reservoir head of 350 feet and to safely withstand a maximum nonoperating head of 561 feet. The gate bodies were designed for the internal pressure to be resisted by the reinforced concrete in which the gates are embedded. To reduce the flow disturbances past the slot opening, the width of the fluidway at the downstream edge of the slot was made 1 inch greater than at the upstream edge, with the sides of the downstream body converging the flow to the original width. This practice was based on a paper published in the Proceedings of the American Society of Civil Engineers.2 The bonnet cover was designed to withstand the hydrostatic pressure plus the maximum thrust developed by the hydraulic hoist when seating the leaf.

The average bearing pressure between the leaf and body seats under the maximum reservoir head of 561 feet was limited to 1,675 pounds per square inch (static), and under the maximum operating head of 350 feet was limited to 1,040 pounds per square inch. A 1/16-inch-thick asbestos sheet packing was installed between the leaf and the leaf seals so that more nearly uniform bearing pressure across the seals could be obtained when the leaf deflects under load.

The hydraulic hoists were designed for an oil pressure of 2,000 pounds per square inch with normal factors of safety; however, the relief valve was set at 2,150 pounds per square inch to limit the maximum system pressure. The maximum pressure required to operate the gates will not exceed 1,950 pounds per square inch, at which pressure the hoists will develop a lifting force of approximately 1,400,000 pounds which exceeds the calculated force of 1,200,000 pounds required to lift the leaf and to overcome downpull and frictional resistance. The upper cylinder head of each hoist was designed with a hanger which can be manually engaged with the piston stem to hold the gate open. Each hanger stud is equipped with a break stud designed to support the leaf and piston weight of approximately 35,000 pounds and to break at approximately 92,000 pounds in case gate closure is initiated before the hanger stud is disengaged.

(2) Controls. -To provide the desired operating time of approximately 30 minutes per gate, a pumping capacity of 13.6 gallons per minute at 2,000 pounds per square inch was necessary. Two pumps were provided so that if one pump or motor

square inch for operating and 1,675 pounds per square inch for nonoperating (static) conditions.

(4) Shear.-Allowable design stresses in shear were not more than 0.6 of the allowable design stresses in tension.

(5) Hoist cylinder. -Allowable design stresses for hoist cylinders were based on the recommendations of the ASME Boiler and Pressure Vessel Code-Unfired Pressure Vessels-Section VIII.

B. STRUCTURAL DESIGN OF DAM

1. Analysis of Stress and Stability Factors for Dam

23. GENERAL. The degree of safety of an arch-type dam is defined by the stresses developed in the dam. These stresses can be obtained through mathematical analyses or by model studies. The design of Glen Canyon Dam was developed using a mathematical analysis, termed "Trial-load Method of Stress Analysis." A brief description of this system is presented in the following section.

24. METHOD OF ANALYSES. The trial-load method of stress analysis assumes that the load applied to an arch dam is divided between horizontal and vertical elements in such a way as to produce equal movements in all directions, at points of intersection of these horizontal and vertical elements. Because the required agreement of all deformations can best be obtained by assuming different distributions of load and computing the resulting movements until the specified conditions are fulfilled, the procedure is logically called the trial-load method.

It may be assumed that the dam to be analyzed is divided into a series of arch and cantilever elements by passing through it a series of horizontal and vertical

The horizontal planes defining the arch elements are assumed to be spaced a unit distance apart, and the vertical radial planes defining the cantilever elements spaced a unit distance apart at the axis of the dam. The sum of the arch elements occupies the total volume of the dam, which is also the case with the cantilever elements. Each arch and each cantilever is assumed to move independently of all others, but at the conclusion of the analysis, geometrical continuity must be restored at all points in the structure.

Instead of investigating a large number of horizontal and vertical elements, only a relatively few representative arches and cantilevers are studied to complete the analysis within a reasonable length of time. If the dam is approximately symmetrical about the maximum cantilever section, only half of the structure needs be analyzed, and five to seven cantilevers may be sufficient. If the structure is nonsymmetrical, however, both sides must be analyzed and approximately twice as many cantilevers are necessary.

The trial-load analysis is carried out in steps, generally referred to as adjustments. Three adjustments-radial, tangential, and twist-are necessary to achieve geometrical congruence. The radial adjustment accounts for radial displacement. Tangential movement is brought into agreement by use of the tangential adjustment. The twist adjustment provides rotational congruency about the tangential and vertical axes. When equality of these linear and angular displacements of the arch with those of the cantilever has been achieved at their points of juncture, the requirements for a solution are complete.

