TC547 Cornell University Library TC 547.C91 Engineering for masonry dams, 3 1924 004 025 213 (Cornell Mniuetaitj} ffithratij 3tf?ara, Sfetn IJatk BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF HENRY W. SAGE 1891 ENGINEERING LIBRARY Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924004025213 ENGINEERING FOR MASONKY DAMS ENGINEERING FOR MASONRY DAMS BY WILLIAM PITCHER CREAGER, C.E. Mexbek, Ajikkicax Socasrr of Cmi. Exgixkebs FIRST EDITIOX >TEW YORK: JOHN WILEY & SONS, Inc. Losdos : CHAPMAX & HALL, Limited 1917 Copyright, 1917 BY WILLIAM PITCHER CREAGER PRESS OF BRAUNWORTM & CO. BOOK MANUFACTURERS BROOKLYN- N. V- PREFACE In reviewing this book the reader will probably be impressed with the fact that many of the fundamental assumptions of design are based on very obscure data. This is particularly true regard- ing uplift pressure, ice pressure, and the distribution of stresses in high dams. It is believed, however, that such assumptions are all on the side of safety ; for each of the recorded failures of masonry dams may be attributed to a violation of one or more of the standard rules on which the theory is based. It is considered that the methods of design described, and the assumptions recommended, represent present conservative prac- tice, and correspond to a proper degree of safety for the average enterprise, and where considerable damage to property and loss of human life would result if failure occurred. Some of the designing assumptions, and particularly those for the usual unit working stresses, may seem ultra-conservative, as compared with those allowed for other masonry structures. It must be remembered, however, that the theory on which the design of masonry dams is based, is not exact, and, moreover, has not been verified by satisfactory experiments. The failures which have occurred have furnished valuable information regard- ing some of the limiting conditions, but not all. For instance, no failure due to crushing of concrete masonry has been recorded, arid it is not known just what margin of safety the usual work- ing stresses afford. Designs of extremely radical tendencies are being made continually, particularly for dams in unsettled regions, and where there are no government restrictions. There is a possibility, therefore, that future failures will furnish much needed information in this respect. For valuable suggestions, criticisms and other help, the author desires to acknowledge his indebtedness to Messrs. H. L. Coburn, A, A, Conger, A. S, Crane, A. D. Flinn, N. C. Grover, R, C. Lat- vii viii PREFACE imer, Daniel Moran, G. S. Thompson, and many of his office associates. Special thanks are due to Mr. J. W. Van Demburg for generous and intelligent assistance in the preparation of examples of design and in research work. Much information was obtained from several branches of the Federal Government, many of the State and Municipal engineer- ing departments, and from engineering literature, as noted in the text. Acknowledgment is also made to Mr. T. J. McMinn and Mrs. W. P. Creager for invaluable assistance in the editorial work of publication. CONTENTS CHAPTER 1 Investigations and Surveys PAGE 1. The Choice of Location 1 2. The Nature of Investigations 3 3. Preliminary Investigations 4 4. Final Investigations 6 CHAPTER II The Choice of Type op Dam 5. General Considerations 11 6. Solid Gravity Masonry Dams 12 7. Hollow Gravity Masonry Dams 13 8. Arched Masonry Dams 14 9. Embankments 14 10. Timber Dams 15 11. Other Types 15 CHAPTER III Forces Acting on Dams 12. Nomenclature 16 13. General Considerations 19 14. External Water Pressure 19 15. Internal Water Pressure. Uplift 25 16. Earth Pressure 33 17. Atmospheric Pressure 35 18. Ice Pressure 37 19. Wave Pressure 40 20. The Weight of the Dam ." 40 21. The Weight of the Foundation 42 22. The Reaction of the Foundation 42 ix x CONTENTS CHAPTER IV Requirements fob Stability op Gravity Dams PAGE 23. Causes of Failure 48 24. Rule 1, Governing the Location of the Resultant 49 25. Rule 2, Governing the Inclination of the Resultant 50 26. Rule 3, Governing Compressive Stresses 52 27. Rule 4, Governing Tension in Vertical Planes 56 28. Rule 5, Governing the Margin of Safety 57 29. Rule 6, Governing Details of Design and Methods of Construction . . 58 CHAPTER V General Equations for Design of Gravity Dams 30. General Considerations 60 31. Equations for Rule 1 64 32. Equations for Rule 2 70 33. Equations for Rule 3 71 34. Equations for Rule 4 75 35. Equations for Rule 5 76 36. Equations for Rule 6 77 CHAPTER VI The Design of Solid Non-Overflow Gravity Dams 37. General Considerations 78 38. Example No. 1. 200-ft. Solid Non-Overflow Dam 79 39. Example No. 2. 102-ft. Solid Non-Overflow Dam 99 40. Comparison of Non-Overflow Dams 104 CHAPTER VII The Design of Solid Spillway Gravity Dams 41. General Considerations 105 42. The Shape of the Crest 105 43. Discharge Capacity HO 44. The Bucket '.'.'.'..'....'... '. 115 45. Example No. 3. 91-ft. Solid Spillway Dam without Ice Pressure .... 117 46. Example No. 4. 87-ft. Solid SpilwayDam with Ice Pressure 118 47. Example No. 5. 30-ft. Solid Spillway Dam .120 48. Comparison of Solid Spillway Dams 130 CONTENTS xi CHAPTER VIII The Design of Hollow Dams PAGE 49. General Considerations 132 50. Example No. 6. Hollow Non-overflow Dam ] 40 51. Example No. 7. Hollow Spillway Dam 145 CHAPTER IX The Design op Arch Dams 52. General Considerations 148 53. Arch Stresses 149 54. Vertical Beam Stresses 157 55. Recommendations for Design 158 56. Details 160 57. Multiple Arch Dams 164 58. Allowed Stresses 165 59. Examples of Arch Dams 165 CHAPTER X Preparation and Protection op the Foundation 60. General Considerations 172 61. Rock Foundations 172 62. Earth Foundations 181 CHAPTER XI Flood Flows 63. General Considerations 194 64. High-water Marks 195 65. Comparison with Other Rivers 196 CHAPTER XII Details and Accessories 66. Masonry for Dams 203 67. Water-Proofing 205 68. Contraction Joints 206 69. Drainage Systems 210 70. Architectural Treatment 211 71. The Regulation of High- Water Surface 214 ENGINEERING FOR MASONRY DAMS CHAPTER I INVESTIGATIONS AND SURVEYS 1. The Choice of Location. A dam is usually a unit in a more or less extensive project involving the construction of a number of structures of various types. The general location, therefore, is fixed by factors varying with the purpose of the project, and is affected by considerations but remotely allied to the text of this book. Bearing in mind the fact- that the general location to be adopted is that which, at reasonable cost, will be best suited to the purpose for which the dam is intended, we have only to consider here those factors which affect the cost and safety of the dam, and the choice of its exact position. The general location having been chosen, the exact position will be fixed after careful consideration of each of the following factors: a. The character of the foundation; b. The configuration of the earth and rock surfaces at the site and its effect on the length of the dam, quantity of material to be excavated, and other factors; c. Availability and character of materials for construction; d. The value of the necessary lands and water rights; e. Requirements as to coffers, pumping, conduits, and other provisions necessary for unwatering the site; /. Transportation facilities and the accessibility of the site; g. Availability of suitable sites for construction equipment and camps; h. The safety of the structure. The foundation * is one of the most important factors in the final location. It should be practically impervious, or capable * See Chapter X. 2 INVESTIGATIONS AND SURVEYS [Chap. I of being made so, and have sufficient strength to sustain the weight of the dam and prevent sliding. The valley should have sufficient width at the crest of the proposed dam for the requisite spillway,* and in some instances for other structures, such as the power-house in a water-power development. A greater width than that necessary for the spillway and other structures is, of course, undesirable, because of the greater cost of a long dam. The configuration of the earth and solid rock at the site affects greatly the cost of the dam, because of its influence on many items of cost, such as excava- tion, coffers, and masonry. The location of sand, and stone or gravel for the masonry, the cost of quarrying, the facilities for transportation to the concrete mixers or stone yard, the quality of such material and its effect on the strength, durability, and appearance of the masonry; the hardness of the stone and its effect on the cost of crushing for concrete or shaping for ashlar facing, all influence the cost of the work and therefore the choice of location. An important item in the cost of the project will be the value of the lands to be provided for the dam, for the reservoir area, for construction equipment and camps, and for the right of way for a construction highway or railroad to the site. The in- crease in the elevation of the water surface by the dam may necessitate the relocation of existing railroads or public highways. In some instances the abandonment of towns and villages has been necessary. The removal of water from the site of the dam, in order to facilitate construction, commonly called " unwatering " the site, often involves a large percentage of the total cost of construction. If the depth of water at the site is considerable, and if it is prob- able that floods will cause great increases in the stage of the river during the period of construction, the cost of the coffer-dams may be excessive. Many otherwise attractive sites have had to be abandoned on account of the probable great cost of unwatering caused by the great depth and velocity of the water and the dif- ficulties to be overcome in constructing and sealing the coffer- dams. If the construction is to be accomplished at a reasonable cost, it is important that the site be accessible for the transportation * See Art. 43. Art. 21 THE NATURE OF INVESTIGATIONS 3 of plant, materials, and supplies. It is often necessary to con- struct a highway or a railroad from the nearest shipping point to the site. The cost of constructing, maintaining, and operating the highway or railroad is a controlling factor in the choice of location. A convenient site of adequate size for the construction equip- ment and camps is essential for economical construction. The area required for this purpose varies considerably, depending on the size of the structure, and the desired rate of progress in con- struction. A typical site for the construction plant is a broad, fiat stretch of land adjoining and just below the dam site, and above maximum high water. If the materials for the masonry are to be delivered to the dam by cars passing over the coffer- dams, the equipment should be nearly at the same elevation in order to permit of moderate grades. If, however, delivery is to be made by cableway stretched across the valley, or by chutes, or compressed air, most of the plant will be more economical at a higher elevation. In cold climates, floating ice sometimes enters the reservoir faster than the slack vater can cany it to the dam. If the quan- tity of ice is sufficiently great, and other conditions are favorable, an ice jam will form, and may be greater than has ever been experienced under the natural conditions of the river. This may cause back-water and damage to property, perhaps for miles above. The probability of such ice jams should be given con- sideration in fixing the site of the dam. In general, a location below, rather than above, a community of considerable size is to be preferred, principally because of the necessity of providing a greater margin of safety where a failure would result in great destruction of property or an appalling loss of life. Another consideration which, for storage dams, affects the choice of general location, rather than the exact position, is the quantity of silt carried by the stream. In some cases this is enormous, and may, in the course of a few years, completely fill the reservoir and destroy its usefulness. Sluices in the dam are never effective in preventing silting of the reservoir, except near the dam. 2. The Nature of Investigations. | In considering a site for a dam, a preliminary investigation or reconnoissance must be made, 4 INVESTIGATIONS AND SURVEYS [Chap. I in order to determine whether or not the project is feasible. This may include considerable surveys, if a preliminary estimate of cost is necessary. The final investigation is made after the project has become active, and includes all necessary studies up to and sometimes after the start of construction. There are often many intermediate steps which partake of the nature of both of these divisions of the investigations, so that a clear distinction cannot be made. The extent to which the pre- liminary investigations should be carried will depend on the nature of the project and the information necessary for a complete report. 3. Preliminary Investigations. The preliminary examina- tion of a dam site should be made with the following objects in view: a. Advising as to the general adaptability of the site to the purpose of the project, and as to the desirability of pro- ceeding with further investigations if other features are favorable; b. Locating one or more possible sites for the dam; c. Determining the character and extent of future investi- gation to be made; d. Obtaining information on which to form a basis for a pre- liminary and approximate estimate of cost. If results of instrumental surveys are not available, as is gen- erally the case, the preliminary investigation should be as com- plete as it is possible to make it, without going into detailed sur- veys, as these will constitute a part of the final investigations which will follow if the preliminary report is favorable. The engineer should see every part of the proposed site or sites, make full notes, and take photographs or everything that may be of interest. Unless he is experienced in such work, he is likely to regret, when it is too late, that he did not record certain features which at the time of the investigation appeared to be of insuf- ficient importance to warrant attention. At each prospective site a study should be made of the mate- rials of the foundation as far as it is possible to make such study from surface indications. If a foundation of rock is expected, the elevation of rock surface, the dip, direction, and character of the Art. 3] PRELIMINARY INVESTIGATION'S 5 strata and the probable quantity of overburden should be esti- mated. It is seldom that the exact character of the foundation can be ascertained from surface appearances, but sometimes a reasonable estimate can be made, especially if the foundation is to be of earth, or if rock is exposed throughout the width of the valley and the stream is small. In case of a proposed important and expensive structure, where an accurate estimate of cost must be made, it will ulti- mately be necessary to complete the examination of the founda- tion by digging tost pits and making borings, in order to ascertain the elevation and character of the rock. In his preliminary inves- tigation, the engineer should decide, therefore, on the number and location of test pits and borings to be made later. Possible locations for construction plant and camps should be sought, and their positions, estimated areas, and elevations recorded. The problem of unwatering the t-ite should be care- fully considered on the ground, and notes made as to the method to be adopted, the approximate depth of water, the estimated height cf coffer-dams, and the effect of the velocity of the cur- rent and the nature of the foundation on their construction and the work of making them tight. Specimens of materials available for construction and to be removed from the foundation should be obtained and preserved. Rough estimates should also be made of the area and value of the land to be used, of the length and cost of railroads and public highways to be relocated, and of the area and cost of clearing and grubbing the reservoir, if required. Search should be made for suitable stone, sand, gravel, timber. and other materials for construction purposes. The suitability of materials to be excavated from the foundations for use in con- struction should be considered. The quantity of timber required for coffer-dams and forms for concrete is generally so large that it is necessary to obtain this material from local sources of supply. if it is possible to do so. In cases where the work is of great mag- nitude, and very remote from the nearest shipping point, it has been found feasible to manufacture locally the cement for the masonry. Search should be made for high-water marks, as an aid in estimating the magnitude of the maximum flood * and the prob- * See Chapter XI. 6 INVESTIGATIONS AND SURVEYS [Chap. I able highest elevation of water below the dam, in order that its effect on the design and on the temporary and permanent works below the dam may be properly evaluated. The accessibility of the site should be carefully investigated. The condition of existing highways, railroads, bridges, and navi- gable waters should be examined, having in mind the transporta- tion of materials and supplies. If it is probable that the exist- ing means of transportation are inadequate, or incapable of being made satisfactory, reconnoissance for one or more possible loca- tions for a highway or railroad should be made. The site should be studied for the purpose of arriving at a tentative conclusion as to the best type of dam to be built. Limited surveys, including cross-sections of the valley, showing water surfaces, rock outcrops, and other information will be nec- essary if a preliminary rough estimate of cost is required. 4. Final Investigations. The final investigations are usually carried on under the supervision of the engineer who has conducted the preliminary examinations, or at least in accordance with his recommendations. The principal items are, a. To determine the relative merits of two or more sites for the dam in question, so that a final location can be adopted; b. To settle beyond a doubt the nature of the foundation, as affecting the safety and cost of the dam; c. To fix the limits of the lands to be controlled for flowage, for the sites of structures, and for other necessary pur- poses; d. To determine the length and character of relocation of railroads and public highways necessary on account of raising the water surface; e. To ascertain the character of the Government regulations to be observed; /. To obtain sufficient information for an accurate estimate of cost; g. To fix the final location of the dam, construction equip- ment, camps, coffer-dams, construction highways, and railroads, as well as the probable source of materials of construction, and all other information needful to the constructing engineer; Art. 4) FINAL INVESTIGATIONS 7 h. To obtain all necessary information affecting the design of the dam. Usually, there are not many sites to consider in the final exam- ination. Of all that are available, an intelligent preliminary examination usually reduces the problem to a consideration of a very few, and, in many cases, where the problem is simple, it is sufficient for a final choice of site. Thus, a rigorous investigation of two or more sites is not always necessary in order to make a selection; in fact, such a requirement is the exception, rather than the rule. At any rate, it is never necessary to conduct the final investigation throughout its whole scope for more than one loca- tion, as the final choice of site can be made before the investiga- tions have proceeded very far. The foundation is one of the most important features to be investigated. If it is to be earth, a series of test pits or borings should be made, in order to determine the nature and extent of treatment which will be necessary to produce a stable and prac- tically impervious footing. The investigations should include tests to determine the probable bearing power of the earth, or the number and length of bearing piles which will be necessary, and the nature and length of sheet-piling to be driven to prevent excessive leakage. Except for low dams, the foundation should be rock. For rock the investigation should fix, not only the depth of the ledge below the surface, but its nature or suitability for a foundation. This is usually accomplished by test pits or wash-borings to rock surface and core drilling into the rock. With the latter it will be possible to determine, from the samples produced, not only the character of the rock, but the location of pervious or soft pockets, seams, and other faults, by the action of the drills during the boring operations. Water forced into the drill holes under pressure will indicate the perviousness of the foundation.* Too much stress cannot be placed on the necessity of making extensive investigations of the foundations before construction begins, if an accurate indication of the construction difficulties, the cost of the work, and the tightness of the foundations is desired. A very good description of modern methods of conduct- ing wash and core borings is to be found in Chester W. Smith's " Construction of Masonry Dams."t * See Art. 61. t McGraw-Hill Book Co., Inc., 1915. 8 INVESTIGATIONS AND SURVEYS [Chap. I The problem of determining the proper location and depth of borings and the interpretation of the results obtained is very dif- ficult. It is impossible to give a general rule for the spacing and depth of borings, as each case presents a different problem. All probable zones of weakness, such as the plane of contact between intrusive igneous rock and rock of sedimentary origin should be thoroughly investigated. It should be remembered that a valley, in the making, will usually be started at the weakest part of the geological formation. One is, therefore, likely to find at every dam site a geological reason for the particular location which the stream has adopted, and such indications will assist materially in the determination of the zones requiring more thorough investigation. The spacings and depth of borings will be governed, not only by the geological formation, but also by the height of the dam, and the extent to which the importance of the work will justify the expense of the investigation. A careful study of the data obtained from the borings as they are drilled will influence the depth and location of additional borings. In important work, particularly for high dams, the services of an expert geologist should be obtained, and his recommendations should be given full consideration. The danger of excessive leak- age is not confined to the vicinity of the dam; such leakage may extend for considerable distances on each side of the dam site, and even through the ridge to an adjacent valley lying at a lower elevation. Surveys should be made of the lands necessary for the site of the reservoir and of the dam, and for various other purposes. The results of these surveys should be indicated on maps of adequate scale, depending on the value of the lands to be obtained and, therefore the accuracy with which such information must be shown. These maps should indicate: a. The original edge of the stream at low, ordinary, and high water; b. The edge of the proposed reservoir at low, ordinary, and high water, usually with a sufficient number of inter- mediate contours to enable an estimate of the storage contents to be made; c. The location of the dam, the sites for construction equip- ment, camps, and other proposed structures; Art. 4] FINAL INVESTIGATIONS 9 d. The property lines of the lands required. For the res- ervoir, a distance from the original shore line to the edge of the survey equal to two or three times the dis- tance from the original to the new shore line will usually be sufficient. However, where part of the land required includes a large percentage of an individual estate, it may be necessary to purchase the entire tract. In this event the surveys should be extended to cover the total area. e. The location of all railroads, public highways, bridges, dams, important structures, estates, enterprises, and other items of importance on and in the neighborhood of the proposed reservoir and dam site; /. The proposed new location of railroads and public high- ways which must be relocated .on account of raising the water surface, showing the boundaries of the lands required for such purpose; g. A brief description of each parcel of land required should be placed on the maps, but a complete description should be given in a supplementary report, including the ap- proximate value of lands, water rights, structures, etc., to be taken over. Government requirements will include the observance of ripar- ian rights, if the water is to be retained or diverted; the necessity of providing log chutes, fish ways, or navigation locks; regulations covering the design and construction of dams, and allied matters. The Federal Government has jurisdiction over all navigable streams; also in most States it is required that the plans for the dam must receive the approval of the State Engineer or other official before construction starts. A map of the dam site should be prepared showing, a. Contours of the natural surface within and near the area to be occupied by the dam, construction equipment, camps, and other proposed structures. The required contour interval will be determined generally by the height of the dam. Usually, an interval of about 5 per cent of the height will be found to be sufficiently close. 10 INVESTIGATIONS AND SURVEYS [Chap. I 6. The location of the river, existing and proposed structures, coffer-dams, test pits, borings, rock outcrops, high- water marks, proposed quarries, sand banks, and all other items of use to the designing and constructing engineers. This map should be accompanied by cross-sections of the site at frequent intervals, showing the depth of test pits and borings, and the estimated elevations of all . underground materials and their nature. For estimating purposes, contour maps and profiles of the rail- roads and public highways to be relocated, as well as the railroad or highway for providing access to the site, may be necessary. The results of the final investigations should be incorporated in a complete report, and this should be carefully preserved, as years may elapse between the time of making such investigations and the beginning of construction operations. CHAPTER II THE CHOICE OF TYPE OF DAM 5. General Considerations. It is not within the scope of this volume to describe in detail all the advantages and disad- vantages of the different types of dams which have been built, but merely to outline briefly the adaptability of each type, in order that the reader may be able to comprehend the relative limitations of masonry dams. The usual types of dams may be summarized as follows: Solid gravity masonry dams, Hollow gravity masonry dams, Arched masonry dams, Earth and rock embankments, Timber dams, and Other types. Many combinations of these types have been constructed. The choice of the type best suited to a particular location or use is a matter on which experienced engineers will often differ considerably, and is quite often purely a matter of judgment and experience. However, an intelligent study of the existing condi- tions and requirements will assist materially in the choice. Safety, of course, is the first consideration. It is impossible to build with safety some types of dams under certain conditions of foundations and other characteristics of the site. Consideration of this question will often decrease considerably the number of possible types from which to choose. The first cost of the structure, as affected by the availability and price of construction materials and other characteristics of the site, is, perhaps, of next importance. The choice of type is often limited by the funds available for construction of the dam and other requirements of the project. It will sometimes be found that the difference in cost between an expensive, permanent dam and an inexpensive structure of short H 12 THE CHOICE OF TYPE OF DAM [Chap. II life and high maintenance charges, if set aside at compound inter- est, will be more than sufficient to provide funds for the higher maintenance cost and a sinking fund to cover the rapid deprecia- tion of the less expensive type. It may be said, however, that, in general, the most permanent dam will be found to be the most economical, and it is usually adopted for ordinary sites, unless the structure is for temporary use, or if sufficient funds are not avail- able. A comparison of the several types of dams follows: 6. Solid Gravity Masonry Dams.* There is no type of dam more permanent than one of solid masonry, nor does any other type require less for maintenance. It is adaptable to all localities except where a sufficiently impervious cut-off at and below the surface is impractical of attainment, where there is danger of con- siderable uplif t, or where the low bearing strength of the foundation prohibits its use. It is imperative that high masonry dams be built on rock foundations. Low dams of this type are some- times built on earth or piles, but such support, for dams more than about 30 ft. in height, should be adopted with caution. The solid gravity masonry type, being the most common of all masonry dams, is the safest, according to the popular idea; and, in this respect, has an advantage when the enterprise is affected, to a large extent, by public opinion, as in municipal or other public works. The difference in first cost between solid and hollow gravity masonry dams is the subject of considerable debate. The solid dam requires less cement per cubic yard of concrete, less form work, less expense in placing concrete, and has no steel reinforce- ment. On the other hand, the hollow dam requires considerably less concrete per linear foot of dam.f It is the author's opinion, based on a number of comparative estimates, that for a remote location, where materials of construction are expensive, the hollow type will usually cost less to build than the solid type; but, in an ordinary location, comparatively near a railroad, where there is a good quarry, and a sand bank is convenient, the reverse is true. The solid gravity masonry dam will usually cost more than a timber dam. However, this may not be the case if a first-class, * See Chapters VI and VII. t Usually from 35 to 40 per cent of the concrete required for a solid dam. Art. 7] HOLLOW GRAVITY MASONRY DAMS 13 rock-filled, timber crib dam is adopted at a site where a coffer- dam is required for its construction, and if timber is expensive. An earth or rock embankment will almost always cost con- siderably less than any form of gravity masonry dam, if materials for the former are found convenient to the site. Therefore, if conditions admit of an embankment, that type of dam is usually to be preferred. The limitations of embankments will be men- tioned later. There is considerably less material in an arch dam than in any other masonry type, and consequently it will cost much less to construct. However, as will be pointed out later, a site suitable for an arched dam is the exception, rather than the rule. 7. Hollow Gravity Masonry Dams. Most hollow dams have been constructed of reinforced concrete of the types described in Chapter VIII. Compared with most methods of construction, reinforced concrete is in its infancy. Although its durability has not been tested for as long a period as plain concrete, it has, thus far, shown itself to be as permanent as can be desired. When the element of time is a governing consideration, the hollow dam possesses some advantage over the solid dam, because, there being less concrete to deposit, it can be constructed in a somewhat shorter period. Turbines and other apparatus have often been placed within hollow dams, thereby making a saving in the necessary housing for such appliances. A hollow dam is often adopted in preference to a solid gravity masonry dam in localities where considerable uplift * under the latter type would be expected. The hollow dam has a distinct advantage, owing to the fact that the narrow walls and buttresses are subjected to a practically negligible uplift pressure, the water under the dam having a direct exit. Another advantage claimed for the hollow dam having an up- stream face with considerable batter, is that it is impossible for it to overturn; as the resultant of all forces, for any depth of water, falls well within the base. However, this advantage is more fanciful than practical, as either type, if properly designed, should give no cause for worry in that respect. Hollow dams, being lighter per square foot of area covered, * See Art. 15. fi ^"U 14 THE CHOICE OF TYPE OF DAM [Chap. II can, by having spread footings, be made to exert less unit pres- sure on the foundation than solid dams. For this reason the former type is sometimes adopted where the requisite support for a solid dam is lacking. 8. Arched Masonry Dams.* This type is adaptable when the length is small in proportion to the height, and when the sides of the valley are composed of good rock which can resist the end thrust. It is the ideal permanent dam, containing much less material than other masonry types, and being equally permanent, it is always adopted where conditions permit. Unfortunately, however, sites suitable for this type are seldom found. The weight of the arched dam is not counted on to assist materially in the resistance of external loads. For this reason there is always sufficient weight of masonry to resist any possible, uplift on the base. Combinations of arch and solid gravity masonry dams are common for sites where the length is thought to be insufficient to permit of the adoption of the pure arched type. Such dams are designed to resist the loading by gravity, but are curved in plan and designed with a smaller margin of safety than if straight and with no arching possible. 9. Embankments. When plenty of materials are convenient to the side, embankments can usually be built for considerably less cost than any form of masonry gravity dam. The use of this type, however, is often limited by the necessity of providing a more suitable spillway for the passage of floods. It is not safe to allow water to spill directly over the embankment, even if it is well paved, unless the volume of the flood per linear foot of crest is small. Therefore a spillway of more suitable character is a necessary adjunct. In some instances such a spillway would require most, if not all, of the available length of the dam; in which case an embankment would be out of the question. The quantity of seepage through pervious material is propor- tional to the distance the water is required to travel. An earthen embankment, having the longest base in proportion to the height, is particularly adaptable to sites having pervious foundations. With proper maintenance, the embankment dam should be as permanent as the best. The necessary maintenance charges are comparatively high during the first year or two, but become rapidly * See Chapter IX. Abt. 11] OTHER TYPES 15 less as the structure settles into its final position and becomes well compacted, tight, and overgrown with proper vegetation to with- stand wash from rains. Earthen dams possess a distinct advantage in landscape work where it is desired to change as little as possible the appearance which Nature has given to the site. 10. Timber Dams. A timber dam is the ideal temporary type; although when well designed, constructed and maintained, it may last fifty years or more. Maintenance charges, however, are very high, compared with other types. Timber dams are seldom very tight. In fact, a small leakage is necessary for the proper preservation of the timber. Such leak- age, however, is of importance only when the value of the stored water is exceptionally high. This type is often used on soft foundations where masonry dams are out of the question, as a slight settlement, which, in the former would be permissible, would, in the latter, be an element of considerable danger. Owing to a scarcity of funds, a timber dam is sometimes adopted with the intention of utilizing it later as a part of the necessary coffer-dam for the construction of a more permanent structure. 11. Other Types. Various other types of dams have been designed and built. These include structural steel dams, pecu- liarly shaped masonry dams, the many forms of movable dams, and others. These, however, may be considered as either struc- tures of unique character, suitable for special conditions not ad- mitting of comparison in the general sense, or types which were the creation of fanciful engineers of radical tendencies. CHAPTER III FORCES ACTING ON DAMS 12. Nomenclature. The following nomenclature will apply, in general, to all parts of the text. Special nomenclature, applic- able to arch dams, is given in Chapter IX. Unless definitely- mentioned, all forces are stated in pounds and all dimensions in feet. W = A vertical force; positive when directed downward; P =A horizontal force; positive when directed toward the left; P t =Ice pressure per linear foot of dam; R = A resultant of forces; 2(W) =The algebraic summation of the vertical components of all forces acting on the dam above a given joint, including uplift, but excluding the reaction at the joint; positive when directed downward; 2(P) = The algebraic summation of the horizontal components of all forces acting on the dam above a given joint, excluding the reaction at the joint; positive when directed toward the left; 2(Wa;)=The algebraic summation of the moments, about a given point, of the forces contained in the summa- tion, 2(TF); positive when counter-clockwise; 2 (Pa;) =The same for the summation 2(P); A =An area, in square feet, or the area of a water-shed, in square miles; a =The distance from the top of the dam to the water surface; C =A constant; c =The percentage of area of joints or base subjected to uplift; e =The eccentricity of a loading (see " Irregular Bases," Art. 22, and Fig. 12) ; e =A subscript used to represent the condition of empty reservoir; 16 Art. 12] NOMENCLATURE 17 r =a subscript used to represent the condition of full reservoir; /' =The actual coefficient of static friction at a given joint or the base; / =The coefficient of static friction of the same materials as indicated by well-dressed test specimens; g = The acceleration of gravity = approximately 32.2 ; h =A vertical distance, a height of masonry, a head of water, etc.; he =The measured head on a spillway crest; h, =The total head on a spillway crest; K =The head corresponding to a given velocity; H =The total height of a dam above a given elevation; I =The moment of inertia of a figure; k =The percentage of voids in earth or silt, expressed as a decimal; L =The top width of dam; = The known length of a horizontal joint; =The unknown length of the horizontal joint next below; = The total length of a spillway crest; = The net, or effective, length of a spillway crest, Art. 43; = The average width of the channel of approach to a dam; m =A distance to the right or left of the center of gravity of a figure; see definition given in " Irregular Bases," Art. 22; N =A flood coefficient, Art. 65; n =The number of complete end contractions on a spill- way crest; p =Unit pressure or compressive stress, in pounds per square foot; p' =The same at the down-stream extremity of the base; p" =The same at the up-stream extremity of the base; (The foregoing system of primes and double primes applies also the following special values of p.) p r =The unit vertical reaction * of the foundation at a joint or the base, exclusive of uplift pressure; p e =The unit vertical compressive stress * at a joint or the base, inclusive of uplift pressure; * See foot-note, p. 43. 18 FORCES ACTING ON DAMS [Chap. HI p, =The unit maximum inclined compressive stress in the masonry or the foundation; p„ =The unit effective uplift * on a joint or the base; p n =The unit normal pressure of water or earth on the face of the dam; Q =The total quantity of water passing over the spill- way crest, in cubic feet per second; Q m =The maximum flood from a given drainage area, likely to be exceeded only once in T years; q =The quantity of water passing over each linear foot of effective spillway crest; r =The radius of a circle; S = A factor of safety, Art. 25; t =A period of time, in seconds; T =A period of time, in years; u =The horizontal distance from the down-stream ex- tremity of a joint to the point of intersection of the resultant, R, with that joint; v = A velocity; in feet per second; w =Unit weight; w\ =Unit weight of masonry; W2 =Unit weight of water; W3 =Unit weight of earth; x =In general, a vertical distance. Also used as the lever arm of both vertical and horizontal forces; y =The horizontal distance from an origin of moments to the up-stream extremity of a joint or the base; z =The horizontal distance from an origin of moments to the point of intersection of the resultant, R, with a joint or the base; a = The angle of repose of earth; 8 = The angle of inclination with the vertical, f of the result- ant, R, of the forces, 2(W) and 2(P) ; =The angle of inclination, with the vertical,* of the face of the dam; ' = The same for the down-stream face at a given elevation; " = The same for the up-stream face at a given elevation. * See Art. 15 and foot-note, p. 45. t This is the common definition, as, in general, the joints and bases are horizontal. For inclined joints or bases, the angles 8 and should be measured from a normal to the joint or base. Art. 14] EXTERNAL WATER PRESSURE 19 13. General Considerations. The first consideration in designing a dam is the determination of the nature of the forces acting on the structure. These forces may be considered as con- sisting of the following: a. Water pressure, b. Earth pressure, c. Atmospheric pressure, d. Ice pressure, e. Wind pressure, f. Wave pressure, g. Weight of the dam, h. Weight of the foundation, i. Reaction of the foundation. The nature of most of these forces, unfortunately, is such that they do not admit of exact determination, and their amounts, direction and location must be adopted by the designer after a thorough consideration of all obtainable facts bearing on the case, and the exercise of his best judgment, based on his experience and that of others who have had to deal with similar problems. It must always be borne in mind that conditions in no two dams are alike,, and that a general theory must never be applied to a particular case without thought as to the possible need of modifica- tion to suit the conditions peculiar thereto. 