Selecting Analytic Tools for Concrete 

Dams Address Key Events Along 

Potential Failure Mode Paths 

FEMA P-1016 / July 2014 

U. S. DEPARTMENT

OF 

HOMELAND SECURITY

FEMA
 

 

 

Selecting Analytic Tools for Concrete 

Dams to Address Key Events along 

Potential Failure Mode Paths 

Federal Emergency Management Agency
 

July 2014
 

ACRONYMS AND ABBREVIATIONS 

2D two-dimensional 

3D three-dimensional 

ACI American Concrete Institute 

DVD digital versatile disc 

FEM finite element method 

FEMA Federal Emergency Management Agency 

FERC Federal Energy Regulatory Commission 

ft feet 

g acceleration in relation to gravity (1 g = 32.2 ft/s2

) 

GT-STRUDL structural design and analysis software 

ICODS Interagency Committee on Dam Safety 

kip 

kip/ft3 

1,000 pounds 

1,000 pounds per cubic foot 

kip-ft 1,000 pound-feet 

lb/ft2 

lb/ft3 

lb/in2 

pounds per square foot 

pounds per cubic foot 

pounds per square inch 

NDSRB National Dam Safety Review Board 

PDF portable document format 

TVA Tennessee Valley Authority 

USACE United States Army Corps of Engineers 

 

 

 

 

 

 

 

 

CONVERSION FACTORS
 

To the International System of Units (SI) (Metric)
 

Pound-foot measurements in this document can be converted to SI measurements by multiplying 

by the following factors: 

Multiply By To obtain 

acre-feet 1233.489 cubic meters 

cubic feet 0.028317 cubic meters 

cubic feet per second 0.028317 cubic meters per second 

cubic inches 16.38706 cubic centimeters 

degrees Fahrenheit (°F-32)/1.8 degrees Celsius 

feet 0.304800 meters 

feet per second 0.304800 meters per second 

gallons 0.003785 cubic meters 

gallons 3.785412 liters 

gallons per minute 0.000063 cubic meters per second 

gallons per minute 0.063090 liters per second 

inches 2.540000 centimeters 

miles 1.609344 kilometers 

pounds 0.453592 kilograms 

pounds per cubic foot 16.01846 kilograms per cubic meter 

pounds per square foot 4.882428 kilograms per square meter 

pounds per square inch 6.894757 kilopascals 

pounds per square inch 6894.757 pascals 

square feet 0.092903 square meters 

square inches 6.451600 square centimeters 

 

STANDARDS 

Bureau of Reclamation 

State-of-Practice for the Non-Linear Structural Analysis of Concrete Dams at the Bureau of 

Reclamation, January 2006 

Design of Small Dams, Denver, Colorado, Third Edition, 1987 

Design of Gravity Dams, Denver, Colorado, 1976 

Design of Arch Dams, Denver, Colorado, 1977 

Publications are for sale at the National Technical Information Service: 

http://www.ntis.gov/ 

Federal Energy Regulatory Commission 

FERC Engineering Guidelines for the Evaluation of Hydropower Projects 

For further information on publications: 

http://www.ferc.gov/industries/hydropower/safety/guidelines.asp 
 

U.S. Army Corps of Engineers (USACE) 

EM 1110-2-1612 Ice Engineering, October 2002 

EM 1110-2-2100 Stability Analysis of Concrete Structures, December 2005. 

EM 1110-2-2104 Strength Design for Reinforced—Concrete Hydraulic 

Structures, August 2003 

EM 1110-2-2200	 Gravity Dam Design, June 1995 

EM 1110-2-2201	 Arch Dam Design, May 1994 

EM 1110-2-6050	 Response Spectra and Seismic Analysis for Concrete Hydraulic 

Structures, June 1999 

EM 1110-2-6051	 Engineering and Design—Time-History Dynamic Analysis of 

Concrete Hydraulic Structures, December 2003 

EM 1110-2-6053	 Engineering and Design—Earthquake Design and Evaluation of Concrete 

Hydraulic Structures, May 2007 

For further information on publications: 

http://140.194.76.129/publications/ 
 

WEB SITES
 

The following Web sites can provide additional information related to organizations associated 

with concrete dams: 

American Society of Civil Engineers: 

http://www.asce.org 

Association of State Dam Safety Officials: 

http://www.damsafety.org 

Bureau of Reclamation: 

http://www.usbr.gov 

Federal Emergency Management Agency: 

http://www.fema.gov/plan/prevent/damfailure
 

http://www.fema.gov/plan/prevent/damfailure/fema148.shtm
 

Federal Energy Regulatory Commission: 

http://www.ferc.gov/industries/hydropower.asp 

International Boundary and Water Commission: 

http://www.ibwc.gov 

International Commission on Large Dams: 

http://icold-cigb.net 

National Performance of Dams Program: 

http://npdp.stanford.edu 

Natural Resources Conservation Service: 

http://www.nrcs.usda.gov/wps/portal/nrcs/main/national/home 

Tennessee Valley Authority: 

http://www.tva.gov 

United States Society on Dams: 

http://www.ussdams.org 

U.S. Army Corps of Engineers: 

http://www.usace.army.mil 

 

 

 

 

 

 

 

 

 

 

 

 

 

PREFACE
 

The primary authors of this document in alphabetical order are Bruce Brand, P.E. (Federal
 

Energy Regulatory Commission); David A. Dollar, P.E. (U.S. Army Corps of Engineers);
 

Husein Hasan (Tennessee Valley Authority); Luis Hernandez (International Boundary and Water
 

Commission, U.S. Section); Larry K. Nuss, P.E. (Bureau of Reclamation); Rex Rowell, P.E. 

(Tennessee Valley Authority); and William A. Wallace, P.E., SECB (U.S. Department of
 

Agriculture, Natural Resources Conservation Service).
 

The technical editor for this manual was Lelon A. Lewis (Bureau of Reclamation). Additional 

technical assistance was provided by Cynthia Fields (Bureau of Reclamation) and Cindy Gray 

(Bureau of Reclamation).
 

Peer review of this manual was provided by Glenn Koester (Federal Energy Regulatory
 

Commission); Robert Hall, David Schaff, Travis Adams, Greg Werncke, and Rick Poeppelman
 

(U.S. Army Corps of Engineers); Jennifer Dickerson (Tennessee Valley Authority); Carlos Pena, 

Jr. (International Boundary and Water Commission, U.S. Section); Gregg Scott and 

Barbara Mills-Bria (Bureau of Reclamation); Stephen Reinsch (U.S. Department of 

Agriculture, Natural Resources Conservation Service); and Dan Hoang and Juan Uribe 

(Nuclear Regulatory Commission). 

The National Dam Safety Review Board (NDSRB) reviewed this manual prior to issuance. The 

NDSRB plays an important role in guiding the National Dam Safety Program. The NDSRB has 

responsibility for monitoring the safety and security of dams in the United States, advising the 

Director of the Federal Emergency Management Agency (FEMA) on national dam safety policy, 

consulting with the Director of FEMA for the purpose of establishing and maintaining a 

coordinated National Dam Safety Program, and monitoring State implementation of the 

assistance program. The NDSRB consists of five representatives appointed from Federal 

agencies, five State dam safety officials, and one representative from the U.S. Society on Dams. 

The authors also extend their appreciation to the following individuals for graciously providing 

information and permission to use their materials in this publication: Gregg A. Scott, 

Barbara Mills-Bria, and Chris Powell (Bureau of Reclamation) for their work upon which 

appendix A3 is based. 

The authors caution the users of this manual that sound engineering judgment should always be 

applied when using references. Users should be aware that certain portions of references cited in 

this manual may have become outdated in regard to design and construction aspects and/or 

philosophies. While these references may still contain valuable information, users should not 

automatically assume that the entire reference is suitable for design and construction purposes. 

The authors utilized many sources of information in the development of this manual, including: 

•	 Published design standards and technical publications of the various Federal agencies and 

organizations involved with the preparation of this manual. 

•	 Published professional papers and articles from selected authors, technical journals and 

publications, and organizations. 

i 
 

 

 

 

 

 

 

 

 

•	 Experience of the individuals, Federal agencies, and organizations involved in the
 

preparation of this manual.
 

