Aula%12%(%Elementos%esta0camente ... · os#eixos#estão#apoiados#em#mancais#...

Prof. Wanderson S. Paris -‐ [email protected] Resistência dos Materiais

Aula 12 -‐ Elementos esta0camente indeterminados carregados com torque.

Prof. Wanderson S. Paris, M.Eng. [email protected]


Conceito

•  Um eixo carregado com torque pode ser classificado como esta4camente indeterminado se a equação de equilíbrio de momento aplicada em torno da linha central do eixo não for adequada para determinar os torques desconhecidos que agem no eixo.


Equações


Exercício 1

O eixo mostrado na figura é composto por um tubo de aço unido a um núcleo de latão. Se um torque T = 250 Nm for aplicado em sua extremidade, faça uma representação gráfica da distribuição da tensão de cisalhamento ao longo da linha radial de sua área de seção transversal.

G(aço) = 80 GPa, G(lat) = 36 GPa.


Solução Exercício 1

Equilíbrio:



Subs4tuido:

Temos:



Deformação por cisalhamento:


Exercício 2


Solução do Exercício 2

© 2008 by R.C. Hibbeler. Published by Pearson Prentice Hall, Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under allcopyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

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Solução do Exercício 2


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Exercícios Propostos

[P48] O eixo de aço A-‐36 tem diâmetro de 50 mm e está preso nas extremidades A e B. Se for subme4do ao momento, determine a tensão de cisalhamento máxima nas regiões AC e CB do eixo.

218 CHAPTER 5 TORS ION

5

A

C0.4 m

0.8 m

300 N!m

B

Prob. 5–77

A

C

D 1 m

1 m

1.5 m

200 N!m

500 N!m

B

Prob. 5–78

5–78. The A-36 steel shaft has a diameter of 60 mm and isfixed at its ends A and B.If it is subjected to the torques shown,determine the absolute maximum shear stress in the shaft.

5 in.

8 in.

12 in.

1 in.

0.5 in.A

B

C

D 500 lb!ft

Prob. 5–79

A600 mm

600 mm

600 mm

B

2 kN!m

4 kN!m

C

D

Probs. 5–80/81

•5–77. The A-36 steel shaft has a diameter of 50 mm and isfixed at its ends A and B. If it is subjected to the torque,determine the maximum shear stress in regions AC and CBof the shaft.

PROBLEMS

5–79. The steel shaft is made from two segments: AC has adiameter of 0.5 in, and CB has a diameter of 1 in. If it isfixed at its ends A and B and subjected to a torque of

determine the maximum shear stress in the shaft.Gst = 10.811032 ksi.500 lb # ft,

*5–80. The shaft is made of A-36 steel, has a diameter of80 mm, and is fixed at B while A is loose and can rotate0.005 rad before becoming fixed. When the torques areapplied to C and D, determine the maximum shear stress inregions AC and CD of the shaft.

•5–81. The shaft is made of A-36 steel and has a diameterof 80 mm. It is fixed at B and the support at A has a torsionalstiffness of If it is subjected to the geartorques shown, determine the absolute maximum shear stressin the shaft.

k = 0.5 MN # m>rad.

5–82. The shaft is made from a solid steel section AB anda tubular portion made of steel and having a brass core.If it is fixed to a rigid support at A, and a torque of

is applied to it at C, determine the angle oftwist that occurs at C and compute the maximum shearstress and maximum shear strain in the brass and steel.Take Gst = 11.511032 ksi, Gbr = 5.611032 ksi.

T = 50 lb # ft

A

0.5 in.

1 in.

2 ft

3 ft

B

CT " 50 lb!ft

Prob. 5–82



[P49] O eixo é feito de aço A-‐36, tem um diâmetro de 80 mm, e é fixado em B, enquanto A é solta e pode girar 0,005 rad antes de tornar-‐se fixo. Quando os binários são aplicados para C e D, determinar a tensão máxima de cisalhamento nas regiões CA e CD do eixo.

218 CHAPTER 5 TORS ION

5

A

C0.4 m

0.8 m

300 N!m

B

Prob. 5–77

A

C

D 1 m

1 m

1.5 m

200 N!m

500 N!m

B

Prob. 5–78

5–78. The A-36 steel shaft has a diameter of 60 mm and isfixed at its ends A and B.If it is subjected to the torques shown,determine the absolute maximum shear stress in the shaft.

5 in.

8 in.

12 in.

1 in.