14. External Water Pressure. The weight of a cubic foot of fresh water has been determined to be 62.42 lb. per cu. ft. at 32° F. 62.26 lb. per cu. ft. at 75° F. 62.00 lb. per cu. ft. at 100° F. The weight usually adopted in the design of dams is 62.5 lb. per cu. ft. The total pressure, P, of quiet water on any vertical submerged plane of area A is P=w 2 Ah 3 , (1) where W2 is the weight of 1 cu. ft. of water and hz is the distance from the center of gravity of the plane to the surface of the water. The force, P, will be horizontal. 20 FORCES ACTING ON DAMS [Chap. Ill In Fig. 1, let 1-2 represent a submerged vertical rectangular plane of unity width, measured perpendicular to the paper, and having its top edge parallel to and a distance, hi, from the surface of the water. As the width of the plane is unity, the length, 1-2 = h, will be a measure of its area, A* "Water Surface . \ ■ , ~" ~ f \ + -5 « ti < to "V 5 \ \ ■ ' : L. 2 \ Fig. 1. The center of gravity of the plane, 1-2, is at the point, 5, mid- way between 1 and 2. The total pressure, P, on the plane, 1-2, may be obtained from ^ (1): *,+*. P = W2Ah3 = W2h ;; — , since A is of unity width. Substituting the value, h = h2 — h\, there results P = w%h% % W2hi 2 (2) The location of the force, P, may be determined in the following way: Lay off the horizontal line, 1-4, equal to W2hi, the unit water pressure at point 1. Lay off the line, 2-3, equal to W2A2, the unit water pressure at point 2. Then a horizontal intercept between any point on the line, 1-2, and the line, 4-3, will equal the unit water pressure at that point, and the total area, 1-2-3-4, will equal * In the design of dams, it is customary to consider a slice bounded by two planes perpendicular to the axis of the dam and 1 ft. apart, thereby simplifying the calculations by allowing the neglect of the third dimension. Tjnless specifically stated, it should be assumed, in all that follows, that such a slice is being considered. Art. 14] EXTERNAL WATER PRESSURE 21 the total pressure, P. The force, P, will pass through the center of gravity of the area, 1-2-3-1, which is a vertical distance above point 2 equal to _3hh+h 2 x = 6/ii+3/i ' In the design of dams it is found convenient to deal with hori- zontal and vertical forces only. If the plane, 1-2, is inclined, as in Fig. 2, the resultant total pressure, R, on the plane may be resolved into horizontal and vertical components, P and W. The horizontal component, P, will be equal to the pressure on the pro- jection, 2-5, of the plane, 1-2, and its amount and location can be calculated from Eqs. (2) and (3). The vertical component, W, will be equal to the weight of water above the plane, 1-2, namely, within the boundaries, 1-2-6-7. The force, W, will pass through the center of gravity of the figure, 1-2-6-7. Water Surface g Water Surface Fig. 2. Fig. 3. Inasmuch as it is possible for a dam to be entirely submerged, with water pressure on every square foot of surface, we will now investigate the forces due to water pressure in the following order: a. On the up-stream face, b. On the top, c. On the down-stream face, d. On the bottom. In Fig. 3, the horizontal component, P, of the total water pres- sure on the up-stream face of the dam is equal to the total pressure on the plane, 3-8, as indicated by Eq. (2). In this case, hi being equal to zero, and fe being equal to h, P = ^# (2a) 22 FORCES ACTING ON DAMS [Chap. Ill The distance, x, from point, 3, to the force P, may be found from Eq. (3). For this case there results h (3a) The vertical component, W, is equal to the weight of water within the area, 2-3-8-7, and passes through the center of gravity of that area. In Fig. 4, the height, h, of the dam is not as great as the depth of water, h 2 , so that the water is constantly spilling over the crest. In this case the dam will not be subjected to horizontal water "Water Snrface 'Fig. 4. pressure on the plane, 1-8, above the crest. On account of the velocity of the water passing the plane, 8-9, which approaches very nearly spouting velocity in well-designed dams, the vertical component of the pressure on the plane, 9-10, and on the top of the dam is always neglected. To be on the safe side, however, the horizontal component of the pressure on the plane, 9-10, is included. Considering, again, vertical and horizontal forces only, we have a vertical force, W, equal to the weight of water within the limits, 7-8-2-3, passing through the center of gravity of that area, and a total horizontal water pressure, P, equal to the pressure on the plane, 11-3, found by Eq. (2), and located by Eq. (3). The heads, hi and fa, should correspond to the measured head Aut. 14] EXTERNAL WATER PRESSURE 23 on the dam; that is, to the water surface in the channel of ap- proach sufficiently remote from the dam to be beyond reach of the surface curve.* An additional head, equal to twice. the velocity head of the water in the channel of approach, should be added to allow for impact on the face of the dam.f If <'i=t^ is the velocity of approach, the average unit impact pressure is approximately p-211^,-2 — =_ (4) and the total impact pressure on the dam is lC-2l'\ 2 h P' = 2w 2 hji = - g where g is the acceleration of gravity (about 32.2), q is the discharge per lineai- foot of crest, and icj is the unit weight of water. Fig. 5 represents a dam containing a barrier, such as flash- boards, projecting above the crest, or closed gates held between piers." In such cases, the weight of the water within the bound- aries, l-2-3~i, should be included in the calculations as well as the horizontal pressure on the plane, 1-2. The stream of water indicated in Fig. 4, flowing down the face of the dam, is, on account of its velocity, considered as transmit- ting no pressure to the dam. Pressure on the dam from water standing in the lower pool, as indicated in Fig. 5, may be treated in the same manner as described for water pressure on the up-stream face. In the case of a spillway, tail-water does not always exert a pressure on the dam, as its depth adjacent to the dam is often practically eliminated by the impact of the water spilling over the crest. J This condition is indicated in Fig. 4. Air. A. H. Gibson, § in Ins discussion of standing waves in non- * See Art. 43. t See Gibson's "Hydraulics," Art. 96; D. Van Nostrand Co., 1908. J See Karl R. Kennison's paper "The Hydraulic Jump in Open Channel Flow at High Velocity," Transactions, Am. Soc. C. E., Vol. LXXX, p. 338. § Gibson's "Hydraulics and its Applications," Art. 86. J. Wiley & Sons, 1908. 24 FORCES ACTING ON DAMS [Chap. Ill uniform channels, states that the conditions under which this phenomenon will occur are and v 2 =>gh 3 , j2h 3 v 2 ,h 3 2 h 3 -V~r + T-2-- hs For spillway dams, v 2 is always greater than gh 3 , hence we may expect a reduction in depth of tail-water, provided hs is less than the value given in the second equation, Water Surface Head-water Fig. 5. A more convenient form may be obtained by writing the second equation as follows: , /2feV ,h 3 2 h 3 But h 3 2 v 2 = q 2 , where q is the discharge, in cubic feet per second, per linear foot of crest. Making this substitution, there results ' /*5 -V 9 2 | h 2 h I6.M3 4 2' (5) Abt. 15] INTERNAL WATER PRESSURE 25 The thickness, hs, of the sheet of falling water for any given head, hi, and height of dam, h, may be found from Figs. 25 and 26: A method of determining the discharge, q, corresponding to the head, hi, is given in Art. 43. If As is greater than the value given in Eq. (5), the water flow- ing over the spillway will expend its energy in creating eddies, and will cause no reduction in tail-water level. If hs is equal to this value, a standing wave, 14-15, (Fig. 4), or " hydraulic jump," will occur at the toe of the dam. If hs is less than this value, the standing wave will occur at a distance down-stream, where the depth, hs, has increased an amount sufficient to fulfill the conditions of Eq. (5). Therefore, if, in any example, it is found that hs is less than the value given in Eq. (5), the pressure of the water on the down- stream face of the dam may, theoretically, be neglected. The foregoing has been based on theory alone. Although Gibson's equation has been fairly well substantiated by experi- ments on a small scale, the actual height, hs, of the standing wave for large structures may differ considerably from that calculated. It is recommended, therefore, that when the depth of tail-water is within 20 per cent of the value given in Eq. (5), the pressure of tail-water should be considered doubtful and the dam tested for stability with and without it. For a practical application of this theory see Art. 47. Obstructions or " baffles," have often been placed in the tail- race, adjacent to the dam to prevent the standing wave from occur- ring below the apron and the swiftly moving water from having access to the unprotected portion of the foundation. Such an arrangement is indicated in Figs. 6 and 7. This feature will be further discussed in Art. 61. 15. Internal Water Pressure. Uplift. If the foundations are pervious, an upward pressure, or " uplift " will occur on the base of the dam, as indicated by the force, W, Fig. 5. The amount and location of this force depends on the relative pervious- ness of the foundation at various points and the details of the dam.* In order to comprehend more clearly the principles of uplift under solid dams, consider a solid dam to be slightly raised from * Such details refer to the difference between solid and hollow dams, and the use of drainage systems. 26 FORCES ACTING ON DAMS [Chap. Ill £ si « I Pi I a 02 CO O Abt. 15] INTERNAL WATER PRESSURE 27 1 | "3. a 28 FORCES ACTING ON DAMS [Chap. Ill its foundations, so that water flows through from the upper to the lower pool. The laws governing the pressure over the base are the same as those for the flow through pipes. In this case, neglecting the inappreciable loss at entrance, the pressure would diminish uniformly from W2I12 (Fig. 5), at point 6 to W2I15 at point 5. The total.uplift would be W = W2l — ~ — . The location of the force, W, can be obtained (for this as well as for other cases to be considered later), by locating the center of gravity of a force diagram constructed in the manner described for Fig. 1. Assume, however, that although the dam is slightly raised, a portion of each square foot of the base has absolute contact with the foundation. The force, W, would then be reduced an amount in proportion to the percentage, c, of such area not in contact, Then, w=CW2 ib+h (6) If there is an impervious barrier at point 5, the pressure over the portion of the base not in contact would be uniform and equal to u>2h,2, and the total uplift on the base would be w = CW2II12 If the impervious barrier is at point 6, the uplift would be uni- form and equal to W = CW2lhs If the barrier is at any point between 5 and 6, the uniform unit pressure above the barrier would average cw-^,2, and below the barrier CW2I15. In practice, where the joint between the dam and the founda- tion, or other horizontal joints in the dam, are conducive to the passage of water, uplift will occur, and the foregoing laws will apply. The conditions are also the same for a horizontal seam in the rock below the base, with the exception that the points, 5 and 6, should be considered located at the entrance and exit of the seam, respectively. Art. 15] INTERNAL WATER PRESSURE 29 Most engineers recognize the likelihood of uplift. The ques- tion of amount and distribution is, of course, impossible to deter- mine exactly, and, of late years, this question has been the source of much debate. In order to bring the subject before the engineer- ing profession for discussion, the late C. L. Harrison, on Decem- ber 20, 1911, presented a paper before the American Society of Civil Engineers, in which he made some brief suggestions regarding the methods of estimating the amount and distribution of uplift on the foundations and horizontal joints of dams. The paper was quite freely discussed by some of the most prominent engineers. It is quite surprising to note the difference of opinion on the sub- ject, varying practically from one extreme to the other. The following are extracts from Mr. Harrison's paper * and his analysis of the discussion thereon: For convenience in discussing this subject, reference is made particularly to masonry dams on rock foundations. The principles involved will apply equally to other foundations and to dams built of other materials. The upward pressure may be due to water getting into the foundation of the dam or into the dam itself. Foundations vary so much in character that it is necessary to study each particular site before deciding to what extent water may get into them. (1) In the case of a foundation of hard, sound rock, without either horizontal or vertical seams, there is no reason to expect that water will get into it and produce an upward pressure, and, in the design, no allow- ance should be made for it. In such cases the junction between the masonry and the foundation can easily be made water-tight. (2) In the case where the foundation is stratified with well-defined horizontal seams, and the dam is located near a fall or rapids in the stream, so that the water may flow from the seams at the toe of the dam as freely as it enters them from the reservoir, the upward pressure will be approx- imately equal to the static head at the heel and gradually decrease to zero at the toe of the dam. (3) Take a foundation similar to the foregoing in every respect except that the water in the seams of the rock cannot escape freely near the toe of the dam, but must flow some distance down-stream through rock or other materials before it reaches the surface of the ground, or must rise ver- tically to the surface. Then the upward pressure at, the heel will be equal to the static head, and that at the toe will be equal to the head required to overcome the resistance to the water escaping at that point. While these three cases present well-defined conditions, it is probable that at most sites the conditions will he between those presented in Case 1 and Cases 2 and 3, that is, the water will not be in the foundation through- * Tramactiotis, Am. Soc. C- E., Vol. LXXV, pp. 142-225. 30 FORCES ACTING ON DAMS [Chap. Ill out its entire area, but will cover only a part of this area. This makes it necessary to study the foundation carefully at each site in order to deter- mine to what extent water may get into it. When this upward pressure exists, weight must be added to the dam by additional masonry to counter- balance it. Generally, it will be found cheaper to make large expenditures to provide a cut-off in the foundation, which will not only reduce the uplift, but will also save the water. Such a cut-off should be located at the heel of the dam. If it is located under the middle of the dam, there would be an upward pressure under the up-stream half of the dam, due to the full head of the water in the reservoir. * * * ,* * * * In order to determine what allowance to make for pressures due to water which gets into the dam itself, one must first decide on the character of the construction. With suitable stone, sand, and cement, it is possible to build a masonry dam which will have no horizontal cracks or seams, and it is also possible to provide against vertical cracks, to » large extent, by expansion joints. Water in vertical cracks, however, does not produce an upward pressure. In such structures very little, if any, allowance should be made for the upward pressure due to water getting into the masonry. If the materials for building water-tight masonry are not to be had at the site of the dam, and it is very expensive to import them, it is generally advisable to adopt a, different class of masonry, which will probably be more pervious and also more difficult to construct without horizontal cracks or seams, thus allowing the water to enter the dam, and resulting in upward pressures. The extent of such pressures will depend on the character of the masonry and the care with which it is built, all of which must be known before an estimate can be made of the extent to which the water will get into the dam. The effect of this upward pressure, however, must be counter- acted, either by increasing the section of the dam or by increasing its height above the water level in the reservoir, or by both. In many cases it may be advisable to provide drainage wells near the up-stream face to intercept the water and carry it off through pipes at the toe of the dam, thus reducing or eliminating its effects in the main body of the dam. After determining the type of masonry to be constructed, it is still a question of judgment, based on observation, tests, and experience, as to what the upward pressure in the dam will be. ******** An analysis of the discussion indicates the following conclusions: 1. For any stable dam the uplift in the foundation cannot act over the entire area of any horizontal seam, and in the masonry it cannot act over the entire area of any horizontal joint. 2. The intensity of uplift at the heel of the dam can never be more, and is generally less, than that due to the static head. Also, this uplift decreases in intensity from the heel to the toe of the dam, where it will be zero if the water escapes freely, and will be that due to the static head if the water is trapped. 3. The uplift in the foundation should be minimised by a cut-off wall, Art. 15] INTERNAL WATER PRESSURE 31 under-drainage,* and grouting when applicable; and in the dam itself by using good materials and workmanship, and by drainage when advisable. 4. The design should be based on the conditions found to exist at each site after a thorough investigation by borings, test pits, and otherwise, and modified if found necessary after bed-rock is uncovered. Mr. Harrison's paper should be read by all who are interested in the design of dams. In practically all the designs of solid gravity dams embodying uplift which have come to the author's attention, the assumption has been a pressure varying uniformly from wohz (Fig. 5), at the heel to N'i/is at the toe, Eq. (6), being directly applicable. The only difference has been the percentage, c, of the base assumed to be subjected to the pressure. In Table I is given a list of some recently constructed high dams in the design of which extreme values of c were adopted. TABLE I Percentage of Area of Base Subjected to Uplift, Adopted for Various Dams Dam. Value of c. Per Cent. Wachusett Cross River. . . . Elephant Butte Olive Bridge . . . Kensico Lock Raven . . . 66 66 33 (approximately) 66 66 66 All the dams mentioned in Table I contain drainage systems, cut-offs, and other features designed to reduce, as far as possible, uplift on the base. It must be remembered, however, that these dams are among the largest and most important structures built in this country, and failure would undoubtedly result in immense loss, not only at the dam but to other structures below it, and, in most cas?s, an appalling loss of life. For these reasons an unusu- ally large margin of safety has been provided, not only in the assumptions of uplift, but in other features. Many dams, includ- * Author's Note : Except for very high dams, comparatively few have been provided with under-drainage. 32 FORCES ACTING ON DAMS [Chap. Ill ing the large New Croton Dam of the New York City Water Sup- ply system, contain no provision for uplift in the design. On the other hand, the failure of dams on very poor foundations, in a number of cases, may be attributed to insufficient provision for this force. The author is of the opinion that a value of c = f is the extreme limit for dams on fair rock foundations, and for the most impor- tant structures; and that ordinarily a much lower value may be adopted. It is impossible to recommend definite values of c for the many classes of foundations which exist, no two of which are alike. The final choice must be made in accordance with the judgment and experience of the engineer after a thorough investigation of the site after the foundations have been exposed. For earth foundations, a value of c = 1.0 is necessary, but the distance which the water must travel may be considered as being the length of the base plus twice the depth of the impervious cut-off.* Little, if any, uplift can exist in hollow dams of the types de- scribed in Chapter VIII. The pressure of water finding a passage through the deck or cut-off will be almost immediately relieved through the sides of the buttresses before it can penetrate any appreciable distance. The author knows of no case in which an assumption of uplift was used in the design of such structures. Mr. Arthur P. Davis f gives the following general rules for the relative perviousness of several classes of foundations: The determination of the perviousness of natural formations is one of the most difficult things in Nature. Any examination of such formations which disturbs them changes the conditions which it is desired to know. For this reason, it is necessary to allow a large factor of safety in any estimates which involve this factor. In general, it may be said that water will more readily follow seams or bedding planes than devious paths through the material of the rock. It follows that it will pass more readily and in larger volume in the direction of stratification than in a direction normal thereto. Similarly, stratified rock will permit percolation more easily and in greater volume than good, massive rock, such as granite. Granular rock, such as sandstone, is. likely to transmit more water through the rock itself than one of denser or finer grain, such as limestone or shale, * See Art. 62. f Transactions, Am. Soc. C. E., Vol. LXXV, page 208. Art. 16] EARTH PRESSURE 33 but no exact rule of this nature can be laid down, because there are many varieties of each kind of rock, with various percolating capacities. In gen- eral, however, the following rules may be taken as a rough guide: 1. Massive or crystalline rocks, such as granite, gneiss and schists, will transmit water less freely than those of sedimentary origin. 2. Stratified rocks will transmit water much more readily in the direc- tion of stratification than transverse thereto. 3. In the direction normal to stratification, sandstone will generally transmit water more readily than limestone, and the latter more readily than shale. 4. Stratification on a plane approximately horizontal is the worst pos- sible condition for introducing upward pressures beneath a dam. Con- versely, the most favorable position in this respect for stratified rock is in vertical beds. It is rational to assume that, in some cases, there may be less proportionate uplift on horizontal joints above the base than on the base. However, it is customary to adopt an uplift on all hor- izontal joints equal, proportionately, to that assumed for the base. This wijl lead to no appreciable error, and will greatly simplify the calculations, because, if the assumption of uplift for the joints were less than that used for the base, the upper part of the section designed would not suit those parts of the structure where the foundation is at a higher elevation. The effect of silt deposits in reducing uplift pressure is dis- cussed in Art. 16. 16. Earth Pressure. Practically all streams transport silt and gravel, particularly during floods, when, in some cases, the quantities are enormous. The construction of a dam across a stream results in a stretch of slack water which causes the material to deposit. It fills the upper end of the reservoir first, and grad- ually advances until it reaches the dam. Quite often, sluices are constructed in the lower part of the dam, and, if periodically flushed in the proper manner, this limits the depth of such deposits adjacent to the dam. An earth fill is sometimes deposited against the down-stream face of the dam, and may or may not be submerged. The horizontal component of the silt or earth pressure, P" (Fig. 5), exerted on the dam may be derived from Rankine's well-known equation: ,_ w 3 h 2 / l-sma \ , ' ' 2 \l+sin«/ K ' 34 FORCES ACTING ON DAMS [Chap. Ill Where w 3 = the weight, in pounds per cubic foot, of the silt or earth, in air or submerged, as the case may be; A=its depth, in feet; and a = its angle of repose. 2h P" is located a distance, — , from the surface of the earth, and, o when it is the result of submerged earth, it is in addition to the water pressure. The uncertain element of static friction between the earth and the dam is usually neglected, as it is on the side of safety. When the face of the dam on which the earth or silt pressure acts is not vertical, as in Fig. 5, the weight of the material ver- tically above the plane (the area, 6-9-10-11), should be included in the forces acting on the dam, in the same manner as described for water pressure. As a matter of fact, the determination of the vertical and horizontal components of earth and silt pressures parallel exactly that described for water pressures, except, that the portion of Eq. (7) in parentheses modifies the pressure in accord- ance with the internal resistance to movement, as measured by a, the angle of repose. If wz is the weight of earth per cubic foot in air, and k is the percentage of voids, then in 1 cu. ft. of the fill there will be (1 — A;) cubic feet of solids, weighing wz pounds. The weight of water displaced when this cubic foot of fill is submerged will be w>2(l — k) pounds. Therefore, the weight, wz, of the submerged fill will be wz=wz'— 102(1 — k) (8) For example, if the material weighs 110 lb. per cu. ft. in air and has 30 per cent of voids, there will be (100-30 per cent) or 70 per cent of solids in each cubic foot. Therefore, 1 cu. ft. of the fill will displace 0.7 cu. ft. of water weighing 0.7X62.5 = 43.75 lb., and the weight per cubic foot of submerged fill will be 110—43.75 = 66.25 lb. =wz- The weight, voids, and angle of repose vary, of course, with the nature of the material, and should be made the subject of investi- gation for each case. Safe values often used for sand and gravel are 103'= 110, A; = 30 per cent, and a = 30°. This corresponds to a value for w 3 of 66.2. For material capable of acting as liquid mud the usual Art. 17] ATMOSPHERIC PRESSURE 35 values are w s ' = 125, k = 0, and a = 0. This corresponds to a value for w 3 of 62.5. Thus it is seen that the usual unit weight of submerged earth is not far from that of water, but the angle of repose for use in Eq. (7) varies from zero for liquid mud to about 30° for sand and gravel. It is not often that silt pressure on the up-stream face and uplift pressure from head-water are assumed to act simultaneously against the dam, unless the silt is of an unusually pervious nature. Ordinarily, the silt deposited by streams is an impalpable clay mixed with very fine sand or similar material, particularly im- pervious to the passage of water. It is the usual practice, where silt of an impervious nature is expected, to use in the calculations for the design that one of these two forces which has the greatest influence on the shape of the section. It has been claimed that silt deposits relieve the water pressure on the up-stream face of the dam, but this is an assumption incon- sistent with conservative design. 17. Atmospheric Pressure. Except as described later, atmos- pheric pressure is exerted on every square foot of the surface of Water Surface Fig. 8. the dam. Fig. 8 represents a section of a spillway where the over- flowing sheet of water is not in contact with the down-stream face of the dam. In such cases the friction of the moving water surface, 1-2-3, entrains the air contained within the boundaries, 1-2-3-4, and carries it away. If the chamber, 1-2-3—1, has free access to the atmosphere, air will be easily supplied as fast as it is removed. It is not always feasible to provide such access, particularly if 36 FORCES ACTING ON BAMS [Chap. Ill the dam is very long and the head, h, on the crest is relatively great. If the chamber, 1-2-3-4, does not have free access to the atmosphere, a partial vacuum or reduction of atmospheric pres- sure will occur therein. An adjustment of conditions, as indicated in Fig. 9, will then occur. The difference in. atmospheric pressure on the two sides of the sheet will cause it to move toward the dam, the difference in pressure being balanced by the force overcome in changing the direction of the sheet. The reduction of atmospheric pressure on the area, 3—4, will result in raising the water surface between the sheet and the dam an amount, hi, sufficient to balance such reduction. The reduc- tion of atmospheric pressure on the face, l-4rAa, of the dam must be balanced by the stability of the structure. The resulting force, Fig. 9. R, on the dam is equal to the trapezoid, l-5-6-4a, in which the distances, 1-5 and 4-6, represent the reduction in atmospheric pressure, hi, over the area, 1-4. Quite often, in such cases, the reduction of atmospheric pres- sure becomes intermittent; that is, a partial vacuum within the chamber, 1-2-3-4, accumulates up to a certain amount, then a break occurs in the sheet admitting air, with a sudden return to normal pressure. The operation is repeated periodically. Such periods sometimes become of very short duration, causing a strong vibration, with a consequent tendency to loosen the dam from the foundation and thus decrease its stability considerably beyond that corresponding to the steady force, R. As an example of the possible magnitude of such vibrations, the late J. P. Frizell * stated that the vibrations set up in this way * "Water Power," by Joseph P. Frizell, 3d ed. J. Wiley & Sons, 1905. Aet. 18] ICE PRESSURE 37 by the water flowing over the old crib dam at Holyoke, Mass., rattled the windows in Springfield, six miles away. On account of the impossibility of determining accurately the amount and effect of the force, R, it is customary to fit the down- stream face of the dam to the lower nappe of the jet corresponding to the maximum head, h, which may occur on the crest of the dam. A method of determining the shape of the jet is given in Art. 42. 18. Ice Pressure. In common with other materials, ice expands and contracts with changes of temperature. In a reser- voir completely covered with ice, a contraction due to a decided reduction in the temperature of the air will take place, opening up large cracks in the ice in which, subsequently, the water freezes solid. When the next rise of temperature occurs the ice expands, and if it is not free to slide up the banks of the reservoir, it will exert considerable pressure on the dam. This pressure usually causes the sheet of ice to buckle or crush. Should the conditions be favorable, the ice may exert an overturning force on the dam. The thrust of ice is impossible of exact determination. It is, of course, limited to the crushing strength of ice which, however, is variously reported between 100 and 1000 lb. per sq. in. The latter value corresponds to the enormous amount of 144,000 lb. per sq. ft., and, where ice attains a thickness of 4 ft., amounts to the absurd value of more than a half a million pounds per linear foot of dam. That ice thrust, under usual conditions, can never approach the latter value is proved by the fact that a great many dams are standing to-day which otherwise would certainly have failed. This, in part, may be explained by the fact that the full thrust of ice cannot be exerted on a dam: 1. If it is in a narrow gorge where the full effect of expansion cannot be felt; 2. If the opposite banks are sloping so as to allow the ice to ride up on them and thus relieve the pressure; 3. If, as has been claimed, the ice next to the dam cannot attain its full crushing strength owing to the greater temperature of the masonry. Mr. C. L. Harrison's paper * on this subject brought out a * Transactions, Am. Soc. C. E., Vol. LXXV, p. 142. 3$ FORCES ACTING ON DAMS [Chap. Ill very free discussion. The following is quoted from his closing remarks in reply thereto: It is generally agreed that this force [ice pressure] should be considered, and allowance made in some cases, but not in others. The limitations of these cases are not definitely stated by those who have discussed the subject. In order to give the matter more definite shape, it may be suggested that, under the following conditions, it is not necessary to provide for ice pressure: 1. For the ordinary storage reservoir with sloping banks, in climates where the maximum thickness of ice is 6 in. or less — for dams with southern exposure this limit may be placed as high as 1 ft. None of the discussions fixes this limit, but it is what the writer has in mind as a reasonable provision. 2. For reservoirs which are filled during the flood season and from which all the stored water is drawn off each year during the low-water season. This would include even the Jarge reservoirs on the head-waters of the Mis- sissippi River, where the ice has a thickness of more than 4 ft., and the atmospheric temperatures reach 50° below zero. 3. For storage reservoirs where the water will be drawn off each year during the winter to a level where the dam is strong enough to resist the ice pressure. 4. For reservoirs where the contour of the ground at the high-water level is such that the expansive force of the ice will not reach the dam. ******* The dams cited in Table II, where a value has been given to the ice pressure in the design, are in the vicinity of New York or Boston, where the maximum thickness of ice may be taken as about 2 ft. No allowance is made for ice pressure in the New Croton Dam, which is in the same climate. TABLE II Ice Pressure Adopted for Various Dams Dam. Location. Allowance for Ice Pressure, in Pounds per Linear Foot. Boston New York New York New York New York New York 47,000 Olive Bridge Kensico 47,000 47,000 30,000 Croton Falls 24,000 New Croton No allowance The stored waters in all the reservoirs inTable II are for domestic supply, and, excepting Olive Bridge and Kensico, are in service. The reasons given for the smaller allowances made for the Croton Falls and Cross River Dams Art. 18] ICE PRESSURE 39 are that local conditions will prevent the full ice thrust from reaching the dams, and also that they are located up stream from the New Croton. If either of these dams should fail, no valuable property would be damaged, and the waters would flow into the New Croton Reservoir. Those respon- sible for the design of the New Croton Dam believed that no allowance should be made at this dam for ice thrust. At first glance, this looks like a wide range in judgment, but it must be remembered that the foregoing statement gives only a part of the facts, and to this must be added the local conditions and the service the dam is to render in each case before judg- ment is passed on the wisdom of the design. Reservoirs for domestic supplies are generally drawn down during the ice period, and the greatest expansion of the ice occurs at the end of this period, thus applying the pressure at a point below high-water level, where the dam is strong enough to resist it. If, however, such reservoirs are to be at high-water level during, and especially at the end of, the ice period, at any time during their service, then the proper allowance should be made for ice pressure in the design. The daily fluctuations in the water level in the forebay at power dams will usually prevent the ice from freezing to the dam, which, therefore, will not be subjected to thrust caused by the expansion of the ice in the pool above the dam. In such cases, the proper course seems to be, not to reduce the allowance, but to omit it altogether. If, however, a storage reservoir is to be filled to the high-water level during the full ice period, at any time during the life of its service, then not a partial, but the full ice pressure should be allowed for in the design of the dam. It is contemplated that the Kensico Reservoir is to be kept at or near the high-water level at all times, and therefore will be subject to full ice pres- sure at high-water level; also, the Olive Bridge and Wachusett Dams may at intervals be subject to this pressure at high-water level. It is entirely possible that it would be proper to allow for ice pressure on a dam in a given locality and also proper to make no allowance for such pressure on another dam in the same locality, depending on the service each is to render. The fact that so many dams have been designed and built without making a specific or separate allowance for ice thrust, and have for years stood the test of actual service without failing,, is an indication that ice pressures may not be as great as sometimes thought, or that the factors of safety allowed for other purposes are sufficient to take this pressure. On the other hand in the cases mentioned in this discussion, there seems to be good and suf- ficient reason for allowing for ice pressure in the designs. Ice floes are capable of exerting comparatively little pressure against a dam. If the velocity of the approaching water is high and the crest is not clear, the most that can be expected is a local thrust, which need not be very large, as the ice in such cases is always soft. Possible ice thrust on the down-stream side of the dam is usually neglected in the calculations. 40 FORCES ACTING ON DAMS [Chap. Ill 19. Wave Pressure. It is seldom necessary to consider the effect of waves on the stability of the dam. Stevenson's formula for the determination of the height of waves is: h = lWF+(2.5-\ / F), (9) where h is the height, in feet, and F is the fetch, or straight length of clear water, in miles. Unless the waves break, due to the relative shallowness of the water near the dam, very little impact results. Except for dams which are unusually small compared with the size of the reservoir the usual dimensions obtained from other considerations are ample to resist the effect of wave action. 20. The Weight of the Dam. The unit weight of masonry varies considerably, depending on the ingredients of which it is composed. Table III, giving weights of masonry, is taken from the " American Civil Engineer's Pocket Book." * TABLE III Weight of Masonry in Pounds per Cubic Foot Ashlar: Granite, syenite, gneiss. Limestone, marble Sandstone Mortar Rubble:, Granite, syenite, gneiss Limestone, marble Sandstone Table IV gives the approximate average weight of plain Port- land cement concrete. Although the weight of concrete is affected greatly by the materials of which it is composed, it is influenced very little by changing the proportions of the ingredients. A wet mixture will ordinarily result in a somewhat lighter concrete than if the ingredients are mixed dry and well compacted. * By Mansfield Merriman, John Wiley & Sons. Art. 20] THE WEIGHT OF THE DAM 41 TABLE IV Approximate Weight of Plain Concrete and Coarse Ingredients, in Pounds per Cubic Foot Coarse Aggregate. Coarse Aggregate (Solid). Concrete. Trap 175-200 150-195 155-175 140-155 150-160 140-160 Granite, syenite, gneiss Limestone, marble, quartz Sandstone, bluestone 135-160 140-145 130-140 If the nature of the ingredients is known, the weight of dry- concrete can be calculated approximately by use of Taylor and Thompson's * table of quantities of materials required for one cubic yard of concrete. Assume trap rock to be used for the coarse aggregate, weigh- ing 175 lb. per cu. ft. in the quarry and containing 45 per cent of voids when broken up with dust screened out. One cubic foot of crushed stone will then weigh 175 (1 — 0.45) = 96.3 lb. Assume the sand to weigh 90 lb. when dry, and that it has the characteristics assumed in Taylor and Thompson's table. Cement usually weighs about 100 lb. per cu. ft. From Taylor and Thompson's table, the weight of 1 cu. ft. of 1:3.6 concrete can be computed as follows, assuming a barrel of cement to contain 3.8 cu. ft., and to weigh 3.8X100 = 380 lb. One cubic foot of concrete requires 1^ bbl. of cement at 380 lb 15.6 lb. 27 0.47 cu. ft. of sand at 90 lb 42.3 lb. . 94 cu. ft. of crushed stone at 96.3 lb . . . 90 . 5 lb. Weight of 1 cu. ft. of concrete 148 . 4 lb. If 20 per cent of plums are used, the weight of the cyclopean concrete masonry composed of this same material would be as follows. * "Concrete, Plain and Reinforced." 42 FORCES ACTING ON DAMS [Chap. Ill One cubic foot of cyclopean concrete requires: 0.8 cu. ft. of plain concrete at 148.4 lb . . . . 118 . 7 lb. 0.2 cu. ft. of plums at 175 lb 35.0 lb. Weight of 1 cu. ft. of cyclopean concrete. . . 153 . 7 lb. This method of computing the weight of concrete is quite ap- proximate, and adopted values equal to 95 per cent of computed values are not too conservative. In important work, an exact analysis of the materials or tests of the actual weight of concrete blocks, should be made. 21. The Weight of the Foundation. Dams have sometimes been tied down to the rock foundation in order to increase their resistance to overturning, sliding, or both. The arrangement has consisted of steel bars grouted into holes bored in the rock and extended into the dam near the up-stream face. This practice, however, has been severely criticised lately, on the ground that a satisfactory anchorage of the bars in the foundation is seldom possible. 