Suggestions for changes, corrections, or updates to this manual should be directed to: 

Bureau of Reclamation 

Denver Federal Center, Bldg. 67 

ATTN: Director of Dam Safety, 86-45000 

6th Avenue and Kipling Street 

Denver, CO 80225-0007 

Please reference specific pages, paragraphs, or figures within the manual, together with proposed 

new material in any convenient format. Sources of proposed new material should be completely 

cited. Submission of material signifies permission for use in a future revised edition of this 

manual, but credit for such new material will be given where appropriate. 

The material presented in this manual has been prepared in accordance with recognized 

engineering practices. The guidance in this manual should not be used without first securing 

competent advice with respect to its suitability for any given application. The publication of the 

material contained herein is not intended as representation or warranty on the part of individuals 

or agencies involved, or any other person named herein, that this information is suitable for any 

general or particular use, or promises freedom from infringement of any patent or patents. 

Anyone making use of this information assumes all liability from such use. Any use of trade 

names and trademarks in this manual is for descriptive purposes only and does not constitute 

endorsement. The information contained herein regarding commercial products or firms may not 

be used for advertising or promotional purposes and is not to be construed as an endorsement of 

any product or firm. 

Various computer codes were used in developing the example problems. The uses of these codes 

are not an endorsement or recommendation for their use. 

ii 
CONTENTS
 

Page 

1.1 Introduction ..........................................................................................................................1
 

2.1 Theory and Types of Concrete Dams...................................................................................1
 

2.1.1 Gravity Dams........................................................................................................ 2
 

2.1.2 Arch Dams ............................................................................................................ 2
 

2.1.3 Buttress Dams ....................................................................................................... 3
 

3.1 Potential Failure Modes........................................................................................................6
 

3.1.1 Defining Failure Criteria....................................................................................... 6
 

3.1.2 Developing Potential Failure Modes .................................................................... 6
 

3.1.3 Sequence of Events............................................................................................. 10
 

3.1.4 Event Trees ......................................................................................................... 12
 

3.1.4.1 Advantages.......................................................................................... 13
 

3.1.4.2 General Principles............................................................................... 13
 

4.1 Relating Analyses to Failure Modes ..................................................................................15
 

4.1.1 Limit Equilibrium Analysis ................................................................................ 15
 

4.1.2 Linear Finite Element Analyses.......................................................................... 17
 

4.1.3 Nonlinear Finite Element Analyses .................................................................... 19
 

4.1.4 Static and Dynamic Analyses ............................................................................. 23
 

4.1.5 The Role of Standards-Based Criteria and Codes............................................... 25
 

5.1 Focusing on Parameters Affecting Analysis ......................................................................25
 

5.1.1 Common Errors................................................................................................... 28
 

5.1.1.1 Unrealistic Boundary Conditions........................................................ 29
 

5.1.1.2 Finite Element Mesh Density.............................................................. 30
 

5.1.1.3 Damping.............................................................................................. 32
 

5.1.1.4 Failure Plane Tension/Cohesion ......................................................... 33
 

5.1.2 Summary............................................................................................................. 33
 

6.1 Uncertainty and Confidence...............................................................................................34
 

6.1.1 Uncertainty.......................................................................................................... 34
 

6.1.2 Confidence .......................................................................................................... 36
 

7.1 Conclusions ........................................................................................................................38
 

Bibliography ..................................................................................................................................41
 

Tables 

Table Page 

4-1 Limitations of structural analysis methods to compute the structural response 

for various sequences of events of a potential failure mode....................................... 16
 

4-2 General differences in various structural analysis methods............................................ 17
 

4-3 Results of limit equilibrium analysis .............................................................................. 18
 

4-4 Forces, centroids, and moment ....................................................................................... 24
 

iii 
Figures 

Figure Page 

2-1 Gravity dam, Bureau of Reclamation. .............................................................................. 2
 

2-2 Arch dam, Bureau of Reclamation.................................................................................... 3
 

2-3 Slab and buttress dam, Bureau of Reclamation. ............................................................... 4
 

2-4 Massive-head buttress dam, Bureau of Reclamation........................................................ 4
 

2-5 Multiple-arch buttress dam, Bureau of Reclamation. ....................................................... 5
 

2-6 Multiple dome buttress dam, Bureau of Indian Affairs. ................................................... 5
 

3-1 Construction photograph of a curved gravity dam showing the many complex
 

features: contraction joints, shear keys, lift lines, thermal expansions, grouting, 

foundation discontinuities, material strengths, penstocks through the dam, outlet 

works, spillway sections, and spillway gates, Bureau of Reclamation......................... 7
 

3-2 Potential failure modes given an initiating event.............................................................. 8
 

3-3 Possible failure mechanisms of a concrete gravity dam. ................................................ 10
 

3-4 Event tree. In this example, increased reservoir load could cause sliding instability..... 14
 

4-1 Limit equilibrium analysis. ............................................................................................. 18
 

4-2 Linear finite element model: principal stress vectors. .................................................... 20
 

4-3 Base stress from linear finite element analysis............................................................... 20
 

4-4 Base stress from nonlinear finite element analyses and limit equilibrium analysis. ...... 21
 

4-5 Irregular base with nonlinear finite element analyses..................................................... 22
 

4-6 Results from irregular base with nonlinear finite element analyses. .............................. 23
 

4-7 Results from applying instantaneous peak dynamic loads. ............................................ 24
 

5-1 Base cracking example. .................................................................................................. 27
 

5-2 Base pressure distribution through a section of a concrete dam..................................... 27
 

5-3 Modular ratio effects (note that this curve is specific for the example cited)................. 28
 

5-4 Buttress dam finite element idealization......................................................................... 29
 

5-5 Fine finite element mesh at reentrant corner................................................................... 31
 

5-6 Stresses from the fine finite element mesh. .................................................................... 31
 

5-7 Coarse finite element model and stress results. .............................................................. 32
 

6-1 Uncertainty bands around a seismic hazard curve (peak acceleration vs. exceedance 

probability).................................................................................................................. 35
 

Appendices 

Appendix 

A1 Increased Reservoir Level Causes Sliding in a Gravity Dam 

A2 Earthquake Causes Sliding in a Gravity Dam 

A3 Earthquake Causes Sliding in Foundation Leading to Dam Failure 

A4 Increased Silt Load Causes Failure of Reinforced Concrete Corbels of a Buttress Dam 

A5 Freeze-Thaw Deterioration Removes Concrete Cover and Causes Upstream Slab to Fall 

B Loads and Loading Requirements 

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1.1 Introduction 

The purpose of this document is to stress the importance of understanding the sequences of 

events leading to failure of concrete dams and selecting analysis methods that address these 

specific events. The selected analysis method may range from straightforward to complex, 

depending on the potential failure mode being analyzed. It is stressed that a less complex 

analysis with less uncertainty is the preferred strategy. Included in this report and the appendices 

are examples of this process. These examples are not intended to be a complete listing of 

concrete dam potential failure modes. Each dam is unique and has its own issues; therefore, it is 

important for the engineer to understand the potential failure modes and sequences of events that 

enable them. 

In this document, failure is considered the uncontrolled release of the reservoir. However, this 

may or may not always be the case given the purpose or hazard of a structure or given an 

agency’s requirements for a structure. How do we determine if a concrete dam can fail? Failure 

results from sequences of events that must follow one upon another. Because a dam cannot fail 

without the full chain of events, conclusively ruling out any event justifies concluding that the 

dam will not fail. This document introduces event trees. Event trees are pictorial representations 

of the sequences of events (called nodes) leading to failure. The possibility of each node 

occurring is evaluated by analyses. 

It is the experience of the authors that engineers often rush to analysis without consideration of 

the failure process. As a result, expensive analyses are done that are not needed, waste time, and 

often do not answer the question of whether a dam will fail. 

This document represents the collective experience of the Bureau of Reclamation, Federal 

Energy Regulatory Commission (FERC), International Boundary and Water Commission, 

U.S. Section, Natural Resources Conservation Service, Tennessee Valley Authority, and 

U.S. Army Corps of Engineers (USACE). 

Note that this document should not be substituted for design criteria such as those established by 

these agencies. 

