resmat2007a
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resmat2007a


DisciplinaResistência dos Materiais II6.442 materiais139.750 seguidores
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cais presos a`s suas extremidades (veja figura). Os fios te\u2c6m o mesmo comprimento e
mesma a´rea de sec¸a\u2dco transversal mas diferentes mo´dulos de elasticidade (E1 e E2).
Desprezando o peso pro´prio da barra , calcular a dista\u2c6ncia d , do ponto de aplicac¸a\u2dco
da carga P ate´ a extremidade A , para que a barra permanec¸a horizontal.
Resposta (d = (LE2)/(E1 + E2))
E1 E2
\ufffd\ufffd\ufffd\ufffd
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\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
P
A B 
d 
L
Figura 3.16: Figura do exerc´\u131cio 11
12. Um dispositivo de tre\u2c6s barras e´ utilizado para suspender uma massa W de 5000 Kg
(veja figura 3.17). Os dia\u2c6metros das barras sa\u2dco de 20 mm (AB e BD) e 13 mm
(BC). Calcular as tenso\u2dces normais nas barras.
Resposta (150,8 MPa em AB, 119 MPa em BC e 159 MPa em BD).
49
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
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\ufffd
\ufffd
\ufffd
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\ufffd
\ufffd
\ufffd
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\ufffd
\ufffd
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\ufffd
0,30m 3,60m
1,20m
0,90m
A
C
B
D
W
\u3b2 \u3b1
Figura 3.17: Figura do exerc´\u131cio 12
13. As barras AB e AC da trelic¸a representada na figura 3.18 sa\u2dco pec¸as de madeira 6
cm × 6 cm e 6 cm × 12 cm, respectivamente. Sendo as tenso\u2dces normais admiss´\u131veis
de 12 MPa a trac¸a\u2dco e 8 MPa a compressa\u2dco, calcular o valor admiss´\u131vel da carga P .
Resposta (P = 60, 8KN).
450
450
\ufffd\ufffd\ufffd
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\ufffd\ufffd\ufffd
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\ufffd\ufffd\ufffd
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\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
 
C
B
A
P
Figura 3.18: Figura do exerc´\u131cio 13
14. As barras da trelic¸a representada na figura 3.19 sa\u2dco de madeira com sec¸o\u2dces retan-
gulares 60 mm × L (BC) e 60 mm × 1,4L (AC). Calcular L para tenso\u2dces normais
admiss´\u131veis de 12 MPa a trac¸a\u2dco e 8,5 MPa a compressa\u2dco.
Resposta (L = 73 mm).
0
60
300
\ufffd\ufffd\ufffd
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\ufffd\ufffd\ufffd
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\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
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\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
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\ufffd\ufffd\ufffd
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\ufffd\ufffd\ufffd
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60 KN
A
B
C
Figura 3.19: Figura do exerc´\u131cio 14.
15. As barras AB e BC da trelic¸a da figura 3.20 comprimento de 3,0 m e a´rea de
sec¸a\u2dco A. Especificados \u3c3x = 220 MPa e E = 210 GPa, calcular o valor de A e o
50
correspondente valor do deslocamento vertical da articulac¸a\u2dco C.
Resposta (A = 170,45 mm
2
e \u2206L = 5,23 mm).
45KN
1.80m
A B
C
Figura 3.20: Figura do exerc´\u131cio 15
16. Na trelic¸a da figura 3.21, as barras sa\u2dco de ac¸o (E = 210 GPa) com tenso\u2dces admiss´\u131veis
de 210 MPa (trac¸a\u2dco) e 166 MPa (compreessa\u2dco). As a´reas das sec¸o\u2dces transversais
sa\u2dco 400 mm 2 (BC) e 525 mm 2 (AC). Calcular o valor admiss´\u131vel de P e os valores
correspondentes das tenso\u2dces normais e deformac¸o\u2dces nas barras.
Respostas:
\u2022 P= 52,19 kN.
\u2022 Barra AC: \u3c3x = 166 MPa e \u2206L = 3,95mm.
\u2022 Barra BC: \u3c3x = 174,8 MPa e \u2206L = 3,33mm.
\ufffd\ufffd\ufffd
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\ufffd\ufffd\ufffd
 
