The Shear-difference method is a conventional method of determining the magnitudes of the principal stresses at a point. It presupposes a complete knowledge of the directions of the principal stresses at all points as well as the magnitudes of the principal stress differences (both of which may be determined from photo-elastic model studies). The iterative method, on the other hand, can be used if only the magnitudes of the principal stress differences (fringes) are known. The accuracy of the method depends primarily on the determination of fringe orders by interpolation. Basically, this method is a numerical solution of Laplace's equation, and, in the limit, it yields values for the sum of the principal stresses at all points considered. This information, together with the values of the principal stress differences at the same points, allows one to determine the principal stresses. While solutions of this type are time consuming if carried-out longhand, it has been found that the use of an I.B.M. 7040 computer greatly simplifies the procedure. For this reason we are now using the method as a standard laboratory procedure. Note- In this paper we use the following conventions: P is always the major principal stress, and Q the minor. Compressive stresses are positive, and tensile stresses are negative. The principal stress difference, (P-Q), is always positive since it is the difference between major and minor principal stresses. The sum of the principal stresses, (P+Q), may be positive or negative depending on the boundary conditions.
Providing that a model plate is subjected to plane stress it may be shown that the sum of the principal stresses at any point is a harmonic function, that is, it satisfies Laplace's Equation. Because of this, the function is uniquely determined at all points in a region providing that the boundary stresses are known. Furthermore, if a square grid is superimposed on the model plate, it may be shown that the value of the function at the centre of any square is a weighted average of the values at the mid-points of the sides (the weighting factors depending upon distances from the central point). The equation used is called the four-point influence equation (Figure 1), and an iterative process is used to obtain values at each point which satisfy it. For a complete treatment of the iterative process the reader is referred to several standard books (1, 2). At any point on a free boundary one principal stress (normal to the boundary) must be zero. Thus, the value of the sum of the principal stresses and, disregarding signs, the value of the principal stress difference will be equal at such points. While the applied loads are known, error may be present in estimating the principal stresses on loaded boundaries. The major source of error is due to the unequal distribution of the load. For the present study, rubber strips between the model and the platens made it possible to apply loads quite evenly.