Examples presented are for instructional purposes only. Assumptions made, material properties 

used, and loadings were selected to illustrate structural analysis methods and failure modes, and 

are not generally applicable. 

2.1 Theory and Types of Concrete Dams 

This document addresses structural analysis of concrete dams related to potential failure modes. 

Embankment dams are not addressed in this document. The three basic types of concrete dams 

are gravity dams, arch dams, and buttress dams. 

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Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

2.1.1 Gravity Dams 

Gravity dams (figure 2-1) derive their strength from their weight. Shear resistance, which is a 

function of weight, friction angle, and cohesion, controls sliding. A gravity dam’s weight, as 

defined by its geometry and concrete density, and its uplift pressures control overturning 

stability. All failure modes must somehow either decrease the net effect of the weight of the 

structure or the resistance against sliding and overturning, or increase the driving forces. In most 

situations, all three of these occur. 

Figure 2-1: Gravity dam, Bureau of Reclamation. 

The majority of gravity dams fall into the category of straight gravity dams. The upstream face 

of these dams is typically straight in plan view and perpendicular to the stream channel. Curved 

gravity dams, a subset of gravity dams, do not rely on the curvature for static stability. However, 

the curvature may increase their ability to withstand changes in loads from reductions in 

drainage, flooding, or seismic events. 

2.1.2 Arch Dams 

Arch dams (figure 2-2) distribute loads vertically to the foundation by cantilever action and 

horizontally to the abutments by arching action. The load distribution depends on the curvature 

of the dam, structural features, and material strength. Because concrete is strong in compression 

and weak in tension, arch dam design is based on maximizing compressive stress distributions 

and minimizing tensile stress distributions. Since the load is distributed to the foundation and 

abutments, the geometry and shear strength of the abutment contact are important. An arch dam 

relies on the abutments for stability. 

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Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

Figure 2-2: Arch dam, Bureau of Reclamation. 

2.1.3 Buttress Dams 

Buttress dams consist of buttresses transverse to the dam axis that support longitudinal elements, 

such as flat slabs, domes, massive heads, or arches that contain the reservoir. The upstream face 

of the dam is typically sloped up to about a 45-degree angle. Therefore, hydrostatic pressures 

that produce the driving forces also provide a downward vertical force that adds to structural 

stability. Overall stability of buttress dams is accomplished by the same means as for gravity 

dams. 

A variety of dams fall into this category. A slab and buttress dam (figure 2-3) typically consists 

of buttresses, flat or tapered slabs, and struts between the buttresses. For a massive-head buttress 

dam (figure 2-4), the geometry of the individual buttresses consists of a downstream stem that is 

widened at the upstream face to form a “massive head.” The massive heads of adjacent buttresses 

are in contact typically with water-stops between them, providing a watertight upstream face. For 

a multiple-arch (figure 2-5) or dome (figure 2-6) buttress dam, the arches are integral parts of the 

upstream side of the buttresses, and reinforcement is continuous from the arches into the 

buttresses. 

The main advantage of a buttress dam is reduced concrete volumes. Many of these dams were 

constructed when labor and steel costs were more than offset by the cost savings of reduced 

concrete volume. At or above the foundation, uplift affects buttress dams less than gravity dams 

because uplift pressures are reduced between the buttresses, and the remaining uplift pressures 

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Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

Figure 2-3: Slab and buttress dam, Bureau of Reclamation. 

Figure 2-4: Massive-head buttress dam, Bureau of Reclamation. 

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Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

Figure 2-5: Multiple-arch buttress dam, Bureau of Reclamation. 

Figure 2-6: Multiple dome buttress dam, Bureau of Indian Affairs. 

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Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

act on the relatively small surface area of the buttresses’ footprint. Historically, this type of dam 

was considered to be more forgiving to variations in foundation conditions. However, it must be 

realized that each individual buttress must be stable on its own. Failure modes associated with 

buttress dams include the typical gravity dam failure modes, as well as modes associated with 

lateral movement of the buttresses and strength-related modes of the reinforced concrete 

elements. 

3.1 Potential Failure Modes 

Identifying, fully describing, and evaluating site-specific potential failure modes and sequences 

leading to failure are arguably the most important initial steps in conducting a structural 

analysis for a concrete dam. The process this document suggests is to: (1) define failure 

criteria, (2) identify potential failure modes, (3) develop a sequence of events (nodes) for a 

failure to transpire, (4) select a node with the most likely chance to circumvent the failure 

process, (5) select a structural analyses method to compute the response of the dam at that node, 

(6) quantify the uncertainty, and (7) build the case that the failure process terminates or does not 

terminate at that node. This chapter explains steps 1 through 3. Appendix A provides five 

examples of the entire process. 

3.1.1 Defining Failure Criteria 

Failure for dams is usually defined as uncontrolled release of the reservoir. This may or may not 

always be the case, and the purpose and hazard of the structure, as well as agency requirements, 

may change the definition of failure. In the examples in appendix A, failure is defined as 

uncontrolled release of the reservoir. 

3.1.2 Developing Potential Failure Modes 

One person can develop potential failure modes, or a multidisciplined team can. It all depends on 

the intended use for the potential failure modes and how comprehensive they need to be. A 

facilitated multi-discipline team is best for developing potential failure modes for a concrete dam 

because concrete dams are complex structures—as shown in the curved gravity dam under 

construction in figure 3-1—and synergy develops in a group. A complete understanding of the 

structure involves team members with specialties in hydrology and hydraulics, seismology, 

concrete construction, concrete materials, structural stability, foundation materials, rock 

mechanics, foundation stability, and operations of the dam. The team would consist minimally of 

a structural engineer, geotechnical engineer, and geologist. The team is greatly enhanced when 

field personnel are included. Materials engineers are included when there are issues with the 

concrete or foundation. Specialists in hydrology or seismology are included when these are 

issues (FERC, 2005). 

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Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

The team should review initial designs and assumptions, construction records, historic 

inspections, as-built drawings, material testing, field investigations, rehabilitations, 

instrumentation data, structural and stability analyses, and current operations. The team 

brainstorms potential failure modes after reviewing historic data and before visiting the site. The 

potential failure modes are then reviewed after a site visit. Site inspections would identify new 

misalignments, deterioration, seepage, plugged drains, and cracking that may not be in the 

historic records. 

Figure 3-1: Construction photograph of a curved gravity dam showing the 

many complex features: contraction joints, shear keys, lift lines, thermal 

expansions, grouting, foundation discontinuities, material strengths, 

penstocks through the dam, outlet works, spillway sections, and spillway 

gates, Bureau of Reclamation. 

Failures start with some initiating event that causes an adverse change in the structure. During 

normal operating conditions, these actions might be clogging of drains leading to increased uplift 

pressures, degradation of the grout curtain leading to increased seepage and increased uplift 

pressures, alkali-aggregate reaction in the concrete leading to reduced concrete strength, or 

corrosion of the reinforcing steel in a buttress dam leading to reduced capacity of the member. 

During a hydrologic event, these actions might be a rising reservoir that increases hydrostatic 

loads on the structure, a rising reservoir that overtops the dam and erodes the foundation, or 

increasing flow through the spillway that causes cavitation. During a seismic event, increased 

inertial forces on the structure might lead to overstressing, or displacement of the foundation 

under the dam could cause overstressing or misalignment. See appendix B for a discussion of 

loads. 

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Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

There are two basic methods of developing potential failure modes. Both should be used to make 

sure all modes are captured. 

The first method starts with initiating events (or loads) like the ones listed in appendix B, and 

then determines the impact of the initiating events on the structure (figure 3-2). For instance: 

•	 Increasing reservoir level: 

o	 Increases stress levels, cracks the dam, and causes the dam to slide 

o	 Overtops the dam, erodes the abutment, and causes sliding 

•	 Increasing uplift pressure: 

o	 Reduces the normal force on slide planes and increases the potential for sliding by 

reducing frictional resistance 

•	 Seismic load: 

o	 Increases stress levels, cracks the dam, and causes the dam to slide 

Figure 3-2: Potential failure modes given an initiating event. 

