PROBLEMA 1

METODO DE LAS RIGIDECES O DE LOS DESPLAZAMIENTOS

PASO 1Determinar la Matriz a

0 0 00 1 0

[a]= 1 0 00 0 10 1 0.707

0.707 0.707 0

0 0 1 0 0 0.7070 1 0 0 1 0.7070 0 0 1 0.707 0

PASO 2Obtener la matriz global de rigidez K

10.5 0 0 0 0 00 10.5 0 0 0 00 0 10.5 0 0 0

k= 0 0 0 10.5 0 00 0 0 0 10.5 00 0 0 0 0 10.5

-150[F] = -50

0

0 0 10.5 0 0 7.42[K] = 0 10.5 0 0 10.5 7.42

0 0 0 10.5 7.42 0

15.75 5.25 0[K] = 5.25 26.25 7.42

0 7.42 15.75

〖 [𝑎〗^𝑇] =

[𝑘]=𝐸𝐴/𝐿

[𝐾]= 〖 [𝑎〗^𝑡][𝑘][𝑎]

PASO 3Calcular la matriz de desplazamiento u

0 0 00 0 00 0 0

-10[u]= 0

0

PASO 4Hallar el vector deformacion e

0 0 0 -10.000 1 0 x 0 =1 0 0 0 3x10 0 10 0 0.707

0.707 0.707 0 6x3

PASO 5Calcular las fuerzas internas de cada barra

10.5 0 0 0 0 00 10.5 0 0 0 00 0 10.5 0 0 00 0 0 10.5 0 00 0 0 0 10.5 00 0 0 0 0 10.5

PASO 6VERIFICACION

0 0 1 0 0 0.7070 1 0 0 0 0.7070 0 0 1 0.707 0

PASO 7REACCIONES EN LOS APOYOS

[𝐹]=[𝐾][𝑢][𝐹]∗〖 [𝐾]〗^(−1)=[𝑢]

[𝑒]=[𝑎]∗[𝑈]

[𝑆]=[𝑘]∗[𝑒]

[𝐹]=[𝑎]^𝑇∗[𝑆]

〖 [𝐾〗^(−1)] =

1 0 0 -1 0 10 -1 0 0 0 20 0 -1 0 0 30 0 0 0 -1 4

0.707 -0.707 -0.707 0 0 50 0 0 -0.707 -0.707 61x 1y 3x 4x 4y 6x5

1 0 0 0 0.707 00 -1 0 0 -0.707 00 0 -1 0 -0.707 0-1 0 0 0 0 -0.7070 0 0 -1 0 -0.707

[𝑎_𝐻 ]=

[𝐻]=[𝑎_𝐻 ]^𝑇∗[𝑆]

0 0 00 1 0

* 1 0 00 0 10 1 0.707

0.707 0.707 0

00

-1000-7 6x1

0 00 0

x -10 = -1050 00 0-7 -74

0 -157x 0 = -52

-105 000

-74

0 0 R0 0

x -105 = 105 3x0 52 4x

5x6 0 52 5x1 ---- 4y-74 6x1

PROBLEMA 2

METODO DE LAS RIGIDECES O DE LOS DESPLAZAMIENTOS

PASO 1Determinar la Matriz a

0 1 0 0 0 00 0 0 0 1 00 0 0 0 0 1

[a]= 0 -1 0 0 0 00 0 1 0 0 00 0 0 0 0 00 0 0 0.95 0.32 00 0 0.71 0 0 0-1 0 0 1 0 0

0 0 0 0 0 01 0 0 -1 0 00 0 0 0 1 00 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 -10 0 0 0 0 00 0 0 0 0 1

PASO 2

Obtener la matriz global de rigidez K

0.25 0 0 0 0 00 0.25 0 0 0 00 0 0.32 0 0 0

[k] = 0 0 0 0.32 0 00 0 0 0 0.17 00 0 0 0 0 0.50 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