0.5 in.A

B

C

D 500 lb!ft

Prob. 5–79

A600 mm

600 mm

600 mm

B

2 kN!m

4 kN!m

C

D

Probs. 5–80/81

•5–77. The A-36 steel shaft has a diameter of 50 mm and isfixed at its ends A and B. If it is subjected to the torque,determine the maximum shear stress in regions AC and CBof the shaft.

PROBLEMS

5–79. The steel shaft is made from two segments: AC has adiameter of 0.5 in, and CB has a diameter of 1 in. If it isfixed at its ends A and B and subjected to a torque of

determine the maximum shear stress in the shaft.Gst = 10.811032 ksi.500 lb # ft,

*5–80. The shaft is made of A-36 steel, has a diameter of80 mm, and is fixed at B while A is loose and can rotate0.005 rad before becoming fixed. When the torques areapplied to C and D, determine the maximum shear stress inregions AC and CD of the shaft.

•5–81. The shaft is made of A-36 steel and has a diameterof 80 mm. It is fixed at B and the support at A has a torsionalstiffness of If it is subjected to the geartorques shown, determine the absolute maximum shear stressin the shaft.

k = 0.5 MN # m>rad.

5–82. The shaft is made from a solid steel section AB anda tubular portion made of steel and having a brass core.If it is fixed to a rigid support at A, and a torque of

is applied to it at C, determine the angle oftwist that occurs at C and compute the maximum shearstress and maximum shear strain in the brass and steel.Take Gst = 11.511032 ksi, Gbr = 5.611032 ksi.

T = 50 lb # ft

A

0.5 in.

1 in.

2 ft

3 ft

B

CT " 50 lb!ft

Prob. 5–82



[P50] Os dois eixos são feitos de aço A-‐36. Os eixos tem 25 mm e os dois estão acoplados pelas engrenagens. As outras extremidades de cada um dos eixos estão engastadas em apoios fixos em A e B. Além disso, os eixos estão apoiados em mancais em C e D, que permitem que eles girem livremente. Se for aplicado um torque de 500 Nm à engrenagem em E, determine as reações em A e B.

5.5 STATICALLY INDETERMINATE TORQUE-LOADED MEMBERS 219

5–87. Determine the rotation of the gear at E in Prob. 5–86.

5

5–83. The motor A develops a torque at gear B of which is applied along the axis of the 2-in.-diameter steel shaftCD. This torque is to be transmitted to the pinion gears at Eand F. If these gears are temporarily fixed, determine themaximum shear stress in segments CB and BD of the shaft.Also, what is the angle of twist of each of these segments? Thebearings at C and D only exert force reactions on the shaftand do not resist torque.Gst = 1211032 ksi.

450 lb # ft,

4 ft 3 ft

B

DC

A

E F

450 lb!ft

Prob. 5–83

*5–84. A portion of the A-36 steel shaft is subjected to alinearly distributed torsional loading. If the shaft has thedimensions shown, determine the reactions at the fixedsupports A and C. Segment AB has a diameter of 1.5 in. andsegment BC has a diameter of 0.75 in.

•5–85. Determine the rotation of joint B and the absolutemaximum shear stress in the shaft in Prob. 5–84.

A

B60 in.

48 in.C

300 lb!in./in.

Probs. 5–84/85

5–86. The two shafts are made of A-36 steel. Each has adiameter of 25 mm and they are connected using the gearsfixed to their ends. Their other ends are attached to fixedsupports at A and B. They are also supported by journalbearings at C and D, which allow free rotation of the shaftsalong their axes. If a torque of is applied to thegear at E as shown, determine the reactions at A and B.

500 N # m

B

50 mm

100 mm

A

C

D

1.5 m

0.75 m

500 N!m

F

E

Probs. 5–86/87

*5–88. The shafts are made of A-36 steel and have thesame diameter of 4 in. If a torque of 15 kip ft is applied togear B, determine the absolute maximum shear stressdeveloped in the shaft.

•5–89. The shafts are made of A-36 steel and have thesame diameter of 4 in. If a torque of 15 kip ft is applied togear B, determine the angle of twist of gear B.

#

#

2.5 ft

15 kip!ft

3 ft

12 in.

6 in.

2.5 ft

A

D

B

C

E

Probs. 5–88/89


Referências Bibliográficas

•  hAp://www.cronosquality.com/aulas/rm/index.html •  Hibbeler, R. C. -‐ Resistência dos Materiais, 7.ed. São

Paulo :Pearson Pren4ce Hall, 2010. •  BEER, F.P. e JOHNSTON, JR., E.R. Resistência dos Materiais, 3.o

Ed., Makron Books, 1995. •  BUFFONI, S.S.O. Resistência dos Materiais, Universidade Federal

Fluminense – Rio de Janeiro: 2008.

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