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HUME DAM SEISMIC ANALYSIS OF SOIL/STRUCTURE INTERACTION

Guy Lund1 Brad Dawson2 Mark Foster3

ABSTRACT

Hume Dam is located near Albury/Wodonga, Australia and was constructed between 1919 and 1936. The reservoir was enlarged in the 1960s to its current capacity of 3,038,000 megaliters (2.5 106 acre-feet). It is the main regulating reservoir on the River Murray System and supplies irrigation water and hydro-electric power. State Water Corporation (State Water) on behalf of the Murray Darling Basin Authority (MDBA) currently manages the dam and reservoir.

The main dam consists of an embankment dam with a concrete core wall and a gated concrete gravity spillway. Spillway discharges flow through the gates, over the ogee gravity section, and into the river through a discharge channel. The flow is trained with large concrete training wall on both the right and left side of the discharge channel. The left, southern training wall (STW) is located between the spillway channel and the main embankment, and retains the embankment fill as well as containing the spillway discharges. The height of the STW varies from approximately 50 meters (165 feet) near the crest of the embankment dam to 18 meters (60 feet) at the downstream end, and is the subject of this paper.

Modifications have been performed on the STW over the last few decades to improve stability due to the increased loads caused by severe deformation of the embankment. The modifications have included installation of sub-vertical post-tensioned tendons and horizontal post-tensioned anchors. However, continued embankment deformation has resulted in the need for additional rehabilitation. In addition, it is understood that the critical loading condition is due to the safety evaluation earthquake (SEE), and a significant portion of the load is dependent on the combined behavior of the embankment fill and the mass concrete wall.

The finite element method of analysis was used to analyze the soil/structure interaction and the behavior of the STW for both static and dynamic loads. This paper summarizes the finite element model, parameter assumptions, and sensitivity studies used to verify the behavior of the model with the actual STW, and the results used to develop the design modification.

1 P.E., Principal Civil/Structural Engineer, URS Corporation, Denver 2 P.E., Civil/Mechanical Engineer, URS Corporation., Denver 3 CP Eng., Project Manager, URS Australia., Sydney, Australia

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PROJECT DESCRIPTION

The main dam consists of an embankment dam with a concrete gravity spillway. The spillway consists of a gated ogee section, flip-bucket type stilling basin, and mass concrete, gravity training walls. The north (right) and south (left) spillway training walls extend downstream from the axis of the dam past the concrete apron to retain the embankment fill and train spillway discharges. The southern training wall (STW) is located on the left side of the spillway looking downstream and is photographed in Figure 1.

Figure 1. Photograph of the Southern Training Wall and Hume Dam, Australia.

Modifications to improve stability were performed in the 1980s and included installation of sub-vertical post-tensioned tendons through the full height of the STW and into the foundation rock. These tendons are spaced at approximately 2 meter (6.5 feet) centers and were designed to have a working load (equal to 65 percent of the minimum breaking load) of 2,900 kN (12.9 kips).

Additional modifications were performed to the STW between 1995 and 2004 to accommodate the increased load due to a rockfill berm constructed on the downstream slope of the embankment dam. These modifications included installation of large diameter, horizontal anchors that extend from the upper part of the STW to a concrete deadman wall located approximately 80 meters (262 feet) within the embankment/berm. A mechanically stabilized earth (MSE) wall was installed to retain the rockfill berm along the crest of the STW.

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Figure 2. Modifications to the STW, including sub-vertical tendons, horizontal anchors, and MSE Wall with rock fill berm.

The Hume embankment dam continues to deform, and has resulted in significant settlement at the deadman wall, and the deformation has again increased the load on the STW. Recently, some of the horizontal anchors have failed, which has increased concern regarding the long term safety of the STW. To compound concern, recent field investigations indicate that the zones of backfill adjacent to the STW is potentially liquefiable, which would add significantly more load on the STW. An external review panel (ERP) for the Hume Dam safety evaluation concluded that additional evaluation was required to assess the capacity of the wall, and design alternatives need to be developed to provide long term serviceability for the STW.

Initial stability studies were performed for several sections of the STW using the simplified gravity method of analysis. The studies identified the most critical sections of the STW regarding stability, and evaluated different alternatives that would stabilize the wall. It is important to note, that due to the embankment deformation and localized failure of selected horizontal anchors, it was concluded that any potential modification should include the decommissioning of the horizontal anchors. Based on the results from the gravity studies, the preferred alternative to stabilize the STW consists of a concrete buttress.

