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Problema 209
Dibujar los diagramas de fuerza cortante y momento flexionante.
Solución:
Por semejanza de triángulos:
y = 4(x – 2)
F = (x – 2) F = 2(x – 2)^2
ΣF H = 0 HA = 0
ΣM (^) B = 0 VA (10) – 50(12) – = 0 VA = 126.67 kg
ΣFV = 0 VB = 50 + – 126.67 = 123.33 kg
(V):
0 ≤ x ≤ 2 V = – 50
2 ≤ x ≤ 12 V = – 50 + 126.67 –
x = 2 V = 76. x = 12 V = –123.
(M):
0 ≤ x ≤ 2 M = – 50x
x = 0 M = 0 x = 2 M = –
2 ≤ x ≤ 12 M = – 50x + 126.67(x – 2) – (x – 2) 2
x = 2 M = – x = 8.19 M = 216. x = 12 M = 0
x
50 kg 40 kg/m
A B
2 m 10 m
F
y 40
x – 2
Problema 210
3.81 m
6.19 m
2 m
V
(kg)
M
(kg-m) 100
Dibujar los diagramas de fuerza cortante y momento flexionante.
Solución:
ΣF (^) H = 0 HA = 0
ΣM (^) B = 0 VA (8) – 30(6.5) – 20(3) – 30(1) = 0 VA = 35.625 T
ΣF (^) V = 0 35.625 + VB – 30 – 20 – 30 = 0 VB = 44.375 T
3 m 2 m
A B
10 T/m
20 T 20 T/m
3 m
Problema 211
Dibujar los diagramas de fuerza cortante y momento flexionante.
Solución:
ΣF H = 0 HA = 0
ΣM B = 0 VA (10) – + 20 = 0 VA = 12.67 T
ΣF V = 0 VB + VA – 20 = 0 VB = 7.33 T
(V):
0 ≤ x ≤ 4 V = 12.67 –
x = 0 V = 12. x = 4 V = – 7.
V = 0 x = 3.
4 ≤ x ≤ 10 V = 12.67 – 20 = – 7.
(M):
0 ≤ x ≤ 4 M = 12.67x –
x = 0 M = 0 x = 3.18 M = 26. x = 4 M = 24
4 ≤ x ≤ 8 M = 12.67x – 20
x = 4 M = 24 x = 8 M = –5.
8 ≤ x ≤ 10 M = 12.67x – 20 + 20
x = 8 M = 14. x = 10 M = 0
4 m 4 m 2 m
A B
10 T/m
20 T-m
Problema 212
Dibujar los diagramas de fuerza cortante y momento flexionante.
Solución:
ΣF H = 0 HA = 0
ΣM B = 0 VA (5) – 20(4) – 40(3) + 20 + 10(3) = 0 VA = 30 T
ΣF V = 0 30 + VB = 20 + 40 + 10 VB = 40 T
(V): 0 ≤ x ≤ 2 V = 30 – 10x
x = 0 V = 30 x = 2 V = 10
2 ≤ x ≤ 5 V = 30 – 20 – 40 = – 30
5 ≤ x ≤ 8 V = 30 – 20 – 40 + 40 = 10
M(T-m)
3.18 m
V(T)
2 m
6 m
0.82 m
A
A
2 m 3 m 2 m 1 m
B
A
10 T/m
20 T-m
40 T
10 T
Solución:
ΣF (^) H = 0 HA = 0
ΣM (^) B = 0 VA (12) – 10(14) – 40(8) + 6 + 20(4) = 0 VA = 31.17 T
ΣF V = 0 VA + V B – 10 – 40 – 20 = 0 VB = 38.33 T
(V):
0 ≤ x ≤ 2 V = – 10
2 ≤ x ≤ 10 V = – 10 + 31.17 – 5(x – 2)
x = 2 V = 21. x = 10 V = – 18.
10 ≤ x ≤ 14 V = – 10 + 31.17 – 5(8) = – 18.
De derecha a izquierda:
0 ≤ x ≤ 4 V = 20
Para: 2 ≤ x ≤ 10 V = 0 x = 6.
(M):
0 ≤ x ≤ 2 M = – 10x
x = 0 M = 0 x = 2 M = – 20
2 ≤ x ≤ 10 M = – 10x + 31.17(x – 2) – 5(x – 2)
x = 2 M = – x = 6.23 M = 24. x = 10 M = –10.
