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

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

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

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

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??

Figure: 13-05-01a

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

Figure: 13-08

Examples: Figure: 13-07-01UN