CMT E C H N IC A L REPO RT NO. 3-777

SOIL MECHANICS DESIGNSTABILITY OF SLOPES AND FOUNDATIONS

m g

February 1952

(Reprinted April 1967)

Sponsored by

Office, Chief of Engineers

U. S. Army

Published by

U. S. Army Engineer Waterways Experiment Station

C O R P S O F E N G IN E E R SVicksburg, Mississippi

library

MAY 2 4 1973

Bureau of Reclamation Denver, Colorado

BUREAU OF RECLAMATION DEN\ IRARY

C ’V ‘WUb3L14

TECHNICAL REPORT NO. 3-777

SOIL MECHANICS DESIGN;STABILITY OF SLOPES AND FOUNDATIONS

February 1952

(Reprinted April 1967)

Sponsored by

O ffice, C hief o f Engineers

U. S. Army

Published by

U. S. Army Engineer Waterways Experiment Station

CORPS O F ENGINEERS

Vicksburg, Mississippi

A R M Y -M R C V IC K S B U R G , M IS S .

92063614

Foreword

The current issue of EM 1110-2-1902, Stability of Earth and Rockfill Dams, was preceded by a manual, dated February 1952, covering Stability of Slopes and Foundations which was originally issued as Part CXIX of the Engineering Manual for Civil Works Construction. The earlier manual in­cluded a comprehensive description of methods for analyzing the stability of slopes and foundations that was later considered too detailed for a de­sign manual but which has proved to be useful as a convenient summary of design methods. Since a comparable description of these methods is not readily available in technical publications, the February 1952 manual has been reprinted as a technical report to make it available for use in design activities of the Corps of Engineers.

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TABLE OF CONTENTSParagraph Page2-01 PURPOSE AND SCOPE_____________________________ _________________ 12-02 NOMENCLATURE AND REFERENCES________________________________ 12-03 STABILITY CONSIDERATION________________ 1

a. Care in Regard to Investigations.__ . _ ___________ ___ _______________ 1b. Factors Influencing Stability___ _____ _______________________________ 1c. Strength of Soil Masses____________ _ _____________ _______________ 1d. Unretained Slopes_________ _________ _________ _____ 2e. Retained Slopes____________________________________________________ 2/ . Physical Properties and Design D ata______________ _________________ 2g. Influence of Seepage________________________________________________ 2h. Influence of Consolidation______________________________ ____________ 3i. Need for Field Observations________ 3

2-04 EARTH FOUNDATION FAILURES____ ______________________________ 3a. Subsidence_______________________ 3b. Spreading_________________ __________ -___________________________- 3c. Piping Failures_________________ 3

2-05 SLOPE FAILURES____________________ 4a. Shear Failures_______________________ _____________ — --------------- 4b. Failures Induced by Foundation Weakness_____________________________ 4c. Liquefaction of Cohesionless Materials________________________________ 5

2-06 ANALYSIS OF EARTH FOUNDATIONS_________________________________ 5a. The Elastic Theory Method of Analysis_______________________________ 5b. Checking Stresses with Photoelastic Models____________________________ 6c. Jurgenson’s Method________________________________________________ 6d. Rendulic’s Method_________________________________________________ 7

2-07 ANALYSIS OF EARTH SLOPES________________________________________ 7a. Circular Arc Method_______________________________________________ 8b. Method of Slices_________________ _______________________________ 8c. ^-circle Method________________ 8d. Moment Method________________ 9e. Modified Swedish Slide Method______________________________________ 9/ . Wedge Method__________________ . _______________________________ 9g. Hydraulic Fills___________________ 9h. Special Conditions_________________________________________________ 10

2-08 DESIGN CRITERIA AND SAFETY FACTORS___________________________ 13a. Types of Safety Factors_____________________________________________ 13b. Specific Values_______________________________________________ - 14

LIST OF SYMBOLS___________________________________________________ - 16LIST OF REFERENCES________________________________________________ 18LIST OF ILLUSTRATIONS______________________________________________ 19

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Paragraph PageAPPENDIX A APPLICATION OF ELASTIC THEORY___________ A-l

B APPLICATION OF JURC.ENSON’S METHOD________ H-lC APPLICA'I'ION OF RENDULIC’S METHOD___________________ C-lD APPLICATION OF SWEDISH SLIDE METHOD._____ _________ D 1E APPLICATION OF «-CIRCLE METHOD_____________________ E-lF APPLICATION OF MOMENT METHOD__________ __ ________ F 1G APPLICATION OF MODIFIED SWEDISH METHOD._. _ . . . . . . G-lII APPLICATION OF WEDGE METHOD______________ H-lI STABILITY CALCULATIONS FOR HYDRAULIC FILL... ....... .. 1-1J STABILITY WITH SPECIAL MOISTURE CONDITIONS_______ J-l

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SOIL MECHANICS DESIGN

STABILITY OF SLOPES AND FOUNDATIONS2-01. PURPOSE AND SCOPE

The purpose of this chapter is to present various methods of analyzing the stability of earth foundations and unretained slopes. Also included are discussions of various critical conditions which influence such studies. Methods pertaining to analyzing the stability of retained slopes are covered in the manual in Part CXXV—Wall Design.2-02. NOMENCLATURE AND REFERENCES

A list of symbols used in formulae discussed herein is placed near the end of the chapter. For the most part, the symbols given conform with the soil mechanics nomenclature contained in the A. S. C. E. Manual of Engineering Practices No. 22.

A list of references in regard to source of subject matter is given near the end of the chapter. These are indicated throughout the text by numbers placed in the exponential position.2-03. STABILITY CONSIDERATIONS

Slopes, embankments and underlying foundations are subject to shear stresses caused by the action of gravity. In addition, shear stresses can be induced by seepage and earthquakes. The capacity of soil structures and foundations to remain stable and retain an unchanged position, or the settlement or dimensional distortions that can be expected to take place in the course of time, can be evaluated with reasonable assurance when the geological and physical properties of their soil composition are known. However, conditions involved in stability problems are at times exceedingly complex and it may be necessary to make simplifying assumptions, in obtaining solu­tions, which will render the results less precise.

a. Care in Regard to Investigations. Thorough explorations of foundations and slopes cannot be overemphasized. Inadequate information may lead to erroneous conclusions and eventual failure of a structure. The need for a complete knowledge of the physical characteristics of soil masses is especially true when the materials are slowly consolidating clay or a relatively unconsoli­dated silty soil. It is false economy and poor engineering to neglect the adequate investigations required for a clear understanding of foundation and embankment conditions.

h. Factors Influencing Stability. The stability of soils is governed principally by the acting forces and the shearing strength characteristics of the materials. Forces which the earth mass must resist include induced stresses caused by the superimposed embankment, unbalanced water pressure, percolating water, earthquake forces and forces induced in the foundation by action of construction. The structural strength of the foundation will, in many cases, govern the design of the cross section of an embankment. Differential settlement may result in large secondary stresses within the structure itself. High seepage forces in foundations may result in “quick” conditions or piping.

c. Strength of Soil Masses. The shearing strength of a soil is dependent upon its cohesion and internal friction and is generally evaluated by Coulomb’s formula which is as follows:

S = C + N tan 4> 1

(1)

The terms C and ^ are not true constants for most soils; however, their use provides a convenient means of approximating strength envelopes of soils for use in stability computations. For example, the strength envelope for a clay may be somewhat curved. There may be occasions when strengths used in stability computations should be determined from such curves; however, ordinarily a straight line can be substituted with minor error. Such substitution greatly simplifies the computations.

For cohesionless soils such as sands and gravels the equation reduces to:S = N tan <f>_____ _____________________________ (2)

The value of <#> may vary for cohesionless materials depending principally upon the initial degree of compaction; however, the degree of variation is generally moderate, In contrast, the value of C and <t> may vary over an extreme range for cohesive soils, principal variations being with changes in water content. I t is therefore of extreme importance that proper consideration be given the selection of procedures and equipment used in testing soils of low permeability to insure proper methods of investigation for critical conditions.

d. Unretained Slopes. Unretained slopes are not supported or contained by any structure other than the soil itself and are entirely dependent upon the strength of the soil comprising the foundation and the mass beneath the slope itself to maintain stability. Unretained slopes may be created by geologic means as found in natural riverbanks and hills, or by construction of cuts and fills. Failure of slopes may result when either the soil within the slope itself or the soil within the foundation is overstressed due to increased loading or decreased strength.

e. Retained Slopes. Retained slopes are supported or contained by structures, such as masonry or reinforced concrete walls, bulkheads, piling, quays. Structures of this nature furnish local support and are discussed in chapters 1 and 2, part CXXV of the Engineering Manual for Civil Works. If structures of this type are entirely involved in a mass movement of the slope and foundation, the methods of stability analysis for unretained slopes, as presented herein, are applicable./. Physical Properties and Design Data. Permeability, consolidation, and shear tests provide b&sic design coefficients used in the analysis of embankment foundations. In practice, engineers arfe principally interested in saturated or nearly saturated soils. For example, a saturated imper­vious soil suddenly placed under stress may develop important pore-water pressures which slowly dissipate as consolidation takes place. Any stress change in a saturated soil mass is therefore associated with the rate of change of moisture content. This factor should be given careful con­sideration in the testing program for all projects of importance.

Two types of tests generally used for determining the shearing strength of soils are the direct shear test and triaxial shear test. Unconfined compression tests are sometimes used for special applications. An envelope is generally used to express the results of these tests. Methods and data relative to various shear tests are discussed in Part CXVIII—Subsurface Investigation, Chapter 3—Soils Investigations.

g. Influence of Seepage. Seepage through the foundation or embankments of a flood-control project need not be detrimental if escape gradients are controlled so piping or boils do not occur. The more common methods of seepage control are cutoff walls and trenches, upstream blankets, relief wells, horizontal drains and filters. Failure to properly control seepage may result in exces­sive hydrostatic uplift at the downstream toe and, as a result, sloughing of the embankment in addition to piping and boils.

Water flowing through the soil undergoes a loss in head due to friction between the soil and water. The frictional forces tend to move the soil in the direction of flow. If water is flowing upward the fluid friction of forces acts upward against gravity forces of the soils which, in addition to a buoying effect, reduces the effective soil weight. This reduction of weight decreases the inter­granular friction and thus reduces the soil shearing strength. If in the downstream toe area the

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shearing strength of the soil is reduced sufficiently by seepage, sloughing or embankment failure will occur.

h. Influence of Consolidation. Since foundations of earth structures are often nearly saturated, consolidation is an important factor. Impervious, compressible foundations increase in strength with the progress of consolidation, the rate of gain being dependent primarily upon the permea­bility and length of drainage path. A highly impervious clay foundation of considerable depth cannot be expected to gain appreciable strength during a normal construction period; however the final safety factor which will be attained several years after completion may be considerably higher than the design value. The critical period for an earth dam on a clay foundation may occur during construction or during the initial rapid drawdown stage; the governing condition being the degree of consolidation that occurs during the construction period and the time required to fill and draw down the reservoir. If the gain in strength due to consolidation is sufficiently great to offset the conditions of rapid drawdown, then, of course, some point during the construction period will govern the design. This in turn depends upon the rate of loading of the foundation.

i. Need for Field Observations. All methods developed for analysis of structures require for solution simplification of assumptions as to structural and physical properties. An analysis by any of these methods is therefore but an estimate of true conditions. The nonuniformity of soils and the fact that soil stresses do not always remain within the elastic range make such assumptions necessary. The results of a theoretical analysis should always be viewed with the variance between assumptions and reality in mind. Field observations of structures during and after construction with the aid of pressure cells, piezometers, and settlement gages provide a check on design methods and assumptions and should be given thorough consideration in all projects of importance. Refer to Part CXXIII-Structural Design, Chapter 1-Earth Dams.2-04. EARTH FOUNDATION FAILURES

Foundations fail due to insufficient shear strength. The most common types of foundation failures are designated as subsidence, spreading and piping failures. These are discussed as follows:

a. Subsidence. Failures by subsidence occur on deep, soft clays and are characterized by deep-seated rotational movement of the foundation mass including part of the overlying slope or embankment.

b. Spreading. Failures by spreading are characterized chiefly by horizontal movements with little or no vertical displacements. Such failures will occur where the foundation contains a weak layer, or where the foundation contains highly stratified soils such as varved clays. I t may also occur when the foundation contains very loose saturated sands. Sudden local stresses may cause a local mass of sand to liquefy which may then spread and involve a much larger area. When overstressing is indicated by analysis, one or a combination of the following methods may be utilized as corrective measures.

(1) Flatten slopes or add berms to spread the load on the foundation.(2) Excavate the weak foundation materials and replace with material of adequate strength.(3) Lengthen the construction schedule to obtain benefit of increased strength due to

consolidation.(4) Use consolidation wells to accelerate consolidation and gain in shear strengths.

c. Piping Failures. Piping occurs when the escape gradient is somewhat in excess of the critical hydraulic gradient and sufficient pressure and velocity are available for the removal of soil particles. Piping starts at the exit usually in the form of boils and springs, then will progress back into the dam or foundation. I t may start at any unprotected point within a structure or its foundation.

Piping through the foundation is influenced to a large extent by stratification and minor geological circumstances which cannot entirely be anticipated, such as the area and form of lens­shaped beds of more pervious material. Conditions conducive to piping exist when seepage water has a relatively free inlet upstream or riverside of a dam or levee and its exit is restricted or rela-

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tively restrained on the downstream or land side. Under these conditions excessive gradients and uplift pressures may be concentrated near the toe. The thickness of an impervious top stratum is the controlling factor in many cases as to whether boils will occur.

If the seepage has free entrance on the upstream side and there are no natural sources of pres­sure relief downstream, essentially the full pressure head will be exerted against the blanket on the downstream side. The foregoing assumption is usually severe and the head acting at the toe may be considered to vary from 50 to 75 percent of the total head when the top stratum extends a con­siderable distance to the riverside. On dams, except in the case of small structures, the effective weight of a downstream blanket is seldom equivalent to the head. One or a combination of the methods discussed for seepage control in chapter 1 of this part may be used effectively for the reduc­tion of head and uplift pressure. Control of seepage must be given very careful study in the design of all levees and earth dams.2-05. SLOPE FAILURES

The most common types of slope failures are designated as shear failures and failures induced by foundation weakness. Occasionally failure results from liquefaction of cohesionless materials. These are discussed as follows:

а . Shear Failures. A failure in a relatively homogeneous coheisve soil often occurs in which the unstable portion of the mass is separated from the stable portion by a failure surface or a narrow failure zone. The shape of this failure surface is approximately circular. A failure in heterogeneous soils, expecially stratified materials, is generally more complex and the failure surface or zone may approximate a combination of circular arcs and planes.

In homogeneous cohesive soils a failure surface generally emerges at the toe of the slope if the angle of the slope to the horizontal is more than 53 degrees. For lesser slope angles, the slide will generally emerge beyond the toe; however, its point of emergence will be influenced by any firmer layer which may be present in the foundation. If such a layer is very close to the horizontal surface at the toe of the slope the failure surface may emerge part way up the slope.

The character of the slide will vary, depending upon the type of soil present. In soft, plastic, unfissured clays, the sliding mass deforms in a progressive manner. . If the clay is very soft there may not be a well-defined failure surface but instead a zone of plastic movement in the soil. How­ever, this zone may be so narrow that a surface of sliding can be substituted therefor for purposes of analysis. Brittle, unfissured clays tend to move as a mass and will have a reasonably well-defined failure surface. Because of its brittle nature this type of clay will experience a very marked drop in shearing strength at any overloaded zone. The failure will usually be progressive although at times may be very sudden with very little or no advance warning. Fissured clays often take a very long time to fail. The fissures, which may have been tightly closed, tend to open after stresses within the clay mass have been altered by construction operations or by shrinkage due to seasonal moisture changes. As the fissures open, the clay adjacent to the fissures expands, especially if free water is present, and a marked softening and reduction in shearing strength results. This progressive softening of the fissured clays may result in slides a long time after completion of construction operations.

Soils such as gravels, sands, and silts are cohesionless when they contain no clay. However, these soils may exhibit some cohesive properties due to the cementing action of iron or lime or capillary action of soil moisture. If cuts are made into these soils which have a minor amount of cohesion they may fail, in a manner similar to cohesive soils, if the excavated slope is too high or steep. On the other hand, if the minor amount of cphesion is destroyed by exposure to the air, the failure will be entirely superficial. Cohesionless soils subjected to seepage may have slide failures that go out as a mass, similarly to cohesive soils, or the soil may flow out similarly to a very heavy liquid, as discussed below.

б . Failures Induced by Foundation Weakness. An embankment may fail dtie to overstressing4

of the foundation beneath the slope, regardless of the strength of the embankment soil. In cases where the weak foundation is a thin layer, the failure will be by lateral spreading, especially if the soil is strongly stratified. The more pervious and more cohesionless soil layers may acquire high water pressures by acting as horizontal drains for the compressing layers and, with increasing strain, a loose granular structure of the more pervious soil layer may collapse and liquefy with consequent loss of shear strength. On the other hand, if the foundation is massively stratified and deep with an especially weak layer at some depth, the foundation may fail by a more or less deep slide in which the slide movement is mainly rotational.

c. Liquefaction of Cohesionless Materials. Dense cohesionless soils tend to expand when sheared. If such a soil is saturated the pore water is put in tension and intergranular forces are temporarily increased by the inwardly directed seepage forces. Loose sand, on the other hand, tends to decrease in volume when sheared and, if saturated, the excess water has a tendency to flow out of the voids.

While the water is being forced out, the seepage pressures reduce the intergranular forces in the direction of seepage. If the shearing deformation occurs very quickly, little or no drainage may occur and part of the intergranular forces is transferred to the water. Because water has no static shearing strength, the soil mass experiences a general loss of shearing strength and, if the loss is sufficient to reduce its stability, the soil may flow out as a fluid mass. This is very likely to occur if the mass of sand is fine-grained, which, because of its low coefficient of permeability, will drain slowly. Because of the semifluid nature of the soil, the failure occurs very quickly and results in a very flat final slope. The analyses of flow failures are beyond the scope of this chapter and are not discussed herein.

2- 06. ANALYSIS OF EARTH FOUNDATIONSThe safety of earth masses used for structural purposes is often a problem of both slope and

foundation stability. In the discussion that follows the application of methods of analysis to embankments will be given.

a. The Elastic Theory Method of Analysis. The stability of slopes and foundations may be studied by means of an approximate application of the elastic theory. The theory of elasticity is based upon a homogeneous mass obeying Hooke’s law. In most analyses the materials are also assumed to be isotropic; however, a few solutions have been developed based upon aeolotropic properties. Soils actually are only partially elastic and, as the maximum strength is approached, plastic phenomena predominate. Stresses calculated by the elastic method then become invalid. In soils that are massively stratified, the modulus of elasticity of each layer is different from that of adjacent layers. The effect of this difference may be accounted for in layered photo-elastic models, but the mathematical solutions are so complex that they are not practicable. In general with this method the difference in elastic properties of the various layers is ignored.

