librium and produce equal arch and cantilever deflections in all directions in all parts of the loaded structure.

10. Total vertical loads, including weights of water on the faces of the dam as well as weights of concrete, are assigned to cantilever elements and are assumed to be transmitted to the foundation without any transfer of load laterally to adjacent cantilevers by vertical arching.

11.

Effects of flow can be adequately considered by using somewhat smaller values of modulus of elasticity than would otherwise be adopted.

4. Accuracy of Assumptions.-The first of the basic assumptions listed above is probably the most uncertain of all, since it is doubtful if rock formations at any dam site are ever sufficiently free from cracks, seams, and fissures to be considered satisfactorily homogeneous. It is also doubtful if such formations are ever uniformly elastic in all directions; so that their movements under applied loads can be accurately calculated by mathematical formulas based on the theory of elasticity, or that the amount of plastic flow or joint slippage can be determined. However, for the present there seems to be no better way to estimate these movements which are important and which, therefore, should be considered in the design of curved concrete dams.

It is not believed that much uncertainty is involved in basic assumptions listed as numbers 2 to 5, inclusive. The uncertainty regarding homogeneity and uniform elasticity of concrete, assumption 2, which may have been important in some of the older structures, is gradually becoming less important, due to recent researches in design of concrete mixes together with improvements continually being made in methods of manufacturing and placing concrete.

Assumption 4, regarding closing of the dam before application of water load, can probably be violated in most cases without causing serious errors in maximum stresses calculated for the full condition of the reservoir. In the case of Boulder Dam, a series of careful trial load analyses for successive stages during the construction period, taking into account actual grouting operations, eccentric vertical loads produced by adding successive increments of concrete weight, intermediate partial-depth water loads, and effects of Poisson's ratio, did not change final maximum stresses more than 10 percent from those determined by the usual trial load analysis for full reservoir loading. The stress distribution was changed somewhat; some of the smaller stresses were altered appreciably, particu

larly in the arch elements; and elevations of maximum arch stresses were shifted slightly at both crown and abutment sections. However, the general effect on magnitude of maximum stresses was not important in either arch or cantilever elements.

As regards assumption 11, recent results of laboratory investigations indicate that, for stresses lower than those usually permitted in dam design, concrete flow is substantially proportional to stress. Consequently, the effect of flow on the action of a dam under continuous full reservoir load is primarily an increase in downstream deflection, although some effect on stress may result from the fact that the flow of the foundation and abutment rock may not be the same as the flow of concrete.

EFFECTS INCLUDED IN ANALYSES

5. Adjustments Considered. In calculating deflections of a curved dam for use in a trial load analysis the various arch and cantilever elements are first assumed free to move independently and are then brought into radial, tangential, and twist agreements by varying magnitudes of loads applied. In the radial adjustment the vertical component of the water load is assigned to the cantilevers, and the horizontal component of the water load is divided by trial into two components, both radial, one acting on the arches and the other on the cantilevers. The desired division. of load is that which gives the arches and cantilevers a certain common deflection at each point. This process involves the use of certain pattern loads, the effects of which combine conveniently, but which set up twisting and tangential influences in other parts of the structure. These influences must be considered later in the analysis by means of other self-balancing torsional and tangential loads.

It is expedient to introduce the new loads in such a manner that torsional and tangential shearing stresses are assigned to the cantilevers only. Arches are subject to twisting, bending, and compression only, and may be treated as though they had no resistance to tangential sliding. Cantilevers must resist bending, compression, twisting, and tangential shear.

At each stage in the analysis it is necessary to determine displacements of arch cross-sections relative to corresponding cantilever cross-sections. These relative displacements are deflections and rotations. New selfbalancing loads must be applied in such a manner that not only relative radial deflections but also relative rotations and tangential deflections of arches and cantilevers are removed.

6. Poisson's Ratio Effects.-The influence of Poisson's ratio is introduced into the trial load analysis, indirectly, in calculations of shear, twist, and tangential deflections, through the use of the shearing modulus of elasticity. Additional Poisson's ratio effects are analyzed by special readjustments after preliminary trial load calculations are completed. Corrections of stress analyses to allow for additional Poisson's ratio effects are not particularly important, inasmuch as the general tendency is toward a more uniform distribution of arch and cantilever stresses without causing changes of more than from 5 to 10 percent at locations of maximum stress.

7. Importance of Supplemental Effects. With the exception of Poisson's ratio adjustments, all supplemental effects included in trial load analyses since the first use of the method, have been found worthy of consideration, the importance of the effects usually varying with the size and height of structure.

Effects of including radial shear in calculating arch deflections were found particularly important in thick arches, as would naturally be expected, the importance increasing rapidly as the ratio of arch thickness to radius of curvature increases. Radial shear deflections in the lower arches of a massive curved concrete dam frequently equal or exceed moment and thrust deflections at practically all locations along the arch ring.

