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U.S. Army Corps of Engineers

Modernization and Optimization

of Existing Dams and Reservoirs

27th Annual USSD Conference

Philadelphia, Pennsylvania, March 5-9, 2007

On the CoverThe Corps of Engineers Beltzville Lake in East-Central Pennsylvania. In this south-facing photo, from the bottom

to the top, features include the project office, the emergency spillway, the 4,560-foot long embankment, the intake

tower, and a series of ridges of the Appalachian front. Beltzville Lake is on Pohopoco Creek, which drains into the

Lehigh River. The Lehigh Rivers water gap through Blue Mountain can be seen in the background of the photo.

(Photo by Anthony S. Bley.)

The information contained in this report regarding commercial projects or firms may not be used for

advertising or promotional purposes and may not be construed as an endorsement of any product or

from by the United States Society on Dams. USSD accepts no responsibility for the statements made

or the opinions expressed in this publication.

Copyright 2007 U.S. Society on Dams

Printed in the United States of America

Library of Congress Control Number: 2007921375

ISBN 978-1-884575-40-2

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U.S. Society on Dams

Vision

To be the nation's leading organization of professionals dedicated to advancing the role of dams

for the benefit of society.

Mission USSD is dedicated to:

Advancing the knowledge of dam engineering, construction, planning, operation,

performance, rehabilitation, decommissioning, maintenance, security and safety;

Fostering dam technology for socially, environmentally and financially sustainable water

resources systems;

Providing public awareness of the role of dams in the management of the nation's water

resources;

Enhancing practices to meet current and future challenges on dams; and

Representing the United States as an active member of the International Commission on

Large Dams (ICOLD).

Three-Dimensional FLAC 285

THREE-DIMENSIONAL FLAC STABILITY AND DEFORMATION ANALYSIS HOW MUCH CAUTION IS NEEDED?

John W. France, P.E.1 Imran S. Gillani, P.E.2

ABSTRACT In 2004 Itasca introduced a three-dimensional version of its popular finite difference analysis program FLAC (Fast Lagrangian Analysis of Continua). The earlier, two-dimensional version of FLAC has been widely used for stability and deformation analyses of embankment dams. The three-dimensional version of the program provides geotechnical engineers a new tool for the evaluation of embankment dams. Dam engineers have been particularly interested in applying FLAC3D to the analysis of cases where liquefiable or weak soils are present in the foundation, but are of limited lateral extent. In such cases, two-dimensional analysis can underestimate factors of safety and overestimate deformations, because any liquefiable or weak soils included in the section are modeled as if they extended infinitely in the direction perpendicular to the two-dimensional model cross section. The three-dimensional program allows the effects of stronger materials at the sides of the liquefiable soils to be considered in the analysis.

In an effort to better understand and calibrate the results obtained from FLAC3D, the authors completed analyses of a simple case, consisting of a 75-foot-high embankment founded on a 10-foot thickness of weak soil. The width of the valley was varied and the variation in factor of safety with valley width was compared with the results of FLAC2D stability analyses and limit equilibrium stability analyses. As the valley becomes wide, the FLAC3D results should be comparable to the two-dimensional results. Some interesting results were found in this comparison. In this paper, the authors will review their modeling results and provide some observations and conclusions on those results, including the observation that the FLAC3D results appear to be inherently unconservative (i.e. higher factors of safety and smaller calculated deformations) when compared to the comparable FLAC2D and limit equilibrium analyses results.

INTRODUCTION Over the past decade the computer program Fast Lagrangian Analysis of Continua (FLAC) has become very popular for numerical analyses of a wide range of geotechnical problems, including dam embankments under both static and dynamic loadings. The program that has been widely used for the past decade is a two-dimensional version of FLAC, referenced in this paper as FLAC2D (Itasca, 1999). In 2002, a three-dimensional version of FLAC, referenced in this paper as FLAC3D, was introduced (Itasca, 2002).

1 Vice President and National Dam Technology Leader, URS Corporation, 8181 East Tufts Avenue, Denver, CO 80112: phone (303) 740-3812: fax (303) 694-3946: john_france@urscorp.com 2 Project Manager, URS Corporation, 8181 East Tufts Avenue, Denver, CO 80112: phone (303) 740-3870: fax (303) 694-3946: imran_gillani@urscorp.com

286 Modernization and Optimization of Existing Dams and Reservoirs

The three-dimensional version of the program is being applied by geotechnical engineers as a new tool for the evaluation of embankment dams.

