= = deflection. II. When the flanges are not equal, and the girder is not parallel, deflection. III. When the beam has no top flange, and the depth varies, = BWC = 163Ch. The formulæ given by Hurst, Handbook, &c. for finding deflection, which occur under Stiffness of beims, are, I. When supported at the ends and loaded in the middle, leet3 W cwts. C - deflection inches. II. For cylinders, 24 d or diam. inches" - deflection inches. 4binches d inches III. If the beam be fixed at one end and loaded at the other, the deflection = 16 times the product. IV. If fixed at one end and uniformly loaded, 6 times. V. If supported at both ends and uniformly load. d, gths. VI. If fixed at both ends and loaded in the middle, ¡th. VII. If fixed at both ends and uniformly loaded, ths. He gives the following: TABLE OF THE RELATIVE STRENGTH OF BODIES TO RESIST DEFLECTION = C. VIII. The deflection of a rectangular beam is to a cylindrical one, as 1 to 17. IX. When the deflection is taken as th of an inch per foot in length (which is considered to be safe under a proof of of the breaking weight) then for a beam supported at both ends and loaded in the middle, =W; &/bd = 1; = C; but for or 2C for XI. For cylinders, ✅/1-772WC = diameter. take ths, as before. PWC 8/12WC=d; = b; b bd3 C 2/bd3 C C 2 12 W bd3 XI. For an uniform load 1630i. The modulus of elasticity, or resistance of materials to stretching, is the term given to the ratio of the force of restitution to the force of compression. It is the measure of the elastic force of any substance. By means of it, the comparative stiffness of bodies may be ascertained. Thus from the following table it will be perceived that a piece of cast Iron is 10-7 times as stiff as a piece of oak of equal dimensions and bearing. Resilience, or toughness of bodies, is strength and flexibility combined; hence any material or body which bears the greatest load, and bends the most at the time of fracture, is the toughest. The modulus is estimated by supposing the material to present a square unit of surface, and by any weight or force to be extended to double. or compressed into one-half the original length; such a weight will represent the modulus. Table of the MODULUS OF ELASTICITY; WITH THE PORTION OF IT (LIMITING THE Cohesion OF THE MATERIAL, OR) WHICH WOULD TEAR THEM ASUNDER LENGTHWISE. 1 120x120 R, from Rankine, Civil Engineering. 163Ck. Hence, the modulus of elasticity being known for any substance, the weight may be determined which a given bar, nearly straight, is capable of supporting. For instance, in fir, supposing its height 10,000,000, a bar one inch square and 10 feet long may begin to bend with the weight of a bar of the same thickness, equal in length to .8225 x × 10,000,000 f、et=571 fect, that is, with a weight of about 120 lbs; neglecting the effect of the weight of the bar itself. If we know the force required to crush. a bar or column, we may calculate what must be the proportion of its length to its depth, in order that it may begin to bend rather than be crushed. (Gregory, p. 382.) 16301. For a rectangular beam supported at both ends and the weight applied in the middle, Gregory, p. 388, gives the formula deflection in inches in the middle. Here M modulus of elasticity in pounds; 7 length in feet; W weight in pounds; bbreadth and d depth both in inches. Fenwick, Mechanics of Construction, gives the formula Mor W 73 = deflection. Hre length is in inches; and I moment of inertia of the sectior, which for a rectangle, 1⁄2 bd3. 48 MI 422 13 W = 4 W B 1630m. As it may often be necessary to calculate the deflection for an arm from that of a beam, or vice versa, we notice the statement made by Barlow, edit. 1897, that “the deflection of a beam fixed at one end in a wall and loaded at the other, is double that of a beam of twice the length, supported at both ends, and loaded in the middle with a double weight.” But by his editor in 1851, the word double was altered to equal Certain experiments made by us on both the beam and the arm, tended to prove that the former was correct (Builder, 1866, p. 124); but scientific investigations show that mathematically the latter is correct; but as they mainly depend upon the perfect manner in which the tail of the arm is secured, the former, or double deflection, is recommended to be anticipated in practice. 1630. There is no such thing as permanent elasticity in any rigid material, and the only possible way of constructing a beam which will return to its original form after the load is removed, is a compound or trussed beam, put together in such a way that the permanent alteration of one material counterbalances that of the other. All beams, without exception, will settle in the course of time, even with the lightest load. Not only the load, but the changes of temperature afford a permanent cause of this sett ing. Facts on this point are difficult to obtain, as the experiments require to be extended over years, and on the same piece of material. Iron rods, one inch square, which may carry 60,000 lbs. before they are torn, stretch permanently by a load of less than 20,000 lbs. The best wrought iron cannot bear more than one-sixth of its load, without being permanently altered. These data apply only where the material is permanently at rest; if motion or accidental increase of burden happens, the above rules and numbers are considerably modified. As elasticity in material varies as much as its strength, and does not follow the same rules as cohesion, it is advisable to make experiments in each particular case where important structures are to depend upon the smallest quantity of material. (Overman). 