304

[F] = 000

〖 [𝑎〗^𝑇] =

[𝑘]=𝐸𝐴/𝐿

050

0 0 0 0 0 00.25 0 0 -0.32 0 0

0 0 0 0 0.17 0[K] = 0 0 0 0 0 0

0 0.25 0 0 0 00 0 0.32 0 0 00 0 0 0.32 0 -0.50 0 0 0 0 00 0 0 0 0 0.5

0.17 0 0 -0.17 0 00 0.57 0 0 0 00 0 0.33 0 0 0

-0.17 0 0 0.46 0.10 0[K] = 0 0 0 0.10 0.28 0

0 0 0 0 0 0.320 -0.32 0 0 0 00 0 -0.16 -0.29 -0.10 00 0 0.16 -0.10 -0.03 0

PASO 3Calcular la matriz de desplazamiento u

38.19 5.35 5.88 32.31 -1.35 0.005.35 4.00 0.00 5.35 0.00 0.005.88 0.00 5.88 5.88 0.00 0.00

32.31 5.35 5.88 32.31 -1.35 0.00-1.35 0.00 0.00 -1.35 4.00 0.000.00 0.00 0.00 0.00 0.00 3.139.53 4.00 0.00 9.53 0.00 0.00

24.28 4.00 5.88 24.28 0.00 0.0012.20 4.00 0.00 12.20 0.00 0.00

259.4936.0470.59

[u]= 241.85-4.04

[𝐾]= 〖 [𝑎〗^𝑡][𝑘][𝑎]

[𝐹]=[𝐾][𝑢]

〖 [𝐾〗^(−1)] =

[𝐹]∗〖 [𝐾]〗^(−1)=[𝑢]

0.0064.20

202.4082.22

PASO 4Hallar el vector deformacion e

36.04-4.040.00

28.1670.5918.029.87

-35.21-17.65

PASO 5Calcular las fuerzas internas de cada barra

9.01-1.010.00

[s]= 9.0112.009.013.16

-11.27-3.00

PASO 6Comprobamos el equilibrio de la estructura

3.000.004.00

[F]= 0.000.000.000.005.000.00

PASO 7REACCIONES EN LOS APOYOS

0 -1 0

[𝑒]=[𝑎]∗[𝑈]

[𝑒] =

[𝑆]=[𝑘]∗[𝑒]

[𝐹]=[𝑎]^𝑇∗[𝑆]

[𝐻]=[𝑎_𝐻 ]^𝑇∗[𝑆]

0 0 -1-0.95 -0.32 0 0

0 0 0.95 -1-1 0 0 00 0 00 0 00 0 -0.710 0 0

-12.00H = -9.01

17.57

[𝑎_𝐻 ]= [𝑎_𝐻 ]^𝑇=

0 0 00 0 00 0 01 0 00 0 0-1 0 10 -0.95 -0.320 -0.71 0.710 0 0

0 0 -10 0 00 0.71 0

0.95 0 10.32 0 0

0 0 00 0 0

-0.95 -0.71 0-0.32 0.71 0

0 0 00 0 00 0 00 0 00 0 00 0 0

0.32 0 00 0.32 00 0 0.17

0 0 -0.17 0 1 00 0 0 0 0 00 0.23 0 0 0 0

0.30 0 0.17 0 -1 0 0.10 0 0 * 0 0 1

0 0 0 0 0 00 0 0 0 0 0

-0.30 -0.23 0 0 0 0.71 -0.10 0.23 0 -1 0 0

0 0 0-0.32 0 0

0 -0.16 0.16 0 -0.29 -0.10 0 -0.10 -0.03 0 0 0

0.82 0 -0.50 0.45 -0.06

-0.5 -0.06 0.69

9.53 24.28 12.204.00 4.00 4.000.00 5.88 0.009.53 24.28 12.200.00 0.00 0.000.00 0.00 0.007.12 7.13 7.127.13 21.21 9.137.12 9.13 9.12

0 -0.95 0 -1 0 0 00 -0.32 0 0 0 0 0-1 0 0.95 0 0 0 -0.71

0 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 0 00 0 0 -1 0 1

0.95 0.32 0 0 -0.95 -0.320 0 0 0 -0.71 0.711 0 0 0 0 0

000

PASO 10 0 0 0 0 0 00 0 0 0 1 0 00 1 0 0 0 0 01 -1 0 0 0 0 00 0 0 0 0 1 0

[a]= 0 0 0 0 0 0.71 0.710.71 0 0 0 0 0 0

0 0.89 -0.45 0 0 0 00 -0.89 -0.45 0.89 0.45 0 00 0 0 0 0 0 00 0 0 0.45 -0.89 -0.45 0.89

