A Treatise On The Principles And Practice Of Dock Engineering

RESULTANT PRESSURE. 321 Another method of procedure, which has the advantage of including both diagrams in a single drawing, is as follows :— From one extremity, K (fig. 256), of the water-bearing surface of the leaf draw, perpendicular to the direction of the lieel-post reaction, a line, K 0, to intersect at O, the centre line of the passage, which itself is perpendicular to the mitre-post reaction. In this way a triangle, KLO, is formed, having its sides respectively perpendicular to the lines of action of the forces and therefore proportional to their magnitudes. And as P, the total water pressure, is measured by the length of the leaf, K L, multiplied by ^ , so the heel and mitre-post reactions are KOx — — and LO x — respectively and indifferently, for they are equal, as we have already seen. The reaction at any section ofthe gate can be obtained by drawing a line from O to that point of the water- bearing surface which lies on the section line in question. The length of this line multiplied by —g - gives the required reaction. It may be convenient to obtain an expression for the value of the heel- post, the mitre-post, and the sectional reactions generally, in terms of the Span and rise of the gates. This can be done with very close approximation as follows: by the span of the gates is to be understood the distance between the centres of the two heel-posts and by the rise, the vertical distance from this line to the centre of the abutting surfaces of the mitre- posts. From an inspection of fig. 256 it will be seen that the line HM joining the centre of the heel-post to the centre of the mitre meeting surface—that is, joining the extremity of the span to the extremity of the rise—is practically and sensibly parallel to the line K L which connects the extreniities of the water-bearing surface. Oonsequently we may imagine the angle H M T equal to the angle KLT without appreciable error. Then from similar triangles, H T M and O S L, we have OL = HM S L - MT’ Kow, S L = one-half the total water pressure on the surface of the leaf 4 ; and O L is to all intents and purposes a measure of the resultant pressure on any section—i.e., R. Again, H M is the length of the leaf minus the radius of the heel-post = I - p■, and M T is the rise of the gate = r. Substituting, we obtain „wh2 l(l - p) K— ----i---1 4r . (47) If we choose to neglect p, which is a very small quantity in comparison itli Z, and to substitute for I2 its component value 7 + r2, we arrive at an 21