Cracking and Sliding

Cracking and Sliding

Deterioration:

Loss of Weight

Loss of Strength

Erosion Undercuts

Dam Causing Loss

of Bearing or

Sliding

Sliding Along Discontinuities

Sliding Along Lift Joint

Sliding Along Contact

Cracking, Crushing, or

Sliding

Thermal Cracking – Loss of

Strength Along Crack

Loss of Bearing Strength

Cracking and Sliding 

Cracking and Sliding 

Deterioration: 

Loss of Weight 

Loss of Strength 

Erosion Undercuts 

Dam Causing Loss 

of Bearing or 

Sliding 

Sliding Along Discontinuities 

Sliding Along Lift Joint 

Sliding Along Contact 

Cracking, Crushing, or 

Sliding 

Thermal Cracking – Loss of 

Strength Along Crack 

Loss of Bearing Strength 

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Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

• Alkali-aggregate reaction: 

o Reduces the strength of the concrete and reduces the load-carrying capacity 

o Expands the concrete mass and binds mechanical equipment 

• Human interaction: 

o Causes a spillway gate failure, causing the reservoir to rise 

o Fails to open the spillway gates, causing the reservoir to rise 

• Landslide: 

o Causes a large rock mass to move into the reservoir, causing the reservoir to rise 

o Causes a large wave in the reservoir that overtops the dam 

• Leaching of foundation material: 

o Reduces the bearing capacity in the foundation, and the dam settles 

The second method starts with possible failure modes and then determines what types of loads 

would cause the failure modes. For example, looking at the concrete gravity dam in figure 3-3, 

the possible modes of failure and the possible loads that might cause that failure are: 

• Sliding along an unbonded lift joint 

o Causes: reservoir water, earthquake, ice, silt, or increase in uplift 

• Concentration of stress at upstream change in geometry results in cracking, then sliding 

o Causes: increase in reservoir, ice, silt, or earthquake 

• Loss of foundation bearing strength 

o Causes: leaching of material from the foundation mass 

o Causes: piping of soil material due to seepage 

o Causes: erosion of the foundation and undercutting during overtopping 

• Sliding along a discontinuity in the foundation 

o Causes: increase in reservoir, earthquake, or increase in uplift 

• Loss of material along the downstream face 

o Causes: freeze-thaw damage or overtopping flows 

Once potential failure modes were identified, the team would rank the modes and identify the 

most critical. 

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Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

Figure 3-3: Possible failure mechanisms of a concrete gravity dam. 

3.1.3 Sequence of Events 

After establishing possible modes of failure, the more critical potential failure modes would then 

be described in detail. The entire sequence of events (nodes) leading to failure is developed: 

(1) to ensure that there is a common understanding of the failure mode, (2) to ensure reviewers in 

the future will have a clear understanding of what was considered, and (3) because an event tree 

can be developed to further explain and show the sequences. 

For instance, during a flood, a dam might fail by sliding. A sequence of events (nodes) for this 

potential failure mode that would need to occur might be: (node 1) flooding with reservoir above 

elevation 960 feet, yet below the top of the parapet walls, causes tension at the heel of the dam 

and potentially causes the concrete to crack; (node 2) uplift pressures increase in the crack, 

causing the crack to propagate through the thickness of the dam; (node 3) the driving forces (and 

reduced normal force on the crack) on the dam are higher than the shear strength along the 

cracked surface, and sliding starts; (node 4) there are no mechanisms that stop the sliding, and 

the dam fails. 

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Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

More importantly, developing the entire sequence of events provides a clearer understanding for 

the structural engineer of what is needed from a structural analysis. In the above example, a 

structural analysis targeted on node 1 would compute the tensile stress at the heel due to flood 

loading above elevation 960 feet, on node 2 would compute crack length and uplift pressures, on 

node 3 would compute sliding factors of safety, or on node 4 would identify mechanisms that 

might stop sliding if sliding starts. 

When describing the sequences of events, it is desirable to have specific details. For example, the 

following shows an insufficient description and a sufficient description of the sequence of events 

for a particular potential failure mode: 

• Without sufficient detail: Sliding along an unbonded lift joint in the dam. 

•	 With sufficient detail: As a result of high reservoir levels, sliding commences on an 

unbonded lift joint in buttress number 26 at elevation 950 feet, identified from a coring 

program in 1995. As the portion of the buttress above elevation 950 feet slides, the 

minimal reinforcing steel along the lift joint yields, and sliding continues. Differential 

movement between the buttresses causes misalignment of the upstream face slab. The 

face slab becomes unseated along the simple support condition along the corbel and face 

slab. The shear key along the face slab and buttress provides minimal resistance to 

sliding. The lift joint between slabs opens and eliminates the 3-dimensional (3D) support 

along the face slab, and the slab fails because of lack of support or overstressing. Breach 

potentially occurs in two bays on either side of the buttress down to elevation 950 feet. 

The following are more examples of initiating events and the sequences that need to occur for 

failure: 

•	 Deterioration of concrete under all loads: This potential failure mode results from 

deteriorating and weakening of the concrete at specific locations. The deterioration of the 

concrete is allowed to progress to the point where the applied loads are greater than the 

strength of the concrete, and failure occurs. The loads might be during normal operating 

conditions, during a flood, or during a seismic event. The critical initiating event occurs 

when the strength of the concrete is less than the applied stress. The structure might fail 

in different ways, depending on whether the structure is loaded in tension or 

compression, or by misalignment. 

1. Concrete deteriorates and loses strength over time at elevation 234 feet 

2. An earthquake occurs on the XYZ fault producing 0.63 g 

3. Structure deforms at least 10 inches, significantly more than usual 

4. Misalignment causes redistribution of load 

5. Structure cannot carry redistributed load 

6. Structure fails 

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•	 Flood loading overloads concrete dam: This potential failure mode occurs during a flood 

when increased water and uplift overload the dam causing cracking of concrete to the 

point that the dam fails by sliding. 

1. Reservoir water surface at or above a given level 

2. Tensile stresses increase on upstream face in block 5 

3. Concrete cracks 

4. Uplift increases in the crack 

5. Dam cracks through the thickness of the dam 

6. Sliding commences, and dam fails 

•	 Uplift increases under concrete dam: This potential failure mode occurs when the drains 

in the dam become ineffective due to plugging and uplift pressures increase under the 

dam. The normal force along a horizontal slide plane reduces the frictional resistance, 

and the dam fails due to sliding. 

1. Drains become less than 25 percent effective, causing an increase in uplift pressures 

2. Reduced normal force on horizontal slide plane decreases frictional resistance 

3. Sliding commences, and dam fails during normal operating conditions 

•	 Human error when operating spillway: This potential failure mode occurs when there is a 

human error when operating a spillway gate. In this case, a mechanism is initiated that 

raises the gates to their full height without the ability to stop the gate from rising. There is 

uncontrolled release through the spillway. 

1. Spillway gates rise to full height because the stop switch is not pushed 

2. There is no intervention possible to stop the gates from rising 

3. Unexpected downstream flow causes consequences 

•	 Earthquake overloads foundation blocks and causes failure of dam: This potential failure 

mode is caused by an earthquake that increases loads from the dam into the foundation 

and also causes inertia forces of the foundation blocks. It has been established that there 

are removable foundation blocks in the abutment. 

1. Earthquakes above 0.5 g acceleration increases forces from the dam into the foundation 

blocks 

2. Foundation blocks become unstable with 40-degree friction and move 

3. Deformations occur in the dam with 4 feet of foundation movement 

4. Dam fails due to load redistribution 

3.1.4 Event Trees 

An event tree is a graphical representation of the potential progression of failure engineer. It 

provides an efficient way to organize the chronological sequence of events for a particular 

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potential failure mode from the initiating cause on the left, through a series of linked events 

(nodes or branches), to the failure or no failure condition on the right (see figure 3-4). Each node 

represents an event or condition with possible outcomes or states that need to exist for failure to 

ultimately occur or not occur. 

Event trees can be developed in a qualitative sense without assigning probabilities and used in a 

failure-modes-and-effects analysis. The same qualitative event trees can be used and expanded 

in a quantitative risk analysis. Event trees are easily understood because they portray the 

chronological sequence of events that must occur for failure to happen. Event trees aid in 

understanding of the steps needed in a potential failure mode. This decomposition aids in the 

structural analyst’s understanding the failure mode and also in briefings to management. 

