Design Standards No. 13 Embankment Dams

Chapter 4: Static Stability Analysis Phase 4 (Final)

U.S. Department of the Interior Bureau of Reclamation October 2011

Mission Statements

The U.S. Department of the Interior protects America’s natural resources and heritage, honors our cultures and tribal communities, and supplies the energy to power our future.

The mission of the Bureau of Reclamation is to manage, develop, and protect water and related resources in an environmentally and economically sound manner in the interest of the American public.

Chapter Signature Sheet Bureau of Reclamation Technical Service Center

Design Standards No. 13

Embankment Dams

Chapter 4: Static Stability Analysis

DS-13(4)-6:1 Phase 4 (Final) October 2011

Chapter 4 - Static Stability Analysis is an existing chapter within Design Standards No. 13 and was revised to include:

 Lowering of minimum factor of safety for reservoir operational conditions

 Current practices in slope stability analysis and computer programs

 Application to slope stability analysis for existing embankment dams

 Guidance papers for static slope stability analysis

 Minor corrections and editorial changes

1 DS-13(4)-6 refers to Design Standards No. 13, chapter 4, revision 6.

Prepared by: askck Ashok Chugh, RE. Date Civil Engineer, Geotechnical Engineering Group 1

Peer Review:

VPAN/ William ngemoen, P.E. Date Geotechnical Engineer, Geotechnical Services Division

Secu "� eview:

Ot' Larry K. s, P.E. Structura Engineer, Structural Analysis Group

Recommended for Technical Approval:

Thomas N. McDaniel, RE. Geotechnical Engineer, Geotechnical Engineering Group 2

Submitted:

7 /1 Karen Knight, P.E. ate Chief, Geotechnical Services Division

Approved:

Lowell Pimley, RE. Date Director, Technical Service Center

Design Standards No. 13, Chapter 4

Contents Page

4.1 Introduction...... 4-1 4.1.1 Purpose...... 4-1 4.1.2 Scope...... 4-1 4.1.3 Deviations from Standard ...... 4-1 4.1.4 Revisions of Standard ...... 4-1 4.1.5 Applicability ...... 4-1 4.2 Loading Conditions and Factor of Safety ...... 4-2 4.2.1 General...... 4-2 4.2.2 Selection of Loading Conditions ...... 4-2 4.2.3 Discussion of Loading Condition Parameters ...... 4-3 4.2.4 Factor of Safety Criteria...... 4-5 4.3 Shear Strength of Materials...... 4-9 4.3.1 General Criteria ...... 4-9 4.3.2 Shear Strength Data and Sources...... 4-9 4.3.3 Shear Strength Related to Loading Condition ...... 4-11 4.3.4 Anisotropic Shear Strength ...... 4-13 4.3.5 Residual Shear Strength...... 4-13 4.4 Determination of Pore Pressures ...... 4-14 4.4.1 Phreatic Surface Method ...... 4-14 4.4.2 Graphical Flow Net Method ...... 4-14 4.4.3 Numerical Methods ...... 4-14 4.4.4 Field Measurement (Instrumentation) Method ...... 4-15 4.4.5 Hilf’s Method ...... 4-15 4.5 Slope Stability Analyses...... 4-16 4.5.1 Method of Analysis...... 4-16 4.5.2 Slip Surface Configuration ...... 4-16 4.5.3 Slip Surface Location...... 4-17 4.5.4 Progressive Failure ...... 4-17 4.5.5 Three-Dimensional Effect ...... 4-18 4.5.6 Verification of Analysis ...... 4-19 4.5.7 Existing Dams ...... 4-20 4.5.8 Excavation Slopes ...... 4-22 4.5.9 Back Analysis ...... 4-22 4.5.10 Multi-Stage Stability Analysis for Rapid Drawdown...... 4-23 4.6 References ...... 4-24

DS-13(4)-6 October 2011 4-i Design Standards No. 13: Embankment Dams

Appendices

A Selection of Shear Strength Parameters B Spencer’s Method of Analysis C Computer Programs D Guidance Papers for Static Stability Analyses

Figures Page

4.5.7-1. B.F. Sisk Dam (near Los Banos, California). View showing upstream slope failure during reservoir drawdown ...... 4-20 4.5.7-2. General cross section of B.F. Sisk Dam in the location of the slide during reservoir drawdown...... 4-21 4.5.7-3. Pine View Dam (near Ogden, Utah). View showing stability berm to improve performance of the dam during a seismic event...... 4-21 4.5.7-4. General cross section of Pine View Dam with stability berm...... 4-22

Table Page

4.2.4-1 Minimum factors of safety ...... 4-8

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Chapter 4 Static Stability Analysis

4.1 Introduction 4.1.1 Purpose This standard provides guidelines for accomplishing a thorough examination and satisfactory analytical verification of the static stability of an embankment dam. This will be accomplished through a discussion of applicable loading conditions, material properties, pore pressures to be considered, and appropriate minimum factors of safety that should be obtained for those loading conditions.

4.1.2 Scope Criteria are presented for the determination of: (a) loading conditions and minimum factor of safety, (b) material strength properties, and (c) pore pressures. Methods for computing embankment stability under static loading are described in appendix B and are illustrated with numerical examples included therein.

4.1.3 Deviations from Standard Stability analyses within the Bureau of Reclamation (Reclamation) should conform to this design standard. If deviations from the standard are required for any reason, the rationale for not using the standard should be presented in the technical documentation for the stability analyses. The technical documentation should follow the peer review requirements included in reference [1].

4.1.4 Revisions of Standard This chapter will be revised as its use indicates. Comments or suggested revisions should be forwarded to the Chief, Geotechnical Services Division (86-68300), Bureau of Reclamation, Denver, Colorado 80225; they will be comprehensively reviewed and incorporated as needed.

4.1.5 Applicability These stability analyses standards are applicable to the design and analysis of embankment dams founded on either soil or rock. While the methods discussed herein are also applicable to cut and natural slopes, analysis of cut and natural

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slopes may involve factors not addressed herein. For design of small dams, refer to Reclamation publication cited in reference [2].

4.2 Loading Conditions and Factor of Safety 4.2.1 General The loading conditions to be examined, for either a new or an existing dam, should be based on knowledge of the construction plan, reservoir operation plan, emergency and maintenance operation plans, and flood storage and release plans of the reservoir along with the behavior of the embankment and foundation materials with respect to the development of pore pressures in the dam and foundation.

Appropriate minimum factors of safety will be assigned for these loading conditions.

4.2.2 Selection of Loading Conditions The loading conditions to be examined are:

 Construction conditions.—For a new dam, the end-of-construction condition must be analyzed. It may also be necessary to analyze stability for partial completion of fill conditions, depending on construction schedule and relationship of pore pressures with time.

 Steady-state seepage conditions.—For either a new or an existing dam, the stability of the downstream slope should be analyzed at the reservoir level that will control the development of the steady-state seepage surface in the embankment. This reservoir level is usually the top of active conservation storage, but may be lower or higher depending on anticipated reservoir operations.

Operational conditions.—For either a new or an existing dam, if the maximum reservoir surface is substantially higher than the top of active conservation surface, the stability of the downstream slope should be analyzed under maximum reservoir loading. The upstream slope should be analyzed for rapid drawdown conditions from the top of active conservation capacity water surface to the top of inactive capacity water surface and from the maximum water surface to the top of inactive storage water surface. The upstream slope should also be analyzed for rapid drawdown conditions from the top of active conservation water surface to an intermediate level if upstream berms are used.

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 Other conditions.—Other loading conditions that need to be analyzed in some cases, if appropriate, include: (a) internal drainage plugged or partially plugged, (b) drawdown due to unusually high water use demands, (c) drawdown for the emergency release of the reservoir, (d) construction modifications , and (e) earthquake loading included in Chapter 13, Seismic Analysis and Design, of this Design Standards.

4.2.3 Discussion of Loading Condition Parameters General guidelines for obtaining reservoir elevation, soil properties, and pore pressure parameters for analysis of different loading conditions are as follows:

 Construction conditions.—The end-of-construction condition can be examined either by effective stress concepts or by undrained shear strength concepts.

o Effective stress shear strength envelope method.—The materials in the dam or foundation may develop excess pore pressures due to loading imposed by the overlying soil mass during construction. The effective stress method requires estimation of the change in pore pressure with respect to construction activities and time. Consideration should be given to monitor pore pressures during construction to ensure that the estimated pore pressures are not exceeded by a large margin [3]. The preferred methods for estimating pore pressures during and at the end of construction loading conditions are:

. Conduct laboratory tests on representative samples of embankment and foundation materials to determine the initial pore air and pore water pressure.

. Conduct laboratory tests to determine the pore pressure behavior with respect to time and applied load for each material.

. Establish the expected construction schedule, determine pore pressure versus time function in the materials for that schedule, and check the stability of the upstream and downstream slopes.

. If necessary, revise the schedule based on actual construction and recheck stability.

o Undrained shear strength envelope method.—This method for determining soil shear strength does not involve measurement of pore air or pore water pressure in the soil sample; thus, it is relatively simpler than the effective stress shear strength envelope method described above. However, analysis in terms of undrained shear

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strength implies that pore pressures occurring in laboratory tests on the material satisfactorily approximate field pore pressures and, therefore, the shear strength of those materials. Shear strength tests should be performed on specimens compacted to anticipated placement water contents and densities. Undrained shear strengths used in analyses should correspond to the range of effective normal consolidation stresses expected in the field [4].

 Steady-state seepage conditions.—The annual reservoir operation plan should be examined to determine the appropriate reservoir water elevation for use in estimating the location of the steady-state phreatic surface. Usually, it is the top of active storage or joint use pool elevation; although, it is possible that under certain operational plans, such an elevation (top of active storage or joint use pool) is reached for only a small fraction of time each year or it is reached in an oscillatory cycle. In either case, the effective reservoir elevation could be taken to be near the midpoint of the cycle. However, use of steady-state phreatic surface in stability analysis of a wide clay-core embankment dam is an accepted practice even though steady-state phreatic surface may not be expected to develop for a very long time.

For an existing dam, the appropriate elevation can usually be determined from operational records, specifically from reservoir operational plots of reservoir elevation versus time. Reservoir operational plots are available on most dams in the Reclamation inventory. Piezometric data, if available, can be used to estimate the phreatic surface in the embankment.

 Operational conditions

Maximum reservoir level.— If the phreatic surface under flood loading is significantly different (higher) from that of the steady-state condition for the active conservation pool, then the stability under this (higher phreatic surface) condition should be analyzed. A phreatic surface should be estimated for the maximum reservoir level. The maximum reservoir level may occur from a surcharge pool that drains relatively quickly or from a flood control pool that is not to be released for several months. The hydraulic properties (permeability) of materials in the upper part of the embankment affected by the reservoir fluctuations should be evaluated to determine whether a steady-state or transient analysis should be made when estimating the position of the phreatic surface. If the phreatic surface is significantly different (higher) from that of the steady-state condition for the active conservation pool, then the stability under this (higher phreatic surface) condition should be analyzed.

o Rapid drawdown conditions.—During active conservation pool stages, embankments may become saturated by seepage. If, subsequently, the

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reservoir pool is drawn down faster than pore water can drain from the soil voids, excess pore water pressure and unbalanced seepage forces result. Typically, rapid drawdown analyses are based on the conservative assumptions that: (1) pore pressure dissipation does not occur during drawdown in impervious material and (2) the phreatic surface on the upstream face coexists with the upstream face of the impervious zone and originates from the top of the inactive capacity water surface.

However, the critical elevation of drawdown with regard to stability of embankment may not coincide with the minimum pool elevation, and thus, intermediate drawdown levels should be considered.

 Other conditions

o Inoperable internal drainage.—If uncertainties exist with regard to the success of internal drainage features or dewatering system designed to control the phreatic surface in an existing dam, then checks should be made using the phreatic surface developed assuming these features are not fully functioning.

o Unusual drawdown.—All reservoir drawdown plans for maintenance or emergency release of the reservoir should be reviewed to determine the appropriate parameters for the stability analysis and the need for any modification of the usual phreatic surface assumption on the upstream face. Drought may cause reservoir drawdown and should be considered as such for the stability analysis. Intermediate drawdown levels would, in general, not be required to be examined.

o Construction modifications.—All excavation plans in close vicinity to an existing embankment should be reviewed to determine the appropriate parameters for the stability analysis of the excavation and that of the embankment. Similarly, all dam raise plans should be reviewed to determine appropriate parameters for the stability analysis of the existing and the raised dam configurations including reservoir operations.

4.2.4 Factor of Safety Criteria For each loading condition described previously, a recommended minimum factor of safety is provided. Deviations either higher or lower from these general criteria may be considered, but should be supported with an appropriate justification. The specific values selected need to consider: (a) the design condition being analyzed and the consequences of failure, (b) estimated reliability of shear strength parameters, pore pressure predictions, and other soil parameters, (c) presence of

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structures within the embankment, (d) reliability of field and laboratory investigations, (e) stress-strain compatibility of embankment and foundation materials, (f) probable quality of construction control, (g) embankment height, and (h) judgment based on past experience with earth and rockfill dams.

For the purpose of slope stability analysis, the factor of safety is defined as the ratio of total available shear strength of the soil to shear stress required to maintain equilibrium along a potential surface of sliding. The factor of safety indicates a relative measure of stability for various conditions, but does not precisely indicate actual margin of safety. In addition, a relatively large factor of safety implies relatively low shear stress levels in the embankment or foundation and, hence, relatively small deformations. The minimum factors of safety for use in the design of slope stabilization should follow rationally from an assessment of a number of factors, which include the extent of planned monitoring of pore pressures and assumptions and uncertainties involved in the material strength.

The factor of safety criteria presented in this standard are based on the slope stability analysis being performed by limit equilibrium method using Spencer’s procedure. A different procedure within the limit equilibrium method of analysis could give a different factor of safety for the same embankment cross section with the same material properties under the same loading conditions.

For the end-of-construction loading condition, excess pore pressures may be induced in impervious zones of the embankment or foundation because these soils cannot consolidate completely during the construction period. If effective stress shear strength parameters are used for the analysis, then excess pore pressures have an important influence on the factor of safety. A minimum factor of safety of 1.3 would be considered adequate if pore pressures are monitored during construction. However, if the effective stress shear strength envelope is used without any field monitoring of pore pressures, the minimum safety factor should be at least 1.4 to eliminate uncertainties involved in excess pore pressures.

A minimum factor of safety of 1.3 would also be adequate when the analysis is carried out in terms of undrained shear strength. However, if an undrained shear strength envelope is used, the laboratory testing performed to define the envelope must satisfactorily model the pore pressure behavior and state of stress anticipated under field loading conditions.

For the steady-state seepage condition under active conservation pool, a minimum factor of safety of 1.5 would be justified to take into account the uncertainties involved in material strengths, pore pressures in impervious material, and long-term loading. In addition, the failure of the downstream slope under a steady-state seepage condition is more likely to result in a catastrophic release of water, which definitely demands a higher safety margin than for the end-of­ construction or rapid drawdown conditions.

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For the operational conditions, a factor of safety of 1.2 for assumed steady-state seepage conditions under maximum reservoir water level during a probable maximum flood event would be justified if the duration of high flood pool is relatively short and the reservoir operations call for draining the flood storage quickly using spillway and outlet works facilities at the dam site, and restoring the reservoir to the active conservation pool. A higher factor of safety (approaching 1.5) might be required if the duration of flood storage above the active conservation pool is long and could potentially result in phreatic surface which is significantly higher than the steady-state phreatic surface under the active conservation pool.

For the rapid drawdown condition from active conservation pool (normal water surface) to inactive conservation pool, or other intermediate level, the loading due to unbalanced seepage forces may render the upstream slope unstable; however, the loading is of short duration, and the reservoir level is reduced during drawdown. Consequently, failure of the upstream slope would not likely release the reservoir. Hence, a minimum factor of safety of 1.3 is adequate, and in some cases, a lower factor of safety is acceptable with justification. Similarly, for the rapid drawdown condition from maximum reservoir surface (following a probable maximum flood) to active conservation pool, a factor of safety of 1.2 is adequate considering the short duration of the flood pool surcharge before returning to the normal pool. For the rapid drawdown below the active conservation pool, the minimum factor of safety of 1.3 applies.

For the other loading conditions, a minimum safety factor of 1.2 for drawdown at maximum outlet capacity, inoperable internal drainage, or failure of dewatering system is justified mainly because of infrequent occurrence and reliance on quick remedial action in the event of inoperable drainages and failure of dewatering system. However, if quick remedial action cannot be ensured in advance, higher factor of safety (approaching 1.3 or higher) should be required. For the construction modification, stability of the temporary excavation slopes and the resulting overall embankment stability during construction should have a minimum factor of safety of 1.3 for a conservative combination of ground water conditions and foundation soils during construction. However, all construction activities shall be well planned and executed under close supervision of qualified personnel.

Table 4.2.4-1 summarizes the minimum factors of safety required for various loading conditions.

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Table 4.2.4-1. Minimum factors of safety based on two-dimensional limit equilibrium method using Spencer’s procedure Shear Minimum Loading strength factor of condition parameters* Pore pressure characteristics safety Generation of excess pore pressures in 1.3 embankment and foundation materials with laboratory determination of pore pressure and monitoring during construction Generation of excess pore pressures in 1.4 Effective embankment and foundation materials and End of no field monitoring during construction and construction no laboratory determination Generation of excess pore pressures in 1.3 embankment only with or without field monitoring during construction and no laboratory determination Undrained 1.3 strength Steady-state Effective Steady-state seepage under active 1.5 seepage conservation pool Effective or Steady-state seepage under maximum 1.2 undrained reservoir level (during a probable maximum flood) Rapid drawdown from normal water 1.3 Operational surface to inactive water surface conditions Effective or undrained Rapid drawdown from maximum water 1.2 surface to active water surface (following a probable maximum flood) Other Effective or Drawdown at maximum outlet capacity 1.2 undrained (Inoperable internal drainage; unusual drawdown)

Effective or Construction modifications (applies only to 1.3 undrained temporary excavation slopes and the resulting overall embankment stability during construction), * For selection of shear strength parameters, refer to appendix A.

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4.3 Shear Strength of Materials Stability analyses of embankment dams and natural slopes require determination of shear strengths of materials involved along any potential failure surface. There are large volumes of information available related to the shear strength of soils; however, it is beyond the scope of this design standard to cover the topic in great detail. It is important to recognize that the selection of shear strengths for use in a numerical analysis of a slope (natural or manmade) should be a deliberate effort and involve the entire design team (i.e., designers, laboratory personnel, and geologists). A valuable reference for shear strength as it relates to slope stability is the textbook by Duncan and Wright [5]. The textbook by Lambe and Whitman [6] is a valuable general reference on the subject.

4.3.1 General Criteria Stability analyses of embankment dams and natural slopes require the determination of shear strengths of the materials involved along any potential failure surface. Based on Mohr-Coulomb failure criterion with effective stress concepts, shear strength “S” (mobilized at failure) can be written as:

S = c′ + (σ - u) tan φ′ where: c' = Effective cohesion intercept φ′ = Effective angle of shearing resistance u = Developed pore pressure on failure surface, at failure σ = Total normal stress on failure surface due to applied load at failure

Based on undrained shear strength concepts, shear strength “su” can be written as:

su = f (σc′ )

which expresses the undrained shear strength as a function of σc′ , the effective consolidation pressure prior to shear failure.

4.3.2 Shear Strength Data and Sources Material shear strengths can be obtained from field testing, laboratory testing, or they can be estimated based on experience and judgment, depending on the design stage of the analysis. The analysis and decision regarding how to obtain shear strength parameters should also consider the sensitivity of the final slope stability on the strength parameters. For example, stability analysis may prove to be insensitive to small features such as gravel toe drains. Varying the strength value

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of the drain material may not have a large impact on the computed stability, and it may therefore be prudent to use typical values in such a case.

For existing dams in Reclamation’s inventory, project related earth material (EM) reports from Reclamation Materials Engineering and Research Laboratory provide site-specific soil strength data. However, caution should be exercised in using these past test data if the in situ stress conditions do not match with the stress conditions that were used in the laboratory tests. Guidance from experienced design staff should be solicited in deciding appropriateness of past test data.

In general, values for shear strength parameters to be used in appraisal or feasibility level design can be estimated based on judgment, previous experience and testing, local geologic data, or from tabulated data such as that included in appendix D from references [2] and [7]. For intermediate and final phases of design, shear strength parameters should be obtained from appropriate laboratory and field tests.

For fat clays, cyclic wetting and drying can reduce their shear strength to a fully softened state. Furthermore, if prior shear deformations have occurred in the foundation materials or material stress-strain behavior is strain softening, the shear strength of the clay may be at its residual strength value. Consolidated undrained (CU) tests with pore pressure measurements are used to measure fully softened strengths. Torsional ring shear and repeated direct shear tests are performed to determine residual shear strengths of clays in foundations, and natural and constructed slopes.

In situ shear strength testing can be performed in the field on foundation materials or embankment materials. Field testing methods include standard penetration test, vane shear test, cone penetration tests, and borehole shear device test. In situ tests are often used to estimate undrained shear strength parameters. References [8, 9] include details of in situ testing.

Laboratory shear strength testing is performed on disturbed or undisturbed samples of foundation and embankment materials to obtain shear strength parameters to be used in stability analyses. Appropriate laboratory shear strength tests include direct shear, repeated direct shear, triaxial shear, torsional ring shear, and simple shear tests. Shear strength tests should be supplemented with one-dimensional consolidation (oedometer) tests in order to ascertain the stress history of the material, particularly when determining undrained shear strength. Refer to appendix A for selection of proper laboratory tests that are compatible with field loading conditions. Reference [8] includes details of laboratory testing. Use of modern (state-of-the-art) sampling and testing equipment and procedures is advised.

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Determination of the shear strength parameters is the most important phase of a stability analysis, yet the most difficult, especially for undrained strengths. It is difficult to obtain representative samples, avoid sample disturbance, simulate external loading and internal pore pressure conditions, and avoid inherent error in the testing methods. It is normally impossible to obtain samples that truly represent the range of materials existing in the field. Therefore, shear strength parameters are generally determined from samples representing extremes, and parameters are selected within the range. Loads and stresses on a sample in the laboratory are different from those on an element of soil located on a failure surface in the ground. Hence, judgment and experience play an extremely important role in the evaluation of test results to ensure that the parameters chosen are representative of the materials in place.

As a general rule, for significant projects, shear strength parameters should be obtained from appropriate laboratory and field tests. Regardless of the source of shear strength data (laboratory tests, field tests, published data), sensitivity analyses by varying the selected strength parameter values should always be performed to ensure safe design. Results of slope stability analyses, including sensitivity analyses, should be discussed with the experienced staff, and their concurrence with the design should be solicited.

4.3.3 Shear Strength Related to Loading Condition The following loading conditions are usually evaluated for stability analysis of embankment dams: (1) end of construction, (2) steady-state seepage, and (3) rapid drawdown. The material shear strength parameters used in the analyses must correctly reflect the behavior of the material under each loading condition.

 End-of-construction loading can be analyzed by using either the undrained shear strength envelope or the effective stress strength envelope concept.

o Undrained shear strength.—Applicable shear strength parameters of saturated fine-grained foundation soils can be determined from unconsolidated undrained (UU) triaxial shear tests without pore pressure measurements conducted on undisturbed samples. Undisturbed samples should be selected and tested from a range of depths in the foundation material. Field vane tests, if used, should also be conducted over a range of depths. Test specimens representing embankment materials should be compacted to anticipated placement densities and moisture contents and tested in UU triaxial compression. The confining pressures used in these tests shall correspond to the range of normal stresses expected in the field. Generally, the undrained shear strength envelopes are parallel to the normal stress axis for fully saturated fine-grained soils, but for partially saturated soils, envelopes have a curved portion in the low

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normal stress range. This curved portion, or an approximation of it, should be used when the anticipated normal embankment stresses are in that range. UU triaxial shear test results approximate end-of-construction shear strengths of embankment zones consisting of impervious soils. These tests, performed on undisturbed samples, are also applicable to impervious foundation materials where the consolidation rate is slow compared to the fill placement rate.

o Effective stress.—Effective stress shear strength parameters are used in conjunction with the estimated embankment and foundation pore pressures generated by placement of the embankment materials. CU triaxial shear tests with pore pressure measurements are appropriate for clays and silts which, because of their low permeabilities, can be assumed to fail under undrained conditions. Consolidated drained (CD) triaxial shear tests or direct shear tests may be used for free-draining foundation and embankment materials. Shear strength of overconsolidated clay and clay-shale foundation materials could be obtained by using CU triaxial shear tests. If these clays have undergone prior shear deformation or the stress-strain behavior is strain softening, the residual strength may be appropriate and repeated direct shear or torsional ring shear tests should be used. When thixotropy is a possibility, shear strength parameters higher than residual may be considered.

 Steady-state seepage loading condition should be analyzed using effective stress shear strength parameters in conjunction with measured or estimated embankment and foundation pore pressures. The use of the CD triaxial shear test or the CU triaxial shear test with pore pressure measurements is appropriate. Sufficient back pressures should be used to effect nearly 100-percent saturation for both compacted samples of embankment materials and undisturbed foundation samples to ensure accurate pore pressure measurements. The use of the direct shear test is applicable for sands. It can also be used for silts and clays; however, the required rate of shearing would be very slow; therefore, it may not be practical. If prior shear deformation has occurred in the foundation materials or material stress-strain behavior is strain softening, residual strengths may be appropriate for these materials. If foundation materials consist of very soft clays, undrained shear strength of these materials should be determined as discussed above under the heading: Undrained shear strength.

 Rapid drawdown loading condition can be analyzed using effective stress shear strength parameters in conjunction with embankment and foundation pore pressures or using undrained shear strength developed as a result of consolidation stresses prior to drawdown. The CU triaxial shear test with pore pressure measurements is recommended for impervious and semipervious soils because such tests will provide both effective stress

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shear strength parameters as well as undrained strength as a function of consolidation stress. Sufficient back pressures should be used to effect nearly 100-percent saturation to ensure accurate pore pressure measurements. CD triaxial shear test or direct shear test can be used if the material is highly permeable (> 10-4 cm/s).

For testing of overconsolidated clay shales, considerations must be given to the geology of the damsite, the existence of bedding planes, and past shear deformations. CU triaxial shear tests with pore pressure measurements, CD triaxial shear tests, or direct shear tests may be used for these materials. Where potential failure surfaces follow existing shear planes, residual shear strengths from repeated direct shear tests or torsional ring shear tests are appropriate.

Shear strengths to be used for undrained strength analysis shall be based on the minimum of the combined CD and CU shear strength envelopes of embankment and foundation materials. Effective stresses used to determine the available undrained shear strength shall be the consolidation stresses developed prior to drawdown. The stability of upstream slopes needs to be analyzed for this loading condition.

4.3.4 Anisotropic Shear Strength The undrained strength of clays varies with the orientation of the principal stress at failure and with the orientation of the failure plane. UU tests on vertical, inclined, and horizontal specimens provide the data to determine variation of undrained strength with direction of compression. Typically, the anisotropic shear strength is expressed by the variation in Su with orientation of the failure plane [5].

4.3.5 Residual Shear Strength Residual shear strength of clays may be determined by testing remolded soil samples via repeated direct shear test or ring shear test. However, because the testing procedures for repeated direct shear are still being standardized, professional guidance should be solicited when conducting these tests.

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4.4 Determination of Pore Pressures 4.4.1 Phreatic Surface Method Pore water pressures for steady-state seepage loading conditions can be estimated as hydrostatic pressures below the steady-state phreatic surface without introducing significant error. The steady-state phreatic surface can be estimated following established procedures developed by Casagrande [10], Pavlovsky [11], Cedergren [12], or others. In general, pore water pressures estimated from the phreatic surface method are conservative for zoned embankments; however, this method may underestimate the values for some special loading conditions such as infiltration of precipitation, artesian pressures in foundation, etc. Other methods of evaluating pore pressures should be used in such cases. It is acceptable to have multiple phreatic surfaces in a zoned embankment, one for each of the zones in the embankment and its foundation.

The phreatic surface method can also be used for determining pore pressures under rapid drawdown conditions. The phreatic surface from the steady seepage condition is modified in accordance with the following conservative assumptions: (1) pore pressure dissipation as a result of drainage does not occur during drawdown in impervious materials and (2) the water surface is lowered instantaneously from the top of active conservation water surface to the top of inactive conservation capacity water surface. The phreatic surface may be assumed to follow the upstream surface of the embankment. For embankments with semipervious shell materials, partial dissipation of pore pressures may be assumed in the shell material.

The phreatic surface method is not appropriate for determining pore pressures under end of construction or during construction loading conditions.

4.4.2 Graphical Flow Net Method Under steady-state seepage conditions, flow net methods of analysis are used on transformed sections of the embankment and foundation to determine pore pressures. The values of horizontal to vertical permeability ratios depend on the method of compaction of embankment materials and/or geologic history of the foundation materials.

4.4.3 Numerical Methods Numerical methods for determining pore pressures are excellent tools for rapid drawdown and steady-state seepage loading conditions for complex geometric

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conditions and are the only way to compute pore pressures in three-dimensional space. However, the added degree of sophistication is not necessary in most cases. Finite element, finite difference, and boundary element are some of the well-known numerical methods that are used to determine pore pressures.

For estimating pore water pressure during rapid drawdown, numerical procedures that reflect the influence of dilatancy on the pore water pressure changes are based on stress changes during drawdown. These procedures are described in reference [5].

Permeabilities of all materials in the embankment and foundation must be known in order to get an accurate estimate of pressures and flow.

If necessary, numerical methods should be considered in the final phase of design.

All of the preceding methods are described in detail in Chapter 8, Seepage, of this design standard.

4.4.4 Field Measurement (Instrumentation) Method The development of pore pressures during construction in either the foundation or in the embankment depends upon the soil properties and the amount of consolidation occurring during construction. Pore pressure observations made during construction should be compared with the predicted magnitudes of pore pressures for effective stress analysis to ensure that the actual pore pressures do not rise to a level that will cause instability. Pore pressure instrumentation (piezometers) to be used for observation during construction should be “no flow devices.”

If it is necessary to confirm stability of the dam during construction, the embankment and foundation should be adequately instrumented to monitor movement and pore pressures at critical sections corresponding to the stability analyses.

Pore pressures obtained from piezometers can be used directly for stability analyses of slopes of the embankment or natural slopes under steady-state or rapid drawdown conditions. Alternately, piezometer data can be converted into phreatic surface(s) in the embankment and foundation materials and used as such in stability calculations.

4.4.5 Hilf’s Method A detailed procedure for predicting a total stress versus pore water pressure curve from the results of a few laboratory consolidation tests and a large number of field

DS-13(4)-6 October 2011 4-15 Design Standards No. 13: Embankment Dams

percolation-settlement tests is given by Hilf [3]. The procedure can be adapted to estimate pore water pressures during construction stages.

4.5 Slope Stability Analyses 4.5.1 Method of Analysis Reclamation’s preferred method of stability analysis of earth and rockfill embankments is limit equilibrium based on Spencer’s procedure [13]. The procedure: (1) assumes the resultant side forces acting on each slice are parallel to each other, (2) satisfies complete statics, and (3) is adapted for calculating factor of safety and side-force inclination for circular and noncircular shear surface geometries. To facilitate stability analysis of embankments, Spencer’s procedure is implemented in computer programs. Use of a particular computer program may be adopted with approval from appropriate line supervisors and managers. Additional comments on computer programs are included in appendix C.

