Here ct 28; and similarly for a cantilever, D = In practice it is seldom such a condition can be found, and if the sectional area and moment of resistance vary, so will the radius of curvature; so it is advisable to see in what way the deflection varies, and then to fill in constants from experiment. Returning to the first equation, R= d E 2 S' and including all terms except the load and known dimensions in the constant a, to be determined by experiment, we find— of a" must be filled in from experiment. 7/3 Where the section of the beam is not a solid rectangle, b d3 is to be replaced by b d3, b' d'3 . . . br d13, as in the determination of moments of resistance; let this quality = m. For iron of good quality, we find the following formulæ by filling in the value of a" from experiments :-/span in feet, W = load in tons, b and d = breadth and depth in inches, D deflection in inches. CAST IRON.-Girder loaded at the centre, D = loaded uniformly over its length, D= lever loaded at the free end, D= W 13 W. 13 14 m Canti 22.4 m WROUGHT IRON.-Girder loaded at the centre, D = This is for sound wrought iron, such as square bars or sound forgings. The rolled iron shows much greater deflection when beam and H sections are reached, and this may be due to the fact that for complex sections the iron must be softer to enable it to follow the form of the rolls. For rolled girder we find Very great variations may be expected in riveted work; but the following expressions are taken from samples of good ordinary work, taken within working limits, the girders having the plates distributed so as to approximate as nearly as possible to uniform strain per sectional square inch, the strains varying from 3 to 5 tons per square inch. W 13 Single web plate girder, loaded at the centre, D = 16. m The double web plate girders come up to the rolled beams given above. Framed structures will distort under strain, and in fact a lattice girder may be said to deflect as a whole, though none of its elements are under bending stress; from the shortening and lengthening of the bars composing the triangles the distortion may be determined. As I have observed above, the formula for deflection are not practically of much use, for although the constants may be determined for bars, yet they cannot be made to include all the varied conditions of material and manufacture; they are therefore chiefly of use for purposes of comparison, and to enable us to determine the requisite proportions of combined girders in a structure, for if one sort of iron is being used throughout a structure, or a series of structures, the coefficient of deflection, whatever it is, should be the same for all; that is to say, its variation should not exceed the limits of variation for differences of strain, and as presumably all parts of the work will be calculated to undergo the same strain per square inch, the coefficient of deflection will become practically constant for the structure under consideration. It is the practice to make girders slightly arched, or cambered, in order that when loaded they may not deflect below the horizontal line, which would give a very undersirable appearance, and the amount of camber allowed is 1 inch to every 40 feet of span, which in all ordinary cases will be found to be ample. Suppose a girder to be strained on top and bottom flange to 5 tons per square inch, and T2 (c+t) take E 7,000 tons, and d = then D= 10.7 (55) 8 X 7000 = 10 8 d. E 7 = or about 1 inch to 47 feet. It is, 560' however, very seldom that the deflection amounts to anything like this amount. CHAPTER VII. IRON ARCHES. B 2 A a T In the iron arch properly designed there exists only one kind of strain, compression, but the arch must be of such a form that the line of strain lies wholly within the depth of the arch, so in the first place the mode of determining the correct shape must be set forth. The thrust at the crown will be found from a formula demonstrated in a previous chapter, showing that the tangential strain on a circle or circular arc at any point is equal to its radius at that point multiplied by the radial pressure d a T b e h j k Fig. 44. at the same point. At the crown of an arch the tangential thrust is evidently horizontal, and the load at that point acting vertically, acts at right angles to the tangent, and F therefore radially. Let R= the radius of the arch in feet, and w = the load per (horizontal) lineal foot on the arch, then if T the thrust at the crown, T = w × R. Let Fig. 44 represent half an arch, and suppose the length to = be divided up into parts, each equal to where the 20' span; let also v = the rise at the centre above the springings. The load on each of the longitudinal divisions will Let Rx 6, then the thrust T6 w. l. W 7 20' Through the centres of gravity of the loads draw the vertical lines a W, b W, &c. The direction of the horizontal thrust at the crown A is shown by the arrow T. Produce the arrow, and from a, where the produced line intersects a W, mark off to scale the horizontal thrust. This will be seen better on the w l enlarged sketch C. Make a d=T, and aƒ= or W; 20 complete the parallelogram ad ef, then e will be a point in the curve of strain. Join a e, and produce it to intersect b W in C; make bg = a e, and bi = or W1; complete in ωι c, the parallelogram bghi, then h will be another point in the curve; produce b h to intersect c W2 make cj = bh, complete the parallelogram cjkl; k will be a third point in the curve, and similarly other points are to be found until the abutment B is reached, and through these points the line of thrust is to be drawn, and the arch so proportioned as to enclose it. If the load is not uniformly distributed, then W, W1, W2, &c., must be made to agree with the respective loads for each part. In designing an arch, all the different loads coming upon it should be taken, and the lines of thrust worked out, and then the arch laid on so as to include them all. |