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