The loads on the arch and cantilever elements that produce deformation agreement may be freely chosen with the provision that the sum of the arch and cantilever loads must equal the external load at every point. The external loads include all external forces acting on the dam. These consist of the weight of the structure, reservoir water, tailwater, water, tailwater, temperature changes, earthquake shocks, and silt loads. These loads are divided between the arch and cantilever system until a satisfactory agreement of radial deflections is obtained.

To complete the deformation agreement, it is necessary to introduce internal tangential and twist loads. These are applied in pairs of equal and opposite loads, one acting against the arch and the other against the cantilever. By this means, arch and cantilever deflections may be brought into tangential and rotational agreement without changing the external load on the structure. The internal loads represent forces set up by the interaction between the assumed arches and cantilevers in the dam.

To facilitate the process of dividing the loads between the horizontal and vertical elements, certain patterns of loads called unit loads have been developed. In the case of the arches, these loads consist of a uniform load over the entire length of the arch and

triangular loads varying from a maximum value at the abutment to zero at the quarter points. These loads may be used to represent radial forces, tangential thrusts, or moments, depending on the adjustment under consideration. The unit cantilever loads are assumed to vary from a maximum at one arch elevation to zero at the arches directly above and below. These loads are used to represent radial shears, tangential shears, or twisting moments on the cantilevers, depending on the appropriate adjustment.

In applying the unit loads, it is advantageous to compute the movements of arch and cantilever elements produced by the unit loads before attempting to divide the external load between the arches and cantilevers. The arches are statically indeterminate structures terminating at elastic abutments. Computations of deflections of arch elements are made by the elementary theory of flexure for curved beams, with the effects of rib-shortening, transverse shear, and yielding abutments included. The arch elements resist radial forces applied at the faces, tangential forces and horizontal moments applied at the centerlines, and twisting moments in vertical radial planes. The cantilevers are elastic units, seated on an elastic foundation. They resist vertical and radial forces applied to the upstream or downstream faces, and tangential forces, twisting and bending moments applied at the centerlines.

The total loads resisted by horizontal and vertical elements are determined by the trial-load adjustments. With the loads, stresses may be computed throughout the dam, provided a definite variation of stress between the upstream and downstream faces of the dam is assumed. Three of the stresses-the vertical stress normal to a horizontal plane, the horizontal stress normal to a vertical radial plane, and the horizontal tangential shear stress acting in a tangential direction on a horizontal plane-are assumed to vary linearly between the upstream and downstream faces. Arch and cantilever shearing stresses are assumed to vary parabolically from the upstream face to the downstream face of the dam. These stresses may be computed using the total arch and cantilever loads. From these stresses, principal stresses or stresses on any plane may be computed throughout the dam.

increase the volume of concrete appreciably in an arch dam. Because of the low strength and low modulus of elasticity of the abutment rock at the Glen Canyon site, the dam will require wider abutments than normal to spread the forces over a larger area, and thus reduce the stresses. The abutments yield in an arch dam when the waterload is applied to the dam. This yielding produces increased compressive stresses in the arches at the crown extrados and abutment intrados, and decreases the compressive stresses at the crown intrados and abutment extrados. A reduction in the amount of yielding by the abutment is accomplished by increasing the area of contact between the concrete and rock.

A gravity-type dam could be designed with an overflow spillway, eliminating the necessity of tunnel spillways through the abutments. However, in a gravity dam uplift pressures due to porosity of the rock would be a design factor and an overflow spillway would make it necessary to put the powerplant underground or along the side of the canyon. An arch dam would require less concrete in the dam than the gravity design, but provision for a spillway around the abutments would be required. The powerplant could be underground, or at the toe of the dam.

With these alternatives under consideration, schemes with both straight gravity and thick arch dams were considered. The gravity dam was studied with an overflow spillway and an underground powerplant. Several thick arch dams were laid out using radii to the axis of 1,000 feet and 1,100 feet, with slopes on the downstream face of 0.55 vertical to 1.0 horizontal. An underground powerplant and a powerplant at the toe of the dam were considered with the arch-type dam. The scheme with a powerplant at the toe of an arch dam proved to be the more economical; therefore no further consideration was given the design for gravity dam or underground powerplant.

