1. ☻ ☻ ☻ ☻ 2.0 Bending of Beams sx 2.1 Revision – Bending Moments
2.2 Stresses in Beams x sx
Mxz P ☻
2.3 Combined Bending and Axial Loading P1 P2 ☻
2.4 Deflections in Beams
2.5 Buckling
(Refer: B,C & A –Sec’s 10.1, 10.2)

2. 2.5 Bars Under Axial Compression - Buckling
(Refer: B, C & A–Sec 10.1, 10.2)
Euler, Leonard (1741)
For bars or columns under compressive loading, we may witness a sudden failure, known as buckling.
This is a form of instability. P v
Consider load, P, plotted against horizontal displacement, v. P P
Point of Bifurcation
Short, Thick
Long, Slender

3. P P P Mxz=-P.v Consider a long slender bar under axial compression: y
From the Engineering Beam Theory:
Let
Euler Equation

4. If A=0, v=0 for all values of x
General Solution:
x=0
Boundary Conditions:
x=L
(Also, )
From (i):
From (ii):
If A=0, v=0 for all values of x
(i.e. The bar remains straight – NO BUCKLING)
If Sin aL=0, then (possible buckling solutions)

5. P v The Lowest Real Value: Since
Where PC is the Critical or Euler Buckling Load.
For buckling (i.e. at point of bifurcation): P
(Iz is the smallest 2nd Moment of Area) P v
Points of Bifurcation v

6. sC Unstable sYield Stable S Let Let Kz2 (Radius of Gyration)
and S (Slenderness Ratio)
Short, Thick
Long, Slender sC S
Euler Stress
Stable
Unstable
sYield
Empirical
Material Yielded

7. P L
Free or Pinned Ends
End Constraints: P L
Fixed Ends
L/2 P L
One End Free
& One End Fixed 2L P L
One End Pinned
& One End Fixed

8. Example:
Find the shortest length L of a Pin-Ended strut, having a cross-section of 60mm x 100mm for which the Euler Equation applies. Assume E=200 Gpa and sYield=250 MPa
Euler Eqn.,
i.e. Iz
Min. value, sC S
Euler
Theory
sYd
=250 MPa
Note: 90