10 ≤ x ≤ 12 M = – 10x + 31.17(x – 2) – 40(x – 6)
x = 10 M = –10. x = 12 M = –48.
12 ≤ x ≤ 14 M = – 10x + 31.17(x – 2) – 40(x – 6) + 6
x = 12 M = –42. x = 14 M = –
De derecha a izquierda:
0 ≤ x ≤ 4 M = – 20x
x = 0 M = 0 x = 4 M = – 80
Problema 214
Dibujar los diagramas de fuerza cortante y momento flexionante.
Solución:
ΣF H = 0 HA = 0
ΣM B = 0 VA (3) – 10 – 20(5) – 15(4)(1) = 0 VA = 56.67 T
ΣF V = 0 VA + V B – 20 – 15(4) = 0 VB = 23.33 T
2 m
2 m
M
(T-m)
2 m
2 m
4 m
V(T)
4.23 m
2 m 3 m 1 m
A B
10 T-m
15 T/m
20 T
Problema 215
Dibujar los diagramas de fuerza cortante y momento flexionante.
Solución:
ΣF H = 0 HA = 0
ΣF V = 0 VA – – 6(0.4) – 4 = 0 VA = 8.2 T
ΣMA = 0 – (2.1) – 6(0.4)(1.7) + 12 – 4(0.5) – M (^) A = 0 MA = 2.14 T-m
(V):
0 ≤ x ≤ 0.6 V = –
x = 0 V = 0 x = 0.6 V = –1.
0.6 ≤ x ≤ 1 V = – – 6(x – 0.6)
x = 0.6 V = –1. x = 1 V = –4.
1 ≤ x ≤ 2 V = – – 6(0.4) = – 4.
2 ≤ x ≤ 2.5 V = – – 6(0.4) – 4 = – 8.
(M):
0 ≤ x ≤ 0.6 M = –
x = 0 M = 0 x = 0.6 M = –0.
0.6 ≤ x ≤ 1 M = – 1.8(x – 0.4) –
x = 0.6 M = –0. x = 1 M = –1.
0.6 m 0.4 m 0.5 m 0.5 m 0.5 m
M A
VA
A
12 T-m
6 T/m 4 T
Problema 216
1 ≤ x ≤ 1.5 M = – 1.8(x – 0.4) – 2.4(x – 0.8)
x = 1 M = –1. x = 1.5 M = –3.
1.5 ≤ x ≤ 2 M = – 1.8(x – 0.4) – 2.4(x – 0.8) + 12
x = 1.5 M = 8. x = 2 M = 6.
De derecha a izquierda:
0 ≤ x ≤ 0.5 M = 2.14 + 8.2x
x = 0 M = 2. x = 0.5 M = 6.
0.6 m 0.4 m 0.5 m 0.5 m 0.5 m
V(T)
M(T-m)
Dibujar los diagramas de fuerza cortante y momento flexionante.
Solución: ΣF (^) H = 0 HA = 0
ΣM (^) B = 0 VA (5) – = 0 VA = 666.67 kg
ΣF (^) V = 0 VA + V (^) B – = 0 VB = 1333.33 kg
2 m 5 m 1 m
A B
500 kg/m
Problema 217
Dibujar los diagramas de fuerza cortante y momento flexionante.
Solución:
ΣM A = 0 V(4) – 10(4)(2) = 0 V = 20 T
ΣF V = 0 V + VA – 10(4) = 0 VA = 20 T
ΣMB = 0 – 20(6) – 2(3) + MB = 0 MB = 126 T-m
ΣF (^) V = 0 VB – V – 2 = 0 V (^) B = 22 T
(V):
0 ≤ x ≤ 4 V = 20 – 10x
x = 0 V = 20 x = 4 V = –
4 ≤ x ≤ 7 V = 20 – 10(4) = – 20 7 ≤ x ≤ 10 V = 20 – 40 – 2 = – 22
Para 0 ≤ x ≤ 4 V = 0 x = 2
4 m 3 m 3 m
A B
10 T/m
Rótula
2 T
4 m 3 m 3 m
A B
V
V
VA V
B
M B
10 T/m
2 T
V(T)
M
(T-m)
2 m
2 m
2 m
2 m
3 m
3 m
3 m
3 m
Problema 218
Dibujar los diagramas de fuerza cortante y momento flexionante.
