Proceeding from the centre towards either end, these moments corresponding to a full load decrease in intensity till they become positive, being everywhere determined by the ordinates of a parabola.* The central chord strains will be : 20,536,880 34.3 598,743 pounds. = The greatest positive moments (compression above and tension below) occur when one arm only is loaded; the greatest negative moments over the pivot pier occur when both arms are fully loaded, and those at the centre of each arm and near the ends of the draw take place when neither arm is loaded.† The maximum moments at the end of each panel are given in the tables in Appendix G, the values being obtained by measuring on a diagram the ordinates of the curves of strain; the corresponding chord strains are likewise given in the tables, their values being found by dividing the moments by the depth of truss at the corresponding point. To find the strains which actually take place in the chords, a correction must be made for the strain which is carried horizontally by the members of the web; these corrected strains are also given in the tables, the tensile strains being distinguished by a negative sign; as the strains carried horizontally by the web are all in tension, being carried by the diagonal ties, the actual amount of compression in the chord materially predominates over the tension.‡ The web is formed in two systems which are connected only through the chords. As each system is supposed to carry but half the load, in calculating the strains on their members the dead load will be assumed to be 480 pounds to the foot, and the live load 560 pounds. The strain in each post will be the same as the weight carried by the tie which depends from it ;§ the strain in each tie is equal to the weight carried, multiplied by the proper coefficient for the inclination of the tie. †The resultant curve of maximum intensity of strain for all parts of the draw, without regard to the sign of the moments, is drawn on the diagram in a plain heavy line. Strictly speaking, plus the weight of the upper chord for one panel; this weight however is small, and the calculation need not be complicated by it. § These corrected chord strains, expressed in tons of 2,000 pounds, are given on the skeleton diagram on Plate XII. When the draw is swung or unloaded, each tie carries the load between the end of the draw and the foot of the tie next inside. The strains and weights which then occur are given in the tables in Appendix G. When the draw is closed and loaded, the pivot pier carries the entire weight from the centre to that point on each arm where the moment of flexure has the greatest positive value; this point is half way between the point of reversal of the moments and the end of the draw; it is nearer the centre when only one arm is loaded than when both arms are loaded. The strains in the ties which carry this load to the central pier should therefore be calculated with reference to the case in which the whole draw is loaded, and the strains in those ties which carry this load towards the ends, with reference to the case in which but one arm is loaded. When both arms of the draw are loaded the general equation for the moment of flexure is 7 being the length of one arm and x the general abscissa. Differentiating :— The strains in the ties which carry this load towards the central post will therefore be calculated as if the centre of the truss was distant 145.25 feet from that post, making the equivalent length of span 290.5 feet. The general equation of the strains in the web under the action of a moving load is S= 12 w-x2 w' х го in which I denotes the total length of span, w the dead load, and w' the moving load per foot. Substituting the values this becomes (w = 480,-w' = 560,-and l = 290.5 S69720 - .964 x2 - 480 x the value of x for each tie being the assumed length of beam (290.5 feet) minus the distance from the centre post to the foot of the tie next inside. The strains in the ties and posts, calculated by this formula, are given in the tables in Appendix G. As the practical centre is distant 145.25 feet from the centre post only when both spans are fully loaded, the results thus obtained are excessive for all but the centre ties, the others being most intensely strained under a partial load when the practical centre will be nearer the centre post and the practical length of beam less. When but one arm is loaded the general equation for the moment of flexure is The strains in the ties which carry their load to the end posts will therefore be calculated as if the centre of the beam was distant 133 feet from the centre post, or 49 feet from the end post, the equivalent length of span being 98 feet. stituting 198 in the general equation for the web strains, it becomes Sub the value of x for each tie being the assumed length of beam (98 feet) minus the distance from the end post to the foot of the next tie outside. The strain in the several ties and posts, calculated by this formula, are also given in Appendix G.* The maximum strain on the centre post will be equal to the total dead and moving load between the points 145.25 feet on each side of that post, excepting the half panels adjoining, the weight of which is carried directly by the pivot. The strain is therefore (290.5-15.5) X 2080572,000 pounds. In like manner the strain on each end post is found to be (496.25) X 2080 87,800 pounds. * The maximum web strains expressed in tons of 2,000 pounds are marked on the skeleton diagram on Plate XII. The practical centres used in calculating these strains are also designated on the same diagram. The calculations of the contractors, by which the draw was framed, were made on the supposition that each arm acts as an isolated truss when the draw is closed; the panel ties are therefore of the same size at the centre and ends of the draw. To make the actual strains agree with these calculations, the lifting jacks placed under the end posts must carry one-half the weight when the draw is fully loaded, the weight carried by each jack being 912080189,280 pounds. The weight carried by each end bearing, under the present arrangement of wedge plates, is making the weight which would actually be lifted by the jacks under each post, after closing the draw, 112,840 pounds. |