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 images is the height of the beam with images which minimizes the energy1

images

where images is the bending stiffness and images is an external vertical force (per unit length). Then if images is any smooth function with images on the boundary, images for any images, thus

images

should have a minimum at and so should be zero. But

so

Here we have ...

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