STRENGTH OF MATERIALS 417 This formula is frequently designated as Gor- don's, having been deduced by Gordon from Hodgkinson's experiments. Multiply the value of #, as given in the table, by 4 for columns rounded or jointed at both ends, and by 2 where fixed at one end, rounded at the other. Connecting rods of steam engines are calcula- ted as pillars rounded at both ends. Piston and pump rods are considered as fixed at one end, free at the other. 11. The collapsing of boiler flues was made the subject of a series of ex- periments by Mr. Fairbairn, and the following formula was deduced: P=806,000 ~, where P = collapsing pressure in pounds per square inch, t = thickness of iron in flues, L = length of flue in feet, and d = its diameter in inches. When the flue is strengthened by angle-iron rings, as is sometimes done with long flues, L is taken as the distance between the rings. This formula has not been verified for short flues of great diameter, or for exceptional pro- portions. A slight deviation from a truly cy- lindrical form considerably reduces the strength of the flues; t* is generally taken instead of 2 ' 19 . Elliptical flues, having a major diameter a and a minor diameter 5, are of equal strength with a cylindrical flue of the diameter 2. 12. The transverse strength of beams may be calculated by the following formulas : _KM 2 KA.d for beams fixed at one end and ~ ~L~~ * ~~T~ loaded at the other. WKM 2 <yKA<2 vhere fixed at one end and uni- =2 - L - and W=2-j-' formly loaded ,i W -19 KA ^ where ft* 6 * 1 at both ends and 12-j- uniformly Joaded Here TV = breaking weight in pounds, K = a coefficient which varies with every change in form of cross section of the beams, d = depth of beam in inches, 5 = breadth in inches, A = area of cross section of the beam at point of rupture in square inches, and L = length be- tween supports in feet. The values of K given in the table, where the beams are of rectangu- lar section, fixed at one end and loaded at the other, are obtained from various sources. 13. For other than rectangular sections the follow- ing may be taken as the values of K for cast iron : Shape, ; value, K = 500. Shape, TT, equal flanges ; value, K = 520. Fairbairn, ^_ ; value, K = 580. Hodgkinson, T ; value, K = 850. The following values are given for wrought iron: rolled rails, * , 600; Fairbairn's riveted beam, , 900; box beam, Jj, 1,000. 14. For the wrought-iron beam, when supported at both ends and uniformly loaded, the formula W^- is used by some American manufacturers. D= depth in feet; a= area of flange in inch- es, a= that of "stem" or web; S= stress per square inch of area, +^, in tons. The .006WL3 deflection, S=/ / , where the load is ap- - plied at the middle, and S'=7 ;T when ( 0+ T> 2 applied uniformly. The depth D is measured between the centres of gravity of the flanges. In such beams it is customary to allow as maxima 10,000 Ibs. per square inch in ten- sion and 6,000 to 8,000 in compression. De- flection should not exceed -fa of an inch per foot of length, in any structure. 15. Torsional strength is computed by the for- mula W = S'^ ; D = 1/^?; where W = rt B' breaking weight in pounds, D = diameter of shaft in inches, and R = length of lever arm in feet. The coefficient S' is very nearly pro- portional to the tenacity of the material, where the torsion is equal in degree. 16. Resilience is a term introduced by Dr. Young. It is measured by the amount of work performed in producing the maximum strain which a given body is capable of sustaining, and is the quality of primary importance where shocks are to be sustained. Mallet's coefficient of re- silience is the half product of the maximum resistance into the maximum extension. But for tough metals it is equal approximately to two thirds the product of the ultimate strength of the material by the distance through which the body yields before the straining force. For very brittle materials it is measured by half that product. No material can resist the shock of a body in motion, unless it is capable of offering resilience equal to the amount of work performed in setting that body in motion at the given velocity ; i. e., equal to the amount of energy stored in the moving mass at the in- stant of striking. In predicting the effect of shock, therefore, it becomes necessary to know the amount of energy stored in the moving body and the resilience of the resisting material. To meet a violent shock successfully, resilience, rather than mere strength, must be secured. As an instance, it is found that wrought iron of comparatively low tenacity but great tough- ness, capable of stretching considerably before fracture, is far superior to steel for armor for iron-clad ships; the latter has much greater strength, but also greater brittleness. Such calculations are not usually made in designing. Immunity from the injurious effect of shock is
