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Beam-Column Equation

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:

Kinematics:
Constitutive:
Resultants:
Equilibrium:       
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:

Kinematics => Constitutive => Resultants => Equilibrium => Beam-Column
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 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,

STEM Career Outlook

Wages, employment opportunities, and growth projections for STEM jobs.

Mechanical Engineers Outlook

Guide for those interested in becoming a mechanical engineer. Includes qualifications, pay, and job duties.

Essentials of Manufacturing

Information, coverage of important developments and expert commentary in manufacturing.

Selecting the Right 3D Printer

Discover how to choose the right 3D printer for your needs and the key performance attributes to consider.