0 0 0 1 0 0 0.710 0 1 -1 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 1 0 0 0 0 00 0 0 0 1 0.71 00 0 0 0 0 0.71 00 0 0 0 -1 0 -0.710 0 0 0 0 0 0.710 0 0 0 0 0 01 0 0 0 0 0 0

PASO 20.1 0 0 0 0 0 00 0.1 0 0 0 0 00 0 0.05 0 0 0 00 0 0 0.05 0 0 0

[k] = 0 0 0 0 0.05 0 00 0 0 0 0 0.05 00 0 0 0 0 0 0.0450 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

〖 [𝑎〗^𝑇] =

[𝐾]= 〖 [𝑎〗^𝑡][𝑘][𝑎]

[𝑎^𝑇 ]*[𝑘]=

0 0 0 1 00 0 0 0 00 0 0 0 00 0 0 0 0-1 0 0 0 [F] = 00 0 0 0 0

-0.71 0.71 0 0 -500 0 -0.89 0.45 35.360 0 0 0 -35.36

0.45 0.89 -0.45 -0.45 150 0 0 0 0

0 0 0 00.89 -0.89 0 0-0.45 -0.45 0 0

0 0.89 0 0.450 0.45 0 -0.890 0 0 -0.450 0 0 0.890 0 0.45 00 0 0.89 0

-0.89 0 -0.45 00.45 0 -0.45 0

0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0

0.045 0 0 00 0.045 0 00 0 0.045 00 0 0 0.045

PASO 1

0 0 0.707 0.707 0[a]= 0 0 -0.707 0.707 0.707

-0.707 -0.707 0 0 0.7070.707 -0.707 0 0 0

0 0 -0.707 0.7070 0 -0.707 -0.707

0.707 -0.707 0 00.707 0.707 0 0

0 0.707 0.707 0

0 0 -0.707 0.707 #N/A #N/A0 0 -0.707 -0.707 #N/A #N/A

0.707 -0.707 0 0 #N/A #N/A0.707 0.707 0 0 #N/A #N/A

0 0.707 0.707 0 #N/A #N/A#N/A #N/A #N/A #N/A #N/A #N/A#N/A #N/A #N/A #N/A #N/A #N/A#N/A #N/A #N/A #N/A #N/A #N/A#N/A #N/A #N/A #N/A #N/A #N/A

-4.50224E+15 -4.50224E+15 -4.50224E+15 4.50224E+15 -9.004479E+15-4.50224E+15 -4.50224E+15 -4.50224E+15 4.50224E+15 -9.004479E+15-4.50224E+15 -4.50224E+15 -4.50224E+15 4.50224E+15 -9.004479E+154.50224E+15 4.50224E+15 4.50224E+15 -4.50224E+15 9.004479E+15

-9.004479E+15 -9.00448E+15 -9.00448E+15 9.00448E+15 -1.800896E+16#N/A #N/A #N/A #N/A #N/A#N/A #N/A #N/A #N/A #N/A#N/A #N/A #N/A #N/A #N/A#N/A #N/A #N/A #N/A #N/A

00

[u]= 000

#N/A

〖 [𝑎〗^𝑇] =

[𝐾]= 〖 [𝑎〗^𝑡][𝑘][𝑎]

[𝐹]=[𝐾][𝑢][𝐹]∗〖 [𝐾]〗^(−1)=[𝑢]〖 [𝐾〗^(−1)] =

PASO 4 PASO 5 PASO 6

00 00 00 00 #N/A

#N/A #N/A#N/A #N/A#N/A

PASO 7

[𝑒]=[𝑎]∗[𝑈] [𝑆]=[𝑘]∗[𝑒] [𝐹]=[𝑎]^𝑇∗[𝑆]

1 0 0 0 10 1 0 0 [F] = 0

[K] = 0 0 1 0 -10 0 0 1 0

0

1.00 0.00 0.00 0.00 -0.500.00 1.00 0.00 0.00 -0.500.00 0.00 1.00 0.00 -0.500.00 0.00 0.00 1.00 0.50

