This document discusses plastic bending and torsion of beams. It describes the assumptions made in plastic bending analysis and the stages of plastic yielding, including incipient, elasto-plastic, and fully plastic yielding. Non-linear stress-strain curves are presented for bending and torsion. An example problem is given to calculate the maximum stress induced in a rectangular beam subjected to a concentrated load based on its stress-strain curve. A second example calculates the twisting couple applied to a circular shaft that yields plastically over its outer 2cm depth.

1. Non Linear bending &
torsion Equation
Dhavalsab.M.L
R15MDD10

2. How to analyze
The formula M/I=Sigma/y= E/R which is applicable to the elastic range
only for the determination of stress in the beam for the plastic range ,
the knowledge of the stress-strain law for the beam material is essential
& also the yield point of the beam material

3. Assumption made during plastic bending of beams
• The beam is initially straight & c/s has at least one axis of symmetry
• The bending load is applied in the plane of symmetry
• Plane section are remain plane after the loads are applied
• The true stress-strain relation for each longitudinal fiber member is
some that obtained from the tension & compression specimen of the
material
• The C/S dimension are very small when compare to the length hence
deformation due to shear is neglected
• The deflection are so small that the curvature formula can be
approximated to the 2nd derivative of the deflection with respect to
the distance along the member

4. Stages of plastic yielding Incipient yielding :
when the loading of the beam causes the
yielding at the outer most fiber which is the
high stressed region. The yielding is called as
the incipient yielding
We know that
The bending moment which is causing
incipient yielding Is designated as M

5. Elasto plastic yielding :
When the bending moments applied to the
beam beyond y=n yield practically such a type
yielding is called elastic plastic yielding then
bending moment due to the Elasto-plastic
yielding is given by
Fully plastic yielding
When the bending moment is applied is such
that the beam yield plastically without any
elastic yielding such a type of yielding is called
fully plastic yielding bending moment required
for fully plastic yielding

6. Non-Linear stress-strain curve
Stress relation for the bending
moment is given by
where H& n are the material constant
.for a beam material subjected to
bending moment S.T
Non linear stress-strain curve for the
beam is given by where H & n are
material constant considering the
bending of rectangular beam of width
b & depth h we have

8. Equation :
A rectangular beam of width 4cm depth 6 cm is 3cm causes a concentrated load of 2kn at the
midspan if the stress-strain curve for the beam material is given by Sigma*= 700
(strain=0.25)determine the maximum stress induced in the beam
B=4cm = 40 mm , H=6cm =60mm, L=3000mm , W=2000n

9. Torsion of Bars
When the prismatic bar subjected
for torque at the end torsional stress
are setup if the torque applied is
increased to such a level that the
shear strain produced became grater
than that of the yield strength of the
bar & then the bar yields plastically .
The yield strength is completed in 3
stages
Incipient yielding
Elastic-plastic yielding
Plastic yielding

10. Non-linear stress-strain curve

12. Equation
A circular rod of 4cm inner radius &
outer radius is 10 cm is subjected to
twisting couple so that the outer
2cm deep yield plastically .
Determine the twisting couple
applied to the shaft yield stress in
shear for the shaft material is
145/mm also determine the couple
for the full yielding's

13. Thank you