Stress Analysis in a Gravity Dam through the Use of a Model: A Report Upon Experiments Performed Citation Robinson, Harlan Baird (1931) Stress Analysis in a Gravity Dam through the Use of a Model: A Report Upon Experiments Performed. Master's thesis, California Institute of Technology. doi:10.7907/9md8-r580. https://resolver.caltech.edu/CaltechTHESIS:09102024-210324493 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Civil Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Civil Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 11 June 1931 Record Number: CaltechTHESIS:09102024-210324493 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09102024-210324493 DOI: 10.7907/9md8-r580 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16712 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 12 Sep 2024 18:42 Last Modified: 12 Sep 2024 18:59 Thesis Files PDF - Final Version See Usage Policy. 10MB Repository Staff Only: item control page

STRESS ANALYSIS IN A GRAVITY DAM THROUGH THE USE OF A MODEL. A report upon experiments performed Harlan B. Robinson June 11, 1931 CONTENTS Discussion of Problem Page 3 Report on Experimental Work 8 Table showing Deflections Obtained 12 Transcript of Original Data Sheets 18 A gravity dam is one which depends for its support upon its own weight, such that each section of the dam is supported entirely by the ground directly beneath it rather than by a foundation to the side, as is true in the case of the various types of arch and dome dams. The assumption is made in the knowledge that it is probably untrue in view of the fact that there will be friction between two adjacent parts of any structure, so that some resisting force will be set up to add to the stability of the individual section. Inasmuch, however, as this assumptive error is on the side of safety, one may proceed without danger on the theory that there is no lateral support. For purposes of analysis, gravity darns are customarily divided into sections perpendicular to the horizontal crossstream axis, it being assumed, as stated above, that each section is self-supporting and independent of any and all others. For analytical purposes the width of such a section is taken as unity, and its volume is therefore represented by the area of the cross section itself, and inasmuch as there are assumed to be no forces acting upon the section at the faces where it joins the adjacent section, either longitudinally or in shear any and all forceswhich act upon the unit section may be represented as acting upon its outer edges. It may be shown that one can represent stresses in a large scale prototype of any structure by means of a model on a smaller scale, due to the fact that the deflection and loads upon substances made of different materials are proportional 2. moduli of elasticity of the model and prototype. Therefore, if the modulus is constant for two given materials - that is, if it does not vary from point to point, and under different conditions of loading, results may be obtained using one as model, the other as prototype. This method of analysis is be- coming more and more popular, both as an easy and rapid way of checking analytical calculations and as a means of original stress determination without mathematical analysis. An example of this may be illustrated by the method of determining reactions of a continuous beam over a number of supports under a given loading, taking advantage of the fact, discussed later in this paper, that deflections at one point are to deflections at another inversely as the loads at the two points. A model beam is set up on a board on which the supports are represented by nails driven into a sheet of graph paper from which distances may be readily scaled. ing the beam at any one reaction a unit By deflect- amount, and measur- ing the ordinates at any given point of loading, in the direction of the load, the proportion of the load carried by that individual support is determined directly. This process is carried out for each suppor•t, accomplishing in half an hour what wou+d be a normal day 1 s work in analytical calculation. The percentage of error can be shown to be about two. In the experime:nts carried on by the writer, a celluloid model of a triangular cross-section of a gravity dam having a base-to-height ratio of 0.70 was used, and known loads 3. were applied to it at certain points in order to find the deflections at various other points under varying conditions of loading with an object of determining the internal work which would be performed by the dam. From a study of the dam section, it is seen that the forces acting upon it, neglecting (as mentioned before) longitudinal and shear stresses transmitted from adjoining sections, are three in number, (a) the horizontal force due to the pressure of thewater behind the dam, (b) the gravitational force acting upon the section in a downward direction, and (c) the reaction of the foundation acting so as to oppose the other two, and hence upward and upstream. Inasmuch as the first two are determined easily, the weight of the section and the pressure of the water being k,nown, it follows that the solution of the problem concerns chiefly the resultant forces opposing motion of the dam. These forces may be divided into components acting upward normal to the under surface of the dam, and those acting horizontally, causing shearing stress between the dam and its foundation. The summation of the shearing stresses at the bottom of 2 the darn equals the summation of the water pressures, or wh , in 2 which his the depth of water at the upstream faceoof the dam and w is the unit weight of water, that is, the weight of a unit cube, inasmuch as thesie tv10 opposing forces are the only ones existing which have a horizontal component. It is the variation of this shear1ng stress along the base which is one of the ultimate purposes to which these experiments are preliminary. 