The three-dimensional program could be particularly valuable in the analysis of cases where liquefiable soils, or other low strength soils, are present but are of limited lateral extent. In such cases, it is generally accepted that two-dimensional analysis can under estimate factors of safety and overestimate deformations, because any liquefiable soils included in the section are modeled as if they extend infinitely in the direction perpendicular to the two-dimensional model cross section. The three-dimensional program allows the effects of stronger soils at the sides of the liquefiable soils to be considered in the analysis.

The authors of this paper are interested in using FLAC3D, but first wanted to compare the results of FLAC3D against more familiar two-dimensional analyses. To make this comparison, stability analyses and earthquake deformation analyses were completed for a simplified embankment configuration. This paper begins with a general description of the FLAC computer program. The simplified embankment dam configuration is then introduced and the results of stability analyses and earthquake deformation analyses are presented. Finally, the authors offer some observations and conclusions regarding the practical implications of the analysis results.

FAST LAGRANGIAN ANALYSIS OF CONTINUA (FLAC) FLAC, in both two-dimensional and three-dimensional versions, is a two-dimensional, explicit, finite difference computer program for geotechnical applications. Because FLAC uses an explicit, time-marching computational routine, it is very well suited for modeling dynamic deformation analyses, which may involve localized or even general instability. The explicit, time-marching computational routine allows for modeling of soils that are at or close to failure much more efficiently than does the implicit finite element approach. The large strain mode of FLAC allows for inclusion of relatively large displacements that would be very difficult to compute using the small strain approach typically used by other programs. Furthermore, the ability to customize the runs and the working of the actual program through a programming language called FISH makes FLAC a very effective program for performing complex dynamic deformation type analyses.

SIMPLIFIED EMBANKMENT CONFIGURATION

The simplified embankment configuration consists of a homogeneous embankment founded on a single stratum of alluvial soil, all underlain by bedrock, all located within a symmetrical valley, as illustrated in Figures 1 and 2. Although this case is simplified, it is not fundamentally different from commonly encountered real cases.

For the simplified example, the embankment height is 75 feet, and the maximum depth of the alluvial soil stratum is 10 feet. The width of the valley was varied in the three-

Three-Dimensional FLAC 287

dimensional analyses to evaluate the effects of the three-dimensional geometry of the structure and the valley on the results of the FLAC3D analyses.

Figure 1. Simplified Embankment Configuration

Figure 2. Maximum-Height Cross Section for Simplified Embankment Configuration

Two different strength characterizations were used in the analyses, as summarized in Table 1. Two types of analyses were completed for the simplified embankment: stability analyses and earthquake deformation analyses.

288 Modernization and Optimization of Existing Dams and Reservoirs

Table 1. Shear Wave Velocities and Strength Characterizations Used in Analyses Strength Characterization

No. 1 Strength

Characterization No. 2 Material Shear Wave Velocity, fps Cohesion,

c Friction

Angle, Cohesion,

c Friction

Angle, Unsaturated embankment 1200 0 psf 34 0 34 Saturated embankment 1200 1600 psf 0 1400 psf 0 Alluvium 1200 900 psf 0 1400 psf 0 Bedrock 2000 Elastic Elastic Elastic Elastic

The = 0 strength characterizations for the saturated embankment soils are representative of those used in analyses of foundation soils that are judged susceptible to liquefaction. Hence, the stability analyses are representative of post-earthquake conditions.

STABILITY ANALYSES

For the stability analysis, the FLAC program and the strength reduction method (Dawson and Roth, 2005) were used. In the strength reduction method, a series of analyses are run with different values of trial factors of safety, Ftrial. In each case, the strength parameters (cohesion = c and friction angle = ) used in the deformation model are reduced according to the following equations:

ctrial = c/Ftrial

trial = arctan(tan/Ftrial)

Ftrial is decreased until calculated deformations suddenly become very large. The value of Ftrial at which the deformations become large is taken as the factor of safety for the case being analyzed. The FLAC program actually includes an internal routine that automates the strength reduction slope stability analysis method.

The FLAC3D model used in the analysis is shown in Figure 3, and a two-dimensional cross section through the maximum section of the model is shown in Figure 4.

The FLAC3D model was used to calculate factors of safety for a series of analyses with the valley width (see Figure 1) varied from 150 feet to 1200 feet. The results for Strength Characterization No. 1 are presented in Figure 5, in the form of a plot of calculated 3D factor of safety versus valley width. The results for Strength Characterization No. 2 are presented in Figure 6.

