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,

The outcome of each of these segments is summarized here:

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,