Airy Stress Function:
The solution of problems in elasticity presents difficulties but the procedure may be simplified by the introduction of a stress function. For a particular two-dimensional case the stresses are related to a single function of x and y such that substitution for the stresses in terms of this function automatically satisfies the equations of equilibrium no matter what form the function may take. However, a large proportion of the infinite number of functions which fulfill this condition are eliminated by the requirement that the form of the stress function must also satisfy the two-dimensional equations of compatibility, and plus the appropriate boundary conditions.
For simplicity let us consider the two-dimensional case for which the body forces are zero. The problem is now to determine a stress-stress function relationship which satisfies the equilibrium conditions of and a form for the stress function giving stresses which satisfy
the compatibility equation
The English mathematician Airy proposed a stress function φ defined by the equations
Clearly, substitution of Eqs verifies that the equations of equilibrium are satisfied by this particular stress-stress function relationship. Further substitution into Eq. (2.7) restricts the possible forms of the stress function to those satisfying the biharmonic equation
The final form of the stress function is then determined by the boundary conditions relating to the actual problem. Thus, a two-dimensional problem in elasticity with zero body forces reduces to the determination of a function φ of x and y, which satisfies Eq at all points in the body and reduced to two-dimensions at all points on the boundary of the body.
We have seen that the elasticity method of structural analysis embodies the determination of stresses and/or displacements by employing equations of equilibrium and compatibility in conjunction with the relevant force-displacement or stress-strain relationships.
A powerful alternative but equally fundamental approach is the use of energy methods. These, while providing exact solutions for many structural problems, find their greatest use in the rapid approximate solution of problems for which exact solutions do not exist. Also, many structures which are statically indeterminate, that is they cannot be analysed by the application of the equations of statically equilibrium alone, may be conveniently analysed using an energy approach.
Further, energy methods provide comparatively simple solutions for deflection problems which are not readily solved by more elementary means.
Generally, as we shall see, modern analysis’ uses the methods of total complementary energy and total potential energy. Either method may be employed to solve a particular problem, although as a general rule deflections are more easily found using complementary energy and forces by potential energy.
Strain energy and complementary energy :
A structural member subjected to a steadily increasing load P. As the member extends, the load P does work and from the law of conservation of energy this work is stored in the member as strain energy. A typical load-deflection curve for a member possessing non-linear elastic characteristics is shown in Fig. 4.l(b).
The strain energy U produced by a load P and corresponding extension y is then and is clearly represented by the area OBD under the load-deflection curve. Engesser (1889) called the area OBA above the curve the complementary energy C, and from Complementary energy, as opposed to strain energy, has no physical meaning, being purely a convenient mathematical quantity. However, it is possible to show that complementary energy obeys the law of conservation of energy in the type of situation usually arising in engineering structures, so that its use as an energy method is valid.
Differentiation of Eqs (4.1) and (4.2) with respect to y and P respectively gives
The bi-harmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows. It is written as
For example, in three dimensional cartesian coordinates the biharmonic equation has the form