22. The Reaction of the Foundation. A. Rectangular Bases. Let 2(TF)* (Fig. 10) be the resultant of all vertical forces acting on the dam above the foundation, including uplift, and 2(P) the resultant of all horizontal forces. The result- ant, R, of 2(1*0 and 2(P) will represent the resultant of all forces. For the dam to be in static equilibrium, the resultant, R, must be balanced by an equal and opposite reaction, it!, of the foundation, consisting of the total vertical reaction, 2 (IF), and the total horizontal shear or friction, 2(P). The masonry of a dam and all classes of foundations, in com- mon with other materials, are known to be elastic. The effect of *2(TF) and S(F) are here used to represent a general condition, and may be applied to either full or empty reservoir. Fig. 10. Art. 22] THE REACTION OF THE FOUNDATION 43 such elastic properties makes the absolute determination of the distribution of compressive stresses very difficult, if not impossible. Because of the lack of knowledge on the subject, the distri- bution is assumed to follow a law of uniform variation, as in the design of beams, although it is known that, on account of such elastic properties, the unit pressures at the extremities of the base will be somewhat less than this method of distribution would indicate. In Fig. 10, p/ represents the unit vertical reaction* at point 1, and p T " that at point 2, due to the load, ~2(W). An ordinate at any point from the base to the straight line, 3—4, will represent the unit vertical reaction at that point. The area, 1-2-3—4, will then represent the total vertical reaction, S(W). To derive an equation for the value of p/, take moments about point 2. Then whence, But, •n "I 2 I 2 ,_ 3Z(TT) 32(W)u p T " Vt ~ I P 2 or, 22W , Pr l Pr. Substituting this value of p r " in the second equation, there results 22W Vt ~ I (*-?) .• • <"*> By taking moments about point 1, there may be derived, in like manner, "v-^fr-O <"> Dividing the base, 1-2, into three equal parts, as in Fig. 11, and calling the middle part the " middle third," it is seen from ■ =0, and p r = — j—*-. (13) When the resultant, R, intersects the base at the middle point, u = - and p r " = pr'= , . When the resultant falls outside the middle third, tension will exist at the opposite end of the base. This is indicated by substituting a value of u less than ■= in Eq. (11), whereupon p r " becomes negative. Also, by substituting a value of u greater 21 than — in Eq. (10), p/ becomes o negative. The former case is indicated in Fig. 11. The unreliability of tension in masonry has led to the common practice of neglecting it as a possible factor in the stability of dams. Moreover, for reasons which will be mentioned later, the resultant for well-designed dams is always made to inter- sect the base within the middle third. The values, p/ and p T ", derived from Eq. (10) to (13), inclu- sive, represent the unit vertical reaction of the foundations on the dam, corresponding to the load, S(Tf). As S(TT) includes the force of uplift and as the water pressure causing uplift is exerted both upward and downward, the corresponding unit vertical pres- sures, p„' and p v ", on both the dam and the foundation, can be Fig. 11. Art. 22] THE REACTION OF THE FOUNDATION 45 obtained by adding to the calculated values of p/ and p T " what- ever effective * unit uplift has been assumed to exist, as indicated in Fig. 10. To sum up: For the resultant within the middle third: p ,> = *Jfl(2-f) +P ; (10a) (^-l)+p u " (11a) 22(TT r )/3w p. - t For the resultant at the down-stream extremity of the middle third: Pw > = Wp.+ Pu ' i (i2a) p," = 0+ Pu " (12&) For the resultant at the up-stream extremity of the middle third: Pv j tPu , (Ida) p,' = 0+pj (136) The foregoing equations are apphcable also to the vertical reactions and pressures on any horizontal plane above the foun- dation. They are simply modifications of the theory of flexure and direct stress as applied to horizontal rectangular sections. B. Irregular Bases. If the base is not rectangular, as in the case of some hollow dams, the foregoing equations should be modified. As indicated in Fig. 12, the total pressure at any point in the base will be the sum of the flexural stress at that point due to the eccentricity of the loading, a uniform direct compressive stress corresponding to the loading, and the uplift pressure. From the principles of mechanics, the flexural stress at any point in the base is, Flexural stress =± j , * p„ is termed "effective unit uplift" as it is the absolute uplift multiplied by the percentage, c, of the area over which it acts, as explained in Art. 15. 46 FORCES ACTING ON DAMS [Chap, m where ra = any distance to the right or left of the center of gravity, in feet; positive on the side of the resultant, 2(TT), and negative on the other side; e = the eccentricity of the loading, in feet; 7= the moment of inertia of the base about an axis through its center of gravity perpendicular to the dam sec- tion, in feet units. The stress will be positive (compression) on the side of the resultant Fig. 12. and negative (tension) on the other side, as indicated by the line, 4-5-3, in Fig. 12. The direct stress is equal to \ , A ' where A is the area of the base, in square feet. This stress is rep- resented by ordinates between the lines, 4-5-3 and 6-7. Ordi- nates from the base, 1-2, to the line, 6-7, will represent the total flexure and direct stress, or the reaction of the foundation. The unit effective uplift, if existing, is laid off below the line 6-7; the line, 8-9, finally represents the total unit pressure in the dam and the foundation. If wl and m" represent distances to the extremities of the base, as indicated, then J 1 — J — r-Vu, .... (14) + Art. 22] THE REACTION OF THE FOUNDATION 47 and p ,, = _2^ + ^) +pu „ (15) C. General. It is well to note here that the vertical unit pres- sures, p/ and p„", are not the maximum compressive stresses in the foundation or the dam, but only the vertical components of * such stresses. The determination of the maximum stresses will be taken up in Art. 33. The distribution of the total horizontal shear, 2(P) (Fig. 10), is probably not possible of exact determination. Many attempts have been made to determine such distribution, with varied results.* Fortunately, it is not necessary to utilize the unit shearing stresses in order to determine the safety of the structure or to design the section. The amount of the total shear, S(P), is sufficient. *See Chapter VIII, "Masonry Dam Design," by Morrison and Brodie, 2d ed. John Wiley & Sons, Inc. 1916. CHAPTER IV REQUIREMENTS FOR STABILITY OF GRAVITY DAMS 23. Causes of Failure. There are two direct ways in which a dam will fail: 1. By sliding (a) on a horizontal joint above the foundation, (b) on the foundation, or (c),on a horizontal or nearly horizontal seam in the foundation. 2. By overturning (a) about the down-stream edge of a hori- zontal joint, (b) the base, or (c) a plane below the base. The direct cause of sliding is the presence of horizontal forces greater than the combined shearing resistance of the joint or base and the static friction induced by the vertical forces. The direct cause of overturning is the presence of horizontal forces, great enough, in comparison with the vertical forces, to cause the resultant, R (Fig. 10), of all forces acting on the dam above any horizontal plane, to intersect that plane outside of the limits of the dam.* A dam may start to overturn, but finally fail by sliding. This is caused by the fact that, when overturning starts, not only is the shearing resistance at the plane of rupture materially reduced, but the admission of head-water pressure to the fissure at once reduces the effective weight of the dam, so that the frictional resistance due to the pressure of vertical forces is also diminished, and sliding ensues. A dam with the resultant well inside the joint may overturn if the toe of the dam fails by crushing or other causes so as to reduce the effective length of the joint or base sufficiently to cause the resultant, R, to pass outside. Theoretically, the dam extends indefinitely below the base, but, the area of the vertical section of the foundation being unlim- ited, stability below the base is usually assured. Failures have been caused, however, by an erosion of the foundation caused by * Tension in the dam at the plane disregarded. 48 Art. 24] LOCATION OF THE RESULTANT 49 water spilling over the crest. In this event, one of two things may happen: the dam may become undermined so as to cause overturning, or the erosion may expose a horizontal seam filled with clay or other practically frictionless material, in which event, the rock ledge down-stream from the dam having been removed, it offers no resistance to sliding and the dam fails in that manner. The gradual disintegration of the dam by weathering and other causes will, of course, finally result in its failure. Modern masonry dams, particularly if of concrete, are practically indestructible if well built and composed of proper ingredients. However, if proper precautions are not observed in the choice of materials and methods of construction, there may be rapid deterioration of the structure. 24. Rule 1. Governing the Location of the Resultant It is obvious that, if the resultant of all forces acting on a dam above any horizontal joint, including uplift, passes outside that joint, the dam will overturn unless the joint is capable of resisting tensile stresses. As the tensile strength is always disregarded as inde- terminate and unreliable, we have as the first consideration that the resultant must intersect the joint. Further restrictions, however, are necessary. It was shown in Art. 22, that, when the resultant falls outside the middle third, tensile stresses are set up at the other end of the joint. If the joint is incapable of resisting these tensile stresses, the elasticity of the masonry will cause a slight opening of the joint. Such an opening is particularly objectionable at the up-stream side when the pond is full, as it will admit full head-water pressure over the entire area not in compression, a condition considerably more severe than usually adopted for uplift. This additional uplift would result in a movement of the resultant toward the toe of the joint, with a further opening, of the joint in tension, and a further increase in uplift. The progression may be sufficient to cause failure. It is customary in most government requirements, and to a lesser degree in practice, to prohibit tension at the down-stream edge of the joints when the pond is empty. The logic of this requirement is open to debate, as one cannot imagine any dam of the usual type overturning up-stream before the water is let into the pond. Parker * states that if tension, reservoir empty, occurs * "The Control of Water." D. Van Nostrand Co. 1913. 50 STABILITY OF GRAVITY DAMS [Chap. IV over 0.04 to 0.10 of the joint, it does not necessarily mean a bad design. However, out of deference to precedent, and the usual govern- ment requirements, the author has adopted this condition. We can now write the first designing rule. RULE 1. GOVERNING THE LOCATION OF THE RESULTANT Tension shall not exist in any joint of the dam, under any condition of loading. For dams with rectangular joints, the re- sultant of all forces acting on the dam above any horizontal joint (including uplift) shall, for full or empty reservoir, intersect the joint within the middle third.* 25. Rule 2. Governing the Inclination of the Resultant. The resultant, 2(P), (Fig. 10), of all the horizontal forces acting on the dam above any horizontal joint has a tendency to slide that part of the dam over the lower part. The shearing and frictional resistance of the joint must be sufficient to withstand this tendency. The planes of weakness are the necessary horizontal joints (including the base), between two days' work. The shearing value of such joints, though sometimes considerable, f is unreliable. It is customary, therefore, to neglect any possible assistance from shear, and rely solely on the frictional resistance due to the ver- tical forces, 2(T7), acting on the dam above the joint. If /' represents the coefficient of static friction of the materials above and below the joint, then/'2(W) will be the frictional resist- ance to sliding. For equilibrium, f'2(W) must be equal to or greater than 2(P). This may be expressed algebraically by: /'2(F) = >2(P), or, |£}=tan *=, and from published records of computed maximum vertical stresses. These may be considered to be the greatest stresses which have thus far been adopted for conservative designs. The table also indicates the maximum stresses corresponding to examples of design of two authorities, and values recommended by others. TABLE V Various Recommended Working Stresses and Existing Stresses in Masonry Dams Dam. Vebtical Stresses. Inclined Stresses. Toe. Heel. Toe. Heel. 17,500 31,000 33,400 20,000 16,800 28,000 16,400 15,600 25,500 32,000 30,800 25,000 21,000 31,000 20,400 20,000 26,800 43,700 37,000 26,000 46,000 52,400 25,000 28,000 35,300 31,600 35,000 Wegmann's Practical Profile No. 3 Morrison and Brodie's Example of Design Recommended by Parker 25,400 31,400 25,000 Recommended by Rankine * Stresses for the maximum section are probably not as great as computed, owing to the narrowness of the gorge between the sides of which the maximum secticn is located. After an extensive study of published actual stresses computed for existingstructures of many types, and the opinions of various authorities, it is recommended that, for maximum inclined com- pressive stress in masonry dams, a working value of one-ninth of the ultimate strength of the masonry should be adopted. This factor of safety should be sufficient to cover the element of uncer- tainty in the design and the relatively lower strength of masonry in large masses as compared with that of test specimens. Table VI, extracted from Taylor and Thompson's " Concrete Plain and Reinforced," indicates the ultimate strength of concrete of different proportions. Art. 26] GOVERNING COMPRESSIVE STRESSES 55 TABLE VI Relative Ultimate Strength of Portland Cement Concrete, in Pounds per Square Foot Proportion. Age One Month. Age Six Months. 1:2:4 350,000 470,000 1 2| :5 310,000 420,000 1 3 :6 280,000 380,000 1 4:8 230,000 300,000 1 5 :10 180,000 250,000 1 : 6 : 12 150,000 200,000 For important structures, tests should always be made, at intervals, to determine whether the concrete, as being mixed, con- forms in strength to the assumptions used in the design. Although the ultimate strength of stone masonry has never been directly determined, a value of 250,000 to 300,000 lb. per sq. ft., for use as herein recommended, may be assumed as the ultimate strength of good rubble. Until recent years, the maximum allowed unit vertical stresses commonly used for masonry dams on rock were those recom- mended by Rankine, for the dams of the City of Bombay, namely, 15,620 lb. per sq. ft. at the toe and 20,000 lb. at the heel. It will be shown later that, for constant vertical stresses, the amount of the inclined stresses in the masonry at either extremity of the base increases with the angle, , which that face makes with the ver- tical. Since, in the ordinary type of solid gravity masonry dams, the up-stream face is considerably less inclined than the down- stream face, Rankine reasoned that greater vertical stresses could be allowed at the heel than at the toe, and arbitrarily adopted these working vertical stresses in the dam on the design of which he was reporting. The designers of recent dams, however, have exceeded these stresses considerably, as indicated by Table V. There is no logical basis on which to make a recommendation of working values of vertical stresses, because, as explained previously, the safety of the dam depends entirely on the amount of the existing inclined pressures which, for constant maximum vertical stress, varies with the shape of the section. It is necessary, however, to adopt 56 STABILITY OF GRAVITY DAMS [Chap. IV values which are within the limits of those commonly used for masonry dams, and which, moreover, will result in a section in which the maximum inclined stresses are not excessive. The fol- lowing values will ordinarily fulfill both these conditions: At the toe of solid dams ^t of the ultimate strength. At the heel of solid dams ^ of the ultimate strength. At the toe and heel of hollow dams x§- 0I the ultimate strength. For the foundations, it is universally agreed that bed-rock suitable for a high dam will in every case be stronger than the masonry placed upon it. For dams on earth foundations, the requirements of Rule 2 necessitate a very small angle of the result- ant with the vertical. It will be shown later that the difference between the maximum inclined and the maximum vertical stresses in the foundation is a function of this angle and, therefore, so small as to be negligible. Common values for allowed stresses on earth foundations are as follows: Clay 8,0001b. per sq. in. Coarse sand. 4,000 to 8,000 lb. per sq. in. Fine silt 2,000 to 4,000 lb. per sq. in. It must be borne in mind that the qualities of such materials vary greatly, and that adopted values, in all cases, must be based on complete investigation at the site. 27. Rule 4. Governing Tension in Vertical Planes. Mr. L. W. Atcherly, * of London University, claims to have proved that, although the resultant, in dams of the usual type, may fall well inside the middle third of the base, there may still be considerable tension in vertical planes near the toe of the dam. Messrs. Mor- rison and Brodie | give an outline of Atcherly's theory, as well as the opinions of a number who have entered into a discussion of the paper. The paper attracted a great deal of attention at the time, and even influenced the design of several important struc- tures. Later studies, however, have led to the general conclusion that such danger is not ordinarily to be feared. Atcherly's theory * In a paper "On Some Disregarded Points in the Stability of Dams " See foot-note, p. 71. ' t " Masonry Dam Design." s Ast. 28] MARGIN OP SAFETY 57 is based on an assumed distribution of horizontal shearing stresses, which remains to be demonstrated,. A great number of existing dams, if investigated according to Atcherly's theory, would be found to be quite unsafe, if not incapable of sustaining their loads. Considering the hundreds of dams which have been conserva- tively designed and built without a thought of tension in vertical planes, not one of which is known to have failed through this source of weak- ness, it must be assumed that there is some error in Atcherly's methods, and that they impose too severe a condition on structures of the usual type. However, one must not lose sight of the fact that tension will certainly FiQ 12o exist in vertical planes if the inch- nation of the down-stream face of the dam is too great to transmit the loads to the foundation, as in Fig. 12a. In such cases there is danger of the toe cracking off on the line, 1-2, even though resistance to shear on that plane is ample. The influence of these contingencies on the design of the dam is taken up in Art. 34. RULE 4. GOVERNING TENSION IN VERTICAL PLANES The inclination, with the vertical, of the down-stream faces of the dam shall be limited to prevent or safely withstand all possible tensile stresses in vertical planes. 28. Rule 5. Governing the Margin of Safety. The usual meaning of the expression " factor of safety," in the design of structures, is the ratio of the loading which will be just sufficient to cause damage or failure to the loading which has been adopted in the design. Factors of safety may be applied to the balance or resistance of a structure to overturning or to unit stresses. In a cantilever bridge it is usual to design the shore span so that it will have a resisting moment about the pier equal to twice the moment of maximum combined dead and live loads on the river end, thus providing a factor of safety of two against overturning. A steel tie-rod designed to be stressed to one-half its elastic limit will have 58 STABILITY OP GRAVITY DAMS [Chap. IV a factor of safety of two against damage (permanent elongation), and a factor of about four against failure. In dealing with unit stresses, the term " factor of safety " is misleading, as it is usually applied to the ultimate strength, whereas in most cases, it is the elastic limit which is the limiting stress. With regard to the balance of a structure, or the location of the resultant of forces, a factor of safety of one is, theoretically, all that is necessary, provided the loading is certain. However, on' account of the usual uncertainty as to the loading, a factor of safety greater than one is necessary. In a dam, the head- and tail- water pressures for any given depth may be accurately calculated. A careful assumption of the weight of the dam should not vary more than 4 per cent from its correct value. Then we must adopt pressures for ice, silt, and uplift, which we are certain will not be exceeded, in order to justify a location of the resultant at the extremity of the middle third. If this is done the dam, if cor- rectly designed, may be considered perfectly safe, provided, also, that the calculated induced unit compressive stresses amount to a safe proportion of the ultimate strength of the materials. The dam will have an additional element of safety on rock foundations, because the adhesion of the concrete to the foundation and in horizontal planes above the foundation, though certainly consid- erable, is always neglected. In fact, to this feature, alone, can be attributed the existence of a number of poorly designed dams. RULE 5. GOVERNING THE MARGIN OF SAFETY All assumptions of forces acting on the dam shall be unques- tionably on the safe side; and all unit stresses adopted in the design shall provide an ample margin against rupture. The margin of safety varies considerably in recent dams, and depends to a great extent on the use to which the structure is to be put, its magnitude, and the probable damage and loss of life which would result from failure. One of the most conservative designs of modern dams is that of the Olive Bridge Dam, of the New York City Water Supply System (Fig. 23). Probably few private enterprises could stand the burden of such conservatism. 29. Rule 6. Governing Details of Design and Methods of Construction. The shape of the section of the dam having been determined in accordance with the foregoing rules, careful atten- Art. 29] METHODS OF CONSTRUCTION 59 tion must be paid to the details of the design and the methods of construction, in order that the structure may be satisfactory in every respect. The location and extent of vertical building joints, passage- ways, and other planes of weakness must be well within proper limits, in order that the stresses used in the design will not be seri- ously increased. Such features as drains and cut-offs, on which 'the assumption of uplift has been based, must be carefully worked up, and other matters of much importance attended to. The engineer should be in a position to insure that the masonry in the structure will be of a quality to withstand safely the working stresses adopted in the design, practically water-tight, and dur- able. The final rule may be written as follows: RULE 6. GOVERNING DETAILS OF DESIGN AND METHODS OF CONSTRUCTION All details shall be adopted so as not to interfere with the assumptions used in the design; the masonry shall be of a quality suited to the working stresses adopted, and shall be practically water-tight and durable. CHAPTER V GENERAL EQUATIONS FOR DESIGN OF GRAVITY DAMS 30. General Considerations. It is impossible to derive a general equation for the direct determination of the shape of the section. The only possible solution is to design the dam, joint by joint, beginning at the top, making -each joint conform to the designing rules given heretofore. Assuming the dam to have been designed from the top to a certain horizontal joint, 3-4, Fig. 13, ^ *--Point of reference, or orgin of momenta Fig. 13. of length, lo, equations can be written for the determination of the location, y, and length, I, of the next joint a distance, h, below, to conform to that one of the foregoing rules which happens to govern at the particular stage of the design. The length of the joint will be mathematically exact, but the width of the section between the joints will be approximate only. By taking the vertical distance, h, small enough, say 15 per cent of the height of the dam above the joint considered, the resulting error will be negligible. The section will have polygonal faces, which, of course, may be smoothed up later, for appearance, with no appreciable change in the stability of the section. Each step in the design is based on one of the rules which is 60 Art. 30] GENERAL CONSIDERATIONS 61 thought to govern. After each joint is calculated, and conforms to that rule, it is necessary to investigate the joint to determine if the other rules are also complied with. It is essential, therefore, for each designing rule, to derive two sets of equations, as follows: 1. " Equations of Determination " for each rule, with which to determine the length and location of successive joints in conformity with that rule; 2. " Equations of Investigation " for each rule, with which to investigate any pre-designed portion of the dam for conformity with that rule. Such equations will be derived for each of the designing rules, it being found convenient to derive the latter set first in each case. Fia. 14. Before proceeding further, however, it seems desirable to indi- cate the influence of each designing rule on the general shape of the section of the dam. In order to do this, reference is made to Figs. 14 and 15, which indicate typical sections of a solid non- overflow and spillway dam, respectively. 62 DESIGN OF GRAVITY DAMS [Chap. V The section of the dam may be divided into a number of zones, as indicated, it being necessary to design each zone in accordance with a different rule or combination of rules.* Water Snrf ace Non-overflow Dams When ice pressure occurs, the quantity of masonry in Zone I, above the water surface of non-overflow dams, is fixed by Rule 2, as sufficient weight must be provided to prevent the portion, l-2-3^i, from sliding. In Zone II the resultants, reservoir full and empty, lie well within the middle third, due to the fact that the width of the top is always greater than necessary to conform to Rule 1. Both up- and down-stream faces, therefore, will remain vertical f until, * As far as the author is aware, Wegmann was the first to use similar divisions of the section of the dam to explain the methods of design. f It is generally recognized that there is no economy in battering the faces of the dam unless that is necessary in order to conform to one of the designing rules. Art. 30] GENERAL CONSIDERATIONS 63 at the plane, 5-6, the resultant, reservoir full, intersects the joint at the exact extremity of the middle third. At the top of Zone III, the down-stream face must begin to batter in order to accord with Rule 1, reservoir full; and the resultant, reservoir empty, still being within the middle third, the up-stream face remains vertical until at the plane, 7-8, the resultant, reservoir empty, also intersects at the extremity of the middle third. Therefore, at the plane, 7-8, the up-stream face must begin to batter, in order to accord with Rule 1, reservoir empty. In Zone IV the resultants, reservoir full and empty, intersect the joint at the extremities of the middle third. The upper limit of Zone V is fixed by the condition of limiting vertical pressures, Rule 3. Usually, the maximum allowed unit pressure is reached at the down-stream face first. Below the plane, 9-10, the length of the joints must be determined by Rule 3, for full reservoir, and by Rule 1, for empty reservoir. This will result, for Zone V, in the resultant, reservoir full, intersecting well within the middle third, and, for reservoir empty, the resultant continues to intersect at the extremity of the middle third. The vertical pressures at the up-stream face, however, will gradually increase, and, at the upper extremity of Zone VI will just reach the allowed working value. In Zone VI, therefore, the length of the joints will be determined entirely by Rule 3, the resultants, reservoir full and empty, intersecting well within the middle third. As the section increases in height, the batters of both up-stream and down-stream faces increase. The down-stream face, in solid gravity dams, always has a flatter slope than the up-stream face. Consequently, at some elevation, such as 13-14, the inclination of the down-stream face from the vertical may reach the maximum allowed value. Zone VII, therefore, represents a portion of the dam in which the inclination of the down-stream face is apt to be greater than the limit fixed by Rule 4. It is unfortunate for the peace of mind of the designer if this happens, for, in that event, as will be indicated later, it will be necessary for him to start the design all over again, and provide a different arrangement in the upper part which will result in a steeper down-stream face. 64 DESIGN OF GRAVITY DAMS [Chap. " Spillway Dams Owing to a deficiency in the weight of the top of all spillway dams, the resultant, reservoir full, within Zone I, will necessarily fall outside the limits of the middle third, in direct violation of Rule 1. This condition must be met by special construction, as explained later. At the extreme top of the dam, where the weight of the masonry is negligible in comparison with the water pressure, the resultant will intersect at a distance infinitely remote, as indi- cated by the trend of the line of resultants, reservoir full, shown in Fig. 15. For the same reason, above the bottom of Zone la, the inclina- tion of the resultant, reservoir full, will make an angle with the vertical greater than the allowed value, in direct violation of Rule 2. At the extreme top of the dam, where the weight is negligible in comparison with the water pressure, there will be no frictional resistance and sliding of the upper part of the dam must be pre- vented by shearing resistance. It is obvious that, above the bottom of Zone la, these two rules must be violated, as it is not feasible to supply enough masonry without obstructing the water spilling over the crest. Fortu- nately, however, the conditions are not severe and can be easily met by special treatment. The conditions fixing the limits of Zones II to VII, inclusive, for spillway dams are exactly as previously described for non- overflow sections. It should be borne in mind that the arrangement of zones indi- cated in Figs. 14 and 15 represent the usual conditions met in the design of solid dams. In hollow dams the arrangement will be different, but the general theory will apply. The equations for design given in the following articles will be derived with respect to solid dams; but they are equally applicable, in general, to hol- low dams. 31. Equations for Rule 1. Equations of Investigation. To derive a general equation for the location of the intersection of the resultant, R (Fig. 13), with the joint, 1-2: Let S(TF) represents the algebraic summation of the vertical components of all forces acting on the dam above the joint, 1-2, including uplift; 2(JF), also rep- Ast, 31] EQUATIONS FOR RULE 1 65 resents the equal and opposite vertical reaction of the foundation; 2(Wa;) represent the algebraic summation of the moments about any convenient joint, 17,* of the forces above the joint, contained in the summation, 2(W r ); positive when counter-clockwise; 2(P) represent the algebraic summation of the horizontal components of all forces acting on the dam above the joint, 1-2; 2(P) also represents the equal and opposite horizontal reaction of the founda- tion; 2 (Pre) represent the algebraic summation of the moments about the point, 17, of the forces above the joint, contained in the summation 2(P); positive when counter-clockwise; Subscript e represents the condition of empty reservoir; Subscript f represents the condition of full reservoir; Other symbols as indicated in Fig. 13. The moment of all forces, acting on the dam, above the joint, about the point 17, is 2(TFa;) + 2(Pa;). The moment of the reac- tions about the same point is 2(TF)z, the moment of the force 2(P) being zero. For equilibrium, these moments must be equal; therefore, for the general case of either full or empty reservoir, we have: 2(Wx)+2(Px) = I,(W)z (17) Solving for z, there results: _ 2(PTx) + 2(Px) Z 2(TF) (18) 21 According to Rule 1, the distance, z-y, must be less than — for o full reservoir and greater than - for empty reservoir. The usual forces acting under the condition of " full reser- voir " are indicated in Fig. 16. Equations for the determination * Point 17 should be on any convenient line of reference, and at the elevation of the joint considered. 66 DESIGN OF GRAVITY DAMS [Chap. V of the amounts and locations of these forces have been given in Chapter III. For " empty " reservoir, the force due to the weight of masonry- is usually considered to be the only one acting on the dam. This is consistent with the loading after completion, but before the coffer- dam enclosure is allowed to fill, and before earth or silt is deposited against the dam. It usually gives the severest condition, as far as Rules 1 and 3 are concerned. Hoto:- Reaclions on the joint, 1-2, not indicated. Fig. 16. Equations of Determination. First. The general case, where the location of the resultant for both full and empty reservoir governs the design of the section, as in Zone IV (Fig. 14). In Fig. 13, suppose the dam to have been designed from the top down to the joint, 3-4, and that it is desired to determine the length, I, of the next joint, 1-2, and its location, y, relative to the convenient point of reference, 17. Art. 31] EQUATIONS FOR RULE 1 67 For full reservoir we may substitute (y+%1) for z in Eq. (18). Adding the subscript, F , there results: v+V ^'tmf*' - <19) For empty reservoir, by substituting y+%1 for z in Eq. (18) and using the subscript, E , there results: I _ S(ITs) g +S(Ps) E ,__* y+ 3~ ^ m Thus we have two equations with two unknowns, y and I. The factors in the second terms of these equations, however, are functions of y and I. By substituting in the second terms the values of the various forces and moments, in terms of y and I, exact equations may be derived from which the values of y and I can be solved directly. Unless, however, the vertical components of water and silt pressure on both faces of the dam are neglected, such equations will be too intricate for practical application.* A sufficiently accurate tentative method of solving for y and I may be used, as follows: From the general trend of the slopes of the up- and down- stream faces, tentative approximate values of y and I may be used for the determination of the factors in the second terms of the equations. If the height, h (Fig. 13), is not more than 15 per cent of the height of the dam above the joint considered, the resulting calculated values of y and I will generally he within 0.5 per cent of the truth, so that usually, a second trial substitution will be unnecessary. * Messrs. Morrison and Brodie, in their "Masonry Dam Design," by neg- lecting the items mentioned, have derived such equations applicable to Rules 1 and 3 and combinations thereof. The author has found, however, by actual comparison, that his tentative method, hereinafter described, is sufficiently accurate, involves no more labor, and has the advantage of not only including all forces acting on the dam, but requires a tabulation of forces, moments, and other features in such a manner as to allow comparatively little opportunity for omissions or misapplications. Parker, in his "Control of Water," has also derived similar equations but states: "I am not aware that these equations have ever been applied in practice, and as the result of actual experiment, I am inclined to believe that a return to first principles and trial "and error is probably more rapid." 68 DESIGN OF GRAVITY DAMS [Chap. V Second. The special case where the location of the resultant, reservoir full, is the only governing condition, as in Zone III. For this case the up-stream face is vertical and, therefore y, in Eq. (19), is known. For this reason, and also because the resultant, res- ervoir empty, is well within the middle third, Eq. (20), is not necessary, and the lengths, I, of successive joints in this zone may be solved from Eq. (19) alone. Third. The special case where it is required to determine the elevation of the joint where the resultant, reservoir full, first inter- sects the down-stream extremity of the middle third, as at the bottom of Zone II. For most types of dams it will be found con- venient to determine this elevation by trial, using Eq. (18). For Water Surface Fig. 17. the non-overflow dam, however, it is possible to derive an exact equation applicable to the usual conditions of loading, namely, the external forces acting on the dam consisting of head-water pressure, ice pressure, and uplift. In Fig. 17 let 1-2-3-4 represent the top of such a dam. Let it be required to determine the distance, h, below the water surface, at which the resultant, R, reservoir full, lies just at the extremity of the middle third. The forces assumed to act on this portion of the dam are indi- cated in Fig. 17, and may be expressed as shown below. The origin of moments in this case is taken at the down-stream extrem- ity of the middle third. Remember that counter-clockwise moments are considered positive. Art. 31] EQUATIONS FOR RULE 1 Item. Force. Moment. Vertical forces: Weight of masonry, -\-wiLH Uplift,* wiLW 6 CW2I1L cw2hL 2 ~~2~ " l 6 v ™ TV cw 2 hL . . it>iL 2 H t cw 2 hL 2 2 ' v " J 6 ' 6 Horizontal forces: Ice pressure, +P { +PJi Water pressure, -\ — %r- -\ — |— 2(P)=P t +^ S(Px)=P^+^ Eq. (18) may now be written: _ -E(Wx) + 2(Px) 6 ^ 6 2 = = — - w\L 2 H , cw2hL 2 . _ . , W2h 3 '-P t h-\ — -T- 2(W) TTT cw 2 hL wiLH y~ Substituting h+a for H, there results: (cw 2 L 2 +6P t -w 1 L 2 )h+W2h 3 = wiL 2 a, . . . (21) from which the value of h may be derived by proper substitutions. Fourth. The special case in which it is required to determine the elevation of the joint, where the resultant, reservoir empty, first intersects the up-stream extremity of the middle third, as at the bottom of Zone III. No equation can be written for this case. It must be solved by trial, using Eq. (18). Fifth. The special case where the location of the resultant, reservoir empty, and limiting pressures at the down-stream ex- tremity of the joint (Rule 3), reservoir full, governs the design, as in Zone V. For this case, the resultant, reservoir full, being well within the middle third, Eq. (19), should be disregarded and Eq. (20) used in conjunction with another equation embodying the * The uplift is assumed to vary uniformly from a value, cw 2 h, at the up- stream end of the joint to zero at the down-stream end. 70 DESIGN OF GRAVITY DAMS [Chap. V condition of limiting toe pressures. This combination will be taken up in the derivation of equations for compressive stresses. 32. Equations for Rule 2. Equations of Investigation. The tangent of the angle, 6, of the inclination, with the vertical, of the resultant, R, Fig. 12, is evidently ^ e =0) c°J According to Rule 2, tan 6, for rock foundations, must be equal to or less than the coefficient of friction, /; and for earth founda- / tions, equal to or less than ". Equations of Determination. As mentioned heretofore, it is nec- essary, in non-overflow dams, to provide sufficient masonry above the water surface to counteract the sliding force of ice. It is not practicable to write an equation for this condition, for, if the width of the top and the superelevation adopted on account of other considerations is not sufficient for this purpose, there is nothing to indicate whether an increase in the top width or an increase in the superelevation would provide more economy for the dam as a whole. A slight increase in either has little influence on the quantity of masonry in the dam, because an addition of mate- rial at this place will effect a greater or less reduction in the lower parts. The choice must be made in accordance with the judg- ment of the designer. In spillway dams it is not practicable to provide sufficient masonry at the top to conform to Rule 2, to provide for ice thrust. In this case it is necessary to violate the requirements of Rule 2 and compensate by some special construction, such as keys, in- clined joints, or monolithic concrete.* Violations of this kind are also sometimes found in the tops of concrete non-overflow dams, where it is possible to provide monolithic concrete within the danger zone. Except as mentioned previously, it seldom happens that Rule 2 is a factor in the design of solid dams on rock foundations. Tan 9 should always be determined, however, and compared with the allowed value, after each step in the design has been completed. If, at any stage of the design, it is found that tan 6 is greater than the value allowed by Rule 2, then several courses of procedure are * See Example No. 4 of Art. 46. Art. 33] EQUATIONS FOR RULE 3 71 open to the designer. It is evident, in such a contingency, that the fault lies in the lack of weight, or, in other words, lack of resultant vertical forces, 2("FF). Additional weight may be ob- tained by adding to the top width, increasing the surcharge, in- creasing the batter of the up-stream or down-stream face, or a combination of these expedients. Each case must be decided on its own characteristics. It will usually be found most economical to increase the batter of the up-stream face, as this will include an additional vertical component of water pressure. One of the principal reasons for the flat slope of the up-stream face of a hollow dam is the need for the large vertical component of water pressure to fulfill the conditions of Rule 2. A hollow dam with a vertical up-stream face would contain, on account of this rule, nearly as much masonry as a solid dam. The batter of' the up-stream face of a hollow dam, if fixed by Rule 2, is found by successive trials, using Eq. (22). 33. Equations for Rule 3. Equations of Investigation.