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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