-0.50 -0.50 -0.50 0.50 1.00#N/A #N/A #N/A #N/A #N/A#N/A #N/A #N/A #N/A #N/A

#N/A#N/A#N/A#N/A#N/A#N/A#N/A#N/A#N/A

00000

#N/A

[𝐹]=[𝑎]^𝑇∗[𝑆]

1-1

a= a= 0000

100000

〖 [𝑎〗^𝑇] =

[𝐾]= 〖 [𝑎〗^𝑡][𝑘][𝑎]

[𝐹]=[𝐾][𝑢][𝐹]∗〖 [𝐾]〗^(−1)=[𝑢]

〖 [𝐾〗^(−1)] =

0 0 0 0 0 0.3330 0 0 0 0 00 1 0 0 0 k= 00 -1 0 0 0 00 0 0 1 0 00 0 0 -1 0 0

-1 0 0 0 00 0 0 0 00 1 -1 0 00 0 0 0 00 0 0 1 -10 0 0 0 0

0.333 -0.333 0 0 0 0 #N/A0 0 0 0 0 0 #N/A0 0 0.4 -0.4 0 0 #N/A0 0 0 0 0 0 #N/A0 0 0 0 0.333 -0.333 #N/A0 0 0 0 0 0 #N/A

#N/A #N/A #N/A #N/A #N/A #N/A #N/A#N/A #N/A #N/A #N/A #N/A #N/A #N/A#N/A #N/A #N/A #N/A #N/A #N/A #N/A

0.666 0 0 0 0 0 #N/A0 0 0 0 0 0 #N/A0 0 0.8 0 0 0 #N/A0 0 0 0 0 0 #N/A0 0 0 0 0.666 0 #N/A0 0 0 0 0 0 #N/A

[𝐾]= 〖 [𝑎〗^𝑡][𝑘][𝑎]

0 0 0 0 00.333 0 0 0 0

0 0.4 0 0 00 0 0.4 0 00 0 0 0.333 00 0 0 0 0.333

1 0 0 0 0 0-1 0 0 0 0 0

a= 0 0 1 0 0 00 0 -1 0 0 00 0 0 0 1 00 0 0 0 -1 0

1 -1 0 0 0 00 0 0 0 0 00 0 1 -1 0 00 0 0 0 0 00 0 0 0 1 -10 0 0 0 0 0

0.333 0 0 0 0 00 0.333 0 0 0 0

k= 0 0 0.4 0 0 00 0 0 0.4 0 00 0 0 0 0.333 00 0 0 0 0 0.333

0.333 -0.333 0 0 0 00 0 0 0 0 00 0 0.4 -0.4 0 00 0 0 0 0 00 0 0 0 0.333 -0.3330 0 0 0 0 0

0.666 0 0 0 0 00 0 0 0 0 0

k= 0 0 0.8 0 0 00 0 0 0 0 00 0 0 0 0.666 00 0 0 0 0 0

〖 [𝑎〗^𝑇] =

[𝐾]= 〖 [𝑎〗^𝑡][𝑘][𝑎]

[𝐹]=[𝐾][𝑢][𝐹]∗〖 [𝐾]〗^(−1)=[𝑢]

〖 [𝐾〗^(−1)] =

-1 1 0 0[a]= 0 0 -1.923 -0.707

0.799 0 -0.602 00 -0.799 0

-1 0 0.799 0 0.51 0 0 -0.799 a -0.7070 -1.923 -0.602 0 -1

0-5 0 0 0

k 0 -5 0 00 0 -5 0 0.50 0 0 -5 0

0

0.200

#N/A#N/A

〖 [𝑎〗^𝑇] =

0.707 1 0 0-0.707 0 1

0 0 1

0.4 0 0 00 0 0 0.707 0 00 1 k 0 0 1 0

0.866 0 0 0 0 10 1

-0.707 -1 00 0.866 01 0 1

-0.499849 -1 0 1.45339324 -0.866 -0.499849 #N/A0 0.866 0 -0.866 0.749956 0 #N/A

0.707 0 1 -0.499849 0 1.707 #N/A#N/A #N/A #N/A #N/A #N/A #N/A #N/A#N/A #N/A #N/A #N/A #N/A #N/A #N/A

#N/A #N/A #N/A #N/A#N/A #N/A #N/A #N/A#N/A #N/A #N/A #N/A