3.1.4.1 Advantages 

Event trees have the advantage of being: 

•	 Well suited for displaying the chronological order of events 

•	 Well suited for displaying dependencies between events and in which order they occur 

•	 Able to facilitate communication about assumptions in developing the model 

•	 Easily understood by managers 

•	 Well suited to display details of the problem 

•	 Drawn on standard computer spreadsheets or specialty software 

•	 Intuitive because the events from left to right are the sequence of events that occur given 

the previous event or condition to the left 

3.1.4.2 General Principles 

Event trees visually show the sequence of events leading to failure. Event trees also follow the 

basic principles of probability theory and can be used to quantify the probability of the failure 

mode if likelihoods are assigned at each node of the event tree. This document describes 

potential failure modes and associated structural analysis. This document does not discuss 

probability or risks, but the following list shows how an event tree can be used to quantify risks. 

•	 Each branch from a node must be mutually exclusive from the other events so that there 

is no overlap in probability between branches. In other words, probabilities for events 

emanating from a single node of an event tree add to 1. 

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Figure 3-4: Event tree. In this example, increased reservoir load could cause sliding instability. 

•	 Probabilities are positive. There is no such thing as a negative probability. 

•	 If probabilities are assigned to a node, the probabilities are between 0 and 1. In percent, 

this is between 0 and 100 percent. There is no such thing as a 110-percent chance of an 

event occurring. 

•	 If probabilities are assigned to nodes, the probabilities along each path can be multiplied 

together from left to right. The product of multiplying probabilities along a path is the 

probability of occurrence of that event. 

•	 If probabilities are assigned to an event tree, adding the products of all the failure paths in 

the event tree for a given initiating condition or event gives the probability of having a 

failure due to the initiating event. 

•	 The occurrence of every event in the tree is conditional on the event to the left having 

occurred. 

An initiating event could produce multiple failure modes. For example, a flood may produce 

sliding or overturning in a gravity dam. Each potential failure mode requires its own tree. 

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4.1 Relating Analyses to Failure Modes 

Analyzing potential failure modes allows the engineer to characterize how a structure might fail 

and identify the sequence of events leading to failure. Structural analysis is used to determine the 

structural response of a given node. For instance, the first event might be that an increased load 

causes the concrete to crack. There are various methods to determine if the concrete will crack, 

such as structural analysis, physical testing, or scale-model testing. Structural analysis uses 

either limit equilibrium analysis or various types of finite element methods (FEM), namely, 

2-dimensional (2D) linear elastic, 2D nonlinear, 3D linear elastic, or 3D nonlinear. 

It must be realized that each one of these analysis methods has a certain level of applicability and 

represents a certain level of reality. Table 4-1 summarizes the applicability of different analysis 

techniques. The technique chosen should be the one that answers the specific question being 

asked with minimum effort. These methods also vary greatly in cost, difficulty, and level of 

effort as generally indicated in table 4-2. Of course, these indicators depend on the availability of 

finite element mesh generators, post-processing capabilities, and the experience of the structural 

analyst. The acceptability of the results depends on the acceptability of the material properties 

being used, the acceptability of the loading condition, and the way the structure was modeled. An 

assumed, yet conservative, material property may be sufficient without testing to satisfy stability 

requirements. However, more importantly, applicability of the analysis method being used 

depends on the question being asked. 

The following examples compare and contrast different analysis techniques as they are applied to 

the same dam. 

4.1.1 Limit Equilibrium Analysis 

Traditionally, analyses of the stability of dams were based on a set of assumed failure modes. 

The failure modes consisted of rigid body motions that were kinematically admissible. If force 

and moment equilibrium could be satisfied within the limits of the foundation and concrete 

strength, the dam was considered stable. 

Figure 4-1 and table 4-3 show the results of a conventional 2D limit equilibrium analysis. In this 

case, stability means that the resultant vector lies within the footprint of the dam, so that rigid 

body overturning will not occur, and that the resultant intersects the foundation at an angle that is 

not more acute than the foundation friction angle, thus preventing sliding. 

When applicable, limit equilibrium analyses provide unambiguous answers to the question of 

dam stability because they presuppose a failure mode and then evaluate its possibility. Their 

results are easily understood and can be presented concisely as shown in table 4-3. They are 

attractive for this reason. 

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Table 4-1: Limitations of structural analysis methods to compute the structural response for various sequences of events 

of a potential failure mode 

Initiating event: Earthquake 

Failure mode: Sliding in a concrete gravity dam 

Sequence of events initiated by seismic loading 

Upstream 

cracks appear 

Cracks 

penetrate 

through 

dam 

Sliding 

begins 

Duration of 

shaking 

fails dam 

Dam fails 

after 

earthquake 

Considerations for types of analyses / capability 

Method 

Can upstream 

stresses be 

computed? 

Can the analysis 

show if the dam 

cracks through? 

Can analysis 

show if the dam 

slides given loads 

and shear 

strengths? 

Can analysis 

show how far the 

dam can slide? 

Can the analysis 

perform a postseismic stability 

study? 

Limit Yes, assumes Yes, pre-supposes Yes, it can Yes, using a Yes, only sliding 

equilibrium plane sections 

remain plane, so 

lacks accuracy 

crack determine if 

sliding commences 

Newmark type 

approach 

factor of safety 

computed 

2D FEM Yes, includes No, computed No, only sliding No, unrealistic No, only sliding 

linear flexibility but no 

3D effects 

stresses can be 

higher than the 

strength 

factor of safety 

computed by 

integrating stresses 

linear strength 

carries all imposed 

loads 

factor of safety 

computed by 

integrating stresses 

2D FEM Yes, includes 2D Yes, but 3D effects Yes, but 3D effects Yes, but 3D effects Yes, but 3D effects 

nonlinear flexibility and 

nonlinear but no 

3D effects 

not included not included not included not included 

3D FEM Yes, includes 3D No, computed No, only sliding No, unrealistic No, only sliding 

linear flexibility but no 

nonlinearity 

stresses can be 

higher than the 

strength 

factor of safety 

computed by 

integrating stresses 

linear strength 

carries all imposed 

loads 

factor of safety 

computed by 

integrating stresses 

3D FEM Yes, best available, Yes, best available Yes, best available Yes, best available Yes, best available 

nonlinear includes 3D 

flexibility and 

nonlinearity 

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Table 4-2: General differences in various structural analysis methods 

Method Cost 

Difficulty: easy (x); 

difficult (xxxxxxxxxx) 

Time to 

complete 

Limit equilibrium $ x Hours 

2D FEM linear $$ xxx Days 

2D FEM nonlinear $$$ xxxxx Weeks 

3D FEM linear $$$$ xxxxxxx Month 

3D FEM nonlinear $$$$$ xxxxxxxxx Months 

Note: Cost, difficulty, and timeframe symbols are general qualitative indicators and not 

directly relatable to each other. For instance, $$$ does not indicate 3 times the cost of $. 

The ratings depend on the level of expertise of the engineer, on the availability of the 

software and preprocessors, and on having the input ready to go. 

The 2D gravity dam analysis shown below is accurate if the underlying assumptions apply to the 

case being studied. A few of these underlying assumptions need to be considered, in particular 

that: 

• The base of the dam is approximately planar 

• Normal stress distribution on the dam base is trapezoidal 

• Tension between the base and foundation is not allowed 

• 2D analysis is appropriate 

4.1.2 Linear Finite Element Analyses 

Linear analyses have a critical role in structural analysis toolboxes. Results from linear finite 

element analyses provide an essential baseline to compare to nonlinear analyses, determine the 

natural frequency of the structure, and provide stresses and displacements in the dam for 

behavior in the linear range of behavior. If results stay in the linear range, it is not necessary to 

conduct nonlinear analyses. Nonlinear analyses run in “linear mode” should produce results 

comparable to linear analysis results. Computed stress levels that are unrealistic or well into the 

nonlinear range of the material justify performing nonlinear analyses. 

Linear finite element analyses in the dynamic realm are useful in determining natural frequencies 

of vibration and mode shapes. This information is useful in selecting site-specific ground 

motions. 

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Figure 4-1: Limit equilibrium analysis. 

