engineering fundamentals Theory: Beam Equation
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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,

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