* In Art. 22, the following general equations were derived for the deter- mination of the maximum vertical compressive stress in the hori- zontal joints and base:* At the toe, 22(F) /_ 3«^ At the heel, p , = 2^Wl^ +pu> (1Qa) 2S W(f-l) +Pu " (Ha) V I The symbols are explained in Art. 12 and indicated in Fig. 10. However, as explained in Art. 26, the vertical compressive stresses are not the maximum stresses in the structure, the latter occurring on planes which are not horizontal. The question of the maximum, or " inclined," stresses in dams has been the sub- ject of considerable discussion, f more particularly abroad than in * Eqs. (12a) to (136) inclusive, being for special cases, have been omitted for brevity, and may be easily included by the reader. t L. W. Atcherly, Dept. of Applied Mechanics, University College, Uni- versity of London, Draper's Co. Research Memoirs, Technical Series II; Sir Benjamin Baker, Minutes of Proceedings, Inst. C. E., Vol. 162, p. 120; L. W. Atcherly, Engineering, Vol. 79, p. 414; W. C. Unwin, Engineering, Vol. 79, pp. 513, 593, and 825; Ottley and Brightmore, Minutes of Pro- ceedings, Inst. C. E., Vol. 172, p. 89; Wilson and Gore, Minutes of Proceedings, Inst. C.E., Vol. 172, p. 107; E. P. Hill, Minnies of Proceedings, Inst. C. E., Vol. 172, p. 134; William Cain, Transactions, Am. Soc. C. E., Vol. 64, p. 208. 72 DESIGN OF GRAVITY DAMS [Chap. V America. It is unfortunate, however, that no general agreement has been reached on this subject. The most commonly accepted theory is that of Enger, as ex- panded by Cain, Hill and others. The result of Enger's theory is an inclined stress at the extremity of the joint equal to Pi=Pv sec 2 the angle of inclination of the face of the dam with the vertical. If a normal pressure, p n , of water or silt acting on the face of the dam, is included, this equation will reduce to Pi=(pv sec 2 —p n tan 2 ) or p n . In the last equation, the two values of p t correspond to the two principal planes of stress; the governing condition being, of course, the greater of the two. Enger's theory, however, is not generally applicable, because the whole argument is based solely on a triangular dam, with vertical up-stream face, loaded with water pressure level with the top. The author has found that the theory will not apply to any other shape of dam or loading; and that, for cases where the angle, 6, of the resultant, R, is greater than , pt, from the fore- going equation, will give results too low. It is, indeed, rational to assume that for all values of greater than , the theory of Bouvier * will apply more closely. This theory gives as the value of the inclined stress, Pt=pv sec 2 0. This equation is probably the most accurate of any for the actual maximum stress in the foundation for all conditions. Combining the two theories, the author proposes the following general equations for the true maximum, or maximum inclined compressive stresses in the dam and the foundation. For maximum stress in the dam: At the toe p/ = (p/ sec V - p n ' tan V) or p„' or p/ sec 2 0. . (23a) * "Calculs de Resistance des Grands Barrages en Maconnerie." Annaks des Ponts et Chausstes, Aug., 1875. Art. 33] EQUATIONS FOR RULE 3 73 At the heel, Pi" = (p„" sec V-P»" tan V) or p." or p„" sec 2 0. (236) For maximum stress in the foundation: At the toe, p t ' = p/ sec 2 (24a) At the heel, p," = p„" sec 2 (246) In these equations the general expression (p„ sec 2 —p n tan 2 ), gives the approximate flexural stress on planes, A, Fig. 12a, lying close to the point, 3, and normal to the face, 2-3. The value, p„, is, of course, the normal pressure of external forces on the plane, 2-3, close to the point, 3. The general expression, p„ sec 2 0, gives the stress between the dam and the foundation, on planes, B, lying normal to the resultant, R. Eq. (236) will not apply to the heel of hollow dams, if the horizontal area of the deck is included in the area of the base (as explained in Art. 49), the equation applying only to solid sections. Fortunately, however, it is known that, in all ordinary types of hollow dams having sloping up-stream faces, the maximum stress at the heel corresponds to the bearing of the deck on the buttresses. For practical application to hollow dams, see Art. 50. Although the foregoing equations and the theories on which they are based are not exact, and, for certain conditions, probably not closely approximate, nevertheless the equations certainly indi- cate a value for the maximum stress considerably nearer the truth than the vertical pressure, p„, generally used. The author does not propose a change in the present practice, in which the shape of the section is fixed by certain allowed maxi- mum vertical pressures, p„, as indicated by Eqs. (10a and 11a). It is recommended, however, that the completed section should always be tested for maximum stresses in accordance with his proposed Eqs. (23) and (24), and altered if found necessary. Equations of Determination. It has been pointed out that it is usual to consider only vertical pressures in the determination of the shape of the section, the section afterward being investigated for maximum stresses and changed if found necessary. First. When the condition of limiting vertical toe pressures, reservoir full, and the condition requiring the resultant, reservoir 74 DESIGN OF GRAVITY DAMS [Chap. V empty, to intersect at the up-stream extremity of the middle third governs the design, as in Zone V. This is a combination of Rules 1 and 3. The location of the resultant is given by Eq. (18), from which, by using the subscript, F , to represent full reservoir, there results, z(Wx) f +?;(Px)f Zf ~ X(W) r The distance, u F , Fig. 13, is: u F =l+y-z F =l+y xCWh ~^' The maximum vertical compression at the joint for full reservoir is at the toe of the joint, and is given by Eq. (10a). Transposing that equation, using the subscript, F , and substituting, for u F the value just given, there results: 42(W) F l-p,'l 2 +Pu'l 2 _ 7 , 2(Wx) f +2(Px)p ' 6Z(TP), ~ ~ t+V 2(TF), Solving for I, there results: Pl^plp _ X(W) Ll = x(W)i ^_ { 2(w x)f+ 2 {Px)f] . . (25) For empty reservoir, Eq. (20) applies: y+ Z~ 2(WS (20) Thus we have two equations, (25) and (20), with two unknowns, y and I, from which the location and length of the joint may be determined. The values, 2(1*0, 2 (Pa;), and 2(T7x), for both full and empty reservoir, are also functions of y and I. Here, as in the equations of determination (first case), applicable to Rule 1, these values, in terms of y and Z, may be substituted in Eqs. (25) and (20) and exact equations derived. However, as in the former case, certain omissions must be made in order to provide equations of practical application.* The author, therefore, proposes the * See foot-note on page 67. Aet. 34] EQUATIONS FOR RULE 4 75 same method of trial substitution of approximate values of 2(W), (2P#), and 2(Wx),' derived by anticipating, from the general trend of the faces of the dam, the location, y, and the length, I. If such substitutions are made in Eqs. (25) and (20) the resulting calculated values of y and I will be, after the first or second trial, sufficiently accurate for all practical purposes. Second. When the conditions of limiting pressures, reservoir full and empty, govern the design, as in Zone VI. The location of the resultant is given by Eq. (18), from which, by using the subscript, E , to represent empty reservoir, there results: ?(Wx) B +'2(Px) B Ze ~ 2(TP)* The distance, u E , is: 2(Wx) E +2(Px) B u E =l+y—z E =l+y—'- i(ff)i The maximum vertical compression at the joint, for empty reser- voir, occurs at the heel, and is given by Eq. (11a). Transposing this equation, using the subscript E , and substituting for u E , the value just given, there results: p v "l 2 +2Z(W) E l-p«"l 2 _. Z(Wx) E +2(Px) E 62(TT) B ~™ 2(W) E Solving for I, there results P"-P*'\ 2_2^^ l= 2 {why _ { 2 iWxh+ z ( p x)El (26) For full reservoir, Eq. (25) applies: P -^^l 2 -^^l = ^(W) F y-{2(Wx) F +i:(Px) F }. . (25) Here we have two equations with two unknowns, y and I, from which the location and length of the joint can be determined by the author's method of trial substitution, as in the preceding case, the discussion of which also applies here. 34. Equations for Rule 4. It was explained in Art. 27 that tensile stresses in vertical planes may be caused by an excessive inclination of the down-stream face with the vertical. It is seldom 76 DESIGN OF GRAVITY DAMS [Chap. V that the shape of the section of the dam is fixed in any way by Rule 4. However, where this rule is a governing feature, there is no way of proportioning the shape of the section except by the cut- and-try method. It is obvious that the direct cause of such tension at the toe is a relatively greater movement of the triangle, 1-2-3 (Fig. 12a), toward the left than that of the rest of the dam. This condition may be brought about by failure of the frictional and shearing resistances of the plane, 1-3, or by unequal inclined settlement of the foundation. The allowed inclination, ', of the face, there- fore, must be a function of the coefficient of friction, /' (shear on 1-3 being neglected, as usual), and the settlement coefficient of the foundation. The author knows of no satisfactory theoretical analysis of this feature. His empirical equations, based on safe approximate assumptions, are as follows - For earth or pile foundations, tan0'=<^, (27) where H is the height of the dam. For rock foundations, tan<£'=<|/ or /io tan<2>'=< A/fi . whichever allows , the greater value ' " '*" ' where / is the allowed coefficient of friction. The value (tan '= <$•/) in Eq. (28) embodies the condition of no tension in vertical planes, but Eq. (27) and the second part of Eq. (28) embodies the condition of tensile stresses so small as to be negligible. Eq. (28), for good rock foundations, does not impose a severe condition. Substituting a usual value of /=0.75, there results <£' = 45°, which is about the limit reached in high existing dams. According to the second part of Eq. (28), a value of ' greater than 45° would be allowed in dams less than 10 ft. high. 35. Equations for Rule 5. Rule 5, governing the margin of safety, is applicable only to the determination of constant assump- Abt. 36] EQUATIONS FOR RULE 6 77 tions used throughout the design, and does not require equations for its correct application. 36. Equations for Rule 6. It is evident that no universal rules or equations can be written to provide against defective details and methods of construction. Details vary considerably with each dam, and depend mostly on the judgment of the designer to guard against infringement of the designing rules. CHAPTER VI THE DESIGN OF SOLID NON-OVERFLOW GRAVITY DAMS 37. General Considerations. As it will always be found convenient to start design at the top of the dam, the shape of the top of the section is the first consideration. A superelevation of the top above high-water surface is some- times desirable in order to get beyond the reach of waves,* for appearance, and for other incidental purposes. When ice pressure is assumed to act at the level of high water, some superelevation will be found necessary in order to fulfill the conditions of Rule 2 at the point where the ice pressure is applied. In any case it is probable that a superelevation of about 5 per cent of the height of solid non-overflow dams is, in general, productive of economy, rather than an expenditure of material, although sufficient proof has not been presented to verify this for all cases. However, there is no evidence to show that the adoption of a slight super- elevation is uneconomical. The most .economical width of top of a solid non-overflow gravity dam is a direct function of the height of the dam, and is dependent on the assumptions used in the design. For dams of fairly uniform height, and designed in accordance with the usual designing assumptions, the most economical top may be assumed at about 14 per cent of the maximum height, f but for dams of considerable variation in height this figure should be somewhat reduced. It is usually considered that a value of 10 per cent is about the minimum advisable. The width of top. of low dams is usually somewhat greater than that dictated by economy, as a roadway or passageway is often necessary, as well as sufficient width to withstand the shock of floating bodies. The details of the top of the dam having been determined, the up-stream and down-stream faces must be designed to conform strictly to the designing rules hereinbefore given, it being remembered that there * For height of waves see Art. 19. tSee "The Economical Top Width of Non-overflow Dams," by the Author, Transactions, Am. Sop. C. E., Vol. LXXX, p. 723. 78 Art. 38] EXAMPLE NO. 1 79 is no economy in battering either face unless necessary to conform to such rules. The determination of the shape of the section is simply a practical application of the general equations for design previously derived. In order to explain the application of such equations to solid non-overflow dams, the following examples are given : 38. Example No. 1. 200 ft. Solid Non-overflow Dam (Fig. 21) . Assumptions : w\ = weight of masonry =145 lb. per cu. ft.; w>2 = weight of water = 62.5 lb. per cu. ft.; p,' = maximum allowed vertical compressive stress at the toe = 18,000 lb. per sq. ft.; p v " = maximum allowed vertical compressive stress at the heel = 25,000 lb. per sq.ft.; p t = maximum allowed inclined compressive stress, in the dam and foundation, at the toe or heel = 42,000 lb. per sq. ft. ; H = maximum height of dam = 200 ft.; c = the area of joints and base subjected to uplift = 50 per cent. The uplift is assumed to vary uniformly from head- water pressure at the heel to zero at the toe (there being, for this example, no tail-water) ; /=the working value of the coefficient of friction of the joints and base = 0.75; L = the width of top = 12 per cent of the height = 24 ft.; o = the distance from the top of the dam to the level of highest water surface = 10 ft.; a' = the distance from the top of the dam to the level of the spillway crest = 20 ft.; P« = ice pressure =40,000 lb. per lin. ft. of dam, assumed to act at the level of the spillway crest, or level 20. * As the foundation is assumed to be rock, the allowed inclina- tion of the down-stream face of the dam may be taken from 4 4X0 75 Eq. (28), or, maximum allowed tan (j>' = -^f= — ^ — = 1.0. There- fore, maximum allowed angle #'=45°. * If a rise of water surface above the spillway crest can be caused only by a freshet in the stream, it is reasonable to assume that, during the period of high water, the ice in the pond will be incapable of exerting pressure on the dam, owing to the larger area of water surface. 80 SOLID NON-OVERFLOW GRAVITY DAMS [Chap. VI It is seen that there are two conditions of loading, for full reser- voir, for both of which the designing rules must be fulfilled. First Condition. — Low water and ice pressure. Second Condition. — High water and no ice pressure. It is necessary, therefore, to determine, at every stage of the design, by inspection or trial, which of these loadings will impose the more severe conditions. It will be found that, for this example, the first condition of loading will govern from the top down to Level 126.0 below which level the second condition must be used. It will be sufficient for the purpose of this example to consider in the explanation only that loading which, for any particular joint, is the determining condition. The width of top and superelevation having been adopted, the first consideration is to ascertain whether there is sufficient ma- sonry above the level of the application of ice pressure to fulfill the conditions of Rule 2. Referring to Art 32, the equation of investigation for Rule 2 is n 2(P) AJbove Level 20: S(P) = Ice pressure = 40,000 lb. ; S (W) = weight of masonry = 24 X 20 X 145 = 69,600 lb. . . 40,000 ._„, tan * = 69lo00 = - 574 ' which is seen to be well within the allowed value of 0.75. The next consideration is to determine the level of the bottom of Zone II, Fig. 14, or at what level the down-stream face must begin to batter. This evidently comes under the third case given in Art. 31, and Eq. (21) will apply. (cw 2 L 2 +6P i -w 1 L 2 )h+w 2 h 3 = w x l?a, (0.5X62.5X24X24+6X40,000-145X24X24)/i-r-62.5/i 3 = 145X24X24X20. This equation must be solved by trial. The final value will be found to be h = 9.27, and the bottom of Zone II will be 9.27 ft. below the water surface, or at Level 20+9.27 = 29.27. Art. 38] EXAMPLE NO. 1 81 It is evident that the resultant, reservoir empty, intersects the center of the joint as long as both faces are vertical. With the resultant, reservoir empty, at the center of the joint and the resultant, reservoir full, at its down-stream extremity, all condi- tions of Rule 1 are met. For Rule 2 we have, from Eq. (22) 2(P) tan0 = ^ ; 2(117 2(P) = Ice pressure 40,000 lb. __ . 9.27X9.27X62.5 ORfir - 1K Water pressure ~ = ^,ooo lb. 42,685 lb. 2(TT) = Masonry 24 X 29.27 X 145 = 101,800 lb. Uplift 9-2-X62.5X24X0.5 = _ 3>4751b . 98,325 lb. which is well within the allowed value of 0.75. The requirement of limiting compressive stresses (Rule 3), is never a governing condition for solid dams less than 100 ft. high, on good rock foundations. As the down-stream face is vertical, Rule 4 is not a governing condition. Therefore, . it is proved that the dam above Level 29.27 is stable, and we are now ready to proceed to the design of lower joints. The height of the dam should now be divided into a num- ber of parts by imaginary horizontal joints, each joint being a dis- tance from the one next above not greater than about 15 per cent of the distance of the former below the top of the dam. Such joints are indicated in Fig. 21, at Levels 29.27, 34.00, 39.0, etc. It now remains to determine the length and location of such joints. The joint at Level 34.0 is evidently in Zone III, and Case 2, of Art- 31, applies. Choosing the point of reference on the vertical 82 SOLID NON-OVERFLOW GRAVITY DAMS [Chap. VI up-stream face of the dam; y, in Eq. (19), is zero, and there results, 21 2(Wx) F +-2(Px)r nq , 3~ Z J « o § o U g g OOiO OtDcO (DO od»o* 3 I * fc, g g S x x x §X5 X 4) O a P-l- ,|x > H CO o 13 S s o O JO V OOO OO-H oo HO oo oo £ I 0, + ft. OQxC OtN ON g g oo 0J3 &£> a a o o 3 a EH <3 I.SW ft. 0) O 'Co 0) O 'CO w 84 SOLID NON-OVERFLOW GRAVITY DAMS [Chap. VI subsequent calculations should, of course, be taken from Table VIII. The two sets of calculations afford a valuable check on one another. To determine the location of the resultant for any predesigned section, we may use Eq. (18). Placing the subscript, E , to repre- sent an investigation for empty reservoir, we have: -2 ,{Wx)e+^{Px)e From Table VIII we get *""' Z(TP), 1,437,510+0 * _ 10/vy z= 119,004 12 - 07 ' The distance from the point of reference to the up-stream extrem- ity of the middle third being 26.17-f-3=j8.72 ft., it is seen that the resultant, reservoir empty, lies well within the middle third. Since the resultant, reservoir full, has been made to intersect at the down-stream extremity of the middle third, all conditions of Rule 1 have been observed. To test for Rule 2, we have, , „ 2(P) P 46,120 _.„._ w= W, = nt2T9 =0 - 407 ' which is well within the allowed value of 0.75. As explained heretofore, the requirement of limiting compres- sive stresses (Rule 3) is not a governing condition for this height of dam. For Rule 4, we have, tan <^=|^ = 0.459, which is well within the allowed value of 1.0. Thus we have designed and tested the dam above Level 34.0. The design is continued in the same manner to the bottom of Zone III. The results of all calculations should be tabulated, as * In every case, that condition of loading which will give the severest requirements for the up-stream portion of the joint should be taken for "empty reservoir." Usually, the assumption that the reservoir is empty will result in the greatest pressures at the heel, and the nearest location of the resultant to the up-stream extremity of the middle third. Aet. 38] EXAMPLE NO. 1 85 a g pa PS H PS o o -J u H 13 O I Ph S o O o o o o o o o o o o 2 o o O O CO o o M N O o o o S o o o d o cm o m t- CO © i-t o °- © i O CO H U50NO lO o CO o w CD M ffl » «1 CO 00 iC J> N CM m t(1 rH in ^ iH m w | ■* CM CM KJ &, ft;, g £ w w W d o '■3 CM rt ■1- 3 W m h O ■*-> b o "3 « a W .1. 2 ' ^ "3 2 io i-i ■* •I- "5 »C d X »n d ■!• .2 io £ revic X14 15 X 5X1 CM cm CO CD X X O X »n ! s From p 0.3X15 50.56 X 8.55X1 CO rH X X as X 3 m CM <& x CO CO d d CM 00 « So , TJ *o q o rf t^ (*: ■d "o -4J 00 00 d a, 2 ™ d d > > "S O m a ai d ft <» j2 -' a b O 3 CD o ft a » t-> d to s CD l-H « ,0. £>& a a o o. m co o s. — ' tj el as .2 "£ > HI H 0. "ft e pressui orizontal head-wa- SS S3 43 W 1 s =3 9 03 g o O o 2 S "C o E>h 86 SOLID NON-OVERFLOW GRAVITY DAMS [Chap. VI in Table XIII, and the location of the resultants, tan 6, and other characteristics plotted for both conditions of loading, as in Fig. 21, to an extent sufficient at least to foresee when it will be necessary to change from one zone to the other, or from one condition of loading to another, in the determination of the shape of the section. Ordinarily, it is not necessary to calculate the compressive stresses occurring in the upper part of the dam. This has been done for Fig. 21, however, for the reader's information as to the variation of such stresses. It is seen, by referring to Fig. 21, that, when Level 87.0 is reached, if the up-stream face is continued vertical, the resultant, reservoir empty, will intersect outside the middle third at the next joint to be designed. Level 87.0, therefore, marks the location of the last joint in Zone III, and the joint at Level 102.0 must be designed in accordance with the principle governing Zone IV. The first case of " Equations of Determination " of Art. 31 will apply. It will now be found convenient to locate the point of reference well outside the base, say 10 ft., as in Fig. 21. Eqs. (19) and (20) will apply to this case. 2l_ 2(Wx) F +2(Px) F V± d 2(F) P ' Uy; l__ 2(Wx) B +2(Px) E m V ^Z~ 2(W) B {M) It is now necessary to adopt tentative values of y and I (Fig. 13), for the derivation of the factors for substitution in the second term of each of these equations. From the general trend of the down-stream face of the dam, and a knowledge that the up-stream face must begin to batter slightly, values of y=9.7 and Z=59.5 will be adopted, as indicated in Fig. 19. Making the proper substitutions from Table IX in Eqs. (19) and (20), there results, 8, 22,460,100 4 y ^S 453,982 y '' l_ 15,675,730+0 y ^3 528,826 y,b '' Solving these two equations, there results: 1 = 59.40, y =9.87. Art. 38] EXAMPLE NO. 1 87 These values of I and y are fairly close to the assumed values, namely 3=59.5 and y=9.7, but closer values, as well as a check on the foregoing calculations, will be obtained by adopting them for new tentative values of I and y for a second set of calculations. I = 69.40 (Second trial) Fig. 19. Substituting valves from Table X in Eqs. (19) and (20), there results, 21 22,452,670 y+ 3 ~ 453,553 =49.50, I 15,679,410+0 y+ 3 = 528,721 = 29 - 68 ' Combining these two equations, and solving for I and y, there results: 1 = 59.46, y = 9.86. These values of I and y agree very closely with the tentative values, namely, £ = 59.4 and ?/ = 9.87, and will be called final. Thus, the joint has been designed in conformity with Rule 1. To test for Rule 2, we have, for the first condition of loading, Tan for the second condition of loading, determined in the same way, will be found to give the greater value, as indicated by Fig. 21. Tan 6, however, for both conditions of loading is seen to be well within the allowed value of 0.75. 88 SOLID NON-OVERFLOW GRAVITY DAMS [Chap. VI 9 O r-T M «3 C. X 00 b- i-H O ^H CM g b- oo *o CO CO ■* CM ID o r " 1 CO »o eq • CO CM -* CM IV O ■* s i-T co >o" »o cm eo" eo" »d" OS CM- i-H t-H rt 1 .-i CM ei fe. fc fe, I I I 0^ w W W + fe ^' >D to n n -* CM t- O CO I (D CO O) CO O a> a> CD O CO i~- os *ra eo o o Cl CM t» ^ CM eo O CN 00 CM w O i-l o o l-[ i-H i-< CO © eo o o o OS O Tft O 00 N N (O ID © m o o o o O i-i CM CO b- CO CO T-l ID O i-H o en oo 00 OS CO eo d d o fa O iH CM CM t- iD -tt< .-< »D ■^ i-H ID W , -tf CM CM Kj & fc, £ fe £ w w h rt .2 "-3 CM $ •[■ 3 ID 01 o 13 CM X uj X h >D tH ID CM + © X CM ID lO »D cl -V oi CM CM CM .2 »D ft CO CD CO ■^ -.■: XX X & s b- »o ■># ft .-H V »H g CO CD CM CO i-t XX »D B> CM 2 °o CO CO X cj x fa O "O 00 i-t i-H d d CN GO m co o o T3 d *o 00 GO 9 m >■ > o o o a 3 o § |3 © 9 s O a> Q. CO a o ,2 jD (a a a o o in m |1 S A '•3 s 3 in U ft 2 pressure rizontal ci tead-water IS > ft 73 ■** ™ » 3 > fa Art. 38] EXAMPLE NO. 1 89 It will be evident, from a little study, that the maximum com- pressive stresses, both inclined and vertical, will occur at the toe of the joint for full reservoir, and at the heel for empty reservoir. For full reservoir we may substitute in Eq. (10a)* of Art. 33, 2(TF) = 2 (lF)r =453,550 from Table X, 1 = 59.46, W = L= 19.82, p„' = 0, as there is no tail-water. The maximum vertical compressive stress for full reservoir is then found to be: , 2X453 ,550/ 3X19.82 X ,„_,,„,„ Q~ [ 2 r-59M~) + ° 15 ' 25 °- 59.46 For empty reservoir, Eq. (11a) * applies, 2(^0=2(1^ = 528,720, m = | Z = 39.64, P*" = 0, as there is, for empty reservoir, no head-water pressure. The maximum vertical compressive stress for empty reservoir is then found to be: 2X528,720 / 3X39.64 \ Vv 59.46 \, 59.46 ^+U-l/,77U. For the maximum inclined compressive stresses in the dam, Eqs. (23a) and (23&) apply. Eqs. (24a) and (246) have no prac- * As this is a special case, with the resultant, reservoir full and empty, at the exact extremity of the middle third, special Eqs. (12a) and (13a) of Art. 22, would be more simple of application. In order to reduce the number of equations to be dealt with, these equations have not been used in the examples, but may be readily applied by the reader, simply by omitting the parts of Eqs. (10a) and (11a) in parenthesis, which, for this special case, are equal to unity. 90 SOLID NON-OVERFLOW GRAVITY DAMS [Chap. VI tical use unless the strength of the foundation is less than that of the masonry. 8 67 At the toe of the dam, tan <*>' =^-=0.577; sec 2 " = 1.00. For full reservoir, tan =0.552; sec 2 =1.305. For empty reservoir, tan =0; sec 2 =1.00. p n is zero at the toe for full reservoir and at the heel for empty reservoir. Using these values in Eq. (23a), the maximum inclined com- pressive stress for full reservoir is found to be: P t '= (15,250X1.330-0) or 0, or (15.250X1.305), P f ' = 20,300, the greatest of these values. In the same way, from Eq. (236), the maximum inclined com- pressive stress for empty reservoir is found to be: P 4 " = (17,770X1-0), orO, or (17,770X1), P" = 17,770, the greatest of these values. All vertical and inclined pressures at this joint are seen to be well within the allowed values, and, therefore, Rule 3 is followed. To test for Rule 4, we have only to observe that the value of tan ', as just derived, is well within the allowed value of 1.0. Thus we have designed and tested the section above level 102.0. The design is continued in the same manner to the bottom of Zone IV. After the joint at Level 120.0 has been reached, it will be seen, by reference to Fig. 21, that the lines of resultants for the two conditions of loading will cross before the next joint is reached. It will be obvious, therefore, that the second condition of loading will govern from here on; namely, water surface at Level 10 and no ice pressure. When the joint at Level 140.00 has been designed, the diagram on Fig. 21 will indicate that, unless proper precaution is taken, the vertical pressures at the toe for full reservoir will exceed the allowed limit of 18,000. Level 140.0, therefore, marks the elevation of the last joint in Zone IV, and the joint at Level 160.0 must be designed Art. 38] EXAMPLE NO. 1 91 in accordance with the principles governing Zone V, namely, Rule 3 for full reservoir and Rule 1 for empty reservoir. Eqs. (25) and (20), of Art. 33, apply to this case. 6 V— 2(If% l = 2,(W) F y-{Z,(Wx) F +2(Px) F }. . (25) l_ V(Wx) E +-2(Px) E y+ 3 ?j(W)e (20) As before, it is necessary to adopt tentative values of I and y. These will be assumed at 107.0 and 6.0, respectively. Zo= 87.01 I .= 107.0 (First trial) I =100.6 (Second trial) Fig. 20. Making the proper substitutions from Table XI in Eq. (25), there results, 0zJM22p_9?if9i=991,900 2/ -74,513,440, which reduces to Z 2 +110.21Z=-330.63?/+24,838. . . (25a) Making corresponding substitutions in Eq. (20), there results: , I 49,580,000+0 An C7 /nn . y+ 3 = 1,212,780 40 - 87 (20a) Combining Eqs. (20a) and (25a), and solving for I and y, there Vesults, 1= 106.5, y=5.37, 92 SOLID NON-OVERFLOW GRAVITY DAMS [Chap. VI « I d O J g O I P § O o a a o s 33,985,000 15,100 12,890,000 2,690,000 49,580,000 145,740 81,650 6,050 O O ■«* O ■* o_ eo" o" W iO 00 ■* 0» O •* rH I O co" CD CO oi" CO o o o o IO I-H io" eo o co" IO ■<* ei fc. , — \ . — * g + h3 os co -i *° N HO N* 00 CM O M CM -h" 1 o o OS Ol Oi o o »o eo" o 65 It, g g h, CD O -*5 O at h a & •3 a •? jf 1 + o n a „ 6 « "*" + o .2 j. X o .1. 5^ '5 ift • • " "in^^n . - 1 .3 oXX x t- "! t~ oo 3 * * K CM •1- iO CM CD X o »o X o »o fH a *> a o © ja ■* -^ _ ^ *o II | £ g & § S a 5-s as g rt -o of « * & -as o Bo .3 -g S CO tt -4J . ._ 2S £ £ a 3 rv en E ° 1 n d> , ts 11 o -a ■BJ K S3 00 o ■* o •* o ■* £ io d o * O 1QOQ0 »o o »rt o IO OONffiH i-i 1*1 r-l t> o N U5 M o as co »o "*- ^ ^ CO CN ■* •^ CO s co cn" n Oi as o" Os" 10 •* CO iH ■* I CO CO t> fcl fc. fc. fc. I I Oh, £ v_^ w w w^ + ^ CO CO Tt* CO CO N o I © as NOW ■* o CO o (OHO CO CO © d r-l "5 O -* w w O O O © o o o o o o o o ONOCD 'O CO CI CI t- o b- o CN C3 «3 0_ OS l> CN CO CN CO •* o (H rJ « N IO iH (OSH tC Os" l> CO o CO "3 tN rH iH rH TJ4 «tfl OS o Pn as cm CN CM CN -J 1 OS ( fc. v^ w w w' as to d ft CN o .|. "-p o US (D 3 P CN 73 « + H "*! g tN O .2 -I. d X CN + O -P es ® "1" U5 W ■**l -P lO ' ^^ O id as O O ■P ■* ■<* a as cd <-• r- < o C Si o u +3 o s 5> >■ as o ft 3 g £ o S ft g d m 5 ® £ 09 8 » S3 fl O hi O ft OS C r- ll 0) H 1- oi as O P ft d J, -p 1— 1 3 fl o o ■p O 13 as tn 03 o3 "■p ►? as ^ > "ft "B5 SS p w cs S ad 9 as •3 o S o O cs "C o [>FS4 94 SOLID NON-OVERFLOW GRAVITY DAMS [Chap. VI These values of I and y are somewhat different from the values, 1= 107.0 and j/ = 6.0 adopted tentatively. ' Closer values, as well as a check on the calculations, will be obtained by using them for new tentative values for a second set of calculations. Making the proper substitutions from Table XII in Eq. (25), there results, 0-18,000 ;2 _ 997 470 ; = QQ? _ 74^430, 6 o which reduces to Z 2 +110.83Z=-332.492/+24,875. . . (25a') Making corresponding substitutions in Eq. (20), there results, "f^SSr— ™ Combining Eqs. (20a') and (25a'), and solving for I and y, there results, 1 = 106.4, y = 5.30; which is sufficiently close to the tentative values of Z = 106.5 and y = 5.37 and will be considered as final. The necessary investigations to determine whether or not the other designing rules are complied with are, in every respect, similar to those described for the upper joints, and need not be repeated. The results are indicated in Fig. 21. The joint at Level 180 is determined in the same manner, and, after the necessary calculations have been completed and plotted in Fig. 21, it will be noticed that, unless proper precautions are taken, the maximum allowed vertical pressures at the heel, for empty reservoir, will be exceeded. The joint at Level 200, therefore, will he within Zone VI, and must be designed by the application of Eqs. (26) and (25), of Art. 33, in the same manner as Eqs. (25) and (20) were applied in the determination of the length and location of the joint at Level 160. The angles in the faces of the designed section may be smoothed up, if desired, as indicated by the broken line for the angle at Level 29.27 . In low dams the down-stream face is usually made straight, as indicated in Fig. 35, in order to save the expense of curved forms, for the concrete. m K Zor el e a £ w BighVi Level 1 Zon ;II j 55/ k 3- Si s of 3 and Ice 29.27 , o^/ 00/ ^r-/ 34 39 ' Zon sill EXAMPLE N0.1 OF ART. 38 200-FOOT SOLID NON-OVERFLOW DAM ST'T?) © / 5cr- / s / ^»/ ■at S- BS 3 g n 3 Pi ?7p ° U 50 57 .» / C » / /T 8 / & Ikl n if/ 5 «/ /&> / «? / Asr ft //co -5 // -t -a fr — ^ fe-g — 65 75 « / ! / */ f O t- / /*V O co 1 ^ //I 1 ft to / / ', or maximum inclined pressures. Although this is usually the case for solid dams not exceeding 200 ft. in height, it will be well to indicate here the proper method of procedure, if, at any stage of the design, it is found that such values are determining conditions. Referring to Fig. 21, it is seen that tan 6 reaches a maximum value of 0.712 at Level 180.0. If it is considered that this value is too high, it will be necessary to provide an additional vertical, downward force, namely; an increased S(W), as it will be seen from Eq. (22) that tan 8 decreases as 2 (IF) increases. This may be accomplished simply by adding more masonry, but, usually, it will be found best to increase the batter of the up-stream face, as by doing this, a larger vertical component of head-water pres- sure will be included. A reduction in the value of tan ' may be made in a number of ways. If the section is redesigned with a greater superelevation or width of top, the resulting value of tan ' will be less; but, in very high dams, the change will be relatively small. The desired reduction may also be obtained by arbitrarily thickening the mid- dle and upper part of the section and redesigning the lower part. Such increase in thickness can be obtained by increasing the batter of the up- or down-stream faces. It is probable that the best remedial expedient, for most cases, is to increase the batter of the up-stream face from about mid-height to the base, and to redesign the lower part of the dam with the new up-stream face as a fixed condition in the calculations. The inclined pressures, for a dam of this type, will always be greater at the toe than at the heel, notwithstanding the fact that the opposite is true of the allowed vertical pressures. It is seen from Eq. (23a) and (24a), that the maximum inclined stress in the masonry at the toe of the dam is proportional to the vertical stress, p,', and to sec ' or sec 8, whichever is the greater; and that the maximum inclined stress in the foundation is proportional to p/ and sec 9. It will be found that a reduction in the inclined stresses in the dam can be obtained to best advantage by reducing sec ', if 4>' is greater than 8, or by reducing p v ', if 8 is greater than (j>'. To reduce the inclined stress in the foundation, a reduction in p/ will usually prove best. 96 SOLID NON-OVERFLOW GRAVITY DAMS [Chap. VI TABLE XIII Results of Calculations fob Example No. 1 of Art. 38 Vertical Forces. Level of Joint. Weight of Masonry. Vertical Component of Head-water Pressure. Uplift. Pressure. Summation of Vertical Forces, Reservoir Full, Reservoir Empty. Loading 1. Loading 2. Loading 1. .oading 2. Loading 1. Loading 2. 29.27 101,800 119,000 138,700 160,000 187,500 222,300 265,900 326,500 409,000 528,700 700,700 931,300 1,212,000 1,551,000 1,957,000 3,475 5,725 8,400 11,380 15,400 20,700 27,700 38,050 53,050 76,100 112,800 163,200 232,500 7,230 12,820 20,500 33,830 60,950 123,600 176,800 249,800 338,300 453,000 98,320 113,300 130,300 148,600 172,100 201,600 238,200 288,500 356,000 454,000 596,000 784,800 1,014,000 94,570 31 39 125,900 44 50 167,000 57 65 232,100 75 87 348,100 102 932 8,096 16,770 34,540 120 140 160 180 7,700 16,780 35,320 56,320 584,800 771,300 997,500 1,296,000 200 103,300 1,607,000 Horizontal Forces. Level of Joint. Horizontal Ice Pressure. Horizontal Component of Head-water Pressure. Summation of Horizontal Forces, Reservoir Full, 2(P)f. Loading 1. Loading 1. Loading 2. Loading 1. Loading 2. 29.27 34 40,000 40,000 40,000 40,000 40,000 40,000 40,000 40,000 40,000 40,000 40,000 40,000 40,000 40,000 40,000 2,685 6,120 11,280 18,000 28,120 42,750 63,300 94,600 140,200 210,100 312,500 450,000 612,000 11,600 42,685 46,120 51,280 58,000 68,120 82,750 103,300 134,600 180,200 250,100 352,500 490,000 652,000 11,600 39 44 26,280 26,280 50 57 50,000 50,000 65 75 94,550 94,550 87 102 185,300 185,300 120 140 160 180 378,000 528,100 703,000 903.000 378,000 528,100 703,000 903,000 200 1.128.000 Art. 38] EXAMPLE NO. 1 TABLE XIII— Continued 97 Momex rs of Vertical Forces Level of Joint. Moment of Weight of Masonry Moment of Vertical Com- ponent of Head-water Pressure. Moment of Uplift Pressure. Reservoir Empty. Loading 1. Loading 2. Loading 1. Loading 2. 29 27 1,222,000 1,438,000 1,706,000 2,018,000 2,451,000 3,050,000 3,869,000 5,137,000 7,098,000 15,680,000 22,930,000 33,980,000 49,410,000 70,680,000 99,570,000 27,800 49,900 79,200 115,000 168,400 247,000 363,000 561,000 894,000 2,257,000 3,688,000 5,965,000 9,480,000 57,850 34 39 120,900 44 50 224,300 57 65 443,500 75 87 1,029,000 102 9,260 75,490 145,700 258,700 120 140 160 180 72,040 145,700 264,600 352,400 4,040,000 6,465,000 10,200,000 15,420,000 200 393,400 22,490,000 Moments of Horizontal Forces. Total Moments. Level of Joint. Arm and Moment of lee Pressure. Arm and Moment of Horizontal Component of Head-water Pressure. Summation of Mo- ments of all Forces, Full Reservoir. 2(Wx)f+2(Px)f Loading 1. Loading 1. Loading 2. Loading 1. Loading 2. 29.27 34 39 9.27 14.00 19.00 24.00 30.00 37.00 45.00 55.00 67.00 82.00 100.00 120.00 140.00 370,800 560,000 760,000 960,000 1,200,000 1,480,000 1,800,000 2,200,000 2,680,000 3,280,000 3.09 4.67 6.33 8.00 10.00 12.33 15.00 18.33 22.33 27.33 33.33 40.00 46.67 8,300 28,600 71,400 144,000 281,200 527,500 950,000 1,734,000 3,133,000 5,741,000 10,420,000 18,000,000 28,600,000 6.42 9.67 13.33 18.33 25.67 36.67 43.33 50.00 56.67 63.33 74,450 1,573,000 1,976,000 2,458,000 3,007,000 3,764,000 4,810,000 6,256,000 8,510,000 12,020,000 22,450,000 33,730,000 50,970,000 74,390,000 1,238,000 254,200 1,839,000 50 57 65 75 666,500 1,734,000 2,894,000 5,159,000 87 102 4,755,000 10,820,000 120 140 160 180 4,000,0 4,800,0 5,600,0 00 00 00 13,860,000 22,890,000 35,150,000 51,220,000 71.450.000 32,820,000 50,560,000 74,630,000 106,800,000 200 148,900,000 . • 1 EXAMPLE NO 1 2 OF ART. 39 u g -o High "Water Le 102 FOOT SOLID NON-OVEKPLOW CAM '—^=-=^r^r^=r Zone I o * i C O a 3 Low Water Zone II ■ / and Ice LereTjjj 19.30 It g cS / ^ / / S5 / c* j /OS~f CS f "**7 23 27 Zone III O / CO / ci/ oaA — s / 3 g /#/ 9> °°fe 31 38 C u c g/„__ £ s/^Z. J'.sM ___ __ / 1 5 / Sf <£ Afa | to / / ft sfs SO \ . CO / OS / cif/ rail /!5 3 J a/ CO / cs / if CO yff 4/ 1 J Zone IV Tail-water mf V s IP -==^====£^^3^ ~^==^r^^~f 1 c 1 ~ I o / « / \ " °f M 1 31 ° Al si / 1 / 3 i-/ . U. 8 j- Fmj o.o Tangent 9 0.4 as 0.3 0.1 Loading 1-Wat*r Surfaoe and lee Pressure at Level ^5 I Tail-water At Level 78 Loading 3 -Water Surfaoe at Level li 140 100 80 60 Cubic Yards per Linear Foot 20 To face page 99 Akt. 39] EXAMPLE NO. 2 99 39. Example No. 2. 102-ft. Solid Non-Overflow Dam (Fig. 22). In this example the assumptions are the same as in Example No. 1, Art. 33, with the following exceptions: H = maximum height of dam = 102 ft.; L = width of top = 14 per cent of height = 14 ft. ; a = the distance from the top of the dam to the level of high- water surface = 5 ft. ; a' = the distance from the top of the dam to the level of the spillway crest = 15 ft. ; a" = the distance from the top of the dam to tail-water surface = 72 ft.; c = the area of joints and base subjected to uplift = 50 per cent. The uplift is assumed to vary uniformly from head- water pressure at the heel to tail-water pressure at the toe. Except that the negative force and moment, of tail-water is present, the method of design to be followed is exactly the same as that indicated for the upper part of Example No. 1, and will not be repeated. The design is governed by low water, with ice pressure down to Level 65.0, and high water with no ice below that elevation. The results of computations are indicated in Table XIV, and are plotted in Fig. 22. It will be noticed that the dam does not extend below Zone IV, the pressures at the base with resultants at the middle third extremities not exceeding the allowed values, as indicated by the calculations which follow. It will be evident, from a little study, that the maximum com- pressive stresses, both inclined and vertical, will occur at the toe of the dam for full reservoir, and at the heel for empty reservoir. For full reservoir we may substitute in Eq. (10a) *, of Art. 33, the following known values from Table XIV. 2(F) = Z(T7 F ) =395,700, 1=68 A, 68.4 _ w = -g— = 22.8, p u '= 0.5X30X62.5 =937.5. * See footnote, p. 89. 100 SOLID NON-OVERFLOW GRAVITY DAMS [Chap. VI TABLE XIV Results of Calculations fob Example No. 2 of Akt. 39 Vertical Forces. Level of Joint. Weight of Masonry X(W)b. Vertical Com- ponent of Head- water Pressure. Vertical Compo- nent of Tail- water Pressure. Uplift Pressure. Summation of Vertical Forces, ReBervoir Full 2(W>. Empty Reservoir. Load- ing 1. Load- ing 2. Load- ing 2. Load- ing 1. Load- ing 2. Load- ing 1. Load- ing 2. 19 30 39,200 47,310 57,350 68,560 84,110 105,100 137,000 183,700 252,400 353,400 497,300 4,000 9,180 12,850 4,640 21,200 940 2,029 3,442 5,075 7,435 10,750 16,150 24,570 39,300 3,130 4,562 6,310 8,240 10,970 14,740 20,740 30,000 46,180 82,570 135,700 38,260 45,280 53,900 63,480 76,680 94,310 120,800 159,100 216,400 36,070 23 42,750 27 51,040 60,320 36 73,140 42 90,320 50 116,200 60 153,700 72 86 3,343 210,200 284,700 102 395,700 Horizontal Forces. Level of Joint. Horizontal Ice Pressure. Horizontal Component of Head-water Pressure. Horizontal Component of Tail- water Pressure. Summation of Horizon- tal Forces, Reservoir Full, 2(P)f. Loading 1. Loading 1. Loading 2. Loading 2. Loading 1. Loading 2. 19 30 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 578 2,000 4,500 8,000 13,780 22,780 38,260 63,300 101,400 6,380 10,140 15,130 21,120 30,030 42,750 63,300 94,500 140,400 205,000 294,000 1 20,578 22,000 24,500 28,000 33,780 42,780 58,260 83,300 21,400 6,380 23 10,140 15,130 21,120 27 31 36 30,030 42 42,750 63,300 94,500 104,400 198,900 265,900 50 60 72 86 6,130 28, 1^0 102 Abt. 39] EXAMPLE NO. 2 TABLE XIV— Continued 101 Moments of Vertical Forces. Level of Joint. Moment of Weight of Masonry, ?