A Treatise On The Principles And Practice Of Dock Engineering

i6o DOCK ENGINEERING. “ The Stability of Loose Earth,” contributed to the Philosophical Transac- tions of the year 1856, viz. : —“The resistance to displacement, by sliding along a given plane, in a loose granulär mass, is equal to the normal pressure exerted between the parts of the mass on either side of that plane, multiplied by a spécifie constant.” The restriction renders the theory somewhat defective in its relationship to ordinary revetment walls with well-consolidated backing, but it is nevertheless apparent that any Fig. 81a. calculations made on this basis will err only on the side of excessive strength. Starting with a definition of conjugate stresses as a pair of stresses acting upon two planes supposed to traverse a point in a body, such that each stress is parallel to the plane upon which the other acts, and, futher, dis- tinguishing as principal stresses those stresses which are mutually normal, * we may go on to show that there are three cases in which the intensity and direction of the resultant stress can be deter- mined, viz. :— 1. When the principal stresses are of the same kind—i.e., either both positive (compressive) or both negative (tensile), with equal intensities. 2. When, with equal intensities the stresses are not of the same kind ; and, 3. When the stresses are of either kind, but with unequal intensities. Case I. —The resultant stress must clearly be of the same kind as the principal stresses, and have an intensity equal to that of either of them. In fig. 82, A B and B C are planes upon which two principal stresses, P and Q, are supposed to act. Since these are, by hypothesis, equal in intensity, heir magnitudes will be proportional to the sides, A B and B C, respec- tively. If, then, from the point of intersection we set off O X to represent P = p x A B, and O Y to represent Q = q (or p) x B 0, O Z will give the magnitude and direction of the resultant, R. Since the triangles, ABC and O X Z, are similar, it follows that R is perpendicular to the plane, A C, and is proportional to the side, A C (i.e., R = rx A 0), and, therefore, that the intensity of pressure of the resultant is equal to the intensity of each of the principal stresses, which is equivalent to stating that r = p - q. Case II.—When the sense of one of the principal stresses is altered, the intensities remaining equal, the effect is to change the direction of the resultant, but not its amount or intensity. In fig. 83 the principal stresses are P and Q, as before, but the sense of P is inverted. By a construction * If two planes, X X and Y Y, be supposed to traverse a point, 0, in any body, and if the direction of the stress, p, on the plane X X be parallel to the plane Y Y, then the direction of the stress, q, on the plane Y Y is parallel to the plane X X, and the two stresses are said to be conjugate. When X X and Y Y are at right angles the stresses become principal stresses (fig. 81a).