Chapter 13 – Buckling of Columns

Failure mode of column = buckling (i.e. not stress)

At Pcr, stress less than yield strength!

Load P can be increased until failure occurs by yield or fracture

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

Derivation of Euler’s Equation:

PMdxdEI 2

2

02

2

EIP

dxd

Homogeneous, second order, linear DE with constant coefficients.

x

EIPCx

EIPC cossin 21

Solution to DE is:

C1 and C2 = constants of integration. =0 at x=0 so C2=0.

x

EIPC sin1

0sin

x

EIP

Which leaves:

Which is satisfied if: nLEIP

..........,3,2,12

22

nLEInP

Smallest value of P??

2

2

LEIPCR

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

2

2

2

/

,

rLE

ArInote

CR

Examples: