99. Triangular Twist Load. The moment due to a triangular twist load, as shown in figure 35-b, is obtained by the formula, 100. Temperature Load.-Load constants for a uniform tempera ture change throughout an arch are D2 =- ct sind, ds = -ctr sin,do, -ctr vers = -C + YA (175) S (176) CAPI D1 =√ ct cos &,ds = ctr cos p,dp, = ctr sinap, = ct XA D3 101. Summary of Load Formulas. Table 4 gives a summary of formulas for moments, thrusts, shears, and D-terms for all unit loads on a circular arch. Integrals involved in the formulas are given in the left part of the table and items for which they are used comprise the remainder of the table. Columns designated "item" give the formulas in their original form, as in equations 131, 132, and 133. Columns headed "designation" give the convenient D-term notation used for the load. constants. The remaining "multiplier" columns give quantities by which the integrals must be multiplied to give desired arch movements. For example, under the headings "radial loads" and "triangular,' which agrees with equation 145 previously derived. The term E in the formulas is the modulus of elasticity of concrete, Ec, the subscript e being omitted for convenience. The evaluation of integrals in the column "trigonometric part" is discussed in chapter IX, which also gives tabulations of their numerical values for angles between 0° and 90°. Tables of numerical values for portions of D-terms that are functions of arch angles only are also given in chapter IX. These include 15 unit load patterns for all angles between 10° and 90°. VARIABLE THICKNESS ARCH 102. General Statement.-Variable thickness arches discussed herein have a constant upstream radius, and variable center line and downstream radii as shown in figure 36. General arch formulas are used in their analysis. By making certain revisions in various terms, as discussed subsequently, uniform thickness circular arch formulas are applicable to variable thickness arches. Each side of the arch is divided into voussoirs by drawing lines radial to the upstream face with each voussoir subtended by a central angle equal to one-fourth the angle from the crown to the abutment. Each voussoir is considered as a segment of a circular arch whose thickness is equal to the average thickness of the voussoir. This makes it possible to use the tabulated arch and load constants for a uniform thickness arch. Although this method is approximate, it is believed sufficiently accurate for a trial load analysis. The method was checked by comparing it with an accurate analysis of an arch in which the thickness varied as the secant of the central angle, the term To sec being integrated and substituted for T in the uniform thickness arch formulas. A relatively close agreement, within 2 percent, was found between the results of the two analyses. The approximate method is probably within the limits of accuracy of the trial load analysis for any variable thickness arch likely to be encountered in dam design. A discussion of the approximate method is given in the following section. 103. Notation. The additional notation needed to discuss a variable thickness arch is given below. T= actual thickness of arch at quarter points of the left or right part. rp radius to arch center line at any quarter point, To actual thickness at crown. Το radius to center line at crown, Ty = average thickness of voussoir, radial to upstream face. radius to center line of voussoir -RU-Ty/2. moment of inertia about center line of voussoir, - T3v/12. eccentricity of voussoir center line from a circular are drawn through the midpoint of the arch thickness at a quarter eccentricity of voussoir center line with reference to the crown center line arc. This is the eccentricity of the applied tangential thrust with reference to the voussoir center line, rory. = ep eccentricity of arch points, midpoints of arch thicknesses at quarter points, with reference to the crown center line arc, 104. Arch Constants.-The method of computing arch constants is apparent from a brief examination of equations 125 to 130. Radius r and thickness T appear therein as constants. In the case of the variable thickness arch, these quantities are functions of and must be included within the integral sign. When the variable thickness arch is replaced by the approximation shown in figure 36, integrals of equations 125 to 130 are replaced by summations of successive integrals. Constants of integration due to abrupt changes in r and T appear and must be included in the computations. These integrals and integration constants may be written in terms of uniform thickness arch constants, a procedure which is followed in equations 177 to 182. The derivation of these equations from a physical standpoint is discussed below. In this discussion, constants of integration appear in terms of eccentricity of the arch center line. The application of arch constants for a uniform thickness arch to the analysis of a variable thickness arch requires the determination of the effects of movements of an assumed uniform thickness voussoir on all arch points between it and the crown of the arch. The contribution of a voussoir to an arch constant at a point is the product of the multiplier for that voussoir, in terms of ry, Tv, and Iv, and the increment of the uniform thickness tabular values for the angles from the point considered to the limits of the voussoir. It should be kept in mind that the tables of constants contain values of the integrals only, the multipliers being computed separately. The tabular values are commonly referred to as prime (') values. In transferring voussoir movements to a distant arch point, the fact that the center lines of the voussoir are eccentric with respect to arch points makes it necessary to include terms involving eccentricities in the constants. For example, consider the contribution of the voussoir between 3/4 and 1/2 points to the B, term at the crown. By definition, B, is the angular movement at a point due to a unit thrust at the point. Applied to this case, the angular movement at Q is desired for a unit thrust at Q. The thrust at Q has an eccentricity, e, with respect to the voussoir considered. This results in a moment e which causes an additional angular ry movement at Q of AA'e. Then the contribution of voussoir, 3/4 Ec ly point to 1/2 point, to B, is: 1 The contributions from other voussoirs are determined similarly and B1 is computed as shown in equation 178 below. Other terms involving e are determined in a similar manner. The following equations for arch constants are for any point and represent the summation of the various voussoir contributions at the point considered. |