Beams: Introduction 
EulerBernoulli Beam Equation 
The outofplane displacement w of a beam
is governed by the EulerBernoulli 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: 
To relate the beam's outofplane displacement w to its pressure loading p, we combine the results of the four beam subcategories 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 y^{2} is the definition of the beam's area moment of inertia I,
allows us to arrive at the EulerBernoulli beam equation,
 
