Beam and Plate Bending
Granville Sewell · 2018
This chapter considers a rigid (1D) beam and a rigid (2D) plate, with external vertical forces. It establishes a beam bending equation with u(x) as the height of the beam, D(x) as the bending stiffness and q(x) as an external vertical force. Since e is an arbitrary smooth function in the interior of (0, L), the factor multiplying e must be zero everywhere, which gives the fourth-order beam equation. The boundary conditions at either end will also make the boundary terms disappear, so these are also possible boundary conditions. The boundary conditions are called “clamped” boundary conditions because they model a beam whose height is fixed and whose slope is “clamped” at an endpoint. “Simply supported” boundary conditions model a beam whose height is fixed at an endpoint, but whose slope is not clamped. The chapter also establishes a derivation of the plate bending equation.