4. Considering the differential cube at the base of the upstream face, it is seen that, inasmuch as there can be no appreciable shear as between water and a solid, the shear along the upstream vertical face is necessarily zero. Inasmuch as the shearing stresses on two adjacent faces of a cube must be equal in magnitude, there is no shear at the base of this small cube; in other words, the shearu.ng stress at the upstream face of the dam is zero. It may then be considered as varying from zero up to a maximum value at the toe, along some curve whose shape is unknown. Considering vertical reactions, if moments be taken at the base directly beneath the resultant of the weight, at which point the weight and the b ase shear have no moment, it becomes apparent that the resisting ground pressure has a result~ht on the downstream side when the reservoir is full, which is the condition assumed. The ground pressures are therefore greatest near the toe of the dam. There can, of course, be no tension be- tween the dam and its foundation, as this would tend to form cracks which would permit a water pressure under the dam te nd ing to overturn it, but for economic reasons, the best dam is one in which the bearing stress at the upstream face approaches zero. For this reason, the bearing stress may also be assumed to follo\11 some curve of unknown shape, fro m zero at the upstream face to a maximrun value at the toe. The ultimate purpose of these experiments, but one which is beyond the scope of this paper, is to find the curves which these two stresses follow. The stress is temporarily assumed to follow a straight line whose end points are determined such that the area under the line represents the force which is resisted. To this straight line is to be fitted a parabola such that an equal area will be above and below the line, and hence the area under the curve as fitted will be the same as that under the line. The parabola will have the form y-= .-l(. ( x2 - a2 ) in which y and x are the ordinate and abscissa, respectively, along the force line assumed, and may then be referred to the same co-ordinates as is the equation of that line. In the analysis it is necessary to consider the principle of least work, which may be used frequently in the solution of problems that may not be solved solely by means of the three fundamental conditions of static equilibrium, as in the simple case of a table or chair with four legs, which having one more unk~o\m than there are equations, is t h erefore .redundant and must be solved by some means involving indeterminate structures. The principle of least work states that the stresses in a redundant structure have values such that the internal energy of all the stresses is at a minimum. The proof of this is given by Merriman as follows: Let equilibrium. P, ., P"t. " P, • • • • • P_ be n forces in Then let a small internal displacement be made without performing work on the system as a whole such that the point of application of each of the fo1•ces move through small distances As no outside work was done the total work is equal to zero, hence A law of mechanics states that fol" cases of stable equilibrium the integral of this expression is at a minimum, so that a minimum in which . . . . e-. are e, the total distances through which the various forces have moved in reach:tng a state of equilibrium. The work done by each of the forces Pin increasing from o to P is therefore Pe, and equals the energy stored by the internal t:, stresses which balance the forces. Therefore the internal energy of the entire system is at a minimum. It may be shown that the internal energy which is stored in a system is exactly equal to the external work to which the system is subjected. By the use of this fact, it is possible to determine the internal energy of the system through an analysis of the external work done upon it. In the case of the dam, this work consists of half the surmnation 6f the loadings times the distance through which each acts. Inasmuch as it is i mpossible to determine the movement of each point when the entire section is under stress, an approximation must be made, even using the model. In this case, points were selected along the two faces of the dam upon which stresses act, fol" determination of the movement of each of the points under various conditions of loading. The de.fiections under different loads are additive; t hat is, if the model has a deflection g at a given point under a load at one point, and a deflection Q under 7. a loading at another, it will have a deflection a~ b under the combined loadings at the same time, and this may obviously be extended to include loading at all points. The law of reciprocal deflection, which was cited briefly in an example given, states that if a given amount of work is done at one place, the sunnnation of the work done at a number of other places is the same if the body remains in equilibrium. Hence if a weight is applied at one point, cauaing a certain deflection, if there are a number of oth er forces acting upon the body the applied work will be equalled by the work done on the other forces. This fact may also be reversed, stating that if a number of different forces do work on a body, work may be done on an applied force at another point. The application of this experimental data is an extension of this latter principle. A load is actually applied at a number . of given points, noting in each case the deflection at other points, as well as the actual work performed by measuring the deflec t ion under the applied load. The reversal of this process is carried out in the analysis. By studies involving the resultants of the water pressure and of the gravitational action at any of the other points, it is possible to arrive at the work which would actually be done upon the model, using the deflection obtained for the horizontal and ve r tical loadings at the base. A simple proportion then exists between the work actually done upon the model, and the theoretical work which would be done upon it under full load, as by summing the amounts of the theoretical work performed a re- 8. sult is obtained which is to the actual work done as the reaction to the theo1"'etical applied loads is to the i!.:oad actually applied to the model. In this way, it is possible to obtain the resultant thrusts and shears at each of the control points for any given dam. By plotting these, a curve may be drawn through t hem, and then related, as previously suggested, to a parabola div~rging from a straight line. Experimental Work. The model set up was a celluloid triangle 15 inches long, 10½ inches along the base, and about 0.075 inches in thi ckness. 