Indeterminate-torsion-problem PDF

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Torsion of statically indeterminate circular shafts (Strength of Materials - II, Final Exam-39-4, 14-06-2005)

Problem :

The solid cylinders AB and BC are bonded together at B and attached to

fixed supports

at A and

C. Knowing that AB is made of steel (G = 77 GPa) and BC of brass (G = 39 GPa), determine for the loading shown (a) the reaction at each support, (b) the maximum shearing stress in AB, (c) the maximum shearing stress in BC.

1. Statically indeterminate circular shafts

Solution : Equilibrium equation

TA + TC

= 12.5

(1)

The compatibility condition

µTL ¶

φB/A + φC/B

GJ

GJ

+

µ T L¶

AB

=

0

=

0

(2)

BC

The polar moment od each section

JAB

=

JCD

=

× 10−5 m4 π 4 −6 4 (0.075) = 3.1063 × 10 m 32 π

32

4 (0.125) = 2.3968

and the internal torques =

TAB TBC The compatibility condition becomes

µ

TA

0.3

77

µ

¶

TA −TC 0.2

¶

× 109 × 3.1063 × 10−6 −7 −6 1. 625 5 × 10 TA − 1. 650 9 × 10 TC

=

0

=

0

TA

=

10.156TC

× 109 × 2.3968 × 10−5

Dr. M. Kemal Apalak

=

− TC

39

(3) 1

TA + TC

=

12.5

10.156TC + TC

=

12.5

TC

=

1.12 kN.m

TA TA

× 1.12

=

10.156

=

11.375 kN.m

The maximum shear stress between A and B

µ

(τ AB )max

=

(τ AB )max

=

T

× c¶

J

µ

Dr. M. Kemal Apalak

=

(τ BC )max

=

AB

29.66 MPa

The maximum shear stress between B and C (τ BC )max

=

¡

× 103 × 0.125 2 2.3968 × 10−5

11.375

T

×c¶ J

BC

13.52 MPa

¢

(4)

=

¡

× 103 × 0.075 2 3.1063 × 10−6

1.12

¢

(5)

2...