(Wx)e. Moment of Vertical Component of Head- water Pressure. Moment of Vertical Component of Tail- water Pressure. Moment of Uplift. Pressure. Empty Reservoir. Loading 1. Loading 2. Loading 2. Loading 1. Loading 2. 19.30 274,400 335,800 422,700 531,100 697,400 949,900 1,390,000 2,144,000 5,992,0~0 4,395 10,980 21,070 34,290 56,100 91,400 158,900 286,000 929,000 14,620 23 24,700 27 38,600 31 55,650 36 82,850 42 115,400 50 203,900 60 349,200 31,630 37,840 S2.420 109,600 1,091,000 282,000 1,449,000 ',415,000 102 14,960,000 4,817,000 Moments of Horizontal Forces. Arm and Moment of Arm and Moment Arm and Moment of Horizontal of Horizontal Com- Joint. of Ice Pressure. Component of Head-water Pressure. ponent of Tail- water Pressure. Loading 1. Loading 1. Loading 2. Loading 2. 19.30 4.30 86,000 1.43 ! 826 4.77 30,400 23 8.00 160,000 2.67 1 5,340 6.00 60,840 27 12.00 240,000 4.00 18,000 7.33 110,900 31 16.00 320,000 5.33 42,640 S.67 183,100 36 21.00 420,000 7.00 : 96,460 10.33 310,300 42 27.00 540,000 9.00 I 204,900 12.33 527,500 50 35.00 700,000 11.67 446,500 | 15.00 949,500 60 45.00 900,000 15.00 950,000 18.33 1,733,000 72 57.00 1,140,000 19.00 1,926,000 22.33 3,135,000 86 27.00 5,535,000 4.67 28,600 102 32.33 9,510,000 10.00 281,200 102 SOLID NON-OVERFLOW GRAVITY DAMS [Chap. VI TABLE XIV— Continued Total Moments. Total Moments. Level of Summation of Moments of all Forces, Keservoir Full. Level of Summation of Moments of all Forces, Reservoir Full. Joint. 2(.Wx)f+'Z(.Px)f. Joint. X{Wx)f+^{Px)f. Loading 1. Loading 2. Loading 1. Loading 2. 19.30 356,800 290,200 50 2,377,000 2,135,000 23 490,200 371,900 60 3,708,000 3,528,000 27 659,600 495,000 72 8,160,000 8,073,000 859,400 1,158,000 1,603,000 658,500 924,800 1,352,000 86 12,830,000 102 20,930,000 42 Location of Resultants. Length of Distance of Heel from Tangent 4>. Level of Distance from Point of Reference. Tangent d. Joint. Base. Point of Reference. Empty Load- Load- 4>" *' Load- Load- Reservoir. ing 1. ing 2. Heel. Toe. ing 1. ing 2. 19.30 7.00 9.32 8.05 14.00 0.538 0.176 23 7.14 10.82 8.70 16.23 0.603 0.486 0.237 27 7.37 12.24 9.69 18.36 0.532 0.455 0.296 31 7.76 13.53 10.92 20.29 0.482 0.442 0.351 36 8.29 15.11 12.64 22.66 0.475 0.441 0.410 42 9.04 17.00 14.97 25.50 0.473 0.457 0.473 50 10.14 19.67 18.37 29.50 0.500 0.482 0.544 60 11.67 23.29 22.94 34.93 0.543 0.523 0.614 72 23.72 37.70 38.35 43.89 9.09 0.076 0.671 0.561 0.667 86 26.53 45.10 55.71 7.96 0.081 0.763 0.699 102 30.10 52.90 68.40 7.30 0.041 0.752 0.672 Maximum Pressures, in Thousands of i Pounds per Square Foot. Level of Joint. At Heel. At Toe. Empty Reservoir. Loading 2. Vertical. Inclined. Vertical. Inclined. 102 14.5 14.6 12.5 18.5 Art. 39] EXAMPLE NO. 2 103 The maximum vertical compressive stresses for full reservoir is found to be: For empty reservoir, Eq. (11a) * applies, S(T7) = S(Tf) B = 497,300, w = 2 ->' =0.566; sec 2 '=——— = 0.75 lo At the heel of the dam, , .„ 0.66 tan<^" = ^g- = For full reservoir, tan = 0.672; sec 2 = 1.455. For empty reservoir, tan = 0; sec 2 0=1.00. Using these values in Eq. (23a), the maximum inclined com- pressive stress for full reservoir is found to be: p/ = (12,500 X 1.568 - 30 X 62.5 X 0.566) or 30X62.5 or 12,500X1.455, p/ = 18,540 or 1,875 or 18,200, pt = 18,540, the greatest of these values. * See foot-note, p. 89. 104 SOLID NON-OVERFLOW GRAVITY DAMS [Chap. VI In the same way, from Eq. (23b), the maximum inclined com- pressive stress for empty reservoir is found to be: Vt" = (14,550-0) or or 14,550X1.0016, p/' = 14,550, the greatest of these values. 40. Comparison of Non-overflow Dams. A comparison of high, solid, non-overflow dams is given in Fig. 23. With the exception of the Olive Bridge final section, the designs of these dams were all made in accordance with the same general theory, the differences in areas and shapes being affected solely by the assump- tions, as indicated in Table XV. The theoretical section, of the Olive Bridge Dam was arbi- trarily increased to its final section, on account of the importance of the structure. TABLE XV Comparison of Solid, Non-overflow Dams, Showing Assumptions Used in Design (See. Fig. 23) Dam. Unit Wt. of Mason- ry, in Lbs. per Cu.ft. Percentage of Area of Base Sub- jected to Uplift. Total Ice Pressure, in Lbs. per Lin.ft. Maximum Allowed Verti- cal Pressures, in Lbs. per Sq.ft. Toe. Heel. Olive Bridge, theoretical section Olive Bridge, final section New Croton Elephant Butte Wegmann's Practical Profile No. 3. . . Morrison and Brodie's example of design The author's Example No. 1 145.8 145. 8 156.2 140.0 145.8 146.0 145.0 661 33J 50 47,000 47,000 40,000 40,000 12,200 33,400 22,000 16,800 28,000 18,000 40,000 23,000 30,800 28,000 20,600 36,000 25,000 The maximum vertical pressures indicated for the Olive Bridge final section probably do not exist, as the maximum section is in a narrow gorge confined on both sides by good rock, between which the dam is wedged, without the possibility of movement. In the upper part of the New Croton Dam the vertical pres- sures were limited to 16,400 and 20,600 lb. at the toe and heel, respectively, and, in the lower part, 33,400 and 30,800. This difference in allowed pressures, in the top and bottom of this dam, has been severely criticised. COMPARISON OF SEC HIGH SOLID NON-OVER SUPERIMPOSED WITH WATER SURFAC '?, -^ # / / / J/ ■/ V> t / "/ /_y To face page 104 CHAPTER VII THE DESIGN OF SOLID SPILLWAY GRAVITY DAMS 41. General Considerations. The general method of design of solid spillway dams differs in no way from that previously described for solid non-overflow dams, except at the crest which, as mentioned in Art. 17, should be proportioned to fit the lower nappe of the sheet of water spilling over the dam during maximum flood. The application of the general equations of design has been described in Example 1, Art. 38, and will not be repeated for the following examples. 42. The Shape of the Crest. It was pointed out in Art. 17 that, if the sheet of water spilling over the crest leaves the face of the masonry, there is danger of the existence of a considerable indeterminate overturning force due to the formation of a partial vacuum under the sheet. This, of course, will not occur if the continuous length of crest is unusually short and free access to the atmosphere is provided at the ends. At any rate, it is very desirable to avoid impact and vibration, and the best practice dictates that the crest and down-stream face should always com- pletely fill the space under the lower nappe of the sheet corre- sponding to the maximum flood to be expected. Experiments have been made to determine the shape of the sheet of water flowing over aerated sharp-crested weirs. The general form of the sheet is indicated in Fig. 24. If the area below the lower nappe is filled with masonry, the shape of the sheet and the discharge will not be changed appreciably. The resulting section will not only fit the sheet of water, but will provide the maximum possible discharge, as the water will pass the masonry crest with no disturbance or unnecessary contraction. This shape of crest has become standard in modern designs and, with few exceptions, differs for individual cases only in the methods used in determining the shape of the sheet. For refer- ence, therefore, it will be designated the " standard dam crest." 105 106 SOLID SPILLWAY GRAVITY DAMS [Chap. VII A comparison of the shape of the crests of several modern dams is made in Fig. 38. Bazin's experiments for the shape of the sheet of water flowing over a sharp-crested weir have been translated by Messrs. Arthur Marichal and John C. Trautwine, Jr.* The experiments for weirs with vertical water faces and those with water faces inclined 45° apply directly to the determination of the shape of the crest of the ordinary types of solid and hollow dams, respectively. The curves indicated in Fig. 25 and 26 to about x = +0.12 for the upper nappe and x = +0.65 for the lower nappe are plotted directly from the experiments. The extension of the experimental data was made by the author in the following manner: The average velocity in any normal section of the sheet of water was found by Bazin to he very close to one-third of the dis- tance from the lower nappe, as at point 4, Fig. 24. The curve, 2-3-4, was drawn through three scaled points of average or result- ant velocities. The curve has the form of a jet spouting with an initial horizontal velocity, v h . Its equation, therefore, is that of a parabola, and was derived as follows: In the time, t, a particle on the curve will have fallen, from rest, a vertical distance from point 1, equal to therefore, x'-*V- X ~ 2' e-K In this same time, t, the particle will have moved horizontally, from point 1, a distance of y' = vj,t; therefore, -(9" Equating these two values of t 2 , there results: „ 2v h 2 , V ~T ' ••••••• (29a) * Proceedings Engineers' Club of Philadelphia, Apr., 1893. Art. 42] THE SHAPE OF THE CREST 107 which is the equation of the curve, 1-2-3-4, referred to the origin at point 1. Water Surface ~~~~ u s^\ "if Ax Axis of is of y' - v > 1 rf- y « , V 61 '4 i v h \ \ t/ 555" >j Great of Weir «' \ 6 II « o .a i Fig. 24. Referred to point 5, the equation is, (29) where x and y are co-ordinates referred to the origin, 5; and a, b and Vn, unknown constants. Substituting in Eq. (29), measured values of y and x from each of the known points, 2, 3, and 4, of the experimental curve, there results three equations with three unknowns, a, b, and v h . Values of a, b, and %, found in this manner for the two types of crests, and the result of the substitution of vj, in Eq. (29a), are given in Table XVIII (first four items). From Eq. (29a), the curve, 1-2-3—4, may be extended indefinitely. It now remains to deter- mine the thickness of the jet at various points on the curve. The vertical velocity, v„ at point 4 is v<, = V2gx'. The horizontal velocity, v„, being known, the resultant velocity, v T , is Br = VwHV = Vv h 2 +2gx'. Measuring the thickness, 6-7, of the sheet from the plot of the experiments, the discharge, q, per linear foot of crest may be cal- culated from the following equation: q=AVr = AV Vh 2 +2gx', (30) 108 SOLID SPILLWAY GRAVITY DAMS [Chap. VII Fig. 25. Art. i'2] THE SHAPE OF THE CREST 109 SHAPE OF CREST AND DOWN-STREAM FACE OF DAM Water inrface Overflowing Water. Dp-Stream Face Sloping 45* 1 „ © 5 4 5 4 I S.5 3 2 1 • / i 5 2.0 1.5 _,' 1.0 0.5 Or..-n ■ i Co-o ■dinates 1 ^Cresfc of Dain and 1 Vxisof y ^ ! //< i I ! * / | 1 x | ! // ! ' 1 | 1 /'// | Masonry Line ^ / // | o 1 / / l Theoretical Sheet 'C - '* / / ■ 7N < 1 i // ' // / ! '/ / .7 ' c ON 'J i / i / t TABLl O-ORCIH : FOOT H i AV11 IVTES n EADON i i I t / '■ V X Masonry line Theoretical Sheet -J •' Upper Nappe tower Nappe 0.0 0.1 0^ 0.3 0.1 0.6 0.8 L0 t2 Lt LT io io 3.0 io to 0.043 0.010 0.000 0.005 0.02S 0.090 0.1S9 0.3a 0.480 0.665 0.993 tS77 SOI S.0S 4.08 5J4 U8 -0.TS1 -0.756 -0.724 4)1639 -H.HS 41552 -0.435 41393 -0.120 0,015 0.138 0.860 1.71 8.76 too 5.43 7.07 0.043 0.010 0.000 0.005 0.033 0.090 0.193 0.333 0.500 0.700 t05 L17 2.34 3.39 4.61 6.04 7.01 ft 1 1 II ft / / / 1 II i f// / / i 1 / / 1 1 1 / / / / 1 1 I I I I I 5.0 Itultinly the quantitie table hy- the head' on ti i in the esua Fig. 26. 110 SOLID SPILLWAY GRAVITY DAMS [Chap. VII where A is the thickness, 6-7. Values of q for the two types of crests, calculated in this manner, are given in Table XVIII. It is surprising to note that the calculated discharge agrees almost exactly with the discharge found by independent experiments for the same types of weirs. The thickness of the sheet corresponding to any value of x' may now be derived from Eq. (30), the values of v h and q being taken from Table XVIII. The area of the sheet lies one-third below and two-thirds above the curve. In this manner the path of the falling sheet was determined and plotted in Figs. 25 and 26. The paths are only approximate, as they are extended from experimental points relatively close to the top of the dam, and, moreover, they may be somewhat affected by irregularities in the masonry crest. Consequently, it is advis- able to provide a margin of safety by extending the masonry line well into the theoretical sheet as indicated. This line is usually formed by a series of arcs of circles, as indicated in Fig. 36. The experiments were made with a negligible velocity of ap- proach. The effect of velocity of approach on the shape of the sheet not being known, the best that can be done is to let h, Fig. 24, include the head corresponding to the velocity of approach and to consider that, when the velocity of approach is large, the results so obtained are correspondingly uncertain. TABLE XVIII Factors in the Determination op the Shape of the Crest, (h =1.0 ft.) Factor. Water Face Vertical. Water Face Inclined 45°. a b Vh 9 +0.261 -0.063 6.63 2.732x' 3.90 +0.249 +0.007 6.52 2.640a;' 3^94 43. Discharge Capacity.* Francis has determined that the "A very complete discussion of the theory and experiments relating to the discharge of water over dams may be found in U. S. Water-supply and Irrigation Paper No. 200,. by R. E. Horton, and should be read by all those interested in the subject. Akt. 43] DISCHARGE CAPACITY 111 discharge of water over dams may be expressed by the following equation: Q = ql„ = CL{(h c +hf 2 -h e 3/2 }, .... (31) where, Q=the total discharge, in cubic feet per second; g = the discharge per linear foot of effective crest; Z»=the net or effective length of crest, i.e., the total length of crest corrected for end contractions due to piers and sharp-cornered abutments; 7k = the actual or measured head on the crest, taken at a point sufficiently remote from the dam to avoid the surface curve; ft«=the head corresponding to the velocity of approach; and C = a coefficient which depends on the shape of the crest and the head on the crest. In determining the head on the crest corresponding to the maxi- mum flood to be expected, the following approximate equation may be used; provided the head on the crest is not greater than the depth of the channel of approach, the error for that head being much smaller than the error to be expected in determining the maximum flood: ' Q = ql„ = Cl n (h c +Kf* (31a) Francis' equation for the necessary correction due to complete end contractions is, ln=l t — 0.lnh e , where, It = the total or gross length of crest between abutment and piers (Fig. 27), and n = the number of complete end contractions. If the crest is obstructed by piers having considerable widths and sharp corners, as indicated in Fig. 27, n represents the number of corners which serve to deflect the water, there being six com- plete contractions in this instance. Usually, however, the piers are relatively thin and are provided with sharp up-stream ends, as indicated in Fig. 28. In such cases the contractions for the piers are not complete, and Francis' equation would give values of h, 112 SOLID SPILLWAY GRAVITY DAMS [Chap. VII too small. The number of experiments has not been sufficient to determine closely the effect of piers on the effective length of crest. Unless the piers are unusually thin, relative to the head on the crest, or very considerably pointed up-stream, they may be con- A N ! 1 1 ili\v Fig. 27. — Complete Contractions Due to Abutments and Large Piers. sidered to offer a partial contraction, probably amounting to not more than 0.04 h c for each contraction, in case the piers are pointed as indicated in Fig. 29, and varying between this limit and 0.1 h c for thick, blunt piers, depending on the degree of sharpness and the relative thickness. Fig. 28. — Partial Contractions Due to Sharp Piers. Francis' contraction equation may then be written, t = &— A<,(C n„-r-C»n» . . . C n n„), . . . (32) where C„, d etc., represent the contraction coefficients applicable to the several different contractions which may be expected, and n a , n t etc., the number of contractions having contraction coefficients, C a , C b , etc., respectively. Art. 43] DISCHARGE CAPACITY 113 Thus, if the piers in Fig. 28 are shaped as indicated in Fig. 29, the effective length of crest would be, ln = l,-h c (0.lX2+0MX-i) = l t -0Mh c . Obviously, the effective length of crest between any two piers of whatever shape or, in the absence of piers, between the abut- ments, can never be less than 0.788 of the clear length of crest, as this reduction corresponds to the contraction of the width of a jet through an orifice in a thin plate. The head corresponding to the mean velocitv, *>i, in the chan- vx 2 nel of approach is -^ . The velocity in the channel of approach, however, is not uni- form, the filaments above the elevation of the crest of the dam sometimes having a velocity con- siderably greater than the mean, depending on the depth and width of the channel, and its surface conditions. The energy of the filaments above the elevation of the crest have a proportionately greater effect in increasing the discharge. The true value, h , may be represented by Fig. 29. — Typical Sharp Nose Pier. hr==Cf Yg' where C depends on the condition of the channel of approach described above. Values of 0„ usually adopted in experiment work vary from 1.0 to 1.5.. However, in view of the fact that the velocity of approach above the elevation of the crest may be materially affected by wind, ice, and other conditions, it would seem advisable to use a value of C = 1.0, particularly as such assumption is on the safe side, when determining the capacity of the dam to pass the max- imum flood. 114 SOLID SPILLWAY GRAVITY DAMS [Chap. VII Values of C, for Eqs. (31) and (31a), applicable to standard dam crests, may be taken from Fig. 30 which, in the absence of authentic experiments on this type of crest, was derived by the author from theoretical considerations and a comparison of a number of experiments on similar shapes of crests. The values of the coefficients, for the head ratio of unity, are probably within 3 per cent of the truth; but, at other points on 4.0 3.9 8.8 &3.7 u ■a .§3.6 8 3.3 6.8 '3.1 #V J f ■/. rf &/ */ nV ' f . 55 65 76 90 106 140,800 198,200 273,000 386,800 542,700 28,200 40,100 56,400 81,200 115,500 112,600 158,100 216,600 305,600 427,200 75,080 109,400 154,400 222,400 315,900 Moments of Vertical Forces. Moment op Horizontal Forces. Total Moments. Level. Moment of Weight of Masonry X(Wx)b. Moment of Uplift Pressure. Moment of Head-water Pressure. Sum of Moments of all Forces Reservor Full 2(Wx)f+ ■2(Px)f. 55 65 76 90 106 1,881,000 3,015,000 4,774,000 7,970,000 13,220,000 338,500 574,000 955,000 1,657,000 2,820,000 1,166,000 2,084,000 3,529,000 6,155,000 10,440,000 2,708,000 4,525,000 7,348,000 12,470,000 20,840,000 Location op Resultants. Length of Base. Distance of Heel from Lin of Reference Tangent <£. Level. Distance of Resultant from Line of Reference. Heel. Toe. Empty Reservoir. Full Reservoir. 55 65 76 90 106 13.4 15.2 17.5 20.6 24.4 24.1 28.6 33.9 40.8 48.8 36.1 42.9 50.8 61.2 73.2 0.53 0.68 0.72 0.74 0.75 Tangent 6. Maximum Pressures, in Thousands of Pounds per Sq. Foot. t At Heel. At Toe. Level. Reservoir Full. Vertical. Inclined. Vertical. Inclined. Dam. Founda- tion. Dam. Founda- tion 55 65 76 90 106 0.667 0.693 0.713 0.729 0.739 14.8 14.8 14.8 11.7 18.3 18.1 120 SOLID SPILLWAY GRAVITY DAMS [Chap. VII has been adopted for the top of the dam, tension exists above Level 45.0 for the first condition of loading. The height of the portion of the dam in which tension exists may be reduced by- increasing the thickness of the top of the dam. The proportions can be settled only by the judgment of the designer. As it would be impracticable to provide monolithic concrete between the crest and Level 45, steel reinforcement must be used to resist the tension. The reinforcement should be computed in accordance with the theory of "flexure and direct stress,"* and a uniform uplift equal to head-water pressure should be assumed to act over the entire area up-stream from the neutral axis of any horizontal plane. Fig. 35 indicates sections of the dam for water storage on the Upper St. Maurice ftiver, Province of Quebec. The temperature of the site of the dam drops to —60 or — 70° F., and the range is about 160°. Ice pressure of 50,000 lb. per lin. ft. was assumed to act at crest level. The lines of resultants in the figure indicate clearly the need of reinforcement at the up-stream face. 47. Example No. 5. 30-ft. Solid Spillway Dam (Fig. 36). Assumptions: H = maximum height of dam = 30 ft. above. good rock; ' c = the area of joints and base subjected to uplift = 30 per cent. The uplift is assumed to vary uniformly from head- water pressure at the heel to tail-water pressure at the toe;f w\ = weight of masonry = 145 lb. per cu: ft. ; w 2 = weight of water = 62.5 lb. per cu. ft.; w>3 = weight of silt in air = 125 lb. per cu. ft.; J a = angle of repose of silt in water = 0°; % K = voids in silt = per cent;{ /=safe value of the coefficient of friction of the joints and base = 0.75; Pi = ice pressure = none; Q m = maximum flood to be expected = 32,400 cu. ft. per sec; It = total length of crest = 200 ft. ; *See "Principles of Reinforced Concrete Construction," by Turneaure and Maurer. 2d Edition. John Wiley & Sons. 1910. f See foot-note, p. 126. J Assumed as liquid mud. See Art. 16, Tangent 9 0.4 0.3 Fig. 34. 120 100 80 60 40 20 Cubic Yard3 per Linear Foot, not Including bucket or cut-oa To face page ISO Art. 47] EXAMPLE NO. 5 Table XXI Results op Calculations for Example No. 4 op Art. 46 121 Vertical Forces. Horizontal Forces. Level. Weight of Masonry ■L{W)e. Uplift Pressure. Summation of Vertical Forces Reservoir Full 2{W)r. Ice Pressure. Head- water Pressure. Summation of Horizon- tal Forces Reservoir Full S(P)f. Loading. 45 52 60 72 86 102 108,700 145,900 192,200 269,700 375,600 520,300 16,540 22,050 29,330 49,700 71,900 103,200 92,160 123,900 162,900 220,000 303,700 417,100 20,000 20,000 20,000 28,120 42,760 63,250 137,100 201,900 290,900 48,120 62,760 83,250 137,100 201,900 290,900 1 1 1 2 2 2 Moments op Vertical Forces. Moments of Horiziontal Forces. Total Moments. Level. Moment of Weight of Masonry, X(Wx)e. Moment of Uplift Pressure. Moment of Ice Pressure. Moment of Head-water Pressure. Sum of Moments of all Forces Reservoir Full, Z(Wx) F + •2{Px) F . Loading. 45 52 60 72 1,476,000 2,160,000 3,083,000 4,803,000 7,569,000 12,110,000 193,500 280,700 407,700 786,000 1,360.000 2,345.000 600,000 740,000 900,000 281,200 527,000 949,000 2.944,000 5,305,000 9,208,000 2,164,000 3,146,000 4,524.000 6,961,000 11,514,000 18.970,000 1 1 1 2 86 2 102 2 Location of Resultants. Length of Base Level. Distance of Resultant from Line of Reference. Loading. Empty Reservoir. Full Reservoir. 45 52 60 72 86 102 13.58 14.79 16.04 17.82 20, 20 23.31 23.49 25.39 27.78 31.65 37.90 45.52 35.23 38.08 41.67 47.47 56.80 68.20 1 1 1 2 2 2 122 SOLID SPILLWAY GRAVITY DAMS [Chap. VII /„ = effective length of crest = /, minus the effect of two com- plete end contractions; lc — width of the channel of approach = 210 ft. Maximum Head on the Crest The velocity of approach, from Fig. 36, is Q n 32,400 »i = 154 4.(*.+8) ~210(M-8) ~~ Ac+8" ELSS3.0 : :S^i> ■ ::A v. NON-OVERFLOW SECTION Fig. 35. — Upper St. Maurice River Dam, Showing Reinforcement to Resist Ice Pressure. (Eng. Record, Vol. LXX, p. 394.) The head corresponding to the velocity of approach is v£ _J / 154 \ 2 = 368 2flf 2X32.2^+8/ (Ac+8) 2 * As there are two complete end contractions, corresponding to two sharp-cornered abutments having crest coefficients of C = 0.1, Eq. (32) may be written,* h = lt-h c (0.1X2), l„ = 200 -0.2fc r . * The effect of end contractions, for this example, is negligible, but is included in order to indicate the application of Eq. (32). Art. 47] EXAMPLE NO. 5 123 The shape of the crest is to be proportioned to fit the sheet of water corresponding to the maximum flood to be expected; there- fore the coefficient of discharge, from Fig. 30, will be C = 3.94, corresponding to a head ratio of unity. Substituting these values in Eq. (31a), there results, 32,400 = 3.94(200 - 0.2k) 368 1 3 '* * c+ (k+8) 2 This equation, being solved by successive trial substitutions, there results, K= 11.0 ft., whence t'i = 8.1 ft. per sec, A„=1.0ft., g = -^=164 cu. ft. per sec. These values are indicated in Fig. (36). The head, h c , is the actual pressure head on the crest, and should fix the water surface to be used in detennining the static head-water pressures on the dam. The head, K+hr, is the total head on the crest as affecting the discharge, and should be used, as hereinafter described, in fixing the shape of the crest. The curve of the upper nappe may be drawn approximately, as indicated, from the plotted curve to the actual water surface. Impact of the Approaching Water From Art. 14 it is seen that the unit pressure from the impact of the approaching water corresponds to approximately twice velocity head, or Unit pressure = 2u>2k=2X62.5Xl.0 = 125 lb. per sq. ft. The pressure is assumed to be distributed uniformly over the area above the silt. Tail-water The depth of tail-water at maximum flood is assumed to be 20 ft. From Art. 14, it is seen that the pressure of tail-water may or may not act upon the dam, depending on its depth relative to 124 SOLID SPILLWAY GRAVITY DAMS [Chap. VII the height of the dam, and the discharge. The determination of the depth, h 5 , of tail-water which is apt to be swept away from the toe of the dam, as indicated in Fig. 4, is obtained from Eq. (5), h= 4. = — ^ . or -\/™ = l-0 or 0.577, tan = 1.0, which is the greater of the two. Therefore = 45 degrees. The necessary calculations, including tail-water, for the deter- mination of the stability of the dam above Level 42.0 will now be given. Calculations excluding tail-water may be made in the same manner. The location of the resultant (Rule 1) may be determined from Eq. (18) of Art. 31. Using the subscripts, E and F , to repre- sent empty and full reservoir, respectively, referring to Fig. 13, and taking the point of reference on the up-stream face of the dam, we have, _ 2(Wx) E +2(Px) E _ 2(Wx) F +2(Px)r * 2(W) F The necessary calculations are indicated in Table XXII. Substituting in the foregoing equations, 1,359,000+0* 102,600 ' 1,994,000 01 . Z * = -937I90- = 2L4 - Plotting these values on Fig. 36, it is seen that the resultants he well within the middle third. * The pressure of silt is neglected for empty reservoir, representing the condition before the water first fills the reservoir. Art. 47] EXAMPLE NO. 5 127 o 1—1 g J EH g o o W r» r- t^ CO 1 CO o o o o o o o o *"" n eo_ o d h o to + O o fM C-4 IN CD CO 00 •* lO © .2 « of* So N O O ^ CO CO N X X CN lO lO Xw N W nXX^ J O rH X CO CO r-l CO & 13 X H £ a* 128 SOLID SPILLWAY GRAVITY DAMS [Chap. VII Art. 47] EXAMPLE NO. 5 129 03 a o a o 6 CO 3 ■a B O PQ o ffl O a 130 SOLID SPILLWAY GRAVITY DAMS [Chap. VII To investigate for safety against sliding (Rule 2), we may use Eq. (22), of Art. 32. Evidently, Rule 2 is fulfilled for empty res- ervoir. For full reservoir, we have, tan5 -2(l0;-937l90- - 534 ' which is well within the allowed value of 0.75. Referring to Fig. 36, it is seen that the more severe condition, as affecting Rule 2, is without tail-water, although tan 8 is still well within the allowed limit. Unit pressures in the masonry and the foundation (Rule 3), are obviously safe for this height of dam, but may be determined, if desired, as explained in Example 2 of Art. 39. Results of calculations, for all elevations, and with and with- out tail-water, are indicated in Fig. 36. Above Level 18.8, (within Zone la, Art. 30), tan 8 is greater than the allowed value; and above Level 14.3 (within Zone I), the resultant, reservoir full, lies outside the middle third. This condition necessitates careful treatment of the upper part of the dam, as described in Example 3 of Art. 45. It will be noted that the shape of the section is governed solely by the shape of the sheet of falling water. In other words, the whole dam lies above the elevation of the bottom of Zone II. Should the calculations indicate a location of the resultant outside the middle third, the dam should be widened, as indicated in Example 3. 48. Comparison of Solid Spillway Dams. The fundamental theory of design, for solid spillway and non-overflow dams, differs only in the upper part, which, in the former, is proportioned to conform to the shape of the sheet of water spilling over the top. A comparison of this feature, therefore, would be of interest, and is given in Fig. 38. In order to make a direct comparison with Example No. 5, the dimensions of each dam were multiplied by a constant, thus reducing each head on the crest to a value of 10 ft. This can properly be done, as it was shown in Art. 42 that the co-ordinates of the issuing sheet, and hence the curve of the top of the dam, are direct functions of the head on the crest. As a matter of fact, the Art. 48] SOLID SPILLWAY DAMS 131 stability of any dam, subjected to head-water pressure only, will not be changed if all dimensions, including the head on the crest, are increased in the same proportion. This, of course, provided the compressive stresses remain within the allowed working values. Water Surface COMPARISON OF SECTIONS OF SOLID SPILLWAY DAMS All reduced to a common head-on crest of 10 feet Feet - Author's Example No.5 of Art-17 - Keokuk Dam of the Mississippi River Power Co. - Dam No.2 of the Appalachian Power Co. Parr Shoala Dam of the Parr Shoals Power Co. {Fig. 37) Ocoee No.l Dam of the Tennessee Power Co. — — — McCaU.Feny Dam of the Pennsylvania Water and Power Co. Fig. 38. The differences in the tops of the dams indicated in Fig. 38 may be attributed chiefly to.differences in the methods used in determining the shape of the sheet of spilling water and the dis- tance the masonry line was extended into the sheet to provide a margin of safety. CHAPTER VIII THE DESIGN OF HOLLOW DAMS 49. General Considerations. Hollow dams may be made in an almost unlimited variety of forms; but, thus far, they have been largely constructed entirely of reinforced concrete, of the but- tressed type, having substantially triangular buttresses set par- allel to one another at fixed intervals across the bed of a stream. Two types of hollow dams have become common. Fig. 47 indicates a typical " Ambursen " hollow dam. The distinguishing feature of this type is the flat reinforced concrete decks. Some- times these decks are arched, as in Fig. 46. Such a small amount arching, however, will not be effective while the reinforcement is intact, as a shrinkage of the concrete due to setting, or a fall in temperature, will result in a slight opening of the joints and destroy all arch action until the reinforcement has failed. Therefore this feature should be considered as limiting complete failure, and not a partial failure, such as a cracking of the concrete and a rupture of the steel. Fig. 43 indicates a typical " Multiple Arch " hollow dam in which the decks consist of series of arches spanning between the buttresses. The arches are often reinforced. For design of the arches see Art. 57. The up-stream edges of the buttresses have a slope of any desired degree; but, in by far the larger number of cases, this slope is about 45°. The slope of the down-stream edges of the buttresses varies usually from zero to about 15° with the vertical for non-overflow dams. For spillway dams, the shape of the top and down-stream edge is usually fixed by the necessity of providing a crest and apron designed to fit the shape of the sheet of water spilling over the dam.* The width and superelevation of the top is fixed by practical considerations, mentioned in Art. 37; but, in the case of hollow dams, a considerable superelevation or top width is not wholly or * See Art. 42. 132 Art. 49] GENERAL CONSIDERATIONS 133 > £ .4 03 -a § 3 O J3 *3 3 134 THE DESIGN OF HOLLOW DAMS [Chap. VIII partly compensated by a reduction in masonry in the lower parts, as in the case of solid non-overflow dams. The apron of the spillway type is a reinforced concrete slab, %'-'Cot. Bar^, lapping Bare laced with No 6,{'° Cor. Bars, (batted at Butt C.L. /-El. 1635 1:3:6 Struts and Deck 1:2:1 Buttress reinforcement 1 °Cor. Bars two in each course, lapping 30." Three inches between steel and face of buttress. Column reinforcement (l°Cor. Bars) made continuous by 24" lap at lift. •% ° Cor. Bars, HI C. t SECTION B-B Fig. 40. — Details of Mathis Dike Dam. {Eng. News, Vol. LXXIV, p. 592.) but, the loading being indeterminate, its thickness and reinforcing are matters of judgment. It is essential to provide ample concrete at the discharge lip, as indicated in Fig. 47, to withstand shocks Art. 49] GENERAL CONSIDERATIONS 135 from floating ice and logs, and the pressure of the jet in changing its direction of flow. On soft foundations the buttresses may be provided with plain or reinforced concrete spread footings, in order to keep the com- pressive stresses within reasonable amounts. In extreme cases, these footings may have a width equal to the spacing of the but- tresses, so that the dam virtually rests on a concrete mattress, as indicated in Fig. 41. Such footings should be provided with large weep-holes at close intervals, in order to preclude the possibility of uplift from head-water. Fig. 41. — Foundation Mattress, Mathis Dike Dam. The buttresses should be braced with horizontal struts at inter- vals, as indicated in the illustrations. The spacing of the struts should not exceed twelve times the thickness of the buttresses when the latter are stressed in compression to their full working value. Additional stiffness is obtained by horizontal steel rein- forcement between struts, as indicated. The reinforcement in the struts is usually continuous through at least three bays, but in some cases it has been carried contin- uously throughout the structure, with no deleterious effects from contraction. The last method was used for the Mathis Dike Dam, Fig. 39, etc. The struts should abut solidly against the 136 THE DESIGN OF HOLLOW DAMS [Chap. VIII buttresses. The horizontal building joints in the buttresses should be at the elevation of the center line of the struts, if possible. The open holes through the buttresses, indicated in the illus- trations, are found convenient for the passage of men and materials during construction. An inspection gallery is usually provided, unless, in low dams, access may be had from the down-stream side at ground level. Provision for draining the interior of hollow spillway dams is usually made, as shown in Fig. 47, where, for this purpose, an opening of considerable size is indicated under the bucket. In Fig. 42. — Down-stream View of Mathis Dike Dam. such cases the high velocity of the water at the end of the bucket will entrain the air and cause a partial vacuum to form within the dam, unless sufficient air inlets are provided. If a partial vacuum is allowed to occur, the loads on the deck and apron will be ma- terially increased, as explained in Art. 17. Large openings through the buttresses and an adequate open shaft at each end of the dam are usually provided. The mistake has sometimes been made of providing solid doors at the entranc.es to the interior of the dam. In hollow dams of the usual type, the resultants for full or empty reservoir always intersect the joints relatively close to the Art. 49] GENERAL CONSIDERATIONS 137 f -j ^ F ~ Z = 5 < = £ " s 138 THE DESIGN OF HOLLOW DAMS [Chap. VIII center of gravity. Rule 1, therefore, is seldom a governing con- sideration in determining the shape of the structure. As the weight of masonry alone is never sufficient to prevent F I shding (Rule 2), it is necessary to batter the up-stream face to include the vertical component of considerable water pressure. Limiting the compressive stresses to safe values (Rule 3) necessitates an adjustment of the length and thickness of the buttresses. Art. 49] GENERAL CONSIDERATIONS 139 I I f a 6 i I 0! Pi o I 1 I a "3 l O e 140 THE DESIGN OF HOLLOW DAMS [Chap. VIH In hollow dams there is some question as to the effective area of base to distribute the pressures on the foundation. It is com- mon practice to neglect the horizontal area of the deck in computing the compressive stresses. This procedure, however, is- not always on the side of safety. The addition of the area of the deck to the area of the base will always reduce the direct stress (Fig. 12), but the eccentricity, e, may, in some cases, be increased a sufficient amount to increase the fiexural stress to a greater extent, resulting in an increased total stress at the extremity of the base. The complete design of hollow dams cannot be made, step by step, as in the case of solid dams. The general outlines of the dam are usually chosen tentatively, in accordance with the judg- ment of the designer, tested for conformity with the designing rules, and adjusted if found necessary. The spacing of buttresses is limited by the economical span of the deck. In general it will be found that a considerably greater spacing can economically be adopted for Multiple Arch Dams than for the Ambursen type. For large dams, a spacing of from 30 to 40 ft. for the former type and from 15 to 25 for the latter, are common. In order to indicate more fully the several features entering into the design, reference is made to the examples which follow. 50. Example No. 6. Hollow Non-Overflow Dam (Fig. 46). Assumptions: w, = weight of masonry = 150 lb. per cu. ft. ; w% = weight of water = 62.5 lb. per cu. ft. ; p,' and p," = maximum allowed vertical compressive stress = 31,000 1b. per sq. ft.; Pt and p" = maximum allowed inclined compressive stress = 50,0001b. per sq. ft.; /= working value of the coefficient of friction of the joints and base = 0.60. The section having been laid out, as indicated in Fig. 46, in accordance with the judgment of the designer, it remains to inves- tigate it for conformity with the designing rules, and make the requisite alterations if found necessary. The methods will be described for the joint at Level 73.0 only. A length of dam of 18 ft. will be considered, Abt. 50] EXAMPLE NO. 6 141 Rule 1. Location of the Resultant The location of the resultant may be found from Eq. (18) of Art. 31, moments being taken about the point of reference, A, at the up-stream extremity of the joint. For empty reservoir, 2(Wx) E +2(Px) B Zb Z(W) e For full reservoir, _ X(Wx) F +2(Px) F Zf ~ 2(W) r The necessary calculations are indicated in Table XXIII. Making the proper substitutions in the foregoing equations, there results: i For empty reservoir, _ 91,550,000 _^ ift **~ 1,943,000 _4 '- 1U - For full reservoir, _ 235,450,000 F 4,938,000 These values are indicated in Fig. 46. Rule 2. The Inclination of the Resultant From Eq. (22) of Art. 32, tane ZtfV,_ 2,916,000 tan eF -sa^"4,938,000-°- 591, This value is close enough to the maximum allowed value of tan 6 = 0.60, and;no adjustment is necessary. The resultant, reservoir empty, is, of course, vertical. ! A change in tan 6 F could best be obtained, if found necessary, by altering the inclination of the up-stream face of the dam, thereby changing most effectually the total force, 2(W) F , on account of (the additional water pressure. It is essential, for economy, to j have as steep an up-stream face as the allowed max- imum value jpf tan 6 F will permit, i 142 THE DESIGN OF HOLLOW DAMS [Chap. VIII 8 § © o o o O O fl O © © o o eo tH Oi o as O i-< CM Tf CM Kl k k fe g *£ H W w m °° h O0 § X X ■ Is "? m O •3 o © 5 x X eg •a « »o Eh « a ca ^ a> a? ° .