For cohesionless materials (c = 0), the infinite slope method can be used to estimate the stability of the slope of an embankment [5].

Continuum mechanics based finite element or finite difference analysis, which satisfies static equilibrium and is capable of including stress changes due to varied elastic properties, heterogeneity of soil masses, and geometric shapes can be used to analyze stability of slopes at the discretion of the designer. Limit equilibrium based analysis, which uses Spencer’s procedure, should also be conducted for comparison of results.

Sample calculations for stability analyses using Spencer’s procedure are included in appendix B. References [5, 14] include useful information on slope stability calculations.

4.5.2 Slip Surface Configuration There are two common slip surface configurations used for stability analysis. The circular arc slip surface is more applicable for analyzing essentially homogeneous or zoned embankments founded on thick deposits of fine-grained materials.

Noncircular slip surfaces, described by linear segments, are generally more applicable for zoned embankments on foundations containing one or several horizontal or nearly horizontal weak layers.

The slip surface resulting from an automatic search should only be regarded as a first approximation of the critical surface. For accurate results, several automatic

4-16 DS-13(4)-6 October 2011 Design Standards No. 13: Embankment Dams

searches should be used, each with a different starting point. Selection of potential slip surface configuration requires experienced judgment in selecting the critical surface considering the stratigraphic conditions and structure of the slope.

4.5.3 Slip Surface Location The location of critical slip surfaces should be related to the location of relatively weaker materials and zones of high pore pressure. The designer should evaluate the overall stability of the sliding mass and determine the location of slip surfaces with minimum factors of safety. In general, the following slip surfaces should be examined:

 Slip surfaces that may pass through either the fill material alone or through the fill and the foundation materials and which do not necessarily involve the crest of the dam.

 Slip surfaces that may pass through either the fill material alone or through the fill and the foundation materials and which include the crest of the dam.

 Slip surfaces which pass through major zones of the fill and the foundation.

 Slip surfaces that involve only the outer portion of the upstream or downstream slope. In this case, the infinite slope analysis may be appropriate for cohesionless materials.

The stability of downstream slopes should be analyzed for steady-state seepage loading condition. The stability of upstream slopes is generally not critical for steady-state seepage loading condition; however, stability of upstream slope for certain configurations of embankment zoning and foundation conditions (e.g., upstream sloping core, a downstream cutoff trench, and weak upstream foundation material that was not removed) should be analyzed.

4.5.4 Progressive Failure Some common conditions that can lead to progressive failure are described in the following along with some possible solutions. There may be other modes of progressive failure [5, 15].

 Because of nonuniform stress distributions on potential failure surfaces, relatively large strains may develop in some areas and peak shear strengths may be reached progressively from area to area so that the total shear resistance will be less than if the peak shear strength was mobilized simultaneously along the entire failure surface. Where the stress-strain

DS-13(4)-6 October 2011 4-17 Design Standards No. 13: Embankment Dams

curve for a soil exhibits a significant drop in shear stress after peak shear stresses are reached (strain softening stress-strain behavior), the possibility of progressive failure is increased, and the use of peak shear strengths in stability analyses would be unconservative. Possible solutions are to: (1) increase the required safety factor and use peak shear strengths or (2) use shear strengths that are less than peak shear strengths and use the typical required factors of safety. In certain soils or shale bedrock materials, it may even be necessary to use fully softened or residual shear strengths.

 Where embankments are constructed on foundations consisting of brittle, sensitive, highly plastic, or heavily overconsolidated clays or clay shales having stress-strain characteristics significantly different from those of the embankment materials, consideration should be given to: (1) increasing the required safety factor over the minimums required in table 4.2.4-1, (2) using shear strengths for the embankment materials at strains comparable to those in the foundation, or (3) using appropriate softened or residual shear strengths of the foundation soils.

 Progressive failure may also start along tension cracks resulting from longitudinal or transverse differential settlements occurring during or subsequent to construction or from shrinkage caused by drying. The maximum depth of cracking, assuming an infinite slope, can be estimated from the equation (2c/γ) tan (45 + φ/2) with the limitation that the maximum depth assumed does not exceed 0.5 times the slope height. Shear resistance along the crack should be ignored, and the possibility that the crack will be filled with water should be considered in all stability analyses for embankments where this condition is possible.

4.5.5 Three-Dimensional Effect Three-dimensional effect should be considered for the stability of upstream and downstream slopes of (i) embankments in narrow canyons, and (ii) the abutment-embankment contact area (groin). For a localized low strength material in the foundation, stress distribution in the longitudinal direction of the embankment and the resulting stability of the slope should be considered. Continuum mechanics based finite element or finite difference analysis of the embankment is recommended for evaluation of the three-dimensional effect. Limit equilibrium based analysis provides an alternative means to account for three-dimensional geometry (of shear surface) effects on factor of safety results.

Reference [16] should be consulted for creating a three-dimensional numerical model. Reference [17] includes useful information on the appropriate boundary conditions for three-dimensional numerical models.

4-18 DS-13(4)-6 October 2011 Design Standards No. 13: Embankment Dams

4.5.6 Verification of Analysis A designer must verify to his/her own satisfaction that his/her stability analysis is correct. The method of verification will depend on the designer’s knowledge and experience. A designer that has a great deal of experience may be able to judge whether the factor of safety from a computer program appears reasonable, while a less experienced engineer may need a more detailed check, such as a hand solution for the critical surfaces.

As a minimum, any engineer doing a stability analysis should research the program that he is using so that he understands the assumptions, theory, methodology, and weaknesses inherent in that program. The engineer should have a good working knowledge of soil mechanics so that he/she can make intelligent selections of shear strength parameters and other properties used as input to the program. The engineer should double check all input data (material strengths, weights, pore pressures, and geometry) and in addition, ensure that input data are independently checked. The engineer should evaluate output from the computer program (computed stresses, forces, pore water pressures, and weights) to ensure that it is reasonable and not blindly accept the validity of factors of safety from the program.

There is a range of methods to check a stability program, such as calculating a solution by the same or different methods on the critical surface by hand or by using a different computer program; using computed stresses, forces, and weights from the output of the program to evaluate equilibrium condition of each slice; or using judgment based on comparing factors of safety and output to past experience. The ultimate goal being that the designer must verify the results of the analysis in a manner that ensures accuracy and with which he/she can submit the analysis to review with confidence.

Guidance on printed outputs from computer programs used for stability analyses is illustrated using sample problems included in appendix B. The printed output should have sufficient details to allow for an independent check and peer review of the stability analyses performed without re-doing the analyses. All input data should be checked for each slice. Pore-water pressure should be carefully assessed, especially when multiple phreatic surfaces are used to define pore-water pressures in different zones, or discrete pore water pressures data from field measurements are used. For a solution to be valid, the following characteristics should be met: (i) the interslice forces should be compressive; (ii) the interslice forces (thrust line) should be in the slide mass, preferably in the middle third; (iii) the interslice force inclination should be within reasonable bounds; and (iv) the normal forces on the base of slices should be compressive. Appropriate user’s manual(s) of the computer program should be consulted for additional guidance to verify that the solution is reasonable.

DS-13(4)-6 October 2011 4-19 Design Standards No. 13: Embankment Dams

4.5.7 Existing Dams Slope stability analysis of an existing dam becomes necessary if the dam is to be raised to increase flood storage or to increase permanent storage of the reservoir. If an existing dam experiences slope failure (e.g., B.F. Sisk Dam in California, figures 4.5.7-1 and 4.5.7-2), slope stability analysis becomes an important part of investigations to understand the cause(s) of the failure and to design remedial measures. Other situations in which slope stability analysis of an existing dam may be necessary include: addition of stability berm to improve performance of the dam during a seismic event (e.g., Pineview Dam in Utah, figures 4.5.7-3 and 4.5.7-4) or if the observed performance of a dam, in some other manner, indicates signs of distress in the dam. In general, for a satisfactorily functioning dam with no performance-based deficiency needing remediation, slope stability analysis is not a required activity [18].

Figure 4.5.7-1. B.F. Sisk Dam (near Los Banos, California). View showing upstream slope failure during reservoir drawdown.

4-20 DS-13(4)-6 October 2011 Design Standards No. 13: Embankment Dams

Figure 4.5.7-2. General cross section of B.F. Sisk Dam in the location of the slide during reservoir drawdown.

Figure 4.5.7-3. Pineview Dam (near Ogden, Utah). View showing stability berm to improve performance of the dam during a seismic event.

DS-13(4)-6 October 2011 4-21 Design Standards No. 13: Embankment Dams

Figure 4.5.7-4. General cross section of Pineview Dam with stability berm.

4.5.8 Excavation Slopes Excavation slope instability may result from: (i) failure to control seepage forces in and at the toe of the slope, (ii) too steep slopes for the shear strength of the materials being excavated, and (iii) insufficient shear strength of subgrade soils. Slope instability may occur suddenly, as the slope is being excavated, or after the slope has been standing for some time. Slope stability analyses are useful in sands, silts, and normally consolidated and overconsolidated clays. Care must be taken to select appropriate shear strength parameters. For excavation slopes in heavily overconsolidated clays, use of fully softened shear strength is considered appropriate. For cohesionless soils, failure surfaces are shallow and have circular configuration; for clays, the slip surfaces are wedge shaped and may be deep seated.

4.5.9 Back Analysis When a slope fails by sliding, it can be used to gain insight into the conditions in the slope at the time of the failure by performing numerical model analyses of the site. The factor of safety at failure is taken to be 1.0. Following a significant slope failure, field and laboratory investigations are generally carried out. These investigations can provide some data for the numerical model (e.g., location and extent of the shear failure, unit weights of soils, ground water and pore water pressure). In numerical models, these data can be used to back-calculate shear strengths for the factor of safety to be 1.0. Depending on the extent of the field and laboratory investigations, and accuracy of the numerical models and methods of analysis, back-calculated shear strengths may provide useful information for

4-22 DS-13(4)-6 October 2011 Design Standards No. 13: Embankment Dams

design of remedial measures. However, much experience and judgment are needed in numerical analyses and in assessing the results of numerical models. Reference [5] should be consulted as it has useful information on back analyses based on case studies of slope failures.

4.5.10 Multi-Stage Stability Analysis for Rapid Drawdown During rapid drawdown, the stabilizing effect of the water on the upstream face of embankment is lost, but the pore-water pressures within the embankment may remain high. As a result, the stability of the upstream face of the dam can be much reduced. The dissipation of pore-water pressure in the embankment is largely influenced by the permeability and the storage characteristic of the embankment materials. Highly permeable materials drain quickly during rapid drawdown, but low permeability materials take a long time to drain.

For high permeability soils, an effective stress stability analysis using the initial and final steady-state phreatic surfaces and the associated reservoir loading is appropriate.

For low permeability soils, a multi-stage stability analysis which uses a combination of effective strength results and total strength (CU) results to estimate a worst-case scenario should be considered.

Two-stage stability computations consist of two complete sets of stability calculations for each trial shear surface: the first set is to calculate effective normal and the shear stresses along the shear surface for the pre-drawdown steady-state phreatic surface and full reservoir loading conditions; the second set is to calculate factor of safety for undrained loading due to sudden drawdown.

Three-stage stability computations consist of three complete sets of stability calculations for each trial shear surface the first two sets are the same as in two-stage stability computations. A third set of computations is performed if the undrained shear strength employed in the second stage computations for some of the slices is greater than the shear strength that would exist if the soil were drained.

Reference [5, 19] should be consulted for proper use of the multi-stage stability analysis for rapid drawdown.

DS-13(4)-6 October 2011 4-23 Design Standards No. 13: Embankment Dams

4.6 References [1] Bureau of Reclamation, Technical Service Center Operating Guidelines, Denver, Colorado, 2005.

[2] Bureau of Reclamation, Design of Small Dams, Denver, Colorado, 2004. (Table 5-1, Average engineering properties of compacted soils from the Western United States, is included in appendix D.)

[3] Hilf, J.W., “Estimating Construction Pore Pressures in Rolled Earth Dams,” Proceedings of the Second International Conference on Soil Mechanics and Foundation Engineering, Rotterdam, pp. 234-240, 1948.

[4] Lee, K.N., and Haley, S.C., “Strength of Compacted Clay at High Pressure,” Journal of Soil Mechanics and Foundation Engineering, pp. 1303-1332, 1968.

[5] Duncan, J.M., and S.G. Wright, Soil Strength and Slope Stability, John Wiley and Sons, New Jersey, 2005.

[6] Lambe, W.T., and R.V. Whitman, Soil Mechanics, John Wiley and Sons, New York, 1969.

[7] Duncan, J.M., P. Byrne, K.S. Wong, and P. Mabry, “Strength, Stress-Strain and Bulk Modulus Parameters for Finite Element Analyses of Stresses and Movements in Soil Masses,” Report No. UCB/GT/80-01, University of California, Berkeley, California, 1980. (Tables 5 and 6 of this report are included in Appendix D.)

[8] Bureau of Reclamation, Earth Manual, Denver, Colorado, 1998.

[9] Clayton, C.R.I., M.C. Matthews, and N.E. Simons, Site Investigation, Blackwell Science, Cambridge, Massachusetts, 1995.

[10] Casagrande, A., Seepage Through Dams, Contributions to Soil Mechanics, 1925-1940, Boston Society of Civil Engineers, Boston, Massachusetts, pp. 295-336, 1940.

[11] Karpoff, K.P., Pavlosky’s Theory of Phreatic Line and Slope Stability, American Society of Civil Engineers, separate No. 386, New York, 1954.

[12] Cedergren, H., Seepage, Drainage, and Flow Nets, John Wiley and Sons, Inc., New York, 1977.

[13] Spencer, E., “Thrust Line Criterion in Embankment Stability Analysis,” Geotechnique, Vol. 23, No. 1, 1973.

4-24 DS-13(4)-6 October 2011 Design Standards No. 13: Embankment Dams

[14] Chugh, A.K., and J.D. Smart, “Suggestions for Slope Stability Calculations,” Computers and Structures, Vol. 14, No. 1-2, pp. 43–50, 1981. (Copy of the paper is included in Appendix D.)

[15] Chugh, A.K., “Variable Factor of Safety in Slope Stability Analysis,” Geotechnique, Vol. 36, No. 1, 1986. (Copy of the paper is included in Appendix D.)

[16] Chugh, A.K., and T.D. Stark, “An Automated Procedure for 3-Dimensional Mesh Generation,” Proceedings of the 3rd International Symposium on FALC and Numerical Modeling in Geomechanics, Sudbury, Ontario, pp. 9-15, 2003. (Copy of the paper is included in Appendix D.)

[17] Chugh, A.K., “On the Boundary Conditions in Slope Stability Analysis,” International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 27, pp. 905-926, 2003. (Copy of the paper is included in Appendix D.)

[18] Peck, R.B., “The Place of Stability Calculations in Evaluating the Safety of Existing Embankment Dams,” Civil Engineering Practice, Journal of the Boston Society of Civil Engineers, Boston, Massachusetts, Fall 1988, pp. 67-80. (Copy of the paper is included in Appendix D.)

[19] Edris, E.V., Jr., and S.G. Wright, User’s Guide: UTEXAS3 Slope Stability Package, Vol. IV, User’s Manual, Instruction Report GL-87-1, Geotechnical Laboratory, Department of the Army, Waterways Experiment Station, U.S. Army Corps of Engineers, Vicksburg, Mississippi, 1992.

[20] Germaine, J.T., and C.C. Ladd, “State-of-the-Art-Paper: Triaxial Testing of Saturated Soils,” Advanced Triaxial Testing of Soil and Rock, ASTM STP 977, Donaghe, R.T., Chaney, R.C., and Silver, M.L., Eds., American Society for Testing and Materials, Philadelphia, Pennsylvania, 1988, pp. 421-459.

[21] Abramson, L.W., T.S. Lee, S. Sharma, and G.M. Boyce, Slope Stability and Stabilization Methods, John Wiley and Sons, New Jersey, 2002.

.[22] Chugh, A.K., “User Information Manual, Slope Stability Analysis Program; SSTAB2 (A modified version of SSTAB1 by Wright, S.G.),” Bureau of Reclamation, Denver, Colorado, February 1992.

[23] Johnson, S. J., “Analysis and Design Relating to Embankments; A State-of-the-Art Review,” U.S. Army Engineer Waterways Experiment Station, Vicksburg, Mississippi, February 1975.

DS-13(4)-6 October 2011 4-25 Design Standards No. 13: Embankment Dams

[24] Wright, S.G., “SSTAB1 -A General Computer Program for Slope Stability Analyses,” Department of Civil Engineering, University of Texas at Austin, Texas, August 1974.

[25] Edris, E.V., Jr., and S.G. Wright, User’s Guide: UTEXAS2 Slope Stability Package, Vol. 1, User’s Manual, Instruction Report GL-87-1, Geotechnical Laboratory, Department of the Army, Waterways Experiment Station, U.S. Army Corps of Engineers, Vicksburg, Mississippi, 1987.

[26] Geo-Slope International, “Slope/W Slope Stability Analysis,” Geo-Slope International, 2007.

[27] Itasca Consulting Group, Inc., FLAC – Fast Lagrangian Analysis of Continua, Minneapolis, Minnesota, 2006.

[28] O. Hungr Geotechnical Research Inc., CLARA-W – Slope Stability Analysis in Two or Three Dimensions for Microcomputers, West Vancouver, British Columbia, 2010.

[29] Itasca Consulting Group Inc., FLAC3D – Fast Lagrangian Analysis of Continua in 3 Dimensions, Minneapolis, Minnesota, 2002.

4-26 DS-13(4)-6 October 2011

Appendix A

Selection of Shear Strength Parameters

Type of Test: Unconsolidated-Undrained (UU) Triaxial Compression test

Loading conditions and Mohr-Coulomb envelope are shown on figure A-I.

Use in Stability Analysis: Unconsolidated-undrained tests approximate the end-of-construction behavior of impervious embankment zones in which rate of consolidation is slow compared to rate of fill placement. Confining pressures used in the tests should encompass the range in total normal stresses which will act along potential failure surfaces through such impervious embankment zones. The UU strength is highly dependent on both compacted dry density and molding water content. Laboratory samples should be compacted to the dry density specified for the impervious embankment zones at water contents wet and dry of optimum within the range of compaction water content to be permitted in the specifications. Because the UU strength is dependent on dry density, it is desirable to obtain samples for testing by trimming cylindrical test samples from a specimen compacted in a Proctor mold. Figure A-2 shows the recommended procedure for a compaction test.

Remarks: Foundation materials which are stiff and fissured may fail under drained conditions (i.e., shear-induced pore pressure equal to zero) even though they are fine grained and subjected to short-term loading such as end-of-construction. Therefore, for these materials, an effective stress analysis assuming zero shear-induced pore pressures is more conservative and should be used instead of an analysis using UU results.

In general, and particularly for foundation materials, CU testing is preferred over UU testing because of the effects of sample disturbance and of high rate of strain in UU testing. These effects tend to offset each other, but not in a predictable way. Reference [20] should be consulted for a more detailed discussion on the relative merits of these two tests.

D8-13(4)-6 October 2011 A-1 Design Standards No. 13: Embankment Dams

t l10. T +°c Saturated samples ~D~ tOe

t l1o. o

OJ= 0c °1=OC+l!J.O.

T Partially saturated samples

NOTE: As degree of saturation approaches 100 percent, failure envelope e1pproaches horizontal

Figure A-1. Unconsolidated-Undrained (UU) triaxial compression test.

Dry density Vd

Laboratory compaction curve

Vdmax Obtain test samples from Proctor mold samples at water contents Min. req'd Vd equal to w dry and wwet

Compaction water content, W

Figure A-2. Compaction test.

A-2 D8-13(4)-6 October 2011 Chapter 4, Appendix A Selection of Shear Strength Parameters

Type of Test: Consolidated-Undrained (CU) Triaxial Shear with pore pressure measurements test

Loading condition and Mohr-Coulomb envelope are shown on figures A-3 and A-4.

Use in Stability Analysis: Consolidated undrained triaxial tests with pore pressure measurements have two primary uses in stability analyses: (1) they furnish effective stress shear strength parameters,

Consolidation pressures used in the CU tests should encompass the range in effective normal stresses which will act along potential failure surfaces prior to drawdown. Because the undrained shear strength derived from the CU tests is highly dependent on both compacted dry density and molding water content, samples should be prepared as discussed for UU tests.

CU triaxial tests can be performed in compression or extension. The loading condition should be based on the expected state of stress at failure. Also, attention should be paid to choosing the consolidation conditions used. Modern computer automation and control allow for anisotropic or Ko consolidation to be employed in triaxial tests. Isotropic consolidation can lead to unconservative strength parameters if the in situ state of stress is anisotropic [20].

Remarks: Because derivation of the effective stress shear strength parameters from the CU tests depends on the accurate measurement of shear-induced pore pressures, samples must be pressure saturated to ensure 1OO-percent saturation. Complete saturation can be verified by applying an increment of all-around pressure with the drainage line to the sample closed, measuring the pore pressure induced in the sample, and computing the B coefficient (observed pore pressure divided by increment of all-around pressure). The B coefficient should be at least 0.95 before conducting the axial loading stage of the test.

D8-13(4)-6 October 2011 A-3 Design Standards No. 13: Embankment Dams

Stage I Stage II (Consolidation) (Undrained compression)

• ~aa • cr'c +a'c

a' 0 a' c c 0 II a' a' • ... c., c => 11- ... =>

tcr'c t a'c t ~cr. At failure:

Total stresses a 3 = a'c ~a a 1= a' C + a

Effective stresses a'3 =a 3 - Uf =a'c -Uf ~a ~a a'1 = a' 3 + a = a' c + a -Uf

where Uf =pore pressure at failure induced by ~aa

Undrained shear strength Su = 0.5 ~aa cos

Figure A-3. Consolidated-Undrained (CU) triaxial compression test with isotropic consolidation.

A-4 D8-13(4)-6 October 2011 Chapter 4, Appendix A Selection of Shear Strength Parameters

For U, =+ (increase) For U, =- (decrease)

T

0', t.o,

T

Undrained strength envelope (Su vs. 0', ) Total stress envelope Effective stress circle I ~ T~tal stress ~-i circle

o .1

Figure A-4. Shear strength envelopes for Consolidated-Undrained (CU) traxial compression test.

08-13(4)-6 October 2011 A-5 Design Standards No. 13: Embankment Dams

Effective stress envelope -- _ -/ (from CD or CU tests) / /" / / / -­ 5® -+------_------/---/-...:::-~-.,,7-"""/'------=,--,--::i::.,~

./ / envelope (constructed SQ) -+------,...... from cu tests) / / /" / / / / a O'N (2)

Effective normal stresses on \ failure surface prior to drawdown

Figure A-5. Shear strength envelopes for Consolidated-Drained (CD) or Consolidated-Undrained (CU) traxial compression test.

A-6 DS-13(4)-6 October 2011 Chapter 4, Appendix A Selection of Shear Strength Parameters

Type of Test: Consolidated-Drained (CD) Triaxial Shear test

Loading condition and Mohr-Coulomb envelope are shown on figure A-6.

Use in Stability Analysis: Consolidated drained triaxial tests are used to obtain the effective stress shear strength parameters,

Because the effective stress shear strength parameters are a function largely of initial dry density for a given soil, samples should be prepared as discussed for UU tests.

D8-13(4)-6 October 2011 A-7 Design Standards No. 13: Embankment Dams

Stage I Stage II (Consolidation) (Axial Compression)

t o'c

o' R o' ~8~

+o'c t ~Oa At failure (all stresses are effective stresses):

T

~oa

O· I: 1 :1 Figure A-G. Consolidated-Drained (CD) triaxial compression test with isotropic consolidation.

A-8 D8-13(4)-6 October 2011 Chapter 4, Appendix A Selection of Shear Strength Parameters

Type of Test: Direct Shear (DS) test

Loading condition and Mohr-Coulomb envelope are shown on figure A-7.

Use in Stability Analysis: The direct shear test can be used to determine drained shear strength parameters for many types of soil, provided the shear box is of adequate size according to the maximum grain size of the soil. Undisturbed or compacted samples can be tested. The test is also appropriate for determining fully softened strength (i.e., normally consolidated peak strength). Consolidation stresses should encompass the range of in situ stresses expected to occur on the anticipated failure surface.

In foundations consisting of highly overconsolidated clays or shales, prior deformations along bedding planes, shear zones, or faults may have occurred in the geologic past such that the maximum available shear strength along these discontinuities is the residual shear strength. Under such conditions, the direct shear test is appropriate for obtaining the residual strength. Samples should be oriented in the direct shear device such that shearing occurs parallel to the discontinuity so that the residual strength is measured. If the residual shear strength is used in analysis, somewhat lower factors of safety than shown in table 4.2.4-] may be acceptable.

D8-13(4)-6 October 2011 A-9 Design Standards No. 13: Embankment Dams

IQC Peak Strength • aN NC Peak Strength (fully softened strength) f

~~t------~~~ L.. III C"""Residual Strength .r:.4> t ell ------=----­ aN•

Shear Displacement

T

Tpulo.-I------'Y'J ~~'~. O"~ =EHoctiVG stress on f

~. =E:Hec1llle angle of shearing resistance a' 0 ... =Residual angle of sMarlng )j r6sist3nce

Figure A-7. Direct shear test.

A-10 DS-13(4)-6 October 2011 Chapter 4, Appendix A Selection of Shear Strength Parameters

Type of Test: Direct Simple Shear (DSS) test

Loading condition and Mohr-Coulomb envelope are shown in figure A-8 [21].

Use in Stability Analysis: The simple shear test imposes constant volume, undrained shearing. The strengths derived from this test can be used in a manner similar to the strengths derived from CD tests.

The simple shear state of stress corresponds to plane strain with constantly rotating principal stresses. The simple shear state of stress occurs in the field directly beneath embankment slopes. It should be noted that the state of stress within the specimen is not as uniform or well defined as for triaxial tests. Although the results are plotted in terms of the shear stress and normal effective stress on the horizontal plane and are useful for engineering purposes, the results cannot be directly compared to effective stress strengths derived from other tests.

I~ O'h sample ~

Initial Conditions With application of shear stress on top and base only

/------'-­

,r--- Circle at failure

Initial circle

0'1f Normal stress, 0'

Figure A-B. Direct simple shear test.

D8-13(4)-6 October 2011 A-11 _len Slo_cIs No. 13: Embonkmon' Oomo

II•• i. SUMi'y Atuly.io. Th< lOI':Oic,o;'. (silt on'u, ''''''''' the c"""__"",,.1 .,00 of ,be 'bear ...... fae<: c""""", durinl: shear .nd """'" Ii><: ."""irocn ooo'inOOll,ly in one dircclion of """i"" f....ny mlIgni''''''' of dioplllCcrllCTl', Th" .llow, l\CIled "n:ngll> i, dclcn",ned by tosting ,lolT)' 'I'"""",n., '1be folly IIOn.ned """nsth i. often tol.on to I>< tl>< "".k >lrcngth of,"" """",.lIy

".

• oc_ Fulyoofl_ (NC> ....k ­ I

FlU"" A-9. TotO~ ring 0110"''''.

Appendix B

Spencer’s Method of Analysis

Figure B-I(a) is the general description of the slope stability problem. For any vertical slice, abdc, the forces acting on it are sho'.vn in figure B-I(b). HL and HR are the hydrostatic forces exerted by the subsurface water on the vertical boundaries of the slice (assumed to be known). Other forces acting on the free­ body diagram of the slice figure B-l(b) are defined at the end of this appendix. The static equilibrium of forces acting on the slice shown in figure B-l(b) can be expressed as:

~ C / b sec tI. - W sin l:t + ~ (W cos Ci - U) tan¢/ F F ZR = ZL + ------.....------­ cos(o - a) [1 - ~ tan(o-a) tan $']

+ P cDs(p-a) [tan(p-a) + ~ tan $' ]

cos(/;-a) [1 - ~ tanio-a) tan $' ]

(1) HL cos a [1 + ~ tan a tan $' ] +

CDS( 0- a) [1 - ~ tan(1i -a) tan $']

~ HR CDSa [1 + tan a tan $' j

cos(li-a) [1 - ~ tanio-a) tan $' ]

08-13(4)-6 October 2011 B-1 Design Standards No. 13: Embankment Dams

The moment equilibrium of the slice shown in figure B-I(b) can be expressed as:

+ !!.- [tanS - tana] [1 + ZLJ 2 ZR

(2) ~ cos~ tan~ + sed) [h 3 - e] ZR

1 + - sed) [HL h4 - HR hs] ZR

B-2 08-13(4)-6 October 2011 Chapter 4, Appendix B Spencer's Method of Analysis

(a) General slope stability problem description.

p

8 HR h2 ~ _H~ l=.::~dC:::::::-::-~s~-i=-:-::_:-:::th=~-=_=- I!4I= u _ a u W b/2 b/2

N

(b) Forces acting on a typical slice.

Figure B-1. Free body diagram of a typical slice.

DS-13(4)-6 October 2011 B-3 Design Standards No. 13: Embankment Dams

The unknowns in equations (l) and (2) are ZL, ZR, hi, hz, 0, and F. The boundary conditions are defined in terms of ZL,I, hl,1 for the first slice and ZR,], hz,1 for the last slice. By using an assumed value for the solution parameters (F, 0) and the known boundary conditions ZL,I and hl,1 for slice], it becomes possible to use equations (l) and (2) in a recursive manner, slice by slice, and evaluate ZR,I and hz,1 for the last slice. The solution of equations (l) and (2) proceeds along the following steps [] 4,22]:

]. Assume some nonzero value for the factor of safety, F, and thrust line inclination, o. (F, 0) are the solution parameters for the problem.

2. Knowing the boundary conditions on the left face of slice], i.e., ZL,I and hl,l, calculate ZR,I and hZ,1 for the right face of slice 1. Because of material continuity between slices along vertical boundaries, ZL,Z = ­ ZR,I and hz,1 = hl,z. Thus, one can calculate ZR,i and hz,i for i =i to n, where n is the number of slices.

3. Compare the ZR,i and hz,i for the last slice against the known boundary conditions ZR,I and hZ,1 and the differences are due to error in values of (F, 0) assumed in step] .

4. Adjust the values of (F, 0) and repeat steps 2 and 3 until the calculated values of ZR,i and hZ,i for i = n agree with the known boundary values ZR,I and hZ,1 to the desired accuracy.

5. The refined value of (F, 0) for which the boundary conditions are satisfied is taken as a solution to the slope stability problem.

There are special nonlinear numerical procedures such as Newton-Raphson algorithm and/or quasi-Newton algorithm for solving transcendental equations which are used in the computer program SSTAB2 [22] for making adjustments to (F, 0) values at the beginning of each new iteration past the first calculation cycle.

The input data for the example problems included herein are formatted for use in the computer program SSTAB2 [22]. A different format for data preparation will be required for use of other computer programs.

