to produce a sound casting free from blow-holes and other imperfections, and in no case less than 2 feet in height. All bolt-holes in the flanges and lugs of cast-iron work to be drilled true to template and to fit the bolts. Wrought Iron.-All plates, bars, angle, and other wrought iron to break under a strain not less than 22 tons per sectional square inch; its fracture to be fibrous, and free from any appearance of crystalline nature; nor is any lateral separation of the iron to be visible except at the place of fracture. The extension of the iron previous to fracture is not to exceed inch per foot of length, nor to be less than inch per foot. The joints of all plates are to be planed so as to butt truly, the planing being done by a planing machine, and not by a rotary cutter. All rivets to have ample allowance for making the head, in no case less than 14 diameters, and the rivets to be solidly closed so as to fill the rivet holes; if collars form at the edges of the rivets, they are not to be cut off. All bolts and screws to be made solid-headed, and the threads, as well as those of the nuts, to be chased with proper chasing tools, and the bodies of the bolts turned; no die-cut threads or ground bolts to be used. The heads and nuts to bear fairly and evenly on the work, and the bolts to be so proportioned that when broken by longitudinal test strain, the fracture shall occur in the body of the bolt, and not by the head pulling off, or the thread of the bolt or nut stripping. The rivet holes are to be either drilled from the solid, or punched inch less than the size of the finished rivets, and then drilled out to the full size, preferably by a pin drill. All sharp edges round the rivet holes to be taken off. These few general remarks will show the nature of the stipulations to be made, but the actual amounts of test weights will be varied according to the nature of the work and the locality of manufacture: it is useless to specify I that which cannot be obtained, but care must be taken to calculate the structure for materials of such strength as may reasonably be had, and the specification drawn accordingly, and rigorously enforced. I have taken this example to illustrate the practical method of calculating structures generally, for I have found students who have acquainted themselves with the theories, at a loss in actually applying them, not clearly seeing how the details are to follow on in proper sequence, or, perhaps, not knowing exactly where to begin; but it is hoped that this one example will serve for all, the general processes being similar, the differences merely occurring in details. I must now turn briefly to the testing of structures, such as iron bridges, when erected complete. Great care must be taken in the erection to prevent any undue straining or wrenching of any of the parts in lifting, or when in place. A competent man, acquainted with the character of the strain for resistance to which the work is designed, should be in charge of the erecting, for a serious and permanent damage may be done to a well-designed and soundly executed structure through the blundering of an ignorant person at this stage, and unfortunately such injury may not be discovered or even suspected until it is made evident by failing, or apparent weakness calls attention to it. The deflection of a riveted or otherwise built-up structure, in my opinion, is not a conclusive test as to its strength; so much of the rigidity will depend upon the quality of the riveting, that the gross result cannot be regarded as a test of the quality of the material, nor can we determine what to attribute to workmanship, and what to metal. This test is accepted, then, as a negative one, showing, where satisfactory, that the structure is not to be condemned, but not proving that it is perfect. This test, to be of any value, should extend over some time. Directly a work is completed, and before any load comes upon it, the relative level to some fixed mark should be determined, then the effect of the first load coming on it noted, and the set taken from time to time to observe if there is any increase in the permanent deflection. It should also at intervals be tested to see if a given load passing on to or over it causes always the same deflection; from such tests we can determine whether the work is permanent, or if it be gradually deteriorating. The maximum load should not in the first instance be put on at once, but gradually, so as to allow the joints to take their bearings steadily, and without any sudden shock or jar, which would act like a blow, and so tend unduly to strain or wrench some part of the work, CHAPTER XIII. ECONOMICAL PROPORTIONING OF STRUCTURES. In all structures it is necessary to study economy of cost, as in other matters; hence we must not be contented with merely determining how to proportion our materials to such strains as may occur, but must examine the variations of strain due to different proportions in the main dimensions of the work intrusted to our charge. In many, perhaps the majority of instances, circumstances determine the ratio of depth to span, and other dimensions may also be strictly limited; but in order that they may be applied where possible, the most economical ratios should be determined. Let us consider the ratio of span to depth of a plate girder. As the depth increases, the strain, and therefore the area and weights, of the flanges will decrease it has, however, been seen that certain requirements of manufacture fix the thicknesses of web in excess of the theoretical thickness, since within certain limits the weight of the web of a girder of given span will increase in direct ratio to the depth of girder. The term web here includes the stiffeners and appendages that practically constitute part of the web structure. If, then, the flanges diminish as the web increases in weight, and vice versa, it stands to reason that there must be some ratio of depth to span corresponding to a minimum total weight, and it is for this ratio that we now seek an equation, and in so doing we shall not be restricted to parallel flanged girders, but if necessary make the girder of varying depth. Let span in feet; d= depth in feet; w = load per lineal foot in tons; s=direct resistance of iron in tons per square inch (mean resistance); a sectional area of both flanges in square inches; a' = vertical sectional area of web (including allowance for stiffeners, &c.) in square inches; t = thickness of web in inches; A = total area of web and flanges. The weights of the parts will vary directly as their sectional area, therefore the ratio giving the minimum total area will also correspond to the minimum total weight. For the area of both flanges the general formula for strain must be multiplied by 2, then at any point distant x from a point of support, a = a = 12 dt; hence a + a' = A = го х d s го х го х wlx w d s ds Also, + 12 d t. Here we find the positive quantity +12 d t, and the wl x ds negative - varying with d, and it is evident that if a " ds point is reached where the value of A is a minimum, and about to change from the descending to the ascending value, making the increment of d very small, the increase of the two qualities must be equal. Let ƒ be an indefinitely small increment of d, then A1 = + 12 t (d+f) wl x 8 (d+f) го х s (d+f) Deducting these from the former values, and equating the differences of the positive and negative quantities, we get |