Table 4-3: Results of limit equilibrium analysis 

Force Fx (kip) Y= (ft) Fy (kip) X= (ft) 

Moment (0,0) 

(kip-ft) 

Dam weight 

Headwater 

Tailwater 

Uplift 

0.00 

281.58 

-3.12 

38.74 

0.00 

131.67 

93.33 

96.30 

-630.00 

0.00 

-2.18 

309.94 

26.55 

0.00 

77.67 

29.63 

16,725.00 

37,074.70 

-121.58 

-5,451.91 

Totals 317.20 -322.24 48,226.21 

Notes: 

1. Resultant location at X = 58.4 ft, Y = 92.7 ft, 19 percent of base cracked 

2. Sliding safety factor = 0.79, 51.6 degrees required for factor of 1.0 

3. Input parameters: 

Concrete unit weight = 0.15 kip/ft3 

Foundation friction angle = 45 degrees 

Head/tailwater elevations = 195/100 ft 

kip = 1,000 pounds, ft = feet, kip-ft = 1,000 pound-feet, kip/ft3 = 1,000 pounds per cubic foot 

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However, linear finite element analysis does not address failure. It answers a different question, 

providing stresses and deflections subject to a different set of assumptions. The assumption 

underlying linear finite element analysis is that the dam and foundation form a continuous, 

linear, elastic solid. Cracking, crushing, or slipping cannot be directly modeled. Because it is not 

sufficient in itself to evaluate a dam at impending failure, it cannot be compared directly with 

limit equilibrium analysis. Tension (sometimes unrealistic tension) is allowed between the dam 

and foundation. However, even with these limitations, linear finite element analysis can provide 

some insight into dam behavior. For dynamic loading, the vibration mode shapes and natural 

frequencies generated by linear analysis are of some interest. For reinforced concrete structures 

such as slab and buttress dams, moment and force resultants from linear models are useful in 

evaluating the adequacy of members. 

A linear finite element analysis of the dam depicted in figure 4-1 is shown in figure 4-2. Because 

there is an assumption of elastic continuity between the dam and the foundation, uplift is 

assumed to vary linearly from headwater to tailwater. Note that in figure 4-2, tensile stress 

develops at the heel of the dam. A stress singularity occurs at the reentrant corner. The question 

that suggests itself is, “Will cracking occur as a result of this tension?” This question was 

disallowed in the limit equilibrium analysis because one of the initial assumptions was a 

trapezoidal base pressure distribution with no tension. 

Answering the question of whether cracking will occur does not specifically address any failure 

mode. All concrete dams have cracks in them, and the fact that an analysis indicates a crack is 

likely at the upstream heel is not in itself conclusive. What is of some interest is the distribution 

of shear and normal stresses that the finite element analysis reveals (see figure 4-3). 

Note that the normal stress is not trapezoidal as was assumed in the limit equilibrium analysis. 

There is a large tensile peak near the heel, as was evident in the principal stress vector plot in 

figure 4-2. Also, there is a compressive spike near the toe. The shear stress variation, which is 

not typically dealt with in limit state analysis, has peaks near the heel and toe of the dam that are 

over twice the average shear stress. If cohesion were relied upon for stability, the cohesive shear 

strength would have to be in excess of the highest peak shear stress, not just the average. Note 

also that the high shear stress near the heel is in a tensile region. 

Limit equilibrium analysis evaluates failure modes (sliding and overturning) subject to a base 

pressure distribution that may not be very accurate. Linear finite element analysis provides more 

detailed information about stress distributions, but does not allow for redistribution of stress due 

to cracking. It also computes where possibly unsustainable tension and shear stresses are likely 

to develop in areas of the dam base. 

4.1.3 Nonlinear Finite Element Analyses 

There are several types of nonlinearity. For this discussion, we will focus on the effects of 

cracks. Cracks can open in the presence of tensile stresses, and sliding can occur along cracks. 

These are the nonlinear effects that will be considered. 

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Figure 4-2: Linear finite element model: principal stress vectors. 

Figure 4-3: Base stress from linear finite element analysis. 

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If a finite element model that allows base cracking and sliding to develop is used to analyze the 

same dam under the same loading condition, the results will be more in line with the original 

limit equilibrium analysis described in section 4.1. Figure 4-4 shows base contact stress as 

determined by a nonlinear finite element model with interface elements. Note the assumed 

trapezoidal base pressure distribution from the limit equilibrium model (green line) plotted for 

comparison. The length of base cracking calculated by the nonlinear finite element solution 

(18 feet) is similar to that calculated by the limit equilibrium method (15 feet). The base normal 

stress distributions are similar with the exception of the area near the toe of the dam, where the 

finite element analysis indicates a local compressive stress peak. 

Figure 4-4: Base stress from nonlinear finite element analyses and limit equilibrium analysis. 

Since there is close agreement between the limit equilibrium and nonlinear finite element 

analysis, one might question the merits of nonlinear finite element analysis. It is more difficult to 

perform, and while it comes closer to modeling the failure state of a dam, as the actual limiting 

condition is approached, the nonlinear finite element solution fails to close. Therefore, a clear 

factor of safety cannot be given. All one can say is that the solution closed at a friction angle 

greater than the minimum friction angle required for a safety factor of 1.0. In this case, the 

analysis failed to close at base friction angles below 53.5 degrees. (Remember that the limit 

equilibrium analysis showed that 51.6 degrees was required for stability.) In this example, limit 

state analysis provided the best results and should be considered perfectly adequate for analysis 

of this dam. 

The value of nonlinear finite element solutions becomes apparent when the failure mechanism 

cannot be reduced to a simple, statically determinate, rigid body movement. Consider the case 

depicted in figure 4-5. Here, an irregular base changes the failure mode. 

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Figure 4-5: Irregular base with nonlinear finite element analyses. 

For the structure to fail, it has to slide up and over foundation irregularities or shear through 

intact concrete or foundation rock. The distribution of shear and normal forces reflects this 

(figure 4-6). Note that between 10 and 65 feet, the dam separates from the foundation because 

of the shape of the contact and the downstream direction of sliding. 

In this case, the friction angle required for solution closure is 18.3 degrees. If the irregularity 

of the dam-to-foundation contact is ignored and a planar failure surface is assumed, 51.6 degrees 

is what is required for stability (see figure 4-1). Clearly, this is a big difference. Note also that at 

incipient failure, the peak normal stress is 250 pounds per square inch (lb/in2

) compression, 

which is small with respect to typical concrete compressive strengths, but is still over four times 

the value given by the limit equilibrium solution with a planar base. 

In reality, failure surfaces are seldom easily analyzed planes or circular arcs. In addition, if 3D 

effects such as curvature in plan and narrow valley geometry are significant, finite element 

analysis with nonlinear contact elements is best for modeling these failure mechanisms. 

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Figure 4-6: Results from irregular base with nonlinear finite element analyses. 

4.1.4 Static and Dynamic Analyses 

Loads that are applied at a rate comparable to the natural frequency of the structure (several 

hertz) are considered dynamic. It is a misconception to refer to short-term static loading as 

dynamic. Sustained loads that last for longer than a minute are static. Therefore, flood 

loading, no matter how short the duration, is static and should be treated as any other static 

load. 

Seismic loading and, in some cases, loading from impact, are the typical dynamic loads that can 

affect a dam. Because of the oscillatory nature of seismic loading, it is not appropriate to apply 

instantaneous peak dynamic loading to the structure as if these loads were static. The example in 

figure 4-7 shows the problem with this approach. The dam from figure 4-1 is loaded with a peak 

horizontal acceleration of 0.365 g. In addition to the normal static loads, the horizontal inertia 

of the dam and the hydrodynamic pressure according to Westergaard are applied, and the 

calculation results are listed in table 4-4. Note that the resultant moves about 98 feet outside the 

base of the dam, indicating that the dam will overturn. 

But this is clearly a ridiculous result, because the dam cannot overturn in the fraction of a second 

that this set of loads implies. A few hundredths of a second later, the dynamic loading would be 

reversed, and the resultant would move well back inside the dam footprint. 

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Figure 4-7: Results from applying instantaneous peak dynamic loads. 