The FLAC3D stability analysis results were compared to two-dimensional stability factors of safety calculated using three other methods: the limit equilibrium method

Three-Dimensional FLAC 289

Figure 3. FLAC3D Model of Simplified Embankment Configuration

Figure 4. FLAC Mesh Through Maximum-Height Cross Section for Simplified Embankment Configuration

using the computer program UTEXAS3, the finite difference method using FLAC2D, and the finite difference method using FLAC3D with a slice model. The slice model in FLAC3D analyzes the behavior of a single row of three-dimensional elements

290 Modernization and Optimization of Existing Dams and Reservoirs

with roller boundary conditions on the sides. The row of elements is located along the maximum height section of the dam. The calculated two-dimensional factors of safety are also shown on Figures 5 and 6, for Strength Characterizations Nos. 1 and 2, respectively.

Figure 5. Summary of Analysis Results for Strength Characterization No. 1

Figure 6. Summary of Analysis Results for Strength Characterization No. 2

Three-Dimensional FLAC 291

The FLAC3D and FLAC2D models were constructed with the same element size, so that element size would not influence the results.

EARTHQUAKE DEFORMATION ANALYSES All of the cases evaluated in the stability analyses were also evaluated in the earthquake deformation analyses. The FLAC models were subjected to an input earthquake motion at the base of the model. The acceleration time history for this earthquake motion is shown in Figure 7.

Figure 7. Earthquake Time History Used for Analysis

For Strength Characterization No. 1, the two-dimensional factor of safety is less than 1.0, indicating that the two-dimensional section is unstable. For the earthquake deformation analysis for this strength characterization, the foundation strength was initially set at c = 1400 psf, = 0, and then reduced to c = 900 psf, = 0, representing a very simple model of foundation liquefaction. For Strength Characterization No. 2, the two-dimensional factor of safety is greater than 1.0, indicating that the two-dimensional section is stable. For the earthquake deformation analysis for this strength characterization, the foundation strength was set at c = 1400 psf, = 0 throughout the analysis.

All FLAC earthquake deformation analyses were run in large-strain mode.

The permanent horizontal and vertical movements of the crest were selected as a basis of comparing the results of the earthquake deformation. In all cases, the permanent

292 Modernization and Optimization of Existing Dams and Reservoirs

horizontal crest movements were in the downstream direction and the vertical crest movements were settlements.

The calculated permanent horizontal and vertical movements for all analysis cases are noted on Figures 5 and 6 and are listed in Table 2.

Table 2. Permanent Crest Movements from Earthquake Deformation Analyses Strength Characterization No. 1 Strength Characterization No. 2

Analysis Case Factor of

Safety

Permanent Horizontal

Movement, ft

Permanent Vertical

Movement, ft

Factor of

Safety

Permanent Horizontal

Movement, ft

Permanent Vertical

Movement, ft

FLAC3D 150-ft valley 1.22 3.03 1.18 1.28 2.99 1.23 FLAC3D 300-ft valley 1.11 3.27 1.29 1.21 2.82 1.07 FLAC3D 600-ft valley 1.00 4.08 1.85 1.16 2.76 1.04

FLAC3D 1200-ft valley 0.96 5.02 2.52 1.13 2.74 1.01

FLAC3D Slice Mode 0.94 4.61 2.33 1.11 2.42 0.85 FLAC2D 0.87 7.99 4.44 1.03 2.91 1.10

UTEXAS3 0.89 -- -- 1.04 -- --

OBSERVATIONS

As can be seen in Figures 5 and 6, the calculated FLAC3D stability factors of safety decrease as the valley width increases; a trend that would be expected. For Strength Characterization No. 1, the three-dimensional factors of safety decreased from 1.22 for a 150-foot valley width to 0.96 for a 1200-foot valley width. For Strength Characterization No. 2, the three-dimensional factors of safety decreased from 1.28 to 1.13 for the same range of valley widths. As the valley becomes wider, the three-dimensional factor of safety approaches the two-dimensional values, coming very close (within about 0.02) to the value from the slice model. However, the three-dimensional factors of safety for the 1200-foot wide FLAC models and for the FLAC 3D slice models are higher than the two-dimensional values from UTEXAS3 and FLAC2D, which agreed within about 0.02 with each other. The differences between the FLAC3D factors of safety and the FLAC2D and UTEXAS3 factors of safety are greater for Strength Characterization No. 1 than for Strength Characterization No. 2. For Characterization No. 1, the FLAC3D factors of safety for the wide valley and for the slice method are about 8 percent higher than the two-dimensional factors of safety. For Characterization No. 2, the FLAC3D factors of safety for the wide valley and for the slice method are about 3 percent higher than the two-dimensional factors of safety.

