sents a moment of P foot-pounds per square foot, applied along the entire center line of the arch. Triangular loads represent moment loads varying from P foot-pounds per square foot at the abutment to zero at the quarter points, as shown in figure 4-c.

36. Concentrated Loads. In addition to the loads just discussed, another set of loads is required by the assumption of elastic foundations and abutments. At the base of the crown cantilever of a symmetrical dam, the foundation is subjected to cantilever loads only. At the ends of the top arch the abutment is subjected to arch loads only. At all intermediate points the canyon wall acts both as an arch abutment and as a cantilever foundation. This is a direct corollary of the assumption that each system occupies the whole volume of the dam.

Both cantilever and arch loads act to deform canyon walls. That is, a part of the arch abutment movement is due to cantilever loads and a part of the cantilever foundation movement is due to arch loads. Figure 1 shows that cantilevers are located on the abutments of all sample arches. By this arrangement, the interaction of arch and cantilever loads on the abutment is brought into the analysis. Concentrated loads, representing shear, thrust and moment, are placed on the arch abutment. Using these loads, total forces and moments on the canyon wall at an arch abutment are determined. Abutment movements due to these total forces and moments are, therefore, abutment movements of corresponding arches and cantilevers. Concentrated radial, tangential, and twist loads are, respectively, a radial shear of P, usually 1,000 pounds, a tangential thrust of P pounds, and a twisting moment of P foot-pounds.

37. Temperature Loads.-Temperature changes in a dam are determined at elevations of sample arches and are assigned to arches as initial loads. A convenient unit of 1° Fahrenheit is used in applying temperature loads. Radial, tangential, and angular movements are calculated for the unit temperature change.

38. Secondary Movements.-Besides producing radial movements, radial loads cause secondary tangential and angular movements. Tangential loads cause secondary radial and angular movements, and twist loads produce secondary radial and tangential movements. In other words, each arch load, radial, tangential, twist, or temperature, produces radial, tangential, and angular movements that have to be considered in the trial load analysis. Consequently each adjustment is disrupted by loads of succeeding adjustments. However, secondary movements converge rapidly

with successive readjustments. More than a few readjustments are seldom required to secure satisfactory agreement of all deflections.

39. Cracked Arches.-Arch elements are ordinarily assumed to carry tension but may be computed as cracked arches. For this condition, concrete in tension is not considered to carry stress. Since this consideration requires the use of the more complicated voussoir summation method of arch analysis, in which tabulated unit loads cannot be used, cracked arches are computed only for unusually important cases.

TRIAL LOAD ADJUSTMENTS

40. General Statement. In a trial load analysis, the dam is replaced by arch and cantilever systems. However, geometrical continuity must exist between the two systems at all points in the dam. This is accomplished by making radial deflections, tangential deflections, and angular movements in arches and cantilevers agree at conjugate points. Radial, tangential, and angular movements are considered respectively in radial, tangential, and twist adjustments.

Loads for which the analysis is made are first applied to the arch and cantilever structures as outlined in the preceding section on load conditions. The arch structure will have initial radial, tangential, and twist deflections due to temperature change. The cantilever structure will have initial radial deflections due to vertical water loads, and initial tangential deflections due to tangential earthquake forces, if included. All other external loads are included in the total radial loads. Starting with this partially loaded and deflected structure, the first adjustment is undertaken.

41. Radial Adjustment. The analysis is started with the radial adjustment. This consists of dividing external radial loads between arches and cantilevers, by trial, with due consideration for effects of tangential shear and twist. The first step is to divide the horizontal component of the total radial load into two parts, one acting on the arches and the other on the cantilevers. In making the division the aim is to distribute the loads between the two systems, so that radial deflections of arches and cantilevers will be reasonably close at conjugate points.

42. Tangential Adjustment.-After radial deflections caused by radial loads have been adjusted, tangential movements are considered in the tangential adjustment. Tangential movements are tangential earthquake deflections of cantilevers and tangential movements of arches due

to temperature and to secondary movements resulting from the radial adjustment. In the tangential adjustment, equal and opposite tangential shear loads, one on the arch and the other on the cantilever, are introduced to remove relative tangential deflections of the elements. The required amounts of the loads are estimated and applied by successive trials to the two systems of elements.

