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TRIAL LOAD METHOD OF ANALYZING

ARCH DAMS

BY IVAN E. HOUK, SENIOR ENGINEER, BUREAU OF RECLAMATION

THE DEVELOPMENT of the trial load method of analyzing arch dams was begun in the Denver office of the Bureau of Reclamation in 1923. Prior to that time most concrete dams had been designed on the assumption that the entire water load would be carried vertically to the foundations by gravity action in the case of both straight and curved gravity dams; and horizontally to the abutments by arch action in the case of arch dams. The use of the trial load method now enables the designing engineer to analyze the load distribution, deflections, and stresses in curved concrete dams of all sizes and shapes, whether of the massive arched gravity type or the relatively thin, monolithic arch type. Furthermore, an adaptation of the method enables him to analyze beam and twist action in straight gravity dams located at sites where steep canyon walls may render such effects important.

In the case of a few arch dams designed prior to 1923, also a few designed by other agencies than the Bureau of Reclamation since that time, stress analyses were made on the assumption that the water load would be divided between the arch and cantilever elements in such a way as to produce equal arch and cantilever deflections in a radial direction at the crown cantilever. However, in such cases the division of load was assumed to be constant from abutment to abutment at each horizontal element analyzed; and no vertical elements were analyzed except at the crown section. This method might, logically, be called the “arch and crown cantilever” method.

The trial load method is, essentially, an amplification of the arch and crown cantilever method. The trial load method, as now used in the Denver office of the Bureau of Reclamation, assumes that the water load is divided between the arch and cantilever elements; that the division may or may not be constant from abutment to abutment at each horizontal element analyzed; and that the true division of load is the one which will cause equal arch and cantilever deflections at all points in all arches and cantilevers, instead of at the crown cantilever only. Furthermore, the trial load method assumes that the distribution of load must be such as to cause equal arch and cantilever deflections in all directions; that is, in tangential and rotational directions as well as in radial directions. Since the required agreement of arch and cantilever deflections

can only be obtained by assuming different distributions of load and calculating resulting arch and cantilever movements until the specified criterion is fulfilled, the procedure is logically called the “trial load” method.

Having determined the true distribution of load between the horizontal and vertical elements of the dam, the resulting arch, cantilever, and principal stresses may be calculated. Arch and cantilever stresses are usually calculated at the upstream and downstream faces of the dam. Arch

stresses are usually calculated at the crown and abutment · locations, but may also be calculated at other sections

along the arch rings. Cantilever stresses are usually calculated at the elevations of the arches and at the foundation ends of the elements. Magnitudes and directions of principal stresses are usually calculated along the lines of contact between the faces of the dam and the rock profiles. Principal stresses are usually found to act in approximately horizontal directions at the abutment ends of the top arch, in approximately vertical planes at the base of the crown cantilever, and in gradually changing directions at intervening locations between the bottom of the canyon and the ends of the top arch. These calculated arch, cantilever, and principal stresses are considered to be the true stresses caused by the assumed condition of reservoir loading.

DEVELOPMENT OF THE METHOD In the first use of the trial load-method the foundation and abutment rock was considered rigid, and both arches and cantilevers were assumed to have constant unit thicknesses. The arch elements were considered to be horizontal layers having a uniform vertical thickness of 1 foot, and the cantilever elements to be vertical slices having a uniform horizontal thickness of 1 foot. In the earlier analyses the calculations included effects of bending and thrust in the case of the arch elements; and bending, thrust, and shear in the case of the cantilever elements. The next step in the development of the method was the introduction of radial shear effects in the arch computations. Since then the method has been gradually amplified until now the formulas consider effects of radial sides of cantilever elements, twist action, tangential shear, and movements of foundation and abutment rock, as well as the more usually considered effects of thrust, shear, and flexure in the

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concrete elements. The influence of Poisson's ratio is introduced into the present methods of analysis, indirectly, through the use of the shearing modulus of elasticity in calculating shear, twist, and tangential movements. In the case of unusually important dams, further effects of Poisson's ratio are analyzed by special readjustments made after the preliminary trial load calculations are completed.

The formulas used in computing the movements of the horizontal and vertical elements are the usual arch and cantilever beam equations derived on the basis of the elastic theory. Since the trial load method brings the arch and cantilever movements into agreement in all parts of the structure, the procedure - automatically makes proper allowances for irregularities in the rock profile and lack of symmetry on the two sides of the canyon.

BASIS OF TRIAL LOAD METHOD The trial load method of analyzing curved concrete dams as now used by the engineers of the Bureau of Recla

mation is based on the following general assumptions:

1. The rock formations which make up the foundation and abutments at the site are homogeneous and uniformly elastic in all directions, and are strong enough to carry the applied loads with stresses well below the elastic limit.

