Computed variation of rise under maximum variation of temperature, . 2".496 o".053 0.00001923 Vertical Computed deformation of the arch due to half the span being loaded. Pressure on timber of caisson, foundation of Pier II., 111.6 pounds to the sq. inch. The method of M. Henri de Dion for computing the resistance and deformation of a beam, either straight or curved, consists in the substitution of measurements by planimetre for the arithmetical computations, sometimes very long, that are otherwise required. The formulæ for deflections due to bending moments are the follow These require that the section with reference to which 4x or sy is calculated, should have an invariable position. When the load is symmetrical the section at the key fulfils this condition. On a line A B, the neutral axis developed, we lay off as ordinates the quantities at each point, and draw the curve through their extremities. μ ΕΙ μ EI' The surface comprised between the line A B and the curve of measured with the planimetre, gives the total angular movement, counting as positive the surface above the line, and negative that which is below. From this surface, and on a plane perpendicular to it, we lay off the values of y (or x) corresponding to each section and we obtain volumes having the surface for base and for heights the ordinates y (or x) — positive when they correspond to positive angular movements and to positive ordiThe sum of the volumes for one curve gives the total horizontal (or vertical) displacement. nates. For the displacement of any other point whose ordinates are x', x-x, and y-y must be substituted for x and y, and the curve of is used only so far as the point xy. ΕΙ If the load is not symmetrical about the key the section at that point does not remain vertical. In that case the arch is supposed to be fixed at one end and the displacement of the different points ascertained as before. But the free end of the arch may come above or below its proper place, and a graphic projection is needed to bring it to its true position. As this method of volumes is not suited to planimetric measurement, MM. Molinos and Seyrig suggest a further simplification—the graphical multiplication of by x or y, as shown by the construction at Section 11 μ ΕΙ of Fig. 1, Plate LXII. The value of x' -- x = (2ax) is plotted at any convenient angle with the vertical, and upon it a length equal to unity is taken. From this point to the corresponding a line is drawn; a par μ ΕΙ μ ΕΙ allel to this line from the end of the oblique ordinate (2a — x) intersects the vertical at a distance (2a-x) from the base. The operation being repeated for each point, the surfaces so formed, measured with the planimetre, give the volumes that would have been obtained by direct computation. μs ΕΙ De Dion's method is not convenient for computing deflections of intermediate points under unsymmetrical loads, as it requires a separate construction for each point. It is better in such cases to recur to direct computation. The method indicated in Fig. 2, gives approximate results. The quantity for each section (which may be measured from the curve given), is laid off from the tangent of the preceding section and extended to the vertical through the free end of the arch rib B. The curve terminates at B', the position the free end of the arch would take under the conditions assumed, and the real deflections are measured from the line A B'. That this is only approximate is due to the necessary distortion of the figure. Deflections here referred to are those due to the bending moments. To them must be added for tangential compression : CONDENSED STATEMENT OF QUANTITIES AND COST OF THE PRINCIPAL CLASSES OF WORK, ETC., IN THE COMPLETED BRIDGE. NOTE. The data available when the original plans were prepared indicated a rock foundation for pier II. at 22 feet below mean high water. An additional price was inserted in the contract to provide for the greater depth made known by subsequent borings. The cost to the city of the foundation of pier II. below the level of mean high water was, for 7,726 cubic yards of masonry and timber sunk to 40.6 feet below high water, $30.64 per cubic yard. |