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ESTRUCTURAS PLANAS CON CARGAS EN LOS NUDOS

Tres coordenadas locales

Ing.: Juan Manuel Urteaga Garca

1 2

3

1 2

3

1 3

2

4.00

3.004.00

Solucionar la estructura mostrada; Trazar los

diagramas de Momento Flector, Fuerza

Cortante y la deformada Viga = (0.30 X 0.60)

Col. = (0.40 X 0.50)

E = 2.1 E +06 Tn/m2

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10.00 T.m. 9.00 T.m.

4.00 T.

CARGAS ACTUANTES

RELACIONES DE COMPATIBILIDAD

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Elemento D1

D2

D3

1

qi 0 0 0

qj 1 0 0

D 0 0 1

2

qi 1 0 0

qj 0 1 0

D 0 0 3/4

3

qi 0 0 0

qj 0 1 0

D 0 0 5/4

0

1

0

1

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

0

-1

0

0

3/4

0

0

-5/45/4 3/4

RELACIONES DE COMPATIBILIDAD

Miembro 1 Miembro 2 Miembro 3

qI,1 = 0 qI,2 = D1 qI,3 = 0

qJ,1 = D1 qJ,2 = D2 qJ,3 = D2

D1 = -D3 D1 = (3/4)D3 D3 = -(5/4)D3

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DETERMINACIN DE LOS COEFICIENTES DE RIGIDEZ

K i,j

PRIMERA DEFORMADA

Desplazamientos

D1 = 1

D2 = 0

D3 = 0

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COEFICIENTES DE RIGIDEZ (PRIMERA COORDENADA)

4EI1/L1

K1,1

K2,1

4EI2/L2 2EI2/L2

K1,1 = 4EI1/L1 + 4EI2/L2 = 4(2.1x106)((0.0042/4)+(0.0054/4)) = 20160 T.m.

K2,1 = 2EI2/L2 = 2(2.1x106)(0.0054)/4 = 5670 T.m.

COEFICIENTES DE RIGIDEZ (PRIMERA COORDENADA)

6EI1/L12

K3,1

6EI2/L22

6EI1/L12 = B

6EI2/L22 = A

53

P3

P2

SFy = 0 => (4/5)P3 = A => P3 = (5/4)A

SFx = 0 => P2+ B = (3/5)P3 => P2 = (3/5)(5/4)A B =>P2 = (3/4)A B (Neg)

K3,1 = 6EI2/L22(3/4)-6EI1/L1

2 = 6(2.1x106)((0.0054/42)(3/4)-(0.0042/52))

K3,1 = 118.125 T.

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SEGUNDA DEFORMADA

Desplazamientos

D1 = 0

D2 = 1

D3 = 0

COEFICIENTES DE RIGIDEZ (SEGUNDA COORDENADA)

2EI2/L2

K1,24EI3/L3

4EI2/L2

K2,2

K1,2 = 2EI2/L2 = 2(2.1x106)(0.0054)/4 = 5670 T.m.

K2,2 = 4EI2/L2 + 4EI3/L3 = 4(2.1x106)((0.0054/4)+(0.0042/5)) = 18396 T.m.

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COEFICIENTES DE RIGIDEZ (SEGUNDA COORDENADA)

K3,2

SFy = 0 => (3/5)B+(4/5)P3 = A => P3 = (5/4)(A-(3/5)B) => P3 = (5/4)A-(3/4)B

SFx = 0 => P2+(4/5)B = (3/5)P3 => P2 = (3/5)((5/4)A-(3/4)B)-(4/5)B

=>P2 = (3/4)A-(5/4)B

K3,2 = 6EI2/L22(3/4)-6EI3/L3

2(5/4) = 6(2.1x106)((0.0054/42)(3/4)-(0.0042/52)(5/4))

K3,2 = -543 T.

6EI3/L32

6EI2/L22

6EI3/L32 = B

6EI2/L22 = A

5337

P3

P2

TERCERA DEFORMADA

Desplazamientos

D1 = 0

D2 = 0

D3 = 1

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COEFICIENTES DE RIGIDEZ (TERCERA COORDENADA)

6EI1/L12

K1,3

6EI3/L32(5/4)

K2,3

K1,3 = 6EI1/L12- 6EI2/L2

2(3/4) = 6(2.1x106)((0.0042/42)-(0.0054/42)(3/4))

K1,3 = 118.125 T.m.

K2,3 = -6EI2/L22(3/4)+6EI3/L3

2(5/4) = -6(2.1x106)((0.0054/42)(3/4)+(0.0042/52)(5/4))

K2,3 = -543 T.m.

6EI2/L22(3/4)

6EI2/L22(3/4)

5/4

4/4

3/4

COEFICIENTES DE RIGIDEZ (TERCERA COORDENADA)

12EI3/L33(5/4)

12EI1/L13

K3,3

SFy = 0 => (3/5)B + C = (4/5)P3 => P3 = (5/4)((3/5)B + C) => P3 = (3/4)B + (5/4)C

SFx = 0 =>P2 = A + (4/5)B + (3/5)P3 => P2 = A + (4/5)B + (3/5)(3/4)B + (3/5)(5/4)C

P2 = A + (4/5)B + (9/20)B + (3/4)C => P2 = A + (5/4)B + (3/4)C

K3,3 = 12EI1/L13 + 12EI3/L3

3(5/4)(5/4) + 12EI2/L23(3/4)(3/4)

K3,3 = 12(2.1x106)((0.0042/43)+(0.0042/53)(5/4)(5/4)+(0.0054/43)(3/4)(3/4))

K3,3 = 4172.77 T.