Table 4-4: Forces, centroids, and moment 

Force Fx (kip) Y= (ft) Fy (kip) X= (ft) 

Moment (0,0) 

(kip-ft) 

Dam weight 

Headwater 

Tailwater 

Uplift 

Dam inertia 

Westergaard 

0.00 

281.58 

-3.12 

62.74 

229.95 

105.00 

0.00 

131.67 

93.33 

94.91 

131.87 

137.90 

-630.00 

0.00 

-2.18 

501.93 

0.00 

0.00 

26.55 

0.00 

77.67 

40.75 

0.00 

0.00 

16,725.00 

37,074.70 

-121.58 

-14,500.99 

30,322,36 

14,480.00 

Totals 676.2 -130.3 83,979.5 

What the above analysis does indicate is that seismic loading would likely damage the contact 

between the dam and foundation, leaving a damaged dam to resist post-earthquake static loads. 

As a result of the instantaneous opening and closing of a foundation crack, post-earthquake uplift 

may be higher. Shear strength between the dam and foundation may be reduced. 

Appendix A2 presents a detailed example of a seismically related failure including all 

appropriate analyses. 

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4.1.5 The Role of Standards-Based Criteria and Codes 

The discussion above focused on failure modes and how to analyze them. All these analyses are 

attempting to determine the load set and downgraded material strength assumptions that would 

bring the dam to a state of incipient failure. All other states are then compared to this limit state. 

Building codes such as the American Concrete Institute (ACI) 318, ACI 350, American Society 

of Civil Engineers 7, and codes from the American Institute of Steel Construction attempt to 

ensure that failure states are never approached, let alone exceeded. Allowable loading is often 

further limited by other considerations such as durability or stiffness. It must be understood that 

exceeding values from code calculations does not necessarily indicate failure. Also, design codes 

presuppose a class of structures. For example, ACI 318 presupposes concrete beams, columns, 

walls, and floor slab-type elements whose behavior is governed by beam theory and KirchhoffLove plate theory. Unreinforced massive concrete structures are not directly addressed. 

Codes are generally used for the design of new structures; the designer uses strength reduction 

factors to attain levels of safety. By contrast, the response of existing structures to initiating 

events for given potential failure modes is assessed without strength reduction factors or load 

factors to compute the “true” strength or limit state of the structure. 

These codes are valuable if the dam or dam appurtenances are of the form that the codes 

envisioned. For example, a slab and buttress dam is more similar to a reinforced concrete office 

building than to a massive gravity dam. In such a case, ACI 318 and ACI 350 are very valuable. 

It would be prudent to evaluate the acceptability of such a dam using the ACI code for the 

strength limit state. 

In short, building codes are helpful when analyzing structures of the form for which they were 

intended. They should be a source of information when assessing the various structural elements 

of concrete dams. 

5.1 Focusing on Parameters Affecting Analysis 

Evaluation of dam structures requires that the engineer make assumptions and use various 

parameters and analysis methods to develop a reasonably accurate representation of structural 

behavior. The impacts of particular parameters on analysis results vary based on the type of 

analysis being performed. Some typical structural parameters are: 

• Joint friction angle and cohesion (concrete and rock) 

• Modulus of elasticity (concrete and rock) 

• Poisson’s ratio (concrete and rock) 

• Coefficient of thermal expansion (concrete) 

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• Unit weight (concrete, rock, and water) 

• Damping (system) 

• Geometry (dam and foundation) 

• Mesh coarseness (dam, foundation, and water) 

• Boundary conditions (foundation and reservoir) 

Accurate determination of parameters is often a daunting task. For example, to determine 

the shear strength to be used in a sliding analysis, a coring program through the joint must be 

undertaken. A sample must be recovered intact and relatively undisturbed. If asperities, 

roughness on a joint plane, are to be considered, the sample must be large enough to capture the 

asperities. One sample can be tested multiple times, but it will only yield peak shear strength on 

its initial shearing. Sufficient samples must be obtained to account for the normal variation. 

Because of the difficulty in recovering and testing statistically significant samples, it behooves 

the analyst to understand what effect parameter variation would have on analysis results. Some 

parameters have little effect on analysis outcome and therefore do not need to be determined 

with great accuracy. Some parameters are important while others will make very little difference 

on the analysis results or ultimately, the risk estimation. 

Consider the effect of the modulus of elasticity on the predicted base cracking in a gravity dam. 

For this specific example, the limit equilibrium gravity analysis of the section in figure 5-1 

indicates that 46 percent of the base would crack. However, an analyst may want to consider the 

effect of dam/foundation interaction. To do this, the moduli of elasticity of both the dam and 

foundation are required. Foundation elasticity is difficult to determine because rock masses are 

jointed and nonhomogeneous. An NX core sample (i.e., a 2.36-inch-diameter core) can be tested, 

but the value of the modulus obtained from such an exercise may not be indicative of foundation 

behavior. While there are several methods to determine the rock mass modulus, it may not be 

necessary, as discussed below. 

One point that should be understood is that the outcome of the analysis is contingent on the ratio 

of the rock modulus to the dam modulus, Eratio. Absolute values do not matter. This means that, 

analytically, one can worry about varying just one parameter instead of two. Analyses can then 

be done varying the ratio with the results discussed below. 

Figure 5-2 shows that when the foundation is soft with respect to the concrete dam (Erock/Econc = 

Eratio = 0.25), the entire foundation contact is in compression. When the foundation is infinitely 

stiff, the base pressure distribution starts to resemble that assumed in the limit equilibrium 

gravity analysis. Figure 5-3 shows the variation of calculated base cracking with respect to 

modular ratio. 

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Selecting Analytic Tools for Concrete Dams to Address 

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Figure 5-1: Base cracking example. 

Figure 5-2: Base pressure distribution through a section of a concrete dam. 

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Figure 5-3: Modular ratio effects (note that this curve is specific for the 

example cited). 

Note that if the foundation is as stiff as the dam or stiffer, predicted base cracking changes very 

little. Therefore, accurate determination of foundation stiffness is not important in this range. 

However, if the foundation is soft with respect to the modulus of the dam, the effect of the 

modular ratio is considerable. In such a situation, accurate determination of the modular ratio 

matters a great deal. 

What has been presented here with respect to modular ratio pertains only to its effect on static 

analysis and base cracking. Absolute values of foundation and dam elastic moduli can be 

important in a dynamic analysis. 

The modular ratio was the only parameter considered in depth here, but what can be noted about 

this parameter’s effect is instructive in dealing with all parameters. 

5.1.1 Common Errors 

Below are presented a few common errors surrounding modeling assumptions. This is not an 

exhaustive list; rather, it is an attempt to pass on what has been learned through particular 

mistakes. 

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5.1.1.1 Unrealistic Boundary Conditions 

Boundary conditions specified incorrectly in finite element models can lead to significant error. 

If the foundation is represented by solid finite elements, the extent of the foundation modeled 

should be validated to ensure that the boundary conditions do not affect the results. The example 

in figure 5-4 is an analysis of a buttress arch dam. 

Figure 5-4: Buttress dam finite element idealization. 

Dams may span entire valleys. Often, a buttress arch dam has dozens of buttresses. The analyst 

can often simplify the task by analyzing one representative bay. Boundary conditions for the 

idealized model must constrain degrees of freedom so that the effects of the bays on either side 

of the idealization are taken into account. A simplified model may not be appropriate in the 

cross-valley direction during seismic analyses because adjacent buttresses provide restraint 

somewhere between a fixed and free conditions. 

For loading in the downstream direction, one would expect that rotations about the crown of each 

arch would be zero; therefore, rotation vectors in the downstream and upward planes must be 

fixed. Also, at the crowns, deflection in the cross-valley direction should be fixed since it is 

resisted by the neighboring bays. If cross-valley loading is to be considered, translation in the 

downstream and the upward directions should be fixed, and cross-valley axis translation should 

be free. Rotation in the downstream and the upward directions must also be free. If both the 

downstream axis and cross-valley axis loading are to be considered, it may be better to model the 

entire dam, since boundary conditions can become mutually exclusive. 

29 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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The base of the buttress should be fixed with respect to all translation; however, the analyst has a 

choice about rotational fixity. If rotation about the upstream/downstream axis is fixed, bending 

moments at the foundation can develop. If the buttress has no physical means of transmitting 

moments into the foundation (i.e., if it is not doweled in), rotational fixity should not be 

assumed. 

