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MODEL OF CROWN SECTION
OF BOULDER DAM

After the completion of the tests on the plaster-celite models of Boulder Dam, the model was analyzed by the trial load method. This analysis determined the distribution of the water load between the arch and cantilever elements. A further check on the calculations was then made by constructing a model of the crown cantilever section and loading it with the computed cantilever component of the water load. The measured deflections were compared with the observed and calculated deflections for the complete model. Also, following the same method as the previous cross-section model tests, measurements of internal strains and stresses were made.

The plaster-celite model of the crown cantilever section was arranged for loading with gravity forces as in the Norris and Grand Coulee models; but the live load, not being linear, was applied by weights acting on plates attached

to the upstream face. The dead loads were calculated assuming the specific gravity of water and concrete to be multiplied by the factor 13.6. The live load represented the portion of the mercury load which was calculated to be effective on the crown cantilever section. This model was tested under the following loadings:

1. Dead load and computed cantilever component of live load.

2. Dead load, computed cantilever component of live load, full reservoir pressure on the upstream foundation and tailwater pressure.

3. Dead load and full mercury pressure on the cantilever and upstream foundation, and tailwater pressures.

This latter loading represented the conditions which would result if the dam should crack, or the abutments deform, to such an extent as to relieve all arch action.

For each condition of loading, measurements were made of the deflection of the dam and foundation in horizontal and vertical directions; and measurements of strains in the

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dam, at the base of the dam, and in the foundation slab.

The results of the deflection tests for the first loading condition gave a very good check on the original deflection tests of the complete model. The cross-section tests gave slightly larger deflections due to the more extensive foundation provided for the cross-section model. The tensile strength of the model was sufficient to carry the live load, with or without the dead load; and it was interesting to note that the same deflections were observed for the live load, whether the dead load was applied or not.

Figure 6 shows the model with apparatus for applying full mercury load. The dead load weights were suspended on piano wires below the platform shown in the photograph. Figures 7 and 8 show typical results from the tests. The deflection diagram shows the model in its zero or weightless position from which the deflections due to dead and live load were plotted to an exaggerated scale. The principal stresses were calculated from strains observed with the optical strain gages.

From the experimentally determined stresses in the foundations, an imaginary boundary near the concrete support was drawn, making all boundary stresses on the model slab known. This provided sufficient data to make an analysis by Airy's function of the stresses in the interior of the slab. When Airy's function stresses were compared

with the experimental stresses, they were in excellent agreement near the base of the dam, but deeper into the foundation the agreement was not so satisfactory. This result was to be expected since the extreme boundary of the foundation slab was bonded to the concrete frame and its deformation was restrained. However, the Airy's function analysis verified the premise based on the principle of St. Venant that the stress distribution at the base of the model would not be affected by the manner in which the foundation stresses were carried to the concrete support as long as the support was remote from the dam section.

GALLERY INVESTIGATIONS

The tests of the model of Norris Dam gave very limited results for the stresses around the gallery, due to the very small scale of the model. Consideration was then given to a model investigation of gallery stresses that could be generally applied to galleries located in any portion of a gravity dam.

In a large dam there usually are considerable lengths of galleries for inspection and drainage. Consequently, in order to obtain an economical use of reinforcement steel and good structural strength and stress distribution, accurate experimental investigations of gallery stresses are valuable.

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The conditions assumed for the gallery tests are as follows:

1. It is assumed that at the point where the gallery is to be located in the dam section, the principal stresses may be calculated, assuming no gallery exists.

2. It is assumed that a square of material is cut out around the region where the gallery is to. be located, with the sides parallel respectively with the directions of the principal stresses acting at the center of the gallery location. If the square is loaded at its edges with these principal stresses acting as direct loads, the same stress conditions will result as in the actual dam.

3. It is further assumed that the dam section is sufficiently large in proportion to the size of the gallery that no appreciable change in stress occurs within the dimensions of the gallery.

4. The square assumed to be cut out from the dam must be sufficiently large that the stresses at the boundary of the square will not be affected by the cutting of the gallery section.

The model on which these experiments were performed was a 3-foot square slab of plaster-celite, supported in a

vertical plane in a reinforced concrete frame. Four rubber bags, fitting each edge of the model slab, were connected to a compressed air supply which provided the loading medium. Previously, several mechanical arrangements were used to load the model, but were not successful because they introduced shear along the boundary of the slab. The compressed air provided uniform pressures which were equal on opposite edges; so that the rubber bags would deform with the slab and no shearing stresses would result.

In conducting the tests on the model, the first step was to obtain the modulus of elasticity directly from the model by observing strains over a 20-inch gage length with definite loads applied. This was done before cutting the gallery. Following this a circular opening, 5 inches in diameter, was cut in the center of the slab. Strains were observed with the tensimeters around the boundary of the circle and on rosettes located on concentric circles, %, 2){, and 4% inches from the boundary of the opening. From these strains were calculated radial and tangential stresses and principal stresses. This was a very important part of the experiment since an exact mathematical analysis could be made of the stresses around the circular opening, thus giving a valuable check on the experimental method. An exact mathematical analysis of the complete gallery with circular top and vertical side walls would be extremely long and laborious; so, having verified the results for the circular opening, the experimental method was accepted as reliable. Figure 9 shows a general view of the model slab. The full gallery opening was 5 inches wide and 7 inches high.

