and therefore the upward supporting force at the other end of the beam, D, which is also the shearing force at E, is given by the equation w (c+x)3 .(1.) = It has already been shown, in Article 290, that the shearing force at a given cross section with an uniform load is F wx; hence the excess of the greatest shearing force at a given cross section with a partial load, above the shearing force at the same cross section with an uniform load of the same intensity, is At the ends of the beam this excess vanishes. At the middle, it consists of the whole shearing force F = 1 wc, or one quarter of the shearing force at the ends; that is, one-eighth of the amount of an uniform load. 314. Allowance for Weight of Beam. -When a beam is of great span, its own weight may bear a proportion to the load which it has to carry, sufficiently great to require to be taken into account in determining the dimensions of the beam. Before the weight of the beam can be known, however, its dimensions must have been determined, so that to allow for that weight, an indirect process must be employed. As already explained in Article 302, the depth of a beam is determined by the deflection which it is desired to allow; and the breadth remains to be fixed by conditions of strength, the strength being simply proportional to the breadth. W Let b' denote the breadth as computed by considering the external load alone, W'. Compute the weight of the beam from that B' provisional breadth, and let it be denoted by B'. Then is the proportion which the weight of the beam must bear to the entire or W' gross load which it is calculated to support; and is the proportion in which the gross load exceeds the external load. Consequently, if for the provisional breadth b' there be substituted the exact breadth, W-B' WEIGHT OF BEAM-LIMITING LENGTH. 347 the beam will now be strong enough to bear both the proposed external load W', and its own weight, which will now be In the preceding formulæ, both the external load and the weight of the beam are treated as if uniformly distributed-a supposition which is sometimes exact, and always sufficiently near the truth for the purposes of the present Article. 315. Limiting Length of Beam.-The gross load of beams of similar figures and proportions, varying as the breadth and square of the depth directly, and inversely as the length, is proportional to the square of a given linear dimension. The weights of such beams are proportional to the cubes of corresponding linear dimensions. Hence the weight increases at a faster rate than the gross load; and for each particular figure of a beam of a given material and proportion of its dimensions, there must be a certain size at which the beam will bear its own weight only, without any additional load. To reduce this to calculation, let the gross working uniformlydistributed load of a beam of a given figure, as in Article 295, be expressed as follows: 1, b, and h being the length, breadth, and depth of the beam, ƒ the limit of working stress, and n a factor depending on the form of cross section. The weight of the beam will be expressed by w' being the weight of an unit of volume of the material, and k a factor depending on the figure of the beam. Then the ratio of the weight of the beam to the gross load is which increases in the simple ratio of the length, if the proportion h is fixed. When this is the case, the length L of a beam, whose weight (treated as uniformly distributed) is its working load, is This limiting length having once been determined for a given class of beams, may be used to compute the ratios of the gross load, weight of the beam, and external load to each other, for a beam of the given class, and of any smaller length, l, according to the following proportional equation: L::L-2: W: B: W-B...............(5.) To illustrate this by a numerical example, let the beams in ques1 tion be plain rectangular cast iron beams, so that n = k = 1, 6' 0-257 lb. per cubic inch; let 40,000 lbs. per square inch be taken as the modulus of rupture, and 4 as the factor of safety, so го that f= 10,000 lbs. per square inch; and let h 1 L= 3,459 inches 288 feet, nearly. Then 316. A Sloping Beam, like that represented in fig. 68, Article 142, is to be treated like a horizontal beam, so far as the bending stress produced by that component of the load which is normal to the beam, is concerned. The component of the load which acts along the beam, is to be considered as producing a direct thrust along the beam, which is to be combined with the stress due to the bending component of the load. 317. An Originally Curved Beam, at any given cross section made at right angles to its neutral surface, so far as the bending stress is concerned, is in the same condition with an originally straight beam at a similar and equal cross section to which the same moment of flexure is applied. Beams are sometimes made with a slight convexity upwards, called a camber, equal and opposite to the curvature which the intended working load would produce in an originally straight beam. The effect of this is to make the beam become straight under the working load, instead of curved, and to diminish the additional stress due to rapid motion of the load, which additional stress arises partly from the curvature of the beam. 318. The Expansion and Contraction of Long Beams, which EXPANSION AND CONTRACTION OF BEAMS. 349 arise from the changes of atmospheric temperature, are usually provided for by supporting one end of each beam on rollers of steel or hardened cast iron. The following table shows the proportion in which the length of a bar of certain materials is increased by an elevation of temperature from the melting point of ice (32° Fahr., or 0° Centigrade) to the boiling point of water under the mean atmospheric pressure (212° Fahr., or 100° Centigrade); that is, by an elevation of 180° Fahr., or 100° Centigrade : (The expansibilities of stone from Adie's experiments.) (Expansion along the grain, when dry, according to Dr. Joule, Proceed. Roy. Soc., Nov. 5, 1857.) Dr. Joule found that moisture diminishes, annuls, and even reverses, the expansibility of timber by heat, and that tension increases it. 319. The Elastic Curve, in the widest sense of the term, is the figure assumed by the longitudinal axis of an originally straight bar under any system of bending forces. All the examples of the curvature, slope, and deflection of beams in Article 300 and the subsequent Articles, are cases in which the elastic curve has been determined with a degree of approximation sufficiently close under the circumstances; that is, when the deflection is a very small fraction of the length. The present Article relates to the figure of the elastic curve for a slender flat spring of uniform section, when acted upon either by a pair of equal and opposite couples, or by a pair of equal and opposite forces. The general equation of Article 300 applies to this case, viz.: I being the uniform moment of inertia of the section of the spring, E the modulus of elasticity, M the moment of flexure at a given point, and r the radius of curvature at that point. When a spring is under the action of a pair of equal and opposite couples applied to its two ends, then, as in Article 304, M is constant, r is constant, and the elastic curve is a circular arc of the radius r. When a spring is under the action of a pair of equal and opposite forces, let A and B denote the two points to which those forces are applied, and A B their common line of action. The figures from 146 a to 146 f, inclusive, represent various forms which the spring may assume, viz.:— I. When the forces are directed towards each other— |