= Let f the maximum compressive stress on the material. due to both direct and bending stresses; f = the maximum tensile stress on the material due to both direct and bending stresses; f C A the skin stress due to bending ; the compressive stress acting all over the section due to the weight W; the sectional area of the column. Then ff + C Load If the column also carries a central load W1, the above become Form of fracture for excentric leading FIG. 461a. Columns loaded thus almost invariably fail in tension, therefore the strength must be calculated on the ƒ basis. We have neglected the deflection due to loading (Fig. 462), which makes matters still worse; the tensile stress then becomesW(X +8) __ W + W1 = A The deflection of a column loaded in this way may be obtained in the following manner :— After the column has bent, the bending moment of course is greater than WX, and approximates to W(X + 8), but & is usually small compared with X, therefore no serious error arises from taking this approximation. The author knows of an instance of a public building in which a column is loaded as shown in Fig. 463; the deflections given were taken when the gallery was empty, and no wind on the roof. The deflections are so serious that when the gallery is full, an experienced eye immediately detects them on entering the building. The following test of a column by the author will serve to emphasize the folly of loading columns in this manner. Thus we see that the column failed by tension in the material on the off side, i.e. the side remote from the load. The discrepancy between this and the tensile strength is due to the bending formula not holding good at the breaking point, as previously explained. CHAPTER XIV. TORSION. GENERAL THEORY. LET Fig. 464 represent two pieces of shafting provided with W FIG. 464. disc couplings as shown, the one being driven from the other through the pin P, which is evidently in shear. Now consider the case in which there are two pins, then W/ Sy + S11 = ƒ‚ay +ƒ„a1у'ı = FIG. 465. W The dotted holes in the figure are supposed to represent the pin-holes in the other disc coupling. Before W was applied the pin-holes were exactly opposite one another, but after the application of W the yielding or the shear of the pins caused a slight movement of the one disc relatively to the other, but shown very much exaggerated in the figure. It will be seen that the yielding or the strain varies directly as the distance from the axis of revolution (the centre of the shaft). When the material is elastic, the stress varies directly as the strain ; hence Substituting this value in the equation above, we have — Thus the inner pin, as in the beam (see p. 356), has only increased the strength by Now consider a similar arrangement with a great number of pins, such a number as to form a hollow or a solid section, the areas of each little pin or element being a, a, a, etc., distant y, y', y, etc., respectively from the axis of revolution. Then, as before, we have- WI = {(ay2 + a.y;" + a.y;2 +, etc.) But the quantity in brackets, viz. each little area multiplied by the square of its distance from the axis of revolution, is the polar moment of inertia of the section (see p. 77), which we will term I. Then W/= = The W/ is termed the twisting moment, M,. f. is the skin shear stress on the material furthest from the centre, and is therefore the maximum stress on the material, often termed the skin stress. y is the distance of the skin from the axis of revolution. |