2Beam and Plate Bending
2.1 Derivation of Beam Bending Equation
In this chapter we consider a rigid (1D) beam and a rigid (2D) plate, with external vertical forces.
To derive the beam bending equation, suppose 
is the height of the beam with 
which minimizes the energy1
where 
is the bending stiffness and 
is an external vertical force (per unit length). Then if 
is any smooth function with 
on the boundary, 
for any 
, thus
should have a minimum at and so should be zero. But
so
Here we have ...
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