TABLE 118.-Of the DEFLECTION and PERMANENT SET of BARS of BLAENAVON CAST IRON, 1, 2, and 3 inches square-continued. (773.) In the Diagram, Fig. 219, these reduced deflections and permanent sets are represented graphically, and show clearly the superior strength and stiffness of small castings. It is important to observe, however, that the permanent set with the safe load of 3rd the breaking weight is about the same with all the bars: thus at A or 3rd the breaking weight of the 1-inch bar, the set is about the same as at B, which is about 3rd the breaking weight of the reduced 2 and 3-inch bars. This shows that the Rule (767) applies to bars of all sizes. That defect of elasticity increases with the size of the casting is manifested in the Diagram by the permanent set and the deflections increasing therewith: it is also shown by the high ratio of the two to one another, which also increases with the size of the bar, as shown by Table 118. Thus the ratio of the ultimate set to the ultimate deflection is with the 1-inch bar 0879 ÷ 01725 5.096: with the 2-inch bar 070480178 = 3.96 and with the 3-inch bar 0761702728 = 2.793. Again: taking from col. 5 the same load as nearly as possible on the reduced 1, 2, and 3-inch bars, we have say 1009, 1020, and 1050 lbs., and might expect the Permanent sets in col. 7 to be all alike, but so influential is size on the strength, and thereby on the set, that col. 7 gives 003078; 007883 and : 01151, or in the Ratio 1.0; 2.56, and 3.74. The fact is that although the loads were nearly identical in actual amount, they were very different in their Ratio to the respective strengths of the bars with the 1-inch bars the load was 10091980·51, or 51 per cent. of the breaking weight: with the reduced 2-inch, 10201372 = 74, or 74 per cent.: and with the reduced 3-inch bars 10501342·78, or 78 per cent. The Permanent Set increasing very rapidly as the Breaking weight is approached, that fact tells on the results, as we have seen. CHAPTER XX. ON IMPACT. (774.) "General Principles.”—The power of bodies to resist an impulsive strain or a blow is second only in importance to their power in bearing a statical load or dead weight. Many of the forces which the Engineer has to deal with in practice are dynamic ones, or forces in motion, and as the laws governing the strength of Materials in that case differ entirely from those for a statical force it will be necessary to investigate those laws somewhat exhaustively. We will take first the case of impact on BEAMS, not only because it is the most important but also as giving the greatest facility for the illustration of general principles. It will be expedient in explaining the theory and leading facts, to take first a very light beam or a case in which the weight of the beam itself is so small in proportion to the strength that it may be neglected without sensible error. The results we thus obtain may be afterwards modified for practical cases in which the weight of the beam or the load upon it is considerable and its effect influential. (775.) Let A in Fig. 186 be a beam or elastic spring, say of steel, fixed at one end, and let its elasticity be such that it deflects 1 inch for each pound at the end, then with W = 10 lbs. as in the figure, the deflection will be 10 inches, or from B to C. Now, a certain amount of power has to be expended in thus bending the beam, which may be expressed in inch-lbs., or pounds falling 1 inch. At first sight it might appear that in our case we have 10 lbs. falling 10 inches, or 10 x 10 = 100 inch-lbs., but the strain or bending weight is not uniform throughout the fall;-at first it is 0 and increases from B to C in arithmetical ratio from 0 to 10, hence the mean weight is evidently (0+10) 25 lbs. which, falling 10 inches, gives 5 x 1050 inch-lbs. as the power required to bend the spring 10 inches. It follows from this, that if the weight instead of being placed steadily on the beam were allowed to fall through the 10 inches deflection, we should require only 5 lbs. to deflect the beam from B to C, instead of 10 lbs. dead weight. If, therefore, we place 5 lbs. at B and suddenly release it, the spring would be deflected by that weight to C, the point due to a dead weight of 10 lbs. ; or, if on the other hand the weight of 10 lbs. had been suddenly released at B it would have deflected the beam not to C only, but to D, 20 inches below B; or in other words, it would have strained the spring to an extent double that produced by a similar dead weight. That this is a fact is proved and illustrated by the experiments of Captain James, R.E., given in Table 119, the mean ratio of the 14 experiments is 1 to 1.938, or very nearly double as given by theory. The same effect would be produced by any weight less than 5 lbs., if the height of fall were proportionally increased; thus a weight of 1 lb. falling 50 inches from E to C would deflect the beam as before from B to C. It should be observed that the height of fall must be measured from C, the point to which the beam is deflected, and not from B, the point where the falling weight first strikes the Beam. RESILIENCE. (776.) The Power of a beam in resisting Impact has been termed its "Resilience," and it will be seen from the foregoing investigation that it may be expressed by taking half the product of the deflection by the weight producing it: hence we have the general Rule: TABLE 119.-Of the DEFLECTIONS by given DEAD LOADS on CAST-IRON BARS, compared with those produced by the same Loads, laid on the Beam and suddenly released. The mean Ratio from the 14 sets of experiments is 1.94 to 1: theory giving 2 to 1. In which the deflection, say in inches: W = the deflecting weight in lbs., tons, &c., and R = the Resilience of the Beam in inch-lbs., or inch-tons depending on W. The most useful values of R are for the two standard loads;-Breaking Weight,— and Working Load; for convenience we may indicate them by R and r respectively. Table 67 gives in cols. 5 and 6, the values of R, r, in inchlbs. for a Standard bar 1 inch square and 1 foot long, which may be termed the "Specific Resilience" of those Materials. Thus for Cast iron R 8 × W × becomes R = .0785 × in col. 5: similarly cols. 3, 4 give = 6.78, as in col. 6. 2063 × 81.0, as = The same reasoning and Rule will apply to cases other than beams, say to the driving of a nail by a falling weight: thus, Mr. Bevan found that to drive a sixpenny nail 1 inch into dry Christiania Deal required a steady pressure of 400 lbs. To do the same work by impact required 4 blows of a hammer weighing 6.275 lbs., falling 12 inches at each stroke, the mechanical work done being 6.275 x 12 x 4 = 301 inch-lbs. by the Rule we obtain R = 1 × 400 × = 300 inch-lbs. : (778.) We can now search for the Laws by which the depth, breadth, and length of a beam govern R, or its power in resisting Impact. By (659) it is shown that d = L3 x W x C hence d3 x b Now by when all are constant except d and W, we have d = hence with (324), the strength of beams or W varies as d2, depths in the ratio 1, 2, 3, 4, the transverse strength for a dead load will vary in the ratio 12, 22, 32, 42, or 1, 4, 9, 16, and with = 2:9 × 1 = 3, and 16 × 4, or in the simple direct 1. hence the ratio of R or 8 x W comes out 1 x 1 = 1: = 4 x 1/ Then, for the breadths: the same Rule (659) shows that W 8 varies as therefore with breadths in the ratio 1, 2, 3, 4, b |