2. If the solid be of the form shown, then calling 6' the internal dimension in the same direction as 6 and A' the internal dimension in the same direction as ; then, n F L = — 6 h ; • awl f = E retaining the preceding notation.
3. In a lamm of the section in the margin, calling 6' the sum of the two deficiencies from the full section, and h their 4 P L height, then also, r L = 6 A ; andf = r ohs 4. The section being circular, and the radius:sr ; then the formulas become r L = 4 P and f= ; when the section is circular and 4, hollow, calling r the external, and r' the internal radius, the formula: become FL = and f = 41't3 4r 5. If now the beans under consideration be carried at a point in its length, and ho acted upon at its extremities by two forces which balance one another upon this point of support, the formula for a pris matic beam of a rectangular section becomes,—(calling m the leverage of the force acting at one end, and n the leverage of the force acting on the other; so that in -sn =L the total length of the prism, and)) a pm+ qn = e the total load ;) : and if m = n or if the pointPL n6/0 of support be in the middle of the length, then 4= 6. If the load applied to a prism, fixed at one of its extremities, iustead of being applied at the other extremity, be evenly distributed over its length, then the load per unity of length being called p, the total load becomes pL ; and the leverage of the total load pr.; the 2 formulte expressing the fundamental conditions of resistance become psa_ nr ; and , retaining the preceding notation for the old terms. From these formulae, it appears that a beam loaded uniformly over its whole length, can resist an effort which would be double the one required to break it if applied only at the extremity the farthest removed from the section of rupture; and in order to produce equal deflections at the extremity, the load, distributed evenly over the whole length, should bear to the load applied solely at the extremity, the proportion of S : 3.
7. In the case of a beans resting upon two points of support at its extremities, if we suppose that the weight of the beam itself can be neglected, and that the weight, or load, P, be placed on the middle of its length ; then as the effect upon the beam would be the same as if it had been fixed in the middle, and loaded at each end, by a weight = the first set of would apply, excepting that P would be re 2 placed by F. E and L by ; to that for a rectangular prism the formulm F. 2 FL become and f = From this it. appears that a beam 4 a supported at the extremities, and loaded iu the middle, is able to support a load four times as great as a similar beam fixed at one end and loaded in the middle ; and that the deflection would be sixteen times less than in the latter case. If the load, instead of being con centrated in the centre, were evenly distributed over the length, and the load per unity of leugth be called p, the total load would be and the formulae would become and f — , 8 'IL 384E1 It has been proved, both theoretically and practically, that a beam fastened at both ends will bear a load applied in the middle of its clear spay, which would be double the one it would be able to support if it merely rested upon two points of support, and that the deflection in the former case would be four times less than it would be in the latter. As it generally happens that iu building operations, beams
and joists have bearings of only about one foot (which is insufficient to constitute an effective fixing of their extremities), it is esseutial to calculate their dimensions on the supposition that they arc merely Learns resting on two points of support.
Before closing these remarks on compression and extension, it may be desirable to add a few practical observations on the resistance of materials used in buildings. These are, 1st, that as the form of greatest resistance is almost always one in which the mass of the elements is concentrated at the top and bottom of a beam, leaving the portion about the neutral axis as light as possible, it follows that with plastic materials it is advisable to make the sections of the beam of a girder shape, that is to say with top and bottom flanges : the material must be distributed in those flanges according to its powers of resist ance to efforts either of compression or of extension. [Gummi 2nd, It is usually considered that a load acting with a shock, or able to produce sudden vibrations, acts in a manner far more injurious than if the same load wore to act steadily ; and iu practice engineers have adopted the rule of never exposing a construction to a rolling weight grater than of the permanent breaking weight. 3rd, It is usually considered that a deflection of of the span, is the maximum which should be tolerated ; but that the safe deflection would only be of the span. 4th, The resistance of cast iron to compression is, com pared to its resistance to extension, as 6I (nearly) to 1; on the contrary the resistance of wrought iron to compression is, compared to its resistance to extension, as 4 to 5.
Torsion. — With respect to the resistance to torsion it may he observed, that in a prismatic body submitted to such an action, the relation of the effort to the angle of torsion is constant for the steno material, so long as the limits of elasticity are not exceeded. Calling this relation c; the effort Q; and the angle of torsion 0, for a rod of a given unity of length and of section ; we have! = 0, which may be 0 called the co-efficient of torsion. If then, we call r the force tending to twist a cylindrical or prismatic solid, in a plane normal to the axis; it, the radius of the leverage with which r acts ; a the angle of torsion ; t, the length of the solid ; and moment of polar inertia ; we have the moment of the force P = r a = LI ; and from this we derive t L