The more critical sections of the STW were further evaluated using sophisticated non-linear finite element analyses. The section locations are described by offset stations (OS) along the axis of the wall. OS zero (0) corresponds to the upstream face of the spillway. OS 31 corresponds to the section located 31 meters downstream of the upstream face of the spillway. Two critical sections were used in the evaluations, OS 31 and OS 75. The upper section, OS 31, simulates the portion of the wall adjacent to the concrete stilling basin, and is the subject of this paper. The lower section, OS 75, simulated the portion of the wall embedded in the rock foundation and loose backfill. The section of the STW at OS 31, spillway apron, and embankment are shown in Figure 3.

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Figure 3. Section of STW at OS 31.

EVALUATION OF THE PRINCIPAL FAILURE MODE

The failure mode of highest concern for Hume Dam consists of the following developments: movement of STW occurs due to either failure of the horizontal tendons or increased loads (i.e. seismic or liquefaction); potential seepage path develops between the embankment fill and the STW; seepage initiates in path opening; seepage initiates piping of materials; hydrostatic load on STW increases causing additional movement and increased path opening; piping results in loss of material causing failure of core; reservoir breaches dam and floods downstream areas.

Potential movement of the STW and the development of a seepage path between the wall and embankment was the primary concern for these structural stability evaluations. After expert elicitation, it was concluded that for these studies, the development of a seepage path would be due to instability of the STW. Instability would be results from one or more of three common failure mechanisms, which are typically used for rigid concrete gravity structures. If the evaluation indicates that the STW has adequate capacity against these three prominent failure mechanisms, then it was assumed that potential for developing a seepage path due to structural instability would be very unlikely. The three prominent failure mechanisms for these studies are summarized below:

Structural Capacity. The results from the analysis were used to evaluate the potential for overstressing of the concrete (i.e., crushing or cracking), and the moment capacity of the STW with the sub-vertical post-tension anchors.

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Overturning. The results from the finite element model were used to assure that moment equilibrium, or rotational stability, was maintained throughout the structure.

Sliding Stability. The results from the finite element model were used to evaluate the potential for sliding failure along different planes through the structure.

Structural Capacity

The structural capacity of the STW was evaluated based on the allowable strength of the mass concrete, and the bending moment capacity at select horizontal sections within the STW.

Concrete Strength

The overstressing evaluation compares the computed stress from the finite element analysis to the estimated strength of the mass concrete. If the computed stress is greater than the allowable strength, then the concrete is expected to crack (tension) or crush (compression). If cracking or crushing of the concrete develops, then there is an increased potential for deformation (movement). On-the-other-hand, without cracking or crushing of the concrete then movement of a rigid body (STW) is very unlikely.

Bending Moment Demand Capacity Ratio (DCR)

The sub-vertical post-tensioned anchors in the STW can be thought of as a type of reinforcement. The capacity of a reinforced section can be evaluated based on the axial-flexural interaction diagram.

The U.S. Army Corps of Engineers use the demand capacity ratio (DCR) method to evaluate the ability of structures to support the dynamic earthquake loads [U.S. Army Corps of Engineers, EM 1110-2-6051, Engineering and Design Time History Dynamic Analysis of Concrete Hydraulic Structures, December 2003.]. The method determines if the behavior of the reinforced concrete structure is linear, or non-linear. If the behavior is linear, then it can be assumed that when the load is removed the structure will return to the initial state. For reinforced concrete hydraulic structures, the maximum bending moment DCR is taken as 2.0, and the shear DCR is taken as 1.0 [USACE, EM 1110-2-6051].

The flexural capacity of the STW at Hume Dam was determined using an interaction diagram, as shown on Figure 4. The relationship of axial vs. flexural load was computed assuming that the maximum compressive strain in the concrete is 0.003, and the maximum tensile strain in the tendons is the yield strain (i.e., yield strength of the tendons was assumed to be 65 percent of the ultimate strength, or 1210 MPa).

Figure 4. Moment interaction diagram for the STW at Hume dam, OS 31.

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For dynamic loads, the computed moment should not exceed the moment capacity for prolonged periods of time. The DCR is allowed to exceed unity (1.0) for short durations during the earthquake, because studies have shown that short term yielding of the steel will not result in a brittle failure. The assumption is considered valid for the STW at Hume Dam, because the tendons are non-bonded, such that any potential elongation will be applied over the full length of the tendon, and not an isolated point such as with reinforced concrete.

