Fundamentals of Structural Analysis | Basic elasticity

Aircraft Structures for engineering students

Basic elasticity

Fundamentals of Structural Analysis

Basic elasticity 

We shall consider, in this chapter, the basic ideas and relationships of the theory of elas-ticity. The treatment is divided into three broad sections: stress, strain and stress–strain relationships. The third section is deferred until the end of the chapter to emphasize the fact that the analysis of stress and strain, for example the equations of equilibrium and compatibility, does not assume a particular stress–strain law. In other words, the relationships derived in Sections 1.1–1.14 inclusive are applicable to non-linear as well as linearly elastic bodies.

1.1 Stress

Consider the arbitrarily shaped, three-dimensional body shown in Fig. 1.1. The body isi n equilibrium under the action of externally applied forces P1, P2, ... and is assumed to comprise a continuous and deformable material so that the forces are transmitted throughout its volume. It follows that at any internal point O there is a resultant force

Fig. 1.1 Internal force at a point in an arbitrarily shaped body.

Fig. 1.2 Internal force components at the point O.

δP. The particle of material at O subjected to the force δP is in equilibrium so that there must be an equal but opposite force δP (shown dotted in Fig. 1.1) acting on the particle at the same time. If we now divide the body by any plane nn containing O then these two forces δP may be considered as being uniformly distributed over a small area δA of each face of the plane at the corresponding point O as in Fig. 1.2. The stress at O is then defined by the equation

The directions of the forces δP in Fig. 1.2 are such as to produce tensile stresses on the faces of the plane nn. It must be realized here that while the direction of δP is absolute the choice of plane is arbitrary, so that although the direction of the stress at O will always be in the direction of δP its magnitude depends upon the actual plane chosen since a different plane will have a different inclination and therefore a different value for the area δA. This may be more easily understood by reference to the bar in simple tension in Fig. 1.3. On the cross-sectional plane mm the uniform stress is given by P/A, while on the inclined plane m-m- the stress is of magnitude P/A- In both cases the stresses are parallel to the direction of P.

Generally, the direction of δP is not normal to the area δA, in which case it is usual to resolve δP into two components: one, δPn, normal to the plane and the other, δPs, acting in the plane itself (see Fig. 1.2). Note that in Fig. 1.2 the plane containing δP is perpendicular to δA. The stresses associated with these components are a normal or direct stress defined as.

Fig. 1.3 Values of stress on different planes in a uniform bar.

The resultant stress is computed from its components by the normal rules of vector addition, namely

Generally, however, as indicated above, we are interested in the separate effects of σ and τ. However, to be strictly accurate, stress is not a vector quantity for, in addition to magnitude and direction, we must specify the plane on which the stress acts. Stress is therefore a tensor, its complete description depending on the two vectors of force and surface of action.