8-4 D8-13(4)-6 October 2011 Chapter 4, Appendix B Spencer's Method of Analysis

Meaning of Synlbols b Width of slice c' Cohesion with respect to effective stress e Eccentricity of external force

F Factor of safety

HL Hydrostatic force on the left side of a slice

HR Hydrostatic force on the right side of a slice h1 Location of left interslice force above slip surface h2 Location of right interslice force above slip surface h3 Location of external force above slip surface h4 Location of left hydrostatic force above slip surface hs Location of right hydrostatic force above slip surface hl,1 Location of boundary force at toe of slide mass h2,1 Location of boundary force at head of slide mass h1,i Location of interslice force on the left face of slice i h2,i Location of interslice force on the right face of slice i n Number of slices

P External force acting on slice

U Force exerted by the pore water on the base of a slice

W Weight of slice Z Interslice force

ZL,1 Force boundary condition at toe of slide mass

ZR,1 Force boundary condition at head of slide mass

ZL Calculated interslice force on the left face of a slice

ZR Calculated interslice force on the right face of a slice a Angle of inclination of base of slice f3 Angle of inclination of top of slice o Angle of inclination of interslice force cp' Angle of internal friction

D8-13(4)-6 October 2011 8-5 Chapter 4, Appendix B Spencer's Method of Analysis

Example Problem A

Example Problem A illustrates an analysis of the upstream slope of a zoned earthfill embankment under end-of-construction loading condition. ]n this example, the embankment and the foundation are described by six types of materials as shown on the figure B-2. The input data for this problem are described and a listing of the input and output data is included at the end of the description.

Profile lines were specified as described in the user's information manual of the computer program SSTAB2 [22]. Shear strengths of six material types are described on the figure B-3. Because of undrained strength analysis concept, pore pressures were not considered. Submerged weights of foundation, materials were used which are below the ground-water table. The factor of safety for a single circular shear surface, shown on figure B-3 is computed. The center of the circle is located at the coordinates x = 222.0, y = 378.0, and passes through the upstream toe of the embankment. The circle is subdivided into slices by using maximum arc length of 30 feet. The initial values assumed for the factor of safety and side force inclination are 2.0 and 25°, respectively. A maximum of 40 iterations is permitted in order to satisfy force and moment equilibrium of the last slice within 100 lb and 100 lb-ft, respectively. The input for the computer program is listed in table B-A.l. The output from the program is illustrated in tables B-A.2 and B­ A.3. A free-body diagram of a typical slice is shown on figure B-3. Horizontal E-forces for a typical slice as obtained from table B-A.3 are divided by cosine of the side force inclination (0) to obtain side forces ZL and ZR. Typical force polygons for two slices are shown on figure B-3 where

D8-13(4)-6 October 2011 B-7 Desig n Standards No. 13: Em bankment Dams

" '

Cl ... ' ' . '"

.. -..,

Cl Cl '"

,t;;\ '._ . ...~ ... . ":.'. .', ~'. ­ "."" .. " .

o o '"

Cl Q '"

Cl ~

Clo N '"

Figure B-2. Cross section of an embankment dam used for example problems.

8-8 DS-13(4)-6 October 2011 Chapter 4, Appendix B Spencer's Method of Analysis

Table B-A.1 Input data for end-of-construction loading

1 Example Problem A 9 6 62.4 3 1 100.0 70.0 460.0 190.0 465.0 190.0 3 2 400.0 70.0 465.0 190.0 470.0 190.0 4 3 405.0 70.0 470.0 190.0 475.0 190.0 535.0 70.0 3 2 475.0 190.0 485.0 190.0 540.0 80.0 3 4 485.0 190.0 490.0 190.0 710.0 80.0 2 2 540.0 80.0 710.0 80.0 5 5 0.0 70.0 100.0 70.0 400.0 70.0 405.0 70.0 425.0 50.0 4 5 515.0 50.0 535.0 70.0 730.0 70.0 1000.0 70.0 2 6 0.0 50.0 1000.0 50.0 0.0 35.0 135.0 0 0 0.0 30.0 130.0 0 0 1200.0 5.0 135.0 0 0 0.0 30.0 125.0 0 0 200.0 2.0 128.0 0 0 2000.0 10.0 63.0 0 0 0 0 0 1 End of construction loading u/s 7 0.0 70.0 100.0 70.0 460.0 190.0 480.0 190.0 710.0 80.0 730.0 70.0 1000.0 70.0 1 1 222.0 378.0 99999 0.0 30.0 100.0 70.0 0.0 2.0 25.0 100.0 100.0 40 0.1 1 2

D8-13(4)-6 October 2011 8-9 m I ~ o Table B-A.2 Output for and-of-construction loading-slica information

S L c £ ~l r 0 R " A r I 0 H x y Y.t!G!-il' CO~ESION F;HCJION' BIroR.. COM"SlQ~ rRIC1"rC~ ~Rl. B,OOHDAR.'V fQRC.:=::S • C:-l};RAC'TE:;~~~jIC SHAP[,S I LB$Inl l PSf ) .... ~",jt.2. STRESS I psr I >"NGL~ f'RESSUnE ,() SUnFN?e or LEFT Ltn lDEGSj ( ~Sf l 1D-1:-G];\ PS f" ) I ..... • ••••• ~ ...... } &" 5 I (J£- f"ORC:Z (':..crOJ( Or ct~nE:Ft Cwn:R f()~Ce-LOCA71 ON WCL1N"1"ION SAr£\'Y PIGHT RIGHT (LBS) X

100.00 '10.00 1.000 I lilO. 00 70.M II. 0,00 0.00 o. 00 O. ,00' 1.000 lOO.OO 70.00 I.OGO ;' 11<.18 "'.Il llaH. :20Q .00 ,.00 o .O~ 200.00 O.OC O. 0.00' 1.000 128.3& &0.23 1.000 J. 1 '1,2. ~J: ~ij. 6:5 1060]6. lOO.OO l.OO ,OD 7.00.00 l.OO 0.00 O. 0, aD· 1.000 l:n .·'B 51. 06 l.OOO ~ 166,.4e ~l.SJ 96,16:2. ~ao. 00 '.00 0,00 -lOO.DO 1,00 0.00 o. 1. 000 m 11\. ,9 50.00 1.000 3 ~ 190.!' 1 )',8:. ~1 :Z:OO:-Ui. >000.00 \0.00 0.00 2000.00 10 .O~ 0.00 0, 0.00' '. 000 e­ 205.J' n.l; 1, 000 el) 6 113.01 ~".n UlDa•. 1000. 00 \ 0.00 ,00 2000. 00 10.00 0.00 o. 0.00· 1.000 :::::I 222.00 ,6. H I, oae 7' ") 236. ~8 n .1,0 ;/6£211. 2000. DO 10,00 , 00 2000 ,00 10.00 0.00 o. 1.000 251. H ~~. 08 1-000 3 ~ fl 2bD. 2:4 1~. D~ 115273-1. >000.00 \0.00 0, 00 20M. Do 10.00 6. 00 O. 0.00­ 1.000 :::l 2.~.5\ 50.00 l.OOC ..... 281. 25 ~2. 'S 308006. 100.00 1,00 0.00 2.00 0.00 o. 0,00· 1.000 o 191.98 '1.55 1.000 ClJ to J11. {I 59. &S 31 HSI. '00.00 2.M 0.00 ;'00.00 ~ .00 0.00 o. 0.00' 1.000 326.3J 63.11 1.000 3 I/J tl .BS.lll 66.8"7 189'1.2. ~oo, 00 -l, aD 0.00 200,00 2.00 0.00 o. 0.00· 1.000 ~~~.OO 70.00 t .000 12 3~1, 61 7{; .15 29;~ 15. 0,00 lS.OO 0.00 0.00 3'.00 C.OO o. 0.00· 1.000 ]1[,35 a:l. ". LOGO 11 30 •. <2 89,&5 n"og~. 0.00 l5.00 0.00 0.00 ,]5.00 0.00 O. 0.00' 1.000 3j:"l"7.;,e g"1.01 l.ODD 14 J~8.'''' ~·1.{l1 2.;·HCl. 0,00 J5,00 0.00 0.00 35.00 O. 00 O. 0.00· 1.000 ~OO .00 :HI.":;:O ,000 \ 5 402.50 100.23 P~60. 0.00 lS.OO 0.00 0.00 J5.0o 0.00 o. 0.00 • ,.000 o '05.00 10\."S ,000 «> H HS,OO J O~ .01 1'8129. 0.00 lS.OO 0.00 0.00 l5.00 0.00 O• 0.00 • 1.000 ..... ,:1',00 \\6.20 , 000 w l' ),25.02 11,.22 315. 0.00 lS.OO 0.00 O. DO 0.00 O. 0.00 • 1.000 ~ 12S .0':' 1 \t;.~-; l.OOO 119.7~ J~.OQ I \8 ·'29.n 1ISO. 0.00 lO. 00 0,00 0.00 0.00 o. 0.00· 1.000 C1> :..lj. 5l~ 123. ~s l ,000 \9 1,41.~B IJJ.H ISll6S. 1200 .00 \.00 0.00 1200.00 5.00 0.00 O. 0.00· I.oao .,56.06 I';). ~ii 1.000 o 20 458.0J 115.56 ;n116. 17.00.00 \.00 0.00 1200.00 ~.OO 0.00 O. 0.00 • 1.000 C"l IJ 60.0D t-P.se 1.000 2~ 161. ,,0 1 SO .lO HG36. 1200.00 ~.OO 0.00 120Q .00 ~.oa 0.00 o. 0.00' I. ~OO o 1j6~ 0­ .Od tS2.:;1 l.OOO ro 2): '161. ~O 1~5. '0 2Jl 0"). 1200.00 ;.00 0.00 1700.00 O.OC O. 0.00' 1.000 ... PO.OO 1IS.)~ 1.000 !'.) 2J 0:/.50 101.'-, 19~09. 1200. 00 ~. 00 0.00 1700.0" 0.00 o. 0.00' 1.000 o..... ·115.00 16'-'J I .oon ~ 0 (j) Tabla B-A.2 Output for end-of-construction load ing-slice information I -> W ~ s c [ J • 0 ~ ~ .. I 0 N ~ NO. ~ v ~t£:.tCH."i' COlU.~ION" rnrC710N" nt1UR. CO(i~S1"'~ ,.IC7JON POR'f eOUNnj\~Y fCRC?.s Cl-lArJ,.C7r:.~ t sr Ie: s""~£5 m (tESJMJ ( ;>Sf I 1>.lJ~~~ S7f

0 ';-H.DD i ~~.1 J l.OOO 0 H ~"17. ~{l l 9~ ..... 1,,'1].Ol 11 ,j, 1.000 17~,n !'.) ?b I.e-:: .00 HI,. O.OQ lO. DO 0.00 0.00 30.00 0.00 o. 0.00' 1.ono 0 ~ s ~. 00 )-;'6.56 LOOO -> 11 ~e5.'1" 11) .S~ 1'~9. 0.00 30.00 0.00 o.Ol 30.00 0.00 o. 0.00- 1.000 ...... ;l.8&. ~j 11ft. ~J J.OOO ]9 • 87. gO 180 .l~ 1 B92. a.oo 30.00 0.00 0.00 ]0.00 0, DO O. 0.00- 1.000 -l8~.O3 In.9' 1.000 :I'> 'B9.~:>' 1 B~. ~~ ,J:j~ , 0.00 ,0.00 0.00 o .On JO .00 O. Do O. 0.00- l.OGO ~ ~O. DO 183.2" 1.000 JO r.90.~J lBJ. ~.1 11B. 0.00 10. 00 0.00 0.00 30.00 0.00 O. 0.00' 1.000 <9\. ~:> 111';.'?1 l.OOO

m I ->...... m, Table B-A.3 Output for and-of-construction loading-solution information [J ~ N :::l FACTOR Of S~fETY ~ 1. J19 ( 0 ITERATIONS) CIl SIDE FORCE I NCLINhTrOill ~-1".B3 D!SGRE?:S or SLICE XAIlG TO'i'At NOMHIAL L I N E 0 r 'I' 11 R (j S l' ~f10RlZON:rAL LENGiP. SHE:i\.R r-ClRCE EFFECTIVE II/WCOSA SHEAR ST/NO~~L ST :::::I r •• ".C~ •• ~~~ •• M~.~~".~;.'VW_~ 0.. NO. NORHAL SHEAR E-FORCE (FEET) (LBSi PHI REQD Q) S7RESS STRESS (LBS) (DEGREES) .., 0.. I~SF') (PSn x YT/'fOT) F'RACTlON < I/J Z 0 100.0 70.0 0.0000 O. 0.000 P 1 100.0 C. O. 100.0 70.0 IL4S30 0, 0.00 0.00 1<1.834 :1 .12Q 0.531 ..... 2 114. ;{ 1519. 192. 128.4 68.9 0.4 .. 95 2029a. 29.99 5759.94 9.566 1,345 0.12& (.,) 3 142.9 40)6, 25$. 157.5 66.) 0.365~ 56146. 29.99 77€0.2B 11 .594 1. 176 0,064 4 16G.5 S747 . 304. 175.5 66.1 0.3560 19925. 18.21 5556.06 12.120 ~.104 0.053 m 7844. 2566. 205,3 7')001.02 11. 8S7 7 0,327 3 5 190.4 62.1 0.2580 ·, 118933. 29.99 l.1 8 e- 6 213.7 alDS. 2683. 222.0 6.2.B 0.251"7 227276. 16.66 IL 410 9.03 12 ..31a ;'.107 0.308 Q) J .237.0 9384. 2774. 252.0 66.2 0.2492 297641 . 29.99 SHe8.35 13.H3 1.0S8 0,296 :::::I 99~8. ·~ 7' 8 260.2 2649. 268.5 68.9 0.2'179 325665. 16.66 47482.64 14.175 L026 0.286 9 283.2 9~85, 416. 298.0 7'L 7 0.3008 28252t. 2<;.99 12498.89 14.285 0.939 0.042 3 · ~ 10 Jl2.4 10115. 420. 326.8 9;'.4 0.3865 211766. 29.99 12592.25 14 .517 1. 00, 0.042 :::l ·~ ..... 11 335.4 9988. 'l]7. 34/, .0 110.8 0.5019 lS

~ [J 12 357.7 0845. 001. 37L~ 113.8 0.4()25 116252. 29.99 14:)%7.52 14.834 0.988 0.531 ClJ 13 38~.4 7758. 4123. 397.5 121. 7 0.3425 16')802. 29.99 123644. H 14 .834 !..007 0,5)1 3 :,00.0 ~· 1" He.7 JU6. 3792. 122.6 0.3368 168022 . 2.98 I L264. 94 14 .834 L. 027 0.531 I/J 1.5 402.5 6924, 3680. 405.0 124.5 0.325l. ~ 16392~. 5.96 2L941.48 14.a:H 1,033 0,5)1 16 415.0 0226. 3309 . 425.0 132.8 0.2677 ~ 140153. 24.62 81443.85 14.834 1. 059 0.531 17 425.0 5682, 3020. 425.0 1]2.9 0.2676 ~ 140692. 0.06 170.64 11.834 i.084 0,531 18 429.5 5573. 2444. 433.9 137.3 0.2406 ~ 122653. 11. 36 27757.93 14 .834 1.136 0.438 19 445.0 4'l22. 1239, 456.1 152.8 0.:2049 50619. 29.99 37116.60 0,000 1.304 0.251 20 459.0 3929. 1172. 460.0 156.0 0.1997 39513. 5.62 658t. 50 0.000 1. 361 0.298 0 21 462.5 3453. 1140. 46S ,0 160.4 0.2031 26960. 1.21 829L.80 0.000 1 . .373 0.330 0 22 467.5 2B61. 110l. 470.0 165.1 1),2122 15671. 1.45 al97.87 0.000 1.37~ 0.385 ...... 23 472.5 2249. 1C~0, 475.0 110.2 0.23~1 e~n6 , 7,64 0100.82 0,000 1. 353 0,411 W 24 417.5 1536. 1016. 480.0 liS.4 0.2611 4444. 7.86 7982.52 0.000 '<.281 0.641 ~ , 25 491.5 966. '175, 483,0 17".9 0.1999 371~. 4.8" 4717.69 14 .834 1.107 LOO9 -C1> 26 484.0 929. 407. 485.0 179 .8 0.2053 .. 2132 . 3.26 1)25.64 14.IHo_~' M_ 01 AooIysls , •• • .. , ,-0 •~ '/ ~! , ,• '. •• •, -I , l , • • , • , ,, 0 ,• • • . I , • @ , I •.. \] .. ! _.. ~ ..... ,

I I I

, I lJ' .. ·" , .{ . • ! • I I , I I 0 , ;~ • • • • • • J I'1 •I I 'I·I . ! ! !!! illi II- .. • •• !Ill ! " ". . I ." .'. ! , ! ! • • ...... Fiw.... B-3. ""''"1iI)I .....,.... of , .... _ ....m ...... "" .....-01""""""ucbon Chapter 4, Appendix B Spencer's Method of Analysis

Example Problem B

Example problem B presents a stability analysis of the upstream slope of the zoned earthfill dam as shown on figure B-2 under rapid drawdown loading condition using effective stress envelope concept. The input data were prepared in accordance with the user's manual of the computer program SSTAB2 [22]. Shear strength of six material types is described on figure B-4. With coarse granular soils, dissipation generally occurs so rapidly that no measurable additional pore pressure actually develops. Thus, the pore pressures are hydrostatic in granular soils like shell material in the example problem. In low permeability medium, excess pore pressures are caused during undrained loading. Effective stress analysis by using pore pressures measured by piezometers assume that the stress path to failure has an A f value of 0.5 (] -sin 0.25 to 0.33) the pore pressures by piezometric method renders too high shear strengths. In this case, excess pore pressures are computed by using the following relationship:

where:

A f and B are the Skempton's parameters, ~u = excess hydrostatic pressure, ~(J3 = change in confining stress, and ~(JJ = change in applied load.

Estimation of shear induced pore pressures can be eliminated by using undrained strength of cohesive materials. In the example problem, CL-ML and CH materials in foundation are assumed to have Af values less than 0.33 and, thus, the piezometric method is applicable to obtain pore pressures. The phreatic surface intersects the upstream slope at the inactive conservation pool water surface at elevation] 00.00 feet (after drawdown reservoir level), then follows the upstream slope (which satisfies Skempton's pore pressure coefficient, B = ]) until it reaches the active conservation pool (before drawdown reservoir level) and then follows the steady-state phreatic surface. If upstream shell material is very pervious, pore pressures should not be considered in that material. The factor of safety for a single circular shear surface is computed. The center of the circle is located at the coordinates x = 220.0, y = 375.0, and the surface passes through the upstream toe of the embankment. The circular arc is subdivided into slices by using maximum arc lengths of] 5 feet. The initial values assumed for the factor of safety and side force inclination are 2.0 and 25°, respectively. A maximum of

D8-13(4)-6 October 2011 8-15 Design Standards No. 13: Embankment Dams

40 iterations is permitted in order to satisfy force and moment equilibrium equations of the last slice within 100 lb and 100 ft-lb, respectively. The input listing is shown in table B-B.l and the output from the program is illustrated in tables B-B.2 and B-B.3. A free-body diagram of a typical slice is shown on figure B-4 to describe internal and external forces. Typical force polygons for two slices are shown on the same figure, where

8-16 D8-13(4)-6 October 2011 Chapter 4, Appendix 8 Spencer's Method of Analysis

Table 8-8.1 Input data for rapid drawdown (effective stress)

1 Example Problem B 12 9 10 0 11 62.4 3 1 430.0 180.0 460.0 190.0 465.0 190.0 3 2 100.0 70.0 430.0 180.0 459.5834 180.0 3 3 459.5834 180.0 465.0 190.0 470.0 190.0 3 4 400.0 70.0 459.5834 180.0 464.5834 180.0 4 5 464.5834 180.0 470.0 190.0 475.0 190.0 515.0 110.0 4 6 405.0 70.0 464.5834 180.0 515.0 110.0 535.0 70.0 3 3 475.0 190.0 485.0 190.0 540.0 80.0 J 7 485.0 190.0 490.0 190.0 710.0 80.0 2 4 540.0 80.0 710.0 80.0 5 8 0.0 70.0 100.0 70.0 400.0 70.0 405.0 70.0 425.0 50.0 4 8 515.0 50.0 535.0 70.0 730.0 70.0 1000.0 70.0 2 9 0.0 50.0 1000.0 50.0 0.0 40.0 125.0 0 0 0.0 40.0 135.0 0 5 0.0 35.0 122.0 0 0 0.0 35.0 130.0 0 5 500.0 28.0 126.0 0 0 300.0 28.0 135.0 0 5 0.0 35.0 125.0 0 5 0.0 25.0 128.0 0 5 400.0 29.0 125.0 0 5 0 0 0 12

D8-13(4)-6 October 2011 B-17 Design Standards No. 13: Embankment Dams

Table 8-8.1 Input data for rapid drawdown (effective stress)

0.0 70.0 1872.0 10 100.0 70.0 1872.0 10 190.0 100.0 0.0 10 0.0 100.0 11 190.0 100.0 11 430.0 180.0 11 459.5834 180.0 11 464.5834 180.0 11 515.0 110.0 11 540.0 80.0 11 710.0 80.0 11 1000.0 80.0 11 1 Rapid drawdown loading u/s (effective stress) 7 0.0 70.0 100.0 70.0 460.0 190.0 490.0 190.0 710.0 80.0 730.0 70.0 1000.0 70.0 1 1 220.0 375.0 99999 0.0 15.0 100.0 70.0 0.0 2.0 25.0 100.0 100.0 40 1 0.1 1 2

8-18 DS-13(4)-6 October 2011 Table B-B.2 Output for rapid drawdown (effactive stress) - slica information

S L £ 1 h' f" 0 R '1 ).. 1 1 0 !{ l( 'ti~.lGHT CQ"I£SlON F"r:1CrlON llIFUR. CO!H·;·S 1c~~ fF.JCTIOU PO~(. e.:::H)~n;"..R·1' f'QaCf;.s . CHJ"RACT ER 157 1C {l..,BS/fT) ( FSf I ),NGL£ STRf-SS I ~F i\t-."0t.E: PRfS$URf. AT SUAUoJ:t Of LITl' E:IT IOF.GSI I ?$F' ) ( ~Sj ~SF" \~'.4 L'" 4"'~'" .~~ ... ~ •• srm.:-r ('tee;. F~\CTOP. O~ c.::nTE"R CE.NjE:R ~ORCL-r..cr...ATl~l r~;CJ....DJ)i.1'lCN /\FE:'!Y RIGH7 R1GIlT ll..B~l \j

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,n Chapter 4, Appendix B Spencer's Method of Analysis

Example Problem C

Example problem C presents a stability analysis of the upstream slope of the zoned earthfill dam as shown on figure B-2 under rapid drawdown loading condition using undrained strength envelope concept. The analysis is done in two steps: the first step includes analysis of the upstream slope at predrawdown level under steady-state seepage condition. The second step involves the analysis of the slope at the after-drawdown level and by incorporating undrained shear strengths of the sliding mass at normal stresses corresponding to steady-state condition from step 1.

The input data were prepared in accordance with the user's manual of the computer program SSTAB2 [22] for predrawdown level. Effective shear strength of six materials is described on figure B-5. Phreatic surface method is used to determine pore pressures at the base of slices. The factor of safety for a single circular shear surface is computed. The center of the circle is located at the coordinates x = 220.0, y = 375.0, and the failure surface passes through the upstream toe of the embankment. The circular arc is subdivided into slices by using maximum arc lengths of] 5 feet. The initial values assumed for the factor of safety and side force inclination are 2.0 and 25°, respectively. A maximum of 40 iterations is permitted in order to satisfy force and moment equilibrium equations of the last slice within 100 lb and 100 ft-lb, respectively.

The input listing is shown in table B-C.] and the output is listed in tables B­ C.2 and B-C.3. Typical slice and force polygons are shown on figure B-5. Force polygons illustrated are for predrawdown condition.

The computer program SSTAB2 is not equipped with the formulation of two-step analysis. Hence, using the normal forces (effective) from the first step, and by obtaining corresponding undrained shear strength as shown by the bilinear shear strength envelope on figure B-5, the factor of safety of the failure mass after­ drawdown condition is computed. The computation is illustrated in table B-CA.

The computed factor of safety is ] .38 and, thus, it meets the minimum factor of safety criteria of the upstream slope under rapid drawdown condition.

D8-13(4)-6 October 2011 8-23 Design Standards No. 13: Embankment Dams

Table B-C.1 Input data for rapid drawdown (undrained strength) 1 Example Problem C 12 9 o o o 62.4 3 1 430.0 180.0 460.0 190.0 465.0 190.0 3 2 100.0 70.0 430.0 180.0 459.5834 180.0 3 3 459.5834 180.0 465.0 190.0 470.0 190.0 3 4 400.0 70.0 459.5834 180.0 464.5834 180.0 4 5 464.5834 180.0 470.0 190.0 475.0 190.0 515.0 110.0 4 6 405.0 70.0 464.5834 180.0 515.0 110.0 535.0 70.0 3 3 475.0 190.0 485.0 190.0 540.0 80.0 3 7 485.0 190.0 490.0 190.0 710.0 80.0 2 4 540.0 80.0 710.0 80.0 5 8 0.0 70.0 100.0 70.0 400.0 70.0 405.0 70.0 425.0 50.0 4 8 515.0 50.0 535.0 70.0 730.0 70.0 1000.0 70.0 2 9 0.0 50.0 1000.0 50.0 0.0 40.0 125.0 oo 0.0 40.0 72.6 oo 0.0 35.0 122.0 oo 0.0 35.0 67.6 o o 500.0 28.0 126.0 oo 300.0 28.0 72.6 oo 0.0 35.0 125.0 oo 0.0 25.0 65.6 o o 400.0 29.0 62.6 oo o o o 1

8-24 08-13(4)-6 October 2011 Chapter 4, Appendix B Spencer's Method of Analysis

Table B-C.1 Input data for rapid drawdown (undrained strength)

Rapid drawdown loading u/s (total stress) 7 0.0 70.0 100.0 70.0 460.0 190.0 490.0 190.0 710.0 80.0 730.0 70.0 1000.0 70.0 1 1 220.0 375.0 99999 0.0 15.0 100.0 70.0 0.0 2.0 25.0 100.0 100.0 40 1 0.1 1 2

D8-13(4)-6 October 2011 8-25 m Table B-C.2 Output for rapid drawdown (undrained strength) ­ slice information I I\.) m S L £ I 1I ,. 0 p, ,., "­ ,. o " .'( y rlE:JGHT COH£:SrON f'"':lCT10~: ar,c>.q. COl\E:HO~ ffilcr"Cll P0R~ GCUNO'>.RY f'ORC~S • C:·{[t;RJ;CTER1Sr i c: S~i.~l?t:S (LOSlf'T) , PS:' I ,\NGL£ ZL'A-ESf. I ~Sf ) ~:'<:;LJ: PRESSURE 1\"'( SURF'ACE: 0, L£f1" L~rr 101:CSI I PS, I 10;,eSI psr I"'" ••• -, . $1 D~-fORC};. f'll<:TOR ()[ r..r;HTt~ CoNTER f'O£:Ct-l.OCA il O~I H,'{:l,J.~7·,ira~ SAf'C'1Y "IGW:­ ~lG~T IWlSI X

