Aula 16 - Elementos esta0camente indeterminados carregados com torque.

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Prof. Wanderson S. Paris -‐ prof@cronosquality.com.br MECÂNICA DOS SÓLIDOS

Aula 16 -‐ Elementos esta0camente

indeterminados carregados com torque.

Prof. Wanderson S. Paris,

M.Eng

.

prof@cronosquality.com.br

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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.

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Equações

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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.

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Solução Exercício 1

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Solução Exercício 1

Subs4tuido:

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Solução Exercício 1

Deformação por

cisalhamento:

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Exercício 2

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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 all

copyright 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.

c05.qxd 9/19/07 8:17 PM Page 198

© 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 all

copyright 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.

198 c05.qxd 9/19/07 8:17 PM Page 198

© 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 all

copyright 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.

198

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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 all

copyright 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.

198

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Exercícios Propostos

[P73] 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 C

H A P T E R

5 T

O R S I O N

5

A C 0.4 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 is

fixed 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 A 600 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 is

fixed at its ends A and B. If it is subjected to the torque,

determine the maximum shear stress in regions AC and CB

of the shaft.

PROBLEMS

5–79. The steel shaft is made from two segments: AC has a

diameter of 0.5 in, and CB has a diameter of 1 in. If it is

fixed at its ends A and B and subjected to a torque of

determine the maximum shear stress in the shaft.

G

_st

_{= 10.8110}

3

_{2 ksi.}

500 lb

#

ft,

*5–80. The shaft is made of A-36 steel, has a diameter of

80 mm, and is fixed at B while A is loose and can rotate

0.005 rad before becoming fixed. When the torques are

applied to C and D, determine the maximum shear stress in

regions AC and CD of the shaft.

•

5–81. The shaft is made of A-36 steel and has a diameter

of 80 mm. It is fixed at B and the support at A has a torsional

stiffness of

If it is subjected to the gear

torques shown, determine the absolute maximum shear stress

in the shaft.

k

= 0.5 MN

#

m

_>rad.

5–82. The shaft is made from a solid steel section AB and

a 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 of

twist that occurs at C and compute the maximum shear

stress and maximum shear strain in the brass and steel.

Take G

_st

_{= 11.5110}

3

_{2 ksi, G}

_br

_{= 5.6110}

3

_{2 ksi.}

T

_{= 50 lb}

#

ft

A 0.5 in. 2 ft 3 ft B

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Exercícios Propostos

[P74] 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

CH A P T E R 5 TO R S I O N 5 A C 0.4 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 is

fixed 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 A 600 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 is

fixed at its ends A and B. If it is subjected to the torque, determine the maximum shear stress in regions AC and CB of the shaft.

PROBLEMS

5–79. The steel shaft is made from two segments: AC has a

diameter of 0.5 in, and CB has a diameter of 1 in. If it is fixed at its ends A and B and subjected to a torque of determine the maximum shear stress in the shaft.

G_st _{= 10.8110}3_{2 ksi.}

500 lb

#

ft,

*5–80. The shaft is made of A-36 steel, has a diameter of

80 mm, and is fixed at B while A is loose and can rotate 0.005 rad before becoming fixed. When the torques are applied to C and D, determine the maximum shear stress in regions AC and CD of the shaft.

•5–81. The shaft is made of A-36 steel and has a diameter

of 80 mm. It is fixed at B and the support at A has a torsional

stiffness of If it is subjected to the gear

torques shown, determine the absolute maximum shear stress in the shaft.

k _{= 0.5 MN}

#

m_>rad.

5–82. The shaft is made from a solid steel section AB and

a 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 of twist that occurs at C and compute the maximum shear stress and maximum shear strain in the brass and steel.

Take G_st _{= 11.5110}3_{2 ksi, G}_br _{= 5.6110}3_{2 ksi.}

T _{= 50 lb}

#

ft A 0.5 in. 1 in. 2 ft 3 ft B C T " 50 lb!ft Prob. 5–82

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Exercícios Propostos

[P75] 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 shaft

CD. This torque is to be transmitted to the pinion gears at E

and F. If these gears are temporarily fixed, determine the maximum shear stress in segments CB and BD of the shaft. Also, what is the angle of twist of each of these segments? The bearings at C and D only exert force reactions on the shaft

and do not resist torque. Gst = 1211032 ksi.

450 lb

#

ft, 4 ft 3 ft B D C A E F 450 lb!ft Prob. 5–83

*5–84. A portion of the A-36 steel shaft is subjected to a

linearly distributed torsional loading. If the shaft has the dimensions shown, determine the reactions at the fixed supports A and C. Segment AB has a diameter of 1.5 in. and segment BC has a diameter of 0.75 in.

•5–85. Determine the rotation of joint B and the absolute

maximum shear stress in the shaft in Prob. 5–84.

A B 60 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 a

diameter of 25 mm and they are connected using the gears fixed to their ends. Their other ends are attached to fixed supports at A and B. They are also supported by journal

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 the

same diameter of 4 in. If a torque of 15 kip ft is applied to gear B, determine the absolute maximum shear stress developed in the shaft.

Aula 16 - Elementos esta0camente indeterminados carregados com torque.