The stresses within an earthen embankment may be determined by dividing it into a number of horizontal layers. If various portions of the embankment have different weights, the individual strips are divided to coincide with zones of constant weight. The stresses caused by each layer are computed by assuming that the soil beneath each layer extends to infinity in all horizontal directions.

Seepage forces in each layer may be divided into vertical and horizontal components. The vertical forces in each layer are averaged and combined with the dead load of the layer. The hori­zontal forces are also averaged and stresses determined) assuming the horizontal forces act on the layer immediately below as tangential shear forces.

If the depth of the foundation is much less than the base width of the embanlanent and over- lies a material more rigid than the foundation soil, then the stresses at the rigid surfaces may be determined. Stresses within the layer above the rigid surface are difficult to determine and the usual practice is to determine the stresses at the upper boundary and at the rigid lower surface

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of the foundation and to sketch in contours of equal stress. I t is pointed out by P ickett** that the equations for stresses at the rigid boundary presented in previous publications are in error. Correct influence values based on Pickett’s equations are shown on Plate No. A5, appendix A.

In all cases the normal and shearing stresses on the vertical and horizontal planes are deter­mined and include the foundation stresses due to the dead weight of the foundation soils. From these component stresses the maximum shearing stresses may be computed as shown in table l and as discussed later.

The maximum shearing stress should be compared with the design shearing strength of the soil; and in all cases the safety factor shpuld be based on a point safety factor rather than average values as was done in the design of Fort Peck Dam 2 so that excessive plastic distortion of the foundation, which would render the stress analysis void, may not occur. An example of the elastic theory method of analysis is presented in appendix A.

In cases where the soil is distinctly stratified, as for example a varved clay where the coarse­grained layers have a higher strength than the finer-grained layers and thus act to limit the spread of strains, the elastic solution devised by Westergaard 3 should be used. When the state of stress in the foundation of an embankment is such that failure by plastic flow is not imminent, the dis­tribution of the stress may be presumed to resemble that in a similarly loaded semi-infinite elastic body. The applicability of the theory of elasticity to soil problems depends upon the extent to which Hooke’s law will describe the stress-strain relationship of the soil. In the elastic method of analysis of a foundation, it is first assumed that the stresses at points within the foundation, when computed by equations based on the theory of elasticity, are valid. The shearing strength of the soil at each point is then estimated and compared with the shearing stress to obtain a point safety factor against the formation of the plastic state. In relatively rigid or brittle materials, an increase in stress at any point beyond the limit of elastic equilibrium will induce failure a t the point through loss of cohesive strength, thus throwing the stress to adjacent materials, which are in turn over­stressed inducing progressive failure. In materials which may deform either elastically or plasti­cally, such an increase in stress causes the formation of a zone of plastic equilibrium. A further increase in stress merely enlarges the plastic zone, and failure does not occur until the plastic zone spreads to the extremities of the foundation. Thus, the relative increase in load necessary for failure over that required to initiate plastic flow at some point in the stressed body will vary widely, depending on the nature of the material. The formation of plastic zones causes a redistribution and reorientation of stresses within the foundation and, except for very simple cases, a stability analysis for an intermediate state between the completely elastic and plastic states becomes im­practicable. For foundations analyzed by the elastic theory method, the criterion for safety should be that no point in the foundation will be overstressed in shear.

b. Checking Stresses With Photoelastic Models. Photoelastic models provide one of the most valuable aids in drawing stress patterns for foundations. Gelatin has been used with the best results for models of embankments and foundations because of the ease and accuracy with which models of the prototype can be built. When use is made of photoelastic models or theoretical stress patterns for the analysis of earth mass stability, consideration should be given to the fact that such studies are applicable only to the extent that the real materials are elastic and homo­geneous. Experience with many projects has demonstrated that the photoelastic method is very helpful and may be actually more applicable to practical foundation analysis than heretofore supposed.

c. Jurgenson’s Method. Jurgenson 4 advanced a method for the analysis of foundations that are overstressed according to the elastic theory. Jurgenson considered the fully plastic state with a spread failure and proposed a formula to calculate the shearing stresses resulting in relatively shallow plastic foundations that are underlain by a rigid material.

•Numbers shown in this manner refer to numbered references listed at the end of the chapter.

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Assumptions. The following assumptions are made in analyzing foundations under embank­ments by this method:

(1) Plastic foundations are homogeneous and isotropic.(2) Loadings applied to foundations are triangular in shape, symmetrical about the center

lines and act vertically.(3) Effects of restraining forces acting in the foundation beyond the embankment toe are

neglected.The method should be used in preliminary analyses or as a check on other methods and is

applicable to relatively shallow foundations only.Description of Method. From the results of an investigation by Hencky, Jurgenson concluded that a triangular loading could be substituted for the actual loading system of most embankments and this applied directly to the surface of the plastic layer. The resulting pressure is considered vertical and proportional to the height of fill. The shearing stress that is developed is constant along the boundaries of the foundation and is determined by the equation:

(3)I t should be emphasized that the above formula expresses the stress in the foundation after

the material has been overstressed. Under this condition, failure is assumed to have occurred. Jurgenson also has expanded the method to include the effects of horizontal pressures due to seep­age forces existing within embankments. These values are applied to the top of the foundation as a distributed shear load. An example of the application of Jurgenson’s method is given in appendix B.d. Rendulic’s Method. A method of determining the shearing stresses at the base of a cohe­sionless fill has been developed by Rendulic, where it is assumed that sufficient yielding of the foundation has occurred so that the cohesionless fill is everywhere in a state of active pressure. The various steps required to determine the normal and shearing stresses on the base of the fill are given in appendix C. The normal pressure is nearly proportional to the vertical height of the fill over any point in question. The maximum shear value lies between the center line of the fill and toe.

If the foundation does not yield enough to create an active condition in the fill, then the ratio K between the vertical and horizontal stress of the fill will be higher than for the active conditions. K is defined as:

K = tan4 ^4 5 “ ................................. - ............— ..........(4)where <£' is an effective angle less than the friction angle. The K-value may be as high as 0.6 with <£' as low as 15°. A selected value of 0' may be used as an effective friction angle and the shear dis­tribution diagram at the base of the fill determined as noted above. The total shear along the base must equal the total horizontal pressure at the fill center line, which is a minimum for the case of active pressure.

With a distribution diagram of shear along the base of fill available, a comparison of shearing stress and strength may be made. The comparison should be made not only for point values but for average values between the toe and points within the base of the fill. In some cases the point stress may be greater than the strength. However, if the average strength between such a point and the toe is greater than the average stress, no failure will occur.2-07. ANALYSIS OF EARTH SLOPES

Soils may exist in nature in any condition between active and passive failure states. I t is possible for a portion of the soil to be stressed to its maximum strength without an overall failure

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occurring if such a zone is contained within an Understressed soil mass. With increasing load the overstressed zone will deform and will not carry any increase in load but rather the load will be passed on by the deformation of the overstressed zone to the understressed zone. The overstressed zone will enlarge with increasing load until ultimately a failure will result. In earth failures there may or may not be a sharp division between the stable and unstable soil masses. In most cases the unstable mass does not remain intact but rather is broken up by numerous failure surfaces. In very soft, plastic soils there often is no sharp division between the stable and unstable soil. Because of these field conditions and also because of the possible difference between in-place and laboratory soil properties, all stability studies are approximate; therefore highly theoretical means of analysis are generally not justified. The principal assumptions of the more common methods of stability analysis are discussed in the paragraphs that follow. Details of computations for various methods are presented in appendixes D, E, F, G, and H.

a. Circular Arc Method. Field experience has indicated that frequently a failure in an earth mass will have a sliding surface which is nearly circular in cross section and of finite width parallel to the slope. For purposes of stability analysis the failure surface can be assumed to be circular in sec­tion and infinite in width. These assumed conditions are slightly on the safe side for slopes that are stable, while if a failed slope is studied to obtain an estimate of the shearing strength of the soil, values slightly on the unsafe side are obtained, due to neglect of the three-dimensional nature.

Basically, circular arc methods of stability analysis involve the determination of forces acting upon the soil mass above an assumed sliding surface and equating the forces tending to produce rotation of the mass to those tending to resist movement. The forces involved are the weight of the mass, seepage or water pressure forces, and the strength of the soil (friction and cohesion) along the failure surface. Factors of safety are determined by assuming the mass to be in equilibrium, computing the soil strengths required for this condition, and comparing them with the soil design* values. Another approach to the analysis is to use the design values of soil strength in the compu­tations and to compare the forces tending to produce movement with those tending to resist move­ment. In either case it is obvious that for any given slope analysis an infinite number of circular arcs could be investigated. One feature of the analysis, therefore, is the determination of the so- called “critical” failure arc having the lowest safety factor, This may be done by successive trials of various arcs, or in the case of homogeneous slopes, by computations illustrated in appendixes D, E, F, G, and H.

b. Method of Slices. In this method a circular arc is assumed to satisfactorily represent the sliding surface. The soil mass above the sliding surface is divided into vertical slices. The forces acting on each slice are earth pressures on the sides of the slice, water pressures on the sides and bottom, intergranular force across the surface of sliding, and cohesion along the surface of sliding. Various assumptions are commonly made as to which forces are acting and the results differ some­what depending upon the assumptions made. Three common assumptions as to forces acting are shown on Plate No. D2, appendix D. The assumption most used in the past has been Case (a) in which the forces on the sides of the slices are assumed to be in equilibrium and only the forces on the base of the slices are considered. Refer to Plate No. D l, appendix D. This assumption is, in general, satisfactory but may be considerably in error in some cases. The factor of safety, regardless of what assumptions are made as to the forces acting, is determined by comparing the moment of resistance offered by cohesion and friction at the failure surface to the moment of the. soil mass about the center of the . assumed failure arc. It can be seen that this method is readily adaptable to slopes where the failure surface may pass through several materials of different strengths.

c. ^-Circle Method. In the analysis of homogeneous slopes by circular arc methods the inter­granular forces across the assumed failure surface act at an angle 0 to the normal forces across the arc, where 0 is the angle of internal friction of the soil, and are tangent to a circle of radius r sin 0, whose center is the center of the assumed failure arc which has a radius r. I t follows that the re­sultant total intergranular force will also act at the same angle to the normal and will be tangent

8

to a circle of radius slightly greater than r sin <£. However, it is assumed, conservatively, that the resultant is tangent to the circle of radius r sin </>. This then is the basis for the graphical procedure for the ^-circle method of stability analysis. As shown on Plate No. E l, appendix E, the method consists of determining the resultant of the weight of the soil mass and the water pressures, and of constructing a force polygon with this resultant and the known directional forces due to cohesion and intergranular pressures. By assuming values for friction and cohesion, the force polygon is made to close, indicating the soil mass to be stable for that condition. The factor of safety is determined by comparing design values of soil strength with those required for equilibrium for an assumed failure surface. Refer to appendix E for illustrative example.

d. Moment Method. Based on the <£-circle method of analysis, D. W. Taylor 5 obtained solutions for the most critical failure arcs in which the relations between height of slope, slope angle, and soil strength are presented on charts, Plate No. F l in appendix F. The charts are prepared for values of friction angles 4> for various slope angles. The results are expressed in terms of a dimen­sionless stability7 number, c/S. F. 7 H , where c is the required unit cohesion, S . F. the safety factor, 7 the unit weight of soil, and H the vertical height of slope. By this method the stability of homo­geneous slopes may be analyzed quite rapidly; however, the method is applicable only to certain limiting conditions of water pressures in the soil mass. Refer to appendix F for further discussion.

e. Modified Swedish Slide Method. The Modified Swedish Slide Method is particularly useful in the analysis of composite embankment and foundation sections. While more laborious in appli­cation than methods described in a, 6, c, and d above, the Modified Swedish Method is recom­mended in all cases in which the embankment or its foundation is composed of zones of different soils having widely varying shearing strengths. In brief, this method consists of locating by trial the most dangerous slide curve, dividing the sliding section'into slices, determining all forces on each slice, and then adjusting both components of strength of all soils involved by the same trial safety7 factor until closure of the force polygon is obtained. The procedure is explained by illustrative example contained in appendix G.

/. Wedge Method. In the wedge method, illustrated in Appendix H, the arcs are replaced by plane sliding surfaces. The factor of safety7 is determined by analyzing various wedges and equat­ing the horizontal resultant of active and passive earth pressures to the soil strength along the horizontal plane.

g. Hydraulic Fills. The embankments of hydraulic fill dams are constructed of a mixture of soil and water pumped to the site and deposited in such manner that segregation takes place whereby7 the fine-grained materials form a central core and the coarse-grained materials an outer shell. The core thus formed is initially fluid and cohesionless. As time passes, it consolidates and compacts, resulting in a high degree of watertightness. The shell, which is composed of coarse heavier material, is generally pervious and provides stability to the structure.

Gilboy 6 obtained the following equations for the stability7 of a hydraulic fill dam assuming a liquid core:

^s _ ( C - A ) ^ í T + W + ^ C ^ ^ f C ^ B ^ T + A i ,v (1+C2) - ( C - A ) ( C - B ) -------------- ------- ------- w

A factor of safety is usually determined by solving for R in the foregoing equation and com­paring that value with the ratio of the actual weights of core and shell materials.

A safety factor with respect to strength of the shell material may be determined by solving for B in the equation and comparing the shell strength for equilibrium with the available strength, as follows: S. F.=jB/cot <t>, where <f> is the available friction angle of the shell. Typical analysis of a hydraulic fill dam is illustrated in appendix I and a nomogram permitting a rapid solution of Gilboy’s equation is given on figure 2, Plate No. II.

9

h. Special Conditions. The danger period of failure in earth dams usually occurs during and immediately following construction or after a rapid drawdown of water wherein embankment materials are subjected to a variety of moisture conditions and seepage forces, which in combi­nation may be the most critical for the structure. These conditions which are the principal factors in stability analyses of fills are discussed as follows:

Moist Condition. Stability analyses in which the soil is considered to be in a moist condition would be typified by those on an embankment immediately following construction and before saturation has had an opportunity to take place. The moist weight of soil is also used in analyses for other conditions where the soil lies above the zone of capillary saturation, or above the water table in the case of foundation soils. The intergranular pressures of soils that are moist result from the gravitational pull on the soil grains and attached water, and the effective weight is there­fore the combined weight of dry soil plus the weight of water held in the soil by capillary action. This effective unit weight, ym is: 7-“7*(lfO(iSo+1)----------- ------- (6)

Capillary Saturation. Analysis for this condition assumes the soil to be completely saturated and that no seepage or evaporation occurs. Such a condition might exist in very cohesive clays, particularly for an excavated slope above the water table. The saturated weight of soil may be used for soils in the zone of capillary saturation, for the condition of sudden drawdown as dis­cussed later, and below tho water surface provided seepage forces are taken into account. The effective weight of a soil saturated by capillarity is equal to the combined weight of soil and water in the voids. The effective saturated unit weight ye„ is:

G+eyc'-y* i+e ( 7 )

Complete Submergence. Slopes which are completely submerged may be analyzed for stability using the submerged or buoyed weight of soil. This weight is also applicable to soils below the water table, where no seepage forces exist. Another method that may be applied to this condition is to use the saturated weight of soil and include the water forces separately. A soil entirely sub­merged has an effective weight equal to that of the saturated soil reduced by the buoyance of the displaced water. The effective unit weight, y b, due to complete submergence is:

76 = 7 «G - ll + e (8)

Partial Submergence. The case of a slope subjected to fluctuations in pool elevation may be analyzed for various pool levels to determine the stability at the most critical pool level. In the analysis, submerged weights are used below the water table and moist or saturated weights above. The safety factor of the slope is a function of pool height versus height of slope, and the minimum value may be obtained as shown on Plate No. J l , appendix J.

Incompletely Consolidated Soils. In the construction of embankments on cohesive foundation soils, it sometimes happens that only partial consolidation of the foundation will take place during the construction period. Since this condition may materially reduce the developed strength of the foundation and thus adversely affect the stability, an evaluation of partial consolidation is usually made in the stability analysis. This is accomplished by first making an estimate of the percentage consolidation to be expected at the end of construction or at the time in the life of the structure under analysis. Assuming the soils to be normally consolidated, the effective weight acting at any point in the foundation will be the weight of foundation soils above the point plus the weight of embankment soils, multiplied by the percentage consolidation. A stability analysis based on this assumption is shown on Plate No. J2, appendix J.

10

Sudden Drawdown. Slopes composed of poorly draining soils may be saturated by high water levels and then the water level lowered at a fairly fast rate. If the speed of drawdown is rapid compared to the rate at which water can escape from the pores of the soil, an unbalanced hydro­static force results which may cause failure of the slope. This is the condition analyzed for the stability of a slope subjected to sudden drawdown. During periods of submergence, the soil within the slope below the saturation line has a net effective unit weight as noted above for complete submergence. When the body of water is removed from the slope the excess water drains out of the soil. The soil above the free water zone then has a moist unit weights The soil below has a saturated unit weight in regard to stress computations; this saturated unit weight is modified by seepage forces or water pressure in regard to strength computations. In general, the computations of slope stability are complicated because of the uncertain location of the free water table within the soil and the uncertain distribution of seepage forces or water pressures. These uncertainties are caused by the rate of drawdown, permeability of the soil, and the soil strength characteristics.

The rate of drawdown within storage pools is governed by discharge capacities of the spillway and outlets. The maximum drawdown is that resulting from maximum pool surcharge with all outlets open. Using these conditions with the hydraulic characteristics of the spillway, outlet, volume of reservoir, and stream inflow, the pool drawdown hydrograph may be obtained. The drawdown of floodwaters against dikes and levees may either be based on experience records or hydraulic computations of the stream. The most severe condition is instantaneous drawdown, which may be approached if the rate of drawdown is fast and the soil has a very low coefficient of permeability.

The pore-water pressures created by drawdown in a soil subject to insignificant volume change when going from the submerged state to moist state can best be studied by use of flow-nets. The pool is lowered a given distance in a certain interval of time and a flow-net is drawn for that condi­tion. The pool is then lowered another increment and the free water surface for the next flow-net is obtained by determining the distance which the free water surface will travel in the increment of time considered. This distance A along any flow line will be approximately:

A=(j[~) i t = k ' i t .................................................. ............(9)Having determined the new location of the upper free water surface in the soil, a new flow-net is drawn and the step process repeated. I t should be noted that the effective permeability coefficient k' may be high even for soils with a low value of k because for such soils m will be low. However, for such soils little water will drain out and the moist weight will be nearly equal to the saturated weight.