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S L c L " f 0 ~ !1 AT x 0 .. X y ~i'I(;H" CO[~esICli nIC"T!Oo 2) ru~. <:OH~S [Q·l: fRlCrION f<)?..E. BOUND;::"1·\):· f-ORC£S • GHP,;:-j\Cf£R lS1lC S~1,~\flLS I L8s/rr; I P$F' I .\~:;-LL STRESS I !Jf' I ;H~:LI:~ PRC;SSU~E: ~j :;URrli.CT:: 0, U:rT 1£" {OrC:;i t f'sr I 10 •.(;$ I .S, I··"·······,· .. SID~-roRC~ fAC70R OF C:::N7t;R CC~1f:R i'Oi\CE- LOC;'\T 1O~ 1:\C:11»;\TIO:.: SAnTY o RICHT ~IG(l1" IL£'S; x o )'iJ.~1 96,~, 1.000 0- J.', 396.7f: 9CJ.02 lH11. C.DO .0.00 0.00 0.00 ~O.oo 0.00 Il. 0.00' ) ,ODe r:Y a> .;00.00 10).0" 1.00l> ..... 15 ~02.50 102.1/ 2.~IO"'. C.OO :0.00 ~.OO 0.00 ~O.OO 0.00 D• 0.00' ) .000 !'.) ,0'.00 10,1./, 1 .000 o }o jll.O~ '08.P , 7J 88. C.De lO.ao C• .QO 0.00 ·10.00 0,00 6. 0.00' ~ ~J1.\g )13.19 1.000 ..... 21 .,21.09 l1tl.23 3·1501. c.o to.OO C.UG O. DC >.U.OO 0,00 o. 0.00' 1.000 ';25.00 119.2"-' 1. 000 1e ~26.50 lZO.I9 )21'5. C.DO lO. 00 0.00 0.00 .10.00 0.00 o. 0.00' ).000 C~8,01 1:'1.71 \.000 2~ ~29_ao 122.5 ezso. C.OO )".00 Q.M O. r) li .~~'. 1)0 D. O. DO' ) .000 <30.00 123.36 1. oeo JO OJ, 65 126. ~o 292 7 0. 0.00 )1.00 0.00 D.OO ~5.DO 0.00 o. 0.00' ).000 0".,0 \29.63 l.aOG ).1 ~ -12. &0 1'3.\. J) ~ I 50\ . 300.00 28.00 G.OO Jon.oo ~S.OO 0.00 o. ~. 00' ). 000 "-lE~. 30 1]51.03 1.000 12 ,~53,~6 )·:~.18 JOO.OO IS,QO : •.1(1 300.00 2f:. r:<." 0.00 O. 0.00' ).000 ~~S. ,7. no.':< 1 .000 33 '39.7.0 l~ .93 JOO. QO ,a. QO 0.00 300.00 n. 00 .'.f." G. 0.00' ) . ,100 '59.~e I~I.)·: l.oeO )...: ~59.1~ )Sl.S6 JOO.OO la.oo 0.00 ,00.00 za.oc 0.00 O. 0.00' Leoo -160.00 ) \1.)'l ) .000 15 ~GZ.~9 154.30 300.00 ,a. DO o. O~ 300.00 2S. DO 0.00 o. 1.000 ~6-t. 58 1~6. e2 1. DOD JC l6~.lj )51.05 JOO.OO 28.00 ::-. fJJ 3eO.OO ~~.OC 0.00 0.00' 1.000 ,~(;~. 00 l,jj. 2.9 1.000 J1 ~6'.50 \60.\1 1·1 ,tg~. JOO.OO '".00 C. QG ;00.00 Z .00 O.DO o. 0.00' 1.000 no.oo IG3.U5 1. 000 )8 ~'1.e2 ~~~.2~ 1 OJ.53. 100.00 2,00 0.00 300.00 2S,00 0.00 o. 0.00' ).000 .\7 .G~ Jt.>)'.C~ 1.000 ); ",en H9.<5 Sll G. 0.00 500.0 2a.OO 0.00 G. C.OO· l. tOO 1173.00 16~.09 1 .000 ,'0 ne.H 111.11 13)62. ~oo.oo 28.00 0.00 500. Q0 2S, DO O.DO G. 0.00' ) .OOLl 181. )8 )11.2, 1.000 .; t .. £3.1 i) IA/~. Q9 0.00 ) \.00 0.00 o. 00 '3~; Cdj 0.00 o. O.QO· 485.00 182.iJ 1.000 ~~ ~Bt.)6 181./r. IJM. a. DC ),.06 O. DO 0.00 )~.Cc 0.00 o. 0.00' 1.000 13".32 185.:>6 1.000 ,j 'S8.b~ 181.~a 9J 2. O.M 15.QO O.O~ 0.00 3~.OO ~.oo o. 0.00' ),000 1~Jl~. 00 18g.) 9' 1. 000 •• l~0.21 181.09 21. 0.00 1\.00 0.00 0.00 J5,00 ~.oo O. 0.00' 1.000 ~go . .(1 18~. i'j 1.000 m [j I I\.) ~ ~ Table B-C.3 Output for rapid drawdown (undrained strength) - solution information ~. (Q ::::l s ~ u T 0 ~ ,­ 0 ~ ~. A T 0 1< f~C'TOR vI' SAr::tl' . :/. <6:1 I 0 I T£PP.t I 0:45! CIl ~ID~ fORC~ tNCLINi"jION ·-16. 3 D~GRB~S or SL1U Xr.'.'C TOT!'e NON I NAL. L ; N 2 f ? , P tI S r -1-l0R1ZONTAl. !.£~(j';li S~EAl\ fORCE ~fT~C1'rv. NI>'CO~" SJlCAR s T I~·?":--J\L ST ::::l ~O. ''0 MAL S~Ei\l\ .t.· .. t· •..•.. !.l •• • •••••• ··.(. "-rORCC (t'r.r::'tl (!.g~n Pill P.£QP 0.. ST\tt3S STRESS (UlSf I()ECR~~SI ..,ell I PSf) I PSfl x '1'1'OTI ri'M~1'IN: 0.. I/) 100. ,) 0.0000 o. ~. c) '0.0 0 Z 100.0 o. a. 100.0 70.0 a. -166' u. 0.00 0.00 JG.~;3 I.ao ~. J11 0 101.0 ... '51. '~L 11.;. ~ &9. ~ Q. ·~135i 36<0. 15.00 l ~'3':' .08 16.l ) l.lO' 0.206 1~1 .~ L~b': . 265. 128.4 f:,? .;: ~. S,O I ~2JO. IS .(\0 3070.00 1 6. ~;) 1.4'11 0.2 m 11•. 9 3~6~. 13~ . J7"'1,6 55 ..< O.3:31~ ;;8068. ",. :l< "I1.:l0 1~.~3J 1. \ '13 o .2~G 18~.O 1~2~j 3 -2 5~2~.SO 1~. 53:' \.01< 0./00 I/) 20 3.... b.~ H25. 1821. J5).e 10·1. 0.361~ 15·1004. !".oo 21·: OU. 72 16.5.,3 1.00S 0.311 2.1 360.6 46B2. )73-1. ~6i.4. lD9.S 0.155\ 1·: jl·-102:. 1S. 00 26045.9" 16. S3J l. OJ S a. 3' I :a 3,)~.0 J.:'~ 0 ~. 1634. :<80.6 J!S. ~ O.31~) \: 32 ~ ~3. t. :1290. (.7!. ~S .a 161.0 O. 9~ lSH1. 15.00 10062. ,5 1.960 1. 3~'" 0.2,3 31 "~9.2 21<2. ·15~. G 16\. 6 0.2666 ,-:))~ . "105.20 1.2~3 3~; -I 63'­ 1.1'­ t. 0.'9 -C1> ).; .; ~9. 8 ~l O~ . 621. .:.; 161. 9 0.~65o 33660. .61 393. \ ~ 6.s8e \. 39~ 0.296 3 .,Z.3 196, . '< . ·;6< .6 J s.~ .2629 26S0~ . 6. a1 ·:(14,1. i ~ 1.917 1.~Ob .30· '6 '6<.B 1029. S61. ·:65.0 165. ~ 0./.625 25991. 0.63 352. ~ 1 O. e ~ 5 1. ·;2S 0.309 0 '67.5 lBO·L ~5J . ':;"10.0 170,0 0.2596 1&210. 7.61 '2'6. n 0.000 \ . ~ . 9 0.309 n J~ 47i .Ii 11S2. 5"'1. '; 13. 6 1?L5 0.2109 12~,90 . 5.69 30~9. 1·1 0.000 I. .19) O.nl 8' 39 ,~7~ . 3: 1599. ~9 7. .. ~" 0 ]7.1. 9 o.277.J 10131. 2.1 S 1~83.43 .000 1.d~7 O.~?) C1" :'0 pa.2 11~6. 49~. ·31.~ 190.7 0.2676 n'H'. 10.1:; 5126.66 t6.5l3 l . .;61 C.425 ....ro 41 483.2 129. 22~. ';S5.0 164.3 0.2711 1634. 6.08 ) 171.';0 l6.S:'2 1.6).2 0.3JO 466.2 ~~O. ;3&. ·:57.3 IS6. B 0.3036 ~2~ . J. ~8 .2 16.~3:i 1. i516 U. :no ~ '2 '5·n ~3 I;B.B. -: 19~. SO). 4 90 .~) _.~ 1. ~B ~17. ~j3: 0 lBB.9 0.0000 ;. 65 - C. l.116 0.310 ..... ,'. ~90. 2B. 'J. .; 90. ~ 189.8 O. 0000 -60. 0.73 • :-!o -!t..~ 1.743 0.3J~ .... 1\\I£.RAG~ FACTOR-or-SAr:;n - 2.262 SUI·j­ O.·)~2.£40& C""""'., "1>".",", B SP*'<"~' _IlO

~ ..... II· . ,>--,,_-';,f-L'.--l.'

Flgw. 8-5. S".lIIty ....."'010 01..,. .."._ olcpo ...... , ..pld d,__n -... 1""....I....d ,,_,n on_po conupll.

,-. Design Standards No. 13: Embankment Dams

Table B-C.4 Computation of factor of safety for after-drawdown condition

Slice ND tan

1 0.02 0.839 0.017 0.0 0.0 2 12.75 0.4663 4.78 -1.65 3 23.13 0.4663 14.32 -4.32 4 32.55 0.4663 23.55 -6.06 5 41.0 0.4663 32.37 -6.89 6 48.5 0.4663 40.7 -6.84 7 18.87 0.4663 82.44 16.33 -2.25 8 58.26 0.5543 6.0 42.01 -4.64 9 63.49 0.5543 6.0 58.66 -4.03 10 56.52 0.5543 5.02 68.79 -1.58 11 71.0 0.5543 6.0 0.2 0.0 12 74.0 0.5543 179.24 6.0 78.19 1.79 13 63.7 0.4663 29.7 8.78 86.72 5.95 14 77.94 0.364 78.39 8.66 15 79.05 0.364 99.94 15.2 16 79.47 0.364 105.65 20.83 17 79.2 0.364 110.3 26.67 18 78.3 0.364 113.85 32.55 19 27.45 0.364 143.40 116.31 38.32 20 72.23 0.839 41.62 14.92 21 69.0 0.839 117.61 45.55 22 65.22 0.839 116.93 50.17 23 60.99 0.839 115.17 54.13 24 29.54 0.839 112.39 57.3 25 22.2 0.839 41.79 22.43 26 51.71 0.839 14.49 7.92 27 31.23 0.839 42.53 23.69 28 11.56 0.839 347.1 91.51 53.23 29 7.78 0.700 38.66 23.44 30 27.72 0.700 24.85 28.09 17.41 31 40.56 0.5317 21.94 13.83 32 36.38 0.5317 64.01 41.54 33 2.54 0.5317 67.85 46.13 34 1.37 0.5317 15.75 55.98 39.9 35 14.37 0.5317 5.69 4.16 36 1.24 0.5317 22.13 16.38 37 15.12 0.5317 18.7 14.12 38 11.27 0.5317 15.08 11.62 39 3.91 0.5317 6.25 13.36 10.53 40 13.72 0.5317 74.69 4.56 3.66 41 5.11 0.7 1.79 1.45 42 2.02 0.7 0.91 0.75 43 1.04 0.7 5.72 0.02 0.02

2:887.2 2:59.8 2:685.99

* Additional resisting moments due to the boundary hydrostatic forces on the lower slices are neglected.

IN tan

8-30 D8-13(4)-6 October 2011 Chapter 4, Appendix B Spencer's Method of Analysis

Example Problem D

Example problem 0 illustrates an analysis of the downstream slope of an embankment dam as shown on figure B-2 under steady-state seepage condition. Selected shear strength parameters and type of triaxial tests to obtain the parameters are shown on figure B-6. The input for the computer program is listed in table B-0.1 and the output is shown in tables B-0.2 and B-0.3. A free-body diagram of a typical slice and force polygons of two slices are illustrated on figure B-6. The maximum arc length of slices is limited to ]5 feet. The factor of safety of the failure mass as shown in table B-D.3 is ] .443. By comparing with the required factor of safety of 1.5 (table 4.2.4-]), the downstream slope did not meet the minimum factor of safety criteria.

D8-13(4)-6 October 2011 8-31 Design Standards No. 13: Embankment Dams

Table B-D.1 Input data for steady seepage condition 1 Example Problem D 12 9 10 0 11 62.4 3 1 430.0 180.0 460.0 190.0 465.0 190.0 3 2 100.0 70.0 430.0 180.0 459.5834 180.0 3 3 459.5834 180.0 465.0 190.0 470.0 190.0 3 4 400.0 70.0 459.5834 180.0 464.5834 180.0 4 5 464.5834 180.0 470.0 190.0 475.0 190.0 515.0 110.0 4 6 405.0 70.0 464.5834 180.0 515.0 110.0 535.0 70.0 3 3 475.0 190.0 485.0 190.0 540.0 80.0 3 7 485.0 190.0 490.0 190.0 710.0 80.0 3 4 540.0 80.0 710.0 80.0 1000.0 80.0 5 8 0.0 70.0 100.0 70.0 400.0 70.0 405.0 70.0 425.0 50.0 4 8 515.0 50.0 535.0 70.0 730.0 70.0 1000.0 70.0 2 9 0.0 50.0 1000.0 50.0 0.0 40.0 125.0 0 0 0.0 40.0 135.0 0 5 0.0 35.0 122.0 0 0 0.0 35.0 130.0 0 5 500.0 28.0 126.0 0 0 300.0 28.0 135.0 0 5 0.0 35.0 125.0 0 5 0.0 25.0 128.0 0 5 400.0 29.0 125.0 0 5 0 0 0

8-32 08-13(4)-6 October 2011 Chapter 4, Appendix B Spencer's Method of Analysis

Table B-D.1 Input data for steady seepage condition

11 0.0 70.0 6864.0 10 100.0 70.0 6864.0 10 430.0 180.0 0.0 10 0.0 180.0 11 430.0 180.0 11 459.5834 180.0 11 464.5834 180.0 11 515.0 110.0 11 540.0 80.0 11 710.0 80.0 11 1000.0 80.0 11 1 Steady seepage analysis 6 0.0 70.0 100.0 70.0 460.0 190.0 490.0 190.0 710.0 80.0 1000.0 80.0 1 1 650.0 375.0 99999 0.0 15.0 730.0 70.0 0.0 2.0 25.0 100.0 100.0 40 1 0.1 1 2

DS-13(4)-6 October 2011 B-33 m Table 8-D.2 Output for steady seepage condition - slice information I W ~ S L C E s r 0 .~ A ~ l 0 'i NO. x "f Xs'!(l.H1 COHE:S t 01-: ,nerlOx Sift) . CO\l~S ro~ f'R rc-r 10" 'PaRr. 1l0lJMDARY ?-ORCE:S • CH:.R.R;.C!ERIST]C: S~APr.s 'LaS/f"T1 ( P'£. \ '''"L£ 'fRE"$r. r PST ) .\~GL£ P;;:ESSUJ:l;: J-S ~UR~At:E:. or UrI 18fT 10£GSI I PSf I I O~GSI ?Sf ) .•• ~ ...•...... ~. So 1O£:- FORCE FAC'TOR Of'" cE'l"TE:R C~:~1'r:R fORCF.-l.OC"..AT 1O~j rXl."'::r.IIU\T!G~ ,s.:,,'F"1:T), RIGHT RIQf·rr ( L5S)

,108.1.1 1 n. 11 1.000 1 ").01 161.01 ~1)0. D.OO 11l.00 0,00 '0.00 3672. ,t P. ~~. 1.000 ..I I'd • 02 Ilil . .',-; \.000 :2 421.;'1 l5/.7"1 1828S. o. 00 10.00 O. 0 O,Do ,0.00 ]]31. .... SIS. 0.0 ~D.I), 1. COO w I' 52 J. ~ , e) . \ 8 I.O

535.00 81."0 I. 000 19 ~37.~0 80 ••5 ~35'6. 0.00 Jr::.OO 0.08 O.OD ;,·;.00 0.00 O. 0.00' J.OOO 5,'0. 00 19. H I. 000 o ~o 5~7.0g 77.o, l~~(~S. 0.00 l5.00 0.00 0.00 ~)..oo O. 0.00' 1,000 C"l 5'54. HI 7 -I. 60 I. 000 21 561.3j "12. <;9 14T761. 0.00 ;'>.00 O,M 0.00 3'.DO o. 0.00' 1.000 o 5iHL 57 70.18 1,000 0­ ro 2:1 169,~~ 10.19 1"1~. 0.00 !S. an O,O~ D.OO ,,>,0 O. 0.00' J.OOO ... S7D,OD 'jO.IJO I. ODD !'.) 2] 5j1.30 63.27 1~)2~~. 0.00 ".00 0.00 2S.00 7 31. ~2 o. O. 00' 1.000 o.... S8·~. &6 6iS. SI. 1.000 ~ Table 8-D.2 Output for steady seepage condition - slice information

S L C E N f 0 R. ,; }. 1" 1 0 l:' X "! rl';~lGHT CO~IES ION f"l'lCTION B) 'v:>.. COHES[O~ ,RlCTlON POFE BOUJ~ ~ARV CO?C:::.S OINU\CH;RIsnc i ~,~15J;i"1 1 PS'­ I ~N(;L~ S":rlf.S$ ~Sf } ANGL£ P'~~S:S(lRC ~ Ai StlRi"'.':"Cr;: 0, tH"r LeFT IO~GSI ( ~sr 10£'3SI IPS. ' ... '"'0.0 ..•..••••. , SIOE-fORC!: fACTOR 0)" CE:N?rR CEHn.-:. rORc~~L02A:IO~ r~CL1NA"j'10)..J Sj\FI.,"'{ i\lGHT ~rGK7 ILB51 x o 0­ o~4.60 66.,4 1. 000 r:Y 2,. " 1. g-; 6S,16 l:no:~o. ~5,OD O.M 0.00 2~. 00 92~.9~ o. 0.00' 1.000 .....a> 599.14 63.18 1.000 !'.) 2:: 601::.7: 62.75 1289)·: . 0.00 ::'>.00 0.00 (LOO 2).00 10'6, J 1 o. 1.000 o ) \ ~ .;>'1] 61.n 1. OOu ~ ..... 2~ 021.6" 61.0, !: 9DO'I. O.DO 25.00 0.00 o.on 25.00 1132.5S O. 1.000 629.j-1 &0.37 1. 000 2', 63~.61 60.06 .07~·ll . 0.00 !5.00 0.00 0.00 ?~. 00 12401.47 o. 0.00' 1.600 6~ I] • 12 5~.7/' 1. 000 '-8 i),~7.06 59.7] 38516. 0.00 ,5.00 0.00 0.00 /.~. 00 126;:;. OJ 0, 0.00 ' 1.0CO (SO,OO 59.6B 1. 000 2~ ~S7.50 ~~. eo 0.00 ,~.oo 0.00 0.00 2~.00 1256.61 o. O.GO' ].oeJ E6~.99 60.0/, i..CiOO 10 0"12.;; Go.se :5.00 0.00 0.00 2,.00 1212..12 ~.GO' 1.0CO c7!L 9~ 61, \1 '.000 31 687.,10 6~.OO 0,00 25,00 0.00 O.CO 2~. 00 )In. ;;:; 0, O.DO· 1.OCJ G94.85 62. B~ 1. 000 32 702.2. a~. 13 3'"15CJ~ , 0.00 lS.OO 0.00 0.00 25.00 0, 0.00' 1.000 709. 64 6:',38 1.000 33 '109,82 65.·j 1 0,00 2S.0a 0,00 o,co 2~"CO u, 0,00' 1.000 710.00 65. -].l \. 000 3~ 711.33 6j. os 0.00 0.00 0,00 2~.oa c, 724.65 6S.6:: 1. 000 J~ "i27. 32 ;;~.32 1<09. 0.00 0.00 J.OO "s.oo 0, 0.00' \.OO~ 1l0.0C 10.00 ,.008 130.00 '0.00 0.00 J~.OO O.OC (). 00 o. 0.00' I.OCO 730,00 70.00 ), 000 37 737.2) 72.06 O,O~ l5,00 o.oc 0.00 35.00 o. 0.00' 1.000 1H.jl 74.LO 1. 000 3~ 751. S', Ib.5i 0.00 15.00 0.00 0.00 "~.Oo 2.t-'.. J: o. 0.00' 1.000 156.61. 78.98 1. 000 )9 759.98 79.0 0.00 J5.00 0.00 0.00 15.00 :;1.81 0.00' 1.000 761.3S 80.0a 1. 000

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Figure 8-6. Stability analysis of the downstream slope for steady-state seepage condition.

08-13(4)-6 October 2011 B-37

Appendix C

Computer Programs

Chapter 4, Appendix C Computer Programs

Two-dimensional analysis using limit equilibrium method

In limit equilibrium method, only static equilibrium of slide mass is considered. In a given soil deposit, the slide mass geometry is defined by a shear surface. The material properties required for analysis are density or unit weight of soils and their shear strengths. Boundary conditions are generally included implicitly in the formulation of the equilibrium equations which are solved using numerical methods [17].

There are several procedures devised to solve a slope stability problem using the limit equilibrium method and some of these procedures have been implemented in computer programs to facilitate computations. The difference between various procedures relate to the assumptions that are made in order to achieve statical determinancy – some of these procedures satisfy only force equilibrium equations or only moment equilibrium equation, and others satisfy both force and moment equilibrium equations. Additional limitations of the various procedures apply to the geometry of the slip surface, i.e. circular, non-circular, log-spiral, etc.

The Bureau of Reclamation prefers use of slope stability analysis procedures which satisfy complete statics and accommodate general slip surface geometry - Spencer’s procedure is one of them. It is similar to but simpler than Morgenstern and Price procedure which also satisfies complete statics but requires additional information (variations in interslice force inclination) not commonly available as a priori – Spencer’s procedure assumes the inclination of interslice forces to be the same (i.e., parallel). The Bureau of Reclamation prefers use of Spencer’s procedure for slope stability analysis.

There are several computer programs which implement Spencer’s procedure – SSTAB2 [22] is one of them, developed at the Bureau of Reclamation by modifying SSTAB1 [24]; others include UTEXAS2 [25], UTEXAS3 [19), SLOPE/W [26], amongst others. All of these computer programs have been used in the Bureau of Reclamation with approval from line supervisors and managers.

Two-dimensional analysis using continuum mechanics method

In continuum mechanics method, force equilibrium and strain compatibility equations of engineering mechanics are considered and the differential equations relating forces and displacements in a solid are solved using numerical procedures. Unlike limit equilibrium based solution procedures for slope stability analysis, continuum mechanics based solution procedures require their adaptation for solving slope stability problems. Also, continuum mechanics based solution procedures require data for deformation properties (for elastic and plastic deformations) in addition to the data for shear strength parameters and material density or unit weight. Boundary conditions are explicitly stated as a part of the input data. In a continuum model, shear surface is not defined as a

DS-13(4)-6 October 2011 C-1 Design Standards No. 13: Embankment Dams

priori – instead, it is determined as a part of the solution. The problem is solved successively using reduced values of shear strength parameters, c' and φ', (by a scalar factor) until the model becomes unstable (i.e. the solution fails to converge). The scalar factor for reduction in shear strength parameters for which the numerical model becomes unstable is taken as the factor of safety and the interface between the elements undergoing displacements and elements with virtually no-displacements is taken as the shear surface.

There are several procedures devised to solve the force-displacement equations of engineering mechanics. In Reclamation, finite element and finite difference based solution procedures are commonly used.

For slope stability analysis, when needed, we use finite difference based computer program FLAC [27] for continuum mechanics based slope stability analysis. The same problem is also analyzed using the limit-equilibrium based Spencer’s solution procedure and computed factor of safety and associated shear surface results are compared.

Engineering judgments are used in deciding relevance of the results to the project and issues of interest/concern.

Three-dimensional analysis using limit equilibrium and continuum mechanics methods

Stability of natural or manmade slopes is affected by its lateral extent in the third dimension. Three-dimensional solution procedures are extensions of their two- dimensional counterparts, and use the same solution strategies. However, the addition of the third dimension increases the complexity of the problem solving methodologies.

For the limit-equilibrium based solution, we use computer program CLARA-W [28] for 3-dimensional slope stability analysis. CLARA-W has 3D extension of Spencer’s procedure. The same problem is also analyzed in 2-dimensions using the limit-equilibrium based Spencer’s solution procedure and results compared.

For the continuum mechanics based solution, we use finite difference based computer program FLAC3D [29] for 3-dimensional slope stability analysis. FLAC3D is a 3D extension of FLAC program.

Engineering judgments are used in deciding relevance of the results to the project and issues of interest/concern.

Note: There are other commercially available computer programs based on limit equilibrium and continuum mechanics methods which can be used for static stability analyses. The programs named herein have been used in Reclamation

C-2 DS-13(4)-6 October 2011 Chapter 4, Appendix C Computer Programs

and their inclusion is for convenience; however, no endorsement or recommendation of their use is implied.

DS-13(4)-6 October 2011 C-3

Appendix D

Guidance Papers for Static Stability Analyses

Part 1 The Place of Stability Calculations in Evaluating the Safety of Existing Embankment Dams

Part 2 Suggestions for Slope Stability Calculations

Part 3 Variable Factor of Safety in Slope Stability Analysis

Part 4 On the Boundary Conditions in Slope Stability Analysis

Part 5 An Automated Procedure for 3-Dimensional Mesh Generation

Part 6 Average Engineering Properties of Compacted Soils from the Western United States

Part 7 Strength, Stress-Strain and Bulk Modulus Parameters for Finite Element Analyses of Stresses and Movements in Soil Masses

Each of these documents is self-explanatory, and no additional comments are considered necessary.

Appendix D

Part 1 The Place of Stability Calculations in Evaluating the Safety of Existing Embankment Dams by Ralph B. Peck

This article was published in the Civil Engineering Practice, Journal of the Boston Society of Civil Engineers Section/ASCE, Volume 3, Number 2, pp. 67-80, 1988. It is reproduced here with permission of the publisher.

CWIL ENGINEERING

)OUR"AI. OfTHI! lIO$To.. SOCJ1!'IY PRACTICE' 01' CIVIl. ESCINHIIS socno""aS(]; VOlUME ~ i'WM••• , --

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E''''ing Embankment Dam S.f.ty Site Analysis

The Place f ta ilit Calculati Evaluati fe f Existing t Dams

"""",,,,,,Uili briu m of sta bility analyses. Every soils student learns about Thorough of them. Sophisticated computer programs exist conditions and constmction for carrying them out. They have an important records should have precedence place in embankment dam engineering, primarily in design, but they can be misleading over analyses in evaluating dam safety if too much depend­ determining the safety of ence is placed on the numerical values of fac­ embankment dams. tors of safety derived from them. Purposes of Stability RALPH B. PECK Analysis in Design HE evaluating the the implications of stability analyses of an existing embankment dam is to with respect to the safety of existing dams can ensure that the catastrophic loss of the considered, the ways in which such analyses reservoir will not occur. Many in which are useful in design should be reviewed. The the safety of existing dams is evaluated discussion herein is [imJted to limit-equi­ the safety to the results ofstabilityanalyses. Yet librium analyses. Other techniques are re­ seismic aside, stability ;:\n,~lv"p", quired for investigating and are often and may even be m1:sje,~d- liquefaction. In practice, slopes for embankment dams To be sure, one the great achievements of are not chosen on the basis of stability calcula­ soil has been the development of tions; they are The

OVJL ENGINEERING PRACTICE FAlL 1988 67 selection is based on the designer's judgment appropriate safety is specified a1!:ainst that takes mto account foundation conditions, an upstream failure resulting from rapid economics, of materials, logis­ do't'm. tics and a whole series non­ All uses of stability are tec.hnical considerations. tentatively legitimate parts They selected both exterior and interior slopes, the knowledge offoundation conditions a.nd of the ae:'lgJt1er carries out to ensure pertinent of the materials in- that conventional are adue'veci, VUIIVrlL!. Th~ from the This tion is a field the are usually in legitimate mUel,J~LI. successive refinement. Furthermore, inasmuch as ment dam di,ffers in some res,pe,ct Stability Analyses of Eristing Dams it is tohavca The application of in the the proposed darn with others whose design phasemakes useof or genera1­ mance is known. Factors ized soil ties, known stalbilil:y anaJY!*'s for the pre_po,sed geometries and ofsliding. In for dams constitute such a contrast, if the of an em­ Moreover, stability analyses are valuable in bankment dam is 10 be determined correctly, comparing efficiencies various arran;ge­ facts rather are needed. Ob­ ments the of the The aniMvses facts is no simple can provide insights regarding the relative both the external and internal merits and economies of placing superior or In­ geometry of the dam must be ascertained, ,..,..."t""n;:lllC: of costs in different which may be difficult if reliable l\S-buiH draw- parts of the embankment. the !lre not available. the Stability assist in avoi,ding properties the need to be deter~ shear failmes during construction. Many dams mined, either from good records as they were have experienced upstream or downstream actually or by investigation. The failures construction because of sw:tac:eof can be weak seams in the Such failures me'a5trr€lme.n ts il or it can be can often be and avoided by estllbli.shE~d n~a.li!;tic1ally from knowJedge the investigation and appraisaJ of the foundation subsurface conditions. The shear strengths conditions and te equilibrium have to be in terms of effec!ive~ stress along the of sliding In addition, faillll'es during constnlction m«lU(:H.nl~ that portion of the surface within the have been known to occur as a of foundation. The on the actual induced in relatively imper­ c,"-t:"",,, ofsliding on potentiaI of vious zones by the of fill. These if the actual one is not known) must be tallurl'''> can predicted by suitable determined as well. librium analyses combined with investigations All these data are at best to pore-pn::SSlue COl~thClents in the relevant evaluate and in many instances rnateria.ls and studies of the rates may to determine. Obtaining These faUuresalso can be avoided by im­ them may necessitate exploratory work within plementing a that enswes and beneath the dam., itself often an that sufficient dissipation detrimentaJ to the safety of the su res occurs aPloropnate llIUI~'~, it is fair to say that if good consl:rUc­ nt>l"if'\rl<: during filling. lion records are unavailable, it may be imprac* Finally, stabilityanalyses are llsed to provide Heal or virtually to adequate assmance that a dam will not under oplUat· {or the factor reliably. ing conditions. A enough factor of safety HO·Wt~e.r, this limitation is only one is to guard downstream slid­ considerations to the conclusion that ml:5je(~a- ing under a full reservoir a.ndJ in addition, an stability may irrelevant or

b8 CIVIL E,' or Baldwin Hills during construction. As the embankment rises, (Californ ia) catastrophes, Non-catastrophic the factor of safety against a slope failure, and failures, which may be expensive, annoying or pa rtieu lady against fou nda tiOll fa il u re, embarrassing, shouJd also be avoided. Owners decreases. Such a slide would not be of dams may be well advised to evaluate the catastrophic unless the pool had been allowed probability of such failures and to take steps to to rise against the embankment as it was being prevent them. Yet, since their consequences faU placed. Under these circumstances, whatever far short of the calamities associated with the pool had been accumulated might escape and flood folJowing a catastrophic failure, they f;tll cause flooding. outside the domains of public safety and the The second critical period is Ule Hrst filling regulatory powers of government, and they do of the reservoir, [f the dam survives the initial not fall within the scope of this study. filling and if there is no blowup at the toe, the Catastrophic failures have one of iour dam can be considered safe On effect, proof­ causes: overtopping, piping by backward test(:'d) against faiJure by piping due to heave. erosion, liquefaction, or downstream sliding at TI,e third critical period is achievement of high reservoir (possibly associated with toe mi'!ximum pore pressure under a full reservoir. failure due to piping by heave). The first three If the dam has not faiJed when this condition of these types of faiJwes - overtopping, back­ ha s been rear hed, its sa fety aga inst ward erosion and liquefaction - cannot be downstream slope fa i1uTe has been predicted by stability analyses. Henc€J the demonstrated. Under many circumstances, in­ legitimate application of stability analyses to duding the presence of relatively thin cores or catastrophic failure is restrided to downstream ample well-drained downstream shells, pore­ sliding, with or without loss of toe support, pressure maxima follow so rapidly after the when there is enough water in a reservoir to do first filling that the survival of the first 6lling catastrophic damage if released. can be considered to be iI demonstration of the Non-catastrophic failures can occur by ultimate safety of the dam under full-reservoir downstream or upstream sliding, including conditions. However, if the impervious section sliding originating in the foundation, when of the dam is thick and impermeable enough to there is no pool orwhen the pool is so smaU that create a time lag betwe-en the rise of the reser­ its release is inconsequential. In addition, voir and the rise of piezometric levels in the faiJures can occur by rapid drawdown, but core or supporting downstream zones, pore­ such failures are not in themseJves catastrophic pressure equilibrium may not occur for severaJ even if the reservoir contains a high pool. years after the reservoir is fixst ruled and the Rapid drawdown has led to significant critical period may be delayed. So-calJed

CMl. ENGINEERING P!