The U.S. Army Corps of Engineers use the demand capacity ratio (DCR) method evaluates the total duration of time that the DCR greater than 1.0. The time duration is plotted on the acceptance curve, shown in Figure 5. If duration of time is greater than 1.0, but within the limits of the acceptance curve (see Figure 5), then the structure is expected to behave linearly and further evaluation is not necessary. If the time duration falls outside the limits of the acceptance curve, then the structure will behave non-linearly, and further studies may need to be performed to correctly evaluate the behavior of the structural system.

Overturning

The rotational stability (moment equilibrium) is satisfied if the summation of moments from the analysis equal zero. The finite element model for the STW at Hume Dam used non-linear contact elements to simulate the condition at the base of the STW. The contact element can develop compression, but will separate rather than developing tension. Therefore, if the analysis shows that the separation of the contact elements stabilized (i.e. there is no further propagation of the crack), then moment equilibrium has been satisfied.

Sliding Stability

The minimum factors of safety required to satisfy sliding stability for the usual, unusual, and extreme load combinations are based on criteria published by the USACE [U.S. Army Corps of Engineers, EM 1110-2-2100, Engineering and Design Stability Analysis of Concrete Structures, December 2005]. The minimum factor of safety for the post-seismic load combination is based on the FERC [Federal Energy Regulatory Commission (FERC), Engineering Guidelines for the Evaluation of Hydropower Projects, Chapter 3 Gravity Dam, 2002.]. If the computed sliding factor of safety is greater than the minimum value set in the criteria, then sliding instability if very unlikely.

Figure 5. DCR Acceptance curve.

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METHOD OF ANALYSIS

The studies for the Hume Dam STW were performed using a two-dimensional plane strain, finite element model using the computer program ANSYS. The geometry of the STW section at OS 31 is shown on Figure 6.

Figure 6. Finite Element Model of STW at Hume Dam, OS31.

To properly simulate the behavior of the STW required the use of several different types of elements in the computer model. The concrete, foundation rock, weathered rock, embankment soil, and alluvium were modeled using 4-node solid elements. Single-node mass elements were used to simulate the added mass due to the hydrodynamic interaction between the tailwater and STW.

Two-dimensional contact elements were used to simulate the STW/embankment and STW/foundation interface, and specialize in surface-to-surface interaction. The contact elements can develop compressive forces, but will separate rather than developing tensile forces. The element also simulates shear along the interface. It was necessary to simulate the behavior of the STW/embankment interface using contact elements, primarily because of the very different dynamic behavior characteristics between the STW and embankment. The different dynamic characteristics results in the two materials oscillating at different frequencies, which results in separation at the interface during the seismic loads. Without the contact element, the interface would develop tensile stress, which would affect the behavior of the STW structure.

The boundary conditions used for the finite element model were defined such that the nodes along edges of the model were restrained against deformation in the horizontal (X) direction, but not in the vertical (Y) direction. The nodes along the base of the model (i.e., base of the foundation) were restrained against deformation in the vertical direction, but not in the horizontal direction.

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The behavior of the concrete, rock and soil were simulated using different material models. The concrete and foundation rock were evaluated using linear elastic material behavior. The weathered rock, alluvium and embankment were evaluated using the Extended Drucker-Prager (EDP) material model, which is an expanded use of the Mohr-Coulomb failure criteria and is used for the evaluation of granular materials, such as rock and soil. The parameters used to define the behavior of the EDP materials are based on the internal friction angle and cohesive strength of the soil.

Dynamic Material Properties

The dynamic behavior of the STW was evaluated using a full transient analysis. Some of the most critical material properties used in this type of studies typically includes the stiffness, Poissons ratio, and damping. The stiffness and Poissons ratio for the concrete material was not modified for the dynamic studies, primarily because the ratio of concrete stiffness to embankment stiffness was so large.