The pore-water pressures created by drawdown in soils subject to significant volume change when going from the submerged to moist condition are not subject to exact study. Soils of this type are clays, especially fat clays, loose silts, and soils having no volume change but trapped air in the soil voids. During submergence the air is compressed by the water pressure. With removal of the pool the air tends to expand and force water out of the soil equal to its change in volume caused by the removal of the pool water pressure. This results in a situation similar to that in which the soil is itself compressible. For soils of this type, saturated unit weights should be used for deter­mining stresses and submerged weights and undrained shear test results should be used for strength determinations.

Correction for Tension Cracks. Experience has indicated that the upper portion of all cohesive slopes is initially in a state of tension. The exact depths to which tension extends cannot be com­puted, except for a very simple case of an infinite slope. For the limiting case of a 90° slope, if the vertical depth of the tension zone is designated as H ey it may be determined as follows:

tan ( 4 5 + 0 ......... ................................................. (10)U

It is believed that this equation would give a reasonable value of the depth of tension in an embank­ment, with the limitation that Hc should not exceed one-half the height of slope. In general, it is not safe to assume that earth can resist tension stress, but rather that failure will occur in the tension zone, resulting in more or less vertical cracks. However, it is possible in some soils for cracks to exist to greater depths than that of the tension zone; these are caused by excessive shrink­age. In any stabilitv study the depth of cracking should be based upon field observations if at all possible.

The effects of these vertical cracks are threefold. First, the length of the failure arc along which cohesion is effective is reduced. Second, the overstressing effect is reduced by the elimina­tion of the volume beyond the crack that would otherwise be included in the sliding mass. Third, if the soil above the crack is not submerged then rainwater can enter and create an added driving force, tending to cause failure. However, in most cases the second and third effects practically compensate each other and stability analysis can be performed using the reduced length of arc for determination of cohesion and neglecting the other factors. Computations for depth of tension cracks are shown on Plates Nos. D l, E l, and Gl in appendixes D, E, and G.

Stratified Slopes. Stratified slopes may be analyzed by means of appropriate methods discussed above using the strength of each layer for that part of the assumed failure surface within it. In all cases the failure surface should have as much of its length within the weakest material as possible; in some cases it may be necessary to use composite surfaces to accomplish this. Consideration should be given to the difference in the stress-strain characteristics of the various layers. I t is possible that in some cases a stronger and more brittle material will fail at less strain than a weaker and more plastic material. The use of the full strength of the soils in these cases would indicate a factor of safety much higher than that which would actually exist.

Earthquake Forces. In geographic regions which are seismically active, consideration should be given to the effect of an earthquake upon the stability of embankment slopes and earth foundations. A sudden movement of the foundation, such as results from a seismic shock, requires that the superimposed structure move in a similar manner if rupture is to be avoided. Consequently, the forces necessary to overcome the inertia of the mass induce additional stresses within the embank­ment and foundation. The magnitude of these additional stresses is determined principally by: (1) the severity of the earthquake; (2) the mass being moved; (3) the elasticity of the mass: and (4) the earthquake effect on the water load.

The magnitude of forces induced by seismic shock is proportional to the acceleration, which has been observed to vary from 0.0037 g to 1.0 g. For design purposes, the acceleration is generally accepted as 0.10 ¿( applied horizontally and neglecting the minor vertical acceleration.

Although Japanese investigators have indicated that critical stresses, induced by shock, occur at a variable height in an embankment, little is known of the exact distribution. The method of including a horizontal force tending to cause failure equal to one-tenth the weight of the mass appeal’s to be the best scheme presently available for introducing earthquake forces into stability analyses. Modification of this figure may be made but should depend upon regional seismic history, proximity of active faults, and character of foundation. Where a pool is being maintained, it may be necessar} to investigate the effect of the increase in water load caused by an earthquake. Westergaard7 has investigated this condition and developed approximate formulas for estimating the magnitude and distribution of these new water forces over the upstream face of a concrete structure.

Earthquake effects have been observed to vary locally according to geological features. Loose, wet earth and artificial fills apparently suffer more damage,-while rock outcrops appear less sus­ceptible to movement and damage. Thus, cohesionless, loose, wet soils may be in danger of liquefaction if near their critical void ratio, and flow slides may be possible. Little, if any, informa-

19

tion of a quantitative nature is available concerning the development of excess pore pressures in fine-grained impervious foundations. In such cases, care should be taken to avoid probability of a dangerous condition resulting.2- 08. DESIGN CRITERIA AND SAFETY FACTORS

Conservative design of load-carrying structures requires that there be a reserve of strength to provide a margin of safety against the unexpected and the unknown. In addition, some allow­ance must be made for possible variations in computed strength resulting from use of approxi­mations and assumptions in analysis. Thus, the margin of safety is seldom absolute, although it may be bracketed within the limiting values.

I t is usually more convenient to express the excess of strength in terms of the principal strength­controlling factor; for example, if stability is dependent on load, it is convenient to consider the ratio of maximum safe load to actual load as a safety factor. Similarly, other safety factors are possible, each of which defines relative stability in relation to certain strength-controlling factors and for definite conditions of stability.

In dealing with solids, there is usually limited knowledge of stress distribution and also it is often problematical how much strain can be tolerated before a condition of failure occurs. Where investigations are based upon an elastic theory, the analysis is not valid when the stresses exceed the elastic limit of the soil. Therefore it becomes necessary to establish the criterion of failure for each method of analysis, and, furthermore, to establish'a minimum acceptable value of the safet}’ factor for each method.

Although failure is implied when the safety factor falls below unity, the true condition of incipient failure may occur at smaller or greater values, dependent upon accidental factors and upon the validity of any additional assumptions and approximations.

While conservative design requires that a minimum safety factor be obtained for a given set of conditions, care must be taken to prevent uneconomical design by introducing a safety factor in too many places. Thus, estimation of soil strength should not be too conservative and stresses assumed abnormally high at the same time. Where little information of soil characteristics and other factors affecting the results of analysis is available, the safety factor obtained should be considered as of a qualitative nature and the final decision regarding the suitability of a given design must be made on a basis of judgment and experience. In a particular area, the knowledge gained by past experience on the behavior of embankments and the analyses of slides which have occurred should be used to the fullest extent in determining minimum acceptable safety factors.

a. Types of Safety Factors. There are, in general, two types of safety factors with regard to the stability of earth slopes. The first is computed in regard to the static equilibrium of the mass above the failure surface. The second type is one which is applied to the strength properties of the soil.

Safety Factor with Respect to Equilibrium. One method of computing the safety factor with respect to static equilibrium which is used principally for circular arc failure surfaces, is to consider the individual forces on the segments of the failure mass. The safety factor is the ratio of the sum of the moments of the forces resisting motion to the sum of the moments of forces causing motion, as follows:

q jr Sum of resisting forces (SC+SiV tan <j>)r ,n vSum of overturning forces”” Wd .................. '

where 2C is the total cohesion on the failure surface, SiV is the sum of the normal intergranular forces, <t> is the angle of internal friction, r is the radius of failure surface, W the effective weight of the mass and d is its moment arm. In this method, design values of soil strength C and <f> are

18

selected and used in appropriate parts of the analysis. The assumption is made that full soil strength is capable of being mobilized at the point of failure. Analyses using this type of safety factor are shown on Plates Nos. D i, J l , J2 and J3, appendixes D and J.

Another method of computing the safety factor with respect to static equilibrium is by con­sidering the resultants of forces on the segments of the failure mass (used principally for sliding planes and wedges). The safety factor is expressed as the ratio of the sum of passive resultants to the sum of the active resultants. For example, consider the case of an active wedge, a sliding block, and a passive wedge as shown on Plate No. H i, appendix H. The safety factor may be ex­pressed as follows:

g jr an ^ (12)

where P p is the passive pressure of the resisting wedge, C + N tan 4> is the soil strength along the base of the sliding block, and Pa is the active pressure exerted by the active wedge.

Safety Factor with Respect to Strength. A safety factor with regard to the strength of the soil may be one of several types. However, in all cases the adopted design strength values are divided by a safety factor with respect to strength and the resulting values are used in the analysis to arrive at an equilibrium safety factor equal to 1.

The safety factor may be computed with regard to total strength of the soil. In this method the same factor is applied to both cohesion and friction, as follows:

cD = S .F . and tan 4>d ztan 4>

"S .F . (1 3 )

where c and tan <f> are adopted design values and cD and tan <t>D are the values used in the stability analysis, solving for different values of safety factors until the condition of equilibrium is reached. This method has been used in the analyses shown on Plates Nos. E l, Gl and H i, appendixes E, G, and H.

Another method is to determine the safety factor with respect to cohesion alone. The friction^ <t>, is assumed to be fully mobilized and the design cohesion value is reduced by the safety factor^

cCd~ S F * Similarly, the safety factor with respect to friction alone may be determined by assum­ing the cohesion to be fully active. In this case the friction value used is tan

Still another method, based on the belief that cohesion should not be relied upon as fully as friction, consists of using different safety factors for cohesion and friction. In this case

cD= cS~ F~. and tan 4>D tan 4>

S.4>F. (14)

where S.CF. and S.<t>F. are safety factors with respect to cohesion and friction, respectively. A higher value is usually used for S .eF. than for S.4>F.

Of the foregoing methods, the safety factor with respect to total strength is to be preferred, since it is doubtful that the soil design strengths selected are accurate enough to warrant special consideration in prorating safety factors.

b. Specific Values. For cases where the shearing strength of the soil is uncertain, a minimum safety factor of 2 is recommended. This applies to brittle soils, subject to progressive failures, and some plastic clays not thoroughly tested, and loose uniform sands that may lose strength due to practical liquefaction.

For cases where the shearing strength of the soil is more certain, a minimum safety factor of 1.5 is recommended where it can be economically obtained.

14

Because the assumption of sudden drawdown is conservative, safety factors not less than 1.25 and 1.00 are recomjmended when drawdown is from spillway level and from maximum surcharge level, respectively. For earthquake forces, a safety factor of 1.25 is recommended. Earthquake forces and sudden drawdown effects need not be combined, but rather should be considered sepa­rately.

For cases of long embankments or cut slopes, it may be more economical to use a safety factor of 1.25 or slightly ifess, providing local failures of the earthen structure during construction will be involved and where the savings created by such a low safety factor will more than offset the cost of replacement of materials. This is especially true of embankments constructed on com­pressible foundations, where a gain in strength will result as consolidation progresses.

15

LIST OF SYMBOLSThe symbols that follow are used throughout this chapter. Only symbols that are frequently

repeated are given. Other terms are defined in proximity to their use within the text.Symbol TermA = cotangent of angle of core slope with horizontal.B = cotangent of angle of internal friction of shell.C = cotangent of angle of outer slope of shell with horizontal.C = cohesion, resultant or total.CD = cohesion, developed per unit area. c — cohesion, per unit area (from test sample). cD = cohesion, adopted design value. d = length of moment arm.E = modulus of elasticity.E = horizontal earth pressure (active) as forces acting on sides of slice.e =void ratio.Fn = normal soil force.F n = normal soil force (effective)./ = frictional force.6 = specific gravity of solids.g = acceleration of gravity, 32.2 ft/sec/sec.H = height, depth or thickness with subscripts denoting location.H c = vertical depth of tension zone (cracking). h = differential head with subscripts.i = hydraulic gradient (along a flow line).i c == critical hydraulic gradient.J = seepage forces.K = ratio between intensities of horizontal and vertical pressures at a point in the embankment.Jc = coefficient of permeability.kv = coefficient of permeability (vertical).kn = coefficient of permeability (horizontal).k! = applicable coefficient of permeability.L = length of arc or length of seepage path.L = one-half the base width of the structure.m = ratio of water drained out of soil to water present at saturation.N = normal load.N f —2l factor derived from the angle of slope and the characteristics of the most critical failure arc. n = porosity.n = variable used in Taylor’s <£-circle method.P = horizontal hydrostatic force.P —maximum unit pressure on foundation.Pa = active wedge drive force.P p = passive resistance (total force). p = intensity of surface strip load.R =ratio of unit weight of fluid core to unit weight of shell. r = radius of circle.S = specific gravity.

16

TermSymbolS = total shearing stress.S. F .=safety factor. s = shear strength per unit area.T = total tangential force. t = any time increment or duration of time of load application. U = upward hydrostatic forces (uplift).W = downward force from weight of material.GreekSymbola — designated angle in graphical solution. a —active wedge critical angle.P = designated angle in graphical solutions.P = passive wedge critical angle.P = angle of obliquity.7 = density (unit weight).y b = submerged soil weight.7 c = saturated soil weight.7 m = moist soil weight.yw = density of water 62.5 lbs./cu. ft.A = increment or small part of.0 wangle of slope with horizontal. t == shear at rock surface.n = Poisson's ratio.n = coefficient of absolute viscosity.<t> = angle of internal friction.<j)d = effective angle of internal friction.<f>D = developed angle of internal friction.<pw = weighted angle of internal friction.S = summation, arithmetic total.

17

LIST OF REFERENCES1. “Stress Distribution in a Loaded Soil with Some Rigid Boundaries,” by G. Pickett, p. 35,

Part II, Proceedings, 18th Ann. Meeting, Highway Research Board, 1938.2. “Notes on Principles and Applications of Soil Mechanics,” Corps of Engineers, Fort Peck,

Montana, June 1939.3. “A Problem of Elasticity Suggested by a Problem in Soil Mechanics: Soft Material Reinforced

by Numerous Strong Horizontal Sheets,” by H.„ M. Westergaard, Contrib. Mechanics of Solids, Stephen Timoskenko, 60th Anniversary volume, New York, The Macmillan Co., 1938.

4. “The Application of Theories of Elasticity and Plasticity to Foundation Problems,” by L. Jurgenson, Journal, Boston Society of Civil Engineers, July 1934.

f5. “Stability of Earth Slopes,” by D. W. Taylor, Journal, Boston Society of Civil Engineers, July 1937.

f6. “Mechanics of Hydraulic-Fill Dams,” by G. Gilboy, Proceedings, Boston Society of Civil Engineers, July 1934.

7. “Water Pressures on Dams During Earthquakes,” by H. M. Westergaard, Transactions, American Society of Civil Engineers, pp. 418-433, vol. 98, 1933.

8. “On the Stability of Foundations of Embankments,” by L. Jurgenson, Paper G-8, Vol. II, Proceedings, First International Conference on Soil Mechanics and Foundation Engineering, Cambridge, Mass., 1936.

9. “Calculation of the Stability of Earth Dams,” by W. Fellenius, Transactions, Second Congress on Large Dams, Vol. IV, Washington, D. C., 1938.

10. “Tenth Progress Report, Cooperative Triaxial Research Program,” by D. W. Taylor.

t Copyrighted m ateria l from th e se re feren ces h a s b een in clu d ed in th is report by p erm issio n of the copyright owners. R eproduction in w h ole or in part i s prohibited ex c ep t by p erm ission . Inquiry re la tiv e thereto should b e d irected to T he Adjutant G eneral, Department o f the Army, W ashington, D. C. 20315.

18

PLATE NO. Al PLATE NO. A2 PLATE NO. A3 PLATE NO. A4 PLATE NO. A5

PLATE NO. Bl

PLATE NO. Cl

PLATE NO. D l PLATE No. D2

PLATE NO. E l PLATE NO. E2

PLATE NO. FI

PLATE NO. Gl

PLATE NO. HI

PLATE NO. II

PLATE NO. J l PLATE NO. J2 PLATE NO. J3

LIST OF ILLUSTRATIONS

APPENDIX A

Elastic Method of Stability Analysis.Influence Lines—Uniform Tangential Strip Load N x.Influence Lines—Uniform Tangential Strip Load N,.Influence Lines—Uniform Tangential Strip Load Txt.Influence Lines—Maximum Shear at Rigid Surface TIt.

APPENDIX B

Stability of Shallow Plastic Foundation on Rigid Boundary.

APPENDIX C

Rendulic Method of Stability Analysis.

APPENDIX D

Swedish Slide Method of Stability Analysis.Circular Arc Analysis Method of Slices.

APPENDIX E

^-circle Method of Stability Analysis.Development of a Flow Net.

APPENDIX F

Moment Method of Stability Analysis.

APPENDIX G

Modified Swedish Method of Stability Analysis.

APPENDIX H

Wedge Method of Stability Analysis.

APPENDIX I

Stability Method for Hydraulic Fill Shells.

APPENDIX J

Partially Submerged Slopes Variation of Safety Factor with Pool Level. Stability Analysis Using Seepage Forces.Sudden Drawdown Stability Analysis with Use of Flow Net.

19

APPENDIX AAPPLICATION OF ELASTIC THEORY

The analysis of the foundation consists of determining the stresses at selected points due to the weight of material above. In the example on Plate No. Al stresses were determined at several points along the junction between the embankment and foundation and at the rigid boundary at the base of the foundation. The embankment was first divided into a number of segments of convenient thickness as shown in figure 1. The strips were further subdivided in horizontal extent at boundaries between different materials and at the saturation line. Considering first the points at the base of the embankment, the normal stresses and shearing stresses due to the influence of each segment were computed for each point. Typical computations for two points on the center line of the dam are shown in tables 1 and 2 placed at the end of this appendix.

The effect of seepage may be estimated as follows. From the lines of equipressure, Plate No. Al, figure 5, the hydraulic pressures on the embankment strips as shown in figure 6 are determined. The pressures along the top and base are averaged and combined with saturated soil weights to obtain the net weight of the segment. The horizontal and hydrostatic pressures are averaged and then distributed over the base of the embankment strips as tangential strip loads. Typical com­putations are shown in figure 6. The stresses N x, N z, Txz induced in the foundation are then determined from the influence charts shown on Plates Nos. A2, A3, and A4.

Computations for tangential strip loads are shown in table 2 for the point 1 on the center line of the dam at the top of the foundation. The stresses due to each of the embankment segments were determined and then the individual values are summed algebraically to give total values N„ Nz, and Txz for each point in the foundation. It is necessary to observe caution about the sign of the individual stress values. N z and N x due to the segment weights are compressive stresses; hence their values are considered positive. The signs of N z and N x due to tangential strip loads depend on the direction of the tangential force and the position of the point with respect to the center line of the segment; compressive forces are positive and the proper sign may be determined by reference to Plates Nos. A2 and A3. Values of Txz due to weight of segment are positive if the point lies to the left of the center line of the segment. Values of Txt due to tangential strip loads are positive for directions of seepage forces showm in Plate No. A4. Oppositely directed seepage forces would produce negative values of Txz.

