Torsion of statically indeterminate circular shafts (Strength of ...

Problem :
Torsion of statically indeterminate circular shafts
(Strength of Materials - II, Final Exam-39-4, 14-06-2005)
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 =77GPa) and BC of brass (G =39GPa), 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.
Solution :
Equilibrium equation
The compatibility condition
The polar moment od each section
and the internal torques
1. Statically indeterminate circular shafts
µ
TL
+
GJ AB
TA + TC =12.5 (1)
φ B/A + φ C/B = 0
µ TL
GJ
BC
JAB = π
32 (0.125)4 =2.3968 × 10 −5 m 4
JCD = π
32 (0.075)4 =3.1063 × 10 −6 m 4
TAB = TA
= 0 (2)
TBC = −TC
The compatibility condition becomes
µ
0.3
TA
77 × 109 × 2.3968 × 10−5 µ
0.2
− TC
39 × 109 × 3.1063 × 10−6
= 0
1. 625 5 × 10 −7 TA − 1. 650 9 × 10 −6 TC = 0
TA = 10.156TC (3)
Dr. M. Kemal Apalak 1

TA + TC = 12.5
10.156TC + TC = 12.5
TC = 1.12 kN.m
TA = 10.156 × 1.12
TA = 11.375 kN.m
The maximum shear stress between A and B
µ
T × c
(τ AB)
max =
J
= 11.375 × 103 × ¡ ¢
0.125
2
2.3968 × 10−5 (τ AB)
max =
AB
29.66 MPa (4)
The maximum shear stress between B and C
µ
T × c
(τ BC)
max =
J
= 1.12 × 103 × ¡ ¢
0.075
2
3.1063 × 10−6 (τ BC)
max =
BC
13.52 MPa (5)
Dr. M. Kemal Apalak 2