the derivatives of must have the same form to cancel out and hence as solution of the form may be expected. This observation leads to the characteristic equation The solutions of this quartic equation are where The general solution of the Timoshenko-Rayleigh beam equation for free vibrations can then be written as Quasistatic bending of plates The defining feature of beams is that one of the dimensions is much larger than the other two. A structure is called a plate when it is flat and one of its dimensions is much smaller than the other two. There are several theories that attempt to describe the deformation and stress in a plate under applied loads two of which have been used widely. These are the Kirchhoff–Love theory of plates (also called classical plate theory) the Mindlin–Reissner plate theory (also called the first-order shear theory of plates) Kirchhoff–Love theory of plates The assumptions of Kirchhoff–Love theory are straight lines normal to the mid-surface remain straight after deformation straight lines normal to the mid-surface remain normal to the mid-surface after deformation the thickness of the plate does not change during a deformation. These assumptions imply that where is the displacement of a point in the plate and is the displacement of the mid-surface. The strain-displacement relations are The equilibrium equations are where is an applied load normal to the surface of the plate. In terms of displacements, the equilibrium equations for an isotropic, linear elastic plate in the absence of external load can be written as In direct tensor notation, Mindlin–Reissner theory of plates The special assumption of this theory is that normals to the mid-surface remain straight and inextensible but not necessarily normal to the mid-surface after deformation. The displacements of the plate are given by where are the rotations of the normal. The strain-displacement relations that result from these assumptions are where is a shear correction factor. The equilibrium equations are where Dynamic bending of plates Dynamics of thin Kirchhoff plates The dynamic theory of plates determines the propagation of waves in the plates, and the study of standing waves and vibration modes. The equations that govern the dynamic bending of Kirchhoff plates are where, for a plate with density , and The figures below show some vibrational modes of a circular plate. See also Bending moment Bending Machine (flat metal bending) Brake (sheet metal bending) Brazier effect Bending of plates