'I'his represented a cross-sectional area through the dam , the vertical face bein g considered as the upstream side. The base, and the vertical edge were each divided into ten parts, the nine divisions being marked by lines ruled on the face of the triangle. Small bridges, a part of the Begg s deforrneter apparatus, were us~d to support the pins which held the two corners of the base. The pin holes were drilled exactly at the intersection of the edges prolonged, at the centers of small circles of 1/5 11 radius. Further supports for the rest of the triangle were pro- vided in the form of small glass blocks which prevented it from excessive sagging. A Bausch and Lomb measure the deflections. microscope, no. 1412, was used to A small dot of ink was placed on the triangle adjacent to the point whose deflection was desired, the cross-hair being brought tangent to this dot of ink for each reading. Five one-pound weights, wired together and connected by means of a flexible trout cord to a wire stirrup which passed around the triangle a--E£1- could be applied at any of the nine points on the base of the triangle, at each of which there was a notch to hold it in place. The line passed from the triangle over a moveable glass rod at the edge of the table, so that the weights were free to hang vertically. A drawing stool was used to rest the weights upon during'no load' conditions. In operation, the microscope was kept at one place throughout each series, the weight being moved from point to point. This was obviously because of the greater ease in moving the weights, although perhaps for experimental purposes the reverse would have been slightly superior because of the advisability of obtaining all results at any one point under approximately the same conditions. The procedure in taking any set of readings was as follows: 1) With no load, the cross-hair was brought tangent to the point. 2) The load was applied by releasing the weights. 3) The reading of the scale was taken and record- ed to tenths of the least reading (To nearest ten-thousandth of one inch. 4) The cross-hair was set on the new position. 5) The load was removed, by placing the weights upon a 10. rest so that there was no tension in the cord. 6) The scale was again read and recorded for load conditions. 7) Same as ( 1 ) etc., etc. This procedure continued until five readings had been taken under load and under no load conditions. The results were inspected, and if no wide discrepancy appeared in the readings, an average was taken for each condition. These averages were then compared to determine the deflection to be recorded. Deflections were first obtained for the normal reactions, measurements being taken for points along the base. Due to the impossibility of making direct readings to give the shear on the base, it was found necessary to apply the loads at an angle and then to correct for the disci--epancy, finding the horizontal component of the angle. Loads were applied at 45°, deflections be- ing taken along the base both horizontally and vertically, and horizontally along the water face, for all positions of the loads. An inspection of the set up will show that multiplication of the deflections by [2 gives the same result as would a corresponding increase of the weight, which, in turn, is equivalent to two forces of the original weight acting, one tangantially, the other normal. By subtracting the deflections already found due to the normal load, it is theoretically possible to obtain the effect of the pure tangential load. However, it was found that due to absolute bending of the model due to the applied load, results obtained in this way as due to the 11. pure tangential load were so nru.ch in error as compared vvi th the probable true deflection that this method was abandoned, and the horizontal components used along, without correction. The data recorded includes a tabulation of the deflections at each of the control points, in ten-thousandths of an inch, resulting in a five pound load applied horizontally and vertically at each of the nine points along the base. This is now in a condition to be used in further calculations for any particular dam, according to the methods as outlined before. It is felt that the results obtained would have been much better had it been possible to obtain them from a set up which did not involve bending in the triangle to so great an extent as did the conditions used, in which it acted largely as a beam supported by the two pins. Some bending is inevitable, and indeed the success of the problem depends upon it, but the writer feels that it was largely due to this fact that it was impossible to obtain more perfect figures for the tangentially loaded deflections, as in a beam of varying moments of inertia, the direction of application of the load becomes more involved than was assumed to be the case. In conclusion the writer wishes to express his appreciation for the help of the staff of the California Institute, and particularly that of Prof. Eugene Kalman, under whose direction the entir•e series of experiments were carried on. O£FL Ee. TION.5 LoAO A-T DUI: TO D!:FI..Ft..,TfoN AT NOR...M4L.. 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