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Schist

FlGURE 2. Longitudinal section through the east end of the Walter Bou..ldin power house and the junction with east wing dam. est...i.blished. They discounted the likelihood of into the schist to reach suitab~e foundation sup­ piping at a lower elevation where il probably port for the power house fa ft. occurred. n'ere was one eyewitness to the events lead­ The essential features of the project lncJuded Lng up to the faiJure: Mr. Sanford, the night an earth claro about 165 feel high across the guard. He was interrogated many times i.n the de~pestpartala valley, flanked on the westand course of the ensuing investigation and east by wing dams founded on Pleistocen.e ter­ recounted a remarkably conslstent series of race deposits al a higher level. A power house recoUectiC5ns, As a non-teclUlicaI person, he had was embedded in lhe downstream slope of the no hypothese5 about the causes of failure and highest part of the dam; th~ roof of lhE' power appare.ntly had no feaSOI'\ to report o~her than house was at lbe same ele.vation as a berm on what he experienced. the downstT'eam slope. The power house's The chronology ofSanforo's activities on the foundation eXlended into Precambrian schist cloudy, moonless night of the failure C1Ul be bed.rock overlain by Crel:3ceoussedim-en ls con­ traced with refere.nce to Figure 1, a simplified sisting largely at slightly cohes.ive sands -and sketch of the power house and dam as seen silts with layers of stiff clay. Beneath the wing from downsb:eam. He went on duty about9:45 dams these sediments were overlain in tum by p,m" made his first routine inspection startirlg the terrace deposits. The relationships are at his office in ~he reception room (A) at the shmvT\ diagrammahcally in Figure 1. Figure 2 northwest corner of the roof of the power depicts a longitudi.nal section of the a.rea house, and rerurned about midnight without through the east end of the power house and its hi! ving observed anything unusual. After read­ junction v.ith I'he east wmg dam; it shows the ing the 5unday paper for some time in !..he e'Xcavahon made through the CrelE\CeoUs soDs reception room, he glanced. out a west window

CIVIL E\:CtNFERtNG PRAcnCE FALL 1938 71 11!.r>Obl. ,"_,h. d,. In luw", '" '" ""'dod '" ",.Il< ""'••'" • '""'" oo-""boo"""" o,mooly ".,,,~ '"It.. _,,,,, ""'"- Hc""fI""I'Sl I. hoo' 01 ,.....".bou...... , """'." ,h... wr<. no wI""""" "" ,'," ...... h"" "'od"'£ ,"'" ....n;! """" ~i" to ,.;, "''''' l>.illd1'S' F~8 , ...... 1.11 "" tb< """'od "" ,.,." '" '''' oqoipm<'" ,N0S ""~ "",."" ",. '"1"""' T.. " ....Pl"'rt"'_ TI"", woo """ 'n .1""",,, 110.. 0", 1'0. ,", _I\<'''''f'k"'«l tuobrid ... '''' US~...... , OVI, ""l"d"'l ,I« On'" col). w,", ...... """'"8 '."'.'" '.. d"'" I".. (Cl. Tho ... I."", ""Pl' p....."" to ""'"P"­ By ,~" tim• .. hO<"""", .01 of'....' ""oobik'ish t~ tho< • SOp,,;,,,,, I",,,,, mol. 'f'!""",,y&"''''''1 hom "" """~­ '",..."'"'",ht...."'""'"...to< w" """,­ I~I_ ..." ,"'." ",,' ... Not ",,,....U Il. i"II ''''''"I~ _d"" """'Sh ,,·"n. no.. ' ,,,. ''''0'' &.p, Sonfm-d·• ..-roo.. " IuIly ,_""I< ..,tt. to ~" ,""'" m ....d_. f""" III ,,, ""8'" 'ho P"'I'""" ron ~ 01 ,I« "",r 01 on (0). A' ")"1'" "'" '" '.. ""'.. op<.... ,,.,. ._""""'"'" h "''''''~ '''' &'...... 1""..' boo' leo. r... m,""'.. '''ab· .,m" 'ho.." .... oltho po , """",....~. ,,"'.. ,,,,, ,htl.1" (C)...""" .... "".."" '. 1...... ""',,,,',colly '" F,~.. I ""- ,ho iM •.., 01 'ht ""'"'" _on"""""'" ,I\< .s..""", 01"", "'nn...... " """,;"""q,,,, htco.'" ""'_, w"" w...... '0 now """,.. pow" """" """f "'..... tlNl" 01 '"' 1­ ,...... oI'..d."'­ ""... '"""'P'''''" by f.1l ;n~ nJ"'p, b.._ FIGURE 4_ Erosion of cut slope in Crtlaceous soils during Bouldin Dam reconstruction. the crest. TIle similarity of the events to the the reservoir to escape, a slide would have had development of ~he erosion tunn.el through to extend upstream at least tltis distance. The Teton Dam is evideJlt (see Figure ). extent of such a slide parallel 10 the axis of the The official reports concluded thai the dam would have had to be ofcomparable mag­ failure started as an upstream slide exlensive nitude, several times greater than the 26 feet enough to breach the crest to a level below the from the end of the power house to the lamp reservoir surface, whereupon the water flowed post at (C), Thus, the crest light would havedis­ through the gap and injtiated the erosion. This appeared with the first earth movement. Yet. it scenario is incompatible with Sanford's ,le­ continued 10 function for at least twenty count. Foremost among the {acts Ihtunnel. Thescenario would notbccom­ muddy water began 10 flow over the roof of the pJete, however, unless a set of physical condi­ power house. Had overtopping occurred tions existed that wowd pennit the initial un­ through a slide-produced gap, the lamp post OJ: detected development of such a tunneL The its power supply cables would have been official reports correctly noted illa! seepage among Ihe first casualties, and the events was expected by the designers and had oc­ described by Sanford would have taken place curred extensively CIt the downstTeam loes of in the dark. Furthermore, the temporary the east and west wing dams where they rested decrease in the flow that he observed when he on permeable terrace deposits. Relief wells had reached the gate is characteristic of blockage been installed, along with other remedial and caused by collapse of material overlying a run­ observation works, to conlTol the seepage. In­ nel, whereas the flow through an open channel deed, the record is replete with references to the would only have increased with time. It is also attention given ro the prevention of piping at noteworthy that the horizontal thickness ofthe

OVlL I)..'GlNEERlNG rRAcna. F.... U 1988 73 se.epage were undoubtedly present. Backwarrl eroSiOll could \.lave gone unobserved below laihvater level lor a long tUne until finaUy an erosion f:uN1el reached the reservoir and per­ nutted concentrated destructive nows to cause rapid enlargement and failure. The course 01 events before the runnel reached the. level of the po\\let: house roof is speculative. but the existence of conditions favorable to the development of a tunnel by backward erosion is 110t. Neither are the events observed after the tunnel reached the level of tl,e power house roof. TheconciusiQn seems in­ e-scapable. Ihat failure acrually occurred by piping, and tha.tany supposed shortcomings in the construction of the embankment, even if they existed, were irrelevant. The official reports postulate an upstream slide without the destabilizing influence of a dr?\vdown. This explanation in itself is logica1· Jy questionable. An earlier shallo\... d.rawdown slide in the steep upstream slope had occurred, however, and had been repaired by dumping crushed stone and rockfill in the affected area. The investigators reasoned that Ihe strength of the clayey materials under the dumped fill had graduaUy deteriorated as their moisture con· FIGURE 5. Erosion lu.nnel at the contact of tent increased, and that at the time of failure the two layers of Creuceous soils in excavation at strength had reduced to theexlent thallheslide the Bawdin diUnSile. Note theverti,cal joint in was r~cti.valed. This hypothesis, like. that 01 the cohesive gravelly material. any upstream slide. is lncompatible with the events recounted by Sanford and with the ex­ possibility of piping in the underlying tento[ a slide that wouJd have been required 10 Cretaceous materials. although these materials lower the top of the dam below reservoir leve-J. cOIHained many slightly cohesive, highly l1I short, Ihe failure of Walter Bouldin Dilrn erodible sands and ');115 (see Figures 4 and 5). occurred because of piping by backward The foundation raftof the power house was erosion. As no 01 her e)(ample of catastrophic cast inside formwork in an excavation extend­ failure and loss of reservoi.r has been attributed ing a few teet into somewhat weathered schist to an upstream sljde, it may be conclUded that underlying the Cretaceous beds. An ap­ il dam that has successfully swvived construc­ proximate section through the edge of Ihe raJl lion will not experience a catastrophic and adjace'Ilt materials is shown in Figure 2 upstream slope failure. Any analysis that ind.i­ The geometry of the backJilled zone would cales otheMise must be e.rroneous. Further, a have favored seepage along its boundaries. stability al1alysis to investigate the s.uety o( the Seepage could Ats<> have d~veloped readUy upstream slope under full reservoir is ir· through tlle )ointE. and more pervious zones in relevant. If an upstream slope does nOl fail lhe Cretaceous deposits.lndeed, lheexcavaHon duri.ng constructi.on, its ~ctor of safety must foy the po\Ver house and intake structure re­ exceed unity under the more favorable condi­ quired dewatering by weU points; three tiers tion of reservoir loading. Rapid drawdoWTI were instalJed, the !owi:;r two of which were en­ may induce a ('!'Iilure, but sud\ a failure is shal­ tirely in the Cretaceous beds. Thus,avenues for low and has never been known to rut back into the embankment far enough to permit overtop- stabi lLty caku 1(1 tions under cenditions of and cause the catastrophic loss of a reser­ decreasing increments of movement is Gar­ voir. diner Dam on the South Saskatchewan River in There is a remote possibility thata dam con­ Canada. The case of this dam illustrates the fine-grained soUs possessing shear limitations of equilibrium stability analyses. str<:mg"ith due to capillarity, or containing stiff collesive materialssusceptible to swelling, may Behavior of Gardiner Dam SI.T,en~~th when submerged if the thickness Conception, and construction of Gar- upstream shell material is inadequate. diner Dam the quarter cen­ up:~tre·:am slopes of dams containing such in which of shear strength materiials are usually fairly flat and failure sur­ was its most revisions, and would tend to be shallow, with conse­ at evolution of the design the qu,eru:essitnil,U'lo those ofdrawdo\'lTl failures. ge<)te<:hnical stul:ties reflect~~d the new frontiers no catastrophic failure of this is followine history is greatly ablbre'vl.ltelandslide topogTaphy givi::ng tes­ pressure w ill increase with time, unJess this fac­ timony to propensity for stability problems tor of safety is so close. to unity that cyclic load­ duri excavation and fill placement (see- ing produced by fluctuations in the level of pool causes strain softening and a critical loss Farm Rehabilitation Ad­ of 11, however, the factor of safety is ministratiDn (PFRA) began studies so close to unity, dO\'lTlslream slope for an irrigation project involving a dam across pt€:ceejed by and 111­ the river. Total stress stability analyses were the c:re~SJJU!: increments ofmovement at successive ruJe at the time, and the initial design wasbased levels. Any calculation a fac­ on two sets of undrained peak-strength o safety appreciably different from unity parameters: (= 15 to 20 psi, Arthur Casagrande, engaged as a con­ creaging. If the contrary occurs, the "J::.lhilihJ sultant, turned the emphasis to a Study of the may be decreasing. Stability c-alculati(:ms slumped slopes in the vicinity rupplemel1ted be usefuJ in judging the influence of various by laboratory tests on a few select samples, by remedial measures, but the cmnpute'd a test drift in which surfaces of sliding in the nitudes of the factor of safety are me~.nin~~es;s, shale could be observed, by instrumentation to An outstanding example of the imcle.'V8Ilce detect movements and by installation of

CML ENGrNEERING PRACTICE FAI.L 1988 75 9".-...... 8o,",,,,,,;oS ,"" ••d'",'~ ...... 1 .... oI.w...s. FS • ',J w., _ """10m ol .... ''''''eo "... "" ...... oido,.d oW_' ... lo~ o

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l"~':,,, . IJ ,.L j ..... ~_-. .' I nGU~U. ~".'" ~ '''b,d.,. e ,~,~ po f..,.. "" "'PP""'" ,""' ...... _ .."" .... _ ...... bolo !.-...... tlw_"'"', ...... d'" ...~,ll~& whKh ruk ...... 011... --.: .. -'ttl .... _ nluo ...... -.-d••'1<1 • _'" wr-' .. ,ht ._...... 1<1, oNly £_ "" ...... "... • _.m I0<._~ '" _-,...... _to." 1.l 00«1 p;,., i< ...... -,_.,-- """_ Ipo _ .""" ,10< , ...... _ ...... __ n.r .... ­ __ Tlot_I....-"""kNTS.l0. ...,.... -...---..,... midwol_ d-'yr.._ ...._ _I'I-u,_ _ n ..• ...... _ __.. -...... -..~ O'l,.- _ d4plo< 01 ""'" I .. 0«IIn0d ""n..~ """'__ ",lcuIo_ ...... _ ... _-, ..._--, _-..-, Iu~ f"X'l .... _".""'_101 pool. ""'_I..-_pIo,...... __ ties of the mass undergDing movement in Gar­ factor of safety is obviously to unity. diner Dam impose inherent limitations on The actual safety can bE' assessed only on the limit-eguiHbrium analyses. The limitations basis of monitoring the movements and as­ have been recognized, and finite-element socjated events. The is ex­ studies have been carried out with sophisti­ emplified by the studies at Darn, cated refinemenls.9 assigning what ap­ where the cruoal observations were those peared to be values to the physical indicating decreasing increments of move­ pn:)peTtu~ of the various materials involved, ment undersuccessivecomparable reservoir by adjusting these values a.c; required to fillings. Nevertheless, if fador achieve agreement between predicted and ob­ safety is approximately umit-eqw­ served behavior, a modeJ was developed that liblnUln calculations in judg­ could reproduce deformations similar to those elf,ectivene:ss of various alternatives in the field, including the pattern of response to increasing the safety. use of equi­ cyclic reservoir operation. Prediction over librium analysis is justifiable, its effec­ many load cycles. however, was not satisfac­ tiveness has been not only tory. Although the study provided valuable in­ with respect to dams, but with to sight into the behavior of the dam, it is dear, natural slopes. It be dear, nevertheless, that no such finHe-elernent study; h01Ne"er, that the absolute value of the fac­ without the calibration afforded by extensive tor of safety of the cal­ observational data, can be depended on to culations is of no signifi:caIlce. indicate the degree of safety of this or probab­ ly any other existing dam. Stability anaJyses are the investigator. Conclusions "with respect to evaluating stability of exist­ The foregoing discussion leads to the con­ dams.n isno! meant that they should never clusion that stability analyses are unreliable be performed. However, l1UJl'\erical values bases for assessing the 5tability of an exi:StiI1i~ for the factor of should little if any dam with respect to catastrophic failure. weight in judging the of the struc­ conclusion strictly applies to staHc ture respect to failure. tions; insight regarding the behavior in an The in 100 much em- earthquake may be gained by analysis, al­ On stability is may though not generally by limit-equilibrium regarded as a for the much more analyses. difficult and expensive investigations and U the pool research needed to establish the real equilibrium reached, the results of character of the structure in question. Some stability analyses may be assessed as follows: dam owners may prefer the relatively small ex­ pendihlre for a perhmctory stability study in 1. U the calcuJated factoT of safety is less contrast to costly and time-consuming field than unity, it must be erroneous. studies. Of greater importance, because of the 2. If the fador of safety is greater potential danger> some regulatory greater than the results merely indi­ bodies may take more comfort in orderly cate the is unneces­ stability calculations based on unsupported or sary to show that the dam is standmg. Fur­ unverifiable assumptions than in qualitative thermore, no conclusion about the judgments basro on gxperience and careful in­ degree of can be drawn from the vestigation. Yet, the former may have uttle or numerical value a computed factor of no relation to the ~al safety of the dam, saiety: hence, satisfying some prescribed whereas the latter are essential in assessing the criterion for va lue is not in itself a Ukelihood or possibiWy of a catastrophic suitable indicator of failure. 3. If movements are occur­ This djscussion guite possibly conveys a ring; a is irrelevant because the negative impression about our ability to deter­ """' ",,",,I>« '" ""' ... , .... "~ .",.,.".",.", RAlI'" 6 PKK _ f/>t 1'1''''' d.m i, d'''''.''''biy "f••,.'n" • 0( C L D.C,£ 10- lo" , ,...,"""~"' f.,J"",. T~.. ".1".....", i' 001 'ho 1'oI~ ;, ''''''''' '" r'Jl ... ""...."'" 01 "", otudy,"", woW<=' JlJl. _jO«hibIe ..""...; """_ • .-:1 ...... "..... ••- ..- ••, t>- ,..,s-" "'''',,­ """ '" """,,,,,,,,,,~,• .-:I ," y ",. 1<,.,,.. Not __'''£''!'_'''f•..J". ,'-'" ;.~."",,,,,, .nd """"'''''''101 . "", ff<1m.';'... W'Eo...,Ci, p«<1vd••• ,""""" "'''''", '" ""..~y , "•.-""t _l.,-~,~"._O<· 01 "" dJ",.nd 'hoy ""y _ provo '" "",swr'" w Boolo '''P<''~vo ()., 'o. ~"" ''''Y ...y .... F~_"' '" ""'- s..-_ ,m ,~ ~ui<1. ...d"',,,,,,,,, ,.,l>­ "'~, ,~" , • f .....'~.~e.- oti "'... lot ,,'"""""t" ijs.boM ... D.." , ..,.,. "d ",~-doO ~""." .. , w.".. "",,'.J. ,. ,",h ...... , ...." iI q... .000'",""""'••; 0Iti<0 01 ""'In<"""" "",,",. pils""'''' ....." HOC, W _.OC.s..._,.,. ~' ~'.' ~ "' ...... """ C"""'bu_ F"""""" _C.","" AOWIB>GThIENfS ~ """"''' """"",. ,""ow","" I, ...... 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Appendix D

Part 2 Suggestions for Slope Stability Calculations by Ashok K. Chugh and John D. Smart

This article was published in Computers and Structures, Volume 14, Number 1-2, pp. 43-50, 1981.

Co"ljlPItf, &. SITliCIOW. Vol. 14. No, 1-2, PO, 4j...j{), 19111 OOil-1949181 IUIlOO4)..()8SO'l,OO/O Printtel in ureat Britain. Perga.... ore" Lid.

SUGGESTIONS FOR SLOPE STABILITY CALCULATIONS

ASHOK K. CHUGHt and JOHN D. SM.ART~ Water and Power Resources Service, Engineering and Research Center, D222, P,O. Box 25007. Denver. CO 80225. U.S.A.

(Rueiued 21 Jal1UtlTY 1980; received Jor publication 3 lilly 1980)

Abstract-Considerations for the selection of potential slide surface geometry in slope stability calculations by the limit equilibrium method are presented. Relations between the solution variables and the Mohr-Coulomb strength nAl"llrneter.' for no failure on the interslice boundaries are derived. Also included are the relations between the ;~iution~~riables for effective and totlll stress considerations. The materials are assumed \0 be "no-tension" type. Significance of evaluating the calculated response for the individual slices making up the potential slide mass is indicated.

NOMENCLATURE Amongst the different methods of analyses, limit equilibrium methods satisfying the three equations of b width of slice statics are now being used extensively for estimating the c' cohesion with respect to effective stress e eccentricity of the e"terna; force P stability of both natural slopes find man-made E horizontal component of interslice force embankments [5-8,12]. The method of slices, considering F factor of safety the interslice forces, as presented by Spencer [10,11,13], H force exerted hy the pore water on the interstice boundary is representative of the modern versions of limit equili­ h height of force above slip surface brium methods. It seeks the solution of the slope stability i ground slope equations, starting with an assumed value of factor of K", coefficient Ijf active earth pressure safety F and thrust inclination 8, satisfying the boundary N force normal La base of slice conditions at the toe and head of the slide mass. The P external force acting on the .slice S total shear force other parameters, such as interslice forces, their mag· T in«:rslice force for the bad wedge location and direction) do not an active role W weight of slice or back wedge in the solution scheme but are as a part of the Z intersli.ce force iteration pracedure. In this presentation, the boundary U force exerted by the pore waler on the base of the slice conditions are considered as being distinct from the a slope of base of slice inters lice forces. P slope of the top of slice The objectives of the present paper are to present sug­ " dope of interslice force gestions lor: 8 backrest angle with horizontal I. Geometrical configuration of a critical segmented failure surface. I!fI'RODUCfION 2. Interrelationship of Ihe mathematical solution for The problem of slepe stability is an important part of tolal and effective stress considerations; and geotechnical engineering. As a result much has been said 3. Limits imposed by the material strength on the and written in Ihe technical literature about various validity of the mathematical solution. methods of analyses-their merits, complexities and simplifying assumptions. justifications for their use in The materials are assumed to be homogeneous and iso­ actual practice, and about comparison of results obtained tropic and obey the Mohr-Coulomb strength hypothesis. from their use[1.4,6-S,12-14U Most of the slaDe An example of stability analysis of a natural slope i., stability analysis procedures have been converted into lI1c:mOleo. The following terms used in this paper 3re computer programs for their fast and accurate de4ined as follows: implementation [3, 15J. Frequent occurrence of slope design problems combined with availability of the com­ Backrest Geometric configuration of the heel of a puters, makes it reasona ble to assert that at present segmented failure surface. almost every geotechnical engineer has an access to one Thrust: Resultant force on interslice boundary. or more of the slope stability analysis computer codes The words "no-tension" and "cohesionless" are implied to and that these codes are frequently used in engineering have identical meaning.

Figure I is a general description of the problem. For any verlical slice, abed, the forces acting are shown in Fig. tCivil Engineer, U.S. Bureau of Reclamation. l(b). HL and H» are the hydrostatic forces exerted by tSupervisory Civil Engineer, U.S. Bureau of Reclamation. the SUbsurface water on the vertical boundaries of the §References included in Ihis paper are representative, but not II slice (assumed to be known), Other forces acting on the complete list, of lhe works on the subiect. free body diagram of a slice are defined in the Nomen­

43 A. K. CHUGH and J. D. SMAIlT

- Polentlal ~Iide tudoce of 10me Qeametri c conf'gllra lio"

101 Fig. I(a). Genual slope Slability problem description.

(b)

Fig. I(b), Forces acting on a typical slice. clature. Considering static equilibrium of the forces: In the derivation of eqns (I) and (2), and elsewhere in this paper, the factor of safety (F) is defined as the ratio of the tolal shear strength available on the slip surface 10 ~ c'b sec a - W sin a +~W cos 11 - U)tan !p' the total shear force required to reach a condition of L=h+ [ I ] limiting equilibrium. cos(8-a) l-p tan(8-I1)tan41' GEOME11UC CONFIGUIATION (~ (~ For circular configuration of potential slide surfaces. p cos - a)[ tan - a) +*tan t/J'] T ­ most computer programs have a routine that optimizes COS(8-a)[I-~tan(8-a)tanl/ll on a circle that gives a minimum factor of safety [3, 15]. This optimization is generally in the neighborhood of the initial estimate for the center of rotation of critical HI.. cos a[ I +~tan atan ¢'] circle pro~ided by a designer. For a segmented geometry of polential slide surfaces, +cos (8 - a)[ 1-~tan(8-a) tan 4>'] generally a discrete calculation is performed for the stability determination along the specified configuration. The critical elements of a slope that should be con­ cos q [ 1+.p Isn 11 tan q,'] sidered in th~ selection of a segmented failure surface (I) geometry are: cos (8 - a)[ I-~ t.10 (8 - a) tan ~,] l. The profile of the weak material responsible for the occurrence of slope stability problem. 1/2'" Z~Z h, +2b [tan 8-tan a] [1 + Z~Z] 2. The profiles of the underlying and overlying rela­ tivel y stronger ma terial. P 3. The pore water pressure distribution. +ZR cos fJ sec 8[ltl tan (3 - eI 4. The length of the potential slide surface. S, The indication of localized weak material zones. I + sec 8[HLh. - Bllkl ] (2) Z. The order in which the above elements are mentioned Slope stability calculations 45 is not important. While these elements inftuence the At the instant of shear failure, from eqn (6) choice of large segments of potential slide surfaces for analysis, they do not, per se, assist in estimating the toe II' sin 8- Tcos(O 8) and heel of the slide surface [2J. hocos illcos £J cos i 0 ') -2 I'h o '(l> ') cos () tan =c SIO-j . ( + sm ,,-I t/!

CRITICAL lIAcK.RFST INCLINATION + T sin (8 - 8) tan 4> The critical backrest of a segmented slide surface ~ 2 should be such as to give the least contribution to the yho cos 8cos i sin (8 - 1/» - rho cos i cos ¢ overall factor of safety against sliding, For a planar T== cos(98-I/»sio(0-;) backrest of the failure surface, the forces acting on the (7) wedge of material are shown in Fig, 2. Pore water pressure is not considered in the following derivation. For a cohesionless material, egn (7) becomes Summing forces along the backrest plane:

s = Wsin e- r cos (e - 5) (3) (8) where Summing forces normal to the backrest f) cos K '" cos i sin (8-£ (9) A cos(8 o-q,)sin(8-i) N =W cos 0+ T5io(8- 0) (4) Differentiating K(8, 8) with respect to 8 and equating it For the Mohr-Coulomb material, shear strength along to zero, one gets the backrest is: (10) shear strength c,AC + N tan I/J hocos i Similarly differentiating K(f), 8) with respect to 8, equat­ = C sin (8 - i) ing it to zero, and substituting eqn (\0), leads to the transcendental equation: ! 2 COS 9cos i +pho sin(8 ncoslJtan,p sin (8 - ¢) sin (8 i) tan 8- sin (t/! - i) == 0 (11) +T sin (e - 8) tan t/! (5) The solutions of the eqn (1 nare graphed in Fig, 3 for The expression for the factor of safety from equations several values of i. The corresponding values of K", the (3) and (5) is: coefficient of active earth pressure, are evaluated from

cho cos i 1 yho2 cos 8cos i , , ell ') +2- '(/} ') cos 8 tan 4J + T Sin (8 ­ 8) tan ¢ F "" sm 0 - I sm - I (6) W sin 8 - T cos (8 - 8) 1 Thus for the backrest to yield the least F, its orientation eqn (9) and are given in Table I, should be such as to give the minimum value for the It may be mentioned that the z.eros of the eqn (\1) thrust, r. were approximated by the one-step linear interpolation For a given material and ground slope, the thrust T formula [91. x,=(XJY2-X2}'1)!(Y2-Yl) where XI to Xl is depends upon Wand 8, Fig. 2. For the planar backrest of the range in which the solution lies, This range was the failure surface, T is a function of (8.8), obtained numerically by incrementing 0 through 1°.

w=~ A8, Aor tall 8 = rr..t.Q.( AO tall i = CE AD DE = i\" h" co, 8 cos i AD = .....,--,--­ sin (8-il W = 1. r h" cos8cos i 2 0 sirUl-;)

Fig, 2. Forces acting on the backrest portion of a slide mass. A. K. CHlJOH and 1. D. SMART 80-,----..------,------.-----,----­

.. GROUND SLOPE ANGLE, i (OJ <1> 60 . ()) 0.0 ...... J Cl 5.0 Z ® <( ® 10.0 l­ ll) 15.0 w 40 0 20.0 u'" ® «'" III ® 25.0 «...J (0 30.0 ::! I- ® 35.0 ~ '" U \!J 40,0 @ 45.0

i I II I o 10 20 30 40 50 ANGLE OF' INTERNAL FRICTION, 16 1°} Fig. 3. Solution of eqn (11) !ll first approximation.

Table l. Values of K"

I I I/J IN DEGREES DEG. SLOPE :5 10 15 20 25 30 35 40 45 0 0 73 :5 .604 .504 .4 2:; .35 :5 .296 .244 .199 160 5 1:11.5 .992 .698 .56 :3 .463 .383 .3 I 6 .258 209 .166 10 j 5.7 970 .654 .5 t 7 418 .340 .275 220 114 .. _--~ 15 1:3.7 .93 ::I .602 467 .371 .295 .234 .1 B3 - .-1'-----1--­ 20 1:2.7 >_...:~83 .54 is .414 322 .250 .193 - f- -- -- 1---­ 25 1:2.1 821 .4 85 .359 273 .207 30 I: I. 7 750 .422 .305 .225 --f----­ 35 I: t.4 _. 671 .359 251 40 1:1 2 587 .297 45 I: 1.0 .500

RELATION OF 'I1lltUSl AND ITS INCUNATION tan (j' = ~:...... ::.=--=~ ( 16) FOR EFFECTIvE AND TOTAL STRESS For the interslice forces to be statically equivalent for the effective and total stress, 4(a, b): From eqns (14)-(16), one gets

T cos S '" T' cos 8' +H (12) TCOSS-!H!b]h " [ 3 h (13) (7) T sin S '" T' sin 8' fl = Teos S H hG-~ T(ho- h) cos 8 T'(ho- h') cos 8' + H( hI) Equations (15)-(17) relate the interslice force, its loca­ (14) tion, and orientation for total and effective stress. However eqns (12)-(17) do nol accounl for the effect of the Equations (12)-(14) ensure the horizontal, vertical. and hydrostatic forces on the intersl ice boundaries on the moment equivalence respectively of the interslice forces calculated factor of safely. F. It is essential, therefore, to for the total and the effective stress. include these hydrostatic forces in the devivation of slope tl''f''I'\f''n ..,," 111\ :l- AUtU .... "1.u \.AJ} stability equations as shown in eqns (l) and (2).

T';: T sin 8 (15) UMITS ON TIIllUST LINE L~CLINAnON sin 8' AND ITS MAGNITUDE Considering the vertical equilibrium of shear force From eqns (12) and (15) acting on the interstice boundary and the mobilized shear Slope stability calculations 41

T

(0 , ( bl

FiS. 4. Tota! and effective interslice forces. strength, Fig. 4{b): ter of the material, it is important that a designer con­ sider the results of this calculation along with the cal­ culated value of the factor of safety. T' sin 8':: ~ [e' ho+T' cos 8' tan 4>'J COMMENTS e'~ tan,p'=Ftanli'-rsecS'. (18) In the derivation for critical backrest inclination presented, it is presumed that the total factor of safety of Ii segmented faIlure surface is composed of the factor of Therefore, in a cohesionless material, (e' = 0), the safety of its various constituents, i.e. necessary condition for no shear failure on the interslice boundaries is: F= Ii 8'$1/>'. (19) where Ii is the contribution of the ith segment and n is The corresponding inclination of the thrust for total the total number of making up the geometry of stress is obtained from eqns (16) and (18); the potential for a slide surface to have the least factor of safety the contribution of each tan 4>' =F 1 tan li. (20) unit should be minimized. I-~ seell In a wedge type of slope stability analysis, where the back and toe wedges of material are replaced by horizontal forces exerted by them on the middle wedge (HJ~) ~ and the factor of safety of the slope is calculated by Since C_ sec 8) musl be I, it follows from eqn considering the equilibrium of the forces acting on the (20) Ihat middle wedge, the backrest angle for maximum horizon­ tal force is greater than the backrest angle that gives the least interslice force, T, used in eqn (6). Since this type of wedge analysis implies occurrence of shear failure Similarly from eqn (16), it follows that: condition on Ihe vertical interfaces between the wedges, it gives a lower varue for the computed factor of safety 8::;;8' (22) of the slope. For stability of natural slopes and embankments, this assumption of shear failure on the and from eqns (15) and (22) that: vertical interfaces of slices is unrealistic and gives un­ duly lower estimates of the factor of safety, and bence results in more extensive remedial treatment(s) than may be necessary to meet a design criterion of faclor of It is perhaps clear that equality in eqns (21)-{23) holds for safety. Alternatively, it could lead a designer into reme­ H=O. a smaller lone, the middle wedge, of a potentially slide mass and thus underestimate the extent of LOCATION Of 'fHlWSY LlN'E essential treamtneL In any case, a wedge type of analysis The slope stability eqns (1) and (2) deal primarily with for slope stability problems is unrealistic and is not the equilibrium of forces acting on the free body diagram recommended for general use. of a typical intermediate slice. Therefore, the location of Since stability problems Renerallv occur in the thrust line on the interslice boundaries in terms of hi geologic formations an{ embankments composed of and hz is of direct significance. If the calculated location different materials and complicated by complex pore of the thrust line for a particular slice is outside the water pressure distribution, it is essential that a designer sliding mass, a tension of some and extent is make a study on the backrest and exit slopes implied. For no tension on the boundaries, it is for a segmented failure geometry. Figure 3 may be used imperative that the thrust line be located within the in making an initial estimate for the backrest angle, sliding mass for every interslice boundary. A further Since there is no way in general to predict what a assumption for normal stress distribution (such as linear) solution to a nonlinear system, while satisfying the pres· on the interslice boundarY shall further define the bounds cribed boundary conditions. may calculate for the (such as middle third) within which the thrust line must various slices making up a slope, il is important to be located for no tension. Since the limit equilibrium keep in mind the physics of the actual problem in solution procedure does not consider the tensile charac· interpreting the calculated response. Assuming com-

CAS V,,1. 14. No. 1-2-0 ,\00 1110t. Ie: ".J t(~,i I

116 o (U.t .... ""rDI f't.tDoVI ••Nrl" 1(0000-U,"'15 0 o 'is 0 tU.' " I~QO ll>Cl~I o .... •••,.LY (LlYS 121$.0 0 " "00 [B .tlll( I~O 0 1000.0 oj

10'[ "iTCI '''(11\,1 II ( OIU.UUf,or (orn ..... I}OO

'000

]'------.l--...L--.l--J...L--l--:~~--.J--::_=----l--,~...,~----.L-~,f.":-,,,-..L-___;_,,~oo;-_'_l..J'·L[_'U_Q-;;,~~.;;t_t_·-.L°?_---;;,,; ..o_L_--;-.~, -;,;.,oo;;-_--,-_--;-,~••;;;._-''-­ ..J..l._-,---~l_ ----L-...J "'" ,- """

Fig. 5. Example problem. Slope stability calculations 49

7.75 1300

1200

725 !IGO

~ <>0 1000 :r z 52 0 1-­ 675':: 900 VJ « w ~ 0:: 'x: ~ u 800 ~ ~ ~ tIl W t-­ 62~~ 700 « ...------­ -' UJ U f­ V) 0:: ::;) 600 0 ~~ a:. u., ~ 1: w Evs8 f­ 575 500 I.t· ~"" 400

1.0 '--_-'--_-'-_~ _"._----..,...... -_::_-::"'::"'"-_='=_-___=_=--1525 300 25 30 35 40 45 50 ~5 e.o 65 7'0 75 90 BACKREST ANGLE 8 (OJ

COULEE DAM N-E AREA; LONG SLIDE SURFACE '# I Fig. 6. Parametric study results, slide surface No. I.