A
B C
P
4,00m
3,00m
Figura 3.21: Figura do exerc´\u131cio 16
17. Uma haste de ac¸o (E = 210 GPa) de 100 m de comprimento, suspensa verticalmente,
suporta uma carga de 55 kN concentrada na sua extremidade livre, ale´m de seu
peso pro´prio (a massa espec´\u131fica do ac¸o e´ 7.850 Kg/m). Para uma tensa\u2dco normal
admiss´\u131vel de 120 MPa, dimensionar a haste (sec¸a\u2dco circular, dia\u2c6metro em nu´mero
inteiro de mm) e calcular o alongamento previsto.
Resposta (D = 25 mm ; \u2206L = 55,22 mm)
18. Calcular a a´rea da sec¸a\u2dco transversal em cada trecho da barra da figura 3.22 , sujeita
a` carga P = 45 kN, ale´m do seu peso pro´prio. Sa\u2dco dados os valores da tensa\u2dco
admiss´\u131vel e da massa espec´\u131fica em cada trecho.
51
\u2022 AB (ac¸o) 120 MPa; 7.800 kg/m;
\u2022 BC (lata\u2dco) 80 MPa; 8.300 kg/m;
Resposta (AB = 382 mm e BC = 570 mm) ;
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
F
10m
12m
B
C
A
Figura 3.22: Figura do exerc´\u131cio 18
19. A haste de ac¸o da figura 3.23 suporta uma carga axial F , ale´m de seu pro´prio peso.
Os dia\u2c6metros sa\u2dco d1 = 18 mm em AB e d2 = 22 mm em BC. Dados a massa
espec´\u131fica 7.850 Kg/m3, o mo´dulo de elasticidade longitudinal 210 GPa e a tensa\u2dco
normal admiss´\u131vel 150 MPa, calcular o valor ma´ximo admiss´\u131vel da carga F e o
correspondente alongamento total. Representar os correspondentes diagramas de
esforc¸os normais e de tenso\u2dces normais.
Resposta (F = 30,18 kN, \u2206L= 477 mm)
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
F
A
B
C
400m
400m
Figura 3.23: Figura do exerc´\u131cio 19
20. A haste de ac¸o suspensa verticalmente suporta uma carga axial F = 15 kN na sua
extremidade, ale´m de seu pro´prio peso. Ha´ uma reduc¸a\u2dco do dia\u2c6metro no trecho AB,
conforme indicado na figura 3.24. Dados \u3c3x = 120 MPa, E = 210 GPa e massa
espec´\u131fica = 8 t/m, pede-se dimensionar a haste (calcular os dia\u2c6metros em nu´mero
inteiro de mm) e calcular o alongamento total. Representar a variac¸a\u2dco da tensa\u2dco
normal ao longo do comprimento (\u3c3x(x)).
Resposta (DAB= 15 mm, DBC = 18 mm, \u2206L = 366 mm);
21. Uma haste de ac¸o suspensa verticalmente tem 1.200 m de comprimento e suporta
uma carga P em sua extremidade. Calcular o valor admiss´\u131vel de P e o correspon-
dente alongamento total da haste, se :
52
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
F
B
C
A
300m
500m
\u3c3
x
Figura 3.24: Figura do exerc´\u131cio 20
\u2022 O dia\u2c6metro e´ constante e igual a 25mm.
\u2022 Sa\u2dco quatro segmentos de 300 m, com dia\u2c6metros 16mm, 19 mm, 22 mm e 25
mm.
\u2022 Considere \u3c3x = 100 MPa, E = 210 GPa e \u3b3 = 7850 kg/m
Resposta:
\u2022 P = 2, 847 kN e \u2206L = 302, 3 mm;
\u2022 P = 15, 371 kN e \u2206L = 482, 5 mm;
22. Calcular o deslocamento vertical do ve´rtice de um cone apoiado na base e sujeito
somente a ac¸o de seu pro´prio peso, sendo a altura igual a L, o peso espec´\u131fico \u3b3 e o
mo´dulo de elasticidade E.
Resposta (\u2206L = \u3b3 L2 /6E);
23. Uma estaca uniforme de madeira, cravada a uma profundidade L na argila, suporta
uma carga F em seu topo. Esta carga e´ internamente resistida pelo atrito f ao
longo da estaca, o qual varia de forma parabo´lica , conforme a figura 3.25. Calcular
o encurtamento total da estaca, em func¸a\u2dco de L, F , A (a´rea da sec¸a\u2dco transversal) e
E (mo´dulo de elasticidade).
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
x
F
Lf
f= kx2
F
Figura 3.25: Figura do exerc´\u131cio 23
Resposta (\u2206L = \u2212FL/4AE);
53
24. Uma estaca de madeira e´ cravada no solo, como mostra a figura, ficando solicitada
por uma carga F = 450 kN, axial, no seu topo. Uma forc¸a de atrito f (kN/m)
equil´\u131bra a carga F . A intensidade da forc¸a de atrito varia com o quadrado da
dista\u2c6ncia z, sendo zero no topo. Dados E = 1, 4 × 104 MPa , L = 9 m e D = 30
cm, determinar o encurtamento da estaca e representar os diagramas (f × z , N × z
e \u3c3z × z).
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
F
f
D
z
L
f
Figura 3.26: Figura do exerc´\u131cio 24
Resposta: \u2206L=-3,069 mm
54
3.2 Solicitac¸a\u2dco por momento torsor
3.2.1 Introduc¸a\u2dco
Neste item sera\u2dco estudas das tenso\u2dces e deformac¸o\u2dces em barras sujeitas a` torc¸a\u2dco. O estudo
a realizado envolve:
\u2022 Barras sujeitas a` Torc¸a\u2dco Pura: Somente o efeito do momento torsor (torque),