CONTINUUM MECHANICS for ENGINEERS

Solution
From Eq 6.7-8 the stress components are
⎛ 2B
D⎞
σr= 2Ar
− + sinθ
3 ⎝ r r ⎠
⎛ 2B
D⎞
σθ= 6Ar
+ + sinθ
3 ⎝ r r ⎠
⎛ 2B
D⎞
τrθ=− 2Ar
− + cosθ
3 ⎝ r r ⎠
The inner and outer radii of the beam are stress-free surfaces leading to
boundary conditions of σ r = τ rθ = 0 at r = a, and at r = b. Also, the applied
force P may be taken to be the summation of the shear stress acting over the
free end θ = 0
b
τ ∫ rθ
] θ=
0
a
dr =−P
These conditions lead to the three equations from which constants A, B, and
D may be determined
2B
D
2Aa
− + = 0
3
a a
2B
D
2Ab
− + = 0
3
b b
2 2
b a
2 2
Ab a B D 2 2
b
( − ) − ln =
− ( − )+
ab
a P
Solving these three equations in three unknowns, the constants are determined
to be
( )
P
Pa b
Pa + b
A = ; B = − ;
P = −
2N2N N
2 2 2 2
where N = a 2 – b 2 + (a 2 + b 2 ) ln(b/a). Finally, use of these constants in the stress
component equations gives