16300. Impact or Collision. A second force, after direct pressure, is that of impact, says Fairbairn, involving a proposition on which mathematicians are not agreed. For practical purposes, we may suppose a heavy case equal to 2,240 lbs. or one ton, falling from a height of 6 feet upon the floor. According to the laws of gravity, a body falling from a state of rest obtains an increase of velocity in a second of time equal to 3 feet and during that period falls through a space of 16 feet. This accelerated velocity is as the square roots of the distances; and a falling body having acquired a velocity of 805 feet in the first foot of its descent, and 6 feet being the height from which a weight of one ton is supposed to fall, we have 6 x 8'05=2·449 × 8·05=19.714 for the velocity in a descent of 6 feet. Then 19.714 × 2240=44.159 lbs. or nearly 20 tons, as the momentum with which the body impinges on the floor. In the present state of our knowledge, this momentum may probably be taken as the measure of the force of impact." On the effects of impact, the deflections produced by the striking body on wrought iron are nearly as the velocity of impact, and those on cast iton greater in proportion to the velocity. The experiments and investigations made for the Commissioners on Railway Structures are extremely valuable. Their results showed that "the power of resisting impact increases with the permanent load upon the beam; the greater the weight at rest upon the beam, the greater must be the momentum of a striking body in order to break it. This is satisfactory, as it diminishes the risk from falling weights in warehouses: the more nearly the weight upon the floors approaches the point at which danger begins, the greater is their power of resisting sudden impacts. Comparing the mean results of the experiments on bars not loaded, "we find that the transverse is to the impactive strength as 2685: 3744, or as 1: 1:39. Similarly, when the bar is loaded with 28 lbs. in the centre, the transverse is to the impactive strength as 2685 4546, or as 1169; and when 391 lbs. is spread uniformly over the bar, the transverse is to the impactive strength as 2685: 5699, or as 1:2.12.”—(Fairbairn, p. 228). 1630p. Tensile strength is that power of resistance which bodies oppose to a sparation of their parts when force is applied to tear asunder, in the direction of their lengths, the fibres or particles of which they are composed. Tredgold's assertions of the principles have been combated by Gregory, to whose work we must refer the student for the reasons he gives. If a piece of No. 10 iron wire bears a tension of 2.000 lbs. before it breaks. ten wires will bear ten times 2,000 lbs. If the sections of 50 wires of this number, form the contents of one square inch, then it will bear a stress of 50 × 2000 lbs. before it is torn asunder, provided the wires are so arranged that each will carry its full weight. But it does not follow that a bar of wrought iron of one square inch will carry an equal weight. not even if the iron be of the same quality. If a solid iron rod of one square inch will carry 50,000 lbs., it does not follow that a rod of 10 square inches in section will carry ten times as much. When welded together, the capacity for resistance appears to be weakened. This observation applies to almost every kind of material, and varies only in degree. The tables of cohesion are generally computed to the tearing of the materiai, but our calcula tions should never go beyond the excess of elasticity, for fear of injuring the material, (Overman.) 1630q. If the strain upon a rod or strut be greatest on any one side, that side must sustain the whole force or break. This consideration is of great practical moment in estimating the value of all kinds of ties, as king and queen posts, &c.—(Tredgold). 1630r. The formula for the strength of tie-rods, suspension bars, &c. is C tons x area of section in inches W tons-a quarter to be taken for safe weight-or Clbs. x area of section in inches W lbs. . C being obtained from one of the columns in Tables I and 11. If the weight to be sustained be given, and the sectional dimensions of the bar be required. divide the weight given by one third or one-quarter of the cohesive strength, and the square root of the quotient will be the side of the square. If the section be rectangular, the quotient must be divided by the breadth. TABLE I., OF THE ABSOLUTE COHESIVE POWER (OR BREAKING WEIGHT) OF MEtals: Sectional area, 1 inch square, 1 foot in length. Tables I. and II. are derived chiefly from the summary in Rankine's Civil Engineering, p. 511, obtained from the experiments of Clay, Fairbairn, Hodgkinson, Mallet, Morin, Napier, Napier and Sons, Rennie, Telford, and Wilmot. Most of the remainder are from Rennie and other authorities. 1630s. English boiler-plates are stated to be of two classes: Yorkshire, and the manu. facture of other districts, classed as Staffordshire. TABLE II., OF THE ABSOLUTE COHESIVE POWER (OR BREAKING WEIGHT) OF THE TIMBERS Sectional area, 1 inch square, I foot in length. USUALLY EMPLOYED, (See 1628i.) 1630t. The tensile strength of cast iron was long very much overrated when Tredgold estimated it at 20 tons. Captain Brown, however, put it at 7:26 tons; G. Rennie (Phii. Trans. 1818) obtained 8:52 and 8.66 tons; Barlow conjectured at least 10 tons from theoretical principles; Hodgkinson made the following experiments more recently; } 6,400 R 5,600 Name and Quality of Iron. Low Moor, (Yorkshire,) No. 3, bore 6 tons. A mixture of irons, a mean of four experiments, gave 7 tons 73 cwt. 1630% The mean of several experiments on the ultimate cohesive strength of a wrought iron bar, 1 inch square section, was: No. 11, experiments by Captain Brown, gave No. 9 by Telford, Mean 61.768 lbs. = 27.575 tons. No. 3 experiments by Brunel, on hammered iron, gave 304, 323, and 30-8 tous 188 bars, rolled, experimented upon by Kirkaldy, 68.848 44,584 72 angle irons 57,555 63,715 37,909 54.729 62,544 37,474 50,787 | =21 60,756 32,450 46,171 f =25 =24 He states that 25 tons for bars and 20 tons for plates, are the amounts generally assumed |