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 y^{2} is the definition of the beam's area moment of inertia I,
We arrive at the beamcolumn equation based on the EulerBernoulli 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 beamcolumn equation expressed with f is therefore,