~·.::t~-"'-" '---_-'--_--'---_...JI_--'-_...J-_ 38 40 42 44'__46 48 50 52 54 BACK REST ANGLE e(0 ) COULEE DAM N-( AREA; LONG SLI[;€ SURFACE "2 Fig. 7. Parametric study resulis. slide surface No.2.

pression to be positive, the calculated negative interslice assumed. Even in actual geologic formations, the limits forces imply the presence of tensile normaJ stresses in imposed by these equations cannot be grossly violated the soil mass. Unless the solution scheme is formulated by a nonlinear solution and still be acceptable. to account for the tensile character of the material, the A poor mathematical solution does not necessarily calculated results for the (P,8) pair can be quite mean­ imply 11 poor nonlinear solution procedure; it can also ingless. Similar comments apply for the solutions that indicate a poor phYsical modeJ. An evaluation of the give thrust line inclination S in violation of the limits intermediate response of a poor mathematical solution to imposed by eqns (191 or (22) for the ideal material a nonlinear system generally reveals tbe bad character of 50 A. K. CHUGH and), D. SMART the physical model. A designer should take into con­ calculations of known past s.lides in the area and labora­ sideration both of the above possibilities in interpreting tory tests. The results of calculated factor of safety as the computed response of a slope stability problem. far as they apply to the specific site are preliminary and It is in these comments, that there is only do not reflect the remedial treatment alternatives under one real value of F and of S that will satisfy both the study to imporve stability. force and moment conditions of equilibrium. SUMMARY SAMPLE PROBLEM The selection of potentiaJ slide surface geometry in Figure 5 illustrates a section located in the Coulee slope stability analysis by the limit equilibrium method Dam northeast area downstream of the Grand Coulee deserves a careful consideration. A close scrutiny of the Dam in the State of Washington. The identification of calculated results of slope stability analysis in terms of potential slide surfaces in the hillside is of interest. For interslice forces-their magnitude, direction. and loca­ slope stability analysis, the geologic makeup of the site is lion is of significant importance and should be con­ assumed to be composed of four materials. Their broad sidered along with the faclor of safety. identification and estimated properties are given in Fig. 5. The pore water pressure distribution in the hillside is shown in Fig. 5. The geometry of the segmented slide REFERENCES surfaces analyzed are also marked in this figure, The L W. F. Chen, Limit Analysis and Soil Plasticity. Elsevier, analyses were performed using the computer code Amsterdam (1975). 2. J. M. Duncan and A. L. Buchignani, All Eng/nming Manual STABLTY[l5] available at the U.S. Bureau of Reclama­ for Slope Stability Studies. Department of Civil Engineering, tion. Engineering and Research Center. This computer University of California (Mar. 1975), program implements the metilod of slices satisfying the 3. D. G. Fredlund. Slope stability analysis. User's Manua/. three equations of static equilibrium and calculates a Department of Civil Engineering, University of Saskatche­ constanf value for the facto'r of safety and a constant wan (1974). inclination for the interslice forces. The effect o[ back­ 4. D. G. Fredlund and J. Krahn, Comparison of slope stability rest angle on the calculated [actor 0[ safety, inclination methods of analysis. Canadian Geotechnical J. 14, 429-439 of the interslice force, and the horizontal component of (1977). the interstice force are shown in Figs. 6 and 7. It should 5. T. C. Hopkins, D. L. Allen and R C. Deen, Effects of Waler 011 Slope Stability. Kentucky Bureau of Highways (1975). perhaps be mentioned that each potential slide surface 6. T. W. Lambe and R. V. Whilman, Soil Mechanics. pp. 352­ was analyzed individually. For the slide surface No.2, 373. Wiley, New York (1969). the calculated least factor of safety corresponds to 7. K. T. Law and P. Lumb, A limit equilibrium analysis of backrest of 48", Fig. 7. The critical backrest piogressive failure in the sUbility of slopes. Canadiaii Q£o~ for the topography and the two material strength techllicaU 15. 113-122 (1978). values, from Fig. 3, are: (J =35° for 4> 11° and (j =58° 8. N. R. Morgnestern and V. E. Price, The analysis of the for c/> =34°. The average of these two critical backrest stability of general slip surfaces. Geofechlliqui IS, 70-93 angles is 46.5". The corresponding values for the critical (1965). backrest angle assuming validity of 45 +q,/2 would be 9. M. G. Salvadori alid M. L Baron, Numerical Methods ill Engineering, Pl'. 18-'20. Prelillce-Hall, New Jersey (1961). 50.5" and 62" with the mean value of 56.25°. Thus, the 10. E. Spencer, A method of analysis of the stability of backrest angle corresponding to the least factor of safety embankm~nts assuming parallel llilerslice forces, Geotech­ tends to agree with the values indicated by Fig. 3 rather lIique, 11-26 (1967). than 45 +q,i2 [2J. i L E. Spencer, Thrust line criterion in embankment stability It should be mentioned that these analyses were per, analyses. Geotechllique, 85-100 (1973). formed using the existing conditions of surface topo­ 12. K. Terzaghi and R. B. Peck, Soil Mechanics in Engineering graphy. interpreted materia! horizons, tested material Practice, pp. 361-459. Wiley, New York (1967). properties for predominantly clay materials (residual 13. S. G. Wright. A study of slope stability and the undrained strength) and estimated material properties for pre­ shear strength of clay shales. Ph.D. dissertation, University of California, Berkeley (1969), dominantly non-clay materials, and interpreted pore 14. H. 1. Hovland, A three-dimensional slope stability analysis water pressure distribution for the steady-slate river melhod. J. Geotechnical Engng Diu. Am. Soc. Civil Engrs, operation (Iailbay elevation 955.0 ft). The residual t03(G19), 971-986 (1977). strength value for materials used in these cal­ 15. S. G. Wright, SSTABI-A general computer program {or culations tends to align with the lower end of the range slope stability analyses. Department 01 Civil Engineering, of strength values obtained to date by both the back University of Texas at Auslin (1974).

Appendix D

Part 3 Variable Factor of Safety in Slope Stability Analysis by Ashok K. Chugh

This article was published in Géotechnique, Volume 36, No. 1, pp. 57-64, 1986.

CHUQH, A. K. (1986l Gear"cnlliq"e J.6, No. 1,57-64

Variable factor of safety in slope stability analysis

A. K. CHUGH-

The use of e variable factor of safety in slope stability procedures commonly used for the slope stability analysis procedures based 011 the limit equilibrium studies are by Morgenstern & Price (1965), method is presented. The proposed procedure consists Spencer (1%7), Sanna (1973) and Janbu (1973). of defining a cnaracteristic that describes the variation Their use in practice is a matter of individual or in the factor of safety along 8 slip surface and tne orgaruzational preference, availability of a partic­ analysis proced ure seeh to determine a scalar factor tnat in combination with the cbal'll.cteristic and the ular computer procedure and past experience. interslice fora: inclination satisfies the boundary condi­ One of the assumptions common to all slope tions in a slope stability problem. A possible form ofthe stability analysis procedures based on tbe limit characteristic for frictional materials is discussed. A c:.ase equilibrium method is a single value factor of study is included to illustrate tbe application of !.he safety F for the entire shear surface, i.e. the factor ideas prC$Cllted. No new assumptions or unknowns are of safety is the same for all locations along the introduced and all equations of static are shear surface. The factor of safety is generally satisfied. defined as the ratio of the total shear resistance available on a shear surface to the total shear L'artide decrit l'utilisBtion d'un faeteur de securiti:: vari­ able dans I'ena]yse de la stabiliti:: des pentes bui:es sur force required to reach a condition of limit equi­ Ill. methode d'equilibrc limite. La methode proposee librium. The factor of safety is thus considered to consiste tI definir une c:aractl:ristique qui dOcrit la varia­ account for uncertainties in the shear strength tion du faeteur de sOcunti:: Ie long d'une surface de g!iss­ values for the materials. emenl., landis que la methode analytique cherche a A shear surface in a typical slope stability determiner un faeteur scalaire qui en combinaison avec problem passes through a variety of distinctly dif­ Ill. caracteristique et J'inclinaison de la force entre les ferent materials------each with different shear tranches remplit les conditions limites dam un prob­ strength characteristics. \\!hile peak shear Ierne de stabilite de pente, On discutc une fonne pos­ strengths for some material..s along a shear surface sible de 18 ClIl'll.ctt~,ristique pour les materiaux may be usable, it is possible to have only residual pulverulenls el on presente un cas coneret pour illustrer I'application des idees exprimi::es. La methode proposee shear strengths available for others for a slope n'introduil aucune nouvelle hypothese ill inconnues et stability problem involving an embankment fill tolltes Ie:! Cquations de I'equilibre statique sont satis­ and its foundations. The level of uncertainty in faites. the shear strengths of materials along a shear surface may not necessarily be the same. Thu.s, KEYWORDS: analysis; case history; dams; failure; the assumption of identical factors of safety every· slopes; stability. where along a shear surface is not realistic (see, for example, Bishop, 1967, 1971; Chowdhury, INTRODUCTION 1978). Slope stability analysis of embankment dams and The objectives of this Paper are natural slopes in geotechnical engineering prac­ (a) to present a procedure for calculating a vari­ tice is usually performed by the equilibrium rimit able factor of safety along a shear surface method. In this method of analysis, sufficient within the framework of the limit oquilibrium assumptions are made to enable the problem to method be solved using only equations for static equi­ (b) to present a possible fonn of characteristic for librium and a failure There is no unique a variable factor of salety for frictional set of assumptions have usually been made. materials Thus, there are different solution procedures (c) to present a geometric interpretation of the available each subscribing to a different set of constant and variable factor of safety assump­ assumptions. The more modern of the solution tion in slope stability analysis by tbe limit equilibrium method, Discussion on this Paper doses 011 1 July t986. For further details see inside back cover, A case study is included to illustrate the applica­ • US Department of !.he Interior, Denver, tion of ideas presented. 57 58 CHUGH

CALCULATlONS FOR FACTOR OF SAFETY (see Fig. I for the meanings of the various The factor of in a slope stability analysis symbols). is defined by The variation in side force indination in equa­ F= tions (2) and (3) is defined as (Spencer, 1973) Available shear strength along a shear surface tan fJ IJ(x) (4) Driving shear stress along the shear surface J(x) (1) where is a predefined characteristic shape function and;' is a scalar factor to be determined. The available shear strength at a point in a soil This formulation for variable interslice force incli­ deposit depends on the shear strength parameters nation does not increase the number of (e', !p') of the soil and the induced effective normal unknowns (Spencer, 1967). stress at that location. In the conventional slope Equations (2) and (3) are recurring relations stability analysis procedures in which the slide and aJJow the boundary value problem to be mass is divided into slices, equation (1) is applied solved lI..S an initial value problem for an assumed for each slice. Typically, the derivation of the value of the factor of safety F, the side force incli­ slope stability equations leads to the following nations fJ and the known boundary conditions at ~re (Chugb, 1984). (The equations included here the left-hand side of the shear surface (Chugh, for the finite formulation of Morgenstern & Price 1982): see Fig. 1. (1965) adapted to Spencer's (1967) procedure. The idea of incorporating a variable factor of However, the ideas presented for a variable factor safety in a slope stability analysis follows very of can be adapted to any other procedure closely the idea used for the variable interslice based on the limit equilibrium method.) force inclination in that a characteristic shape for For static equilibrium of forces acting on the its variation along a shear surface is predefined slice shown in Fig. l(b) and the solution procedure is required to calcu­ ZIi.= late a scalar factor which scales the characteristic. cos (01.. - a)[1 (IjF) tan (61.. - a) tan ¢'] Thus, for equation (1) x cos (Oil - a)[1 - (ljF) tan (b. - a) tan ¢1 T g(x) = + [(ljF);::'b sec a: - W sin a Available shear a slice base + (ljF)(W cos et - U) tan ¢'] shear stress along the base (5) x {cos (bR- 11)[1 - (I/F) tan (oJ! - IX) tan ¢1}-1 P cos (p - oc)[tan (jJ - oc) + OfF) tan ¢'] where T is the unknown scalar factor and g(x) is the characteristic shape for the variation in factor + cos (OR a)[l - (IfF) tan (b - et) tan ¢'] ll of safety along the slip surface. For g(x):= I, HI.. cos et[l + (I/F) tan et tan ¢'] equations (1) and (5) lead to T F. The defini­ tion of the variable factor of safety according to + cos (8R -IX)[l - (t/F) tan (Oil IX) tan 4J'] equation (5) introduces only one unknown-the H a cos a[l + (l/F> tan IX tan 4>'] same as for the constant factor of assump­ COS (bR - a)[l - (l/F) tan (Oli. a) tan ¢'] tion. The solution procedure for T (2) from equation (5) is similar to the procedure for calculating F from equation (1). The effect of For moment equilibrium of the forces acting on using eq uation (5) in the slope stability analysis is the slice shown in Fig. t(b) to change the distribution of induced shear stress and normal stress along a shear surface. h = ZL cos °L hI 2 ZR cos ha CHARACTERISTIC SHAPE FOR VAR.IABLE b 1 i. «) FACTOR OF SAFETY + - . L'sIn OR - a; 2 cos rt. cos CIa The shear stress distribution in materials along a shear surface, for the constant factor of safety + ZL sin (b - IX)] Z1 L assumption, depends on the normal stress dis­ tribution and the values of the shear strength p parameters, i.e. induced shear stresses are higher + - C01.l fJ sec 01(h 3 tan P- e) Za in materials with higher c', ¢' values-all else being equal. Thus, if a" at two points in two 1 different materials were the same, the present + - sec 1)1(HLh. - Hahs) (3) Za analysis would indicate two different values of fACTOR OF SAFETY IN SLOPE STABILITY ANALYSIS 59

Groufld surface Last slice B

a c

Polential shea, surface 01 some geometric b d configuration

(a)

/b)

Fig. 1. GeD«al d8crlpdoo of the limit equilibrium lIIu1ysls of 11 soil deposit: (I) dope !iUMliI:)' problem; (II) Corea I1cting 00 It lypical !like

shear stress and their would depend of safety assumption and a preselected inter­ on the values of c', ¢' for materials. Intu- slice force indination assumption, and finding itively though, it would be expected that the shear ((in', r)lnduced at the base of each slice. Now g(x) is stresses at these points would depend on the coef­ redefined as (r/O'.')I.do..d and the calculations are ficient of lateral stress and inclination of the shear repeated-all else being the same. The procedure surface at these locations. Ir these were nearly the is repeated until (../cr.')lndueed is constant along the same, the induced shear stresses would be shear surface. If the calculated value of the expected to be about the same as well. induced shear stress is greater than the available 11 has been observed in the analysis results of shear strength then only a shear stress equal to several problems that the normal stress distribu­ the available shear strength need be used in cal­ tion along a shear surface is generally smooth; culating g(x). The numerical procedure is gener­ the difference in normal stress distribution along ally able to achieve convergence in two or three a shear surface is relatively smaU for different iterations. The value of T g(x) then defines the interslice force inclination shear factor of safety along the slip surface. strength parameter values (or the soils in a deposit and methods of analysis. However, the induced shear stresses differ appreciably along the GEOMETRIC INTERPRETATION OF shear surface. The results of deformation analysis FACfOR OF SAFETY by the finite element method indicate that the Constantfactor afsafety ratio of induced shear stress to induced effective Equation (1) is the commonly accepted defini­ normal stress aJong a shear surface is reasonably tion of tbe factor of safety in slope stability constant. This observation needs to be further analysis by the limit equilibrium method. By this confirmed with additional studies, definition, the normal and shear stresses induced

If ('r/l1n'};ndoc'd were to be held constant for fric­ along a shear surface are such that their plOl tional materials along a shear surface, then it maintains a constant proportion (equal to the could be used to define the characteristic shape factor of safety value) to the function g(x) in equation (5). The procedure then normal and shear stress strength plots at every consists of making a slope stability analysis for point along the shear surface. Thus, for the com­ g(x) ::::: 1·0 in equation (5), i,e. a constant factor puted F > I the (0'.', ')Iftdead plots below the 60 CHUGH

c' - elF Ian ;;;; - (Ian ¢')/F @ Mohr-COulomb strength envelope

/ ®© (~". r) dis!·'tlulio'l / along Ihe shear surlace ® IIF > 1·0 / © itF < I-a / --- --®

Normar slress ".' (al

j nduced stresses lor conSlant faclor of safety 4000 8i1urcaliOJ'l point 10' bilinea, strenglh envelope

t: 3000 ~ r; ;U ~ 200°1 Induced sI'eSS6S for variable lacror of salely

1000 ~

a 2000 4000 6000 eeoo 10000 Normal st,ess "n' (b)

FIg. Z. Geomelri<: iOlerpreutioo of the flctor or ~fety; (a) COO5taol rl~fOl' of safety; (b) ~ari.bIe fllCfor of gfery

(II.', ')'''0.1'1> data; for computed F < I the Variablefactor ofsafety (a;. r};nduecd plots above the (an', r)ll.o.aU> data; Equation (5) is proposed to define a variable for computed F = I the (11;, r)ind~ced plots on the factor of safety along a slip surface. According to (11.', ')"''''''''11 data (Fig. 2(a»). this definition a distribution of induced normal Although F < I is routinely calculated in slope and shear stresses is sought along a slip surface stability analysis of earth slopes for assigned that plots as a single continuous curve in the values of material strengths and pore pressure (a;. r) plane (see Fig. 2(b)). The F value at any conditions, the calculated (0,', t)iJ>' deg Ib/ll> CD 135 0 20 © 134 0 25 Q> 130 0 15 @ 130 0 35 8erm constructed in respoflsa ® 140 0 40 to movement along shear @ 125 0 12 slJr1ace uode' study 0 125 0·75 17 Shea. Siress distribution (0' .. variable 'act()r 01 safety i::' 500 .2 00 -', ~ 400 / / _'i-!I-~=.....j..._----"") , , '" 300L-__• (d)

of Y g(,lcllor constant laclor 01 salelV .2 2.0 g(xl lor variable laClor 01 salely "§ T - F - 1·926 ~ 0:: 1 ~ '·0 T:-5~0~C-d t [--- -r- ---, !" U5 a "_x o· t 67-fc.=--==-=-=-=-='====·""'-~-!:!-=-~-!:!·~==:::l fbi (el

LL Essenlially same distribution for constant 81'\cl Y y variable lactor of safely '" Elfeclive normal slress at .,' 20 base al shear suriace __:::-<:'f"--.J'-----.L, '" u;e 10 F'arewalar pressure 0; J----l-----,--m--n ~ ~'_..ll E n::Ln) _ I I ..... }( 0 0 200 4 0 600 800 1000 1200 - 0 200 400 600 800 1000 1200 z PfOjecled distance along snear sur1ace: It F'rojeC1ed distance along sMar sunace: h (e) (0 Malerial properl'es Material Unit weight: c" 10/il12 ¢': deg Ib/lt! ...... ­ <: Shear stress distribution 101 ::: 500 .~ , C o '" '" /I ­ .~ 400 -""!I!i~;;~~ii.;;~~~~~;'-'""i ~ 1·0 ".I '" ~ ... .!! [ '~1 ~--~ I.LJ ;U ~, 300 x <='" 0 • x (a) rn (d) r g(x) lor constant laclor 01 safely ~c 7 _ F _ 1.545 L 9(xl tor variable faclor 01 salely 2.D}. [----1------;- 3~;3Y ~ 1·0 -a '" ..'" ..._.,c "" -- J fj, Ibl 0-" (e)

Essenfially same dislr'buhon for constant and variable (aclor or salery Elle<:live normal stress al base 01 shear surlace

Porewaler pressure ~2"'O!-:0'---~-4"'0'-::O'---~----::670"'O------::8~OO,,-----:1-;::O~OO::-"'-...... ,.1-:!20"'D=-- x- -x Projecled d,slance along shear surliieEl: II Ie)

Fig. 4. s.mple problem (reJefYolr ....ler elenriOQ, <480 fl) FACfOR OF SAFETY IN SLOPE STABILITY ANALYSIS 63

CASE STUDY shear surface geometry has the sharpest change in Figure 3(a) shows a part of an embankment direction. The computed factors of safety along dam cross-section and its foundation zones. The the shear surface are shown in Figs 3(1) and 4(1) shear strength properties of differen t materials in for the conditions analysed. The computed factor the embankment and its foundations are shown of safety in the 12· material for the drawdown in the table included in the figure. There was a condition is (·01, It is higher in other materials. slip failure at this site. As a result, the upstream The shear surface under the berm has a factor of berm was constructed and the facility put back: to safety of 3·33. For this sample problem, the use. results indicate that the local shear failure should Recently, during the reservoir drawdown, inter­ occur in the 12° material. Once this happens, nal sliding at the old slip surface has been movement should occur in the lr material which detected by the instrumentation; some cracking will cause tensile stresses in the materials above at the crest of the dam has occurred, but there the back rest portion of the shear surface. Thus, has not been a gross movement of the embank­ the tension cracks at the crest of the dam are a ment materials or a noticeable change in the consequence of internal failure in the 12° material embankment geometry. Thus, the problem needs and define the extent of the active wedge. Since to be analysed to seek answers to questions like the factor of safety along the shear surface under the berm area is substantially higher than 1,0, (a) is there a failure? gross movement along tbe shear surface could (b) where is the failure localized? not have occurred. Filling of the reservoir, (c) how safe is the upstream portion of the darn? without any repair work, should only improve (d) can the reservoir be filled without any major the stability of the embankment because of the rehabilitation work? bUllressing effect of the water. The shear surface geometry and the pore pressure data for use in this are shown in Fig. 3. CONCLUSIONS The slope stability analyses were performed The use of a variable factor of safety in slope using the computer program SSTA82 (Chugh, stability analysis of problems of practical impor­ 1981). This computer program is based on the tance is generally of considerable interest in limit equilibrium method; it satisfies all the statics seeking answers to questions that influence design equations; it provides for use of a constant or a decisions. The proposed procedure provides a variable interslice force inclination and a constant means to calculate variations in the factor of or a variable factor of safety aloog the slip surface safety along a shear surface within the framework according to the ideas presented in this Paper. of the limit equilibrium method. The choice of the The conventional slope stability analysis of the characteristic function g{x) defining the relative sample problem, using constant interslice force safety factor along a shear surface is a dimcult inclination and constant factor of assump­ issue and needs further study_ The proposed pro­ tions, yields F = 1-93 for high level steady state cedure of making g(x) equal to the ratio of shear reservoir operation and F = 1·55 for the stress to effective normal stress for purely fric­ reservoir drawdown. The normal stress, the shear tional materials is reasonable. The resulting shear stress and the ratio of shear stress to effective stress distribution along the shear surface is gen­ normal stress aJong the shear surface, as obtained erally smooth and more likely to occur in nature through these calculations, are shown in Figs than that implied by the constant factor of safety 3(c)-3(e) and 4(c)-4{e) for the high level steady assumption. state reservoir operation and reservoir drawdown REFERENCES conditions respectively, It is not possible to draw Bishop, A. W. (1967). Progressive failure-with special any inference of actual or impending distress reference to the mechanism causing it. Proc. Geo­ along the shear surface from these results of con­ technical COil! Oslo 2, !42-154. ventional slope stability calculations. Bishop, A. W. (t 971). The. influence of progressive A similar slope stability analysis of the sample failure on Ihe choic~ of the method of stability problem using the variable factor of safety ideas analysis. Geotechllique 21, No. 2, 168-172. presented in this Paper, all else being the same, Chowdhury, R. N. (1978). Slope analysis. New York: give the stress distributions shown superimposed Elsevier. Chugl!, A. K. (1981). User irifOr/1Ullion manual for slope on their counterpart results for the constant F ~rabilily arla/ysi, program SST A 82. D<::nver: Engin­ assumption. There is no appreciable difference in elering and Research Center, US Bureau of Recla­ the normal stress distribution, but the shear stress mation. distribution is much improved and more as Chugh, A. K. (1982) Slope stability a.naJysis for earth­ would be expcxted considering the shear surface quakes lnl. J. Nutria. Analyl. Merh. Geomech. 6, geometry. i.e. peak shear stress occurs where the No.3,307-322. 64 CHUGH

Chugh. A. K. (1984), Var;aoJe imer,lice jorce ;ndill(}lion of the stability of general slip surfaces. Geotechnique in -,lope stabi/iry analysis. Internal Report, Engineer­ 15, No. I, 79-93. ing and Research Center. Bureau of Reclamation, Sarma, S. K. (1973). Stability analysis of embankments Denver. and slopes. Geolechniqul! 23, No, 3, 423-433, Janbu, 1'1, 11973). Soil stability computations, In Spencer, E, (I967). A method of analysis of the stability Embankment darn engineering, Casagrande volume, of embankments assuming parallel interslice forces. pp, 47-87 (eds R, C. Hirschfeld and S, J. POlllos). Geocechnique 17, No. I, 1 J-26, New York: Wiley, Spencer, E. (1973), Thrust line criterion in embankment Morgenstern, N. R. & Price, V, E. (1965), The analysis. stability analysis. Geotechnique 23, No. 1,85-100,

Appendix D

Part 4 On the Boundary Conditions in Slope Stability Analysis by Ashok K. Chugh

This article was published in International Journal for Numerical and Analytical Methods in Geomechanics, Volume 27, pp. 905-926, 2003.

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS Int. J. Numer. Anal. Meth. Geomech., 2003; 27:905-926 (001: J0.1002jnag.305)

On the boundary conditions in slope stability analysis

Ashok K. Chugh*,t

u.s. Bureau of Reclamation, Denver, Colorado 80225, U.S.A.

SUMMARY Boundary conditions can affect computed factor of safety results in two- and three-dimensional stability analyses of slopes. Commonly used boundary conditions in two- and three-dimensional slope stability analyses via limit-equilibrium and continuum-mechanics based solution procedures are described. A sample problem is included to illustrate the importance of boundary conditions in slope stability analyses. The sample problem is solved using two- and three-dimensional numerical models commonly used in engineering practice. Copyright © 2003 John Wiley & Sons, Ltd.

KEY WORDS: boundary conditions; slope stability; two- and three-dimensional analysis

J. lNTRODUCTlON

Slope stability problems in geomechanics are boundary value problems and boundary conditions play an important role in the development of internal stresses in the medium and hence influence the calculated factor of safety (FoS). Some slope problems can be analysed adequately via two-dimensional (2-D) numerical models, while others require a three­ dimensional (3-D) model for a correct assessment of the slope performance. Use of appropriate boundary conditions is important in both 2-D and 3-D analyses. In geotechnical engineering practice, slope stability analysis is generally performed using 2-D and 3-D computer programs based on limit equilibrium method. 2-D analyses are more common than 3-D analyses, and the 2-D FoS results are generally considered to be conservative. Geotechnical literature is rich in limit-equilibrium-based 2-D analysis papers; however, 3-D analysis papers are relatively few [J-3]. References included in this paper are representative and not a complete list of works on the subject. Duncan [4] gives a current state-of-the-art in slope stability analysis and includes an extensive list of references on the subject. Arellano and Stark [3] used a commercially available limit-equilibrium-based computer program CLARA [5] to show that for a translational shear surface of sliding in 3-D introduction of shear resistance along the two sides of the slide mass that parallel the direction of movement can cause significant difference in the computed FoS values. Arellano and Stark [3] introduced approximations to include the shear resistance along the two sides of the slide mass to overcome some of the limitations with CLARA, and suggested use of a continuum-mechanics-based analysis procedure which provide an effective alternative means to solve slope stability problems in 2-D and 3-D [6-8].

*Correspondence to: A. K. Chugh, u.s. Bureau of Reclamation, Denver, Colorado 80225, U.S.A. t E-mail: [email protected]

Received 25 October 2002 Copyright © 2003 John Wiley & Sons, Ltd. Revised 10 March 2003 906 A. K. CHUGH

The objectives of this paper are to describe: (1) boundary conditions implicit in the limit­ equilibrium-based 2-D and 3-D solution procedures and (2) boundary conditions commonly used in continuum-mechanics-based 2-D and 3-D solution procedures. The significance of boundary conditions on the computed FoS is illustrated using one of the parametric slope model cases from Arellano and Stark [3] and solving it using commercially available continuum­ mechanics-based explicit finite difference computer programs FLA C [9] and FLAC3D [10] in 2-D and 3-D, respectively. The work reported was carried out as a sequel to Arellano and Stark [3]. It should be noted that the two methods of slope stability analysis referred to in this paper are: (J) the limit equilibrium method and (2) the continuum mechanics method. Within the limit equilibrium method, there are several procedures, e.g. Bishop, Janbu, Morgenstern-Price, Spencer among others (each makes different assumptions to render the problem statically determinate). Within the continuum mechanics method, finite difference, finite element, and boundary element are different procedures (each uses a different solution strategy). For ease of presentation, the nomenclature shown in Figure 1 is used in this paper. A slope is considered to lie in the xz plane and the width of the slope is in the y-direction. Displacements are expressed using the symbols u, v, w for the x, y and z faces, respectively. Stresses are expressed using the stress symbols shown in Figure 1. According to this convention, tension is positive, compression is negative, and the shear stresses shown in Figure 1 are positive.

2. CONCEPTUAL MODELS AND BOUNDARY CONDIT10NS

The conceptual model of a slope stability problem is different in the limit-equilibrium and continuum-mechanics methods. Boundary conditions need to be consistent with the conceptual model. Also, emphasis on prescribing boundary conditions is different in the two methods. Therefore, the conceptual model, model size, and boundary conditions for each method are descri bed separately.

2.1. Limit-equilibrium-based conceptual slope model Figure 2 shows a schematic of a slope model in both 2-D and 3-D. ln 2-D, the model of failure usually envisaged consists of a train of vertical blocks resting on a curved slip surface. These blocks are attached to each other and to the slip surface with a rigid-plastic glue which conforms to the Terzaghi-Coulomb shear strength criterion. The blocks themselves are considered to be rigid and their properties are not related to those of soil. It is in the nature of this model that no deflection occurs prior to failure and that when failure does take place all the blocks begin to slide slowly downwards together-without accelerating. Strictly, the model is applicable only if the radius of curvature of the slip surface is constant; variation in the radius would produce distortion in the blocks which are assumed to be rigid [11,12]. However, this limitation is commonly disregarded. ln 3-D, the 2-D conceptual model extends in the y-direction to the natural boundaries such as end-walls of the slide mass or abutments, and the 2-D vertical blocks become 3-D columns.