The value of maximum shear Tmax is computed for each point in accordance with the following formula:

Tm„ = (N z- N xy+ T *xz__________________ _________ (15)Computations for a point on the bottom of the foundation at the center line of the dam are shown in table 2. Values of Txz due to the weight of each segment are determined from the influence charts on Plate No. A5. Values of Txz due to tangential strip loads are determined from the in­fluence chart on Plate No. A4; this latter chart is for a semi-infinite foundation; therefore, values computed at the rigid surface will be approximate. All values of Txz due to segment weight and tangential loads are added algebraically to obtain the maximum shear stress, Tmax, at each of the selected points.

Using the values of maximum shear stress computed for the various points along the top and bottom of the foundation, contours of equal shear stress are drawn as shown in figure 1, Plate No. Al. If necessary, aid in stress contour sketching may be obtained from photoelastic studies of gelatin models. It will be noted that stress contours are shown in figure 1 for the embankment as

À-1

well as for the foundation. These were obtained by means of stress calculations at numerous points that were made in the manner described in the preceding paragraphs for points 1 and 2.

In the example on Plate No. Al, the effect of any shear stressés that might exist in the foun­dation before construction has been neglectèd. Such stresses will exist if the coefficient of earth pressure at rest, K 0, for the foundation soils is not unity. If necessary, these stresses can be esti­mated and included in the analysis. Also, if the shear stresses at the base of fill are estimated to be greater than that given by the above elastic strip method, then the difference between the esti­mated and computed shear stresses can be applied to the foundation surface as surface tangential loads.

The ability of the foundation soil to resist rupture depends on its shearing strength. If the embankment is more rigid than the foundation, sufficient deformation of the foundation can occur by yielding to cause failure of the embankment without a foundation failure. Furthermore, any overstressed zone in the foundation will yield plastically and transfer the stresses which it cannot resist to the less stressed surrounding areas. Therefore, stresses determined by this method are Valid up to about the yield strength of the soil and for this reason it is only possible to determine the safety factor (shearing stress divided by soil strength) at a point.

À-2

A-3

T a b l e 1.— Typical computation for elastic theory method of stability analysis computations for maximum shearing stresses

P O IN T NO 1.—T O P OF F O U N D A T IO N —E L E V A T IO N 600.0

Segment No.

1

HW idth of seg­m ent ‘ b ” F t

2

zF t

3

XF t

4

Seg­m entnet

weightkips

5

Vertical uniform strip loads extending from — OO to + OO Tangential uniform strip loads extending from —» tO + oa

lOzb

6

lOxb

7

n ,

8

n ,

9

T x t •

10

N,(5)XC8)

11

N ,(5) X (9)

12

Tz.(5 )X(10)

13

zfb

14

xfb

15

n ,

16

n x

17 18

U nithori­

zontalloadkips

19

AT. (19) X

(16)

20

N x(19) X

(17)

21

Nx,(19) X

(18)

22

1 35.0 87. 5 0.0 1.400 25.0 0.0 0.46 0.02 0.00 0. 645 0.028 0.0002A ....................... ............. ............ 21.5 75.0 - 3 8 .5 1.042 34.9 - 1 7 .9 0.23 0.07 - 0 . 11 0. 240 0.073 - 0 . 115 3. 49 1. 79 0.11 0.03 - 0 .0 6 0.236 + 0.026 + 0.007 -0 .0 1 4R 23.0 75.0 + 6 .0 1.400 32.6 + 2 .6 0. 36 0.03 +0. 03 0. 504 0.042 + 0.042o .............- ........... 14.0 75.0 + 43.0 1.375 53.6 + 30.7 0.13 0.05 + 0.07 0.179 0.069 +0.0963A __________ ______________ 45.5 62.5 - 4 5 .5 0.974 13.7 - 1 0 .0 0.46 0.16 - 0 . 22 0. 448 0.156 - 0 . 214 1.37 1.00 0. 21 0.15 - 0 .1 6 0.214 + 0.045 + 0.032 -0 .0 3 4B . _____ . 16. 5 62.5 +16. 5 1.400 37.9 + 10.0 0. 29 0.02 + 0.07 0.406 0.028 + 0 098O 28.5 62. 5 + 61.0 1.375 21.9 -1-21 4 0.17 0.13 + 0.15 0. 234 0.179 +0. 2064A .......... ................... ................. . 71.5 50.0 - 5 4 .0 0.873 7.0

i *1. i- 7 . 6 0.69 0. 29 - 0 . 24 0.603 0. 253 - 0 . 210 0. 70 0. 76 0.24 0.35 - 0 .3 0 0.202 + 0.048 +0.071 -0 .0 6 0

B 53. 5 50. 0 + 71.0 1. 375 9.4 +13. 3 0. 28 0. 26 +0. 24 0. 386 0. 358 +0. 3315A ................................................... 64.5 37.5 -1 0 0 .5 0.841 5.8 - 1 5 .6 0.10 0. 27 - 0 .1 5 0.084 0. 227 - 0 . 126 0. 58 1.56 0.16 0.60 - 0 . 25 0.136 + 0.022 + 0.082 -0 .0 3 4B ____ _____ _____ ____ ______ 34.5 37.5 - 1 . 5 0.917 10.9 - 0 . 4 0.79 0.15 -0 .0 1 0. 725 0.138 -0 .0 0 9 1.09 0.04 0.00 0.02 - 0 .1 6 0. 323 0.000 + 0.006 -0 .0 5 20 65. 5 37. 5 + 98.0 1.375 5.7 + 15.0 0.11 0.29 + 0.17 0.151 0.399 + 0 2346A ................................................... 77.0 25.0 -1 3 2 .0 0.760 3.3 - 1 7 .1 0.02 0.18 - 0 .0 5 0.015 0.137 - 0 . 038 0. 33 1.71 0.06 0. 73 - 0 .1 6 0.074 + 0.004 + 0.054 -0 .0 1 2B ..................... .............................. 48.5 25.0 - 6 . 5 0.910 5.2 - 1 . 3 0.95 0.43 - 0 .0 3 0. 864 0. 391 - 0 . 027 0.52 0.13 0. 03 0.10 - 0 .4 1 0.390 + 0.012 + 0.039 -0 .1 6 0O .............. 83. 5 25.0 +125. 5 1.375 3.0 + 15.0 0.03 0.23 + 0.08 0.041 0. 316 +0.1107 A .................................................. 93.5 12.5 -1 6 3 .0 0.742 1.3 - 1 7 .4 0.00 0.08 - 0 .0 2 0.000 0.059 -0 .0 1 5 0. 13 1.74 0.02 0.79 - 0 .0 6 0.055 +0.001 + 0.044 -0 .0 0 3B .......... ................. ....................... 57.0 12.5 - 1 3 .0 0.841 2.2 - 2 . 3 0.99 0.72 - 0 .0 1 0. 833 0. 605 -0 .0 0 8 0. 22 0. 23 0.03 0. 25 -0 .7 1 0.411 + 0.012 + 0.103 -0 .2 9 20 106.0 12. 5 +150.0 1.019 1.2 +14. 2 0.00 0.13 + 0.04 0. 000 0.132 +0.0418A .......... ................... ................... 126.5 0.0 -1 79 . 5 0.740 0.0 - 1 4 .2 0.00 0.00 0.00 0.000 0.000 0.000 0.00 1.42 0.00 1.10 0.00 0.059 0.000 + 0.065 0.000B .................................................... 35.0 0.0 - 1 8 .0 0. 760 0.0 - 5 .1 1.00 1.00 0.00 0. 760 0. 760 0.000 0.00 0.51 0.00 0. 70 - 1 .0 0 0.280 0.000 +0.196 -0 .2 8 00 ............. ............................ ......................... 15.5 0.0 + 32.5 0.680 0.0 + 21.0 0.00 0.00 0.00 0.000 0.000 0.000 0.00 2. 10 0.00 0.64 0.00 0.782 0.000 -0.0.50 0.000D 129.0 0.0 + 177.0 0.780 0.0 +13. 7 0.00 0.00 0.00 0. 000 0.000 0. 000

Columns 8, 9, and 10 were determ ined from influence charts in + 7.118 +4. 350 +0. 396 +0.170“ Notes on Principles and Applications of Soil M echanics," F ort Peck D istrict, C E .

Columns 16, 17, and 18 were determ ined from influence charts,

+0,170 +0.649 - 0 . 941

Trr>» = ( ^ |^ y + ( - 0 . 5 4 5 ) * = 1 . 6 1 kips7. 288 +4. 999 - 0 . 545

Plates No. A3, A2 and A4, respectively. - 4 . 999

2. 289 =- N .- N *

T able 2.— Typical computation for elastic theory method of stability analysis computations for maximum shearing stressesPOINT NO. 2 —BOTTOM OF FOUNDATION—ELEVATION 550.0

Segment No.

(1)

^ width of seg­ment “b” Ft

(2)

ZF t

(3)

XFt

(4)

Vertical uniform strip loads from — » to + » Tangential uniform strip loads extending from— 00 to + 0»

x —b

(5)

x-\-b

(6)

x -bz

(7)

j+ 6z

(8)

Tzj-blPz(9)

z(10)

HP(9)-(10)

(ID

Segm en tnetw eigh tkips(12)

Txt(1UXQ2)

(13)

z/6

(14)

x\b

(15)

Tr,(

(16)

Unithorizontalloadkips(17)

T(16)X(17)(18)

1 35.0 137.5 0.0 -35.0 +35.0 0. 254 0. 254 0.409 0. 409 ; o.ooo 1.400 0.0002A.......................... ................ 21.5 125.0 -38.5 -60.0 -17.0 0.480 0.136 0.343 0. 431 -0 . 088 1.042 -0.092 5.81 -1.79 0.02 0.236 -0.005B 23.0 129.0 +6.0 -17.0 +29.0 0.132 0. 224 0.432 0.415 +0. 017 1.400 +0. 024fj 14.0 125.0 +43.0 +29.0 +57.0 0. 232 0. 455 0.414 0. 351 +0. 063 1.375 +0. 0873A................................................. 45.5 112.5 -45.5 -91.0 0.0 0.810 0.000 0. 238 0.441 -0 . 203 0. 974 -0.198 2.47 -1.00 0.06 0.214 -0.013B ............... 16.5 112.5 +16. 5 0.0 +33.0 0.000 0. 293 0. 441 0. 398 +0.043 1.400 +0.0600 ...................... 28.5 112.5 +61.0 +32.5 +89.5 0. 289 0. 795 0. 400 0.242 +0.158 1.375 +0. 2174A................................................. 71.5 100.0 -54.0 -125.5 +17.5 1.255 0.175 0.137 0. 424 -0 . 287 0.873 -0 . 250 1.40 -0.76 0.13 0.202 -0.026B ..................... 53.5 100.0 +71.0 +17.5 +124.5 0.175 1.245 0. 424 0.140. +0. 284 1.375 +0.3906A................................................. 64.5 87.5 -100.5 -165.0 -36.0 1.880 0.412 0.068 0. 365 -0 . 297 0. 841 -0 . 250 1.35 -1 . 56 0.19 0.136 -0.026B....... ................................... . .... 34 . 5 87.5 -1 .5 -36.0 +33.0 0.412 0. 380 0. 365 0. 374 -0 . 009 0.917 -0.008 2.54 -0.04 0.02 0.323 -0.006c ..................... 65.5 87.5 +98.0 +32.5 +163.5 0. 370 1.870 0. 374 0.068 +0. 306 1.375 +0.420

s*, 6A._....... .................................... 77.0 75.0 -132.0 -209.0 -55.0 2. 790 0. 730 0. 029 0. 263 -0 . 234 0.760 -0.178 0. 98 -1.71 0. 23 0.074 -0.017| B .......................................... 48.5 75.0 -6 .5 -55.0 +42.0 0. 730 0.560 0. 263 0.318 -0.055 0.910 -0.050 1.54 -0.13 0.08 0. 390 -0.031

If** C - _____ _____ 83. 5 75.0 +125.5 +42.0 +209.0 0. 560 2. 790 0.318 0.029 +0. 289 1.375 +0. 3977A........................................ . .... 93. 5 62.5 -163.0 -256.5 -69.5 4.100 1.110 0.010 0.164 -0.154 0. 742 -0.114 0. 67 -1.74 0. 22 0.055 -0.012B — ....................- ...................--- 57.0 62.5 -13.0 -70.0 +44.0 1.120 0. 700 0.164 0. 272 -0.108 0. 841 -0.091 1.10 -0 . 23 0.15 0.411 -0.061C ................................ - 106.0 62.5 +150.0 +44.0 +256.0 0.700 4.100 0.272 0. 010 +0. 262 1.019 +0. 2678A....... ................. ...................... 126. 5 50.0 -179.5 -306.0 -53.0 6.120 1.060 0. 002 0.174 -0.172 0. 740 -0.127 0. 39 -1.42 0. 28 0.059 -0.017B — ............................ ...............- 35.0 50.0 -18.0 -53.0 +17.0 1.060 0. 340 0.174 0. 386 -0.212 0. 760 -0.161 1.43 -0.51 0.11 0.280 -0.031C - ................................................ 15.5 50.0 +32.5 +17.0 +48,0 0. 340 0. 960 0. 386 0.197 +0.189 0.680 +0.129 3.22 +2.10 0.08 0.782 +0.062D ......... .................. 129.0 50.0 +177.0 +48.0 +306.0 0.960 6.120 0.197 0.002 +0.195 0. 780 +0.152

+0. 624 -0.183-0.183

Columns 9 and 10 were obtained from influence charts on Plate No. A5.Column 16 was obtained from influencé chart on Plate No. A4.N ote.—For dimensions of dam, locations of points 1 and 2, and other details, see Plate No. Al,

4-0.441 kips= Tmmm

CONTOURS OF MAXIMUM SHEAR STRESS BY ELASTIC METHODKPS/SO. FT.

FIGURE 3.

FOUNDATION SHEARING STRENGTHBased upon quick shear strength test S =1.80 + p tan. 7°Where 5- Shear strength — Kips/sq.ft.

p= Normal strength " " "HYDROSTATIC POTENTIAL AND PRESSURE DIAGRAM

FIGURE 2.

NOMENCLATURE FOR DETERMINING STRESSES

At a point X,Z beneath a uniformly loaded long strip FIGURE 4.

POINT FACTOR OF SAFETYAverage factor of safety along Une a - b

Total strenth along a-b Total stress along a-b

= L46

TAB LE OF ADOPTED DESIGN DATAMATERIAL DENSITY-LBS PER CU FT COHESION

TONS PER SO FTTAN (|>

MOIST SATURATEDA 112 121 0.15 0.36B IIO 125 0.0 0.70C - 118 1.80 0.12

FIGURE 6.

CALCULATION OF EFFECTIVE WEIGHT AND TANGENTIAL FORCE FOR TYPICAL SEGEMENT

Weight o f saturated Material "A " Water pressure on top segment

Total effective uplift Total effective weight

195. Kips333.5528.5 420.0108.5 Kips

Weight per square foot - 841*

Unit effective horizontal pressure = P3~ P4= ('5 0 ~225)xJ25x62.5) = ,3 6 #/a '

Effect o f horizontal seepage force =(P3 P4) considered to be a uniform tangential strip toad o f !36*/sq. ft. for segement 5A applied at base o f segment, acting to the right.

STABILITY OF SLOPES AND FOUNDATIONS

ELASTIC METHOD OF STABILITY ANALYSIS

s c a l e : AS SHOWN _______ ______________________

WASHINGTON ,D.C.

OFFICE OFTHE CHIEF OF ENGINEERS CIVIL WORKS

ENGINEERING DIVISION JANUARY 1951

E M C -II9 -2 -A I PLATE A I

NOTE:Uniform tangential strip load o f intensity "p " extending

from — oo to -too, normal to chart

bv ___^ ___^ __^ i S N * OO 2 3 4 5

I I z A V\ \ \ Y. V T 7 I 7\ \ v \s \ f t X 1 7 | y<ò, T t 7 / 7 7

1AA \ Y 7 y A jL t t/ / 7J A Y z / 6, 7 / / tV \ 0 / 7 /_ t \ V \ s , / Tx \ s. C z /111 N /\ y s ^ A 7

\ X X1 T \ 'S y\ \ y Y

s (D.3y \ * N Y\ \ \\ \ \V YA \ \ As P .P

V- v Xo \ \ Y Vc. \ \ s ,\ \

\ \ xV \ YVs. \ X\ V Y Ns o\ \ N /

X\ Yz , \ sb 3 \ \ Y

\ \ S\ \\ \ V\ sY<or

k &\ \ \

\ \Nx /| \/n Iiiûc \ 7

P YV \_L_ I 1 1 \ X YM \ \1 1 ^IMx N > X \ \

\ n \ncl X N s

Tension Compression X Y(negative) ( positive) \ Y

R - \ \ X\ \\ Y

\ \ vs\

\A sAc _ \D \

-Influence values are symmetrical about £ except for signs.

t 5

10

IO II 12 13 14 15 16

NOTE:Foundation assumed to be a homogeneous, isotropic

semi-infinite elastic mass having a Poisson's ratio of 0.50

L J \ 7— 1

1/ 1 r -

-1 / i/ ! /

T~r

O y '-j.

// T

x / j 7o O Z / / f -

o V / //

/ 7y / 7~

Xs 7

O X / N x / p V a luesX /X /Y

Y /Y

As, /Z

7"3

0N O A\

YL\

NX

V X v «\ Y

Y

\ Ay\ k

v YY x

\ V- X\ X Y_

\ X Z

\ Yk

YY Y

SY_Y Y

\ ss O .n E

\

\7

- ___7

\\

YY

Y X Y,Ope-1 "Y

r

STABILITY OF SLOPES AND FOUNDATIONS

s c a l e : a s s h o w n

INFLUENCE LINES - UNIFORM TANGENTIAL STRIP LOAD Nx

W A S H IN G T O N , D .C .

OFFICE OFTHE CHIEF OF ENGINEERS CIVIL WORKS

ENGINEERING DIVISIONJ A N U A R Y 1951

EMC -119-2-A2 PLATE A 2

NOTE:Uniform tangential strip toad o f intensity "p", extending

from — oo to +oo normal to chart.

\s7 V"V-/ / 7 N sT / / / / f 7 s\ \ 7

7 T / A i \ s Nr - / 7 0 3V \ \ \7 f j i \ 7 V N\ s. \ Ns\ \ 7 7

i? ) \ \ 7 s sX p Y 7 \\ \ s N o

\ \ 'Si V s J r 7 V 7

T \ \ \ \ 7\ 77 h \ 7<7

' f \ \ \L' T X \

\ 7 \ / \ 7' 1 \ \ r \ 7" i \ r \ \ sJ > \ \V y7 \ \

V A 07:y 7 \1 t \ > \1 \ 7 7'1 7 V \i \ 7r V VV1 t 7 s, \

\ \r 7 \■ t \■ t V s 11 T \ L■ 7 1 o 7 \ tVo

\ N \ s vO\ o\ro 0>lo» iT V71 V N;7 \/n 111 pct; p VUIUCo

VVV\ 77 \ \\_ Compression \Tension

(negative) (positive)

Influence values are symmetrical about (£ except for signs.