Four series of tests were run on the model, using four different model slabs with the gallery cut out at angles of zero, 15°, 30°, and 45°, respectively, with the direction of the vertical principal stresses. In the interior of a gravity dam the direction of the maximum principal stress does not vary more than 45° from the vertical. Consequently the test results are within 7%° of any actual case.

For each of the four inclinations of the gallery, six different combinations of loads were applied in the following proportions, the first figure being the horizontal load and the second the vertical load: 1-0, 1-1, 1-2, 1-3, 1-4, 0-4. Strain measurements were made around the entire edge of the gallery and on three gage lines outside the gallery at distances of ){, 2%, and 4)% inches from the boundary of the gallery. From these strain measurements, radial, tangential, shear, and principal stresses were calculated for each of the four inclinations of the gallery under the six loading conditions making a total of 24 different stress conditions investigated.

At present the experimental data are being condensed and simplified in order to make the results of the tests available for use in actual designs. It is of interest to note that a similar investigation was performed with photoelastic models and the results obtained by the two methods are in substantial agreement.

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MODEL OF GRAND COULEE DAM

Among many other technical studies on the Grand Coulee Dam, an analysis was made of the twist action produced by the sloping abutments. A trial load analysis similar to that used for the arch dam studies was made, except that the horizontal elements were fixed-end beams rather than arches. This analysis showed that a considerable portion of the water load near the abutments would be carried by twist or horizontal beam action. To verify this result, a model test was made.

In the actual construction of large gravity dams, the concrete is ordinarily cast in vertical sections, and due to the shrinkage of the concrete, there is very little transfer of stress from one vertical section to another until the dam is grouted. Therefore, the dead weight of the structure affects only the individual vertical sections. If the dam is grouted before the reservoir is filled, the structure will behave as a monolith as regards live load and will be

capable of carrying a portion of the live load by horizontal beam action. Considering this condition, it was unnecessary to include dead loads when designing the mode!, providing the model had sufficient tensile strength.

The model scale was 1 inch equals 20 feet, giving a model 22.88 inches high and 17 feet 2 inches long. Figure 10 shows a cross section of the completed model and figure 11 shows the downstream face with strain gage pins set in place. Mercury was the loading medium for this model as with previous plaster-celite models.

Calculated stresses for this model were 15 pounds per square inch tension on the upstream face and 16 pounds per square inch compression on the downstream face, the dead weight of the model having very little effect. Since the material had a tensile strength of 30 pounds per square inch, any cracking in the model would be due to stress concentrations from twist action and not from ordinary gravity action.

The average base of Grand Coulee Dam was assumed to

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be elevation 850, the top of the spillway is elevation 1,260, and the top of the abutment section is elevation 1,308. The elevation of the reservoir surface during maximum flood was taken at elevation 1,290. The model was loaded to an elevation corresponding to elevation 1,290 on the dam during the first series of tests. On the fourth application of load a diagonal crack developed on the downstream face as shown on figure 12. The location of this crack corresponded to the calculated position of maximum horizontal stress in the beam elements and it was evident that the crack was due to horizontal stresses since vertical stresses due to the reservoir pressure were compressive. The model material had a tensile strength of 30 pounds

per square inch and the Q factor was 17.65, so the corresponding tension in the actual dam would be 529.5 pounds per square inch. Of course, in the dam, the construction joints would open up to relieve this tension, but there is danger of a series of diagonal cracks developing across the individual sections. Diagonal cracks near the abutments have been observed on existing gravity dams.

After the diagonal crack developed on the downstream face, additional horizontal load was thrown on the abutment sections of the dam which caused tension cracks along the abutments at the upstream side. The model did not fail, even after cracking, due to the arch action which developed, consequently a series of deflections tests were run on the cracked model.

It was proposed to relieve the twist effects in the dam by constructing three vertical slots, 8 feet wide, in each abutment section. The slots would be watertight on the upstream face but free to deflect downstream independently. When the reservoir is full, the slots are to be filled with concrete.

The model was rebuilt to include these slots and a series of tests were run on the slotted model. Then the slots were filled with the model under load, restoring the model to

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monolithic conditions with the twist effect partially relieved. Following the routine tests on the model with slots filled, a final test to failure was run. The model with twist relieved carried 47 percent more pressure and developed cantilever stresses 90 percent higher, than the original monolithic model at the two conditions of failure. The model, therefore, was very useful in solving the problem of twist action in long, gravity dams.

MODEL OF ARCH ELEMENT OF BOULDER DAM

In the lower portion of Boulder Dam the arch sections are very thick in proportion to their length and some yield in the abutments is to be expected, resulting in stresses and deflections of an extremely complex nature. After the removal of the model of Norris Dam, the concrete supporting frame was adapted to a test of a model of the arch element at the 900-foot elevation in the Boulder Dam. This model included a considerable portion of the foundation. It was set in a vertical plane in the testing frame for convenience of loading and making observations of strain and deflection. The computed arch component of the reservoir pressure was applied to the arch section and the full reservoir pressure was applied to the canyon walls. The arch load was a minimum at the crown and increased toward the abutments; but the variation was small. Compressed air supplied the uniform pressures and the nonuniform component of the arch load was produced by weights and levers.

When both the arch and canyon wall loads were applied to the model, cracks were produced at the abutments, starting at the sharp corners where the arch intersected the abutments and extended into the abutments. A complete set of deflections measurements were taken on the cracked

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