A Treatise On The Principles And Practice Of Dock Engineering

RESULTANT PRESSURE. 319 assumed to act through the axis of the heel-post. These three forces being in equilibrium, the triangle of forces, ors (fig. 254), can be drawn, having its sides parallel to the forces, P, Rp and R2 re- spectively, and, since the magnitude of P is R— ^ „ known, the magnitudes of the other two may readily be determined. TV- If, now, it be required to find the position ’Vf / and amount of the resultant stress across any /^ section, A B (fig. 253), of the gate, we proceed À// as follows: — Join the point, A, to each of the r Fig 254 two extremities, K, L, of the water-bearing surface of the leaf; bisect these lines at U and V respectively, and draw perpendiculars to represent the total water pressure on each section. Each section is in equilibrium under the action of these forces: in one case, the water pressure, the heel-post reaction, and the stress across AB; in the second case, the water pressure, the mitre-post reaction, and also the stress across A B, acting, of course, in the opposite direction. Since this stress must have the same line of action in both cases and must pass through the points of intersection of each of the other pairs of forces in order to fulfil the conditions of equilibrium, we have obviously only to join the two points of intersection to obtain the line of action of the resultant pressure at the section A B. Its magnitude may be determined by drawing a parallel line in the force diagram from the point, 0, and complet- ing the triangle by drawing the line representing the water pressure on either surface of the gate. Thus, in fig. 254, the mitre-post reaction being already determined, rq is the water pressure on the surface of the gate between the point, A, and the mitre-post, and 7 o is the stress across the section A B. Similarly, for the heel portion, a line, qs, might have been drawn parallel to the water pressure on that section. Thus the point, q, is not only obtained, but confirmed. By taking a series of sections in this way, it will be found that the locus of the point q is sensibly the arc of a circle, and therefore that, except perhaps in the case of very flat gates, the resultant pressure is so nearly constant as to be justifiably considered so without serious error. Also, it will be found that the line of pressure is a circular curve. This is perfectly true for all gates which present the form of a continuous arc when closed. It is also approximately and practically true for all segmental gates varying between the straight line and the continuous arc, provided the versed sine or rise of the gates (T M, fig. 253) do not exceed one-fifth of the span. For a greater ratio of rise to the span the divergency of the line of pressure from the circular arc begins to be appréciable, and ultimately, in the case of the flatter gates, becomes very marked, so that it is necessary to find by trial a series of points through which the curve may be drawn. Fig. 255 shows the curves of pressure in a segmental gate for a central reaction at the mitre- post and also in case of nipping on the inner or outer edges of the mitre-post.