.245 0.5 0 -1.9601 0 0.1165 0 1.00 0.0473 0 0.0274 0 0.009 0 0.0055 0 0.0017
0.4646 0.5 8 -3.7169 0.55 0.221 -0.1008 0.0897 -0.022 0.052 -0.0189 0.0171 -0.0099 0.0104 -0.0036 0.0033
0.0009 0 0.0005 0 0.0002 0 -0.876
0 0.0011 0 0.0003 0 0.0002 -1.751
-0.0021 0.002 -0.0007 0.0006 0 0.0004 5.071
0.0016 -0.0014 0.001 -0.0008 0.0003 0 -8.877
-0.0011 0 0 0 0 0 0.207
0 -0.0007 0 0 0 0 0.413
Resolucion del estado con desplazamiento en el primer piso (Estado 1)
1.90 0.2904 0.5 -1111 92.95 213.68 -76.151 -74.684 22.454 29.348 -15.197 -9.3273 4.5719 5.0899 -2.9293 -1.56 0.8851 0.9484 -0.5602 -0.2878 0.1695 0.1799 -0.1069
0.3846 0.5 -1111 427.35 46.475 -149.37 -38.076 1.19 58.696 11.227 -18.655 -7.5986 10.18 2.286 -3.12 -1.4647 1.8968 0.4426 -0.5757 -0.2801 0.3599 0.0847 -0.109
0.6154 0.5 0 683.76 341.88 -239 -114.53 93.914 37.275 -29.847 -18.869 16.288 5.826 -4.992 -3.467 3.0349 1.0542 -0.9211 -0.6555 0.5758 0.1986 -0.1743
0.6154 0.5 0 683.76 341.88 -229.07 -119.49 74.55 46.957 -37.738 -14.924 11.652 8.1439 -6.9339 -2.496 2.1084 1.5174 -1.3111 -0.4605 0.3971 0.2879 -0.2496
0.1897 0.5 -1111 60.712 213.68 -3.3009 -71.584 28.734 23.297 -8.0216 -11.793 6.2475 3.6412 -1.8603 -2.1669 1.2261 0.6589 -0.3696 -0.4097 0.2355 0.1241 -0.0712
0.3846 0.5 -1111 427.35 30.356 -143.17 -1.6505 1.19 46.594 14.367 -23.586 -4.0108 7.2825 3.1238 -4.3337 -0.9301 1.3177 0.6131 -0.8194 -0.1848 0.2482 0.1177 -0.156
-921.643
-771.359
771.359
758.580
-872.136
-758.580
0.3035 0.5 0 97.14 74.36 -5.2815 -60.921
0 0 791.02 0 25.613 0 -1.3926
0.16 0.5 791.02 51.226 0 -2.7851 0
1.90 0 0 791.02 0 39.213 0 -32.126
0.245 0.5 791.02 78.426 0 -64.252 0
0.4646 0.5 0 148.72 48.57 -121.84 -2.6407
0 9.4726 0 -6.4113 0 1.9288 0 -1.2358 0 0.3734 0 -0.2363 0 0.0715 0
1.00 18.945 0 -12.823 0 3.8576 0 -2.4716 0 0.7468 0 -0.4727 0 0.143 0 -0.0902
35.926 22.987 -24.316 -6.4173 7.3151 4.998 -4.687 -1.4882 1.4162 0.9809 -0.8964 -0.2957 0.2711 0.1884 -0.1711
45.975 17.963 -12.835 -12.158 9.9961 3.6575 -2.9764 -2.3435 1.9618 0.7081 -0.5914 -0.4482 0.3767 0.1356 -0.1139
0 12.122 0 -3.3841 0 2.6357 0 -0.7848 0 0.5173 0 -0.1559 0 0.0993 0
1.00 24.244 0 -6.7682 0 5.2714 0 -1.5696 0 1.0346 0 -0.3119 0 0.1987 0 -0.0601
802.07
813.02
108.62
154.61
826.29
861.50
Resolucion del estado con desplazamiento en el segundo piso (Estado 1)
1.90 0.2904 0.5 1111.1 -322.65 -213.68 111 96.77 -48.888 -37.138 21.391 16.51 -9.4418 -6.9238 4.0908 3.0628 -1.8065 -1.3138 0.7809 0.5805 -0.3449 -0.2503 0.149
0.3846 0.5 1111.1 -427.35 -161.32 193.54 55.502 1.19 -74.276 -24.444 33.02 10.696 -13.848 -4.7209 6.1257 2.0454 -2.6276 -0.9033 1.161 0.3905 -0.5007 -0.1725 0.2212
0.6154 0.5 0 -683.76 -341.88 309.66 137.62 -118.84 -61.407 52.832 25.308 -22.156 -11.206 9.8011 4.7863 -4.2041 -2.1152 1.8575 0.9113 -0.8011 -0.4026 0.3539
0.6154 0.5 0 -683.76 -341.88 275.23 154.83 -122.81 -59.421 50.616 26.416 -22.412 -11.078 9.5726 4.9005 -4.2305 -2.1021 1.8226 0.9288 -0.8051 -0.4005 0.3476
0.1897 0.5 1111.1 -210.74 -213.68 89.485 86.01 -45.66 -38.38 20.007 15.818 -8.9549 -7.0037 3.948 2.9914 -1.7194 -1.322 0.7591 0.5696 -0.3285 -0.2516 0.1451
0.3846 0.5 1111.1 -427.35 -105.37 172.02 44.742 1.19 -76.759 -22.83 31.635 10.003 -14.007 -4.4775 5.9829 1.974 -2.6441 -0.8597 1.1391 0.3796 -0.5032 -0.1643 0.2172
723.022
703.646
-703.646
-724.237
802.804
724.237
1.90 0 0 0 0 -136.12 0 46.83 0 -20.625 0 9.0245 0 -3.9833 0 1.7258 0 -0.7621 0 0.3295 0 -0.1455 0
0.245 0.5 0 -272.23 0 93.66 0 1.00 -41.249 0 18.049 0 -7.9665 0 3.4516 0 -1.5243 0 0.6589 0 -0.291 0 0.1257
0.4646 0.5 0 -516.23 -168.6 177.61 71.588 -78.221 -36.528 34.226 16.005 -15.107 -7.164 6.5453 3.1584 -2.8905 -1.3755 1.2495 0.6073 -0.5518 -0.2628 0.2384
0.3035 0.5 0 -337.19 -258.12 143.18 88.803 -73.056 -39.11 32.011 17.113 -14.328 -7.5534 6.3169 3.2727 -2.7511 -1.4452 1.2146 0.6247 -0.5257 -0.2759 0.2321
0 0 0 0 -88.908 0 37.751 0 -19.263 0 8.4403 0 -3.7779 0 1.6656 0 -0.7254 0 0.3202 0 -0.1386 0
0.16 0.5 0 -177.82 0 75.503 0 1.00 -38.526 0 16.881 0 -7.5557 0 3.3312 0 -1.4507 0 0.6405 0 -0.2772 0 0.1224
-103.72
-207.32
-515.70
-441.59
-64.64
-129.15
Calculo de desplazamientos totales Kl=
2629.2 -1231 -1231 1094
R=
Calculo de momentos finales en las barras Barra 1-2 2-3 4-5 5-6 7- 8 2-5 3-6 5-8
Mij
Mji
(Tn-m) (Tn-m) 4.49 2.05 0.20 0.99 6.41 5.89 7.08 7.93 6.70 6.48 -2.24 -14.60 -0.99 -7.93 1.63 -6.48
3.2945 6.7515
D=
0.0088 0.016
8.5362
-8.5362
R2= -6.7515
-2.5362
1.7848
2.5362
-1.7848 2.2846
-2.2846
3.8757
-3.8757
-0.8208
0.1936
0.5812
0.8208
-0.1936
-0.5812
R1 y R2 representa las reacciones en el estado restringido Barra 1-2 2-3 4-5 5-6 7- 8 2-5 3-6 5-8
2.17 0.3468 0.5 4.375 1.2572 1.4968 -0.2303 -0.2151 -0.0502 0.0394 -0.0433 0.0086 -0.0225 0.0074 -0.0082 0.0039 -0.0049
R1=
