1. Chapter 13 – Buckling of Columns
Figure: 13-00CO

2. Failure mode of column = buckling (i.e. not stress)
At Pcr, stress less than yield strength!
Figure: 13-01a

3. Load P can be increased until failure occurs by yield or fracture
Figure: 13-04a
At Pcr, column is on verge of being unstable so small lateral force, F will cause column to buckle.
When F is removed, stays buckled. If P reduced to below Pcr will straighten out, if P increased beyond Pcr will increase lateral deflection
Load P can be increased until failure occurs by yield or fracture

4. Derivation of Euler’s Equation:
Figure: 13-05a
Homogeneous, second order, linear DE with constant coefficients.

5. Solution to DE is: Which is satisfied if: Smallest value of P??
C1 and C2 = constants of integration. u=0 at x=0 so C2=0.
Which leaves:
Figure: 13-05b
Which is satisfied if:
Smallest value of P??

6. Figure: a

7. Euler’s Equation
PCR = critical or maximum axial load on the column just before it begins to buckle.
E = modulus of elasticity for the material
I = least moment of inertia of the column’s cross-section
L = unsupported length of the column, whose ends are pinned

8. Figure: 13-08

9. Examples:
Figure: UN