The out-of-plane transverse displacement w of a beam subject to in-plane loads is governed by the equation,
where p is a distributed transverse load (force per unit length) acting in the positive-y 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 beam-column 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 beam-column equation can be simplified to,
|Origin of the Beam-Column Equation|
Similar to the Euler-Bernoulli beam equation, the beam-column equation arises from four distinct subsets of beam-column 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 out-of-plane displacement w to its pressure loading p, we combine the results of the four sub-categories in the following order:
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 y2 is the definition of the beam's area moment of inertia I,
We arrive at the beam-column equation based on the Euler-Bernoulli 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 beam-column equation expressed with f is therefore,