£ X & O v Q £ « X X 13 O Pn B Art. 50] EXAMPLE NO. 6 143 In this connection, it should be noted that a much steeper up- stream face is possible in the upper part of hollow dams; tan 6, for this example, being very small at the upper levels. A few recent hollow dams have been shaped in this manner. Rule 3. Compressive Stresses a. Reservoir Full. As the base of the dam is not rectangular, the vertical compressive stresses must be found by the method indicated in Art. 22 for irregular bases, and Eq. (14) and (15) will apply, p u ' and p u " being zero for hollow dams. At the toe, p/~S(W) r \\+ At the heel, p„" = 2(WV A ' I \ 1 em" A I The following values for Level 73.0 are easily found: A=313.1; 7 = 291,000; e = 14.20; m' = 62.71; ra" = 33.29. Using the value of 2(W)p from Table XXIII, and making the proper substitutions in the foregoing equations, there results: P /^4,938,000(^+ I4 | 9 ° i y o 2 71 ) =30,870, ^"=4,938,000(3^-^^9)^,740. The maximum vertical stress is seen to be at the toe of the dam, where it equals, approximately, the maximum allowed value adopted. It will be found, upon investigation, that by properly reducing the thickness and increasing the length of the buttresses, and at 144 ; THE DESIGN OF HOLLOW DAMS [Chap. VIII the same time slightly reducing the area of the joint, the vertical stress at the toe will remain the same and that at the heel will be increased. This adjustment is possible, owing to the resultant decrease in the eccentricity, e, of the loading, with a corresponding reduction in the flexural stress. On account of the decrease in the area of the joint without an increase in the maximum vertical stress, the resulting arrangement would be more economical. However, the thickness of the buttresses is limited by practical considerations, and, as it is considered that, in this example, they are as thin as it is advisable to have them, a more economical arrangement, in this respect is not possible: For the maximum inclined compressive stresses in the dam, Eq. (23a), of Art. 33, applies. Eq. (24a) has no practical use unless the strength of the foundation is less than that of the masonry. At the toe, tan<£' = 0.333; tanV = 0.111; sec V = 1-153. For full reservoir, tan 9 = 0.591; sec 2 6= 1.352. p n is zero at the toe. Using these values in Eq. (23a), the maximum inclined com- pressive stress for full reservoir is found to be: At the toe jH r = (30,870X1.153-0), orO, or 30,870X1.352, p/ = 35,600 or 41,750, pi = 41,750, the greatest of these values. It is noted in Art. 33 that the greatest stress at the heel, for this type of dam, is the reaction of the deck on the buttresses. This will amount to, p/ , = ^5X72X18 =37300 The maximum inclined stress is seen to be within the adopted limiting value of 50,000 lb. per sq. ft. Stress In Founds per Square Foot. Note. — Stresses have been calculated only at the levels indicated by circles. Unless indicated in the diagram, the horizontal area of the deck has been included in the area of the joints. Fig. 46. To face page 145 Art. 50] EXAMPLE NO. 6 145 b. Reservoir Empty. As, for empty reservoir, both the eccen- tricity, e, and the weight, 2(W) E , is less than for full reservoir, it is evident that the stress, for this condition, will not govern. The results of all these calculations are plotted in the diagram, Fig. 46, together with corresponding results for the upper joints. It will be noticed that the pressures grow rapidly less in the upper joints. To be consistent with theory, they should be constant at each joint. However, the thickness of the buttresses near the top of the dam is usually fixed by practical considerations. At the top of the dam a minimum thickness of 12 in. was considered advisable. At Level 49.0 a minimum thickness of 19 in. was required in order to keep within the allowed maximum vertical pressure of 31,000 lb. per sq. ft. Between these elevations the thickness of the buttresses was increased uniformly. In order to indicate the difference in calculated stresses, with and without the horizontal area of the deck included in the area of the base, the vertical stresses, corresponding to the latter con- dition, have also been calculated and plotted in the diagram. It is seen that, at Level 73, to include the area of the deck, results in an increase in the maximum vertical stress from 23,600 to 30,870 lb. per sq. ft., and a corresponding increase in the maximum inclined stress. As mentioned in the last part of Art. 49, it is probable that the area of the deck is not wholly effective in dis- tributing its portion of the loads, on account of not being directly bonded to the buttresses, and consequently the stresses may not be as high as calculated. It is certain, however, that they exceed those calculated without including the area of the deck, and it seems best to adopt the more conservative method. Rule 4. Inclination of Down-stream Face The maximum allowed inclination of the down-stream face is seldom a governing feature in the design of hollow dams. In the first place, it is never necessary to provide a very flat face, and in the second place, the usual horizontal reinforcement in the but- tresses is in a position to resist possible tensile stresses in vertical planes. 51. Example 7. Hollow Spillway Dam. Fig. 47 represents a typical hollow spillway dam of the Ambursen type. In adopting 146 THE DESIGN OP HOLLOW DAMS [Chap. VIII the general outlines, consideration should first be given to the shape of the crest and apron, which can be established at once, as described in Art. 42, Fig. 26 being used for the approximately 45° slope of deck. The radius of the bucket can be taken from Fig. 32. The designing methods used in Example 5 apply directly to this case, and will not be repeated. The area of the horizontal joints should not include that of the apron. This is a necessary assumption; for, as indicated in Section B-B there is not a sub- stantial bond between the apron and the buttresses. The assump- tion is on the side of safety. Assumptions: Wi = weight of masonry = 150 lb. per cu. ft. ; u)2 = weight of water = 62.5 lb. per cu. ft.; pj and p v " = maximum allowed vertical compressive stress = 29,0001b. per cu. ft.; Pi and p t " = maximum allowed inclined compressive stress = 40,0001b. per cu. ft.; /= working value of the coefficient of friction = 0.60. The necessary calculations for the determination of the loca- tion of the resultants, at Level 64.0, are indicated in Table XXIIIa. The results of all calculations are plotted in the diagram of Fig. 47. It will be noted that the stresses vary considerably at the different levels, and approach the maximum allowed values only at Level 64.0. This could be remedied only by decreasing the thickness of the buttresses. above Level 64.0. However, they are as thin as it is desirable to have them, so that no adjustment, in this respect, is possible. Some economy would result if the slope of the up-stream face were steepened slightly, as it is seen that the maximum value of tan is less than the allowed value of 0.60. Tangent © EXAMPLE NO. 7 OF ART. 51 TYPICAL REINFORCED CONCRETE HOLLOW SPILLWAY DAM . : ;° : i'> ; ::-;.<>; Stress in Pounds per Square Foot Note. — Unless indicated in the diagram, the horizontal area of the deck has been included in the area of the joints. Stresses have been calculated only at the levels indicated by circles. Fig. 47. Htgh Water LeVel 0.0 EXAMPLE NO. 7 OF ART. 51 TYPICAL REINFORCED CONCRETE HOLLOW SPILLWAY DAM Levell2.0 I * .*. » ' * - » . ; » » I* ,"» = -" k^Wi/^=-i ; :^illkv:V:;W-;x«:;v:V.'.'o;::i;'v'.;: •:■.?;;.•.••<(•::•.■■.•:•.• Level 16.0 Fig. 47. To /ace jwj/e 1JB Art. 51] EXAMPLE NO. 7 147 o E 1-3 .a 3 P. -^> 3 O 03 a O 75,930,000 29,490,000 20,410,000 © O o 6" eo 00 io" 26,380,000 18,250,000 o o o o CO CO ■<# 1 o o CO I + I >■ O CO O U5 CO «5 OC 00 b-" » CO o CO © r^ cd O h O o o o o o o o o o «3 OS N CO O Ttt 1Q (O £N O o o to OS co~ o o o o o o w o U5 t*. o o o CO CM « H o « ■4^ a » 2 § X 1 3s 3 *= C=— — =all constants; C2 k, k', k " = constants; A = horizontal area of an arch dam, between abutments, at any elevation; 5= central angle between abutments (Fig. 52); s=span of arch at any elevation; x= 3.1416. 53. Arch Stresses. In order to indicate the approximate distribution of stresses in an arch dam, we will first consider the simple case of a dam of constant radius, confined between vertical abutments, and having a horizontal base. It will be assumed that: o. The arch radius is very long, compared with the thickness of the dam; 6. The dam is homogeneous throughout; c The thickness at any elevation is proportional to the depth of water, resulting in a triangular section with its crest at water surface; d. The arch stresses are not influenced by temperature changes and other indeterminate considerations to be discussed later. The shaded area, 1—2-3, in Fig. 49, represents a section of the unloaded dam at the crown of the arch. The unit water loads are represented by abscissae to the line, 1-7. Considering the deflection of the loaded dam to be unrestrained at the base, the ordinary equation for the arch stress at any eleva- tion is P=^- (33) This equation is not correct when the arch radius is small, 150 ARCH DAMS [Chap. IX compared with the arch thickness; but it is close enough for all practical purposes. Fig. 48. — Arch Dam of the Shoshone Project, Wyoming. As j and r„ are assumed to be constant for this case, p will also be constant, and will be represented by abscissae to the line, 8-9. Abt. 53] ARCH STRESSES 151 The late R. Shirreffs has shown * that, for constant radius and span, the crown deflection of an unrestrained arch may be rep- resented by, Cjq t ' _=2ra j LOADING DIAGRAM m=- (34) DEFLECTION DIAGRAM ARCH STRESS DIAGRAM 0.2 0.4 0.6 0.8 Unit Arch Loading, in Terms otWiB w 2 H Fig. 49. Ti- ll V l /Arch in restrained dam with, elastic beam of apprecj — able weight Arch ptress in restrained dam with inelastic vertical beam _ofi.no weigj Arch etr unre trained Arch Stress, In Terms of p where G is a constant depending on the radius, span, and other characteristics of the arch- By assumption, and, t = C 2 h, (35) q=W2h, * In his discussion of a paper on Lake Cheesman Dam by Harrison and Woodard, Transactions, Am. Soc. C. E., Vol. LIII, p. 155. 152 ARCH DAMS [Chap. IX therefore, t W\ i— C 2 h m = C^V2h = Cw2 (36) Therefore m at any elevation is constant, and the dam under load will assume the position, 4—5-6. A restraint at the base, however, on account of vertical beam action, will result in a redistribution of the loading taken by the arch. If we assume the vertical beam to be incapable of deforma- tion, of no weight, and free to rotate about the base, the load car- ried by the arch may be represented (as proved later), by the curve, 1-10-11-3, and the arch stress by the line, 13-14. The deflection of the dam will be as indicated by the lines, 15-2-3, the section still being a true triangle on account of the inelasticity of the vertical beam. The area between the vertical, 1-3, and the curve, 1-10-11-3, represents the total load now being carried by arch action, and is less than the total water load, 1—7-3-1, the remainder being taken directly to the foundation by shearing and frictional resistance at the base. All of this can be proved, as follows: Shirreffs' equation applied to the restrained arch is, m!=^i (34a) Substituting the value of t from Eq. (35), there results, m _Ciqi_Cqi For static equilibrium, the moment, about the base, of the loading carried by restrained arch action must be equal to the corresponding moment of the total water load. Therefore, C B qi (H-h)dh=^ (38)' From Eq. (37), ^ = -q- (39) And since, by assumption, the up-stream face of the dam is still straight, m *= -jf (40) Art. 53] ARCH STRESSES 153 Substituting these values of qi and mi in Eq. (38), there results, I " M j(H~h) 2 h _ W2IP EC h ~ 6 • Integrating between the limits and H, and solving for M 1 we have, as the top deflection of the restrained dam, M 1 =2Cw 2 (41) Comparing the top deflection of the restrained arch, as given by Eq. (41), with that of the unrestrained arch, from Eq. (36), it is seen that the former is just twice the latter under the assump- tion of no deformation of the vertical beam. To derive the equation of the loading curve, 1-10-11-3, we have, from Eq. (39), (40) and (41). mih MiiH-h) mi= K -jj L , M 1 =2Cw 2 , qi = 2MH-W > (42) Therefore, which is the equation of the curve, 1-10-11-3. The equation of the stress line, 13-14, may be derived as fol- lows: At the base of the dam we have, from Eqs. (33), (35) and (42), for the restrained arch, qir u qir a 2(H-h)w 2 r u . Vl= T = C2h = — Wc 2 — (43) Also, at the base of the unrestrained dam, Eq. (33) gives, _ W2Hr u _ W2Hr u P t CW or, „ w 2 r u V 154 ARCH DAMS [Chap. IX Substituting this value of C2 in Eq. (43), there results, Pi-^5^. W which is the equation of the stress line, 13-14. It will be noted that, for the restrained dam, the arch stress, pi, is equal to the constant unrestrained arch stress, p, when h is one-half of H. In the upper half of the dam the arch stress for the restrained dam is greater than that for the unrestrained dam, reaching a maximum of twice the latter value at the extreme top. In the lower half, the restrained arch stresses are less, reaching a minimum value at the base, where the arch deflection is zero. Thus far we have been limited by_ assumptions, some of which are impossible and others of which are unusual. In order to arrive at a closer indication of the arch stresses, we must investi- gate the effect of: a. Elasticity of the vertical beam; b. The weight of the dam; c. Uplift water pressure; d. Varying span; e. Varying radius; /. Possible variation of the modulus of elasticity at different elevations; g. Expansion and contraction due to temperature changes; h. Expansion and contraction due to changes in moisture con- tent; i. Contraction of the concrete due to setting of the cement. On account of the indeterminate effect of most of these items, it is the opinion of the author that it is impossible to write exact equations indicating the arch and vertical beam stresses in arch dams, although many attempts have been made.* Fortunately, a number of dams have been constructed which, being virtually experiments on a large scale, serve to indicate * Transactions, Am. Soc. C. E., Vol. LIII, p. 155, paper by Harrison and Woodard, and discussion; Transactions, Tech. Soc. Pacific Coast, Vol. VI, p. 75, paper by Vischer and Wagoner; Transactions, Am. Soc. C. E., Vol. LXXVIII, p. 685, paper by L. R. Jorgensen; Engineering News, June 9, 1910, article by J. S. Eastwood; "Masonry Dam Design," by Morrison and Brodie. 2d edition. John Wiley & Sons. 1916. TABLE XXIV— TABULATION OF ARCH DAMS Arranged in Order of Height a Name of Dam. Location. O bi etc. = corresponding width of canyon, measured in a straight line between haunches of arch. Widths, radii, etc, vary approximately uniformly between points indicated. * Figures from Equation (33), Art. 53, the effect of vertical beam action, temperature changes, and other indeter- minate factors being neglected. t The enlarged base or pedestal indicated by broken lines in Fig. 50, common to some existing dams, has been neglected in the tabulation. The base of the arch dam has been considered at the elevation of the top of the pedestal. Such pedestals occur where a dam was started as a gravity seotion and finished as an aroh; where an enlarged base has been constructed to facilitate future extensions; and for other purposes. t Spillway crest is 9.7 ft, lower, and has crest projecting up-stream to keep sheet of spillway water from leaving the masonry. See Art. 56. Fig. 50.— Explaining Notation in Table XXIV. TABLE XXIV— TABULATION OF ARCH DAMS Arranged in Order of Height co 'S3 he X a Xb Xc Xa Xe V a Vol W> Vc yd V" Radius of the Up-etream Face. W a Wb Wc. Wd We II ^ 4- «'« a 6 Character of Foundation. Material of the Dam. Reinforcement. r a Tb r c ra Te (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34 ) (35) (36) (37) (38) (39) (40) (41) 305 220 210 168 120 37 37 11 31 31 10 16.5 10 6 19 71 71 108 63 16 155 225 155 331 158 192 192 236 186 147 162 200 380 70 80 108 119 94 47 43 +10 243 Granite Lava Granite Concrete Concrete Rubble masonry Concrete Rubble masonry 16.5 350 260 230 195 2 18 28 6 333 294 243 550 205 510 390 280 80 23 93 90 87 81 75 15 4.5 12 3.5 4.3 4 9.4 12 3.5 4.3 33.8 31 24 12.8 14 200 222 100 76 359 200 237 100 76 359 338 46 24 12.8 14 +2 Gneiss Concrete Concrete Concrete Concrete Concrete 15 5.3 13.5 23 9.5 222 -3 -4.5 Sandstone Andesite Porphyry 115 345 111 47 85 76 65 20 g" 4> hor. rods 15"-18" cts.; 12-lb. vert, rails 4' 6" cts. in each face 72 65 64 61 60 11 18.5 2 3.5 3.2 3 3 27 9 9 3 5.2 324 62 335 250 70 324 62 346 268 70 350 80 235 27 9 20 21.5 5.2 -3 +4 -2 -2 Concrete Concrete Rubble masonry Concrete Concrete 3.5 3.2 3 3 8.3 5.2 3 57 30 Sandstone Granite Granite Lava 5.25 7.25 1.2 340 250 342 351 None None 130 68 35 60 55 55 52 50 4.4 5.3 4.3 4.8 4 4.4 7 100 286 77 160 250 100 Dense slate Concrete Concrete Concrete Concrete Concrete 14.5 11.5 15 15.5 1 5.6 6 4 7.1 6 11 14 11.5 14 15.5 77 161 250 115 95 75 40 20 1 1 160 161 -2 -2 Sandstone Sandstone None 195 155 62 50 50 48 46 42 10 3 3.5 3 3 3.5 3 15.5 3 3 3.5 3.8 16.1 18 253 288 150 253 211 253 288 150 263 211 410 390 362 25 18 -1 -5.5 -2 -1 -1 41 Altered shale Concrete Concrete Concrete Concrete Concrete 4.4 3 3.7 10 3 11.6 10 13 11.6 Conglomerate Sandstone Basalt None 0.4 253 253 410 950 350 300 310 100 130 None 39 38 33.5 32 31 6 2 3.5 3 3 2 3.5 3 3 5.3 11.4 7.5 8.7 80 103 300 90 68 80 103 306 90 68 114 113 110 62 85 46 52 5.3 11.4 13.5 8.7 Concrete Concrete Concrete Concrete Concrete 4.4 6.9 -3.5 —5 -2 Sandstone Granite Quartzite Shale None 300 None 3.6 None 92 90 See reference 31 28 25 21 4.8 7.1 2.5 3 318 120 70 220 8.6 13.6 4 8 Concrete Concrete Concrete Concrete 13.6 120 -10 -3 -1 Sandstone Shale Sandstone None (Dam to be raised 14') 90 35 ¥' hor. rods 12" cts. in each face 3 8 220 None (Ultimate height 50') ha, hb, etc. xa, xb, etc. Va, Vb, etc. ra, Tb, etc. wa, wb, etc. = depths of given horizontal planes below the top of the dam. he is the total height of dam. = corresponding distances from line of reference to up-stream face of dam. = corresponding distances from line of reference to down-stream face of dam. = corresponding arch radii, measured to up-stream face of dam. = corresponding width of canyon, measured in a straight lino between haunches of arch. Widths, radii, etc., vary approximately uniformly between points indicated. * Figures from Equation (33), Art. 53, the effect of vertical beam action, temperature changes, and other indeter- minate factors being neglected. t The enlarged base or pedestal indicated by broken lines in Fig. 50, common to some existing dams, has been neglected in the tabulation. The base of the arch dam has been considered at the elevation of the top of the pedestal. Such pedestals occur where a dam was started as a gravity section and 8nished as an arch; where an enlarged base has been constructed to facilitate future extensions; and for other purposes. t Spillway crest is 9.7 ft. lower, and has crest projecting up-stream to keep sheet of spillway water from leaving the masonry. See Art. 56. To face page 155 Art. 53] ARCH STRESSES 155 what may be expected from structures of this type. As far as the author is aware, there are no records of the failure of an arch dam. This, though fortunate from an economic point of view, is lamentable, as there has been furnished no direct indication of the limitations of such structures. Of the existing dams, a few have been designed in accordance with intricate although — in the author's opinion — questionable theory; but by far the greater number have been treated simply as sections of rigid cylinders subjected to external water pressure, i.e., thickness at any eleva- tion determined from Eq. (33), without consideration being given to possible restraint at the base. Table XXIV gives the general characteristics of a number of representative arch dams. The arch stresses indicated in Column 9 were calculated from Eq. (33). In order to indicate the relative stability of future designs, in comparison with existing structures, there is given the following brief discussion of the general effect, on stability, of the items listed above. Under the original assumptions, we have seen that, for an unrestrained arch dam, the arch stress may be represented by the line 8-9 (Fig. 49), and that, when the dam is restrained and the vertical beam is inelastic and weightless, the arch stress changes to 13-12-14. The effect of the elasticity of the vertical beam is to decrease the deflection at the top of the dam and increase the deflection at lower elevations. The effect of the weight of the structure and a resistance to rotation at the base is to increase the proportion of the load carried directly to the foundation and to reduce the arch stresses at all elevations. The arch stress curve under these combined conditions will have the general shape indicated by the line, 18-16-14, the exact amount of stress at the different eleva- tions being dependent on the relative influence of the imposed conditions. The approximate loading diagram and the deflection of the dam corresponding to the arch stress curve, 18-16-14, are indicated by the lines, 1-17-3 and 19-2-3, respectively. The effect of the uplift pressure due to head-water on hori- zontal joints tends only to lessen the effective weight of the dam and consequently the influence of the weight on the distribution of arch stresses. Uplift may be neglected in structures of this type, as it has practically no influence on stability. 156 ARCH DAMS [Chap. IX Other things being equal, the smaller the span or the radius, the greater will be the load taken by an arch for a given deflection. Most sites are V-shaped, resulting in relatively shorter spans at the lower elevations; moreover, as will be shown later, the great- est economy of material results for V-shaped sites when the arch radius gradually decreases toward the base. The effect of these two conditions on the distribution of stresses is similar, although more pronounced in the latter. A shortening of either the span or radius in the lower part of the dam results in a less deflection of the vertical beam and a greater arch stress. Consequently, the stress line, 16-14, in the lower part of the dam will approach the line, 12-9. For the usual shape of site, the reduction of deflection in the lower part of the dam causes also reduced deflections in the upper part, which part, still having the original radius, suffers a decrease in arch stress, or a movement of the stress line, 18-16, toward the line, 8-12. The net result, therefore, of a shortening of either the radius or span, in the lower elevations, is a reduction in the effect of vertical beam action at all elevations. Thus far, the conditions considered permit of approximate mathematical determination of the arch stresses, distribution of arch and vertical beam loading, and the deflection of the dam. Such calculations, however, are long and intricate; but, as mentioned heretofore, they have been made, and the results used in the design of arch dams. A theoretical dam, designed as a pure arch, using Eq. (33), a constant arch stress, 8-9, and neglecting the effect of vertical beam action, will, if checked by such rigorous investigations, be found to have reduced arch stresses in the lower part and increased stresses in the upper part, as indicated roughly by the curve, 18- 16-14. However, the increased stresses in the upper part will approach, but not exceed, double the stress assumed, and are amply compensated for by the factor of safety embodied in the usual working stresses. It is probable, indeed, that, for thin arch dams, such as would result from the use of the customary working stresses in Eq. (33), the vertical beam action will not greatly affect the stresses near the top. Moreover, a practical design embodies an appreciable thickness at the water surface, resulting in a considerable reduction in the stresses in the upper part of the dam. Abt. 54] VERTICAL BEAM STRESSES 157 Rigorous, intricate investigations, for the determination of the arch stresses due to a combination of arch and vertical beam action, therefore, are hardly justified, particularly in consideration of the sometimes compensating and sometimes cumulative effect of stresses due to temperature changes', changes in moisture con- tent, and other indeterminate factors which are sufficient to nullify practically the basic assumptions in such investigations. A glance at the deflection curves of the Barren Jack Creek Dam (Fig. 60), will indicate at once the utter futihty of endeavoring to arrive at an exact determination of the effect of vertical beam action. Another condition which affects materially the results to be obtained from rigorous investigations is the probability of varia- tion of the modulus of elasticity of the masonry at different ele- vations. It is hardly reasonable to assume that, with the vary- ing materials which must be used in different parts of a concrete structure, the modulus would have a great degree of uniformity, particularly when plums are used only in the lower and thicker part of the dam. A considerable reduction in temperature results in an opening of vertical building joints which must be closed by deflection under water pressure before arch action takes place. This has the effect of increasing the load carried by the vertical beam and reducing the load carried by the arch. An increase in tempera- ture will have the reverse effect. Swelling of the concrete, due to saturation as the water rises in the reservoir, will have the same effect on the distribution of the loading as a rise in temperature. It is probable that such swelling is nearly compensated by the previous shrinkage of ihe concrete due to the setting of the cement. 54. Vertical Beam Stresses. As pointed out heretofore, a close determination of the stresses in the vertical beam is diffi- cult if not impossible if all influencing conditions are considered. Fortunately, such stresses are not important, except in very high dams. Near the base, the effect of the water load is to increase the vertical compressive stresses near the down-stream face. Con- crete, in common with other materials, expands laterally when subjected to vertical compression. Being confined between the walls of the canyon, the direct result is an initial negative arch 158 ARCH DAMS [Chap. IX deflection in an up-stream direction which must be eliminated by- water pressure before the vertical beam can deflect down stream and the vertical compression be increased at the toe. Principally to this is attributed the fact that in no existing arch dams has the deflection of the vertical beam resulted in a rupture of the masonry due to excessive vertical compressive stresses, although several arch dams more than 200 ft. in height have been constructed. The question of horizontal shear in the vertical beam, though equally indeterminate, has no influence on the ultimate strength of the structure, as a failure in this respect, though objectionable as affecting leakage, would result in a better distribution of the arch loading. 55. Recommendations for Design. It has been mentioned that the influence of vertical beam action is to increase the arch stresses in the top of the dam, but that the ratio of the resultant stress to the arch stress, computed from Eq. (33), cannot theo- retically exceed two. In view of the fact that the upper part of the dam, which theoretically can reach zero thickness at the water surface, is for practical reasons considerably in excess of the computed thickness, such increase in the calculated stress is inap- preciable in comparison with the factor of safety embodied in the usual working compressive arch stresses. It is recommended, therefore, that the influence of beam action be neglected, and that the thickness of the dam be determined from Eq. (33). The effect of temperature changes in combination with other influences is indeterminate, and, therefore, must be also compensated for by the factor of safety in the unit stresses. A discussion of working stresses is given in Art. 58. Fig. 51, plotted from Table XXIV, indicates the relation between maxi- mum arch stresses and the height of existing structures. The arch stresses were calculated from Eq. (33). It is significant that the arch stresses are much lower for the higher dams. This may be attributed to the adoption of a larger margin of safety in the higher structures, and also to the desire for an excess of thickness, near the base, on account of the greater vertical pressures. On account of the uncertainty as to the actual stresses in arch dams, the design should be carefully compared with similar exist- ing structures, and due allowance made for any variation in shape and local conditions which may make for increased stresses. Until our knowledge of the subject has considerably increased, arch Akt. 55] RECOMMENDATIONS FOR DESIGN 159 dams differing materially from those already built must be con- sidered in the nature of experiments. Although dams of the arch type have proved their reliability, on account of the great number now in successful use, without a 300 • 1 ■ '■ ARCH STRESS, IN EXISTING DAMS, IN POUNDS PER SQUARE FOOT. CALCULATED FROM EQUATION 33 2 300 3* • 4 ] !• • i i 5 wo Num items )ers re in Tab er to le XXI V 7« • 8 .6 • 9 12* 14 • • 11 ,15 t t .10 >13 • IS • " I 1 * 23* I..22 • 24 17 • 32. •27 • 29 26 • ,33 25 • 28 • 31 31 §' s Arch Stress Fig. 51. single failure, such structures can hardly be expected to be received at present with equanimity above communities of large size, on account of their apparent stability being so much less than that of the well-knpwn gravity types. Succeeding years will probably 160 ARCH DAMS [Chap. IX bring increased confidence in this perfectly safe and most econom- ical type of masonry dams. 56. Details. The outlines of most of the existing arch dams have not been proportioned according to any standard practice, thus resulting in many types. Most of them have a vertical up- stream face and a constant radius. Mr. Lars R. Jorgensen * has recently shown that, theoretically, the greatest economy of material in an arch is obtained when it is subtended by a central angle, 6, between abutments, of 133° 34' (see Fig. 52.) Fig. 52. This can be proved as follows: At any elevation, the area, A, of any arch slice between abut- ments is, — q«o° '"«'"} From Eq. (33), From which, t = t = V V qr m v- * Transactions, Am. Soc. C. E., Vol. LXXVIII, p. 685. Akt. 56] DETAIIS 161 As p is the constant working stress, and q is a constant at any given elevation, t = k"r a . Also, r m = -, and s is a constant. 2sm 2 Substituting these values of r m and t in the first equation, there results, sin-^ Differentiating this expression with respect to 5, and equating the differential coefficient to zero, there results for a minimum value of A, 5 = 133° 34'. If the cost of form work and similar items were included in the derivation of the most economical central angle, its value would probably not exceed 120 degrees. As a matter of fact, the pecu- liar configuaration of the site usually limits the value of the maxi- mum central angle which can be adopted economically, it often being necessary to follow certain well-defined ridges in order to increase the average elevation of the foundations. In some in- stances, where the canyon narrows rapidly toward the bottom, a strict application of a constant angle would result in less thick- ness near the bottom of the dam than at higher elevations, which is impracticable. Consequently, in most cases, the economical proportions can be found only by trial, adopting successively different values of central angles at various elevations; always being guided by the fact that, unless affected by peculiar condi- tions, a value of about 120° will be most economical. This method will prove tedious, but will be justified in view of the saving in cost which may be obtained. It will be noted that, for a constant central angle in a V- shaped canyon, the arch radius must decrease toward the bottom of the dam. It has been shown already that, for spillway dams having large flood discharges per linear foot of crest, it is not good practice to 162 ARCH DAMS [Chap. IX Fig. 53. allow the falling sheet of water to leave the face of the masonry. For such cases the constant central angle theory will not apply, as it usually results in a practically vertical down-stream face (Fig. 56). For large discharges the down-stream face of an arch dam may be shaped to fit the sheet of falling water for its entire height; then, providing a vertical up-stream face, a varying radius may be adopted to keep the arch stress within the limits desired. In order to reduce the throw of the sheet of water relative to the up-stream face of the dam, the lip of the crest may be made to overhang, as indicated in Fig. 53. The thickness of the top of the dam, of course, should be proportioned to resist ice pressure, if assumed to exist. The ice pressure may be considered as taken by a portion of the arch equal in height to at least twice the thickness of the arch at the elevation of the ice thrust plus the thickness of the ice, and usually considerably more, depending on existing conditions, the quantity of vertical reinforcement, if used, and the location of the nearest horizontal building joint. The theory is identical with that of the distribution of floor-slab concentrations. It should be noted that, in very few of the dams listed in Table XXIV, was it necessary to provide for ice thrust. Steel reinforcement has been used to some extent in arch dams, but in most cases has been insufficient in quantity to affect materi- ally the stiffness or the distribution of stresses. The use of con- tinuous horizontal reinforcement is not to be recommended, as it is impractical to anchor the rods thoroughly into the rock of the abutments, and there would be a tendency to draw the dam away from the abutments due to temperature changes when the reser- voir is empty. In this respect, the conditions in an arch dam are somewhat different from those in a reinforced arch bridge, as the latter is never without load. Horizontal reinforcement, if used, should not extend through the vertical building joints of the structure, such joints being placed at intervals in radial planes to localize cracks and prevent their formation at too great an incli- nation to the line of arch thrust. Vertical reinforcement may be useful to distribute concen- Art. 56] DETAILS 163 trated loads, such as ice thrust, and will serve also to stiffen the relatively thin upper part of the dam if the length is considerable in comparison with the thickness. An arch dam is in reality a long column receiving lateral sup- port only through its connection with the base. It will be noticed that the ratio of curved length to thickness (" ratio of slenderness") at the top of most of the existing dams, is greater than usually allowed in long concrete columns. This is justified on account of the lateral support which the top of the dam receives from the relatively thicker lower portions, and the fact that the arch Fig. 54. stress in the upper part of the dam is much less than that adopted for other parts of the structure. It is good practice to provide a ratio of slenderness, at mid- height, not greater than 25 and, at the top of the dam, a ratio not greater than 75. The ratio at the top may be somewhat in- creased if the thickness increases rapidly toward the lower ele- vations, and the ratio at mid-height is proportionally reduced. This is particularly true if considerable vertical reinforcement is used. In the Salmon Creek Dam (Fig. 56), the ratio at the top is more than 100, but this fact is compensated for by ample 164 ARCH DAMS [Chap. IX thickness at lower elevations, which, at mid-height is less than 10. In order to prevent sliding, the rock surface at the abutments should be excavated to the proper inclination to the line of thrust. In Fig. 54, let W represent the total weight of masonry and other vertical forces between the vertical, radial planes, 5-1 and 4r-3. Let P represent the total thrust of the arch slice between the two horizontal planes passing through the points, 1 and 3. The rock should be excavated to such lines that the resultant, R, of the forces, W and P, will have a maximum inclination, 0, with a normal to the finished rock surface not greater than the angle of repose of masonry on rock. The angles, 0' and 0", represent the vertical and horizontal projections, respectively, of the angle, 6. The principle is identical with that of Art. 25 providing for the resistance of gravity dams to sliding. The number of steps and the depth of excavation to be pro- vided will depend on the character of the rock, and particularly on the inclination of the stratifications. Probable shearing of the rock on a direct line between points, 2 and 6, should be guarded against. 57. Multiple Arch Dams. The theory of arch dams is applica- ble only in a general way to the arched decks of hollow multiple Fig. 55. arch dams, described in Art. 49. T n this type the loading on any arch slice is not uniform, and Eq. (, 53) does not apply. In Fig. 55, it is seen that, for the arch slice, 1 -2-3-1, the unit water pressure at the haunch is greater than at the crown. The percentage in variation of the loading gradually decreases, of course, toward the base of the dam; but, in the upper part of the structure, the non-uniformity of the loading is pronounced. 168-FT.DAM' FOR SALMON CREEK, ALASKA ALASKA GASTINEAU MINING CO. JUNEAU, ALASKA Transactions Am. Soe. C.E., Vol. LXXVIII, Paper No. 1322. To face page 165 Abt. 59] EXAMPLES OF ARCH DAMS 165 Therefore the arches in the upper part of the dam should be circular in horizontal planes, or well reinforced. In other words, the decks of such dams should be designed as arches subjected to non-uniform loads, the theory of which may be found in many treatises on the subject, such as " Principles of Reinforced Concrete Construction " by Turneaure and Maurer. 58. Allowed Stresses. For the larger and thicker arch dams, 1:3:6 cyclopean concrete has been commonly used. In the thinner structures the mixture has been 1 : 2| : 5, and even 1 : 2| : 4§, and in the very thin arch decks of hollow gravity dams 1:2:4 concrete, without plums, is customary. The stresses in a number of existing arch dams have been cal- culated from Eq. (33), and are indicated in Table XXIV and Fig. 51. It is seen that, in four cases, a stress of 56,000 lb. per sq. ft. is exceeded, and in two cases the stresses amount to about 120,000 lb. per sq. ft. These excessive stresses indicate, in a limited way, the factor of safety embodied in the more conservative designs; but they are not in any sense indicative of the stress which should be adopted, for, though the dams have not failed, there is no indication of how close to their ultimate strength they are stressed. Data on the ultimate strength of masonry have been given in Art. 26. Although the total water load to be carried can be de- termined quite accurately, there is much uncertainty in the determination of the induced stresses in the dam, due to such loading in combination with temperature changes and other inde- terminate factors hereinbefore described. For this reason the stress, calculated from Eq. (33), should not exceed from one- eighth to one-twelfth of the ultimate strength of the masonry, depending on the height and importance of the structure, and the probability of destruction of other property and human life in case of failure. In very high dams the arch stresses near the base are reduced considerably in order to provide additional width to distribute the weight of the structure. 59. Examples of Arch Dams. Fig. 56 shows details of the arch dam of the Alaska Gastineau Mining Co., on Salmon Creek, Alaska. This dam indicates in a general way the application of the constant central angle theory discussed in Art. 56, the radius gradually diminishing toward the base. To have kept the central 166 ARCH DAMS [Chap. IX angle constant at all elevations would have resulted in the struc- ture overhanging in places. The 10-ft. triangular piece at the toe was added, in order to distribute more effectually the vertical loads due to the great height of the structure. Fig. 57 indicates the conservative design of the North Crow Dam for the water-supply system of Cheyenne, Wyo. The computed arch stress is only 23,000 lb. per sq. ft., and the ratio of slenderness is only 11 and 35, at mid-height and top, respectively. A 1 : 2j : 4^ concrete mixture was used throughout the dam, and steel reinforcement was placed in the upper 40 ft., as indi- CROSS-SECTION OF DAM AS CONSTRUCTED CROSS-SECTION OF DAM AS COMPUTED Fig. 57.— North Crow Arch Dam. (Eng. Record, Vol. LXVII, p. 149.) cated. A spillway was provided through a channel excavated in the rock at one end of the dam. Fig. 59 is a typical example of thirteen arch dams built by the Public Works Department, New South Wales. These dams are all described in Table XXIV, but they do not include the Barren Jack Creek Dam. In all these dams the concrete was mixed in the following proportions: 4J parts of Portland cement, 11§ parts of sand, 10 parts of " shivers " of f-in. to |-in. gauge, 13 parts of " metal " of l|-in. gauge. Art. 59] EXAMPLES OF ARCH DAMS 167 1 O 168 ARCH DAMS [Chap. IX Except in Lithgow No. 2 Dam, the concrete was cyclopean, con- taining a maximum number of two-man plums. Many of these PLAN T.W.L. 8.6 f SECTION Fig. 59. — Wollongong Arch Dam, New South Wales. (Eng. News, Vol. LXIII, pp. 588-589.) dams are noted for their slenderness, which greatly exceeds that of usual American practice. Art. 59] EXAMPLES OF ARCH DAMS 169 In Fig. 60 is indicated the result of measurements of the deflection of the Barren Jack Creek Dam under variations in pond level and temperature of the air. It is probable that the temper- ature of the water has more influence on deflection than that of Movement,!!! Inches. lard rf CraUO*.05*j'.06*.10'.16!2o:2&*.80:85:4ol45*.5or55!oO* Lines Show Movements of Dam. PLAN Fig. 60. — Barren Jack Creek Dam, Australia. (Eng. Bee, Vol. LXI, p. 664.) the air, and it is unfortunate that it was not also recorded. The deflection is seen to be much less than would be expected for a structure of this type. Fig. 61 shows a plan and section of the arch dam of the Agua Pura Co. of Los Vegas, N. Mex. The dam is quite rigid, the ratio 170 ARCH DAMS [Chap. IX 1 EU'JO *f 7— - Concrete Bubble , Sections ' ate'f Cut-ofl , to Sound ^Impervious Rock ELEVATION Fig. 61. — Arch Dam at Las Vegas, New Mexico. (Eng. News, Vol. LXIV, p. 446.) Art. 59] EXAMPLES OF ARCH DAMS 171 of slenderness at mid-height and top being only 17 and 52, respec- tively. In this dam the radius might have been reduced consid- erably with a reduction in quantity of masonry for the same arch stress, the central angle being, for the proposed extension, only 88° at the top of the dam and much less at lower elevations. For the existing structure, the central angle at the top is only 48° CHAPTER X PREPARATION AND PROTECTION OF THE FOUNDATION 60. General Considerations. A good foundation is of ample strength to withstand the weight of the structure, sufficiently rough to provide ample margin against a sliding failure, tight enough to prevent excessive leakage and uplift, and as clean as possible, if rock, to insure the maximum effectiveness of the bond * between it and the dam. Absolute tightness is difficult, if not impossible, to obtain. It is possible, with first-class rock, to provide against any appreciable leakage or uplift; but, in the design of dams, the engineer has to contend with foundations varying from solid rock, through all grades of stratified and ruptured rock, to alluvial deposits of a very porous nature. Considerable preparation is always necessary in order to pro- vide the requisites of a good foundation. It is probable that more than 90 per cent of all failures of masonry dams has been caused by faulty foundations. It is of the utmost importance, therefore, that this feature of the design should receive the proper amount of attention. It is unfortunate that the designer is not always the builder, as many of the assumptions used in the design will depend on the extent and character of the treatment which the founda- tion receives. In all cases the designer should prepare the speci- fications for the construction of the dam, and preferably have supervision over the work. 61. Rock Foundations. It is not within the province of this book to enter into a discussion of construction problems in connection with the proper preparation of rock foundations. For this, the reader is referred to C. W. Smith's " Construction of Masonry Dams "f and works of like character. There is pointed out, however, in addition to the necessary designing features, the * Bond between rock foundations and the dam, although neglected on account of its unreliability, always exists to a certain extent, and undoubtedly adds considerably to the stability of the structure. t McGraw-Hill Book Co., Inc., 1915. 