For Strength Characterization No. 1, the calculated crest movements for the FLAC3D cases increase as the valley widens and the three-dimensional factors of safety decrease, as would be expected. For the FLAC3D model for the 1200-foot wide valley, the calculated crest deformations are within 10 percent of those for the FLAC3D slice model, although curiously the deformations for the full FLAC3D model are slightly greater than those for the slice model. The calculated crest deformations for the FLAC2D model are significantly greater than those from either the 1200-foot valley FLAC3D model or the

Three-Dimensional FLAC 293

FLAC3D slice model. Comparing the deformations for the FLAC2D model to those for the FLAC3D model for the 1200-wide valley, the horizontal deformations are 58 percent higher for FLAC2D and the vertical deformations are 76 percent higher for FLAC2D.

The results for Strength Characterization No. 2 indicated some unexpected results. The calculated crest movements for the full FLAC3D models actually decreased as the valley width increased and the factors of safety decreased. All of the horizontal movements from these models were within about 10 percent of each other and all of the vertical movements were within about 20 percent of each other. These results would generally be considered to be essentially the same within the expected accuracy of geotechnical deformation calculations, however, there is no clear explanation of the unexpected trend in these calculated deformations. For the FLAC3D model for the 1200-foot wide valley, the calculated crest deformations are higher than those for the FLAC3D slice model, with the horizontal deformations about 13 percent higher and the vertical deformations about 19 percent higher. The calculated crest deformations for the FLAC3D model for the 1200-foot wide valley are also less than those from the FLAC2D model, with the horizontal deformations about 6 percent less and the vertical deformations about 8 percent higher.

CONCLUSIONS It would be anticipated that the FLAC3D results for the 1200-foot wide valley models would agree reasonably well with the results of the FLAC2D and UTEXAS3 analyses. The height to width ratio of these FLAC3D models exceeds 12, for which it would be anticipated that three-dimensional effects would be minimal. Yet, in this case the FLAC3D model appears to be producing somewhat higher factors of safety than the two-dimensional analyses. Further, the difference appears to be greater for the case where the two-dimensional factor of safety is less than 1.0 (Strength Characterization No. 1), which may be the case for which engineers would be most interested in using the program to analyze three-dimensional effects. The fact that the FLAC3D slice method also produces factors of safety greater than those from the two-dimensional method suggests that there may be something inherent in the mathematical formulation of FLAC3D that produces results that are unconservative relative to two-dimensional analyses.

Similarly, it appears that for this case deformations calculated in the FLAC3D 1200-foot valley model are less than those calculated in FLAC2D, and the difference is more pronounced for the case of an initial factor of safety less than 1.0. One unexpected and as yet unexplained result was the observation of increasing calculated crest deformation with decreasing valley width for Strength Characterization No. 2. This reverse trend is perplexing, but may not be of large significance, because the calculated crest deformations for all of the valley widths for Strength Characterization No. 2 are essentially equivalent within the limits of accuracy of geotechnical deformation calculations.

Considering all of the results presented in this paper, the authors conclude that FLAC3D can help to provide some insights on three-dimensional behavior of cases like that

294 Modernization and Optimization of Existing Dams and Reservoirs

analyzed for this paper. However, at this time the results should be applied with caution, because it seems that they are likely unconservative to different degrees when compared with the results of two-dimensional analyses. This is significant because comparisons between two-dimensional analyses and observed performance have provided the profession with the basis used to establish acceptable factors of safety and calculated deformations. In cases when FLAC3D analyses are used for dam embankment stability and deformation analyses, it would desirable to complete companion FLAC3D slice analyses and FLAC2D analyses to help gauge the magnitude of the differences in results between FLAC3D and FLAC2D.

REFERENCES Itasca Consulting Group. 1999. FLAC Version 4.0 Fast Lagrangian Analysis of Continua, Users Guide.

Itasca Consulting Group. 2002. FLAC3D Fast Lagrangian Analysis of Continua in 3 Dimensions, Users Guide.

Dawson, E. and Roth, W. 2005. 3-D Slope Stability Analysis By Strength Reduction. 25th Annual USSD Conference, Salt Lake City, UT, June 6-10, 2005.