4. CURVED MEMBERS IN BENDING
The distribution of stress in a curved flexural
member is determined by using the following
assumptions.
The cross section has an axis of symmetry in a
plane along the length of the beam.
Plane cross sections remain plane after bending.
The modulus of elasticity is the same in tension as
in compression.
6. Where…
b = Radius of outer fiber
a = Radius of inner fiber
l = Width of section
ro= Radius of centroidal axis
M=Bending moment applied
7. In the absence of body forces equilibrium
equations are satisfied by stress function υ(r,θ)
for which stress components in radial and
tangential directions are
σr = (1/r) (∂υ/∂r) + (1/r2) (∂2υ/∂θ2)
σθ = (∂2υ/∂r2)
τrθ = (1/r2) (∂υ/∂θ) - (1/r) (∂2υ/∂r∂θ)
8. boundary conditions
1 at r = a , σr = 0
2 at r = b , σr = 0
3 τrθ = 0 for all boundaries
at either end of beam circumferential normal stresses
must have a zero resultant force and equivalent to
bending moment M on each unit width of beam
4 ∫ σθ dr = 0 ∫ σθ r dr = M
9. Standard Relations…
from BC's 1 and 2
(B/a2) + 2C + D(1+ 2ln a) = 0 and
(B/b2) + 2C + D(1+ 2ln b) = 0
from BC 4
υab = B ln (b/a) + C (b2 -a2) + D (b2 ln b - a2 ln a) = -M