43. Twist Adjustment. In the next step of the analysis, rotations, or angular movements, of arch and cantilever elements about vertical axes are considered in the twist adjustment. A new set of loads is introduced, consisting of equal and opposite twist loads that act on arches and cantilevers. While these are termed twist loads, they are really horizontal couples which cause twisting of the cantilevers and bending of the arches.

Twist loads are required to remove relative angular movements of elements caused by radial and tangential loads. The amounts of the twist loads are estimated and gradually adjusted by trial. It will be noted that the twist adjustment corrects rotations about the vertical axis only. However, it can be demonstrated that tangential twisting moments are equal to vertical twisting moments. Also, if the structure is brought into twist adjustment in one direction and radial adjustment is maintained, then it is automatically in twist adjustment in the other direction. It has been found simpler to make the twist adjustment for rotations about the vertical axes. Horizontal twist effects on cantilever bending may be computed as shown in chapter VII.

44. Sequence of Adjustments. After the twist adjustment is completed, as described in the preceding paragraph, radial movements due to tangential and twist loads are considered in the first radial readjustment. A tangential readjustment is then made to correct for discrepancies caused by twist loads and the first radial readjustment loads. Likewise, a twist readjustment and subsequent radial, tangential, and twist readjustments may be required, because each readjustment produces movements that have to be considered in succeeding readjustments. In most dams secondary effects converge rapidly; so that only a few readjustments are needed for the complete analysis. In exceptional cases, such as long flexible arches with short stiff cantilevers, the convergence is slow, and the process of adjustment may become long and tedious.

It has been found possible to materially lessen the number of readjustments required. After considerable experience, secondary movements can be estimated with fair accuracy before undertaking an adjustment. Estimated movements can then be allowed for in bringing arch and cantilever

structures into congruence; so that the complete series of adjustments will leave the structure in satisfactory agreement. Using this method, more than a single radial readjustment is seldom needed after completing the first radial, tangential, and twist adjustments. It should be mentioned that this procedure applies particularly to the normal case, where general characteristics of arch and cantilever structures are well known from preliminary studies.

When a completed structure is analyzed, another method may prove more effective. This consists in carrying each adjustment only to the point where errors in adjustment are within the range of estimated secondary movements. A saving in time results, even though several readjustments may be necessary, since approximate adjustments may be made rapidly. This method is often used for dams of unusual shapes, where no experience to aid in estimating secondary movements is available.

45. Adjustments for Poisson's Ratio.-Effects of Poisson's ratio may be analyzed after adjustments for external loads are completed. Poisson's ratio produces radial, tangential, and angular movements in both arches and cantilevers. Consequently, additional loads are needed to restore continuity throughout the structure. Separate adjustments for Poisson's ratio effects are usually made, using the same procedure of adjustments and readjustments as before.

RESULTS OF ANALYSIS

46. Movements.-Radial, tangential, and angular movements, determined by trial load adjustments, give displacements and rotations in the dam for assumed conditions of loading and temperature change. Effects of water soaking of the concrete at the upstream face, plastic flow, and nonlinear distribution of normal stresses are not included.

47. Stresses.-Trial load adjustments determine total loads resisted by arch and cantilever elements. For these total loads, stresses throughout the dam may be computed, providing a definite variation of stress between upstream and downstream faces of the dam is assumed. To this end, three of the stresses, the vertical cantilever stress normal to a horizontal plane, the horizontal arch stress normal to a vertical radial plane, and the horizontal tangential shear stress acting on a horizontal plane, are assumed to vary linearly between the upstream and downstream faces. As was implied earlier, a nonlinear distribution of stress across the dam might be assumed. In that case it would be necessary to carry the

same assumption through a complete additional series of adjustments, calculating arch and cantilever movements on the basis of such an assumption.

By means of the linear assumption, a direct computation of the three stresses mentioned above may be made, using total arch and cantilever loads. From these three stresses either principal stresses or stresses on any desired plane may be computed throughout the dam. Chapter VIII contains the formulas needed for such computations.