_x.b

NOTE:Foundations assumed to be a homogeneous, isotropic

semi-infinite elastic mass having a Poisson’s ratio of 0.50

STABILITY OF SLOPES AND FOUNDATIONS

INFLUENCE LINES - UNIFORM TANGENTIAL STRIP LOAD N 2

s c a l e : as sh o w n

WASHINGTON, D.C.

OFFICE OFTHE CHIEF OF ENGINEERSCIVIL WORKS

ENGINEERING DIVISIONJANUARY 1951

EMC - 119 -2 -A 3 PLATE A 3

_x_b

NOTE:Foundation assumed to be a homogeneous, isotropic

sem i-infin ite elastic mass, having a Poisson’s ratio o f 0.50

STABILITY OF SLOPES AND FOUNDATIONS

INFLUENCE LINES - UNIFORM TANGENTIAL STRIP LOAD T xz

s c a l e : a s sh o w n

WASHINGTON ,D.C.

OFFICE OFTHE CHIEF OF ENGINEERSCIVIL WORKS

ENGINEERING DIVISIONJANUARY 1951

E M C -II9 -2 -A 4 PLATE A 4

0.09

0.08

0.07

■ k 0.05

7.0 8.0 !9.0 10.0

S T A B IL IT Y OF SLO PESAND FOUNDATIONS

INFLUENCE LINES - MAXIMUMSHEAR AT R IG ID SUR FAC E T XZ

O F FIC E OF THE CHIEF OF E N G IN E E R S

WASHINGTON, D.C.C IV IL W O RKS

ENG INE ER IN G D IV IS IO N JANUARY 1951

s c a l e : a s shown

E M C - 1 1 9 - 2 - A 5 P LA TE A 5

APPENDIX BAPPLICATION OF JURGENSON’S METHOD

A case which may be considered to fall within the category of stratified conditions is that of an embankment resting on a shallow plastic foundation as shown on Plate No. Bl. Shearing stresses are built up in the foundation due to the weight of the embankment and due to the horizontal earth pressure for the active Rankine condition. The shearing stresses are resisted by the shearing strength of the material, which in the case analyzed is the cohesion of the foundation soil, according to Jurgenson’s 8 Case A maximum tension which may be developed at the center line of the embank­ment equal to 1/2 Pa where P =maximum unit pressure on the foundation and a =1/2 depth of foundation. When the tension at the center line of the dam is less than the horizontal active earth force, E} then no tension will develop in the dam if the full value of the shear stress is developed along the base of dam. The foregoing condition, if satisfied, becomes equivalent to Jurgenson’s 8 Case D for a hydraulic fill dam with horizontal core pressure. Therefore, the maximum shear stress Sxt to the base of the foundation is:

o _Pa . E& x z - L - r L - -G6)

The case analyzed on Plate No. Bl is to determine what embankment slope will be stable for the foregoing condition. The shear stress is determined by means of the above formula and compared to the shear strength in the foundation and the value of L for equilibrium of the two is computed Knowing L and the height of the dam, the maximum allowable slope can readily be determined.

If the tension at the center line of the dam is greater than the horizontal earth pressure force, then the foregoing analysis does not apply and the condition approaches Jurgenson’s 8 Case B where the embankment can resist only a portion of the full shearing stress along its base.

B -l

PLA

TE NO.

OD

earth forces

Computation o f ho rizon ta l earth fo rce (Active cond ition ) Embankment Design ValuesU nit weight o f embankment = y - ¡2 5 # / cu. ft.

c r h 2 l- s in # I 2 5 x l0 0 2 1 -0 5 „ E ffe c tiv e angle o f in te rn a l fr ic t io n - <p - 3 0 °E = — * i + simp ' -----------1 ------------ * T T a T " 2 0 8 -3 k ips p e r f0 0 t o f dam-

Maximum possible tension at center line o f dam

I D O x IP 'S x P '5 n itp a - — = - ^ - 156.3 k ips per fo o t o f dam where p is the maximum embankment p re ssu re on foundation.

paAnalysis: p a < E, so tha t no tension w ill exist in dam i f f u l l value o f shearing s tre ss , > is deve loped along base o f dam.

This is equivalent to Jurgenson's case "o " see Paper G~6, Proc. Int. Conf. on S o il Mech. and Found. E ngrg .} Vo/. H - /9 3 6

Then fo r a dam with no water load the shearing stress in the foundation , S x z - ^ ^ - k — - 15 6 -3 *2 + 2 0 8 .3 _ 52J_ k jp s / f t

Assume the shearing strength o f d a y - IO O O # / sq. ft.

5 2 /Then fo r e q u ilib riu m , shear s tress equals shear strength, o r IOOO = —j — and L - 521 ft.

5 2 !Maximum a llow ab le s lope fo r s ta b il i ty is then - 5 .2 ! , o r approxim ate ly I on 5

STABILITY OF SHALLOW PLASTIC FOUNDATIONON RIGID BOUNDARY

APPENDIX CAPPLICATION OF RENDULIC’S METHOD

This method is based on the assumption that the embankment is in a state of active pressure. In the analysis shown on Plate No. Cl, shearing stresses at the base of the embankment are deter­mined by evaluating the horizontal component of the active pressure. This is accomplished by first passing vertical planes through the embankment at several locations. Typical planes are shown as lines MO and aE on figure 1. Consider first the analysis for plane aE—if wedge-shaped segments to the left and right of aE are taken, such as dEb and aE/, the forces acting on each wedge will be as shown in figure 2. These forces are: the wreight of wedge W; the soil reaction R, acting at an angle <t> to the normal of the slope of the wedge, and the earth forces E. To satisfy the con- dition of equilibrium across plane aE the earth forces must be equal, opposite, and colinear. This condition is shown in the force polygon, figure 3. The problem then is to investigate the condition which will result in equilibrium across the plane aE and thus arrive at the magnitude and direction of the E forces. The construction known as the “Engesser envelope” is used to analyze the condition. Several wedges, both to left and right of aE are taken, the weights of each wedge determined, and the direction of the R forces established. A convenient weight scale is selected and laid off on a vertical line, as shown in figure 4. Weights of wedges to the right of aE are laid off above the zero point on the scale and weights of wedges lying to the left of aE are laid off below the zero point. At each point denoting the weight of wedge, a line is drawn in the direction of the soil reaction R> for that wedge. Thus, the resulting construction is a series of lines representing the soil reaction forces for the wedges on either side of the vertical plane. Smooth curves are now drawn tangent to the soil reaction forces for the left and right wedges. The intersection of the two curves determines the magnitude and direction of the earth force E as shown on figure 4; the horizontal component EH may be readily obtained as indicated.

Similar computations are made for a number of vertical planes through the embankment and the horizontal component of the earth pressure, EHl is determined for each. In the analysis shown on Plate No. Cl, a portion of the embankment is composed of cohesive spil (material A). Vertical planes, such as MO on figure 1, are analyzed similarly to those just described except that additional forces are acting on the wedge and must be included in the analysis. These additional forces are the water pressures U2, and Uz and the cohesion, C, along the inclined face of the wedge. Water pressure diagrams for a typical wedge, MOX, are shown on figure 1. The condition of equilibrium across plane MO is shown by the force polygon in figure 5. The determination of the E force is made by means of the “Engesser envelope” similarly to that described before, except that the water pressure and cohesion force are included in the force polygon to determine the location of R forces.

A curve showing the variation in magnitude of the horizontal earth pressure, Eff, over thebase of the embankment is shown in figure 6. The horizontal shearing stress, òE g

òx is the slopedEof the Eh curve at any point. Values of ^ may be readily determined by laying off tangents

to the Eh curve and computing the slope. The dotted line on figure 6 shows the computed dis­tribution of horizontal shearing stresses along the base of the embankment. These values are summed up cumulatively from the toe of the embankment and the dashed curve as shown in figure 7 is prepared.

The next step is to determine the shearing strength at the base of the embankment. In the example shown on Plate No. Cl, the embankment rests on a relatively weak clay foundation.

C -l

Therefore, it is assumed that sliding will take place just within the foundation and the strength of the clay is used in the analysis. Normal loads are computed equal to the weight of material in the embankment, with proper compensation for water forces. The shearing strength is evaluated according to Coulomb’s equation (1).

It may be desirable to include the vertical shearing stresses ( ^ ) in some analyses, buttheir effect is usually small and they have been disregarded in this example. Cumulative shearing strengths are summed up over the embankment starting at the toe. A curve of cumulative shearing strength is shown as the solid line on figure 7.

The stability of the embankment is based on the condition that no failure by spreading can occur unless the cumulative horizontal shearing stress at some point in the base of the embankment is greater than the cumulative shearing resistance. Therefore, the factor of safety may be deter­mined at any point by comparing values of shearing stress and shearing strength on figure .7. The variation in factor of safety over the base of the embankment is shown by the curve in figure 8. For the case analyzed the minimum factor of safety was 2.1 at a distance of 30 ft. from the center line of dam.

C-2

120r- 100" T -

80I

20~ T “

DISTANCE FROM Ç. IN FEET

80 100 120 140 2 00 220 240 260 280 3 00 320

EMBANKMENT SECTION

- Cumulative shearing strength in day foundation (Neglecting vertical shear in embankment)

CUMULATIVE SHEARING

-Minimum factor of safety-2.1

140 160 180 200DISTANCE FROM <L IN FEET

FIGURE 7

STRESS AND STRENGTH FROM TOE OF EMBANKMENT

140 160 180 200 220DISTANCE FROM £ IN FEET

FIGURE 8

FACTOR OF SAFETY

240 260 300 320 3 40 360

ENGESSER ENVELOPE TO DETERMINE "E" FORCE

ACROSS PLANE “a E"

FIGURE 3

FORCE POLYGON FOR EQUILIBRIUM ACROSS VERTICAL PLANE

IN EMBANKMENT

FIGURE 2

FREE BODY DIAGRAMS FORCES ACTING ON

TYPICAL WEDGES IN EMBANKMENT

FIGURE 5

FORCE POLYGONFOR EQUILIBRIUM ACROSS PLANE "MO"

IN EMBANKMENT

140 160 180 200 220DISTANCE FROM <£ IN FEET

FIGURE 6

DETERMINATION OF HORIZONTAL SHEARING STRESSES

STABILITY OF SLOPES AND FOUNDATIONS

RENDULIC METHOD OF STABILITY ANALYSIS

SCALE.AS SHOWN

WASHINGTON, D.C.

OFFICE OF THE CHIEF OF ENGINEERS CIVIL WORKS

ENGINEERING DIVISION

E M C - I I 9 - 2 - CI PLATE C I

APPENDIX DAPPLICATION OF SWEDISH SLIDE METHOD (METHOD OF SLICES)

This method is illustrated on Plate No. Dl wherein arc ADB represents one trial circle with the center at Oi. The assumed sliding section above the arc is divided into a number of vertical slices as shown in figure 1. The forces which are considered to act on each slice are represented in figure 2. The weight W is the total weight of the slice of one unit thickness. Force C equals the length of arch HI multiplied by the unit cohesion, c. Force Fn is the effective normal soil force and equals the total normal soil force Fn, minus the hydrostatic force U. The frictional force/ at the base of the slice is equal to Fn times tan <£. The hydrostatic force U, acting at the base of the slice is obtained from the equipressure lines as described below. The neutral pressure in the pore water uW) acting on the sliding surface is computed for a number of points along the arc according to the equation: uw—ywhw where yw is the unit weight of water and hw is the piezometric head, from equipressure lines. The neutral pressures at the selected points are plotted to scale normal to the failure surface and the neutral pressure line GD’D is drawn. To compute the hydrostiatic force U on each slice, the area bounded by the slide arc, the neutral pressure line, and radial lines from the slice boundaries is determined. The example shown on Plate No. D l assumes that the earth and hydrostatic forces on the sides of the slices balance, which corresponds to assumption (a) of Plate No. D2.

The weight vectors W, drawn through the center of gravity of the slices, are each resolved into normal components Fn and tangential components T. The tangential forces T are resisted by the frictional forces / = (Fn— U) tan <t>=Fn tan <t>, and the forces of cohesion C. An expression of safety factor may be written thus:

s F - J i r L .............. ...................................................... ............ ( 17)

when <t> and C are the angle of internal friction and cohesion, respectively. The safety factor mayalso be expressed as an- ^ ==-^, where 4>D and CD are the values required for a safety factor at 1.0. tan <j>D L d

When it is desired to include either hydrostatic forces or hydrostatic and earth forces on the sides of the slice this may be readily done using the approach described for the modified circular arc method in appendix G. An alternate method is to make reasonable assumptions for the earth forces on the side and to incorporate them in the force polygon for each slice, thus obtaining a revised driving weight and revised effective normal force for each slice as illustrated in Plate No. D2. These forces are then summed and inserted in the expression for the factor of safety used in the example on Plate No. D l.

D -l

2:H ydrostatic equipressure tines in fee t o f head. Only those tines pertinen t to the analysis are shown. For fu ll de­velopment o f equipressure line see Plate No. E 2

Figure I

EMBANKMENT SECTION

Scale in Feet

SUMMARY OF FORCES-TONS

SLICE W TNORMAL FORCES

CFn U Fn

1 31.1 22.6 21.8 3.5 18.3 5.22 49.9 30.5 39.3 6.7 32.6 4.23 58.1 29.5 50.2 6.7 43.5 3.94 61.3 24.7 56.4 5.7 50.7 3.65 61.8 18.5 59.2 4.6 54.6 3.36 62.5 12.2 61.4 3.6 57.8 3.27 55.1 5.3 54.8 2.6 52.2 3.28 47.3 -0 .3 47.3 1.0 46.3 3.39 33.3 -3 .6 33.0 0 33.0 3.310 16.8 -3 .3 16.6 0 16.6 2.31 1 13.0 -3 .6 12.6 0 12.6 3.8

TOTAL 132.5 452.6 34.4 418.2 39.3

®

Safety foe,or , 4 1 8 .2 , 0 4 0 * 3 9 .3 __ / 5S

DEPTH OF CRACKINGComputation for the maximum depth of cracking

in cohesive soil in the tension zone.

Hc ph Tan (4 5 ° + 4 } 17 & f >■ I ? ™ *7 .4 'In slice (T) cohesion is assumed acting only over arc DF. No changes are mode in weight or hydrostatic forces due to tension cracks.

W = T = U -- uw- Fns Fn -C --

NOMENCLATURE

Total weight of material in slice Tangential component of total weight in slice Resultant of hydrostatic forces on base of slice Pore water pressure Total normal force on base of slice Total normal soil force on base of slice (Fn =Fn - U) Total cohesion on base of slice Unit cohesion in tons per square foot Unit weight of soil intons per cubic foot Angle of internal friction Maximum depth of cracking

ADOPTED DESIGN DATA

EMBANKMENTShearing strength

Tan <i=0.4 C = 0.I5 t. per sq. ft. Unit weights

Moist = 0.0600 t. per cu. ft. Saturated = 0 .0625 t. per cu. ft.

Figure 2

FORCES ACTING ON A TYPICAL SLICE

STABILITY OF SLOPES AND FOUNDATIONS

SWEDISH SLIDE METHOD

OF STABILITY ANALYSISscale: as shown

WASHINGTON, D.C.

OFFICE OFTHE CHIEF OF ENGINEERS CIVIL WORKS

ENGINEERING DIVISIONJANUARY 1951

EMC-119-2 -D 1 PLATE D 1

SECTION THROUGH SLOPE FORCES ACTING ON SLICES

CASE (a)

NOTE:In this method various assumptions may be made concerning the distribution of forces acting and the nature of the forces acting. The forces shown are for case (a) with no hydrostatic forces acting. Various assumptions poss­ible are:

(a) Water and soil forces on sides of slice are in balance, apply hydrostatic force on base normal to failure arc.

(b) Same as case (a) except that water forces on sides are not assumed equal but are computed and taken into account.

(c) Neither soil nor water forces on sides are assumed equal and are taken into account. Apply hydrostatic force on base normal to failure arc.

FORCES ACTING ON SLICES

CASE (b)

FORCES ACTING ON SLICES

CASE (c)

STABILITY OF SLOPES AND FOUNDATIONS

CIRCULAR ARC ANALYSIS METHOD OF SLICES

SCALE: AS SHOWN

OFFICE OF THE CHIEF OF ENGINEERS CIVIL WORKS

Wa s h i n g t o n , D.c._______ENGINEERING DIVISION____________ J a n u a r y 1951

EM C- 1 1 9 - 2 - D 2 PLATE D 2

APPENDIX EAPPLICATION OF 0-CIRCLE METHOD

The procedure for the 0-circle method is illustrated on Plate No. E l for one trial circle AB with the center at Oi. The weight vector w represents the total weight of soil plus water content of a unit thickness of the sliding section ABE, and it acts vertically through the center of gravity of the section. The hydrostatic equipressure lines on figure 1 are determined by a. flow-net, Plate No. E2. The neutral pressure or pressure in pore water, uw, acting on the slide arc is computed for a number of points along the arc as discussed in appendix D for the Swedish slide method. The neutral pressures at the selected points are plotted to scale on radial lines 1, 2, 3 . . . along the base of the failure arc and the neutral pressure line GD’D is drawn. The slide arc from G to D is then divided into a number of parts and the neutral force acting on each segment is computed and plotted normal to the arc as shown by vectors Ui to Ug. The resultant neutral force, U, which must act through the center (h of the slide arc, is determined in magnitude and direction by the force polygon of figure 2.

The resisting forces due to cohesion are resolved into force C which acts parallel to chord ATt arc ATat a distance from the center Oi equal to cbord AT ma£nitude ^ equals the unit cohesion

times the length of chord AT. Forces W and U are combined vectorially into their resultant R, figure 3, which is projected to point M on the line of action of C, figure 1. The remaining force F, the resultant of the effective normal and frictional forces on arc AB, must act through point M and is assumed tangent to the 0 -circle which has its center at Oi and a radius equal to r sin 0 when r is the radius of the sliding arc. It should be noted that the value of 0 used in computing the radius of the 0-circle may be; either a variable, or developed value hereinafter denoted by 0^ or the applicable design value for the soil involved, depending upon the method used in calculating the safety factor. If the first method is used, a trial safety factor is applied to both the design values of tangent 0 and c to obtain trial values of (¡>D and CD. The 0-circle is drawn using this trial value of <f>D. Since W , U and CD have been determined in direction and magnitude and the direc­tion of F established, the force polygon of figure 3 may be constructed. If the polygon will not close, a new trial safety factor is selected and the above procedure repeated until closure is obtained. If the second method is used, the 0-circle is drawn using the design value of 0, thus establishing the direction of F. The force_pol}rgon in figure 3 is drawn with known magnitude and direction of R and known directions of F and C. The magnitude of force C necessary to close the polygon is determined and this value compared with the magnitude of C computed from the design value of cohesion. This gives a factor of safety with respect to cohesion. It is believed that the factor of safety computed by the second method is less descriptive of the degree of safety than that resulting from the first method in which both components of the shearing strength are considered.