12EI3/L33(5/4) = B

12EI1/L13 = A

5337

P3

P2

12EI2/L23(3/4)

12EI2/L23(3/4) = C

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COEFICIENTES DE RIGIDEZ

K1,1=20160 T.m. K1,2=5670 T.m. K1,3=118 T

K2,1=5670 T.m. K2,2=14868 T.m. K2,3=-543 T

K3,1=118 T.m. K3,2=-543 T.m. K2,1=4173 T

DETERMINACIN DE LAS CARGAS

Q i

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D1 = 1

D2 = 0

D3 = 0Q1 = 10.00 (D1+ D2+ D3) T.m.

Q1 = 10.00 (1 + 0 + 0 ) T.m.

Q1 = 10.00 T.m.

D1 = 0

D2 = 1

D3 = 0Q2 = -9.00 (D1+ D2+ D3) T.m.

Q2 = -9.00 (0 + 1 + 0 ) T.m.

Q2 = -9.00 T.m.

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D1 = 0

D2 = 0

D3 = 1

3/4

Q3 = 4.00 (D1+ D2+ D3) T.

Q3 = 4.00 (0 + 0 - 3/4 ) T.

Q3 = -3.00 T.

DESPLAZAMIENTOS GLOBALES

Di

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20160 D1 + 5670 D2 + 118 D3 = 10.00

5670 D1 + 14868 D2 - 543 D3 = -9.00

118 D1 - 543 D2 + 4173 D3 = -3.00

D1 = 7.0670 X10-4 Rad.

D2 = -7.3167 X10-4 Rad.

D3 = -8.3410 X10-4 m.

DESPLAZAMIENTOS GLOBALES

DESPLAZAMIENTOS LOCALES

qi

qj

D

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DESPLAZAMIENTOS LOCALES

Elemento 1

qi = 0 Rad.

qj = 7.07E-04 Rad.

D = 8.34E-04 m.

Elemento 2

qi = 7.07E-04 Rad.

qj = -7.32E-04 Rad.

D = -6.26E-04 m.

Elemento 3

qi = 0 Rad.

qj = -7.32E-04 Rad.

D = 1.04E-03 m.

ESFUERZOS LOCALES

Mi Mj V

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Elemento 1 Mi = 4EI1/L1qi + 2EI1/L1qj - 6EI1/L1

2D

Mi = (2.1x106)(0.0042)(2(7.07x10-4)/4 - 6(8.34x10-4)/16)

Mi = 0.36T.m.

Mj = 2EI1/L1qi + 4EI1/L1qj - 6EI1/L12D

MJ = (2.1x106)(0.0042)(4(7.07x10-4)/4 - 6(8.34x10-4)/16)

MJ = 3.47 T.m.

V = -6EI1/L12qi - 6EI1/L1

2qj + 12EI1/L13D

V = (2.1x106)(0.0042)(-6(7.07x10-4)/16 + 12(8.34x10-4)/64)

V = -0.96T.

ELEMENTO 2

Mi = 4EI1/L1i + 2EI1/L1j - 6EI1/L12D

Mi = (2.1x106)(0.0054)(4(7.07x10-4)/4 + 2(-7.32x10-4)/4 - 6(-6.26x10-4)/16)

Mi = 6.53T.m.

Mj = 2EI1/L1i + 4EI1/L1j - 6EI1/L12 D

MJ = (2.1x106)(0.0054)(2(7.07x10-4)/4 + 4(-7.32x10-4)/4 - 6(-6.26x10-4)/16)

MJ = -1.63T.m.

V = -6EI1/L12i - 6EI1/L1

2j + 12EI1/L13 D

V = (2.1x106)(0.0054)(-6(7.07x10-4)/16 - 6(-7.32x10-4)/4 + 12(-6.26x10-4)/64)

V = -1.22T.

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ELEMENTO 3

Mi = 4EI1/L1i + 2EI1/L1j - 6EI1/L12D

Mi = (2.1x106)(0.0042)(2(-7.32x10-4)/5 - 6(1.04x10- 3)/25)

Mi = -4.79 T.m.

Mj = 2EI1/L1i + 4EI1/L1j - 6EI1/L12D

MJ = (2.1x106)(0.0042)(4(-7.32x10-4)/5 - 6(-1.04x10- 3)/25)

MJ = -7.37 T.m.

V = -6EI1/L12i - 6EI1/L1

2j + 12EI1/L13D

V = (2.1x106)(0.0042)(-6(-7.32x10-4)/25 + 12(-1.04x10- 3)/125)

V = 2.43 T.

DIAGRAMASCortante / Momento Flector / Deformada

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0.96 T.

1.22 T.

3.47 T.m.

0.36 T.m.

1.63 T.m.6.53 T.m.

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8.34E-04 m.

7.07E-04 Rad

7.07E-04 Rad-7.32E-04 Rad

-6.26E-04 m

1.04E-03 m

7.32E-04 Rad