Direct Problems for Plates and Shells
William G. Litvinov · Birkhäuser Basel eBooks · 2000
Consider a homogeneous thin plate of variable thickness (see Fig. 4.1.1). We suppose that the plate has a midplane such that the plate is symmetric with respect to it. Take the midplane of the plate to be the ( x , y ) plane, and let the z axis be directed downwards. Denote displacements of points of the midplane along the z axis by w and assume that the so-called Kirchhoff hypotheses hold: Normals to the midplane before the bending go over into the normals to the midsurface after the bending, and their length does not change during the bending. Inside of the plate, the stresses normal to the midsurface are small as compared to other stress components, so that they can be neglected in relations between the stresses and strains (the plane stress state hypothesis). Under bending, elements of the midsurface of the plate are not subject to tension and pressure.