2.1.1. Model size and boundary conditions. ln limit-equilibrium-based 2-D and 3-D solution procedures, the size of the model needs to cover the slide mass including the shear surface but

Copyright © 2003 .fohn Wiley & Sons, Ltd. Inl. 1. Numer. Anal. Melh. Geomech. 2003; 27:905-926 BOUNDARY CONDITIONS IN SLOPE STABILITY ANALYSIS 907

Uzz 7:'zx

/yy 7:'xz

Uyy z 7:'xz

L X 7:'zx

Uzz (a) 2 - dimensional

Uzz

l:yz x

(b) 3 - dimensional Fignre 1. Description of state of stress on an element of soil (stresses shown are positive). any extension of the model past this requirement is only for the user's reference and convenience. Also, boundary conditions are built into the solution procedure, and the user is not required to specify them explicitly in a data file. The boundary conditions implicit in 2-D and 3-D solution procedures are as described below.

2.1.2. 2-D model boundary conditions. Figure 2(a) shows a sketch of a 2-D slope of a unit width in the y-direction and the slide mass divided into vertical slices. The boundary conditions apply at the head and at the toe of the slip surface, and at the two faces of the slope in the y-direction.

Copyright © 2003 .fohn Wiley & Sons, Ltd. Inl. 1. Numer. Anal. Melh. Geomech. 2003; 27:905-926 908 A. K. CHUGH

P d

A

Forces acting on a typical slice

(a) 2· dimensional

Direclion oj slide y x movement z~ ~ tu Typical 'I - 'Z section Forces aCling on 8 typical column

(b) 3 - dimensional Figure 2. Limit-equilibrium based slope models and boundary couditions.

The applicable boundary conditions are: (a) Applied force ZL and its location hI at the toe of the slip surface. (b) Applied force Zn. and its location h2 at the head of the slip surface. In the case of a tension crack with water, ZR is the force exerted by the water in the tension crack, and h2 is the loca[ioo of [he waler force. If [here is 00 water in the ten~ioD crack, Zn and h2 are zero, aDd the bottom of the tension crack becomes the end of the shear surface. The water force acts horizontally. (c) Plane strain in the y-direction. This implies the out-of-plane displacement, v, and shear stresses, r y.>: and r yz, on the y-faces are zero. However, the normal stress, (J yy, on the y-faces is D()t zero. Since equilibrium of forces and moments in the xz plane j~ ()f interest, this ()u{­ of-plane force is not considered in the equilibrium equations-however, it is presenl.

2.1.3. 3-D model boundary conditions. Figure 2(b) shows a sketch of a 3-D slope and the slide mass divided into vertical coJL.l1nn~. The boundary conditions (a) and (b) of the 2-D case are

Copyright © 2003 John Wiley & Sons, Ltd. 1m. J. Numer. ,4]1(1.1. Melh. Geomech. 2003; 27:905-926 BOUNDARY CONDITIONS IN SLOPE STABILITY ANALYSIS 909 extended in the y-direction and the boundary condition (c) of the 2-D case is applied to the end boundaries in the y-direction. However, in order to include the shear resistance along the parallel sides of the slide mass, Arellano and Stark [3] added an external horizontal and vertical side force equivalen t to the shear resistance due to the at-rest earth pressure acting on the vertical sides at the centroid of the two parallel sides in calculating 3-D FoS results using the computer program CLARA [5]. One of the objectives of this paper is to study the effects of different boundary conditions at the slope-abutment contact on the computed FoS using a continuum-mechanics-based 3-D analysis procedure.

2.2. Continuum-mechanics-based conceptual slope model Figure 3 shows schematics of a slope model in 2-D and 3-D. The conceptual model of failure in 2-D and 3-D is a deformable, bounded material body with Terzaghi-Coulomb yield strength. Under the action of gravity and externally applied loads, the material body deforms causing

---,r- Likely region of slope failure .­ -

.' H x - face ..--"

z ~ / \ x z-face / l,x _face L

(a) 2 - dimensional

y - face ,.­ 1----1- I-- x- face I----...- ~ --­I"\. - - \ Likely region of f,.---- .. -r-:_ slope failure 1 J / - --<- w ~ ­ ­ -- ~ ---­1 y z

x \ y - face '--z- face L

(b) 3 - dimensional

Material horizon lines Grid discretization Displacement boundary conditions apply at the X-, yo, and z- faces Figure 3. Continuum based slope models and boundary conditions.

Copyright © 2003 .fohn Wiley & Sons, Ltd. Inl. 1. Numer. Anal. Melh. Geomech. 2003; 27:905-926 910 A. K. CHUGH relative deformations that produce strains and stresses in the material body. If stresses at a location in the material body exceed the Terzaghi-Coulomb yield strength, the excess of stresses i~ ~hared by the neighbouring under-stressed locations. When stresses at a sufficient number of locations reach their yield strength, a mechanism is formed along which continuous movement or sliding occur~. Thu~, in a continuum model, a shear surface develops as a part of the solution, (lnd is always the one with the lowest FoS. The deformational behaviour of the bounded mass is controlled by the geometry of the medium, deformational properties of the materials, gravity and external loads, and boundary conditions. The FoS is determined using a strength redoction technique (13) in which the slope problem is solved repeatedly using reduced soil shear strength values until the numerical model becomes unstable (indicating slope failure), and the resulting FoS is the r(ltio of the soil's initial shear strength to the reduced shear strength at failure.

2.2.]. Model size and boundary conditions. For a continuum-mechanic~-based solution to be meaningful, the slope model needs to extend P(lst the location where slope failure is likely to occur. Also, all of the exterior of the slope model constitutes its boundary and boundary conditioos need to be expressed io tenns of applied forces or displacemeots in the input data. It should be mentioned that at anyone location on the body, either a displacement or a stress condition can be prescribed, but not both. In continuum-mechanics-based solurion procedureI', it is common to set every point free to displace and free of all stresses, and the user defines rhe non-zero stress and/or restrained displacement boundary conditions via input dara. A displacement rel\traint is either nil (completely free) or full (completely fixed)-partial restraintl\ are generally not allowed.

2.2.2. 2-D model boundary conditions. Figure 3(a) shows a sketch of a continuum model of a slope with a unit width in the y-direction and the material body divided into a grid. Every node in the grid including the boundaries has two degree~ of freedom, i.e. displacements u and win the x- and z-directioos, respectively. The commooly used boundary conditions are:

(a) No displacement in the x-direction at the ends of the slope model (u = 0 at x = 0 and at x = L). These boundaries are placed far enoogh from the region where slope failore i~ likely to occur. (b) No displacement at the base of the slope model (u = w = 0 at z = 0). This boundary is placed far enough from the region where slope failure is likely to occur. (c) Plane strain in the y-direction. This implies the out-of-plane displacement, v, and shear stressel\, '( >" and '()'Z, on the y-faces are zero. However> the normal I\tress, (Jyy, on the y-facel\ is not zero.

2.2.3. 3-D model boundary condifions. Figure 3(b) shows a sketch of a continuum model of a slope in 3-D and the material body divided into a grid. Every node in the grid including the boundaries has three degrees of freedom, i.e. displacemenr u, v, and w in the x-, y-, and z­ directions, respectively; and a user can constrain any or all components of displacement at any location in the model including the boundaries. The commonly used boundary condition~ are: (a) No out-of-plane displacement in the x-direction at rhe model ends (u = 0 on the end yz planes in the x-direction). These boundaries are placed far enough from the region where I\lope failure is likely to occur.

Copyright © 2003 John Wiley & Sons, Ll.d. till. J. Numel'. Allo/. Me/h. Geomech. 2003; 27:905-926 BOUNDARY CONDITIONS IN SLOPE STABILITY ANALYSIS 911

(b) No displacement at the base of the slope model (u = v = w = 0 for the xy plane at z = 0). This boundary is placed far enough from the region where slope failure is likely to occur. (c) Displacement constraints in the y-direction at the model ends (v = 0 or u = v = IV = 0 for the xz planes at y = 0 and y = Ware the commonly prescribed conditions). The di~placement v = 0 boundary condition i~ u~ed to represent a contact with a rigid, smooth abutment chat can provide a reacting thrust but no in-plane shear restraint. The displacement boundary condition u = v = HI = 0 is used to represent a rigid concact with no po~sibility of movement.

2.3. Appropriate 3-D model and boundary conditions The option of explicitly specifying boundary conditions is available in continuum-mechanic~­ based solution procedures. Selection of appropriate boundary conditions for a slope problem should be derived from the field conditioos being analysed. For example, the boundary conditions at che y-faces for ala boratory model of a slope builc in a wooden concainer with glass walls are different than the boundary conditions at the y-faces for a slope with rock or soil abutments commonly encountered in the field. The boundary conditions described for the x- and z-directions of a 3-D model are appropriate so long as the boundaries are placed far enough away from the region where slope failure is likely to occur. The following recommendations are suggested for establishing the boundary conditioos in the y-directioo of a 3-D model: (I) Exteod the 3-D cootinuum model past the ends of the slope to include the presence of abutments. (2) Place the model boundary conditions at the ends of the exteoded continuum model. (3) Introduce interfaces between the slope and the abutmeots to allow for relative movements at the slope-abutment contact. (4) Use the displacement condition of u = 0, v = 0, and HI = 0 at the extended model boundarie~.

3. SAMPLE PROBLEM

Figure 4 shows the parametric slope model of AreJlano and Stark (3). Arellano and Stark [3) used three slope inclinations (If!: 1V; 3H: 1V; and 5H: 1V); for each slope inclination, seven width (W) to height (N) ratios (WIff = 1, ] .5,2,4,6,8, and ]0); and for each pair of the slope inclination and WIH ratio, four combinations of !.f>lJPperlf.{Jlower values (f.{JlJPperlf.{Jlowel' = 1,1.5,3, and 3.75). The 5H: 1V slope was selected to illustrate the effects of boundary condition~ on computed factor of safety results for the four combinations of

Copyright © 2003 John Wiley & Sons, Ll.d. fill. J. Numel'. Allo/. Me/h. Geomech. 2003; 27:905-926 912 A. K. CHUGH

Shear Resisti'lI"lCII Along the Scarp

//////

L Shear Rasistance ~DirecjiOn Along Verlical Side / LVertical Si e

3D View of Slope Model

x !

l

I y~ ..L L­ -.J

A' ,-:w------1

Plilllview Cross-section A·A' Figure 4. Sample problem showing details of tlle parametric slope model used by Arellano and Stark [3J (reproduced by pel11lission of the publisher, ASCE). comparison purposes, the problem was also solved using a limit-equilibrium-based 2-D analysis procedure SSTAB2 [14] which implements Spencer's procedure [15J. The material properties that were used to perfonn the FLAC, FLAC3D, and SSTAB2 analyses are shown in Table 1. In all of these analyses, the following slope geometry parameters were used: height (H) = 10 m. length (L) = 58.8 m and width (W) = 1001. In the continuum models of the sample problem, the geometric space covered is 176.4 m (3 times the slope length L) in the x--direction, 20 m (2 times the slope height H) in the z--direction, and 10 m (l time the slope width W) or 22 m (slope width W plus two 6-m wide end blocks for abutments) in the y-direction. For WIH = 2, 5, and 10, the continuum model included two 6-m wide end blocks for abutments and the model width in the y-direction was 32, 62, and 112 m, respectively.

3. J. ArellanD and SlIlrk resulrs The 2-D and 3-D values oC FoS calculated using CLARA without applying the Arellano and Stark [3) modification are shown in Table II. Also included in Table II are the 3-D values of FoS calculated using CLARA with the Arellano and Stark [3) modification [or WIH = I, 2, 5,

Copyrigh.t lS> 2003 Joh.n Wiley & Sons. Lid. Inl. J. ,1I,'lImel'. A1111/. Me/h. Geomech. 2003; 27:905-926 BOUNDARY CONDITIONS IN SLOPE STABILITY ANALYSIS 913

Table 1. Material properties for stability analyses of the sample Problem. Material Density Material strength Elastic constants

p (kgJm 3) c (Pa)

J Unil \Vc:igh! )1 (Njm ) = Densily x 9.81; N/A-not applicable:.

Table II. FoS reslllts" for the sample problem from Arellaoo and Stark (31.

WJH= J WJH=2 WJH= 5 WJH= 10 8° 0.90 0.90 2.85 1045 1.05 1.00 10° 1.00 1.00 3.18 1.63 1.23 1.13 20° 1.70 1.70 4.80 2.58 2.00 1.82 30° 2.50 2.50 6.58 3.58 2.85 2.57

*Tbe FoS values given are scaled from (be graphical presentation of resullS in Arellano and Stark: (3).

3.2. SSTAB2 results

The FoS and interslice force ioclination, (5, results from SSTAB2 for the four valueI' of c,olower are given in Table Ill. The shear surfaces used in the SSTAB2 analyses were [he same as those used by Arellano and Stark [3J.

3.3. FLAC results Figure S f;hows the FLAC 2-D model of the f;lope with end extemions in the x- and z-directions, the W are shown in Table III. The FLAC values of PoS are in general agreement with those from SSTAB2 and CLARA 2-D (Table IT). Figure 6 shows contour plots of maximum shear strain rate and the velocity vectors at the instant of numerical instability for each of the values of c,olower analysed. The maximum shear straio rate and velocity vector plots are helpful in identifying the location and shape of failure surface. However, in Figure 6, the shear strain rate plots are hidden from view because of the superimposed velocity vector plots; the location and geometry of the associated f;hear surface for each case is takeD to be along (he

Copyrighl © 2003 John Wiley & Sons, LL.d. fill. J. Nume,.. Allo/. Melh. Geomech. 2003; 27:905-926 914 A. K. CHUGH

Table III. FLAC (Figure 5) and SSTAB2 FoS results for the sample problem. FLAC 2-D SSTAB22-D FoS 0.85 0.78 12.55 l.0] 0.93 12.69 l.64 1.62 13.02 2.22 2.37 !3. !3

Table IV. FLACJD model No.1 (Figure 7) FoS results for the following boundary conditions (W /H = I). »ound:ny cOllstraint(s) used at the y-faces of the oumerical model Fix y Fix x, y Fix x,y,z 0.89 1.42 IAI 1.04 1.57 1.57 1.71 2.26 2.26 2.32 2.82 2.82

z Material group 50 bottom lower upper g 30 Water Table J: 1: .Q> 4l J: 10

·10

10 so 70 00 110 130 150 170 Modellenglh (m) Figure 5. FLAC model for the sample problem. path where the velocity vectors essentially vanish as it marks the boundary between stable and unstable portions of the deposit.

3.4. FLAC3D model No. J resulls In this model, the width W is 10 m in the y-direction. Figure 7 shows the FLAC3D model of the slope with end extensions in the xz plane, the water surface, the viewing information, and the boundary conditions used. For each of the values of If!lowe,, three sets of analyses were conducted [or W/ H = I using the boundary conditions of v = 0 (fix y); u = v = 0 (fix x, y); and u = L' = W = 0 (fix x, y, z) at the ends of the slope in the y-direction. Application of each of these y-direction boundary conditions represents a particular condition: (a) the v = 0 (fix y) boundary

Copyrigh.t lS> 2003 .loh.n Wiley & Sons, Lid. Inl. J. ,1I,'lImel'. ,11111/. Me/h. Geomech. 2003; 27:905-926 () 0 .", ~ lower =e" 4> lower =10" ':Si ';;' Max, S e1l{ slml/;"l-[:J:i8 Max sl'.e[lol[ .o:;~rarn-r..;ltl~ 90 O,OOelOO 90 0.00E~00 ~ 2.0 (;: OJ 5.00E-07 la~ d,OO" OT 1.00E-Oo Co 70 0,0 12·07 70 IS01;;--06 0'" w a,ooE-07 2.0llE-Qe '­ 1,001,,:-06 2,5ol;-oo 0 0r­ 50 Veloc,ly veClolS 50 2.00<::-00 =' Max Vecto .::. 2...r~ 17"E-OG ~ VelO<'Oy vecto •. '"0 n Max V(:eO~or := 7 B5~)E"{:S '< 30 30 c: fi;o a ;.­ 0 ""=' 10 10 ::c 1" -< r' () ~ 0 -10 Z 10 30 50 70 00 110 130 150 170 10 30 50 70 90 110 130 150 170 0 =i (a> FoS =0.85 (b> FoS =1.01 0 Z 1/J Z • lower =20° 'I> lower =30" 1/J M~K shear 'fj'Ha1n·rel·E r Ma~ .Gfleiflr strein-mla 0 ~ 90 o ooE...oo 90 o OOEtOO "'0 1.00E--os 2.50E-07 tT1 ~ 5.00E-07 1/J 2-00E-os -l ~ 70 3.C0E·OS 70 7 SOE'OJ ;.­ ·'C0E S 100E·OO CD ~ 5.00E-oS ' 25E·06 r 1.5010·06 ::". 50 GooE-os 50 =i 700E·OS I 7SE-oe -< ~ eooE--os 2.COE-06 ;.­ 2.25E·06 Z :t 30 Ve ..c:il\! VGctoiS 30 ;.­ "~ Max Vector -:- 1 ~2~·O.1 Valoc,ly veelc", r Ma>; V FoS =1.64 (d) FoS =2.22 '6'" 0 ,~ I Fignre 6. FLAC model rcsults for thc four valucs of {flo,",'." '-0 !-' ..... ""0­ V, 916 A. K. CHUGH

Male/ial Group bottom !ower I upper Water Surface

Rotation: X: jO.OOO Y: 0.000 2:300.000 Boundary condition varied Mag.: 1 on thili y-face Ans-: 22.500

Fix.on / this .·face (u = 0)

Fi•• on Ihis .-Iace (ll =0)

Figure 7. FLAC3D model No. j for the sample problem (W /H = I). condition is similar to the implicit boundary condition in CLARA 3D; (b) the u = v = 0 (fix x, y) boundary condition should model the conditions imposed by the modification proposed by Arellano and Stark [3] to include side shear resistance; and (c) the u = v = IV = 0 (fix x, y, z) boundary condition is used commonly in practice. The FoS values for each of the assigned boundary conditions in the y-direction for the four values of ~~Io\\'cr are shown in Table IV. FLAC3D model No. 1 results are compared with Arellano and Stark [3] values of FoS (Table II). For the fixed J' boundary conditions, FLAC3D FoS values (Table lV) are similar to those from CLARA 3D shown in Table II. For the fixed x, y boundary conditions, FLAC3D FoS values differ from those of Arellano and Stark [3] 3-D shown in Table JI for W jH = 1. For the fixed x, y,z boundary conditions, FLAC3D FoS values are similar to the FLAC3D values for the fixed x, y boundary conditions shown in Table IV. However, this does not mean that the two sets of boundary conditions (fixed x, y and fixed x, y, z) are the same or that they will always lead to the same results. Figure 8 shows contour plots of maximum shear strain rate and the velocity vectors at the instant of numerical instability for each of the values of 010wcr and the three boundary

Fig.me 8. FLAC3D model No. I results for the sample problem with IV jH = I and the boundary condition of l' = 0 (fix y) in Ihe y-direction: (a) FoS = 0.89, (b) FoS = 1.04, (c) FoS = 1.71 and (d) FoS = 2.32. FLAC3D model No. j results for the sample problem with W/H = I and the boundary condition of 11= lJ = 0 (fix .1:, y) in the y-direction: (e) FoS = ].42, (I) FoS = 1.57, (g) FoS = 2.26 and (h) FoS = 2.82. FLAC3D model No. I results for the sample problem with W/H = I ~nd the boundary condition of II = /' = W = 0 (fix x, Y,:) in the y-direction: (i) FoS = 1.41, (j) FoS = 1.57, (k) FoS = 2.26 and (l) FoS = 2.82.

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'"~ 0 •CoOlll1ll of SlQr SnLo IQIO •CaroOllf of S"-SIniA lUte ." ~ ..~-o~ __ ~~o.OIll)o·OOOO r"I "'~ GnIdonI~ ~ O,.. a.oo.s·~OlO ~_ ~'IO 1._·006 -l 1._1II:t.~ ~ Z.~IO~ "" 2.0000-~ III ;).QIlllOeo(lll5 > ~ '.~tl:>.,_ c= 4._10'_ :a.ClllOO*-4QoIO .,~ ;:;; ,.~ 6.00CCo- I~"",,~ 'Z ~ '.I~",- t.~ .. "_ 1'-11>'_ •. 1_1O"~ > '~,,',Al)ol'o-¢OI 1_"",_ ~ ?­'" 111lClMlI .. ,lIIHlCll l.3OCJClo.(lG.tall._ -< 1.4OllCiooOO"lI> 1._ ~ Velacilr -­ 1nI...... I,_ ~ Vdodly __ .. --~ ...... -t._ "" ~ ;>­ ~ ;-,> ~ ~ '-" I \.::> '<:> ,... c-. -J .low8' _ .Iowef _ 10° l) eO '0 Col>Jlw orSl=t S1D1Il Rolle Collll>W' Dr Sb<:llr SIDiD Rau: 0 loII(Ia< - 7~ 4._., •._ 7--=11> '..IlCIXIIO.(J)< e.t.I:IoDoHIaS 10 Il.llOOIl&oOOS CO' 1~"''- 8.00000.(06 '" '.(lOOOe.(lDO '.2S£lQO.(I)( II> 1~ ',_",,~ ~ '_",1.~ 1~ 10 1.40000-004 0 '.71QOo000< II> :uxDle-OOO 1,_10''­ '-' 1.~IOI.~ 2JIXlOO.OOo( "' ~ ,_= 2.1llt4:l5 '­ ~ID~ 0 :z.-IO 2.-.004 go 1nl_.~._ Vdoclly -­ Vdocil}' -­ ...... -5..1",lHJO( 5­ Mmn.m ­ a.t7(.,~ ~ '"'"P.o Ul 0 :0 !" r­ ~

Ie) FoS =1.42 (0 FoS = 1.57 > r- l) ~ lower _ 20° • /OWl!f ~ 3IJ" :I: e-oarof Sb<:at SInin R.>/l: e.:.u..ur or Shcu Slh.Io R..., C Cl ~ ~ - OJl((le.oOOO ~ -1UlOOo-ltlOO 0.-.CaIa.IIolD'> O_~Ir:IJa''''' :I: .... -ll.4.-..oDl to 2.11C1OOlO-OO5 ·1.1lQlI5&.OOIl1l 6.ClCIli:Il>-Q:lG 2.Q0000-0Q51O ~ 6~101~ '.0C0Q0.005 CD 6.0lIlX\04lll 1.0Cllll»000!I,., 1.!llllJ:l8.OO5 3: ',00000.00510 e.oOOO,,-

0 0"" ~... ;.c'"' 0 '-" I (g) FoS" 2.26 (h) FoS = 2.82 ,'<:>... c>­ Figmc 8. (Col)linucd) BOUNDAR'i CONDITIONS IN SLOPESTABILlTY ANALYSIS 919

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ROblllon: Boundary condition varied X: 10.000 ¥: 0.000 on this y·faoe Z: 300.000 Mag.: 1 Ang..: 22.500 :;. Fixxoo r this x-lace n (u '" 0) :I: c: a :I:

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Rxxon this x-face (u =0)

Figl1r~ Q. FLAC3D modd NO.2 ror th~ sample: proble:ffi (WjH = I). BOUNDARY CONDITIONS IN SLOPE STABILITY ANALYSIS 921 conditions analysed; the viewing infonnation for the 3-D model is the same as used in Figure 7. As mentioned for the FLAC results, the location and geometry of the associated shear surface for each case is interpreted to be along the path where the velocity vectors essentially vanish.

3.5. FLAC3D model No.2 results

Figure 9 shows the suggested 3-D model for the slope. It has the same end extensions in the xz plane as in the FLAC3D model No.1 (Figure 1); in addition, in the y-direction, it has two end blocks to represent abutments, and the contacts between the slope and abutments are represented by interfaces. The viewing infonnation for the 3-D model, the water surface, and the boundary conditions used are shown in Figure 9. Three sets of analyses were conducted for WI H = 1 using the boundary conditions of v = 0 (fix y); u = v = 0 (fix x, y); and u = v = )II = 0 (fix x, y, z) at the far ends of the abutments in the y-direction. The conditions represented by each of the y-direction boundary condition are the same as described for the FLAC3D model No. 1. The property values for the end blocks and interfaces used to perform the FLA C3D analyses are included in Table I. FLAC3D FoS values for this model are given in Table V. Figure 10 shows the contour plots of maximum shear strain rate and the velocity vectors at the instant of numerical instability for each of the four values of !Plower and the three boundary conditions analysed; the viewing infonnation for the 3-D model is the same as used in Figure 9. As mentioned before, the location and geometry of the associated shear surface for each case is interpreted to be along the path where the velocity vectors essentially vanish.

3.6. Additional FLAC3D model No.2 results The FLA C3D model No.2 was also used to analyse the sample problem for WI J-J = 2, 5, and 10 for the boundary condition of u = v = )II = 0 (fix x, y, z) at the far ends of the abutments in the y-direction. The FLAC3D FoS results for all four values of WIH are shown in Table VI.

Table V. FLAC3D model No.2 (Figure 9) FoS results for the following boundary conditions (W /H = I).

Table Vl. FLA C3D model No.2 (Figure 9) FoS results for tbe u = v = w = 0 (fix x, y, z) boundary condition at tbe y-faces of the numerical model for the following W/H values.

Copyright © 2003 .fohn Wiley & Sons, Ltd. Inl. 1. Numer. Anal. Melh. Geomech. 2003; 27:905-926 I) 0 "'0 ~ lA' ~ a IV 0 0 w '­ 0 CT "-< ;:; '< R.o V> 0 :3 .Y r­ <>.

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(d)

Figure 10. FLACJD model No. 2 r~sul1s for th<,- sample probkm with W/ H = I and the boundary condition of I' = 0 (tix y) in (he y-

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(,) FoS =1.74 (j) FoS • 1.8Q )­ 7­ t) 0r .2Qr. o'OW6r= jO' • S~ :r e-tolSIal sm!IllW< o...c.u oJSbaoi 11.0.. ,...... ~ """"""-~ C ~~ ..-­ :r ~.,~ ~ 4..~~4~ ~'~'tOI,~ ,_",t.~.. ~",1._'­ < r=::::t= § '~'CI~ '~"'l~ t=::~:~~"'~-- ~ ,.AOCOf.(:6t ~ I ~ '~~,A~ ~l:t) • ..il~ I_·~ _-5._ ~ vdoc:jry -­ vclodly-­ ~ _-3-~ --~ oS:. ~ ~ C') '" 5'" ~ 0:­ N 0 (I) FoS 3.01 <:> (X> = ;,.> ... Figure 10. (ConliOll<:d) \b .....<:> I ,-'<.D 0> BOUNDARY CONDITIONS IN SLOPE STABILITY ANALYSIS 925

4. COMMENTS ON ANALYSIS RESULTS

For WIH ratio greater than 5, the differences between 2-D FoS and 3-D FoS values tend to lose significance (Tables II, III and VI), i.e. the 2-D FoS results approximate the 3-D FoS results reasonably well. For WI H ratio less than 5, the differences between 2-D and 3-D FoS values are significant. For the 3-D FoS results, the choice of an acceptable answer depends on the physical conditions being analysed via the numerical model. For a laboratory model with smooth but rigid walls, the fixed y boundary condition seems appropriate, and the FoS values from CLARA 3D (Table II) and FLAC3D (Table IV) are about the same. However, for field conditions where the end walls are rigid and rough abutments, U = v = w = 0 (fix x,y,z) boundary conditions are more appropriate, and FLAC3D results (Tables IV and V) will be more reflective of the slope behaviour. Between FLAC3D model No.1 and FLAC3D model No.2, FLAC3D model No.2 is more representative of field conditions; therefore, use of FLAC3D model No.2 results in Table VI should be appropriate.

5. SUMMARY AND CONCLUSIONS

(1) Limit-equilibrium-based slope stability analysis procedures use different assumptions to render slope stability problems statically determinate. Some solution procedures only satisfy moment equilibrium or force equilibrium equations or statics, while others satisfy both Corce and moment equilibrium requirements of statics. For 2-D FoS calculations, solution procedures that satisfy complete statics, e.g. Spencer [J 5], are usually used in dam engineering practice. A similar trend is observed in the development of 3-D FoS solution procedures [16], and their use in engineering practice shall follow the advisory for 2-D procedures. In either case, it is essential that the user of these procedures and corresponding software understand the theories, assumptions, and calculations implemented. (2) If spatial variations of geometry, pore-water pressure, and/or material properties indicate that 3-D effects may be significant, it is suggested that the problem be analysed using a 3-D analysis sortware. (3) In a 3-D slope stability analysis, contribution of shear resistance along the two sides of a slide mass that parallel the direction of movement to the FoS is an item of interest. (4) It is easier to visualize model displacement boundary conditions than stress boundary conditions from the physical boundaries of a slope problem. However, for stress boundary conditions, one needs to think through the state of stress at a point, especially the shear stresses which are complimentary and exist in pairs. Inconsistencies in stress boundary conditions can lead to unpleasant consequences in computed FoS results. In continuum­ mechanics-based solution procedures, use of displacement boundary conditions is recommended. (5) Initial stresses in a continuum model can be introduced via applied loads, boundary displacements, or by specifying their values. Any consistent state of stress can be present in the continuum body. However, the model must be in equilibrium under the applied loads, initial stresses, and boundary conditions. (6) In 3-D analyses, the failure surface geometry and location shall likely be different at different sections in the y-direction.

Copyright © 2003 John Wiley & Sons, Ltd. Inl. J. Nume,.. .11101. Melh. Geomech. 2003; 27:905-926 926 A. K. CHUGH

ACKNOWLEDGEMENTS

The author would like to express his sincere thanks to Professors D. Vaughan Griffiths, o. Hungr, and Timothy D. Stark for their interest and several conversations on the subject matter of the paper. REFERENCES I. Hovland HJ. Three-dimensional slope st2bility analysis method. Journal of the Geotechnical Engineering Division, ASCE, 1977; 103(GT9):971-986. 2. Stark TD, Eid HT. Performance of three-dimensional slope stability methods in practice. Journal ofGeotechnical and Geoenvironmental Engineering 1998; 124(11): 1049-1060. 3. Arellano D, Stark TD. Importance of three-dimensional slope stability analysis in practice. In ASCE Geotechnical Special Publication No. /0/: Slope Stability 2000, Griffiths DV, Fenton GA, Martin TR (eds). ASCE: New York, 2000; 18-32. 4. Duncan JM. State of the art: limit equilibrium and 6nite-element analysis of slopes. Journal of Geotechnical Engineering, ASCE, 1996; 122(7):577-596. 5. Hungr O. CLARA: Slope stability analysis in two or three dimensions for IBM or compatible microcomputers. O. Hungr Geotechnical Research Inc.: West Vancouver, British Columbia, 1988. 6. Griffiths DV, Lane PA. Slope stability analysis by finite elements. Geotechnique 1999; 49(3):387-403. 7. Zetter AH, Poisel R, Roth W, Preh A. Slope stability analysis based on the shear reduction technique in 3D. In Flac and Numerical Modeling in Geomechanics, Detournay C, Hart R. (eds). Balkema: Rotterdam, 1999; 11-21. 8. Dawson EM, Roth WH, Drescher A. Slope stability analysis by strength reduction. Geotechnique 1999; 49(6): 835--840. 9. Itasca Consulting Group. FLAC-Fast Lagrangian Analysis of continua. Itasca Consulting Group: Minneapolis, Minnesota, 2000. 10. Itasca Consulting Group. FLAC3D-Fast Lagrangian Analysis ofcontinua in3 dimensions. Itasca Consulting Group: Minneapolis, Minnesota, 2002. II. Spencer E. Personal communications, 1984. 12. Peck RB. Personal communications, 1989. 13. Zienkiewicz OC, Humpheson C, Lewis RW. Associated and non-associated visco-plasticity and plasticity in soil mechanics. Geotechnique 1975; 25(4):671-689. 14. Chugh AK. User information manual for slope stability analysis program SSTAB2. U.S. Bureau of Reclamation: Denver, Colorado, 1992. 15. Spencer E. A method of analysis of the stability of embankments assuming parallel interslice forces. Geotechnique 1967; 17(I):J 1-26. 16. Huang C-C, Tsai C-C, Chen Y-H. Generalized method for three-dimensional slope stability analysis. Journal of Geotechnical and Geoenvironmenlal Engineering 2002; 128(10):836-848.