The importance of making proper allowances for radial sides of cantilever elements increases with the size of the dam and sharpness of curvature. The downstream base width of a cantilever with radial sides, one foot apart at the upstream face, was only 0.496 feet at the crown section of Gibson Dam, a 199-foot, constant radius, arch dam, constructed on the Sun River Project, Montana; and only 0.490 feet at the crown section of Owyhee Dam, a 421-foot, constant radius, arch dam, constructed on the Owyhee Project, Oregon. Proper allowances for radial sides tend to increase flexibility of cantilever elements, thus throwing more load on arch elements. Consequently, the net result of such allowances is a new load distribution with slightly increased radial deflections and slightly higher maximum stresses in both arch and cantilever elements.

Allowances for foundation and abutment deformations naturally tend to increase calculated deflections of the structure. The maximum downstream movement of the foundation rock calculated for the base of the crown cantilever of Owyhee Dam amounted to about 10 percent of the maximum radial deflection at the top of the dam. The corresponding movement of the foundation rock at the base of the crown cantilever of

Boulder Dam amounted to about 30 percent of the maximum radial deflection. The general result of including foundation and abutment movements in trial load calculations seems to be somewhat lower arch and cantilever stresses along foundation and abutment locations, without material stress changes in central and upper portions of the structure.

The general result of considering tangential shear in trial load analyses is a decrease in radial deflections near the crown section, an increase in radial deflections near the abutments, and a slight increase in arch stresses at the abutments without appreciable stress changes at the crown. The importance of tangential shear effects varies primarily with the ratio of length of dam to height. Such effects are not important in an arch dam built in a narrow canyon, as in Shoshone Dam, where the length of the top arch is less than the height of the structure. However, they are important in a relatively long arch dam like Gibson, where the top length is about four and one-half times the height of the structure.

The general effect of considering twist action is a decrease in radial deflections at practically all locations, a decrease in maximum arch stress, a decrease in cantilever stress at the downstream face of the dam, and an increase in cantilever stress at the upstream face of the dam. Twist effects are important in most high dams of the curved gravity type, such as Owyhee and Boulder. They are also important in arch dams of more usual size.

8. Temperature Effects.-Effects of temperature changes in the interior of the dam are included in trial load analyses by adding temperature deflections of arch elements to water load deflections before adjusting arch and cantilever movements. No temperature corrections are necessary in cantilever elements since uniform temperature changes in vertical sections do not cause appreciable horizontal movements. Increases in temperature of arch elements cause upstream movements which work against the water load, while decreases in temperature cause downstream movements which work with the water load. Consequently, the maximum probable decrease in temperature, below the temperature existing at the time of grouting construction joints, is the change which usually must be included in the arch analysis. Of course, proper allowances must be made for any setting heat which may not have been generated or dissipated at the time of grouting the joints.

Temperature changes in arch elements vary with the thickness of the arch and temperature changes at the faces of the dam. For the purpose of trial load analyses, temperature changes in the interior of the structure

are assumed to be uniform from abutment to abutment and from upstream to downstream face of dam at each elevation analyzed. In unusually important structures, such as Boulder Dam, effects of nonuniform temperature changes in arch elements may be made by special analyses after trial load studies are completed.

TRIAL LOAD ANALYSES IN DAM DESIGN

9. Procedure. Several trial load analyses usually are required in designing an arch dam by the trial load method. This is particularly true if the site possesses a shape definitely different from any section. previously investigated, or if the profile at the site contains irregularities which cannot economically be removed in the base excavation and which, therefore, may cause stress concentrations in the finished structure.

Ordinarily, the procedure is to adopt a tentative cross-section and make a preliminary plan of the dam, introducing as much horizontal curvature as possible; and then to calculate maximum stresses in the structure by a simplified trial load method. The simplified method may assume tension stresses to transmit internal forces without cracking the concrete, neglect effects of uplift, tangential shear, and twist action, and make an adjustment of arch and cantilever deflections in the radial direction only. Usually a total of five or six cantilever elements and a like number of arch elements, spaced fairly uniformly throughout the dam, is sufficient in preliminary trial load studies. If at all possible, it is desirable to smooth out irregularities of the canyon profile in assuming excavation lines, and to adopt a design symmetrical about the crown section. Deep holes or relatively narrow gorges in the bottom of the canyon can sometimes be considered as plugged with concrete and treated as parts of the foundation instead of parts of the dam.

When the first trial load analysis is completed, tentative dimensions of the structure can be modified as indicated by preliminary stress calculations, and the modified design reanalyzed. This procedure can be continued, using the simplified, radial adjustment method, until dimensions of the structure are such as to give satisfactory stresses in both arch and cantilever elements. The final design can then be carefully analyzed by the amplified trial load method, including effects of uplift, tangential shear, twist action, cracking in the tension areas when calculated tension stresses are more than about 50 pounds per square inch, and, if necessary, effects of unsymmetrical conditions at the site. In the final trial load studies it usually is desirable to analyze about eight or ten cantilever elements and about the same number of arch elements.