1. Buckling of Slender Columns (10.1-10.4)
MAE 314 – Solid Mechanics
Yun Jing
Buckling of Slender Columns

2. Buckling of Slender Columns
Introduction
Up to this point, we have designed structures with constraints based on material failure.
Find the state of stress and strain due to applied loads for certain simple structures.
Compare the stress state to a maximum stress criterion or the strain state to a maximum strain criterion and determine at what load the structure will fail.
In this chapter, we learn how to design structures with constraints based on structural failure.
No point in the material reaches the maximum stress or strain criterion.
The structure at equilibrium becomes unstable.
Any small change in the loading or any imperfection in the structure will not remain at equilibrium.
This is generally due to compressive loading.
Buckling of Slender Columns

3. Example: Pinned Column
Column in Equilibrium
What happens as P increases?
The point A moves downward and the beam shortens.
At a certain value, PCR, the beam “buckles” and suddenly changes shape.
This new shape may not support the full load.
roller
supports
deformable
column
Equilibrium state for P<PCR
Equilibrium state for P=PCR
Will a structure experience material failure or structural failure first?
Depends on the material, geometry, etc.
In general:
Long slender columns typically fail first due instability.
Short wide columns typically fail first due to the material.
Buckling of Slender Columns

4. Example: Pinned Column
Why does this occur?
To find out, let’s consider a simpler system.
When is the deformed equilibrium position stable?
Move point C a small amount to the right.
If the system moves back to the vertical position it is stable.
If the system moves away from the vertical position it is unstable.
Spring moment: M = K (2Δθ)
Draw a free body diagram of the top bar.
Sum moments about point C.
Assume
rigid bars
torsional spring
angle of rotation of spring
Buckling of Slender Columns

5. Example: Pinned Column
What happens for different values of P?
For P < 4K / L, ΣMC < 0
Point C moves back to the left.
Stable Equilibrium
For P = 4K / L, ΣMC = 0
Point C does not move.
Neutrally Stable Equilibrium
For P > 4K / L, ΣMC > 0
Point C moves to the right.
Unstable Equilibrium
We define PCR to be the critical load P at which the system moves from a stable equilibrium to an unstable equilibrium.
Buckling of Slender Columns

6. Buckling of Slender Columns
Euler’s Formula
This chapter uses slender pinned column as its basis because it is a commonly used member.
The same concepts can be applied to other structures.
Next step is to derive Euler’s formula for pin-ended columns.
Solve for deflection curve of buckled beam.
Moment at some distance x:
Define p2 = P / EI
Solution:
Buckling of Slender Columns

7. Buckling of Slender Columns
Critical Load
Boundary conditions
y(0) = 0:
y(L) = 0:
The lowest value of P corresponds to n = 1.
We cannot solve for the amplitude (A) using a stability analysis.
Other values of n correspond to higher order modes.
n=1
n=3
n=4
n=2
Buckling of Slender Columns

8. Buckling of Slender Columns
Critical Stress
Knowing the critical load, we can now calculate the critical stress.
Evaluate at section of beam with smallest I.
For a rectangular beam, I = bh3/12.
Typical plot of σCR vs. L / r for steel (r is the radius of gyration)
Critical stress depends on the slenderness ratio L /r .
yield stress determines
failure
buckling determines
failure
Buckling of Slender Columns

9. Rectangular Cross-Section
What happens when the column cross-section is rectangular?
i.e. In which direction does it buckle?
So the beam buckles about the 2-2 axis first.
Buckling of Slender Columns

10. Buckling of Slender Columns
Other End Conditions
We can extend Euler’s formula to columns with other end conditions.
Replace the length L with an “equivalent” or “effective” length Le.
L is the actual length of the beam & Le is the length for use in PCR.
Buckling of Slender Columns

11. Buckling of Slender Columns continued
Example Problem
Knowing that the torsional spring at B is of constant K and
that the bar AB is rigid, determine the critical load Pcr.
Buckling of Slender Columns continued

12. Buckling of Slender Columns continued
Example Problem
Determine the critical load of an aluminum tube that is 1.5 m long and has a 16-mm outer diameter and a 1.25-mm wall thickness. Use E = 70 GPa.
1.25 mm
16 mm
Buckling of Slender Columns continued

13. Buckling of Slender Columns continued
Example Problem
Column AB carries a centric load P of magnitude 15 kips. Cables BC and BD are taut and prevent motion of point B in the xz-plane. Using Euler’s formula and a factor of safety of 2.2, and neglecting the tension in the cables, determine the maximum allowable length L. Use E = 29 x 106 psi.
W10 x 22
Buckling of Slender Columns continued