Torsion:

Saints Venant’s Principle:

In the examples of Section 2.3 we have seen that a particular stress function form may be applicable to a variety of problems. Different problems are deduced from a given stress function by specifying, in the first instance, the shape of the body and then assigning a variety of values to the coefficients. The resulting stress functions give stresses which satisfy the equations of equilibrium and compatibility at all points within and on the boundary of the body. It follows that the applied loads must be distributed around the boundary of the body in the same manner as the internal stresses at the boundary. Thus, in the case of pure bending the applied bending moment must be produced by tensile and compressive forces on the ends of the plate, their magnitudes being dependent on their distance from the neutral axis. If this condition is invalidated by the application of loads in an arbitrary fashion or by preventing the free distortion of any section of the body then the solution of the problem is no longer exact. As this is the case in practically every structural problem it would appear that the usefulness of the theory is strictly limited. To surmount this obstacle we turn to the important principle of St. Venant which may be summarized as stating:

that while statically equivalent systems of forces acting on a body produce substantially different local effects the stresses at sections distant from the surface of loading are essentially the same.

Thus, at a section AA close to the end of a beam supporting two point loads P the stress distribution varies as shown in Fig. 2.3, whilst at the section BB, a distance usually taken to be greater than the dimension of the surface to which the load is applied, the stress distribution is uniform.

We may therefore apply the theory to sections of bodies away from points of applied loading or constraint. The determination of stresses in these regions requires, for some problems, separate calculation.

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Semi-inverse method

In elasticity the first application of the semi-inverse method is due to Saint- Venant. He was the first to study the problem of linear electrostatics for a right long cylinder free from volume forces and loaded only at the bases by unspecified tractions. This problem was later on called the problem of Saint-Venant (Saint-Venantsche Problem) by Clebsch.The starting point of the application of the semi-inverse method in order to solve the problem is that some components of the stress vanish. In particular, it is assumed that the normal tension on every section parallel to X3, the axis of the cylinder, be zero:

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When this assumption is made, it is possible to find a closed-form solution of the problem by the use of the linear equilibrium equations (1.50), of the linear constitutive equations (1.48), of the compatibility Beltrami conditions (1.63), and of the prescribed boundary conditions. The displacement field for the points of the cylinder turns out to depend linearly on four constants; these represent kinematic parameters to be specified at one base of the cylinder. Each of them characterizes a simple mode of deformation of the cylinder: extension, bending, torsion, and flexure, and it may be shown that the four kinematic parameters are linear functions of the resultant actions on the bases.

The semi-inverse assumption on the field of stress is of fundamental importance to find the approximate analytical solution. Although this assumption is suggested by the geometrical and boundary surface tractions, it is justified afterwards by the existence of the solution found. The Kirchhoff principle shows then the unique of the solution.

Kirchhoff, 1859 If either the surface displacement or the surface tractions are given, the solution of the problem of equilibrium of an elastic body is unique in the sense that the state of stress (and strain) is determinate without ambiguity, provided that the magnitude of the stress (and strain) is so small that the strain energy function exists and remains positive definite.

Several applications of the semi-inverse method can be found in the literature on nonlinear elasticity. Of course it is not possible to list all such results because a survey aiming at completeness would require a whole book. Here we present some representative examples in order to underline some aspects of the semi-inverse method, and other useful examples for our discussion are given in the next chapters.

First, we recall something just discussed on simple extension, but here we modify the problem a little bit. Then we discuss a problem of anti-plane shear. Finally, some others remarks are discussed for the radial deformation problem.

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