Bending of thin plates

Here we will develop the general equation for bending of thin plates whose thickness is small compared with other two dimensions. When such a plate is bent, it is stretched in the convex part and compressed in the concave side. The stretching decreases as one penetrates into the plate till becoming zero at the neutral surface where there is no stretching or compression, at the other side of the neutral surface, the deformation changes sign.

Let us consider a co-ordinate system with origin on the neutral surface and the -axis normal to it. The -plane is that of the un-deformed plate. Say, is the vertical displacement of a point on the neutral surface along the coordinate. Here, we consider small bending of the plate so that is small with respect to the thickness of the film. Other components of the displacement in the -plane are negligibly small relative to . Thus on the neutral surface the components of displacement vector are: .