The stiffness and Poissons ratio for the embankment materials were modified for the dynamic evaluations. Typically, the Poissons ratio for soils under static loads will range between 0.3 and 0.4 [Lambe, T. William, Robert V. Whitman, Soil Mechanics, Massachusetts Institute of Technology, John Wiley & Sons, Inc. New York, 1969.]. The assumed Poisson ratio for the embankment materials varied depending on moisture content and loading conditions. The Poissons ratio for dry embankment soil was assumed equal to 0.30 for both static and dynamic loads. The Poissons ratio for saturated soils was assumed equal to 0.30 for static loading conditions and 0.45 for dynamic loading condition. The higher ratio better simulated the relationship between elastic and shear modulus during the seismic loads, due to increased pore pressures. When a soil liquefies the Poissons ratio will approach 0.50; however, mathematical limitations within the finite element code will not solve for a Poisson ratio of 0.50. Therefore, for the liquefied saturated soils the Poissons ratio was set equal to 0.49.

The elastic property of the soil was simulated using various values for Youngs modulus. Initial studies assumed elastic values of Youngs modulus for the alluvium, backfill, and rockfill materials based on test data [URS, Hume Dam Remedial Works Geotechnical Investigation, Volumes I and II, Report prepared for State Water, dated June 22, 2009 The stiffness of the materials was then computed based on the relationship with Poissons ratio and shear modulus.

The material damping constants used in the analysis conservatively assumed 5 percent damping for the concrete, 5 percent in the foundation rock, and 10 percent in the weathered rock and embankment materials.

Sensitivity Studies

Several sensitivity studies were performed to verify properties for the embankment materials in the finite element model. The sensitivity studies evaluated the effective horizontal pressure coefficient of the embankment, and vertical settlements of the

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embankment. The results from the finite element analysis were compared to previous studies using Slope/W and field investigation data, as shown Table 1.

Table 1. Horizontal Pressure Coefficients

Model Description Effective Horizontal Pressure Coefficient

Limit Equilibrium Geotechnical Study with Slope/W 0.61 ANSYS Model 0.62

Soil Deformation Studies

The settlement of the embankment at Hume Dam has been measured at various locations since the rockfill berm was constructed. For example, the measured settlement of the embankment near the deadman wall has reached 550 mm at select locations. The measured settlement was compared to the computed displacements from the finite element model. For example, the Youngs modulus for the embankment material adjacent to the STW (backfill) was assumed equal to 4 MPa (580 lb/in2) based on results from these sensitivity studies.

STW Deflection Studies

The horizontal displacements at the crest of the STW have been recorded since the construction of the rockfill berm. The measurements indicate that the STW has deflected into the embankment between 4 and 9 millimeters (0.16 to 0.36 inches). It has been postulated that the catenary action due to the settlement of the embankment has caused the horizontal anchors to increase in tension and pulled the STW into the embankment.

The settlement of the embankment would potentially result in an increase in the tensile load on the horizontal anchors, which would cause the STW to deflect towards the embankment. Studies were performed crest deflection of the STW with increased load from the horizontal anchors. The results indicate that an increase in the horizontal anchor load of 20 percent would result in the measured horizontal deformation in the STW.

RESULTS FROM STRUCTURAL ANALYSES

Usual Load Combination (USLC)

This usual loading condition, USLC, evaluated the dam for static loads due to gravity, embankment soil pressures, hydrostatic pressure (due to the normal embankment phreatic water level, uplift, and tailwater), post-tensioned anchors, rockfill berm, construction of the buttress, and decommissioning of the horizontal anchors. Each individual load was applied to the finite element model through a series of load steps. The change in the horizontal crest displacement of the model due to the loads is shown in Figure 7. The purpose for evaluating the individual loads steps was to provide engineers a better understanding regarding the contribution each load has to the overall behavior of the

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structure. It also provides a basis for comparison of the behavior between the total usual, extreme, and post-earthquake loading conditions.

Figure 7. STW crest deformation versus load steps.

Figure 8. Vertical stress contours for Load Step 2, original design load.

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Load Step 2 Original Design Load Condition

The vertical stress results due load step 2 are shown in Figure 8. The results show that tensile stress develops on the embankment face of the STW. These tensile stresses are greater than the allowable tensile strength (0.33 MPa, or 50 lb/in2), but less than the estimated breaking tensile strength for intact concrete (1.0 MPa, or 145 lb/in2). An important note; the historical performance of the STW for decades under this loading condition suggests that the STW behaved as the designers intended. Therefore, the following hypothesis regarding the known behavior of the STW and the results from the finite element analysis were discussed as a part of the evaluation process:

The finite element model representation of the load on the STW is relatively accurate, and the concrete strength is greater than assumed.

The structural capacity of the wall is greater than assumed in the finite element model.