172 Art. 61] ROCK FOUNDATIONS 173 results which should be obtained from such preparation, in order to justify the adoption of the designing assumptions herein rec- ommended. Surface rock is usually badly weathered, and unsuitable for the support of a dam. It is sometimes necessary to excavate to considerable depths before rock of an acceptable nature is uncov- ered. In the excavation of rock foundations, it is always necessary to take particular care in order that good rock directly beneath the blasting charges is not unnecessarily shattered. It is often specified that the last foot or two of the excavation shall be barred and wedged loose. The proper method will suggest itself to the experienced builder when it is borne in mind that no part of the final foundation should be disturbed from its original position and that no stratifications should be jarred loose. It has been pointed out heretofore that there should be as much resistance to sliding below the surface of the foundation as at other planes. If, therefore, the foundation contains loose horizontal or nearly horizontal stratifications on which there is danger of sliding, the excavation should be deep enough to obtain a " toe hold " on the rock below the dam, in order that some, if not all, of the horizontal loading may be carried to the rock by direct compression through the vertical plane at the toe between the dam and the rock. In order that the masonry of the dam shall have the maximum possible adhesion to the foundation, it is necessary that the final rock surface should be absolutely clean, and the presence of flowing water should be rigidly guarded against. There is nothing better for cleaning rock surfaces than jets of clear water, under consid- erable pressure, and the thorough use of stiff wire brooms. The jet is particularly adaptable to cleaning out vertical seams and pot-holes of considerable depth. Such seams should be well plugged with mortar. For concrete dams it is often specified that the finished foundation shall be covered with a thin coat of rich mortar immediately before the concrete is poured. Springs in the foundation are not usually plugged before the masonry is started, the practice being to allow them to discharge through pipes until a sufficient mass of masonry has been placed to balance any pos- sible uplift from them. The springs are then grouted under pressure. 174 PROTECTION OF THE FOUNDATION [Chap. X It is essential that the plane of contact between the founda- tion and the masonry should be as rough as possible, in order to assist in the resistance of the dam to sliding. In the case of clearly defined horizontal stratifications, it is 'not always possible to obtain a rough surface, and a toe hold on the rock below the dam must be provided, for reasons explained above. Some leakage or seepage through the foundation is to be ex- pected. Aside from the sometimes serious waste of water, leakage is objectionable because it provides a possible means of uplift and, in certain classes of foundations, scour. Foundations, in general, have a tendency to tighten gradually, particularly if silt is present in the water. The exception is in certain classes of limestone foundations, where the water has a solvent action on the rock, and where there is not sufficient silt in the water to plug the leaks thus formed. It must not be supposed that, because the stream in its original state carried large quantities of silt past the dam site, this condition will obtain after the dam is constructed. In large storage reservoirs silt will not reach the dam, even during the greatest floods, until perhaps after a great number of years, when the whole volume of the reservoir has become silted up and its usefulness destroyed. In order to confine the leakage to a reasonable quantity, it is necessary, with poor foundations, to provide a cut-off or artificial impervious barrier under the heel of the dam. For rock founda- tions there are two general types of cut-offs; first, a trench filled with concrete, and second, holes drilled at frequent intervals and grouted under pressure. The first type is much to be preferred if it can be constructed at reasonable cost. Before the use of grouted cut-offs was com- mon, the first type was sometimes carried, in exceptional cases, to depths of 50 ft. or more. Its advantages are twofold, in that it not only provides perfect inspection of the vertical area to be improved, but is a more tangible and sure method of obtaining the desired end. For the excavation of the trench, even more care must be taken not to shatter the surrounding rock, particularly at the bottom. For the cut-off trenches of a number of important dams, where the excavation has been difficult, holes have been drilled a short distance apart to form planes of weakness to which the sides of the trench break without serious disturbance outside of the limits desired. The excavation has sometimes been made Art. 61] ROCK FOUNDATIONS 175 with a channeling machine, but this method, of course, is very expensive and hardly justified for any case, unless the rock is easily cut. The concrete for the cut-off, in rock foundations, is usually built under the same specifications as for the rest of the dam, and provided with the same system of joints. Cut-offs of the second type have been more common of late years. Typical examples may be found at the Estacada Dam, in Oregon; * the Lahontan Dam, of the Truckee-Carson project, U. S. Reclamation Service; f the Horseshoe Falls Dam on Bow River; J and the Arrowrock Dam, of the U. S. Reclamation Service.* A method of grouting which seems to have given the best results consists in drilling a primary series of holes in a row under, the heel of the dam, from 10 to 15 ft. apart on centers, and of a depth depending on the nature of the foundation and the head of water to be sustained. The holes are usually about 3 in. in diameter. The primary holes are first subjected to water pressure, prefer- ably from a tank placed at the same height as the crest of the dam, or a little above it, and the rate of leakage from each hole recorded. The flow of water also serves to clean out the earth seams in the vicinity of the holes in preparation for the grout which is to follow. After the primary series of holes is grouted, a second series consisting of an equal number of intermediate holes in the same row is then drilled, tested, and grouted. If necessary, a third series is drilled and treated, thus reducing the spacing to a quarter of that for the first series. The result of a test of any hole is con- sidered an indication of the relative tightness of the foundation between the two adjacent holes previously grouted. The process, therefore, is continued until the tests indicate that the leakage has been reduced to a satisfactory extent. The upper ends of all holes to be grouted should be provided with threaded pipes with which to make connection to the grout- ing machine. These pipes must be anchored or weighted to pre- vent a blow-out during the process of grouting. This is sometimes * For description see paper by Harold A. Rands, Transactions, Am. Soc. C. E., Vol. LXXVIII, p. 447. f For description see article by D. W. Cole, Engineering News, Apr. 3, 1913. t For description see article by H. S. Johnson, Engineering Record, Dec. 12, 1914. 176 PROTECTION OF THE FOUNDATION [Chap. X done by cementing the connecting pipes into the drilled holes, or placing them in a concrete cut-off carried far enough into the rock to provide ample grip. The drilling, in the latter case, is usually- done through the pipes. When it is desired that the drilling shall not interfere with the erection of masonry, the pipes may be car- ried up along with the masonry and the operations of drilling, testing, and grouting conducted from whatever elevation the masonry has reached. If it is apparent from the borings that there is an open seam close to the surface, care should be taken that there is sufficient weight above the seam to balance the grouting pressure. Part of the dam may be previously built to assist in this respect. The initial grouting pressure, for each hole, should be that which is necessary to force the grout slowly into the hole. The pressure is raised gradually, as the hole tightens, so as to disturb the natural formation as little as possible. The grout usually consists of a mixture of neat cement and water, of proportions varying to suit the character of the founda- tion. For porous rock, with fine seams, a mixture as thin as 1 of cement to 8 or 10 of water has been found satisfactory. Where the seams are large, and other voids exist, a thicker mixture must be used, gradually changing to a thinner mixture as the hole tight- ens to refusal. The grouting operations are usually started from one or two ends of the site, the holes being treated successively. Each hole should be capped, if necessary, as soon as it has been grouted, in order that the grout to be forced into the next hole will not flow back through the completed hole and be wasted, instead of passing on to the ungrouted portion of the foundation. If grout issues freely from an untreated hole, indicating an open seam between it and the hole being grouted, the untreated hole may be capped and the grouting operation considered as serving the two holes. The object is to supply grout to each hole in sufficient quantities and at the desired pressure, and it is immaterial whether it is supplied at the top of the hole or by way of an open seam from another hole. Should the grout from an untreated hole issue sluggishly, indicating an indirect or only partly free connection, the hole should be capped, but should receive its share of treatment in due course. When thick grout is being used, care must be taken that grout Art. 61] ROCK FOUNDATIONS 177 from one hole does not partly fill an adjoining hole and set before the latter can be treated. When thick, grout sets much more quickly than when thin. It is sometimes advisable to provide sufficient shifts of men to conduct the grouting operations con- tinuously. All holes should be gone over the second time after the grout has set. If the grout is thin, considerable settlement in the hole will be observed, and it is often possible to inject an additional quantity. No one method of grouting will apply for more than one site; in fact it is usually found advisable, for each case, to change the adopted method several times during a course of treatment, due to experience constantly being gained as the work progresses. The following is an abstract from the previously mentioned article on the grouting of the Lahontan foundation. The dam is an earth embankment 124 ft. high with a core-wall, but the process of grouting would apply equally well to a masonry dam. The grouted portion of the foundation consists principally of a formation resembling red sandstone, varying from solid rock to a tough red clay or even an unstable sandy clay, containing many intermixtures and stratifications. In the concrete cut-off trench (Fig. 62a), which was 30 ft. deep, were previously set 5-in. well casings in two rows 2 ft. apart. The casings were 3 ft. apart in each row, and were staggered, so that there was a casing for every 18 in. of length of cut-off, as indicated in Fig. 62. Core drillings, with 2f in. outside diameter bits, were made through the well casings and extended at least 30 ft. below the bottom of the concrete cut-off. Some of the holes were carried 70 ft., and the average was 40 ft., below the trench, or 70 ft. below the river bottom. The work was divided into two sections, on account of the necessity of reserving half the width of the site for the diversion of the stream. In the first section, alternate holes, 6 ft. apart in the upper row, were drilled, tested, and grouted; the intention being to complete the upper row, and then the lower row, in successive steps. Those of the first series were designated " primary holes." Testing was done under a head of 127 ft. The rate of flow through each hole was recorded. After the holes of the first series were grouted, the interrnediate, 178 PROTECTION OP THE FOUNDATION [Chap. X or " secondary holes," in the upper row, were drilled, tested, and grouted, thus completing the grouting in the upper row. Fig. 62 indicates the results of all tests, and the number of bags of cement used in each hole. It will be noticed that the grouting of 1 5 9 13 17 21 oooo»ooo» o o o_o - Section 1 29 33 37 41 45 19 63 57 .OOOOOOOOOO ooo o 61 65 69 > • O O O O - Section 2 - 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 187 141 OOOOOOODOO»OoOO«OOOOOOoOoOOOO« #SBass •••• a. .00- ?80- |60- «j 40 = a20- « o o- ^100- g sol- as' o 40- ■3 go- to 01= -% $ », ^ «> ii- ,4 y 4 T nr -2 -- — J. =fS?- z~- - — ■=- - 1 *" ""■"pr- 1 — / \ X 2n J iV ti irj 4 I \ V << A f \ .& 7 \ t g c [<* 1. V is w* - ^lBt & 2i d jTe :ti ir ^^-^ -100 80 60 40 30 -100 80 60 40 20 Fig. 62. — Gallons of Leakage Observed and Sacks of Cement Necessary to Grout Different Borings in Lahontan Dam Foundations. (Eng. Record, Vol. LXVII, p. 340.) the " primary holes " reduced the average leakage per hole from 33.1 to 5.5 gal. per min. It was anticipated that the leakage would be further reduced by the grouting of the " secondary holes," so that very few of the holes in the lower row were drilled. It is Art. 61] ROCK FOUNDATIONS 179 claimed that the quantity of cement which the holes took was not indicative of their relative tightness, on account of caving of the holes resulting in varying volumes to be filled, and the escape of grout into adjoining well casings. The results obtained for the first section of the dam led to an alteration in the procedure for the second section. In the latter the holes were drilled, tested and grouted in the following order: Fia. 62a. — Grout Pipes in Cut-off Trench of Lahontan Dam. (Eng. Record, Vol. LXVII, p. 340.) " Primary holes " Nos. 78, 86, 94, etc., 12 ft. centers throughout; " Secondary holes," Nos. 74, 82, 90, etc., 12 ft. centers throughout; "1st Tertiary holes," Nos. 76, 84, 92, etc., 12-ft. centers through- out; "2d Tertiary holes," Nos. 80, 88, 96, etc., 12-ft. centers throughout. The average leakage per hole for the second tertiary holes was 6.4 gal. per min., indicating the necessity of grouting very few holes in the second row. All grouting was done with a duplex-cylinder, air-stirring, Caniff 180 PROTECTION OF THE FOUNDATION [Chap. X grouting machine; a pressure of 25 lb. being used at the begin- ning of each operation, increasing gradually until finally the grout was driven home at a pressure of 100 lb. per sq. in. A mixture of 1 part of cement to 7 or 8 parts of water was found most desir- able, although the mixture was thickened considerably when the flow appeared to be too free. Uplif t pressure from head-water cannot be eliminated entirely unless the cut-off is absolutely tight. With a deep, well-built, masonry cut-off or a grouted cut-off, constructed under favorable conditions, the uplift may be negligible. Where it is thought that the effectiveness of the cut-off is not thoroughly reliable, and in other cases where unusual precautions are desirable, it is the practice to provide drains in the foundation to facilitate the escape of whatever water finds its way past the cut-off. For this purpose holes may be drilled at intervals across the site in a row just below the cut-off and provided with a free con- nection to tail-water. The holes should be drilled to or a little above the elevation of the bottom of the cut-off. For grouted foundations the holes should be drilled after all grouting operations have ceased. The holes are usually 10 or 15 ft. apart; but the spacing de- pends greatly on local conditions. Other things being the same, the proper spacing should vary directly as the depth of the holes, and directly as the perviousness of the foundation relative to that of the cut-off. The holes also serve the purpose of indicating the extent of leakage past the cut-off, and, for this reason, should discharge a little above tail-water, in order that the quantity of flow may be observed. Box drains, open-jointed pipe, and other types of drains, used without drilled holes, have often been placed between rock foun- dations and the dam, and may be effective in eliminating uplift at that elevation; but they are not adaptable to rock foundations containing nearly horizontal open seams near the surface, as they have no direct connection with such seams, and may be sepa- rated therefrom by a horizontal layer of very impervious rock. For the usual spillway section, the smooth thin sheet of swiftly moving water issuing horizontally from the bucket at the toe of the dam, may, under certain conditions, retain a uniform depth Art. 62] EARTH FOUNDATIONS 181 ;uid velocity until, at a point, perhaps, considerably remote from the dam, it will suddenly increase in depth to tail-water level, forming a standing wave, as indicated in Fig. 4. and as described in the latter part of Art. 14. Under such conditions the river bed below the dam will be subjected to a velocity much higher than normal. The impact from swiftly moving water is known to be very destructive to rough, soft, rock surfaces, particularly if there are seams into winch the jet may enter. In the latter case the velocity head may be converted into a pressure head capable of lifting large masses of rock. In all cases, where the rock surface is known to be soft or stratified, an apron of concrete, or other suitable material, should be provided to afford protection to the foundation at the toe of the dam. However, as explained above, there may be danger, under certain conditions, of the water not returning to normal tail-water level until after it has passed the end of the apron. The likelihood of undermining the toe of the apron, in such cases, is evident, and systems of baffles forming a partial obstruction to the flow, have sometimes been built to insure the return of the sheet of water to tail-water level before it has reached the end of the apron. The concrete baffles used for the Gatun spillway dam are indi- cated in Figs. (3 and 7. Dining large flows, the jet, when it reaches the baffles, is about 6 ft. thick, and flows with a velocity of about 60 ft, per sec. The baffles are 9 ft. high. The effect of the baffles, as indicated in Fig. 7, is to increase the depth to tail-water level or about 20 ft. deep, and to reduce the velocity to IS ft, per sec. The baffles piers are heavily reinforced, and are protected on their up-stream faces with thick iron castings. Where the profile of a spillway dam site is not horizontal, the water passing the ends may flow along the toe of the dam toward the middle with high velocity and destructive force. Fig. 63 is a view of the down-stream face of the Estacada Dam. It will be noted that training walls are provided to conduct the water to a safe distance. 62. Earth Foundations. Masonry dams on earth * founda- tions are numerous; but their use has been practically limited to structures not more than 50 ft. high for good hardpan and 30 ft. for less resisting earth. This limitation in height may be attrib- * Earth, as here used, may be considered as all kinds of material riot usually classed :is bed-rock. 182 PROTECTION OF THE FOUNDATION [Chap. X g bO c3 H a ? (3 Art. 62] EARTH FOUNDATIONS 183 uted to the fact that the treatment of earth foundations, to prevent erosion and excessive seepage, requires an expenditure far in excess of that necessary for the foundations of dams on rock. In fact, the cost of foundation treatment for dams on earth is often the major part of the total cost of the structure. Consequently, for moderate and high dams it will be found best to adopt another type of structure, or change the site. There have been few precedents for dams higher than noted above, although struc- turally there would seem to be no reason for a limit to the height, provided sufficient funds are available to meet the unusual expense. It is essential that there be no excessive, unequal settlement of the dam, as the tightness of the structure is dependent on the absence of settlement cracks. The preparation of the foundation for a dam on earth must be made with four objects in view: a. To prevent excessive seepage under the dam; 6. To prevent scouring by the water passing over the dam; c. To provide ample bearing strength; d. To prevent sliding. Excessive seepage, or Underflow, through the foundation is objectionable, not only on account of the waste of impounded water, but principally because of the danger of movement of the particles of the foundation due to its erosive power. Where practicable, seepage should be prevented by carrying a tight cut-off, under the heel of the dam, to an impermeable forma- tion. The cut-off may consist of a concrete diaphragm or a series of grouted holes, as described for rock foundations; interlocking steel sheet-piling, or tongued and grooved wood sheet-piling. Sheet-piling, unless driven under favorable conditions and with extreme care, is very apt to leak badly; and this is a condition to be avoided, as recent experiments have indicated that very slight leakage is sufficient to destroy its effectiveness. Wooden sheet- piling for cut-offs should never be used where the foundation con- tains boulders which are large enough to cause the piles to buckle or deflect. Even steel sheet-piling has been made useless under very heavy driving. Figs. 39 to 42, inclusive, show details of the Mathis Dike Dam, of the Tallulah Falls water-power development. The dam is built on sand mixed with very little clay and decomposed mica 184 PROTECTION OF THE FOUNDATION [Chap. X schist. A concrete cut-off wall was provided, extending from 30 to 57 ft. to rock, except at one end where, as indicated in Fig. 39, holes were driven to rock and effectually grouted under a pressure of 70 to 85 lb. after the reservoir was filled. Before grouting, these holes were opened to the lower side of the dam, and were found to leak from 0.5 to 12.0 gal. per min. The cut-off wall is ..composed of 1:3:6: cyclopean concrete, with tongued and grooved vertical contraction joints. The upper 15 to 20 ft. of the wall is reinforced in order to prevent shrinkage cracks. The sheet-piling indicated in the illustration was an experiment, but, owing to the nature of the foundation, was abandoned after 50 ft. had been driven. Where it is impracticable to carry the cut-off to an impermeable formation, undermining may be prevented by providing a path of enforced percolation of sufficient length. The velocity of the underflow uas been found by experiments to be directly propor- tional to the hydrostatic head * and inversely proportional to the length of the path of percolation. Its erosive force depends on its velocity. The required length of path of enforced percolation, therefore, is a direct function of the head on the dam, and depends on the nature of the material in the foundation. W. B. Bligh f gives the following empirical formula: l = Ch n> (44) where Z = the length of the path of enforced percolation in feet; see Fig. 64, in which l = h+l2+h+h+la+h; h n = the net head on the dam, in feet; and C = a coefficient which depends on the character of the material in the foundation. For C, Mr. Bligh recommends the following values, which are based on the dimensions of a number of existing dams on earth foundations: C= 18 for mud or silt; C= 15 for fine micaceous sand; * Experiments by D'Arcy, Hagen, Hazen, and others, indicate that the flow of water through fine sand and gravel is similar to the flow through capillary tubes, the velocity being directly proportional to the first power of the head. t"Dams, Barrages, and Weirs on Porous Foundations," Engineering News, Dec. 29, 1910. Art. 62] EARTH FOUNDATIONS 185 C = 12 for coarse-grained sand ; C= 9 for a mixture of sand and gravel; C = 5 to 9 for clay, shale, or a mixture of sand, gravel, and boulders. M i Mafe ---^- o 03 w a o ■■3 ■a 11 186 PROTECTION OF THE FOUNDATION [Chap. X Care should be exercised in the application of these coefficients, as it is impossible to convey an exact description of the character- istics of the foundations on which they are based. Therefore, ample margin should be allowed, unless careful comparison has been made between the materials in the foundation and those in existing structures of like nature. It has been proved experimentally that the length of the path of percolation, as affecting uplift pressure and erosive force, is not the shortest distance between head- and tail-water, but the length of the actual plane of contact between the structure and the earth, including all cut-offs, provided the cut-offs are not closer together than twice their depth. In other words, the effective length of the path of percolation, as indicated in Fig. 64, is the summation of the distances, h, to h, inclusive. The proper length of the path of enforced percolation may be obtained by providing a down-stream apron, an up-stream apron, one or more cut-offs, or a combination of these parts, as indicated in the accompanying illustrations. Many types and combina- tions have been proposed and constructed. There seem to be no standards in this respect. The choice will depend, to a large extent, on local conditions. A single masonry cut-off at the heel is commonly provided for good earth foundations; but it is often found advisable to substitute an up-stream or down-stream masonry apron, which is theoretically as efficient as a cut-off of half its length. The Jamrao and Barra type of weirs, indicated in Figs. 65 and 66, were designed for localities where the foundations are com- posed of very pervious material, requiring a long path of enforced percolation. Fig. 67 is a section of a portion of the Granite Reef Dam of the Salt River Project. The dam is founded on a formation of gravel and boulders. The coefficient, C, from Eq. (44), was only 4.2, the length of the path of percolation being measured on the under surface of the concrete from the river bed at the heel to the first drain hole in the apron. When the dam was first used, con- siderable water passed under the cut-off and out through the drains in the apron. Fortunately, this flow was soon stopped by the large quantity of silt which was carried by the river and deposited at the heel of the dam, forming an increased and sufficient length of path of percolation. Abt. 62] EARTH FOUNDATIONS 187 I T I F-t ' A.. ± .a Ph 03 1 5 £ O Ba l-H a 4 -#*! a 188 PROTECTION OF THE FOUNDATION [Chap. X In all classes of earth foundations, the river bed below the dam must be protected from the wash of the water passing over the crest, in order to prevent undermining the structure, from this The object is to provide, as far as possible, a means for cause. the stream to regain its normal velocity, corresponding to the Fig. 67. — Granite Reef Dam, Salt River Project, Arizona. ("The Design and Construction of Dams," Wegmann.) flow, before the end of such protection is reached. It is common practice to provide a concrete or rock-filled timber crib apron for a short distance below the dam, the end of the apron being protected by an extension of rip-rap, as indicated in the accompanying illustrations. The lower end of the apron is often further pro- Fig. 68.— Power Dam on Au Sable River. (Eng. Record, Vol. LXVI, p. 247.) tected by a vertical diaphragm of concrete or sheet-piling, which serves to retain the foundation, under the apron, if the rip-rap is washed away. Similar protection has been used at the toe of the dam to provide for a possible failure of the apron. The usual type of spillway section, passing large flows, is not adaptable to dams on earth foundations unless the tail-water has Abt. 62] EARTH FOUNDATIONS 189 sufficient depth to break the force of the falling water,* or unless baffles are provided, as described in the last part of Art. 61. A type of apron frequently used is indicated in Fig. 68. The water is dropped, in one or more stages (two in this instance), into pools of sufficient depth to destroy the velocity. In case the bed of the river is higher at the ends than at the middle, the water passing over each end, if not taken care of, will flow parallel to the dam toward the main channel. For steep slopes, this flow may acquire velocities sufficiently great to scour the foundation at the toe of the dam or the end of the apron. This condition should be avoided by the construction of stone fill, masonry, or other suitable training dikes at intervals extending from the dam to a point down stream far enough from the dam to obviate the possibility of damage to that structure. Sometimes the training dikes are supplemented by a system of canals, parallel to the river, in order to provide a gradual descent, f Such con- struction is obviously expensive, particularly for large flows, and often necessitates the limitation of the length of spillway to the width of the level portion of the river bed. Proper unit compressive stress on the foundation is usually obtained by spreading the footing of the dam, although supporting piles have been used in some cases. Many types of wood and con- crete piles have been used for this purpose. The subject of piles is too lengthy for proper treatment here, and the reader is referred to one of the many books on foundations of that type, such as " Foundations of Bridges and Buildings " by Jacoby and Davis. J The weight of the hollow Mathis Dike Dam was distributed over the foundation by a mattress covering the entire base, as indicated in Fig. 41. The weight of solid dams may be distributed through the aprons, which are often reinforced for that purpose. Where sufficient length of enforced percolation is provided by a cut-off at the heel of the dam, the base is usually drained, to pre- vent the possibility of uplift; unless, of course, the weight of the dam is great enough to sustain the uplift pressure, as hereafter discussed. The best type of drain for this purpose consists of an inverted filter surrounding open tile pipe, although many other * See Art. 14. fSee "The Laguna Dam," Engineering News, Feb. 9, 1905, and Feb. 27, 1908. J McGraw-Hill Book Co., 1914. 190 PROTECTION OF THE FOUNDATION [Chap. X types have been used. The main drain should be placed imme- diately below the cut-off, at the heel of the dam, as shown in Fig. 70. Drainage is often provided, even though the cut-off extends to impervious material. Where it is possible to reach impervious material, it is common practice to place the only cut-off at the heel of the dam, although lower rows of auxiliary sheet-piling and even concrete diaphragms are sometimes provided, in order to prevent undermining from water passing over the crest, in view of a possible failure of the apron or apron extension. Examples of such auxiliary piling and diaphragms are indicated in Figs. 70 and 67, respectively. In such cases they should be well perforated or drained, in order to =jSjTi5i , 5 ' >!< 3'5^>i< £ 8 or over JLU .. Fig. 69. — Diversion Dam of the Rio Grande Project, New Mexico. ("Irri- gation Practice and Engineering," Vol. Ill, Etcheverry.) hold back no leakage which may pass the upper cut-off and main drain. Intermediate drains are also desirable, if the area of the base of the dam or apron is large. In brief, to prevent uplift on the base of the dam, the cut-off at the heel should be as tight as possible, and the foundation below the cut-off as pervious as pos- sible. The Rio Grande Dam was drained by providing a mattress of well-rammed broken rock under the entire base and apron. Leak- age is discharged through the pipe over the lower row of sheet- piling indicated in Fig. 69. The mattress of the Mathis Dike Dam was well perforated. Fig. 70 is a section of the spillway of the Coon Rapids power development on the Mississippi River. This dam is built on a Art. 62] EARTH FOUNDATIONS 191 deep bed of glacial drift, and is carried on a pile foundation. The sheet-piling at the heel was driven to penetrate at least 5 ft. into impervious material, which varied from 2 to 25 ft. below the sur- face. The filter, drain, and tunnel are intended to provide for the escape of any flow which may pass the heel, thus relieving the dam from upward pressure. A similar row of sheet-piling was Operating Bridge -$' Z 7 *i E1.84S.5 vrvrfcacnaroKDrrcutnEroa? W ± -BI- SECTION B-B 'e ;'.,»>■ o i ■„ " •i r' . ■* . :.ks'lO>- .K : '. ; ':;7-'':V;.?''- .■"■' ■ ■■-S;ev SECTION C-C CROSS-SECTION OF SPILLWAY SECTION Fig. 70. — Coon Rapids Dam on the Mississippi River. Vol. LXXVII, p. 118.) (Eng. News, driven at the toe of the dam, which, in the author's opinion, should have been perforated or drained. The sheet-piling at the end of the apron was carried down 8 ft., is intended to prevent the scour from undermining the apron, and is drained as indicated. Should the base of the dam or the apron form part of the required path of enforced percolation, as indicated in Figs. 64 to 67, inclusive, drainage, of course, may not be allowed, and uplift pressure must be provided for. 192 PROTECTION OF THE FOUNDATION [Chap. X The pressure head of the underflow, for earth foundations, is usually considered to vary uniformly, along the path of percola- tion, from full head-water pressure at the up-stream end to full tail-water pressure at the down-stream end. The hydraulic gradient may be drawn, as in Fig. 64, by considering the pressure head, K, at any point, A, on the path of percolation, to be equal to the vertical distance, h h , from that point to head-water level, less the friction loss, h /t between the up-stream end of the path of percolation, and the point, or, s Ku = ilh — "/■ The friction loss, h f , is proportional to the relative distance through which the water has traveled and the net head, hn, on the dam, or, , _, h+h+k+k+k+h '~ "h + h + h + h + h + h' The uplift pressure, p u , is, for earth foundations, considered as effective over the entire area, therefore, p u = W2h u = W2(h h —h f ), or, The uplift pressure, p u , under the apron must be balanced by the weight of the apron and the weight of water on the apron. To allow a factor of safety, the thickness of the apron is usually made at least 30 per cent greater than theoretically necessary. It was shown in Art. 14 that tail-water may not always be counted on to assist in balancing uplift, in which case the presence of tail- water should also be neglected in Eq. (45). The uplift pressure will vary with the changing relative elevations of head- and tail- water, and the weight of the apron and dam should be proportioned for the severest condition. The thickness of the apron may be reduced, if properly anchored to bearing piles. When the friction of the foundation is not considered suf- ficient to prevent sliding, piles may be used, as in Fig. 69, although Art. 62] EARTH FOUNDATIONS 193 it is obvious that vertical piles cannot withstand lateral pressure without deflecting to some extent. Therefore some movement of the dam may be expected, depending on the number of piles pro- vided. For this reason, some of the piles are often inclined in the direction of the resultant pressure. In the Mathis Dike Dam, Fig. 40, the longitudinal ribs, at the toe and the middle of the base, are intended to assist in preventing the dam from sliding. In some cases increased resistance to sliding has been obtained by thoroughly anchoring the dam to a deep cut-off wall at the heel of the structure, CHAPTER XI FLOOD FLOWS 63. General Considerations. The magnitude of the prob- able maximum flood to be expected at a dam site has a direct bearing on the safety of the structure. This is particularly true if, as is often the case, the masonry dam is supplemented at one or both ends by an earthen embankment. Although, for a ma- sonry dam, a flood somewhat greater than that for which the structure has been designed may, because of the margin of safety, be passed without failure or even damage, an earthen embankment will almost invariably be breached soon after the water has begun to flow over the top. As the magnitude of a flood is affected by an infinite variety of conditions, the chances of the maximum possible flood from a given catchment area occurring within the life of the structure is infinitely remote. It therefore remains to adopt, as the design- ing condition, a flood which is reasonably certain of not being exceeded; it being remembered that a flood which would be con- sidered " reasonable " for one dam may be unreasonably large or small for another dam having exactly the same size and character- istics of catchment area. The probability of loss of life and property, of interest on the investment, of the use for which the dam is intended, and other factors, must be considered in deciding on the flood to be accommodated. In other words, as it is imprac- ticable to design any, except the most important of dams, for the maximum possible flood, the problem is resolved into a considera- tion of how much chance it is reasonable to take. It is safe to say that very few dams in America have been designed to accommo- date a flood proportionally as great as that which occurred on the Miami water-shed at Dayton, Ohio, in 1913 (see No. 72 of Fig. 71), which was one of the largest floods this country has ever known. The maximum flood to be accommodated from a given catch- ment area may be determined approximately by the following 194 Art. 64] HIGH-WATER MARKS 195 methods, it seldom, if ever, being possible to make a direct dis- charge measurement on a record flood at the time of its occurrence: 1. By a study of record high-water marks on the stream in question; 2. By comparison with known record floods from other catch- ment areas of about the same size and characteristics. 64. High-water Marks. Authentic Federal and State Gov- ernment records of high-water, extending over long periods, may be obtained for many streams. Such records are also often avail- able from mill operators and the officials of municipalities. In the great majority of cases, however, the determination of the ele- vation of record high-water must be from the observations and traditions of residents, and from physical indications on the banks of the stream. Observations and traditions of residents should be regarded with caution. Individual reports of untrained observers are sub- ject to great error and, strange to say, are often of doubtful verac- ity, as the desire to report a high-water a little higher than that reported by a neighbor is often, among certain classes, greater than the love of the truth. Unfortunately, also, reports are sometimes biased by a desire to give an impression of great or small maximum floods, whichever, in the opinion of the observer, will better serve his interests. However, credence may be given when a number of observations closely agree and are referred to definite objects, such as sills of doors and windows or nails driven for reference. Confirmation may be obtained from the elevation of deposi- tions of brush, logs, or alluvial matter, scars from floating bodies on banks and large trees, and whatever other indications of high- water may be discovered. High-water, in an alluvial valley which has been formed from the sediment deposited from floods, is, of course, always higher than the surface of the valley. The elevation of record high-water having been fixed, there are three methods by which an estimate of the corresponding dis- charge can be made: 1. By determining the corresponding head on a dam which existed at the time of such high-water, from which the discharge over that structure can be computed from one of the well-known weir formulae.* * See U. S. Geol. Survey Paper No. 200 by R. E. Horton. § S§?S8S 000,000 600,000 1700,000 600,000 600,000 400,000 — - 800.000 200,000 - 100,000 90,000 70,000 60,000 60,000 a § u <0 y* 'j S Mc /Mj 40,000 y* Uh •^ *a. / 60,000 3 *S .Mi &* '»n * 2 M (H -2 jr" a ) ** 'on "116 wc p "V^ II -p- ^ • j0 Mo „C2 27° < t 10,000 a ooo o> ^r 33 oil •' ,36 OC7 •10 8.000 ,51 / ?,000 100 *5C ll" 0,000 {^000 113 o 3 4ooo ^- 1 n ■"Sii 59 8000 • 32 8,000 1,000 4 = IW! r 1 8 s %. 3 S g O C o 5 i" § § II ■*« ,« •* » » *- oo ofe> S 8 tf 8" S" g" S 88 — — 900,000 . ,'WV > 113* *00 — 000,000 - .._ - - ~ - ^ ^ - _•' s lilt - - — i - — — — — -- — ._._ — loo.ooo ' < ?v ^li ? SI • < IS 5 • ST SB — - ~_ — - - _ — — J > 1 — _i y » s ?/^- • *5 S iS\ °;s l /\ %/i\ 1WO "■ m> 1 ;« ir« u i H .1 .>' c U>6 c.1* 2S S5 1 ; "Tlso" ~° _'a> - — - - - & IS b 3 S 41 13 If. • M nxlin mi low oM axlin 11U aiou r j ver be flow -- - 7,000 1 1 | UNUSL AL F -OOD DISC HARGES UN Ye ) ST/ iTES RIVERS 2 000 i* bin 5qi are] Bles J, 000 1 fro i. 71 ■ 1' K 1 ^J f ' § i !§ r t? 8 s r of a 5f £ 1 s s s To fa ce 11 ■page o o 196 Abt. 65] COMPARISON WITH OTHER RIVERS % TABLE XXV Unusual Flood Discharges, United States Rivers * 197 Refer- River. Place Square Miles of Discharge, in ence, t Drainage Area. Second-feet. 1 Black Warrior Tuscaloosa, Ala. 4,900 141,000 2 Salt McDowell, Ariz. 6,260 138,000 3 Malibu Creek Calabasas, Cal. 97 6,800 4 Mojare Victorville, Cal. 400 13,400 5 Sacramento Jellys Ferry, Cal. 9,300 254,000 6 McCloud Gregory, Cal. 608 41,000 7 Stony Creek Fruto, Cal. 601 29,300 S Feather Oroville, Cal. 3,640 187,000 9 Yuba Smartville, Cal. 1,220 111,000 10 Bear Vantrent (above Wheat- land), Cal. 263 25,800 11 American Fairoaks, Cal. 1,910 105,000 12 Putah Creek Winters, Cal. 805 30,000 13 Kings Sanger, Cal. 1,740 43,900 14 Tuolumne La Grange, Cal. 1,500 52,000 15 Stanislaus Knights Ferry, Cal. 935 57,200 16 Connecticut Hartford, Conn. 10,200 205,000 17 Requonnock Conn. 25 4,000 t 18 East Branch Del. 920 72,000 t 19 Toccoa Blue Ridge, Ga. 231 12,300 20 Broad (Ga.) Carlton, Ga. 762 47,200 21 Oconee Greensboro, Ga. 1,100 68,200 22 Chattahoochee Oakdale, Ga. 1,560 48,800 23 Chattahoochee West Point, Ga. 3,300 88,600 24 Etowah Canton, Ga. 604 19,300 25 Rhine Macon, Ga. 2,570 96,500 t 26 Savannah Augusta, Ga. 7,300 310,000 t 27 Carrabasset North Anson, Maine 340 13,700 28 Androscoggin Rumford Falls, Maine 2,090 55,500 29 Kennebec Waterville, Maine 4,270 157,000 t 30 Piscataquis Foxcroft, Maine 286 22,200 t 31 Westfield Mass. 356 53,000 t 32 Fomer Above Reservoir, Holyoke Mass. 13 2,840 t 33 Nashua Mass. 109 11,400 t 34 Merrimac Lawrence, Mass 4,640 90,000 t 35 Patapsco Woodstock, Md. 251 11,000 36 Potomac Point of Rocks, Md. 9,650 219,000 37 Gunpowder Md. 302 25,000 t 38 Lake Roland Near Baltimore, Md. 39 9,000 t 39 Potomac Great Falls, Md. 11,500 470,000 t 40 Rock Creek D. C. 78 9,800 t 41 St. Mary Near Babb, Mont. 177 7,980 42 Tuckaseegee Bryson, N. C. G62 38,700 t * Compiled principally from Fuller's " Flood Flows," Transactions, Am.Soc.C.E., Vol. LXXVII. t See Fig. 71. X Maximum flows. Other discharges are twenty-four hour average flows. 