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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 \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 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 \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 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 \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 \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 \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 \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 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 \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 \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 \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 \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 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 \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 \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 \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 \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 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),