Beam Theory

A novel micromorphic beam theory that considers the exact shape and size of the beam’s microstructure is developed. The new theory complements the beam theories that are based on the classical mechanics by modeling the shape and size of the beam’s microstructure. This theory models the beam with a microstructure that has shape and size and exhibits microstrains that are independent of the beam’s macroscopic strains.

How can we derive dynamic beam equation from navier equation in elastodynamic? I tried with the displacement vector field:

u=-z*w(x,t)

v=0

w=w(x.t)

and put it in navier equation, but it does not work.