Rótula
3 T 3 T 3 T
A B^
C
2 m 2 m 2 m 2 m 2 m 2 m
(M):
0 ≤ x ≤ 4 M = 20x – 10
x = 0 M = 0 x = 2 M = 20 x = 4 M = 0
4 ≤ x ≤ 7 M = 20x – 10(4)(x – 2)
x = 4 M = 0 x = 7 M = –
7 ≤ x ≤ 10 M = 20x – 10(4)(x – 2) – 2(x – 7)
x = 7 V = – x = 10 V = –
Problema 219
Dibujar los diagramas de fuerza cortante y momento flexionante.
Solución:
ΣM A = 0 V(3) – 12(2)(1) = 0 V = 8 T
ΣF V = 0 V + VA – 12(2) = 0 VA = 16 T
12 T/m 6 T^ 8 T
M B
V B
V B
V
VA
2 m 1 m 1 m^ 1 m^ 1 m
12 T/m
Rótula
6 T 8 T
B
A
2 m 1 m 1 m 1 m 1 m
De derecha a izquierda:
0 ≤ x ≤ 2 M = 1.5x
x = 0 M = 0 x = 2 M = –
V(T)
4 m 2 m
2 m 2 m^ 2 m
M
(T-m) 2 m 2 m
2 m 2 m 2 m 2 m
ΣM A = 0 V(3) – 12(2)(1) = 0 V = 8 T
ΣF V = 0 V + VA – 12(2) = 0 VA = 16 T
ΣMB = 0 – 8(3) – 6(2) – 8(1) + MB = 0 MB = 44 T-m
ΣF (^) V = 0 – 8 – 6 – 8 + VB = 0 VB = 22 T
(V):
0 ≤ x ≤ 2 V = 16 – 12x
x = 0 V = 16 x = 2 V = – V = 0 x = 1. 2 ≤ x ≤ 4 V = 16 – 12(2) = – 8 4 ≤ x ≤ 5 V = 16 – 12(2) – 6 = – 14 5 ≤ x ≤ 6 V = 16 – 12(2) – 6 – 8 = – 22
(M):
0 ≤ x ≤ 2 M = 16x – 12
x = 0 M = 0 x = 1.33 M = 10. x = 2 M = 8
2 ≤ x ≤ 4 M = 16x – 12(2)(x – 1)
x = 2 M = 8 x = 3 M = 0 x = 4 M = –
4 ≤ x ≤ 5 M = 16x – 12(2)(x – 1) – 6(x – 4)
x = 4 M = – x = 5 M = –
5 ≤ x ≤ 6 M = 16x – 12(2)(x – 1) – 6(x – 4) – 8(x – 5)
x = 5 M = – x = 6 M = –
1.33 m
V(T) 0.67 m
M
(T-m)
3 m
1 m 1 m 1 m 1 m
3 m
Problema 221
Dibujar los diagramas de fuerza cortante y momento flexionante.
Solución:
ΣM (^) B = 0 VA (6) – 12 – 6(5) – 20(3)(1.5) = 0 VA = 22 T
ΣF (^) V = 0 VB + VA – 6 – 20(3) = 0 VB = 44 T
V(T)
2.24 m
M
(T-m)
2 m
1 m
De derecha a izquierda:
0 ≤ x ≤ 1 M = – 4x – 10
x = 0 M = 0 x = 1 M = –
1 m 2 m 3 m
A B
20 T/m
6 T
12 T-m
(V):
0 ≤ x ≤ 1 V = 22
1 ≤ x ≤ 3 V = 22 – 6 = 16
3 ≤ x ≤ 6 V = 22 – 6 – 20(x – 3)
x = 3 V = 16 x = 3.8 V = 0 x = 6 V = –
(M):
0 ≤ x ≤ 1 M = 22x
x = 0 M = 0 x = 1 M = 22
1 ≤ x ≤ 3 M = 22x – 6(x – 1) – 12
x = 1 M = 10 x = 3 M = 42
3 ≤ x ≤ 6 M = 22x – 6(x – 1) – 12 – 20
x = 3 M = 42 x = 3.8 M = 48. x = 6 M = 0
V(T)
2.2 m
1 m
M
(T-m)
2 m 0.8 m
