Hydraulics & Pneumatics

For design and manufacturing engineers involved in buying or specifying fluid power components and systems.

Machine Design

How-to, in-depth technical articles for machine design engineers

Essentials of Manufacturing

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

Negotiate Your Salary

Learn the best principles to negotiate the salary you deserve!

more free magazines
 
Choose a Boundary Condition and Calculate!
Cantilever Beam
Cantilevers
Mixed Support Beam
Mixed (Fixed-Simple)
Simply Supported Beam
Simply Supported
Euler-Bernoulli Beam Equation
The out-of-plane displacement w of a beam is governed by the Euler-Bernoulli Beam Equation,

where p is the distributed loading (force per unit length) acting in the same direction as y (and w), E is the Young's modulus of the beam, and I is the area moment of inertia of the beam's cross section.

If E and I do not vary with x along the length of the beam, then the beam equation simplifies to,

Origin of the Beam Equation
The Euler beam equation arises from a combination of four distinct subsets of beam theory: the kinematic, constitutive, force resultant, and equilibrium definition equations.

The outcome of each of these segments is summarized here:

Kinematics:
Constitutive:
Resultants:
Equilibrium:        
To relate the beam's out-of-plane displacement w to its pressure loading p, we combine the results of the four beam sub-categories in the order shown,
Kinematics -> Constitutive -> Resultants -> Equilibrium = Beam
Equation

We'll demonstrate this hierarchy by working backwards. We first combine the 2 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,

allows us to arrive at the Euler-Bernoulli beam equation,