B -l

FIGURE I

EMBANKMENT SECTION

COMPUTATION FOR THE POSITION OF THE RESULTANT OF COHESION ON THE ARC AD

The resultant of cohesion is parallel to the cord of ore AT The distance from the center of the circular slide path to the resultant of cohesion

arc AT chord AT ■ x 209 .0 = 2 2 3 .5 '

COMPUTATION FOR THE MAXIMUM DEPTH OF

CRACKING IN COHESIVE SOIL IN THE TENSION ZONE

Depth of cracking

Hc - ym * tan <4 5 ° + */2 > - 2o °o6 * ,A 7 7 ° " 7 A ‘

Cohesion is assumed acting over arc AT. No changes are made in weight forces due to tension crocks.

FACTOR OF SAFETY

g _ tamp _ QJ5 . 0.4 CD " ton 0D '0 .087 ' 0.233

SCALE IN FEET

0_______20 40

ADOPTED DESIGN DATA EMBANKMENT

Shearing strength Tan 0= 0 .4 0 c =0.151/sq.ft.

Unit weightsMoist 0 .0 6 0 0 t /c u .ft. Saturated 0 .0 6 2 5 t/cu. ft.

NOMENCLATURE

W = Total weight of sliding section U = Resultant of hydrostatic force on bose of section

U/-9 = Components of hydrostatic forces on base of sliding section u w = Pore water pressure

R = Resultant of weight and hydrostatic forcesF = Resultant of normal soil and friction forces on base of sliding section c = Unit cohesion in tons per sq.ft.C = Resultant of total cohesion on base of sliding sectionCo = Resultant of cohesion developed for equilibrium on base of sliding secti*0 = Angle of internal frictionVo = Developed angle of internal friction for equilibrium r = Radius of circular slide path

He = Maximum depth of cracking S./T = Softy factory = Unit weight of soil in tons per cu. ft.

- STABILITY OF SLOPESAND FOUNDATIONS

SCALE: AS SHOWN

0 CIRCLE METHOD OF STABILITY ANALYSIS

OFFICE OF THE CHIEF OF ENGINEERSCIVIL WORKS

WASHINGTON, D. C._________ ENGINEERING DIVISION____________ JANUARY 1951

E M G - I I 9 - 2 - E l PLATE E I

NOTE:Normal dam section showing equipressure

lines in fe e t o f head. EQUIPRESSURE LINES ON A NORMAL SECTION20 10 0________20________40

SCALE IN FEE T

ADOPTED DESIGN DATA

MATERIALCOEFFICIENT OF

PERM EABILITY IN FEET/DAY

(A) 100

® 0.230

© 25

100

© 0.071

© IMPERVIOUS

NOTES:i Pervious foundation stratum and natural blanket are

assumed to be continuous.2. The ratio o f horizontal to vertical permeability in the

ro lled f i l l is 9 to I. Therefore the transformation ra tio in the transformed section is n / kv l~T~ j_

h i " V 9 " 3

FLOW NET AND EQUIPRESSURE LINES ON A TRANSFORMED SECTION50 25 0 50 100 150

H O R IZO N TAL

10 0 10 20 50 40 50

VE R T IC A L

S C A LE IN FE E T

STABILITY OF SLOPES AND FOUNDATIONS

DEVELOPMENT OF A FLOW NETSCALE: AS SHOWN ______________________________________________________________

OFFICE OF THE CHIEF OF ENGINEERSC IV IL WORKS

WASHINGTON, D.C.____________ ENGINEERING DIVISION_______________JANUARY 1951

E M C - I I 9 - 2 - E 2 PLATE E 2

APPENDIX FAPPLICATION OF MOMENT METHOD

This method is used for the analysis of homogeneous slopes, either saturated or submerged, and having uniform slopes. Consider first a slope and foundation consisting of a clay with uniform cohesive strength and no angle of friction. The center of most critical failure circles passing through the toe of the slope may be located by means of the diagram shown in figure 1, Plate No. F l. In this figure the moments of the sliding and resisting forces about the' center 0 must be equal for equilibrium of the sliding mass, as follows:

Wd=CDLr__________________________________(18)For this condition, Fellenius9 has derived the following expression for required cohesion for stability:

CD= ^ N , -------------------------------------------------- (19)Values of N f for different angles of slope for most critical failure arcs through the toe of slope are given in figure 3, Plate No. F l. The factor of safety by this method is determined by computing the required cohesion CD for equilibrium and comparing it to the design value of cohesion, c.

However, Fellenius9 points out that for the homogeneous condition assumed the most critical circle passes through the toe of the slope only for slope angles greater than 0=53°. When the slope angle is less than 53° the most critical circle emerges beyond the toe of the slope as shown in figure 2. The central angle at 0 is approximately 133%°, the radius is theoretically infinite, and 0 is located over the midpoint of the slope. The infinite depth case is not attained in practice since the depth of the most critical circle is ultimately limited by an underlying stronger layer, such as hard pan or sand. Taylor6 has analyzed the case of a homogeneous embankment and foundation with cohesive strength and with <¡>=0 for limiting depths of most critical circles where the failure arc emerges beyond the toe of slope, as well as for the case where the arc cannot pass beyond the toe. The results of his analysis are presented in figure 5, Plate No. F l for slope angles less than 53°. The stability analysis consists of determining from the chart the stability number, N, for a given slope angle, 0, and depth factor, D. The safety factor maj be determined by inserting appropriate values for height of slope and characteristics of the soil into the following equation:

N = S .F . yH ------ ------------------------------------------ (20)For the condition where a soil has both friction and cohesion, Fellenius lias indicated that the

center of the most critical circle will move outward from that of the purely cohesion case along a line such as 0-0' on figure 1. The construction of this line is shown on the diagram. Based on the <£- circle method of analysis, Taylor5 has prepared a chart (figure 5) for determining the stability

Qnumber iV=^ ^ for various slope angles and angles of internal friction of the soil. Usually adeveloped friction angle is used; that is, <f>D=<l)/S.F. such that the safety factor attained is that with respect to the shearing strength of the soil.

Assumptions in the foregoing methods are necessarily simplified and do not strictly conform with the actual conditions in the field. However, analyses made in this manner have the advantage of being performed rapidly and may be used for preliminary investigations and for guidance in

F-l

making later and more detailed studies. Limiting conditions of saturation and submergence may be studied by using appropriate values of the unit weight of soil. The drawdown condition may also be approximated in Taylor’s method by using the total weight of soil and water and a weightedvalue for internal friction, Broken slopes may often be replaced by an average smoothslope and soil strengths of different strata may be averaged for purposes of approximate analyses.

F~2

SLOPE 9 cf ß

1:058 CD O 0 29 ° 40°|:| 45° 000CVI 37°|:| 5 3 3 ° -4 l ' 26° 35°1:2 2 6 °-34 ' 25° 35°1:3 1 8°-26 ' 2 5° 35°l; 5 1 1 ° - 1 9' 2 5° 37°

FIGURE I

LOCATION OF CENTER OF MOST CRITICAL CIRCLE THROUGH TOE OF SLOPE

DEPTH FACTOR, D

CHART SHOWING EFFECT OF DEPTH LIMITATION, D HON STABILITY NUMBER

FOR 0 = 0 °(S.E)yH

O

FIGURE 2

LOCATION OF CENTER OF MOST CRITICAL CIRCLE SLOPE ANGLE< 5 3 ° FIGURE 3

CURVES SHOWING THE FACTOR OF COHESION FOR DIFFERENT ANGLES OF SLOPE IN COHESIVE SOIL

WHEN THE SLIDING SURFACE PASSES THROUGH THE TOE OF THE SLOPE

SLOPE ANGLE 0 FIGURE 4

CHART FOR STABILITY NUMBER

From Stability of Earth Slopes, Donald W. Taylor, Reprinted by permission of the Boston Society of Civil Engineers.

STABILITY OF SLOPES AND FOUNDATIONS

MOMENT METHODSCALE:AS SHOWN OF STABILITY ANALYSIS

WASHINGTON, D.C.

OFFICE OF THE CHIEF OF ENGINEERS CIVIL WORKS

ENGINEERING DIVISION JANUARY 1951

EMC - 1 19-2 - Fl PLATE F I

APPENDIX GAPPLICATION OF MODIFIED SWEDISH METHOD

In brief, this method consists of locating by trial the most dangerous slide curve, dividing the sliding section into slices, determining all forces on each slice, and then adjusting both components of shearing strength of all soils involved by the same trial safety factor until closure of the force polygon is obtained.

Trial curves of circular shapes are selected so that the greater portion of the curve will lie in the weaker soils or in zones in which hydrostatic pressures are most unfavorable. These curves may be circular arcs or combinations of arcs of different radii connected by tangents, depending upon the uniformity of the section. Since both the slope and position of the curves are determined by trial, a number of solutions must be made before the most probable critical curve may be approx- mated. One such trial curve AB is shown in figure 1 of Plate No. G l.

The sliding section above curve AB is divided into a number of slices, the widths of which are selected roughly in proportion to the radius of curvature of AB and to facilitate computation. As an approximate guide the slice width should usually be chosen such that the chord subtended by that portion of the curved failure surface is essentially the same length as the arc. The forces which are presumed to act on each slice are shown in figure 2 of Plate No. G l. The weight force W is the total weight of soil plus water in a slice of unit thickness. The hydrostatic forces Uu U2, and Uz acting on the vertical faces and base of the slice are computed from the equipressure lines shown on figure 1. The interface soil forces E are unknown in direction and magnitude. However, some latitude may be exercised in arbitrarily fixing the direction of these forces without greatly influencing the computed safety factor. Although the lines of action probably vary considerably at the different interfaces, a satisfactory simplifying assumption is that the E forces act approxi­mately parallel to or make a somewhat greater angle with the vertical than the embankment slope. The resisting force of cohesion, CD) assumed to act_parallel to chord GH, figure 2, is equal to chord GH times the developed unit cohesion cD. Force F acting at angle <t>D with the normal to GH is the resultant of the effective normal force at the base and the developed frictional force. The design values of c and tan <f> for all soils through which curve AB passes are divided by a trial safety factor to obtain cD and tan <t>D. The known forces W, Uu U2, and U3 for each slice are resolved into force R as on figure 3. A trial force polygon, figure 4, is then drawn using for each slice the known magni­tudes and directions of forces R and CD} the assumed directions of the E forces, and the directions of the F forces as determined by the angle <t>D. If this polygon fails to close, another trial safety factor is chosen and the procedure repeated until closure is obtained. It will be noted in the example shown in figure 4 that for a trial safety factor of 2.0 the force polygon did not close; a second trial with a safety factor of 1.75 produced closure in the force polygon. Therefore, for this analysis the assumed failure surface AB has a safety factor of 1.75.

The preceding method of determining the safety factor is merely one means of expressing the two conditions for equilibrium that summation of vertical and of horizontal forces must each be equal to zero. Application of the requirement that summation of moments of external forces acting on the sliding mass be zero is not applicable since in this modified method there is no one center of rotation. In this modification of the circular arc method various assumptions as to the forces acting on the sides of the slices, as discussed under the circular arc method, are possible. However, the assumptions used in the example, that neither soil nor water forces on the sides are in equilibrium, are preferred.

G-l

Another method of determining the safety factor in this method of analysis is to determine from the force polygon for each slice the horizontal force required for stability of that slice, consider­ing all forces shown in figure 2, Plate No. Gl, except that hydrostatic and earth forces on the sides are combined into one unbalanced horizontal force. The safety factor is then defined as the ratio of the horizontal forces tending to resist movement to the ratio of the horizontal forces tending to cause movement. This safety factor is not equivalent to that shown on Plate No. Gl, which is preferred.

G-2

FORCE POLYGONSC ALE IN TONS

IQ 0 IO 2 0 30 4 0 50

Factor of s a fty fo r closure = 1.75 Shear s tren th values fo r closure

Ton 0 = 0 .2 2 2 CD = 0 .0 8 3

EMBANKMENT SECTIONSCALE IN FEET

10 0 10__________________50

S U M M A R Y O F F O R C E S -T O N S

SLICE w U, u 3 u 2 Co*

1 7.3 0.4 0.4 0.3 1.4

2 30.1 0.3 2.9 2.9 2.2

3 33.8 2.9 4.8 4.1 1.4

4 51.8 4.1 5.8 4.1 1.7

5 57.8 4.1 5.3 2.9 1.7

6 59.1 2.9 4.4 1.5 1.8

7 80.2 1.5 3.5 0.2 2.5

8 73.8 0 .2 0 .4 0 3.3

9 29.5 0 0 0 2.6

10 12.0 0 0 0 2.4

FIGURE 2

FORCES ACTING ON A TYPICAL SLICE

FIGURE 3

RESOLUTION OF KNOWN FORCES

ON A TYPICAL SLICES C A LE IN TONS

10_______ 0_______ IQ______ 20

* For S .F = 1.75

DEPTH OF CRACKINGC om putation fo r maximum depth o f crack ing in cohesive soil in tension zone.

- f m tan (4 5 ° + 4/* ) f r f f g 5 * 1-4770 - 7,

In s lice (7) cohesion is ossumed acting only over arc CD. No changes are made in w eight or h y d ro s ta tic fo rc e s due to tension c ra ck .

TABLE OF ADOPTED DESIGN DATA

MATERIAL DENSITY-TONS/CU. FT. SHEARING S5TRENTHDRY MOIST SAT. COHESION * TAN 0

A 0 .0 5 0 0 0 .0 6 0 0 0 .0 6 2 5 0 0 .5 0

B 0 .0 5 0 0 0 .0 6 0 0 0 .0 6 2 5 0.15 0 .4 0

C 0 .0 5 0 0 0 .0 6 0 0 0 .0 6 2 5 0 0 .6 5

D 0 .0 4 0 0 0 .0 5 0 0 0 .0 5 5 0 0 0 .5 0

E 0 .0 4 0 0 0 .0 5 0 0 0 .0 5 5 0 0 0 .4 0

F — — — 0.25 0 .2 5

NOMENCLATURE

X = Unit weight o f soil in tons per cubic fo o t W = Total weight of m ateria l in s lice Ui = Total h yd ro s ta tic fo rce on le ft face o f s lic eU2 = To ta l h y d ro s ta tic fo rce on r ig h t fa ce o f s lic eU3 = Toto l hyd ro s to tic fo rce on base o f s liceR - Resultant o f weight and hyd ro s ta tic fo rce s on s liceF, = Total fo rce app lied to soil on side o f sliceF = R esu ltan t o f normal soil fo rce and fr ic t io n developed on base o f slice c - Unit cohesion in tons per square fo o t

Co = Developed un it cohesion is C-r S.FCD = Totol cohesion developed fo r e q u ilib r iu m on base o f slice = Cp x A s 4 = Angle o f in ternal f r ic t io n of so if

*Pp = D eveloped angle o f in te rna l f r ic t io n fo r equ ilibrium (tan 0d = tan 4 -f S .F) Hc = Maximum depth of cracking S.F = Softy fa c to r

STABILITY OF SLOPES AND FOUNDATIONSMODIFIED SWEDISH METHOD

OF STABILITY ANALYSISSCALE: AS SHOWN

WASHINGTON. D.C.

OFFICE OF THE CHIEF OF ENGINEERSCIVIL WORKS

ENGINEERING DIVISIONJANUARY 1951

E M C - I I 9 - 2 - G I PLATE G I

A P P E N D IX HAPPLICATION OF WEDGE METHOD

The wedge method of analysis is particularly adaptable to the determination of the stability of embankments on foundations containing weak strata of considerable extent or zones of low shear strength due to slow consolidation. In this method the soil mass under analysis is divided into an active wedge, a passive block, and a passive wedge. The most critical combination of active wedges, passive blocks, and passive wedges is found by trial, and the safety factor is expressed as the ratio of available shear strength to that required for stability. Reference is made to Plate No. H i for illustration of the wedge method.

For any position of the vertical face of an active or passive wedge, there is one most critical position of the slide plane which will give the greatest thrust or least passive resistance. The critical wedges are found by trial, and examples of such trials are the active wedges GNF, GNE, and GND and passive wedges HMJ, JMK, and JM L of Plate No. H i. The forces considered to act on the active wedges are:

(1) The total w eight W of soil and water in the wedge.(2) The hydrostatic forces UHL, UHR, and Uv which are, respectively, the horizontal com­

ponent on the left, or inclined face of the wedge, the horizontal force on the vertical right face, and the vertical component or uplift on the inclined face. These forces are computed from the pressure diagrams, figure 2, which are plotted for the pressures given by the hydrostatic equi- pressure lines of figure 1.

(3) The cohesion C acting along the slide plane, equal to the effective length of the slide plane, L , times the unit cohesion c for the material. The effective length of the slide plane is the length that passes through cohesive soil, adjusted to compensate for possible cracking. The developed values of cohesion CD and the vertical and horizontal components C Vd and C Hd are used in computing the “ adjusted shearing strength” safety factor and are the adopted design values divided by the trial safety factor. _

(4) The effective soil force F acting at an angle <t>D with the norm 1 to the slide plane.Angle <I>d is a developed angle of friction obtained from the relationship tan 0 ^ = ^ ^ factor* ^slide plane of the wedge passes through several soils with different ultimate angles of friction <£, the various tan <£ values are weighted in proportion to the estimated effective soil loading along the slide plane, and the weighted value tan <l>w is divided by the trial safety factor to obtain tan <j>D. A typical computation is shown on Plate No. H i.

(5) The direction of the active wedge driving force PA is assumed horizontal and the magni­tude is determined by a graphical solution as shown in figure 3, Plate No. H i. This is accomplished by first assuming a trial safety factor and determining CD and the direction of F for a given wedge. A force potygon is constructed using the weight of wedge W> the hydrostatic forces U, the developed cohesion CD, and the direction of force F. The intersection of force F with the horizontal line PA gives the active pressure for that wedge. Since the maximum active pressure is desired it is neces­sary to analyze several wedges. By superimposing these graphical solutions for the various trial wedges and drawing a curve tangent to the F vectors as in figure 3, the maximum PA is readily found.