198 FLOOD FLOWS TABLE XXV— Continued [Chap. XI Refer- Square Miles of Discharge, in ence, t River. Place. Drainage Area.- Second-feet. 43 Gadkin Salisbury, N. C. 3,400 130,000 t 44 Catawba Morganton, N. C. 758 32,200 45 Catawba Catawba, N. C. 1,530 61,000 46 Raritan Bound Brook, N. J. 800 28,500 47 Delaware Riegelsville, N. J. 6,430 177,000 48 Delaware Stockton, N. J. 6,850 . 255,000 t 49 Raritan Bound Brook, N. J. 879 52,000 X 50 Ramapo N.J. 118 12,500 t 51 Passaic Dundee Dam, N. J. 820 35,000 J 52 Delaware Port Jervis, N. Y. 3,250 108,000 53 Delaware, W. B. Hancock, N. Y. 680 33,700 54 Buffalo Cr. N. Y. 420 23,000 t 55 Chemung Elmira, N. Y. 2,050 138,000 t 56 Nine Mile Cr. Stittville, N. Y. 63 7,820 t 57 Sawkill 45 m. above mouth, X. Y. 35 8,000 t 58 Six Mile Cr. Ithaca, N. Y. 46 8,500 X 59 Traut Br. Brooksport, N. Y. 25 3,950 t 60 Hudson Mechanicsville, N. Y. 4,500 108,000 61 West Canada Cr. Twin Rock Br., N. Y. 364 34,300 62 East Canada Cr. Dolgeville, N. Y. 256 12,100 63 Catskill Woodstock, N. Y. 210 21,000 t 64 Croton Croton Dam, N. Y. 339 30,000 t 65 Esopua Saugerties, N. Y. 417 55,000 t 66 Schoharie N. Y. 930 49,600 J 67 Salmon Pulaski, N. Y. 260 10,500 68 Genesee Rochester, N. Y. 2,360 50,000 69 Olentangy Columbus, Ohio 520 51,000 70 Upper Scioto Waterworks Dam, Ohio 1,030 68,000 71 Lower Scioto Columbus, Ohio 1,570 119,000 72 Miami Dayton, Ohio 2,450 246,000 t 73 Rogue Tolo, Ore. 2,020 48,300 74 Umpqua, S. F. Brockway, Ore. 1,800 70,700 75 Willamette XI. F. Jasper, Ore. 1,450 122,000 76 Willamette Albany, Ore. 4,860 188,000 77 Willamette, Coast Fork Goshen, Ore. 690 31,300 78 McKenzie, Springfield, Ore. 960 37,900 79 Yamhill Sheridan, Ore. 290 18,100 80 Kiskiminetis Avonmore, Pa. 1,750 77,700 81 Youghiogheny Confluence, Pa. 435 24,000 82 Casselman Confluence, Pa. 450 20,800 83 Clarion Clarion, Pa. 910 39,300 84 Monongahela Pa. 5,430 207,000 % 85 Youghiogheny Pa. 782 . 46,000 % 86 Ohio Pa. 19,000 440,000 X 87 Allegheny Kitt arming, Pa. 8,700 241,000 X 88 Spring Cr. Pa. 11.6 3,000 X 89 Stony Cr. Johnstown, Pa. 428 30,000 X 90 Susquehanna McCalls Ferry, Pa. 26,800 671,000 X t See Fig. 71. % Maximum flows. Other discharges are twenty-four hour average flows. Art. 65] COMPARISON WITH OTHER RIVERS 199 TABLE XXV— Continued Refer- Place. Square Miles of Discharge, in ence, t Drainage Area. Second-feet. 91 Tohickon, Cr. Point Pleasant, Pa. 102 14,100 X 92 Susquehanna Danville, Pa. 11,100 305,000 t 93 Susquehanna Harrisburg, Pa. 24,000 700,000 t 94 Susquehanna W. B. Williamsport, Pa. 5,640 164,000 95 Schuylkill Philadelphia, Pa. 1,920 82,200 96 Junia Newport, Pa. 3,480 118,000 97 Br. Conemaugh Johnstown, Pa. 49 10,000 t 98 Neshaminy Cr. Below forks, Pa. 139 19,000 t 99 Perkiomen Frederick, Pa. 152 17,600 t 100 Plat R. I. 61 7,350 X 101 Catawba Rockhill, C. S. 2,990 151,000 102 Broad (Car.) Parr, S. C. 4,570 252,000 % 103 Tugalloo Madison, S. C. 593 21,900 104 Little Tennessee McGhee, Tenn. 2,470 70,000 105 Hiwassee Reliance, Tenn. 1,180 55,200 106 Ocoee McCays, Tenn. 374 18,000 107 Little Tennessee Judson, Tenn. 675 57,500 X 108 Tennessee Chattanooga, Tenn. 21,400 735,000 X 109 New Radford, Va. 2,720 138,000 110 Shenandoah S. F. Front Royal, Va. 1,570 76,800 111 James N. F. Glasgow, Va. 831 37,200 112 Roanoke Roanoke, Va. 388 18,100 113 Yakima Martin, Wash. 56 6,150 114 Yakima Cle Elum, Wash. 500 25,600 115 Yakima Umtanum, Wash. 1,540 41,000 116 Clealum Lake Roslyn, Wash. 205 17,700 117 Cedar Ravenusdale, Wash. 170 10,800 118 Black Neillsville, Wis. 675 23,100 119 Ohio Wheeling, W. Va. 23,800 480,000 120 Shenandoah Millville, W. Va. 3,000 140,000 t See Fig. 71. X Maximum flows. Other discharges are twenty-four hour average flows. This equation furnishes a means for comparing the maximum floods to be expected from water-sheds of different areas but hav- ing the same characteristics. Aside from the difference in area of the water-shed, two streams may have materially different flood tendencies accounted for by a difference in the characteristics of the water-shed. The flood coefficient, N, in Eq. (46), which is necessary on account of such differences in characteristics, depends on three general con- ditions, there being several subdivisions, as explained later. 1. The prevailing conditions of rainfall; 2. The storage capacity of the water-shed, or its ability to retain temporarily and distribute the maximum rainfall. 200 FLOOD FLOWS [Chap. XI 3. The capacity of the water-shed suddenly to release stored waters. The annual rate of rainfall on any water-shed is not always an indication of the maximum rate or intensity of precipitation which may be expected. A detailed study of the records of the U. S. Weather Bureau for rain-gauging stations on the catchment area and vicinity will prove of material assistance in determining the maximum rate and duration of precipitation to be expected. Storage, of whatever nature, has a tendency to reduce the size of floods. The storage capacity of the water-shed may be divided into the following items: 1. Storage in reservoirs, lakes, and swampy places; 2. Storage below the ground surface; 3. Storage above the ground surface. It is seldom that storage below the normal flow line of artificial reservoirs is effective in reducing the peak of large floods, because at such times the reservoirs are very sure to be full due to the excessive flow preceding the peak. Storage above the normal flow line of all bodies of water is always available, as such bodies of water must suffer an increase in surface elevation in order to provide sufficient head at the outlet to accommodate the increasing flow. The percentage of area of reservoirs, lakes, and swampy places has considerable influence on the value of the flood coeffi- cient. The magnitude of moderate floods is always less on rivers draining large sandy areas, in which the storage capacity below the ground is considerable. Unless, however, such sandy areas are large and extend to the higher elevations of the catchment area, little if any reduction in large floods will be effected, because, at such times, the voids below the ground surface are apt to be com- pletely filled as the result of the excessive precipitation preceding the peak of the rainfall. Storage below the ground, therefore, is usually considered only as increasing the interval between floods. Storage above the ground is affected by the nature of the vege- tation, the shape and slope of the catchment area, and the char- acteristics of the river bed and banks. It is evident that those characteristics which will permit of rapid run-off of the precipi- tation to the site of the dam will result in large floods. Rocky Art. 65] COMPARISON WITH OTHER RIVERS 201 slopes, devoid of vegetation are conducive to quick discharge. Conversely, areas covered with dense vegetation will prove effec- tive in holding back the water and smoothing out the peak of the flood. Heavy underbrush is particularly effective in this respect, as the rivulets are held back by friction in passing around and among the stalks of the plants and such branches as haA T e been beaten down to the ground surface. Practically no water, at the peak of the precipitation, is held back by adherence to leaves and branches above the ground surface. For this reason it is the opinion of many engineers that it is the removal of the dense under- brush rather than the large trees which has increased flood ten- dencies in districts which have been deforested. Steep slopes, of course, will produce rapid run-off. Therefore floods from mountainous districts are relatively severe. In rivers having tributaries extending fan-shaped from a given point, and of approximately the same size, the peak of the flood from each of the tributaries is apt to reach the main stream and the dam at approximately the same time, resulting in relatively large floods. Conversely, when the catchment area is relatively narrow, with tributaties of different sizes discharging into the main stream at regular intervals, the peak of the run-off from the trib- utary areas will reach the dam at different times, resulting in relatively small floods. A large number of tributaries is also pro- ductive of rapid run-off. Rivers and tributaries which have frequent restricted cross- sectional areas, rough bottoms, and are relatively shallow in com- parison with their widths may also be said to have a moderately retarding effect on the rapidity of run-off. The capacity of the catchment area to release stored water suddenly may be indicated by: 1. The frequency and magnitude of ice and log jams, with consequent danger of release of the impounded waters at or near the peak of the flood. 2. The presence of other dams of questionable strength im- pounding large volumes of water. A number of well- designed dams have failed on account of the destruction of defective dams above, with a resultant enormous increase in the run-off due to the sudden release of im- pounded waters. 202 FLOOD FLOWS [Chap. XI 3. Temporary partial blocking of the flow of the stream, due to lodgment of debris against submerged bridges, and subsequent failure of the bridges, with a release of the impounded waters at or near the peak of the flood. 4. Storage in the form of snow which may be suddenly released by a record precipitation accompanied by a rise in temperature. Little or nothing has been accomplished which would indicate a definite relation between flood coefficients and characteristics of catchment areas. Until more information has been obtained the matter must be left to the judgment of the engineer. It is probable that, in general, the maximum rate and duration of rain- fall, the steepness of the slopes, the shape of the catchment area, and the arrangement of tributaries will, in the order given, have the most influence on the flood tendencies of the stream. The items mentioned as affecting the capacity of the stream to release stored waters suddenly cannot be included in a general classifi- cation, as their effect on floods is too uncertain. An indication of the relative flood tendencies of two streams may sometimes be gleaned from a study of the records of maximum yearly floods when available. Mr. Fuller, in his paper on flood flows, has given a unique approximate method of determining directly the value of the flood coefficient, N, from a study of the maximum yearly floods on the stream in question. CHAPTER XII DETAILS AND ACCESSORIES 66. Masonry for Dams. During the latter part of the last century, rubble masonry was used extensively for the construc- tion of dams; but in recent years this type has been practically superseded by concrete and cyclopean concrete masonry. Cyclopean concrete masonry consists of plain concrete con- taining a large percentage of irregular stones, or " plums." These stones should be as large as can be economically quarried, trans- ported, and handled; and they should comprise as large a per- centage of the mass as possible, consistent with good work. Spalls are rammed into the concrete between the plums. Plums and spalls, of course, are intended to effect a saving in concrete, and, if the work is properly done, should not reduce appreciably the strength of the masonry. For a small structure, the greatest dimension of any plum should not exceed 20 per cent of the thickness of the structure, provided the masonry is to be stressed to a point approaching a reasonable working limit. A wet concrete is essential to the proper setting of the plums. Fig. 72 shows the cyclopean concrete masonry of the Olive Bridge Dam in the process of construction. As much as 30 per cent of plums can be used advantageously, but a larger precentage is apt to result in ineffective packing of the concrete. The usual percentage is from 18 to 22. It has been claimed that a 1 : 2| : 5 or even a 1 : 2\ : 4 mix for cyclopean concrete, on account of its greater fluidity, will not only give decidedly better masonry than a 1 : 3 : 6 mix, but will permit of the addition of many more plums and spalls, thus causing an actual decrease in the quantity of cement per cubic yard of masonry. It is also claimed that an aggregate limited to a maximum size of 2 in. will, for the same reason, give better results than the usual limit of 3 in. or greater. The results to be attained, however, will depend decidedly on local conditions and the per- sonal equation of the superintendent, it often being the case that, 203 204 DETAILS AND ACCESSORIES [Chap. XII Fig. 72. — The Olive Bridge Dam During Construction. Showing Plums Projecting above Horizontal Joints. Art. 67] WATER-PROOFING 205 through the lack of material, or by carelessness, the maximum quantities of plums and spalls are not used, and then the richer concrete is of no economical value. The following proportions of concrete have been commonly used for the various types of concrete dams: Solid gravity dams and large arch dams, 1:3:6 cyclopean concrete; Thin arch dams, 1 : 2| : 4§ to 1 : 2| : 5, plain or reinforced concrete; Slab decks of hollow dams, 1:2:4 reinforced concrete; Arched decks of hollow dams, 1:2:4 plain or reinforced con- crete; Buttresses of hollow dams, 1 : 2\ : 5 to 1:3:6 reinforced concrete; Struts of hollow dams, 1:2:4 reinforced concrete. To the customary requirements for first-class concrete masonry should be added, for masonry dams, the special provision for the maximum possible bond at horizontal building joints. The importance of this feature, as an additional element of safety, has already been pointed out. The proper treatment in this respect also serves to reduce greatly the leakage through such joints, with a resultant decrease in uplift pressure and prevention of unsightly discolorations of the down-stream face. Wherever new concrete is to be laid on old, the surface of the latter should be thoroughly cleaned by using stiff brushes and streams of water; the dead cement and laitance should be scraped from the old surface; and the latter should be thoroughly wet just before the new concrete is placed. It is good practice to spread a f-in. layer of mortar on the old surface immediately before pouring the new concrete. Care should be taken to have a considerable number of plums project above the surface of horizontal building joints, in order to increase the frictional resistance to sliding. In the absence of plums, it is frequently the practice to groove the building joints, as indicated in Fig. 41. 67. Water-proofing. Except as stated in Art. 69, no special provision need be made for seepage through the masonry of solid gravity and thick arch dams, constructed of 1 : 3 : 6 or richer concrete, other than cutting and working the concrete well against 206 DETAILS AND ACCESSORIES [Chap. XII the forms in order to insure a skin of cement at the face. Such dams will leak to a limited extent, but are tight enough for usual purposes. Leakage usually ceases after a few years, due to the filling of cracks by silt, or by effervescence of magnesia or lime from the cement. For thin arch dams, where a mixture not richer than 1 : 2\ : 4§ is used, it may be necessary to provide an impervious coating on the up-stream face.* The 1:2:4 concrete for the decks of hollow dams should be practically water-tight. This, however, necessitates a careful proportioning of the aggregate, a wet mixture, complete incor- poration of the materials, careful placing, and thorough puddling and spading. Tight forms are absolutely necessary. For high heads, the horizontal building joints of the decks have some- times been coated with a layer of tar, asphalt or similar material, from | to J in. thick. In general, as much depends on the quality of the labor and superintendence as on the methods specified and the materials used. The author knows of cases where, with skillful workman- ship, thin walls of 1 : 2\ : 4|, and even 1 : 1\ : 5 concrete have been made practically tight without the addition of any water- proofing material or any special treatment of the finished surfaces. 68. Contraction Joints, f All monolithic masonry structures of considerable length will crack because of restrained shrinkage when reductions in temperature occur. Such reductions may be due to atmospheric conditions or to the cooling of the cement which, when setting, attains a high temperature. Cracks are objectionable, not only on account of their unsightly appearance, but because of their tendency to follow planes of least resistance lying at almost any inclination to the lines of stress. It is common practice, therefore, to construct the dam in comparatively short sections, thus providing vertical contraction joints in order to localize such cracks and confine them to planes normal to the axis of the dam. * For descriptions of water-proofing methods see: "Report of Com- mittee on Masonry," Bulletin of the American Railway Engineering Asso- ciation, Vol. XV, No. 163; "Waterworks Handbook," by Flinn, Weston and Bogert, McGraw-Hill Book Co.; "Concrete, Plain and Reinforced," by Taylor and Thompson, John Wiley & Sons, Inc. t Sometimes erroneously called "expansion joints," Art. 6S] CONTRACTION JOINTS 207 It has been found that temperature cracks will occur in small walls about every 50 or 60 ft. Experiments show that the inter- nal temperature changes in any masonry structure will vary in- versely as its magnitude,* and, for this reason, contraction joints are usually placed farther apart in large dams than in small ones. For solid dams, a spacing about equal to the mean thickness of the structure, but not less than 40 or 50 ft., seems to have become standard practice. For this there is no logical reason, as the height of the structure is probably also a governing condition. In every case, however, it has apparently proved satisfactory. Two series of contraction joints have sometimes been used in very high dams, one series extending from the top to about mid- height and the other completely to the foundation. Such an Top of Dam. EL 4414 EL 4150 Him,. ^^i^Mk; -»«M0 L DiUnngft Gallerr St*, i + oo a + oo s + 00 i + oo 6+00 6 + 00 7 + 00 s + 00 9+00 10 + 00 11 too l^JKOO * Put la taring lata winter ELEVATION OF ELEPHANT BUTTE DAM showing contraction joints. Fig. 73. arrangement is indicated in Fig. 73. In order to minimize con- traction, the sections marked by asterisks were specified to be built in the coldest season. The usual type of contraction joint is indicated in Figs. 74 and 37. The size and shape of the vertical grooves, or "keyways," vary with the size of the structure, and are essentially a construc- tion feature. It is sufficient only to provide as large a number of corners as is practicable, in order to restrict the flow of water and entangle as much as possible of the sediment and other matter which may be available to tighten the joint gradually. The contraction joint indicated in Fig. 75 is unusually elaborate, and was designed to prevent, to as great an extent as possible, * "Temperature Changes in Mass Concrete," by Paul and Mayhew Transactions, Am. Soc. C.E., Vol. LXXIX, p. 1225. 208 DETAILS AND ACCESSORIES [Chap. XII all leakage through the joint. The use of concrete blocks for one face of the joint is not usual practice. Rectangular keyways, as here shown, are designed with the idea of providing less area of waterway, on the up- and down-stream sides of the keys, should the joint open a considerable distance. It is thought, however, that the gripping of the keys by the shrinkage of the subsequently poured concrete on the other side of the joint, may destroy its effectiveness and cause cracks through the base of the SECTION B-B SECTION A-A. Fig. 74.— The Usual Type of Contraction Joint. keys, if not elsewhere in the dam. Of the copper water-stop and the inspection or drainage well, more will be said later. In flat-deck hollow dams, a contraction joint is usually pro- vided at each buttress. In multiple-arch dams, the arches are usually reinforced continuously through the structure, contraction being taken care of by arch deflection. Contraction joints should be coated with suitable material in order to prevent adhesion of the masonry and to assist in checking leakage. The author has used a' thick coating of coal-tar pitch with complete success. The contraction joints of the Mathis Art. OS] CONTRACTION JOINTS 209 OOi}^ mmuisnAiOG^ 210 DETAILS AND ACCESSORIES [Chap. XII Dyke Dam, Fig. 40, were painted with asphaltum. To insure water-tightness, a 6-in. strip of hair felt, laid in hot asphaltum, was placed between the deck and the haunches. In the con- traction joints of the Farnham Dam, one face was covered with paper between two coats of pitch, before the concrete on the other side was poured. 87.75-t ' way f? , Upper inspection -gallery Floy line 13.355 -, TYPICAL SECTION OF KENSICO DAM 20 10 Fig. 76. 60 ft. 69. Drainage Systems. In solid gravity and large arch dams, the usu^rl requirements for first-class masonry and a treatment of horizontal and contraction joints such as previously described will result in a practically impermeable dam. Leakage through contraction joints of the type indicated in Fig. 74 is never exces- sive, particularly after a few years have elapsed, and may be prac- tically eliminated at once by using metal water-stops, such as shown in Fig. 75. The use of these stops has been confined prin- Abt. 70] ARCHITECTURAL TREATMENT 211 cipally to dams in which a discoloration of the down-stream face, due to leakage, would be objectionable, as affecting the appear- ance of the structure. Also, for the same reason, seepage through the masonry and horizontal joints has been prevented by special water-tight con- struction at the up-stream face, backed by a series of vertical drains to intercept and carry away all water which enters the dam. The drains may also be arranged to receive any leakage which finds its way past the metal water-stops. Except in very high dams, drainage systems in the body of the dam have seldom been used solely for the purpose of preventing uplift on horizontal planes. Unless the head is excessive, properly treated horizontal building joints are sure to be capable of resisting, in adhesion, the pressure of what little water finds its way into the masonry. The conditions within the body of the dam are ideal in comparison with those of the usual foundations, where drainage systems are often necessary. (Art. 61.) The elaborate system of drainage for the Kensico Dam is indicated in Figs. 75 and 76. The up-stream face of this dam was made relatively less permeable by the use of solid concrete blocks. The drainage and inspection wells were formed by laying up blocks of hollow porous concrete. 70. Architectural Treatment. Except in public works, the architectural appearance of dams is usually neglected. In private enterprises the need for strict economy usually will not warrant the outlay necessary for esthetic treatment; particularly if, as is usual, the project is in a sparsely populated district. Perhaps the most extensive treatment for architectural effect is that of the Kensico Dam, of the New York City Water Supply System,* (frontispiece). In this case the dam, while serving a useful purpose, was also intended to afford a monumental expres- sion of the magnitude and importance of the largest municipal water-supply system ever constructed. The entire down-stream face of the dam is covered with a layer of granite masonry, arranged in panels and surmounted by a continuous cut-stone cornice. A close view of the facing is shown in Fig. 77. Extensive land- scape work, with terraces, ramps, and artificial pools, imparts a pleasing appearance to the structure. ♦"Architecture of Kensico Dam," by A. D. Flinn, Engineering News, Vol. LXXIV, p. 433. 212 DETAILS AND ACCESSORIES [Chap. XII To facilitate construction, the concrete heart of the dam was completed before the setting of the face work was started. The concrete was built in steps, arranged to receive the stone facing. The style of ornamentation to be adopted must be in keeping with the dignity of the structure. The most appropriate treat- ment also depends, to a large extent, on the appearance of the neighboring landscape. ' Fig. 77.— Granite Facing of the Kensico Dam. Considerable opportunity for esthetic treatment was afforded at the site of the City Reservoir, No. 3, at Portland, Oregon, Figs. 78 and 79. The type of architecture is admirably in keeping with the beauty of the surrounding park. On the other hand, the rugged character of the sites of the Salmon River and Elephant Butte Dams, Figs. 80 and 81, neces- sitated a more simple, massive, and dignified treatment. That of the Salmon River Dam is thought to be the better of the two. Art. 70] ARCHITECTURAL TREATMENT 213 For a solid gravity dam, the weight of copings, pilasters, etc., at the top, being effective in adding to the stability, should not be Fig. 78. — Near View of City Reservoir No. 3 Dam, Portland, Oregon. considered an extra expense, except as to the increased cost of forming and placing the necessary material. In a hollow dam, the weight of water, and not masonry, offers the chief resistance to failure, so that a greater expense is attached to such embellish- 214 DETAILS AND ACCESSORIES [Chap. XII ments. An increased thickness, at the top of an arch dam, in the form of copings or cornices, adds materially to the stiffness of the arch. A water-tight structure is essential, if a pleasing appearance is desired. The slightest leak will not only result in a discoloration of the down-stream face, on account of the darker color of damp Fig. 79. — Distant View of City Reservoir No. 3 Dam, Portland, Oregon. masonry, but will pave the way for a later deposit of laitance, which is even more objectionable.* 71. The Regulation of High-water Surface. The land to be purchased or controlled for a reservoir must include the area which will be covered at the time of maximum flood. Obviously, the water stored between the elevation of the crest of the dam and that of highest water seldom serves a useful purpose, and many devices have been used to keep the maximum rise of water surface as small as possible. * See Fig. 78. Art. 71] REGULATION OP HIGH-WATER SURFACE 215 21G DETAILS AND ACCESSORIES [Chap. XII r Art. 71] REGULATION OP HIGH-WATER SURFACE 217 Flood regulators, in almost unlimited varieties of forms,* have been used. Brief attention will be given here to only a few of the types which have become common in America. Figs. 82, 82a and 83 show typical examples of crest gates. The gates are operated from an elevated platform, the openings being maintained at all times at an area sufficient to provide practically constant water surface elevation in the reservoir. The gates, when entirely raised, should have sufficient capacity to discharge the entire flood, and, if fragile, should be high enough to be clear of large trees and other heavy objects which may be brought down by floods. The piers may be designed as separate gravity segments down to the rock surface, or as reinforced concrete cantilevers relying for stability on the excessive weight of the adjacent overflow portions. Vertical grooves are usually provided in the piers, a short dis- tance up-stream from the gates, to facilitate the placing of stop- logs when inspection or repairs of the gates become necessary. The rectangular type (Fig. 82), varies considerably in detail, according to local conditions and the judgment of the designer, not only in the nature of the materials used, but in the types of bearings, operating mechanism, and the size of openings. Designs and estimates may be obtained from a number of manufacturers who make a specialty of work of this class. Probably the most common type of crest gate, at the present time, is the sector or " Taintor " gate, Figs. 83, 70 and 68. In this type the pressure of the water, passing through the pivots, causes no resistance to opening or closing, except that of the in- considerable friction at the pivots and the sealing strips at the sides. Practically, the only force to be controlled by the operating mechanism is the weight of the gate. Small gates of this type have been made of wood, but for the larger sizes, a steel framework with a steel or wood water face is usually adopted. Water-tightness at the sides of the Great Falls gates (Fig. 83), was obtained by fastening to them strips of 8-in. five-ply rubber belting which slide on the faces of the concrete piers. Local smoothness of the concrete was obtained by nailing J-in. smooth steel plates to the inside of the forms. The plates were removed •See "Improvement of Rivers," by Thomas and Watt, 2d Edition. John Wiley & Sons, Inc., 1913. 218 DETAILS AND ACCESSORIES [Chap. XII -Spindle Housing Operators Cab. is Counterweight Truss Top of gate when in raised position\ 11,537 E1.525 ^Normal Water OUTLINE OF TRAVELER KEOKUK DAM SLUICE GATES MISSISSIPPI* RIVER POWER CO. Fig. 82. Art. 71] REGULATION OF HIGH-WATER SURFACE 219 Fig. 82a. — Up-stream View of Keokuk Dam Crest Gates, Showing Concrete Counterweight and Steel Gate. 220 DETAILS AND ACCESSORIES [Chap. XII Fig. 83.— Taintor Gates for Great Falls Power Dam, Caney Fork River, Tennessee. Art. 71] REGULATION OF HIGH-WATER SURFACE 221 with the forms. For the bottom seal, the lower edges of the gates were planed to a sharp edge. In lowering, this edge cuts through irregularities in the sills, small pieces of wood, and other debris, and the gates come to a continuous bearing. The pivot pins are usually made to cantilever out from the piers, although they sometimes bear on girders spanning from pier to pier, if danger from floating debris is not feared. The bearing boxes are usually babbited or lined with bronze, in order to prevent them from rusting against the steel pins. In a plant where a large number of Taintor gates are used, operation is usually effected by one or more traveling hoists to which motors are attached. In cold climates steam pipes should be provided in order to prevent the gates from freezing tight. Objections have been raised to all form of crest gates which are not entirely automatic in operation, particularly if subjected to ice conditions, on the ground that constant vigilance is necessary for their successful operation. Such antagonism, however, has not prevented the use of non-automatic crest gates in a great many cases, particularly where a sufficient force of men is always avail- able for emergency operation. A few automatic crest gates have been adopted, but their use has not become common. Fig. 84 shows a typical sluice through the body of the dam. Such sluices may serve a variety of purposes, and are sometimes relied on to pass considerable of the flood flow and thus reduce, in a measure, the rise of water surface during floods. It is seldom found economical to provide a sufficient number of sluices to take the maximum flood flow, as in the case of crest gates. The Stevens Creek sluice gates are made so as to split, as indi- cated in Fig. 84, to facilitate removal to the passageway through the water-tight bulkhead. The gates are made accessible by low- ering a weighted timber stop gate between the projections on the up-stream face of the dam. If the leakage through the sluice is too great to permit of the stop gate being lowered against the flow, it may be lowered in a horizontal position to an elevation slightly above the top of the sluice entrance and then allowed to swing down over the opening. The sluice is protected from erosion near the gate by a cast-iron lining. Should tail-water rise above the level of the passageway floor, suction at the contracted throat of the sluice will effectually remove all leakage through the drain- age pipe provided for that purpose. 222 DETAILS AND ACCESSORIES [Chap. XII fftStifc Vr.fi-Wj' •Drainage Pipe Fig. 84. — Arrangement of 8'X8' Sluice-gates, Stevens Creek Development on Savannah River. Art. 71] REGULATION OF HIGH-WATER SURFACE 223 Gates of this type should be made unusually strong in every particular. The disturbance of the water at part gate opening seems to impose a duty on the several parts far greater than the capacity of the manufacturer's usual stock patterns. Considerable trouble may be caused by logs lodging in the sluice and preventing the gate from closing. Rack bars of 6- to 12-in. spacing are usually placed above the gates. They should be as far as possible from the gates, as cases have been known where logs, although having caught on the racks, have extended through them far enough to rest on the gate sill. For this reason some engineers incline to the opinion that racks are a menace rather than a protection, because, without the racks, the logs would have been carried through the sluice. The use of racks, however, is still quite common. There is a great deal of room for improve- ment in this feature, as the gates are relatively inaccessible, and, therefore, expensive to maintain and repair. Fig. 85 shows a typical system of flash-boards with which the normal spillway crest is lowered automatically at times of excess flow. In this country, this is the most common of all devices designed to control the elevation of flood-water surfaces. Although, in certain cases, flash-boards are rather expensive to maintain, owing to frequent renewals, they have the advantage of relatively small first cost and the certainty of automatic operation. The flash-boards are supported by round steel pins which are inserted loosely into sockets set in the masonry crest of the dam. The boards are usually fastened loosely by wire nails bent around the pins, or by wire loops passing around the pins and fastened to nails driven in the boards. The pins are designed with a reason- able factor of safety, for water at full storage level, but are cal- culated to bend over and clear the masonry crest, if the boards are not removed, when a severe flood occurs. After the flood passes, the pins are heated, straightened and replaced, and new boards put in place. The boards are usually fastened loosely enough to be removed by the flood, and are seldom recovered. When increases of flow can be anticipated and sufficient time is available, the boards and pins may be removed by hand before the water surface rises high enough to bend them over. In order to facilitate the handling of a barge for removing and restoring the flash-boards, it may sometimes be found desirable to provide sockets at intervals into which mooring pins may be set. These 224 DETAILS AND ACCESSORIES [Chap. XII a -a 1 I Abt. 71] REGULATION OF HIGH-WATER SURFACE 225 are indicated in Fig. 85. The boards and pins are sometimes manipulated from an overhead platform or bridge provided for that purpose. Where it is possible to remove the boards in advance of floods* they are usually built in panels, as indicated, and provided with handles. Handles of the type shown are not permissible if con- siderable drift is anticipated, as this may collect and cause pre- mature failure. The boards are sometimes provided with planed edges in order to reduce leakage, but more often are unplaned, and ashes or sim- ilar caulking materials are swept along the joints. As indicated in Fig. 85, Let d = the diameter of solid pins, in inches; s = the spacing of solid pins, in feet; hi — the height of solid pins, in feet; ^2 = depth, in feet, of water above the masonry crest; Tim = the depth, in feet, of water above the masonry crest corresponding to the maximum flood; /= the bending stress in the pins, in pounds per square inch; W2 = weight of one cubic foot of water = 62.5 lb. The bending moment in each pin for the water surface at full storage level is M = ^|^ = 10.42/n 3 s . . . ft.-lb. o Within the elastic limit, the moment of resistance of the pin, in foot-pounds is, 0.00818/d 3 . Equating this to the bending moment, and solving for d, there results, , 3 ll275hi 3 s The use of this equation will provide for the pins having a rea- sonable stress, say two-thirds of the elastic limit, when the pond is at full storage level. 226 DETAILS AND ACCESSORIES [Chap. XII Before the pins are completely bent over, they are stressed far beyond their elastic limit, and on this account the ordinary theory of flexure is quite inaccurate. Because of this and other uncer- tainties, it is impossible to write an -accurate equation to indicate at what stage of water surface the pins will bend over, and, there- fore, we must rely on experimental data. The author has found that, if pins of medium steel are stressed theoretically to about two-thirds of their elastic limit when the water surface is at full storage level, they will ordinarily not withstand a head, h,%, above the masonry crest greater than twice their height, and that they usually bend over for heads equal to from 1.5 to 1.75 of their height, depending on the vacuum which forms under the overflow.* To be amply safe, the height of the flash-boards should not or- dinarily be more than about 40 per cent of the head, hm, corre- sponding to the maximum flood. This will leave a margin of about 0.2hi between maximum flood level and the highest level at which it is thought the pins will fail. The pins may then be expected to accommodate, without fail- ure, a flood equal to about one-sixth to one-eleventh of the maxi- mum expected flood, and, if not removed when necessary, will usually bend over several times in the average year unless a part of the moderate floods can be accommodated through sluices in the dam, through turbines, or by other means, or unless the reservoir is exceptionally large. If a greater degree of accuracy is desired, it becomes necessary to test out samples of the pins under actual operating conditions. Experiments of this character are inexpensive, if a log chute, fish way, or wasteway has been, provided in the crest of the dam. In any event, a considerable expenditure will be warranted if a large reduction in land to be purchased can be effected. For greater accuracy, the pins are sometimes grooved to exact diam- eter at the elevation of the masonry crest. Pipes have sometimes been used in place of solid pins. If the pipe pins are short in proportion to their diameter, they may fail by buckling at their supports, and go out sooner than solid pins having the same initial stress. Many types of automatic hinged flash-boards have been used, in which actual failure of the material does not occur when the flash-boards collapse. Their use, however, has not become com- * See Art. 17. Abt. 71] REGULATION OF HIGH-WATER SURFACE 227 mon, except on some of the very low river improvement dams built by the Federal Government. A radically different type of regulator, which seems to be gain- ing favor, is that of the siphon spillway. Fig. 86 shows the siphon spillway of the Tennessee Power Company on the Ocoee River, Tennessee.* In a siphon spillway, the water flows through a closed conduit, producing a suction head that largely increases the velocity, and consequently the discharge per unit area. Under normal conditions, the water, at the Ocoee Siphons, stands at El. 1089.2, or slightly trickles over the lip or crest of the spillway. When the water surface rises, the discharge over the crest increases and the water strikes the far side of the Fig. 86. — Siphon Spillway, Ocoee River Development, Tennessee. lower siphon legs. When the water surface reaches an elevation of about 1089.55, or slightly above the top of the air inlets, the air thus confined in the top of the siphons, as well as that in the lower legs, is quickly entrained and ejected and the siphons prime. The suction thus produced increases the velocity to that corresponding to an effective head equal to the difference in ele- vation between the water surface in the pond and the center line of the siphon outlets, less the head expended in friction within the siphons. If the full discharge of the siphons is greater than the total flow, the water surface in the pond will recede. When the upper parts of the air inlets become exposed, the air drawn in by the suction reduces the efficiency of the siphons until the dis- charge is automatically diminished to that required for stationary * Engineering Record, Vol. LXXII, p. 567. 228 DETAILS AND ACCESSORIES [Chap. XII Pig. 87. — Down-stream View of Ocoee River Siphon Spillway. Fig. 88. — Up-stream View of Ocoee River Siphon Spillway. Art. 71] REGULATION OF HIGH-WATER SURFACE 229 water surface in the pond. If a sufficient quantity of air is drawn through the inlets, however, the siphonic action will be broken. The water surface in the pond will then rise again, and the oper- ation of priming will be repeated. By virtue of their large capacity, relative to that of a simple weir, of the same crest head, siphons may be used to limit the rise of the water surface in the pond during floods, and are particularly serviceable in cases where the fluctuations in discharge are rapid, as at the end of a long flume supplying hydraulic turbines. If properly proportioned, they will prime quickly enough for all practical purposes. The discharge of a siphon of this type may be computed from the ordinary equation for flow through short tubes, Q = CAV2gh, where ft = the gross head on the siphons, in feet from the water surface to the center line of the discharge; A =the area at the throat, in square feet;