STRENGTH OF MATERIALS 419 case, and triangular in the second. Beams of uniform depth, supported at the ends and load- ed at the middle point, are in plan a pair of triangles with a common base at the load. If uniformly loaded, the plan is a pair of parabo- las with their bases at the middle of the beam. When supported at the ends and uniform in breadth, they are in vertical section a pair of parabolas, in the first case with vertices at the ends and bases meeting at the load, and in the last case semi-ellipses extending between the points of support. In building bridge girders, economy of material is secured by the use of isosceles bracing set at angles of 45. In ver- tical and diagonal bracing, the proper angle for diagonals is 55 measured between the diago- nal and the vertical. The amount of resistance of a cylinder to rupture by torsion is nearly double that to breaking across. Bolts exposed to shocks and sudden strains, as when used as armor-plate fastenings, are found to resist much more effectually where resilience is se- cured by turning down the shank to the diam- eter of the bolt at the bottom of the thread, or otherwise creating a uniform area of section between head and nut. Punching rivet holes weakens plates of hard iron and steel. The latter are injured so seriously that steel plates are never punched by careful engineers. (See STEEL.) In hard iron the reduction of strength is often considerable (15 per cent, as shown by some experimenters) ; and in many cases, in boiler work, for this reason, the rivet holes are all drilled, notwithstanding the increased cost. Where the iron is very soft and ductile, punch- ing produces less injury. 22. Elasticity is that quality by the possession of which the strain, or distortion of form, produced in any body by stress, is wholly or partially removed on the re- moval of the stress. All bodies have more or less elasticity, and, when perfectly homogene- ous and free from internal strain, are perfectly elastic within a certain limit, which is called the limit of elasticity. Within this limit, the displacement produced by any force is directly proportional to that force. Beyond the limit of elasticity, the strain produced by stress is not wholly removed on the cessation of the stress. The permanent change of form so produced is called the " set." This set takes place on the application of the slightest force where the material is not uniform in character and free from internal strain. Hodgkinson found that in iron, far within the elastic limit, the lightest loads produced slight set. Beyond the elastic limit the set becomes nearly proportional to the distortion, the resistance also increasing up to the point at which rupture begins, but in a far higher ratio. Repeatedly straining a piece beyond its elastic limit produces " fatigue " and ultimate fracture. This may occur by the ap- plication of force far less than that producing immediate rupture. 23. The modulus of elas- ticity, sometimes called the coefficient of elas- ticity, is the quotient obtained by dividing the measure of the force producing distortion by the measure of the distortion produced by it. Its value varies with every material. The or- dinary values of the modulus are given in the table. _ These values, as is proved by auto- graphic strain diagrams, are liable to variation, within ve.ry wide limits, by every circumstance which affects the physical character of the ma- terials. It has no fixed relation to the ulti- mate strength. It will be seen that this quan- tity may be defined as the measure of that force which, supposing no limit to elasticity, would shorten or lengthen a bar, originally a unit in length, to the extent of one unit. Thus, a bar of ordinary forged iron, one foot long, would be altered in length ^^ by a force equal to UflJ^AfA =2500 Ibs. per square inch of section. 24. Testing Machines. The strength of materials is determined by means of testing machines. 25. Fig. 1 represents a machine for determining longitudinal resistance, as built by the Messrs. Riehle of Philadelphia. It con- sists of a weigh-beam, accurately made and FIG. 1. Kiehte's Longitudinal Testing Machine. nicely poised upon knife edges. At its outer end it sustains a scale pan upon which weights measuring 2,000 or 4,000 Ibs. are placed. In- termediate weights are measured by a poise, not shown in the figure, which traverses the beam, the latter being divided into parts of 10 Ibs. each, similarly to the steelyard bal- ance. The specimen is secured at the upper end by wedges or clamps, in a strong collar which is hung from two knife edges, one on each side the knife edge carrying the scale beam. These knife edges are placed at slightly differ- ent distances from the knife edge supporting the beam, thus making the latter a " differen- tial lever," and permitting the measurement of a very great force without compelling the use either of large weights or of a series of levers. A similar collar below takes the lower end of the specimen to be tested. This second collar is secured to the head of a hydraulic press which is placed within the lower part of the frame of the machine. A small pump, worked