The resisting force of the passive wedge PPW is found in a similar manner. _The sliding resistance of the passive block P P B i is the sum of the frictional resistance FHB,

the developed cohesion CD along the base and the difference between hydrostatic forces UHR and

UHL at the ends of the block. The frictional resistance FHb equals the product of the effective weight of the block (W— Uv) and the developed tan (¡>D of the material at the base.

The resistance of the passive wedge PPW plus the resisting force of the block PPB comprises the total passive resistance PP. If the active driving force PA equals the total resistance Pp, the trial safety factor selected is correct, but if PA and PP are not equal, then another complete trial must be made and the process repeated until a balance is obtained.

The complete analysis by this method will require the investigation of both active and passive wedges with vertical faces at several positions along the embankment slope.

H-2

/ 1 , , ■/ " / / 'r7777T, '' PASSIVE . / A *

WEDGE

/

j

/ UHr \

^C u rve to de te rm j ng. m i n i mum passive «edge ' drawn tangent to F l i n e s . Minimum pass ive

p ressure i s i n t e r s e c t i o n o f curve w i th

NOTE: Normal dam s ec t ion showing equipressure l i n e s fe e t o f head. For comp le te development o f eq' j i - p ressure l i n e s see P l a te No.E 2

FIGURE 4GRAPHICAL SOLUTION FOR MINIMUM PASSIVE WEDGE

SCALE IN TONS

F. : 1.53 SCALE IN FEETHYDROSTATIC PRESSURE IN FEET OF HEAD Ta b le of adoptéd design data

ANALYSIS OF FORCES FOR SAFETY FACTOR

ORWEDGE

WEIGHT

W (Tons)

TAN V U H R u v C C y C„TRIAL

SAFETYFACTOR

TAN 4>0ACTIVE

WEDGE1®P\SS i YE WEDGE=jS

Cp CVD c HD Fy f h F „ B F P4p»

P?B Pp REMARKS

ACTIVEWEDGES

u N r 377 C.U.Cc b Ì 49 48 19.0 16.0 1 l . l i 1.54 0 .3186 I70 l|0 i 35° 30' 12. / 10.4 7 . 4 319 230 398 250G N E 1*97 U.4889 76 49 70 Id .4 14.3 13. 1 1.54 0.3181 I 70 39' 420 ¿ 0 ' 12.6 J .3 0 .5 41 8 239 482 258 MOST

CRITICALG N ü 61*1 u . 4 a 4 1 87 49 *01 19.6 12.6 15.0 1.54 C. 3208 170 47' 50° 00' 12.7 3 .2 9 .7 532 217 575 2456L0CK G N al H 9o3 TAN 4> =.25 49 31 267 46 1.54 0. 162 9° 12' 30 1 13 125

PASSIVEWEDGES

H H J 192 ■ 0 .641b 31 30 66 c 1.54 3 .4166 22° 37' 65° 30' 0 126 136 185 *35 260H M K 226 0. 6456 31 30 82 0 1.54 0.4192 22° 45' 70° Ou U 144 133 196 132 257h II L 280 0.6479 21 3U 1 10 1.54 0.4207 22° 49' 75° 00 ' 0 170 132 215 131 256 LEAST TOTAL

PASSIVE RES.Block GNMH and passive wedge HML have a t o t a l pass ive r es is ta nc e which i s 2 t'>ns less than the t h r u s t o f the most c r i t i c a l a c t i v e wedge GNE There fo re SF. - | .b 4 i s too h igh .

ACTIVEWECGET

G N F 377 C. 4306 67 49 48 19.ö 16.0 1 1.4 1.53 0.3207 17° 47' 35° 30' 12.8 10.5 7 .4 3 19 238 398 ¿49G N E 497 0.4899 76 49 70 19.4 14. 3 13. 1 1.53 0. 320 2 17° 45' 42° 35' 12.7 9 . 3 a . o 4 13 ¿30 48 1 256 MÏÏST

CRITICALo M Û 54 1 9 .494 1 87 49 10 1 19.5 12.6 lo.O 1. 53 0 .3229 17° 54' 50° 00 ' 12.8 8 . 2 9 . 3 532 ¿16 5/4 245BLOCK J M M il 963 TAN 4> 25 49 31 267 4o 1.53 0. 163 3° 15' 30 1 14 126

PASSIVE

WEDGES

H M J 192 0.6145 31 30 DO 0 1.53 0.4193 22° 45' 65° 30' 0 126 136 185 135 26 1H M K 226 0. 6466 3 1 30 82 0 1.53 0.4220 22° 33' 70° 00' 0 144 m; 197 133 259H ,w L 280 0 .6479 3 1 30 1 10 0 1.53 0. 4235 22° 57' 75ô 00' 0 ¡79 132 215 13 1 257 LEAST TOTAL

PASSIVE RES.

Block GNPH and passive wedge HML have a t o t a l passive r e s is ta nc e which i s 1 ton more than the t h r u s t o f the most c r i t i c a l a c t i v e wedge GNE, There fo re S.F. = 1. 53 is too low.

Fac tor o f s a f e t y fo r t h i s a n a l y s i s thus l i e s between 1.53 and 1.5*4; fu r t h e r re f in ement o f the va lue i s not war ran ted here.

M a te ri al Dens i t y in tons per c u b ic foo ohear iun strenci tnDry Moi st ' ì a t j r M e H Jo n e «5 i 0 n T?n

0.0500 9.0600 0.0625 ) . 500.0500 0.9600 0.0625 0 . 15 0.40

© 0.0500 0.0600 0.0625 0 0.65

© 0.0400 0 .0 TOO 0.0550 0 0. 50

© 0.0400 0.0500 0.0550 0 0.40© 0 .2 5 0. 25

NOMENCLATURE: i g n t o f m ate r ia l i n s l i d e , w or block

: fo rc e on l e f t

r i ont

EXAMPLES OF COMPUTATIONS FOR VARIOUS COMPONENTS IN THE ABOVE TABLE

COHESIONLength o f s l i d e plane in fe e t over which cohesion i s e f f e c t i v e x u n i t cohesi on.C cos cr

C Sin or

T r ia l Safety fac to r ; Cy

T r ia l S. F.

FH B = (Wt-up I i f t ) Tan <f>0

f = vrv~fv

P R B+^D -Pp : PpB+ PpW

^FORCE POLYGON FOR ACTIVE WEDGE GND

1___FIGURE 3

GRAPHICAL SOLUTION FOR MAXIMUM ACTIVE WEDGE

50 0 50 100 150

SCALE IN TONS

SOIL FORCES: Wt - u p l i f t - Cvo For ac t i v e wedqe: Wt - u p l i f t + Cvo For passive wedge

Fv-=--r—r --------------For ac t i v e wedoeTan ( <f>D + a )

’■___________ !v_ For pass ive wedgeT an ( |Q — >q )

NOTEWhen the s l i d e p lane o f the ac t i ve o r pass ive wedges cu ts th rough more than one m a t e r i a l , the "Tan<£" used in ana lyz i ng the wedge i s a weighted average va lue o f the Tan <f> o f the separa te m a te r i a l s th rough which the s l i d e p lane passes. The va lues o f Tan<£ fo r the va r i ou s m a te r i a l s are weighted in p r o p o r t i o n to the e f fe c t i v e weigh t upon th a t p o r t i o n o f the s l i d e pi ane.

WEIGHTED AVERAGE TAN 0(OR TAN 0 W) FOR ACTIVE WEDGE GNE

Total we ight o f v/edge = 406.8 tons Weight o f y y1 NG = 183.6 tons Tota l u p l i f t on wedge = 69 .8 tonsUp I i f t on y1 N = 29.9 tons

: p I ane o f wedoe

fo r equ ¡ l i t r i

Tota l h o r i z o n t a l component o f h y d r o s ta t i c fo rc e c o f wedge o r MockH y d r o s t a t i c u p l i f t on base o f wedae o f block Tota l cohesion along s l i d e plane o f wedge or Mock V e r t i c a l component o f cohes ion along s l i d e plane o f wedge o bl ockHo r i z on ta l component o f cohes ion along s l i o r blockDeveloped cohesion fo r e q u i l i b r i u m U n i t cohes ion in tons per sq. f t .Developed v e r t i c a l component o f conssic Developed ho r i z o n t a l component o f cone:

e q u i I i b r i urnTo t a l e f f e c t i v e fo rce on tne s l i d e plat V e r t i c a l component o f e f f e c t i v e fo rce c H o r i z on ta l component o f e f f e c t i v e fo rce <H or iz on ta l f r i c t i o n a l fo rc e i Angle o f i n te r n a l f r i c t i o n o f soi lDeveloped angle o f i n te rn a l f r i c t i o n fo r e q u i l i b r i u m Tangent o f weighted angle o f i n te r n a l f r i c t i o n Ac t i ve wedge d r i v i n g fo rce Pass ive wedge r e s i s t i n g fore-.-R e s i s t in g fo rce o f block ( coho si on + f r i c t i o n - h y d r o s t a t n fo rce di f fe rence)To ta l pass ive res i s ta nce Ac t i ve wedge c r i t i c a l an i l s Pass ive wedqe c r i t i c a l angle U n i t weignt o f mate r ia l

s l i d e pi ane i de p I ai

b lock .

S. F .- Sa fe ty f a c t o r = c :

Tan (f> w = [( 183.5-29 .9 ) .6 5 j + [ ( 3 13. 3 - 3 9 . 9) .40J (U96.8 - 69.8)

STABILITY OF SLOPES AND FOUNDATIONS

WEDGE METHOD OF STABILITY ANALYSISSCALE: AS SHOWN_____________________

WASHINGTON, D. C.

OFFICE OF THE CHIEF OF ENGINEERSC IV IL WORKS

ENGINEERING DIVISION JANUARY 1951

E M C - I I9 - 2 - H I PLATE H

APPENDIX ISTABILITY CALCULATIONS FOR HYDRAULIC FILL

Plate No. I l illustrates the principal steps to be followed in the section design of a hydraulic fill dam. For simplicity and to avoid repetition of analysis procedures pertinent to foundation stability and described elsewhere in this chapter, stability (*) analysis is restricted to the embankment section alone.

Stability analysis. When the shell material is cohesionless and the angle of internal friction of the material approaches the angle of repose as shown on figure 1, Plate No. Il, the steepest possible outer slope is 37° or 1 on 1.327. This is in excess of the actual shell slope of 1 on 3, and indicates the shell slopes will be stable for static conditions. However, the stability of shells against core pressure, and stability of outer slopes at drawdown must be considered. These analyses are dis­cussed in the following paragraphs.

Stability of shells against core pressure. In this analysis the assumption is made that the core acts as a heavy liquid and pushes against the shell, tending to cause it to move out. A diagrammatic section of the dam showing the forces acting for this case is presented in figure 3. The case has been analyzed by Gilboy on the basis of the allowable ratio between effective weights of core and shell materials. The equation used for the analysis is shown in figure 3 and a nomogram for its rapid solution is presented in figure 2.

A typical dam section to be analyzed is*shown on figure 4, Plate No. I l ; pertinent character­istics of the materials concerned are presented on figure 1. The first case analyzed is that where the dam is just completed, the core is assumed liquid and exerts full hydrostatic pressure, and the effec­tive weight of the shell is the moist weight due to partial saturation by seepage from the core. From Gilboy’s 6 equation the allowable ratio of core and shell weights for stability is 1.18. The actual ratio of core and shell weights for the assumed conditions is 0.928. Therefore, the safety factor is1.18

0.928 1.27.The second case analyzed is based on the extreme assumption that immediately after comple­

tion of the dam the reservoir pool is filled and then drawn down rapidly. I t is also assumed that the shell material does not drain completely after drawdown and that the effective shell weight is the average of the moist and submerged (buoyed) weights. For this condition the ratio of effectiveweights of core and shell materials is 1.153 and the safety factor is 1.18

1.153 1.02.I t is pointed out that the latter safety factor may be low due to the extreme conditions assumed

and to the fact that some consolidation of the core will take place and it will thus develop some shear strength. The true safety factor may be somewhere between the two cases analyzed.

Stability of outer slopes. Stability of the outer slopes will be most critical a t the time of rapid drawdown. Assuming a cohesionless shell and restricted drainage, the stability of the outer slopes can be determined by use of a fictitious angle of internal friction as suggested by Taylor.10 An analysis of this type is shown on Plate No. II. Since the assumptions and various steps are shown thereon in detail, they are not repeated here. A safety factor of 1.05 was obtained which is ample considering the severity of the assumptions used.

1-1

MATERIALS CHARACTERISTICSra G C 0 K

HYDRAULIC SHELL 85 2.72 0.0 37° 175:750

HYDRAULIC CORE 120 2.72 0.0 17° 0.1

yp = Average unit dry weight — */c.f.6 = Specific gravity C = Unit cohesion — T/sf

= Angle o f internat friction — degrees K = Coefficient o f permeability — cm/sec x t0 ~ 4

FIGURE I

CHARACTERISTICS OF MATERIALS

L , K 2 .0 +_J z O ° p ,Z M < OC u_ O 2.2 o x

lu 2.3 - -

1 ^ 4 +

° lu 2.6it o.° 3 2.7iL. <«° or 2.8 - -

^ 5 2.9 - - 3 o3 u_ 3 .0 - -

Values o f R=Ratio of the unit weight o f core material to the unit weight o f shell m aterial for various angles of internal friction o f shell materials.

From Soils Mechanics and Foundations, Plummer and Dore, by permission of the Pitman Publishing Company.

FIGURE 2

NOMOGRAM FOR DETERMINING STABILITY OF HYDRAULIC FILL SHELLS

ANALYSIS OF SHELL STABILITY

Assuming that due to construction, the hydrouticked core may be as flat as 2 on t, the factor o f safety against rupture of the outer shell may be bracketed os follows:

CASE IDam just completed; shell moist, full hydrostatic pressure exerted by core.

^ ' r 2

CONDITIONS OF EQUILIBRIUM

+ 62.4 = 8 5 ^ + 6 2 . 4 =U 58*/c.f.yc = density o f saturated core = T d f ^ j + oc.-r - ^ 2 7 2 ^

7s m - density of moist shell=120x1.04 = 1 2 4 .8 * /c.f.R, =H5.8 + 124 .8 =0 .92 8From stability equation, (Fig. 3 ) and nomograph chart, (Fig. 2 ) R = U 8

U 8 ' C

CASE HSame as case I except reservoir pool f ille d and then suddenly lowered. Assume 50% of shell to be moist and 50% saturated (buoyant weight) fc-density o f core as above =H 5.8*/c .f. , - , , _ ,/Sb - effective density of saturated shell = / p \ - q -J s 120 =76 * / c

/SE = effective over-all density o f shell = - 100 .4 */c. f.R2 = 115.8+100.4 = 1.153From stability equation, (Fig. 3 ) and nomograph chart, (Fig. 2 ) R = I.I8

1.18 ' 1 /5 3 'SF=r %=t.02

SHELL STABILITY

FIGURE 3

STABILITY OF HYDRAULIC FILL DAM

STABILITY OF OUTER SHELL SLOPES FOR DRAWDOWN CASE

{. Assumptionsa. Shell m ateria l is cohesionless.b. Reservoir suddenly lowered from maximum pool.

2. Stability analysis: See "Stability o f earth slopes", D. W. Taylor, Journal o f the Boston Society o f Civil Engineers, Vol XXIV,No. 3, July, 1937.

Tan f = S F Tan rp , or approximately Y>d "~s f where<f> = actual angle o f internal friction o f shell material.4>p= effective angle o f internal friction o f shell material.SF= factor o f safety

G~>6 + e ^

where = weighted value o f f at instant o f drawdown.6=specific gravity o f soil particles = 2 .75e = void ratio o f shell m aterials =0.613

Assuming that for the slope to be stable ¡ * * m , where / = angle of the slope with the horizontal, it may be demonstrated from the foregoing equations that: ( 6 -1 \

S F -for this analysis then, .

_ 3 7 ° \ i2.72 -

2.72+0.613 1 8 .4

kL / 9 .3 °!8 .4 °~ 1 0 5

STABILITY OF SLOPES AND FOUNDATIONS

STABILITYs c a l e : a s s h o w n

METHOD FOR HYDRAULIC FILL SHELLS

W ASHING TO N , D.C.

OFFICE OF THE CHIEF OF ENGINEERS C IV IL W O R K S

E N G IN E E R IN G D IV IS IO NJA N U A RY 1951

E M C - I I9 -2 - I I PLATE I I

APPENDIX JSTABILITY WITH SPECIAL MOISTURE CONDITIONS

Moist, saturated, and submerged soils. No special analyses are presented for these conditions, since they are covered by other analyses presented in this chapter. In general, for cohesive soils, the moist weight has been used above the saturation line in the embankment and the saturated weight below; this arrangement has been used in the analyses in appendixes A, C, D, E, G, and H. Submerged weights have been used below the pool level for the analysis shown on Plate No. J l . Usually, where seepage forces are small and are essentially horizontal, the submerged weight may be used in place of the saturated weight and the seepage forces disregarded; this was done on the Rendulic analysis, Plate No. Cl, appendix C, for the portion of the embankment lying to the left of point N and below the saturation line.

Partial submergence. This analysis investigates the condition where the' reservoir pool may be at different elevations against the upstream slope and determines the critical pool elevation which gives the lowest factor of safety. Static conditions are assumed; that is, the pool remains at a given elevation until a steady hydrostatic state exists. For convenience in analysis, the saturation line is assumed horizontal at the pool elevation. The moist weight of soil is used above the saturation line, the submerged weight below. A typical embankment section analyzed in this manner is shown on figure 1, Plate No. J l. The method of analysis shown is best adapted to homo­geneous sections but may be modified if different materials are crossed by the assumed failure arc. First, an assumed failure arc is drawn. At selected locations along the arc, vertical lines W are drawn between the arc and embankment slope. These lines represent the moist weight of soil above the arc at each point. The vector length of W is related to the ^weight of material as follows:

W =hym- ............ ........................... ............. ...............(21)where A= height of embankment above the arc and 7TO= th e moist unit weight of soil. The weight forces W are resolved into components normal to N and tangential to T, the arc as shown in figure 1. On a convenient horizontal base line, figure 2, the normal forces N are laid off vertically at horizontal locations 2, 3, and 4, equivalent to their position on the arc. A smooth curve is then drawn through the N forces to obtain the curve lab 6 shown on figure 2. In this example the vertical scale in figure 2 has been doubled for convenience in drawing. A similar procedure is followed for the tangential forces T and the curve XT is obtained as shown in figure 3. I t should be noted that the tangential forces acting to the left are positive and those acting to the right are negative and are plotted accordingly. The two summation lines of normal and tangential forces are for the entire embankment in the moist condition.