2. The concrete in the dam is homogeneous and uniformly elastic in all directions and is strong enough to carry the applied loads with stresses well below the elastic limit.

3. The dam is thoroughly keyed into the foundation and abutment rock throughout its contact with the canyon profile; so that the arches may be considered as fixed with relation to the abutments, and the cantilevers as fixed with relation to the foundation.

4. The vertical construction joints in the dam are grouted, or the open joints filled, before the water load is applied so that the structure may be considered to act as a monolith and arch action to begin as soon as the reservoir begins to

fill.

5. The horizontal water load is carried by two systems of

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structural elements; namely, a system of horizontal arch elements and a system of vertical cantilever elements.

6. The horizontal elements have a constant vertical thickness of one foot from abutment to abutment and from the upstream to the downstream face of the dam.

7. The vertical elements are bounded by vertical radial planes a horizontal distance of 1 foot apart at the upstream face in cases where the upstream face is vertical, or a horizontal distance of 1 foot apart at a vertical circumferential plane passing through the upstream edge of the top arch in cases where the upstream face is built on a batter.

8. Both horizontal and vertical elements may be considered free to move under reservoir loads, without restraint from adjacent elements, and then brought into geometric continuity with adjacent elements by the application of selfbalancing tangential shear and twist loads or couples, one set of loads or couples being applied to the arch elements and the balancing set of loads or couples being applied to the cantilever elements.

9. The total horizontal water load is divided between the two systems of elements in such a way as to satisfy the conditions of equilibrium and produce equal arch and cantilever deflections in all directions in all parts of the loaded structure.

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

11. Effects of flow can be adequately allowed for by using somewhat smaller values of the modulus of elasticity than would otherwise be adopted.

TRIAL LOAD CALCULATIONS

During the early work on the development of the trial load method, arch and cantilever deflections were calcu

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lated in the radial direction only. Then the tangential shear deformations were analyzed, and next the twist deformations. During some of the earlier analyses, tangential shear and twist effects were evaluated by successive trial load adjustments and readjustments after the arch and cantilever movements had been brought into radial agreement. However, tangential shear and twist adjustments are now being made simultaneously with the radial adjustments.

In the early use of the trial load method the arches were divided into voussoirs, and the cantilevers into vertical increments, and the total loads, moments, shears, slopes, and deflections, calculated by summation methods. Vertical increments and summation methods, much amplified, are still used in analyzing cantilever elements. However, mathematical formulas, based on circular curves at the upstream and downstream edges of the horizontal elements, are now being used in analyzing arch elements. If the extrados and intrados curves are not concentric, so that

the arch thickness varies from the crown to the abutment, the half arch is divided into four segments, a uniform thickness assumed for each segment, and the analysis made by the use of special formulas derived for the shorter sections. If appreciable tensile stresses are indicated in the arch elements, so that the investigation of cracked arches is considered advisable, analyses may be made by the original summation method. During the last 4 years, analyses of uncracked arches have been greatly facilitated by the compilation and use of tables of arch constants. Analyses of both arch and cantilever elements have also been greatly facilitated by calculating effects of unit loads of rectangular and triangular shape, covering different parts of the loaded surface; so that the effects of practically any shape of arch or cantilever load desired can be obtained by combining the effects of unit loads already calculated.

Concrete stresses are computed on the assumption of a straight line distribution of stress from the upstream to the downstream face of the dam in both arch and cantilever

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elements. In the case of unusually important dams, such as Boulder, effects of the nonlinear distribution of stress are determined by subsequent trial load analyses which bring the horizontal and vertical slopes and deformations in the interior of the arches and cantilevers into agreement in all parts of the dam. Effects of nonlinear distribution of stress have been found important at locations near the foundation and abutments. However, they seem to be negligible in the upper central portions of the dam; that is, in the portions extending from the crown section horizontally to about the quarter points in the case of the arch elements, and from the top of the dam down about half way to the foundation rock in the case of the cantilever elements.

use of the method have been found worthy of consideration, the importance of the effects usually varying with the size and height of the structure. Poisson's ratio adjustments were not found to have important influences on maximum stresses even in the case of Boulder Dam.

Effects of including radial shear in calculating arch deflections were found to be particularly important in the case of thick arches, as would naturally be expected, the importance increasing rapidly as the ratio of the thickness of the arch to the radius of curvature increases. Not infrequently, the radial shear deflections in the lower arches of a massive curved concrete dam are found to equal or exceed the moment and thrust deflections at practically all locations along the arch ring.

The importance of making proper allowances for the radial sides of the cantilever elements increases with the size of the dam and the sharpness of curvature. The downstream base width of a cantilever with radial sides, 1 foot apart at the upstream face, was only 0.496-foot

SUPPLEMENTAL EFFECTS With the exception of the Poisson's ratio adjustments, all the supplemental effects included in the trial load method of analyzing curved concrete dams since the first

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