A Treatise On The Principles And Practice Of Dock Engineering

ANALYSIS OF RESULTANT PRESSURE. 325 Now, let x be the point of application, in the curve of pressures, of the résultant, R, acting on the plane of section, L M. Then, if the angle between the line of action of R and the sectional line L M be designated ô, the force, R, may be resolved into two component forces—viz., R sin / parallel to the axis, A A, and R cos 6, at right angles to it. The former is a direct compressive stress and the latter a shear. By introducing two opposite forces at the point O, each equal to R sin / a step which in no way interferes with the equilibrium of the section, we may conceive the line of action of R sin 6 transferred to the axis, A A, provided that this change be taken in moment of which is R a: sin à, and which tends to turn in a clockwise direction. The section, L M, is thus subjected to the action of the following forces :— 1. A shearing stress, R cos Ô, along LM. 2. A direct compressive stress of uni- form intensity throughout the section, the total amount of which is R sin Ô. conjunction with a couple, the Fig. 260. 3. A bending moment, R æ sin 0, producing compression from 0 to L and tension from O to M, both these stresses varying in intensity and increasing with the distance from the neutral axis at O. The force R cos 0, being a simple shear, calls for no further comment. With regard to the force R sin 6, it will simplify the notation in the ensuing investigation if, from this stage, we symbolise it by the letter F. If the area of section be A, the uniform intensity of direct compression is F P —, and if the A vertical depth be taken as unity, it is where b is the breadth of section = L M. The intensity of stress at any point in the section due to the bending moment, Fis, may be obtained from the well-known relationship, M y where y is the distance from the neutral axis of the fibre sustaining the stress intensity, / M is the moment of resistance, equal to the bending moment, F x, and I is the moment of inertia. The greatest intensity of stress is manifestly that in the outermost fibres, at L, where the maximum compressive effect of the bending moment is added to the direct compression. Indicating the distance, O L, by the letter p, and taking the depth as unity, the intensity of stress due to direct .compression is