CONTINUUM MECHANICS for ENGINEERS

These conditions result in the following equations which are used to evaluate
the constants A, B, and C. The inner and outer radii are free of normal stress
which can be written in terms of boundary condition (1) as
A/a 2 + B(1 + 2 ln a) + 2C = 0
A/b 2 + B(1 + 2 ln b) + 2C = 0
No transverse loading is present on the ends of the curved beam which may
be written in terms of boundary condition (2) as
Evaluation of this integral at the limits is automatically satisfied as a consequence
of boundary condition (1). Finally, the applied moments on the ends
may be written in terms of boundary condition (3)
Because of condition (1) the bracketed term here is zero and from the integral
term
or
∫
a
σθ
dr
b
= ∫a∫ a
b
φ φ
r dr
∂
=
∂ r
∂
2 ⎤
⎥ = 0
2
∂ ⎦
r
r dr
r r
b
= dr M
∂
a r ∂
2
φ ⎡ φ ⎤ ∂φ
⎢ ⎥ − =−
2
⎣ ∂ ∫ ⎦ ∂
b ∂
φ B – φ A = M
A ln b/a + B(b 2 ln b – a 2 ln a) + C(b 2 – a 2 ) = M
This expression, together with the two stress equations arising from condition
(1) may be solved for the constants A, B, and C, which are
M
A
N ab
4 2 2
=− ln
2M
B =−
N
b −a
where N = (b 2 – a 2 ) 2 – 4a 2 b 2 [ln(b/a)] 2 . Finally, the stress components may be
written in terms of the radii and applied moment by substitution of the
constants into Eq 6.7-12
b
a
b
a
2 2 ( )
[ ( ) ]
C M 2 2 2 2
= b − a + 2 b lnb−a ln
a
N
b
a