Copyright © 2003 John Wiley & Sons, Ltd. Int. 1. Numer. Anal. Meth. Geomech. 2003; 27:905-926

Appendix D

Part 5 An Automated Procedure for 3-Dimensional Mesh Generation by A.K. Chugh and T.D. Stark

This article was published in Proceedings of the 3rd International Symposium on FLAC and Numerical Modeling in Geomechanics, Sudbury, Ontario, pp. 9-15, 2003.

An auton1ated procedure for 3-din1ensional n1esh generation

Ashok K. Chugh Bureau ofReclamation, Denver, CO 80225-0007, USA Timothy D. Stark University ofIllinois, Urbana, IL 61801-2352, USA

ABSTRACT: An automated procedure is presented to generate a 3-dimensional mesh for numerical analysis of engineering problems. The procedure is simple, effective and efficient, and can be applied to represent complex geometries and material distributions. A listing of the program that was used for the sample problem of a landfill slide is included.

1 INRODUCTION geologists in constructing visual models of complex geologic sites where a number of 2-D cross-sections One of the essential tasks in a 3-dimensional (3-D) are used to represent the field conditions. In these numerical analysis is to represent the geometry and representations, linear variations between material distribution of materials in the numerical model. horizons in consecutive 2-D cross-sections are used FLAC3D provides means to facilitate mesh to depict the 3-D spatial variability of a site. The generation and the built-in programming language accuracy of the representation is improved by using FISH can be used to develop and implement closely spaced 2-D cross-sections. additional program instructions during execution of The 3-D mesh generation procedure presented a data file. herein follows the conventional practices used by In geotechnical engineering, surface geometry, engineers in constructing 2-D numerical meshes by distribution of materials, and water table conditions hand for geotechnical problems to be solved using usually vary from one location to the next and pose a methods other than FLAC3D. For example, in the difficult set of conditions to represent in a numerical creation of a 2-D numerical model of a slope to be model. In order to facilitate the analysis of analyzed using a limit-equilibrium based procedure, landslides, a simple procedure was devised to it is a common practice to define profile lines via a represent complex surface geometry, subsurface set of data points followed by specifications of their material horizons, and water table conditions. The connectivities. Also, in the creation of a 2-D model objectives of this paper are to present: (1) a simple of a continuum to be solved by a finite-element method to describe field geometry and conditions based procedure, it is a common practice to for a 3-D numerical model of a slope problem; (2) a discretize the continuum into a network of zones; simple procedure for automatic generation of a 3-D assign identification numbers to the grid points; mesh; and (3) an illustration of the use of the define the coordinates of the grid points; and then procedure for analysis of a large slide in a landfill. A specify the connectivity of grid points. listing of the program for the landfill slide is Thus, in the conceptual model for the generation included in the paper. This program listing is in the of a 3-D mesh in FLAC3D, use is made of defining a FISH language and uses some of the functions series of 2-D cross-sections at representative available in the FISH library. locations of a site; defining each of the 2-D sections as an assemblage of data points with line-segment connections; and organizing the data for an efficient 2 CONCEPTUAL MODEL and effective discretization of the volume.

The conceptual model for the generation of a 3-D mesh follows the conventional procedure of 3 WATER TABLE portraying spatial variations of materials in 3-D via a series of 2-dimensional (2-D) cross-sections. This The water table surface is specified using the water technique is commonly used by engineers and table data of individual 2-D cross-sections and through the use of 3-point planar polygons between points that are of significance in defining the consecutive 2-D cross-sections. This scheme allows profile lines in all of the 2-D cross-sections. incorporation of non-coplanar variations in the water Tabulate the x-coordinates of these control table surface in the entire 3-D model. points in increasing order. For reference purposes, this table is referred to as Table 100. (b) Use of the 'Interpolate' function expands the 4 DESCRIPTION OF THE PROCEDURE 2-D cross-sectional data of step 1(c) by linear interpolation for all of the control points listed in In geotechnical engineering, the ground-surface Table 100 for all of the profile lines and stores geometry is obtained using contour maps that are the data in separate tables; assigns Table prepared from land or aerial survey of the area. The numbers in increasing order starting with the subsurface material horizons are estimated from user specified starting number and incrementing geologic data and information obtained from it by 1; assigns an identification number to each exploratory boring logs. The subsurface water point; and positions the points in the 3-D model conditions are estimated from field observations, space. These tables contain the (x,z) coordinates piezometers installed at various depths, and/or from of expanded 2-D cross-sectional data. A sample water levels in borings. Subsurface data are used to listing of the 'Interpolate' function and its develop contour maps of the subsurface geology and dependency function 'zz' in FISH language is water conditions. given in Figure 1. The starting table number From these contour maps, the region-of-interest, used in the sample problem data file is 200. and the locations of significant cross-sections are identified; information for 2-D cross-sections are 3. The following steps are used for creating zones read and tabulated; and 2-D cross-sections are drawn in the 3-D model space: for an understanding of the site details and preparation of input data for a 2-D analysis. In (a) Tabulate the y-coordinates of the 2-D cross­ general, the cross-sectional data for a site varies sections in increasing y-direction. For reference from one location to the next - these variations may purposes, this table is referred to as Table 101. be caused by changes in the ground surface and (or) The number of entries in Table 101 should equal in subsurface material horizons, discontinuity of the number of 2-D cross-sections marked in step some materials, or a combination of these or some I(b). other variations. (b) Considering the spacing of x-coordinates of the In the proposed procedure, the following steps are control points in step 2(a), select the number of followed: (For ease of presentation, 2-D cross­ zones desired for each interval in the x-direction. sections are assumed to lie in x-z plane and the x,y,z Tabulate these values for all of the intervals in coordinate system follow the right hand rule.) the increasing x-direction. For reference purposes, this table is referred to as Table 102. 1. The following steps are used for creating an The number of entries in Table 102 should be orderly assemblage of field data for 3-D one less than those in Table 100. discretization of the continuum of the region-of­ (c) Considering the spacing between the 2-D cross­ interest: sections in the y-direction, select the number of zones desired for each interval in the y-direction. (a) On the site map, select values of x, y, and z Tabulate these values for all of the intervals in coordinates that completely circumscribe the 3-D the increasing y-direction. For reference region-of-interest; purposes, this table is referred to as Table 103. (b) Mark locations of all significant 2-D cross­ The number of entries in Table 103 should be sections oriented in the same and preferably one less than the number of 2-D cross-sections. parallel direction; (d) Considering the spacing of the profile lines in the (c) For each 2-D cross-section, tabulate (x,y,z) z-direction, select the number of zones desired coordinates of end-points of all line segments for for each material horizon in the z-direction. each profile line and the water table (for parallel Tabulate these values for all of the intervals in 2-D cross-sections, y-coordinate shall have same the increasing z-direction. For reference constant value between two consecutive cross­ purposes, this table is referred to as Table 104. sections). The number of entries in Table 104 should be one less than the number of profile lines. 2. The following steps are used for creating similar (e) Use of the 'Fill_grid' function generates a brick sets of data at each ofthe 2-D cross-sections: mesh and assigns a group name to each 3-D volume zone. A sample listing of the 'Fill_grid' (a) From the data in step I(c) above, select control function in FISH language is given in Figure 2. def zz 5 COMMENTS zz~table(t n,xx) end (1) Use of a Brick mesh with an 8-point description def interpolate loop j (js, je); profile line #s ­ is versatile and allows for creation of ; js is for the bottom, je is for top degenerated brick forms through the use of dt_n~dt_n_s+j; dt_n is destination table number loop i (is,ie); is is the first interpolation #, multiple points with different identification ; ie is the last interpolation # numbers occupying the same (x,y,z) coordinate xx~xtable (lOO,i); x-coordinate of the location in the 3-D model space. ;interpolation point command (2) During the development of the grid, it is possible set t_n~j to assign group names to different segments of end command table(dt_n,xx)~zz the model. This information can be useful in id_pt~id_pt+l modifying the generated grid. x_pt~xtable(dt_n,i) y_pt~y_pt (3) Expanding the (x,y,z) location data for all 2-D z_pt~ytable(dt_n,i) cross-sections to a common control number of command locations via interpolations facilitates the generate point id id_pt x_pt y_pt z_pt end command programming of the automatic grid-generation endloop procedure. endloop end (4) In engineering practice, it is generally desirable to analyze a few 2-D cross-sections at select Figure 1. Listing of the 'Interpolate' function and its locations prior to conducting a 3-D analysis. dependency function 'zz' in FISH language. Because development of data for 2-D cross­ sections is one of the steps for use of the def fill_grid i_n~table_size(102) proposed procedure, it is relatively easy to j_n~table_size(103) conduct a 2-D analysis using the 2-D cross­ k_n~table_size(104) loop jy (l,j_n) sectional data and the program FLAC. ny~xtable(103,jy) (5) The program instructions listed in Figures 1 and pO_d~(jy-l)*(i_n+l)*(k_n+l) loop kz (l,k_n) 2 can be modified to accommodate geometry nz~xtable(104,kz) and other problem details that are different or if kz~l then more complex than those encountered in the material~'shale' endif sample problem described in Section 6. if kz~2 then material='ns'; gative soil endif if kz~3 then 6 SAMPLE PROBLEM material='msw'; ~unicipal ~olid ~aste x_toe~xtable(105,jy) endif The problem used to illustrate the proposed 3-D loop ix (l,i_n) mesh generation procedure is the 1996 slide in a if kz~3 then xx_toe~xtable(lOO,ix) waste containment facility near Cincinnati, Ohio if xx toe < x toe then - - (Stark & Eid 1998, Eid et al. 2000). Figure 3 is an material='mswt' endif aerial view of the slide. Figure 4 is the plan view of endif the landfill and shows the location of the sixteen nx~xtable(102,ix) pO_d~pO_d+l cross-sections used to construct a FLAC3D model of p3_d~(pO_d+i_n+l) the site (the project data shown are in Imperial p6_d~(p3_d+l) pl_d~(pO_d+l) units). There are three material horizons bounded by p2_d~( (i_n+l)*(k_n+l)+pO_d) four profile lines, and a liquid level present p5_d~(p2_d+(i_n+l)) at this site. Figure 5 shows the 2-D cross-sectional p7_d~(p5_d+l) p4_d~(p2_d+l) views of the site at the 16-locations prior to failure command (the available project data were converted to SI units generate zone brick size nX,nY,nz ratio 1,1,1 & pO~point (pO_d) p3~point (p3_d) & and this conversion lead to numerical values with p6~point (p6_d) pl~point (pl_d) & fractional parts). Figure 6 shows a partial listing of p2~point (p2_d) p5~point (p5_d) & the data file for the sample problem with the p7~point (p7_d) p4~point (p4_d) group material end command following details: if kz~3 then material='msw' endif Table 100 lists the x-coordinates of the 22 control end_loop points considered significant from the sixteen 2-D pO_d~pO_d+l end_loop cross-sectional data. end_loop end Table 101 lists the y-coordinates of the sixteen 2-D Figure 2. Listing of 'FILL_GRID' function in FISH cross-section locations. language. Table 103 1ists the number of zones desired in each of the 15 segments in the y-direction.

Table 104 lists the number of zones desired in each of the 3 material horizons at the site.

Table 105 11sts the x-coordinates of the toe locations of the top profile line in the 2-D cross-sections in the increasing y-direction.

For each cross-section, x- and z-coordinates for data points defining the profile lines are recorded in individual tables numbered as Table 1 for profile line I data, Table 2 for profile line 2 data, Table 3 for profile line 3 data, and Table 4 for profile line 4 data in the data file shown in Figure 6. Profile lines are numbered from 1 to 4 in the increasing z­ direction and each profile line uses a different number of data points to define the line. For cross­ sections where the top profile line temlinates in a Figure 3. Sample problem - Aerial vIew of Cincinnati landfill fallure (from Eid et al. 2000). vertical cut at the toe, the top profile line was (Reproduced by pennission of the publisher, ASCE). extended to x = o. For each cross-section and for each of the fOUf profile lines, the x-coordinate locations identified in Table 100 are used to create data by interpolation at each of the 22 control points. For the sample

o "" problem, this amounts to 88 pairs of (x,z) - coordinates per cross-section, and the y-coordinate of the data points is read from Table ]0 I. Thus, the x-,y-, and z-coordinates for all of the points defined and (or) interpolated are known. Each point

" , is assigned a numeric identity number (id #) starting i' ! with one and incrementing by one. The data points are located in the 3-D model space using their id #

, ... and x-, y-, z-coordinates. This task is accomplished , '1 using the 'Interpolate' function and its listing in FISH language is given in Figure I. At the end of I~ this task, all of the defined and (or) interpolated I I points with an assigned id # have been located in the 3-D model space. The connectivity of data points to define volume discretization is accomplished in the function named 'Fill....,grid' . For each interval in the location of cross­ sections in the y-direction (Table 103), and for each material horizon between the profile lines in the z-direction (Table 104), and for each interval in the x-direction (Table 102), the values of number of .-" zones desired in the x, y, and z-direction and the id I #s of points in the 3-D model space are used in the I / 'GENERATE zone brick pO, pI, ... p8' command of FLAC3D for a regular 8-noded brick mesh. TIle Figure 4. Plan view of the sample problem showing material between the profile lines is assigned a locations of selected 2-D sections. group name for ease of modifying the grid and for convenience in assigning material properties and/or Table 102 lists the number of zones desired in each addressing them for some other reason. This task is of the 21 segments in the x-direction. also accomplished in the function named 'Fill_grid' 3.5 3.5 (a) Cross-section at y=O m (i) Cross-section at y= 164.29 m 3.0 _ 3.0 N <­ <­ '"0 0 2.5 ;: 2.5 ;:. z z 1.5 :2 1.5 2 j*iO '" 2} (Y10"2) 3.5 3.5 (b) Cross-section at y=15.24 m U) Cross-section at y=201.47 m 3.0 3.0 N N <­ <­ ~ 25 ~ 2.5 ':...,. z

1.5 2 ~o .5 1.5 2 2.5 3 (*10'" 2) (*10"'2) 3.5 3.5 (k) Cross-section at y=234.09 m 3.0 N N <­ ~ I 2.5 ~ 2.5 ;:. :..... z 2.0 2.0 xO .5 1.5 2 2.5 3 3.5 xO .5 1.5 2 2.5 3 3.5 (*10"'2) (*10'" 2) 3.5 3.5 (d) Cross-section at y=28.96 m (I) Cross-section at y=253.29 m I 3.0 3.0 N N <­ ~ ~ 2.5 ¥ 25.:...,. z z 2.0 xO 1.5 2 2.5 xO .5 1.5 2 2.5 3 3.5 (·10'" 2) (*10" 2) 3.5 3.5 (e) Cross-section at y=42.06 m (m) Cross-section at y=268.B3 m J.O 3.0 N N <­ <­ 0 2.5 ~ .2.5 [. z z 2.0 2.0 x 0 .5 1.5 2 2.5 3 3.5 xO .5 1.5 2 2.5 3 3.5 (*10"'2) j*10'" 2) 3.5 3.5 (f) Cross-section at y=62.48 m (n) Cross-section at y=287.43 m 3.0 3.0 N N <­ <­ 2.5 ~ 2.5 ~ :..... ~ z z 2.0 1.5 2 3 xo .5 1.5 2 2.5 3 3.5 (*10'" 2) (*10" 2) 3.5 3.5 (g) Cross-section at y=96.93 m (0) Cross-section at y=293.83 m 3.0 3.0 N N < -< <:> 2.5 ~ 2.5 C ~ z xO 1.5 2 1.5 :2 (*10"'2) j*10 "2} 3.5 3.5 (h) Cross-section at y=138.07 m (p) Cross-section at y=307.85 m 3.0 3.0 N N < <­ 2.5 ~ 2.5 ~ ~ ~ z z 2.0 xO .5 1.5 2 2.5 3 xO .5 1.5 2 2.5 3 3.5 (*10"'2) (*10" 2)

Legend: • Shale • Natural soil bI Municipal solid waste - Water table Figure 5. 2-D cross-sectional views of the sample problem. ; Rumpke landfill site; Data are in metric units table 4 0,261.08 29.87,265.18 185.93,268.22 set g~0,0,-9.81 table 4 348.08,307.24 interpolate ; table 100 is for the x-coordinates of ; the desired 3-D grid table 100 0,1 13.11,2 15.54,3 22.86,4 34.75,5 table 100 42.67,6 49.07,7 57.61,8 63.70,9 delete range group mswt table 100 64.92,10 72.54,11 78.94,12 92.66,13 table 100 100.89,14 107.90,15 115.21,16 ; water surface table 100 158.50,17 199.64,18 284.38,19 water den~l table & table 100 318.52,20 337.72,21 348.08,22 face 0,0,228.60 0,15.24,228.60 & 332.54,15.24,268.22 & ; table 101 is for y-coordinates of the face 0,0,228.60 332.54,15.24,268.22 & ; 2-D cross-section locations 348.08,15.24,268.22 & table 101 0,1 15.24,2 20.73,3 28.96,4 42.06,5 face 0,0,228.60 348.08,15.24,268.22 & table 101 62.48,6 96.93,7 138.07,8 164.29,9 348.08,0,268.22 & ;interval # 1 table 101 201.47,10 234.09,11 253.29,12 face 0,15.24,228.60 0,20.73,228.60 & table 101 268.83,13 287.43,14 293.83,15 340.77,20.73,268.22 & table 101 307.85,16 face 0,15.24,228.60 340.77,20.73,268.22 & 348.08,20.73,268.22 & ; table 102 is for the number of zones face 0,15.24,228.60 348.08,20.73,268.22 & ; desired in the x-direction 332.54,15.24,268.22 & table 102 2,1 1,2 1,3 2,4 1,5 1,6 1,7 1,8 1,9 face 332.54,15.24,268.22 348.08,20.73,268.22 & table 102 1,10 I,ll 2,12 1,13 1,14 1,15 5,16 348.08,15.24,268.22 &;interval # 2 table 102 5,17 10,18 4,19 2,20 2,21

; table 103 is for the number of zones ; desired in the y-direction face 0,293.83,259.08 0,307.85,259.08 & table 103 2,1 1,2 1,3 2,4 2,5 3,6 4,7 3,8 4,9 63.70,307.85,259.08 & table 103 3,10 2,11 2,12 2,13 1,14 2,15 face 0,293.83,259.08 63.70,307.85,259.08 & 348.08,307.85,268.22 & ; table 104 is for the number of zones face 0,293.83,259.08 348.08,307.85,268.22 & ; desired in the z-direction 63.70,293.83,259.08 & table 104 5,1 3,2 10,3 face 63.70,293.83,259.08 348.08,307.85,268.22 & 348.08,293.83,268.22;interval # 15 ; table 105 is for the x-coordinates of the ; receding toe table 105 0,1 0,2 15.54,3 22.86,4 34.75,5 table 105 49.07,6 57.61,7 64.92,8 78.94,9 table 105 92.66,10 100.89,11 107.90,12 Figure 6. Partial listing of the data file for the table 105 115.21,13 63.70,14 0,15 sample problem for FLAC3D. set is~l ie~22 set js~l je~4 set id_pt~O and its listing in FISH language is given in Figure 2. set dt n s~200 Table 105 data are used to assign a group name ; Station at y~O 'mswt' to the zones past the vertical cut which are set y_pt~O later deleted using the DELETE command with the table 1 -100,200 500,200 table 2 0,223.60 154.23,223.60 307.24,238.84 range defined by the group name 'mswt'. At the end table 2 348.08,239.14 of this task, a 3-D grid of specification exists in the table 3 0,228.60 154.23,228.60 307.24,243.84 table 3 348.08,244.14 region-of-interest. For the sample problem, the table 4 0,260.00 66.45,280.42 98.15,283.46 generated 3-D grid is shown in Figure 7. The table 4 156.67,286.51 187.15,289.56 table 4 348.08,332.54 representation of continuity of the vertical cut at the interpolate toe of the slope (as seen in 2-D cross-sections, Figure 5) in the 3-D model can be improved by ; station at y~15.24 m set y_pt~15.24 increasing the number of 2-D cross-sections. table 2 erase table 3 erase table 4 erase set dt n s~dt n --- 7ADVANTAGES OF THE PROPOSED table 2 0,223.60 163.07,223.60 306.02,238.84 table 2 348.08,240.67 PROCEDURE table 3 0,228.60 163.07,228.60 306.02,243.84 table 3 348.08,245.67 table 4 0,251.46 91.14,280.42 107.90,283.46 (1) The proposed procedure for describing 3-D field table 4 144.48,286.51 169.77,289.56 conditions utilizes 2-D cross-sections which are table 4 194.46,292.61 332.54,338.33 essentially the same as commonly used by table 4 348.08,338.33 interpolate geologists and engineers to describe the field conditions. Linear variation in geometry, material horizons, and groundwater descriptions station at y~307.85 m between known data points is generally set y_pt~307.85 table 2 erase accepted. table 3 erase (2) Changes in field data can be incorporated in the table 4 erase s~dt numerical model by updating the affected tables. set dt--- n n table 2 0,254.08 348.08,254.08 table 3 0,259.08 348.08,259.08 OJ ~,~ ,"...-...... '""" '"" '" ;,,"""~"" '" ,,'" ,,,-,,,,,,,,,,, dd,O«! .,.J , nb,"i'~ '1",..1 '0''''';>11 of <1>" '"' ).[) ~, t"lQww by ,""",n.._ of ,"'" ""'''''''lm'y " > >im~l, yet po~y or ,­ronSlNClini> J·D ...m..",,1 mode' f""y'" (5) 111< ~ pooc.,J"", rmd'"'<'> "'S""" w'"h """"pt>blc Il''''....''''ie<. i.,.. 110 "",noct< in ronnect,,'",y (6) Cllan8C> '" d"<1<1'..,"", _ 10 0""'8C> '" fi< ,.._.. fIKOCo.Lr< _­ i",lod"'''' d,,,,,,,""'"'" ~1'<1'o«I, (1) N",,_ of d,,,,,"'Z«i VQ~""" U"'" on Mf"","' part> "f ,'" n,,,..,.;c.1 modol .. "",mo,od", ,'" ,,"," of'''' prul>i«n iiOI;...~ .1fu<1. Ir j, boo"""", ~ 10 <~< or "'~n< ,II< d""'.. >oJ ,'" """c-RIO "I:..i. on updaIcd3·Dn.... \~) II bl< ~hIl 00 pcrr""" ...."",;"" "'1)',;, wool. The pro;t..... ,ns'",,",oo, ,on II< ",~,In,n ,n "'he, 1"000ra""",nl ""'lI"'l<' 9 RHLR[.Io;CI'S

! SUMMARY •.., >I.T.. Sart, T.O., r,_ "'"0 "...... ,. H. XIOO. _ ..., ""'" ~_ ""'" too..... " ...... ". ....~,,~ J_"-~'''' <_~ To f..ih J.1l ...Iy", /-'UGD,~ oo..... ""'"g "_\'" ,:>\. "" ',R' '00"", """"' ,n .."""",,-l"ll ..rialioo.. '" g"""""I' ..-.l d"tribulion or "",,,riab '" 3·D

Appendix D

Part 6 Average Engineering Properties of Compacted Soils from the Western United States

This table is from reference [2]: Design of Small Dams, Bureau of Reclamation, Denver, Colorado, pp. 96-97, 2004.

Table 5-1 Average engineering properties of compacted soils [2] Specific gravity Compaction Shear strength Laboratory Index unit weight Avg. placement Effective stress Optimum Maximum moisture Moisture No. 4 No. 4 unit weight, content, Max., Min., Unit weight, content, c' φ' Unified classification Soil type minus plus lb/ft3 % lb/ft3 lb/ft3 lb/ft3 % lb/in2 ° Values listed GW Well-graded clean gravels, 2.69 2.58 124.2 11.4 133.6 108.8 - - - - Average of all values gravel-sand mixture 0.02 0.08 3.2 1.2 10.4 10.2 - - - - Standard deviation 2.65 2.39 119.1 9.9 113.0 88.5 - - - - Minimum value 2.75 2.67 127.5 13.3 145.6 132.9 - - - - Maximum value 16 9 5 16 0 Total number of tests GP Poorly graded clean 2.68 2.57 121.7 11.2 137.2 112.5 127.5 6.5 5.9 41.4 Average of all values gravels, gravel sand 0.03 0.07 5.9 2.2 6.3 8.3 7.2 1.2 - 2.5 Standard deviation mixture 2.61 2.42 104.9 9.1 118.3 85.9 117.4 5.3 5.9 38.0 Minimum value 2.76 2.65 127.7 17.7 148.8 123.7 133.9 8.0 5.9 43.7 Maximum value 35 12 15 34 3 Total number of tests GM Silty gravels, poorly graded 2.73 2.43 113.3 15.8 132.0 108.0 125.9 10.3 13.4 34.0 Average of all values gravel-sand-silt 0.07 0.18 11.5 5.8 3.1 0.2 0.9 1.2 3.7 2.6 Standard deviation 2.65 2.19 87.0 5.8 128.9 107.8 125.0 9.1 9.7 31.4 Minimum value 2.92 2.92 133.0 29.5 135.1 108.1 126.9 11.5 17.0 36.5 Maximum value 34 17 36 2 2 Total number of tests GC Clayey gravels, poorly 2.73 2.57 116.6 13.9 - - 111.1 15.9 10.2 27.5 Average of all values graded gravel-sand-clay 0.08 0.21 7.8 3.8 - - 10.4 1.6 1.5 7.2 Standard deviation 2.67 2.38 96.0 6.0 - - 96.8 11.2 5.0 17.7 Minimum value 3.11 2.94 129.0 23.6 - - 120.9 22.2 16.0 35.0 Maximum value 34 6 37 0 3 Total number of tests SW Well-graded clean sands. 2.67 2.57 126.1 9.1 125.0 99.5 - - - - Average of all values gravelly sands 0.03 0.03 6.0 1.7 6.0 7.1 - - - - Standard deviation 2.61 2.51 118.1 7.4 116.7 87.4 - - - - Minimum value 2.72 2.59 135.0 11.2 137.8 109.8 - - - - Maximum value 13 2 1 12 0 Total number of tests SP Poorly graded clean sands, 2.65 2.62 115.6 10.8 115.1 93.4 103.4 5.4 5.5 37.4 Average of all values sand-gravel mixture 0.03 0.10 9.7 2.0 7.2 8.8 14.6 - 3.0 2.0 Standard deviation

Table 5-1 Average engineering properties of compacted soils [2] Specific gravity Compaction Shear strength Laboratory Index unit weight Avg. placement Effective stress Optimum Maximum moisture Moisture No. 4 No. 4 unit weight, content, Max., Min., Unit weight, content, c' φ' Unified classification Soil type minus plus lb/ft3 % lb/ft3 lb/ft3 lb/ft3 % lb/in2 ° Values listed 2.60 2.52 106.5 7.8 105.9 78.2 88.8 5.4 2.5 35.4 Minimum value 2.77 2.75 134.8 13.4 137.3 122.4 118.1 5.4 8.4 39.4 Maximum value 36 3 7 39 2 Total number of tests SM Silty sands, poorly graded 2.68 2.18 116.6 12.5 110.1 84.9 112.0 12.7 6.6 33.6 Average of all values sand-silt mixture 0.06 0.11 8.9 3.4 8.7 7.9 11.1 5.4 5.6 5.7 Standard deviation 2.51 2.24 92.9 6.8 88.5 61.6 91.1 1.6 0.2 23.3 Minimum value 3.11 2.63 132.6 25.5 122.9 97.1 132.5 25.0 21.2 45.0 Maximum value 149 9 21 17 Total number of tests SC Clayey sands. poorly 2.69 2.17 118.9 12.4 - - 115.6 14.2 5.0 33.9 Average of all values graded sand-clay mixture 0.04 0.18 5.9123 2.3 - - 14.1 5.7 2.5 2.9 Standard deviation 2.56 2.17 104.3 6.7 - - 91.1 7.5 0.7 28.4 Minimum value 2.81 2.59 131.7 18.2 - - 131.8 22.7 8.5 38.3 Maximum value 88 4 73 0 10 Total number of tests ML Inorganic silts and clayed 2.69 - 103.3 19.7 - - 98.9 22.1 3.6 34.0 Average of all values silts 0.09 - 10.4 5.7 - - 11.5 8.9 4.3 3.1 Standard deviation 2.52 - 81.6 - - 80.7 11.1 0.1 25.2 Minimum value 3.10 - 126.0 34.6 - - 119.3 40.3 11.9 37.7 Maximum value 65 0 39 0 14 Total number of 10.6 tests CL Inorganic clays of low to 2.71 2.59 109.3 16.7 - - 106.5 17.7 10.3 25.1 Average of all values medium plasticity 0.05 0.13 5.5 2.9 - - 7.8 5.1 7.6 7.0 Standard deviation 2.56 2.42 90.0 6.4 - - 85.6 11.6 0.9 8.0 Minimum value 2.87 2.75 121.4 29.2 - - 118.7 35.0 23.8 33.8 Maximum value 270 3 0 31 Total number of tests MH Inorganic clayey silts, 2.79 - 85.1 33.6 ------Average of all values elastic silts 0.25 - 2.3221 1.6 ------Standard deviation 2.47 - 82.9 31.5 ------Minimum value 3.50 - 89.0 35.5 ------Maximum value

Table 5-1 Average engineering properties of compacted soils [2] Specific gravity Compaction Shear strength Laboratory Index unit weight Avg. placement Effective stress Optimum Maximum moisture Moisture No. 4 No. 4 unit weight, content, Max., Min., Unit weight, content, c' φ' Unified classification Soil type minus plus lb/ft3 % lb/ft3 lb/ft3 lb/ft3 % lb/in2 ° Values listed 10 0 5 0 0 Total number of tests CH Inorganic clays of high 2.73 - 95.3 25.0 - - 93.6 25.7 11.5 16.8 Average of all values plasticity 0.06 - 6.6 5.4 - - 8.1 5.7 7.4 7.2 Standard deviation 2.51 - 82.3 - - 79.3 17.9 1.5 4.0 Minimum value 2.89 - 107.3 41.8 - - 104.9 35.3 21.5 27.5 Maximum value 74 0 36 0 12 Total number of 16.6 tests

Appendix D

Part 7 Strength, Stress-Strain and Bulk Modulus Parameters for Finite Element Analyses of Stresses and Movements in Soil Masses, by J.M. Duncan, P. Byrne, K.S. Wong, and P. Mabry

This report was published as Report No. UCB/GT/80-01, University of California, Berkeley, California, 1980. Tables 5 and 6 of this report are reproduced herein with permission of the first author, Prof. J. M. Duncan, currently the Distinguished Professor Emeritus, 1600 Carlson Drive, Blacksburg, VA 24060.