The material parameters used to simulate the behavior of the embankment is overly conservative in the finite element analysis.

After review and discussion of the studies, it was concluded that both the two-dimensional finite element model and soil parameters used to simulate the behavior of the embankment are likely overly conservative. However, if the studies were to conclude that the STW has adequate capacity for all assumed loading conditions, then the conservative model would be acceptable and further refinement of the modeling assumptions would not be necessary. At the end of the studies, the stability against overturning and sliding for load step 2 was considered to be stable based on the historical performance.

Load Step 5 Current Load Condition

The evaluation for load step 5 included the effects of gravity, embankment pressures, hydrostatic pressures, post-tension anchors forces, and the added load due to the rockfill berm. This is considered the current condition of the STW. The vertical stress results in the STW are shown in Figure 9.

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Figure 9. Vertical stress contours for Load Step 5, current loading condition.

As expected, that additional load due to the rockfill berm results in an increase in tensile stress on the embankment face of the STW. As previously discussed, the installed sub-vertical post-tension tendons add moment capacity to the STW. The actual (demand) moment on the STW was compared to the moment capacity, as computed using the moment-interaction diagram shown on Figure 4. The results for the critical section of the STW are shown in Figure 10. The demand capacity ratio (DCR) for concrete is defined as the ratio of moment on the structure (demand), divided by the flexural capacity of the post-tensioned section, and is equal to 0.72. Based on the results, the potential for failure due to overstressing is very unlikely due to load step 5.

Figure 10. Moment interaction diagram for STW at EL. 150.6

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The potential for instability due to overturning and sliding for load step 5 was considered very unlikely, due to the historical performance for the STW.

Load Step 7 Final Design Load Combination (concrete buttress with decommission horizontal anchors)

The evaluation for load step 7 includes the effect of gravity, embankment pressures, hydrostatic pressures, post-tension anchors forces, rockfill berm, concrete buttress, and proposed decommissioning of the horizontal post-tension anchor. The vertical stress results are shown in Figure 11.

Figure 11. Vertical stress contours for Load Step 7, final loading condition

The results show that there is an increase in crest displacement towards the spillway of 5.5 millimeters due to the release of the horizontal anchor (see Figure 7).

The decommissioning of the horizontal anchor increases the load on the concrete buttress, as shown by the compressive stress contours in Figure 11. The results from the study showed that the concrete buttress will resists the movement of the STW as it rebounds towards the spillway after the decommissioning of the horizontal anchor. The stresses in the buttress are compressive, and below the allowable compressive limit of the concrete, and the demand moment is less than the capacity as shown in Figure 12.

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The results could indicate that the embankment side of the STW would undergo cracking, due to the tensile stresses. However, if cracking were to develop the load would be transferred to the vertical post-tension anchors. Therefore, to evaluate the structural capacity of the STW must include the effect of the anchors.

The bending demand and capacity for the STW are shown in Figure

12. The computed moment is within the moment-axial capacity curve, which means that the DCR is less than 1.0 (computed to be 0.78). Therefore, the wall is considered to have adequate structural capacity against overstressing.

Overturning is not a concern due to the post-tension anchors. In addition, since the DCR is less than one the potential for overturning is eliminated.

The computed sliding factors of safety for the sliding planes are all greater than or equal to the required sliding stability factor of 2.0, so the potential for sliding instability is unlikely.

Upper Section (OS 31) Extreme Load Combination (EXLC)

The extreme load combination was evaluated using several different seismic events to simulate possible ground motions at the dam site. The safety evaluation earthquake (SEE) corresponds to a 1-in-10,000 year event. Two of the scaled time-history accelerations from Coalinga and Mammoth Lake seismic events were used to simulate the SEE. The operational basis earthquake (OBE) was assumed to correspond to the 1-in- 2,500-year frequency event, and was simulated using a scaled record from the Landers seismic event.

The results from the seismic analysis showing the horizontal crest deflection of OS 31 to the STW is shown in Figure 13. From the results, the following observations can be made from the results:

The results from the 10,000-year seismic events indicate that the maximum amplitude of crest deflection will be approximately 18 millimeters, and there will be approximately 3 millimeters of permanent deformation towards the spillway.

Figure 12. Moment interaction diagram for STW at EL. 150.6

Figure 13. Crest deflection time-history for the seismic analysis

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The results from the 2,500-year seismic event indicate that the maximum amplitude of crest deflection will be approximately 7 millimeters, and there will be approximately 2.0 millimeters of permanent deformation towards the spillway.

