LIMITS OF THE CENTRE OF PRESSURE. a с A 75. Let P be the total normal pressure upon any joint AB of a structure; 7 the length of the joint AB; p the pressure per unit of surface at the extrados A; p' the pressure per unit of surface für at the intrados B; and d the distance of the centre of pressure of the section from the extrados. B C Then if we assume with Navier and other Fig. 36. high authorities on this subject, but especially in reference to the arch, that the pressure on the joint varies uniformly from the intrados to the extrados, it will follow that the pressure at any point of the joint will be represented by an ordinate of a certain trapezoid, the extreme ordinates of the trapezoid representing the extreme pressures. The resultant of all the pressures on the joint will pass through the centre of gravity of the trapezoid ABCD, the pressure p at A being denoted by AD, and p' at B by BC; and, moreover, the total pressure P and the extreme pressures p and p' will be connected by the equation Let a, b, and c, be the projections on A B of the centres of gravity of the triangles BAD, BDC; and the trapezoid ABCD. Then, because the triangle ABD is to the triangle BCD in the ratio of AD to BC, the line ab is divided in c so that and Hence we get the proportions ab ac p + p' : p', abcb :: p + p : P ; from which we obtain by means of (1) the following values But cb Bc Bb = l − d — fl = l − d; ac = Ac - Aa = = Hence, knowing the values of d, l, and P, we shall be able to calculate the extreme elementary pressures p and p'. We see, then, that in proportion as P shall approach the edge B, the pressure p' will diminish, and that it will be reduced to zero for 3d; the trapezoid will then be a triangle, for BC 7 = will be zero. = Hence dl corresponds to the extreme limit of the centre of pressure of the joint AB in order that there may be a pressure at every point of the section. E A D 76. Suppose the pressure to be nothing at the intrados B (fig. 37), and to increase uniformly from B to the extrados A; then, by the preceding article, the pressure at any point along BA will be represented by the corresponding ordinate of the triangle BAD. Let d, the distance of the centre of pressure from the extrados, be less than AE, E being within the joint. Then the whole pressure of the joint comes upon AE. B Fig. 37. 17, so that d = In such a distribution of pressure along the joint as this, we see at once that the structure would be weakened. We hence conclude that for permanent constructions the centre of pressure of each joint must be kept as much as possible within the joint. The centres of pressure at the joints of a structure are sometimes called centres of resistance, and the line traversing all those centres of resistance has received from Mr. Moseley the name of the "line of resistance." 77. The following empirical formulæ for the limits of the centre of pressure of a joint in certain cases are given in Rankine's Manual of Applied Mechanics, p. 227 : Let q denote the ratio which the distance of the centre of pressure of a given plane joint from its centre of figure bears to the diameter or breadth of the same joint, measured along the straight line which traverses its centre of pressure and centre of figure; so that if d be that diameter, qd shall be the distance of the centre of pressure from the centre of the figure. Then the ratio q is found in practice to have the following values : In retaining walls designed by British engineers, q = §. CHAPTER II. ROOFS. 78. WE usually give the name of Roofs to the upper parts of a building designed to preserve the lower parts from the effects of rain; but in scientific works, the term is restricted to frames of wood or of iron, or to vaults, which support the covering of a building. We will first apply the principles of equilibrium to determine the straining forces in roofs, and thence deduce the formulæ for estimating the strength and dimensions of each part of a structure or frame constituting a roof, by the principles of resistance developed in the preceding chapters. Section 1. COMMON ROOFS WITH RAFTERS AND TIE-BEAMS. B Fig. 38. called a tie-beam. The horizontal thrust in this kind of beam is sustained by the tie-beam. 80. PROP. To find the compression or thrust along each of two rafters AB, BC, equally inclined to the horizon, and the horizontal thrust on the tie-beam AC (fig. 38). Denote the length of each rafter by 7, the equally distributed load upon each unit of length of the rafter (including the weight of the rafter) by p, and the inclination of each rafter to the horizon by a. Now the load pl being equally distributed along each rafter, will in general have its resultant passing through the middle of each rafter. Resolve pl into two equal and parallel components, each equal to pl, acting through the ends of each rafter. Wherefore pl is to be considered as directly supported at A, pl at C, and pl + pl = pl, at B; that is to say, the load pl acts at the joint B. Let P be the component of the load pl at B along each rafter; then, by the resolution of forces, or denoting the total weight of the roof and its covering by W = 2pl, This is the compression or thrust along each rafter. Hence, if H be the horizontal thrust along the tie-beam, HP cos a = pl cot a = W cot a (2). STRENGTH AND DIMENSIONS OF EACH RAFTER. 81. In this kind of roof, each rafter is subjected to a transverse strain, and also to a compression in the direction of its length. The moment of flexure being greatest (Art. 33) at the middle of the rafter, let S be the tension per unit of area on the section at the middle of the rafter produced by flexion, I the moment of inertia of this section about a horizontal axis through its centre of gravity, and x the distance of the extreme fibre of the section |