The next step is to determine the normal and tangential forces for the submerged portions of the embankment. Assume the water to be at B, figure 1. The saturation line will be as shown on the drawing. If a vector scale is chosen such that W / represents the weight of submerged material lying below the saturation line at point 4, figure 1, then the normal and tangential com­ponents will be N / and T /. These values are laid off on figures 2 and 3, respectively, vertically downwards from XN and XT curves. A succession of similar constructions with the pool at dif­ferent elevations will define the curves labeled “Water at A ” “Water at B ,” etc. With the water at a given elevation, the area under the XN and XT curves to the left of the line marked “Water a t _ ” represents the summation of forces due to the submerged weight and the area to the right represents the summation of forces due to the moist weight.

Computations of total resisting and driving forces for the pool at various elevations are shown on figures 4 and 5. Areas under the 2N curve for moist and submerged weights are determeined (a planimeter is useful here), then each area is multiplied by a conversion factor to obtain 2Ar in kips per ft. The method of determining the conversion is shown on Plate No. J l . The total resisting force is equal to the sum of friction and cohesion (equation (1)). Therefore, the 2Ar value is multiplied by tan <j> of the material and total cohesion along the length of arc is added to obtain the total resisting force. Similarly, for the driving T forces, the areas under the T curve are determined and multiplied by conversion factors as was done for the N forces. Factors of safety are determened by dividing the sum of the resisting forces by the sum of the driving forces for each elevation of the pool. The change in factor of safety with pool level is readily seen by plotting those values as shown in figure 6. From the plot of factor of safety versus pool level the minimum factor of safety for the critical pool elevation may be determined for the assumed failure arc.

Incompletely consolidated soils. The condition analyzed in this case is where an embankment is constructed upon a compressible foundation which has not completely consolidated. Usually the most critical condition is shortly after construction when the embankment has been built to full height and the only consolidation in the foundation is that which has occurred during construction. For this case the entire embankment and foundation weight (with due regard to submergence or saturation) is assumed to contribute to the driving forces, whereas only a fraction of the added (embankment) weight contributes toward mobilizing the shearing resistance. An example of this type of analysis is shown on Plate No. J2. The analysis is combined with that for the case of steady seepage using seepage forces, which is discussed in a succeeding paragraph; therefore, only the portions pertinent to foundation consolidation are presented here. On Plate No. J2 the founda­tion is assumed to be clay, 50 ft. thick, with a drainage face at the ground surface, but none at the base of the layer. Other studies have determined that the foundation will have an average of 25 percent consolidation under the weight of the embankment at the time the analysis is'to represent. From consolidation data a plot is made of variation in percentage consolidation with depth as shown in figure 1. For purposes of analysis it is assumed that frictional strength mobilized in the founda­tion is dependent on: (1) the full weight of foundation material overlying the assumed failure arc, and (2) the weight of embankment material times the percentage consolidation in the foundation in the segment being investigated. Since seepage forces are used in the analysis on Plate No. J2, the vertical component of the seepage force is added to the embankment weight before multiplying by the percent consolidation.

As an example consider slice 12 on figure 1. The vertical component of the seepage force is 5.8 kips, the weight of embankment is 212.1 kips; therefore, the total weight above the foundation is 217.9 kips. From the consolidation diagram the average consolidation on the base of slice 12 is 21 percent. Then, the effective weight of embankment materials is 217.9 X 0.21=45.8 kips. To this weight is added the weight of foundation material, 27.3 kips, to obtain the final effective weight used to determine the normal force, N, which is multiplied by tan <f> of the clay to obtain the fric­tional resistance of the slice. Similar computations are made for all slices where the failure arc cuts into the foundation.

Sudden drawdown. In the example of analysis of an upstream slope for the condition of sudden drawdown, it was assumed that the reservoir was lowered instantaneously from the maximum pool to the elevation shown on Plate No. J3. A flow-net has been drawn assuming that full seepage will develop with no drawdown of the saturation line due to drainage. This is an extreme assumption and is not likely to be encountered in actual practice. Since the embankment analyzed is homo­geneous, the anafysis is made using the method described in preceding paragraph in regard to partial submergence. Crixer methods, described under the discussion of the circular arc methods, could also be used. However, in the sudden drawdown case described here, allowance must be made for water pressures acting on the base of the assumed failure arc. The embankment soils are assumed

J-2

to be saturated and the full saturated weight contributes to the driving (tangential) forces. The normal forces, however, are reduced by the hydrostatic pressures. A typical force diagram is shown on figure 2. For the method of analysis used it is convenient to select a vector scale for the weight forces such that the length represents the weight force of an equivalent height of dam having the unit weight of water. This is readily accomplished by multiplying the height of embankment

above a given point on the failure arc by the ratio, ^ * 5 where 7cs=the saturated weight of soil.

This may be done graphically by the use of proportional dividers. Normal and tangential forces are laid off as shown on figures 1 and 2. The hydrostatic forces are evaluated by drawing a stand­pipe curve from the flow-net, representing the distribution of water pressure on the base of the failure arc. The construction for reducing the normal force by the amount of the hydrostatic pressure is shown on figure 2. The normal and tangential forces are then laid off on a base line and the envelop­ing curves drawn as shown in figure 3. The total normal and tangential forces on the failure arc are equal to the areas under the respective curves times conversion factors for the scale of the draw­ing and for the unit weight of water (assumed as equivalent soil weight in this case). Total resist­ing forces are cohesion C and friction N tan <t> and are divided by the total tangential force T to obtain the safety factor.

The preceding method cannot be applied to soils subject to significant volume changes when subjected to a change in stress, since the shear strength of these soils cannot adjust itself to the stress conditions after drawdown. The shear strength of such soils after sudden drawdown is no greater than immediately prior to drawdown, and the method of analysis used must take this fact into account.

Analyses using flow nets. In the preceding examples of stability analyses the flow-net has been used to evaluate water pressures on the bottom and sides of the segments or wedges comprising the failure mass. In most cases the flow-net was drawn, from which a system of equipressure lines was constructed as shown on Plate No. E2, appendix E. Either the basic flow-net or the equi­pressure line system may be used to determine pressure distribution and the results will be identical; the latter system is often more convenient to use. Analyses using the flow-net or equipressure system to determine water pressures are shown on plates in appendixes A, B, E, G and H and have been described previously. In all of these the water pressures acting along a segment boundary have been evaluated and then combined vectorially with the saturated weight of soil below the saturation line to determine the effective soil weight which is acting.

Another approach to the evaluation of seepage effects is by the use of seepage forces as illus­trated on Plate J2. A flow-net is first drawn for the condition investigated; in the example shown the case of steady seepage through the embankment is analyzed. An assumed failure arc is drawn and the soil above the arc divided into appropriate segments. In any segment the seepage force equals the product of the mean hydraulic gradient across the segment, times the area of the segment below the saturation line, times the unit weight of water. The force acts through the center of the area in the mean direction of the flow lines passing through the segment.

The seepage force J is computed as shown in figure, 2. Through the midpoint of each side of a segment, i. e., the portion below the saturation line, equipotentials are drawn to interesect the free water surface. The difference in elevation of the equipotentials at the free surface is Ah. Then, the mean hydraulic gradient, i=Ah/L, where L is measured between the segment boundaries through the center of gravity of the submerged portion and parallel to the nearest flow line. The seepage force J = area of the submerged segment x 62.5 x i. It acts through the center of gravity of the submerged portion of the segment in a direction parallel to the flow' line through that point. In the example shown on Plate No. J2, seepage forces have been computed only for the impervious portion of the embankment, Material A; in the foundation and in Material B the hydraulic gradient is practically zero so the seepage forces have been neglected.

J-3

Other features of the analysis shown on Plate No. J2 are somewhat different from those de­scribed previously and they will be discussed below. In all cases moist weights have been used above the saturation line and submerged weights below. Total weights have been used to compute driving forces; for that portion of the arc lying within the foundation, only a part of the weight is utilized in developing frictional resistance, since the foundation is not fully consolidated. This construction has been described in preceding paragraph on incompletely consolidated soils above so will not be discussed here. To compute the overturning moment the weight of each segment is multiplied by its moment arm to the center of the failure arc. The moment arm of each seepage force is determined and multiplied by the force in a manner similar to the weight forces. The summation of all weight and seepage moments, with due regard to direction, gives the total over­turning moment for the arc.

The computation of resisting forces is accomplished as shown in figure 1, Plate No. J2 , and the table on the right side of the drawing. Once the effective weight for a segment has been de­termined, the normal force is computed as follows. A circle with a diameter of unity, to some convenient scale, is drawn with one end of the diameter passing through the center of the failure arc and the other end of the diameter vertically below the center of the arc. The length of any chord on this circle, passed through the center of the failure arc, is then equal to the cosine of the angle subtended between the chord and the vertical. Since the relation between the weight of a segment, TT, and its normal force, N, is N = W cos/3 where is the angle between the two forces, the unit circle may be used to compute cos/3 for any weight force and thus determine N. For example, take slice 3 on figure 1. A line drawn from the center of the failure arc to the intersection of force W3 with the arc, crosses the unit circle at a distance of 0.91 = cosj53. The effective weight W3= 3 8 .2 kips, so the normal force ATZ=38 .2 x 0 .91=34.7 kips. The frictional resistance is com­puted by multiplying the normal force, N, times tan <t> of the material through which the arc passes. The N tan <t> values are summed up for all segments and added to the total cohesion along the failure arc to obtain the total shearing resistance. The moment of resisting forces is the total shearing resistance times the radius of the failure arc. The safety factor is determined by dividing the resisting moment by the overturning moment.

Tension cracks. Corrections to be made to stability analyses for tension cracks in cohesive soils have been described in paragraph 2-07A, in the text. Examples of analyses where this factor has been taken into consideration are shown in appendixes D, E, and G. Since the referenced description and computations on the plates are believed sufficiently clear, no further presentation of the method is made here.

Stratified slopes. As discussed previously, stratified slopes nmy be analyzed by most of the methods presented herein, the main factor to consider is to have as much of the failure surface lie within the zone of weakest material as is possible. This may be accomplished by composite sliding surfaces of arcs having different radii or by a series of planes. An analysis using planes where the horizontal sliding surface is in a weak layer of material is shown on Plate No. H i. T?he Rendulic analysis on Plate No. Cl is also a case where horizontal sliding in a relatively weak foundation layer was considered.

J-4

-C enter o f assumed fa ilu re arc

FIGURE 3SUMMATION OF TANGENTIAL FORCES

HORIZONTAL SCALE: l" = IO'VERTICAL SCALE: l" = 5'

UNIT WEIGHT COMPUTATIONS

Dry Weight- f§ * - - , 0 0 * / f t . 3

Saturated Wt.-_ G+e!+ e “ Wsaf -

2 .6 9 + 0 .6 8 1.68 x 6 2 .4 = 1 2 5 * / f t 3

Moist Weight <& ¡5% w = 1 00 x 1.15 = 11 5 * / f t .3

Submerged Wt. - /2 5 - 6 2 .4 = 6 2 . 6 * / f t 3

VECTOR SCALE CONVERSIONS

Moist Weight (See Columns 4 and 16 below)Scale of Figs. 2 and 3 equals l"= 10' horizontal and l"= 5‘ vertical, therefor Isq.in = 5 0 sq.ft, each square inch = 50 sq.ft, x 115 Ibs./cu.ft.= 5.75 Kips / ft.

Submerged Weight ( See Columns 5 and 17 below)Each square inch = 50sq.ft. x 62.6 Ibs/cu.ft. = 3.13 Kips/ft.

COMPUTATIONS OF RESISTING FORCES© © © © © © © ©

W a te r X N Forces M o is t Wt Submerged Wt ® x 3.13

T o ta l Wt. F r ic t io n a l T o ta la t FIG. 2 M o is t Subm erged

A re a© x 5 .7 5 © + ©

R e s is ta n c e ® x T a n <t>

K ip s / f t .

R e s is ta n c e © + L c K ip s / ft.Moist Sub ­

merged A re a K ip s / f t . K ip s / f t. K ip s / f t.

0i - — 29.10 0 167.0 0 167.0 55.7 104.7

A I 18.20 10.90 104.5 34.1 138.6 46.1 95.1

B 9.65 19.45 55.5 60.9 116.4 38.8 87.8

C > >3.98 25.12 23.0 78.6 101.6 33.8 82.8

D 2 :¿ ' 0

0.70 28.40 4.0 88.9 92.9 31.0 80.0

E — 0 29.10 0 91.1 91.1 30.3 79.3

tan. <f> = 0.333C = 0.4 Kips/sq.ft.

L c = 122.5 x 0.4 = 49 Kips/ft.

COMPUTATIONS OF DRIVING FORCES© © © (14) ©

XT Forces

- M o is tA re a

0.35

N etM o is tA re a

13.05

I 1.45

8.65

4.70

1.65

-Submerged A re a

4.75

8.70

II .75

13.40

- Submerged A re a

0.35

0.35

0.35

0.35

0.35

N e tSubm erged

A re a

1.60

4.40

8.35

11.40

13.05

©M ois t Wt. @ x 5 .7 5

K ips/ft.

Submerged Wt. © x 3.13

K ips/ft.

7 5.0

65.9

49.8

27.0

9.5

5.0

I 3.8

26.2

35.7

40.8

©Total Wt.© + ©K ips/ft.

75.0

63.6

53.2

45.2

40.8

S. F

1.39

1.34

1.38

1.56

1.77

1.94

FIGURE 4

VARIATION OF SAFETY FACTOR WITH POOL ELEVATION

S C A L E I" = 10'

S C A L E l"= 5'

STABILITY OF SLOPES AND FOUNDATIONSPARTIALLY SUBMERGED SLOPES

VARIATION OF SAFETY FACTOR WITH POOL LEVELS C A L E A S SH O W N___________________________________________________________________________________

OFFICE OF THE CHIEF OF ENGINEERSCIVIL WORKS

W ASH INGTO N, D.C.___________ ENGINEERING DIVISION_______________ JA N U A R Y 1951

EMC- 119-2-Jl PLATE J I

COMPUTATION OF OVERTURNING MOMENT

Unit• Weight E */Cu.Ft.

Seepage Forces Area x 62.4 x

COMPUTATION OF FRICTIONAL RESISTANCE

© © © © ® ® © ©

< +

0 .5x29 (8 + 16.5) = 0.5x19x29 = 0.5x25 (15 .5+22) =0 5x25(19 + 31) = 2 5 x 2 0 = 0 .5 x 2 5 (2 + 8 5 ) = 0.5x 25(31 + 40) =2 5 x 2 0 =0 5 x2 5 (85 + 16) = 0 .5 x 2 5 (4 0 + 4 6 ) = 2 5 x 2 0 =0.5x2 5(.I6 + 23) =0 5 x2 5 (46 + 4 9 ) = 2 5x 20 =0 .5x25 (23 + 30) = 0 .5 x 25 (4 9 + 50) = 2 5 x 2 0 = 0 .5 x 2 5 (3 0 + 38)= 0 .5 x 2 5 (5 0 + 4 7 5 ) = 25 x 20 = 0 .5 x 2 5 (3 8 + 4 6 .5 )= 0 .5 x 2 5 (4 7 ,5 + 43) = 2 5 x 20 =0.5x 25(46 .5 + 55) = 0 .5 x 2 5 (4 3 + 36) =2 5 x 20 =0.5x 25(55 +6 75 ): 0 .5 x 2 5 (3 6 + 2 6 ) =

AW =Weight of

Embankment Materials

(£= Ratio Normal

Force to Weff.

ADOPTED DESIGN DATA

Material Moist Weight Ibs/cu. ft.

Submerged Wt Ibs. / cu. ft 0 Tan 0 Cohesion

Kips/ sq ftA M2 59 20° 0 .3 6 4 0 .3 0B HO 63 35° 0 .7 0 0 0C — 56 12° 0.213 0 .4 0

Tanff

468625

500 306

1075 500 488 1186 500 662

1236 500 856

1218

5001270995

0 .7 9

SHEARING RESISTANCE ALONG ARCFriction in "A”and "C" 237 .4Cohesion in "C” = 0 .4 0 x 3 2 0 = 128 0Cohesion i n MA " = 0 .3 0 x 1 0 8 * 32.4

Sofety Foctor3 I .5

I 16 7 6 3 .3

21 1.8

Resisting Moment Overturing Moment

3 97 .8 x 250= 9 9 ,4 0 0 Ft KipsResisting Moment

9 9 ,4 0 0 ^ „161,815 = a 6 1

9 ,0 0 0 Total Frictional Resistance = 2 37 .36C irc le 5

Most dangerous circle

„ Circle with diometer o f unity

Saturation line

Equipotentiaf fine -

Flow lin e ----------- ^

M id-point o f a b

Distance L token through center of gravity o f abed and p a ra lle l to flow lit e.

Assumed failure arc

Mean Hydroutic Gradient i = = j—

Seepage Force J - A reo abed x 6 2 A x i

Center o f gravity abed" -Equipotentia! line

"M id-point o f c d

- Flow line

FIGURE 2

COMPUTATION OF SEEPAGE FORCES

STABILITY OF SLOPES AND FOUNDATIONS

STABILITY ANALYSIS USING SEEPAGE FORCES

impervious------------- ' mmmmtmmm. mmma SCALE: AS SHOWN

FIGURE I

EMBANKMENT SECTION WASHINGTON, D C.

OFFICE 0F THE CHIEF OF ENGINEERS CIVIL WORKS

ENGINEERING DIVISION JANUARY 1951

SCALE I = 20

EM C -N 9-2-J2 PLATE J 2

I N = Norm al fo rce p e r fo o t o f dam = x ^ 2 -4 x ^ q q s 2 5 1 . 5 Tons2000

ADOPTED DESIGN DATAEMBANKMENT SOILS

S aturated u n it weight, y saf - 122 tb /c u .ft. Cohesion, C, = 0 .i5 Tons/sq. f t Angie o f in te rn a l fric tio n , <f>,= 2 0 °, tan. =0.364

STABILITY ANALYSIS OF UPSTREAM SLOPESUDDEN DRAWDOWN OF RESERVOIR POOL

20SCALE IN FEET

0 20 40 60

SCALE: l" = 20 ' ISq. In. = 4 0 0 Sq. Ft.

STABILITY OF SLOPES AND FOUNDATIONS

SUDDEN DRAWDOWN STABILITY ANALYSIS WITH USE OF A FLOW NET

S C A LE :AS SHOWN

WASHINGTON, D. C.

OFFICE OF THE CHIEF OF ENGINEERS CIVIL W O RK S

E N G IN E E R IN G D IV IS ION JANUARY 1951

EMC - 119-2 - J 3 PLATE J 3