The early preliminary designs were made for a reservoir capacity of 30,000,000 acre-feet. With this capacity, the normal reservoir water surface was at elevation 3725 and the top of the dam at elevation 3740. Base of the maximum section was assumed to be at elevation 3040. Estimated costs of a dam and powerplant were based on a preliminary design.

In 1956, soon after Congressional approval of the Colorado River Storage project more information on the abutment rock was obtained and a refined reservoir capacity study was prepared. As a result, the normal reservoir water surface elevation was set at 3700, with the top of the dam at elevation 3715. The base of the

maximum section was assumed to be at elevation 3010. Laboratory tests indicated a value of 750,000 pounds per square inch for the modulus of elasticity of the abutment rock and 0.15 for Poisson's ratio. At the abutments a limiting compressive stress of 750 pounds per square inch for a loading condition, including the effects of earthquake, was tentatively set. In the interior of the dam a compressive stress of 1,000 pounds per square inch was the limiting stress established by design criteria.

(a) Design A-6.-Design A-6, the first of the new layouts, is shown on figure 33. The shortest usable radius to the axis of the dam was found to be 900 feet. By thus reducing the radius, more load is taken in arch action and less in the vertical direction. Consequently, the vertical section could be thinned. To reduce tensile cantilever stresses on the downstream face near the top of the dam, the upstream face was curved in a vertical plane. The crown cantilever analysis in design A-6 revealed the presence of compressive stresses in excess of 750 pounds per square inch at the abutments of the top half of the dam. The need for thicker abutments in the upper portion was evident.

(b) Design A-7.-By reducing the intrados radii in the upper portion of the dam in design A-7, the abutment thicknesses were increased. The crown section of design A-6 was retained. A plan and maximum section for design A-7 is shown on figure 34.

A radial adjustment analysis showed excessive abutment stresses at the top of the dam. By leaving that portion of the dam above elevation 3665 ungrouted, thus assuming no load above this elevation to be carried by arch action, stresses from the complete trial-load analysis were found to be well within the limits set at the time. The arch and cantilever stresses parallel to the faces, along with the loading conditions and assumptions used in the complete trial-load analysis, are shown on figure 35.

In reevaluating the strength of the rock, a limiting compressive stress at the abutments of 600 pounds per square inch was established. To reduce the stresses of design A-7 to an acceptable limit, several steps were considered other than increasing the abutment thickness. Temperature in the dam at the time of grouting had been assumed to be 45° F. at all levels of the dam. By varying this temperature from 40° F. in the bottom to 55° F. at the top, the bottom part could support more of the load, while relieving some of the load in the top. The other measure taken was to formulate a construction program that was realistic and

would force the arches and cantilevers in the lower part of the dam to support more of the load. This construction program assumed the dam constructed to elevation 3550 and grouted to elevation 3500. As construction is continued, water would be stored in the reservoir, and when the dam is topped out, the water in the reservoir would have been raised to elevation 3500. The grouting would then be completed, and the water in the reservoir would be allowed to rise to a normal level of 3710.

A comparison of stresses including the effects of the construction program and omitting them is shown on figure 36. At the abutments the arch stresses are reduced from 20 percent to 30 percent in the upper portion of the dam. Although these stresses were improved, they were not considered entirely satisfactory.

To further reduce the critical abutment stresses, the abutments had to be thickened. Since further reduction of the intrados radii did not appear to be feasible, the alternative was to add concrete on the upstream face.

28. SPECIFICATIONS DESIGN-DESIGN A-8. In design A-8, as shown on figure 37, 15 feet of concrete was added to the maximum section at the base and extending up to elevation 3300. From here the face was curved in a vertical plane to the axis at elevation 3710. Stresses listed at the bottom of figure 37 are estimated final stresses based on stresses resulting from a Crown Cantilever Analysis. Since these stresses were acceptable at the time, specifications for construction of Glen Canyon Dam were based on this layout.

29. SOME DESIGNS PREPARED SUBSEQUENT TO SPECIFICATIONS ISSUANCE. After specifications were issued, based on design A-8, a number of additional layouts were made in an effort to reduce the volume of concrete in the dam and produce a more acceptable stress distribution on the abutment rock. Additional tests of the abutment rock resulted in a lower value of the modulus of elasticity, which in turn tended to increase the compressive stresses at the abutment intrados. Stresses in other portions of the dam were conservative and well below the allowable limits. The problem of design resolved into how to increase the abutment thickness while maintaining or reducing all other thicknesses. This could be accomplished by using uniform thickness sections in the central portion of the dam, terminating in short radii fillets on the downstream face.