pure-bending of curved bar
TRANSCRIPT
![Page 1: Pure-bending of curved bar](https://reader033.vdocuments.us/reader033/viewer/2022042507/55961ef61a28ab7b0e8b47c3/html5/thumbnails/1.jpg)
Pure Bending of curved bars
ByPratish Bhaskar Sardar
(122090025)
![Page 2: Pure-bending of curved bar](https://reader033.vdocuments.us/reader033/viewer/2022042507/55961ef61a28ab7b0e8b47c3/html5/thumbnails/2.jpg)
CONTENTS
Pure Bending of Curved Bars.
Boundary conditions of the problem.
Numerical Examples
![Page 3: Pure-bending of curved bar](https://reader033.vdocuments.us/reader033/viewer/2022042507/55961ef61a28ab7b0e8b47c3/html5/thumbnails/3.jpg)
Pure Bending of curved bars
![Page 4: Pure-bending of curved bar](https://reader033.vdocuments.us/reader033/viewer/2022042507/55961ef61a28ab7b0e8b47c3/html5/thumbnails/4.jpg)
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.
![Page 5: Pure-bending of curved bar](https://reader033.vdocuments.us/reader033/viewer/2022042507/55961ef61a28ab7b0e8b47c3/html5/thumbnails/5.jpg)
Basic concept
![Page 6: Pure-bending of curved bar](https://reader033.vdocuments.us/reader033/viewer/2022042507/55961ef61a28ab7b0e8b47c3/html5/thumbnails/6.jpg)
Where…
b = Radius of outer fiber
a = Radius of inner fiber
l = Width of section
ro= Radius of centroidal axis
M=Bending moment applied
![Page 7: Pure-bending of curved bar](https://reader033.vdocuments.us/reader033/viewer/2022042507/55961ef61a28ab7b0e8b47c3/html5/thumbnails/7.jpg)
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)
![Page 8: Pure-bending of curved bar](https://reader033.vdocuments.us/reader033/viewer/2022042507/55961ef61a28ab7b0e8b47c3/html5/thumbnails/8.jpg)
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
![Page 9: Pure-bending of curved bar](https://reader033.vdocuments.us/reader033/viewer/2022042507/55961ef61a28ab7b0e8b47c3/html5/thumbnails/9.jpg)
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
![Page 10: Pure-bending of curved bar](https://reader033.vdocuments.us/reader033/viewer/2022042507/55961ef61a28ab7b0e8b47c3/html5/thumbnails/10.jpg)
B = (4M/Q)a2 b2 ln(b/a)
C = M/Q 2(b2lnb-a2lna) +b2 –a2
D= 2 M/Q(b2-a2)
Where,
Q =(b2-a2)2 -4a2b2ln(b2/a2)
![Page 11: Pure-bending of curved bar](https://reader033.vdocuments.us/reader033/viewer/2022042507/55961ef61a28ab7b0e8b47c3/html5/thumbnails/11.jpg)
Radial Stress
• σr = (1/r) (∂υ/∂r) + (1/r2) (∂2υ/∂θ2)
=B/r2 +2C+D(1+lnr)
• circumferential stress
σθ = (∂2υ/∂r2)
= -(B/r2)+2C+D(3+2lnr)
![Page 12: Pure-bending of curved bar](https://reader033.vdocuments.us/reader033/viewer/2022042507/55961ef61a28ab7b0e8b47c3/html5/thumbnails/12.jpg)
NUMERICAL EXAMPLES
![Page 13: Pure-bending of curved bar](https://reader033.vdocuments.us/reader033/viewer/2022042507/55961ef61a28ab7b0e8b47c3/html5/thumbnails/13.jpg)
![Page 14: Pure-bending of curved bar](https://reader033.vdocuments.us/reader033/viewer/2022042507/55961ef61a28ab7b0e8b47c3/html5/thumbnails/14.jpg)
![Page 15: Pure-bending of curved bar](https://reader033.vdocuments.us/reader033/viewer/2022042507/55961ef61a28ab7b0e8b47c3/html5/thumbnails/15.jpg)
Thank You …