A Treatise On The Principles And Practice Of Dock Engineering

STRENGTH OF COLUMNS. 383 The equivalent length of a column fixed at both ends is one-fourth of that described above, and if we substitute for and reduce to I, its value — -, 4 unit area, we obtain the following ultimate strength per square inch for a column of circular section, with radius r■. P=E^~(76) This formula (75), having originated with Euler, is known by his name. Its efficacy depends on three conditions, two of which, at least, cannot be guaranteed in practice, viz. :— (“) The uniformity of the modulus of elasticity (E) for all fibres throughout the section ; (ß) The absence of any initial deflection ; (7) The axial position of the load. Furthermore, it will be noticed that no allowance is made in the equation for the possible failure of the material by direct crushing, so that for short columns the calculated strength is greatly in excess of the compressive limit. Professor Claxton Fidler shows* that if the opposite sides of a pillar whose moduli of elasticity are Ej and E2 respectively, be subjected to the same amount of compressive stress, one side will be shortened more than the other in the proportion of - to g-, and, consequently, of the total deflection produced by a given load, one portion causes no difference of stress and, therefore, no moment of resistance, while the remainder alone is the measure of the real moment of resistance. The total deflection is found to be ei ~ P 2 ej + e2'p-p where q and e2 represent — and — respectively; p (=’2e’';) is the resilient force of the ideal column in pounds per square inch; r, in this case, is the radius of gyration, and p is the actual load-intensity. The bending moment AI = Ps, and the extreme stress in the fibres at a distance, y, from the neutral axis due thereto, is Foy = P^y I r2 (7«) Inserting the equivalent for 3 from (77) and giving to - . ^ . 81 ~ ^ 2 r ■ 61 + e2 its approximate value -4, the maximum compressive stress on the concave side of the column, /=p+/=pfl +—-) \ P - p) (79) * “Bridge Construction,” chap. x.; vide also Min. Proc. Inst. C.E., vol. Ixxxvi.