La aJcrj ff ii Vax ay J

A similar equation can be obtained for the plane stress case, namely:

If the body forces X and Y have constant values the same equation holds for both plane stress and plane strain, namely:

This equation is known as the "Laplace differential equation" or the "harmonic differential equation." The function (oxx + oyy) is referred to as a "harmonic" function. It is interesting to note that the Laplace equation, which of course incorporates all the previous equations, does not contain the elastic constants of the material. This is an important conclusion for the experimentalist since, providing there exists geometric similarity, material isotropy and linearity and similar applied loading of both model and prototype, then the stress distribution per unit load will be identical in each. The stress function, previously defined, must satisfy the 'Laplace equation' (8.86). Thus:

dx4 dx2dy2 By4

Alternatively, this can be re-written in the form

indicating that the stress function must be a biharmonic function. Equation (8.87) is often referred to as the "biharmonic equation" with <p known as the "Airy stress function".

It is worth noting, at this point in the development, that the problem of plane strain, or plane stress, has been reduced to seeking a solution of the biharmonic equation (8.87) such that the stress components satisfy the boundary conditions of the problem.

Thus, provided that a suitable polynomial expression in jc and y (or r and 9) is used for the stress function cj> then both equilibrium and compatibility are automatically assured. Consideration of the boundary conditions associated with any particular stress system will then yield the appropriate coefficients of the various terms of the polynomial and a complete solution is obtained.

8.27.1. Forms of Airy stress function in Cartesian coordinates

The stress function concept was developed by Airy initially to investigate the bending theory of straight rectangular beams. It was thus natural that a rectangular cartesian coordinate system be used. As an introduction to this topic, therefore, forms of stress function in cartesian coordinates will be explored and applied to a number of fairly simple beam problems. It is hoped that the reader will gain confidence in using the approach and be able to tackle a range of more interesting problems where cylindrical polars (r, 9) is an appropriate alternative coordinate system.

(a) The eqns. (8.81) which define the stress function imply that the most simple function of <j> to produce a stress is <j> = Ax2, since the lower orders when differentiated twice give a zero result. Substituting this into eqns. (8.81) gives:

Thus a stress function of the form 4> = Ax2 can be used to describe a condition of constant stress 2A in the y direction over the entire region of a component, e.g. uniform tension or compression testing

For this stress function d2<)> 97

Thus is a linear function of vertical dimension y, a situation typical of beam bending.

(c) 4> = Ax2 + Bxy + Cy2. In this case d2<t>

dxdy and the stress function is suitable for any uniform plane stress state.

(d) <t> = Ax3 + Bx2y + Cxy2 + Dy3. Then d2<j) df


<7yy = = 6AjC + 2ß>' d2(p tw = —r-%- = -2Bx - 2Cy dxdy and all stresses may be seen to vary linearly with x and y.

For the particular case where A — B = C — 0 the situation resolves itself into that of case (b) i.e. suitable for pure bending.

For many problems an extension of the above function to a comprehensive polynomial expression is found to be rather useful. An appropriate technique is to postulate a general form which will adequately represent the applied loading and boundary conditions. The form of this could be:

+ Hx4 + i.rv + Kx2y2 + Lxy3 + My4 + Nx5 + Px4y + Qx'y2 + to2/ + Sxy4 + 7V + • • • (8.89)

Any term containing x or y up to the third power will automatically satisfy the biharmonic equation V4(<p) = 0. However, terms containing x4 or y4, or higher powers, will appear in the biharmonic equation. Relations of the associated coefficients can thereby be found which will satisfy V4(0) = 0.

Although beyond the scope of the present text, it is worth noting that the polynomial approach has severe limitations when applied to cases with discontinuous loads on the boundary. For such cases, a stress function in the form of a trigonometric series - a Fourier series for example - should be used.

8.27.2. Case 1 - Bending of a simply supported beam by a uniformly distributed loading

An end-supported beam of length 2L, depth 2d and unit width is loaded with a uniformly distributed load w/unit length as shown in Fig. 8.32. From the work of Chapter 4f the reader will be aware of the solution of this problem using the simple bending theory sometimes known as "engineers bending". Using this simple approach it is possible to obtain values for the longitudinal stress <ja and the shear stress rxy. However, the stress function provides the stress analyst with information about all the two-dimensional stresses and thereby the regions of applicability where the more straightforward methods can be used with confidence. The boundary conditions of this problem are:



? =



= 0

for all values of x,



y =


Cf y y

= — w

for all values of x,



y =



= 0

for all values of x.

wper unit length


wper unit length


Fig. 8.32. The bending of a simply supported beam by a uniformly distributed load w/unit length. The overall equilibrium requirements are: -

(iv) crxxy ■ dy = w(L2 — x2)/2 for the equilibrium of moments at any position x,

(v) f_d<Jxxdy = 0 for the equilibrium of forces at any position x.

The biharmonic equation:

To deal with these conditions it is necessary to use the 5th-order polynomial as given in eqn. (8.89) containing eighteen coefficients A to T.

0 0

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