Buckling: Beam Columns 
BeamColumn Equation 
The outofplane transverse displacement w of a beam subject to inplane loads is governed by the equation,
where p is a distributed transverse load (force per unit length) acting in the positivey direction, f is an axial compression force, E is the Young's modulus of the beam, and I is the area moment of inertia of the beam's cross section. The above equation is sometimes referred to as the beamcolumn equation, since it exhibits behaviors of both beams and columns. If E and I do not vary with x across the length of the beam and f remains constant, denoted as F, then the beamcolumn equation can be simplified to,

Origin of the BeamColumn Equation 
Similar to the EulerBernoulli beam equation, the beamcolumn equation arises from four distinct subsets of beamcolumn theory: kinematics, consitutive, force resultants, and equilibrium. The outcome of each of these segments is summarized as follows: 
In the equilibrium equations, N is the
axial force resulting acting in a tensile manner (opposite in
direction to the compressive resultant f).
To relate the beam's outofplane displacement w to its pressure loading p, we combine the results of the four subcategories in the following order: 
Kinematics  =>  Constitutive  =>  Resultants  =>  Equilibrium  =>  BeamColumn Equation 
This hierarchy can be demonstrated by working backwards. First combine the two equilibrium equations to eliminate V:
Next replace the moment resultant M with its definition in terms of the direct stress s:
Use the constitutive relation to eliminate s in favor of the strain e, and then use kinematics to replace e in favor of the normal displacement w:
As a final step, recognizing that the integral over y^{2} is the definition of the beam's area moment of inertia I,
We arrive at the beamcolumn equation based on the EulerBernoulli beam theory,
Since columns are usually used as compression members, engineers may be more familiar with the axial compression resultant f than the tensile resultant N. Let f = N. The beamcolumn equation expressed with f is therefore,

Copyright © efunda.com