Boundary conditions at the bottom of the arches are similar to those at the bottom of the 

buttresses. Rotational fixity in the downstream and cross-valley planes should only be assumed 

if the connection of the arch to the foundation is capable of significant moment transfer. 

In general, boundary conditions must represent real field conditions as much as possible. Fixity 

should not be assumed in a model when it does not exist in the real dam. Freedom should not be 

assumed when it is inhibited. 

5.1.1.2 Finite Element Mesh Density 

Poor mesh geometry can cause significant error in finite element analysis. An element’s 

displacement field dictates how well it can assume the required deformed shape for a given 

loading condition. All finite element methods will converge to the theoretical deformed shape 

and corresponding stress field as element size approaches zero. It is tempting for the modeler to 

use extremely fine meshes. Postprocessors produce stresses and deflections in graphical form, so 

the analyst is no longer required to wade through hundreds of pages of numerical output. As 

computer speed and available memory increase, one could argue that “the finer the mesh, the 

better” since there is less and less of a penalty to be paid for models with a large number of 

degrees of freedom. 

The analyst should still be aware of the way mesh geometry can affect results. In figure 5-5, a 

uniform tension is applied to a member. Constant strain triangles used in the model are not as 

compliant as higher order elements, but this will illustrate the point more vividly. In areas of very 

high stress gradient, elements need to be small. Where stress does not change much with respect 

to location, elements do not need to be small. 

One would expect stress concentration near the reentrant corners; for this reason, the mesh near 

the corner is very fine. Results are as expected. Stress near the reentrant corner rises to about 

three times that of the applied stress (see figures 5-5 and 5-6). If a coarse mesh is used, the stress 

concentration near the corners is completely missed (see figure 5-7). 

The effects and importance of mesh geometry, like any other parameter, should be understood 

before any analysis is performed. However, the appropriateness of the mesh density is only 

determined by comparing analyses with different mesh densities. 

30 
 

 

 

1 

STRESS MAG. 

3 

2.5 

2 

1.5 

0.5 

0 

0	 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
 

DIST. FROM CORNER
 

 

Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

Figure 5-5: Fine finite element mesh at reentrant corner. 

Figure 5-6: Stresses from the fine finite element mesh. 

31
 
 

 

 

 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

Figure 5-7: Coarse finite element model and stress results. 

5.1.1.3 Damping 

Damping in dynamic models is the result of energy dissipation due to several poorly understood 

physical effects. In mass concrete structures, these phenomena include wave radiation (back into 

the foundation and reservoir), sliding friction (along existing joints and cracks), and the complex 

process of material degradation (damage). Damping is mode-shape dependent, driving-force 

dependent, and a function of time as structural damage progresses through the seismic event. It 

has been shown that, for small vibrations, viscous damping in concrete dams is 5 percent or less. 

Small vibrations are defined as those producing stresses well within the linear range of the damreservoir-foundation system. 

It is often tempting, when doing linear modeling, to use high damping values of 10 to 20 percent 

of critical. One rationale for using these damping values is based on a theoretical expectation of 

large radiation damping in the foundation rock materials. While this may be theoretically valid 

for a uniform, elastic half space, foundation nonuniformities and discontinuities, and increases in 

density and modulus with depth, can significantly reduce the amount of radiation damping that 

actually occurs. Large values of damping need to be justified and are normally not appropriate. 

Another argument for using excessive damping is based on the assumption that cracking and 

crushing of concrete and frictional sliding will dissipate large amounts of energy, justifying high 

damping assumptions. However, when large damping values are used, the dynamic response of 

the dam is so muted that the results do not show the potential for damage that was assumed to 

justify the high damping in the first place. This errant, circular reasoning must be avoided. 

32 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

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It is more appropriate to use low viscous damping of 5 percent or less and a rational basis of 

estimating damage when doing linear modeling. If the result of this assumption is the prediction 

of high stresses indicative of excessive damage, it may be appropriate to do a nonlinear time 

domain analysis. If the results of a linear analysis using higher levels of damping indicate low 

stresses, be aware that the use of higher levels of damping in linear analyses presupposes 

damage. The effect of energy dissipation due to damage accrual is automatically calculated in a 

nonlinear time domain analysis. Radiation damping is also directly accounted for if the 

foundation model includes the mass and the foundation boundaries are nonreflecting, thereby 

accounting for wave propagation. 

5.1.1.4 Failure Plane Tension/Cohesion 

Dam failures often involve cracking of concrete resulting in failure planes where sliding or 

overturning can occur. If the tensile strength of the concrete is not exceeded, no failure plane can 

form, and failure by this mechanism can be ruled out. Many analyses evaluate calculated tensile 

stresses with respect to published values of allowable tensile strengths of intact concrete. It is not 

uncommon for intact concrete to have significant tensile strength (several hundred lb/in2

). 

However, the analyst must be aware that stress calculations only result in stresses produced by 

the applied loads. There are sources of stress that are not typically modeled, for example: 

•	 Gravity load stress influenced by construction sequencing 

•	 Stresses associated with concrete curing and cooling 

•	 Stresses associated with differential foundation deformation 

For this reason, the exact value of in situ tensile stress is often not knowable, and the difference 

between existing tensile stress and ultimate strength is not clear. In addition to the problem 

above, there is uncertainty due to the random variation of the material, and the fact that lift joints 

and preexisting cracks do not exhibit the same tensile strength as an intact laboratory specimen. 

Because of these uncertainties, the analysis should not rely on tensile strength perpendicular to a 

postulated failure plane for the stability of a dam. In the limit equilibrium analysis, the dam 

should be stable as a series of independent blocks. 

5.1.2 Summary 

In general, the following principles can be observed concerning choice of parameters: 

•	 Understand how a parameter affects analysis conclusions. Parameters that have a big 

effect on the outcome need to be accurately determined or bounded. Parameters that do 

not have a big effect can be loosely assigned. 

33 
 

 

 

 

 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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•	 If a parameter has a significant effect with a particular analysis type, and the required 

accuracy of the parameter is not realistically obtainable, consider a different type of 

analysis where the effect of this parameter is not as significant. For example, tensile 

strength of concrete is difficult to obtain. One could perform an analysis that assumes a 

no-tension criterion across a preexisting crack. 

•	 Conservative assumptions need less verification unless they lead to expensive actions. If 

a dam can be shown to be stable with conservative parameters, there may be decreasing 

need to justify those parameters with testing. For example, the case of a gravity dam on a 

competent rock foundation, assuming 35-degree friction angle and no cohesion, would 

not require significant justification, whereas a 65-degree friction angle and 200-lb/in2 

cohesion would likely require more justification than reasonably obtainable. 

•	 Analysis techniques that require few parameters are preferable. 

•	 It is preferable for stability to be demonstrated using simple analysis techniques. 

•	 The analyst needs to know how the data will be used and the effects of that data on the 

analysis before it is determined how much effort and cost to spend to get the data. 

6.1 Uncertainty and Confidence 

6.1.1 Uncertainty 

Decisions about the safety of a dam are based on the understanding and perception at a point 

in time, knowing that the physical world has variability and knowing there are limits to our 

understanding and modeling of the physical world. Uncertainties about our understanding can 

result from the variability in the loads applied to the structure, the material properties of the 

structure, and in some cases, the geometry of the structure (Was the structure really built like it 

was shown on the drawings?). Many times, the level of uncertainty may not be explained to, or 

understood by, decisionmakers. Knowing the limitations and uncertainty will foster better 

decisions and increase the defensibility of the actions being taken. 

Analyses are more valid if accompanied by a presentation of the degree of confidence in the 

results, as well as areas of possible uncertainties and consistency. Discussions would include the 

uncertainty associated with the parameters used in the structural analysis and how the parameters 

were obtained. The parameters might have been assumed based on, for example, material 

properties from dams with similar concrete characteristics, limited laboratory tests, construction 

records, or an extensive field exploration program. The analysis methods would be explained in 

terms of the simplifying assumptions used and their possible effect on the results and gaps in 

technology and the state of the art. The variability in the loads would be developed to show 

confidence bands around mean load levels. