ments resting thereon. From figure 55 it may be seen that at the center line the width of a cantilever, resting on an arch of unit height, is equal to tan 4. Since the assumed unit cantilever used in the analysis is one foot wide at the axis, the radial shear at the base of a cantilever is multiplied by Raris r tan to give the correct amount of concentrated radial load to be applied to an arch one foot high. For the same reason, tangential shears and twisting moments at cantilever foundations are corrected by the factor Raris r tan to determine, respectively, concentrated tangential and twist loads for the arches. In the adjustments, abutment movements of the arches are added algebraically to deflections of corresponding cantilevers, since cantilever foundations move with arch abutments. PREPARATORY STEPS 157. General.-Considerable delay in the analysis may be avoided by completing all preliminary work before starting adjustments. After selecting sample arches and cantilevers, each arch is drawn to a suitable scale. Since arch formulas are based on radial abutments, actual surfaces of contact between arches and abutment rock must be modified somewhat. Radial abutment planes are drawn, approximating actual planes of contact. A radial plane through the point where the centerline of the arch intersects the abutment ordinarily is sufficient. The abutment angle, 4, is taken to the nearest whole degree, in order to use the tables of arch and load constants without interpolating. At the same time, the angles of abutment planes, 4, are determined and tabulated on the foundation form, figure 14, chapter III. 158. Foundation. The foundation form, figure 14, chapter III, is filled out for each arch abutment and each separate cantilever foundadation. Reference should be made to the text of chapter III for cases where b is replaced by b/2 in the evaluation of foundation constants. The foundation constants are then transferred to figure 26, chapter IV, for the cantilever analysis, and to figure 40, chapter VI, for the arch analysis. 159. Cantilevers. The following tabulations are made for uncracked cantilever elements. 1. For each sample cantilever, requisite data are entered in figure 24, chapter IV. This form is completed with the aid of the curves in figure 16. Data are tabulated for the elevation of each sample arch. 2. Radial-side cantilevers are transformed to parallel-side cantilevers by the graphical method shown in figure 17. From the equivalent parallel-side cantilevers, weights and moments of concrete and vertical water loads are obtained. A mechanical integrator may be used for this purpose. Weights and moments are tabulated on figure 25, using separate sheets for concrete and vertical water loads. 3. Shears and moments are computed for earthquake inertia forces from data on figure 24. These shears and moments are tabulated for each cantilever, in the same manner as weights and moments of concrete and vertical water loads. 4. Loads so far considered are those initially applied to the cantilever structure. From them, initial deflections of the cantilevers are computed using equations 59 to 63, chapter IV. Computations of deflections are conveniently made on the forms shown in figure 26, using the headings at a for radial loads and at c for tangential loads. Care should be taken to include foundation movements as directed in the text accompanying equations 59 to 63. 5. Again using the forms of figure 26, cantilever deflections are computed for each unit radial, tangential, and twist load. Moments and shears for unit loads are calculated as indicated in figure 18, chapter IV. 6. A table of deflections is prepared, giving the deflection of each cantilever at each arch elevation for each unit load. 160. Arches. The procedure outlined below constitutes the preparatory steps necessary for analyses of arch elements. 1. Plans of arches are examined to determine whether they are to be classed as uniform thickness, variable thickness, fillet arches, or other types. 2. If of uniform thickness, arches are ordinarily analyzed as symmetrical. In this case the necessary data are tabulated on figure 40, chapter VI, and carried through subsequent computations on the general arch forms as directed for symmetrical arches in chapter VI. Arch constants require computation but once. Load constants require a separate set of computations for each load. For the lower arches of a dam, all unit loads are not usually required. Often a uniform load, a number 5, and possibly a number 3 load are sufficient. After a little experience, the loads required may be estimated from the number of cantilevers intersecting the arch. Other unit loads are not computed unless difficulties in adjustment show the need for additional loads. 3. Arch deflection curves, similar to figures 48 to 53, chapter VI, are plotted as a check on the computations. The other checks listed in section 127 are also made. 4. A table of moments, thrusts, and shears at the crown and abutments is prepared for the unit loads considered. Unit load deflections are also tabulated at all arch points. 5. Temperature deflections are computed for the temperature changes assumed. These deflections are initial deflections of the arch elements. 6. When the arches are of variable thickness or other nonuniform types, the method to be followed is outlined under the proper heading in chapters V and VI. Except for the computation of arch and load constants, the procedure for variable thickness arches is the same as that given for uniform thickness arches. Before entering general arch forms, voussoir dimensions are tabulated on figure 44, chapter VI. Computations of arch and load constants may then be made on figures 44, 45, and 47, with the aid of table 9 and the accompanying text. Evaluated constants are inserted in the general arch forms and subsequent operations carried on as before. RADIAL ADJUSTMENT 161. Procedure. The radial adjustment may now be started. At the beginning, arches and cantilevers have initial deflections due to initial loads. The horizontal radial load must be divided by trial between the arch and cantilever systems, so that agreement of radial deflections is secured. Since cantilevers are not usually located at arch quarter-points, the division may be made graphically. Horizontal radial loads are plotted on the developed arch center lines as shown in figure 56. Cantilevers and arch quarter-points are indicated by full and dotted lines respectively. A trial arch loading is chosen and plotted on the loading diagram. The cantilever loading, which is the difference between total load and arch load, is measured directly on the diagram at each cantilever. Both cantilever and arch loads are made up by summing unit loads. Below the developed arch center line, initial deflections of the elements are plotted, using different symbols to distinguish between the arch and cantilever as in figure 56. As the loading is changed, deflections are replotted from the initial positions. For the first estimate of load distribution, crown cantilever deflections are adjusted to crown arch deflections. After the crown cantilever is adjusted, deflections of other arch and cantilever points are computed and plotted. By comparing these deflections, additional radial loads can be estimated and applied to the structure, until resulting arch and cantilever deflections are approximately the same. at conjugate points. After a few trials, deflections are usually in close agreement, which completes the first stage of the radial adjustment. Upon completion of the radial adjustment, stresses are computed. If excessive tension stresses are found in the cantilevers, the adjustment is repeated, assuming cracking in the cantilever elements. Figures 27 and 28, chapter IV, are used for computing deflections of cracked cantilevers. Elevations at which cracking occurs may be determined from values of the ratio F tabulated in figure 27. The method, which is simply an extension of the middle third theorem to radial-side cantilevers, is given in sections 76 and 77, chapter IV. 162. Readjustments. Subsequent readjustments of radial deflections are necessary because loads introduced in tangential and twist adjustments produce additional radial arch and cantilever movements. Final radial loads and deflections for Boulder Dam, as determined by adjustments and readjustments, are shown in figure 56. The radial adjustment constitutes the most important step in the trial load analysis. Although tangential and twist adjustments are usually advisable, they are of secondary importance when compared with the radial adjustment. The reason for this is evident from the fact that radial movements are much larger than tangential or angular movements. An analysis based on an adjustment of radial deflections only is often made as a preliminary study. Such an analysis has been found to give a fair indication of maximum stresses. This type of analysis is used to compare preliminary designs for a dam, the complete analysis being used to determine stresses in the adopted design. TANGENTIAL ADJUSTMENT 163. General. The radial adjustment is followed by the tangential adjustment, by which movements of arches and cantilevers are adjusted in the tangential direction. At the beginning of the tangential adjustment, tangential movements consist of initial cantilever deflections |