The results from the finite element analysis show that the Coalinga earthquake event was the most severe seismic load and that the most critical time was 11.58 seconds into the earthquake. The vertical stress contours at time 11.58 seconds are shown on Figure 14. The results show an isolated area of tensile stress develops on the embankment face of the STW. The maximum stress is less than the assumed tensile strength of the concrete, which suggests that the structure has adequate capacity for this assumed load. However, if the concrete in the STW does not have any tensile strength (e.g. lift joints) then the section would crack and the load would be transferred to sub-vertical post-tensioned anchors.

Figure 14. Vertical stress results for the Coalinga seismic analysis.

The results from the analysis were also evaluated for the bending moment DCR. The DCR results during the Coalinga earthquake event for selected elevations in the STW are shown on Figure 15.

The critical section regarding moment demand versus capacity is at EL. 150, which corresponds to the top of the spillway slab. The spillway slab effectively fixes the base of the STW against significant rotation, which is why the maximum moment develops at that elevation. The computed DCR is less than 1.0 for all of the sections of the STW during the 10,000-year earthquake events. Based on these results, the STW will be expected to behave as a linear system, and instability due to overstressing is very unlikely.

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Figure 15. Bending Moment DCR for Coalinga seismic analysis.

Overturning is not a concern because of the post-tension anchor loads. The computed sliding factors of safety are greater than or equal the recommended value of 1.1. Therefore, the stability of the STW at OS 31 is considered adequate for the SEE.

Post-Earthquake Load Combination (PELC)

The post-earthquake load combination assumed that the saturated soil in the embankment will liquefy after the 10,000-year seismic event. The embankment properties were modified to simulate long-term static strengths in the dry soil, and liquefied strengths in the saturated soils. In addition, the concrete core wall in the embankment dam was assumed to be damaged during the seismic event, and breach through the core wall would increase the hydrostatic load on the STW.

Additional studies were performed assuming that a crack forms in the concrete at the most critical section to determine if there was a potential for instability of the STW. The results from the analysis showed that a potential crack would stabilize and would not propagate through the thickness of the STW. The results indicate that as cracking develops the load is transferred to the buttress, and is well within the capacity of the concrete buttress.

CONCLUSIONS

The major conclusions from the studies are summarized in the following list:

The finite element method was successfully used to simulate the horizontal embankment loads on the STW structure using the extended Druker-Prager material model.

The behavior of the STW during the seismic loading condition was evaluated using demand-capacity ratio, as defined by the USACE methodology. The results indicate that the structural behavior will remain linear and the potential for development of a

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seepage path between the STW and embankment material due to instability is very unlikely.

There was confidence in the results from the finite element model due to the sensitivity studies, which were able to verify the computed displacement from the model with the measured field deformations.

ACKNOWLEDGEMENTS

URS wishes to acknowledge State Water for the opportunity to work on this project. URS also wished to acknowledge the participation of the Hume Dam Expert Review Panel (ERP) for their expertise and reviewing of the study.

REFERENCES

1. American Concrete Institute, "Mass Concrete (ACI 207.1R-96), 1996.

2. FERC Guidelines. Federal Energy Regulatory Commission, Office of Hydropower Licensing, "Engineering Guidelines for the Evaluation of Hydropower Projects", Washington, D.C., October 1999, Chapter 11-Arch Dams. 3. U.S. Army Corps of Engineers, EM 1110-2-2100, Engineering and Design Stability Analysis of Concrete Structures, December 2005.

4. Lambe, T. William, Robert V. Whitman, Soil Mechanics, Massachusetts Institute of Technology, John Wiley & Sons, Inc. New York, 1969.

5. ACI 207R, Mass Concrete, Farmington Hills, MI

6. U.S. Army Corps of Engineers, EM 1110-2-6051, Engineering and Design Time History Dynamic Analysis of Concrete Hydraulic Structures, December 2003.

7. Federal Energy Regulatory Commission (FERC), Engineering Guidelines for the Evaluation of Hydropower Projects, Chapter 3 Gravity Dam, 2002.

8. Author: Sheldon Imaoka, ANSYS Revision 11.0, Memo Number STI:08/02, March 15, 2008, Subject Sheldons ANSYS.NET Tips and Tricks: Drucker-Prager Model.