0.6843 0.5 -4.375 2.9937 0.6286 -0.4301 -0.1152 0.0788 -0.0251 0.0172 -0.0216 0.0148 -0.0113 0.0077 -0.0041 0.0028
0 0 0 0 0.6907 0 -0.0992 0 0.0182 0 0.004 0 0.0034 0 0.0018 0
0.3157 0.5 0 1.3813 0 -0.1985 0 1.00 0.0364 0 0.0079 0 0.0068 0 0.0036 0 0.0013
Mij
Mji
(Tn-m) (Tn-m)
-0.876 -3.320 0.207 1.851 0.620 5.071 3.528 6.613
-1.751 -3.528 0.413 2.968 1.240 -8.877 -2.968 -1.240
0.0014 -0.0016 0.0008 -0.001 0.0003 0 6.613
-0.0024 0.0017 -0.0008 0.0005 0 0.0003 -1.240
0.0006 0 0.0004 0 0.0001 0 0.620
627.04
0 0.0008 0 0.0002 0 0.0002 1.240
-627
R2=
-627
-604
627.04
603.97 -1132
1131.8
-1231
-2263
2263.2
504.72
527.43
366.01
-504.72
-527.43
-366
R1 y R2 representa las reacciones en el estado restringido Barra 1-2 2-3 4-5 5-6 7- 8 2-5 3-6 5-8
2.17 0.3468 0.5 0 111.02 -270.63 -6.036 -18.991
R1=
0.6843 0.5 0 -541.26 55.509 -37.982 -3.018
0 0 791.02 0 -124.88 0 -8.763
0.3157 0.5 791.02 -249.75 0 -17.526 0
Mij
Mji
(Tn-m) (Tn-m) 802.07
813.02
-921.64
-771.36
826.29
861.50
-872.14
-758.58
654.11
517.13
108.62
154.61
771.36
758.58
-143.96
-517.13
52.543 1.0325 -14.668 -8.9882 11.424 2.5092 -3.4017 -1.9543 2.2421 0.5819 -0.6759 -0.3835 0.4306 0.1156 -0.1302
2.0651 26.271 -17.976 -7.334 5.0184 5.712 -3.9085 -1.7008 1.1638 1.121 -0.7671 -0.338 0.2313 0.2153 -0.1473
0 0.4764 0 -4.1474 0 1.1578 0 -0.9018 0 0.2685 0 -0.177 0 0.0534 0
1.00 0.9529 0 -8.2948 0 2.3156 0 -1.8035 0 0.537 0 -0.354 0 0.1067 0 -0.068
-143.96
-517.13
654.11
517.13
-528.4
528.4
R2=
528.4
565.57
-528.4
-565.57 625.6
-625.6
1094
1251.7
-1252
R1=
-97.2
-60.557
20.717
97.2
60.557
-20.717
R1 y R2 representa las reacciones en el estado restringido Barra 1-2 2-3 4-5 5-6 7- 8 2-5
Mij
Mji
(Tn-m) (Tn-m) -103.72
-207.32
723.02
703.65
-64.64
-129.15
802.80
724.24
22.07
44.23
-515.70
-441.59
3-6 5-8
2.17 0.3468 0.5 0 -385.36 0 163.63 65.922 -83.493 -27.991 36.584 14.283 -16.375 -6.2582 7.2193 2.8012 -3.1441 -1.235 1.3881 0.5378 -0.6008 -0.2375 0.2653
0.6843 0.5 0 0 -192.68 131.84 81.815 -55.983 -41.747 28.566 18.292 -12.516 -8.1874 5.6023 3.6096 -2.4699 -1.572 1.0757 0.694 -0.4749 -0.3004 0.2055
0 0 0 0 0 0 30.418 0 -12.916 0 6.5905 0 -2.8877 0 1.2925 0 -0.5698 0 0.2482 0 -0.1096 0
0.3157 0.5 0 0 0 60.836 0 1.00 -25.832 0 13.181 0 -5.7754 0 2.5851 0 -1.1397 0 0.4963 0 -0.2191 0 0.0948
-232.07
-44.23
22.07
44.23
-703.65
-724.24
-232.07
-44.23
-3.2945
2629.2
-1231
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