(111) other total arch constants are calculated in a similar manner, as indicated by Figure 87. Calculations for x and y are shown in Figure 88. 112. Load Constants for Variable-Thickness Arch. Computations for D' and D" values for D-terms D1, D2, and D3 for a Radial Load No. 1 are given in Figures 89 to 94, inclusive. Tabular values for D-terms for the first column of each sheet are taken from the "Load Constant" tables at the end of the subchapter, using Table 9 as a guide. Table 9 gives fractions of circular arch D-terms to use for prime values. The formulas in Table 9 are clarified by Figure 95, which illustrates the method followed in determining formulas for Radial Load No. 5 at the 1/2 point. Since D-terms are deflections due to load at the left of the point considered, it can be seen that the tabular D-term at the 1/2 point, for the load between the 1/2 point and the abutment, is the tabular value at the 1/2 point for Load 5 for angle Ø (see Figure 95a). Likewise, the tabA ular D-term at the 1/2 point for the load between the 1/2 and 3/4 points, Ø2 to Ø3, is equal to 3/4 of the D-term at the 3/4 point for Load 4 for angle A. This statement is evident if it is assumed that the half-arch is moved to the right, with the load stationary; so that it extends from the 3/4 point to the crown. This assumption makes the load ordinate at the 3/4 point equal to 3/4 of P, and the true 1/2 point corresponds to the 3/4 point of the assumed position of the arch. Then the tabular D-term for voussoir A to Ø3 is D', A to Ø3 = (D for Load 5 at 1/2 point) and the tabular D-term for voussoir 3 to 2 is These values may now be inserted directly in Equations (158) to (166). Then, by following similar procedures, formulas can be determined for D-term values, or prime values, at other arch points for other unit loads. In order to calculate total D-terms, values for ML, HL, and V must be determined as explained in the last paragraph of Section 93, using Equations (170) to (172), inclusive. ML, HL, and V1 values for Radial Loads 1 and 5 are shown in Figure 96. These were calculated by means of Equations (109) to (111) and (117) to (119), using data given in Figures 83, 84, and 27 and trigonometric tables at the end of this subchapter. Values for other loads are calculated by equations given in Sections 78 to 81, inclusive, except that the moments due to eccentricity of tangential loads require calculations for H1e' quantities on the computation form shown in Figure 97. The computations shown on that figure are for Tangential Load No. 1 on the fillet. Values for other tangential loads on the fillet and on the central section are computed in a similar manner. The H terms at the bottom of Figure 97 are the tangential thrusts at the center of the voussoirs; each H1 is the thrust due to the tangential load between the center of the voussoir and the edge of the previous voussoir. Thus, H, is the tangential thrust at the 5/8 point, due to the load between that point and the 1/2 point (see Figure 98). In the upper part of Figure 98 is a tabulation of formulas for the H1 terms for various loads. These equations are based on Equations (126) and (134) of Sections 78 and 79. Calculations for M', H1', and V1' values for Tangential Loads 1 and 5 on the fillet are shown in Figure 99. The equations used are tabulated in Table 6. 1 8 . Figure 100 shows computations for ML, HL, and V1 for the transfer of D-terms using Equations (170) to (172) in Section 93. Values of x and y are given in Figure 88 and ML, HL, and V1 values are given in Figure 96. COMPUTATION FORM FOR ARCH CONSTANTS FOR VARIABLE-THICKNESS ARCHES Crown Section O-2 AND D-TERMS DAM RADIAL LOADS D-No. I LOAD. VERQ COMPUTED BY JH.B. DATE 9-22-38 |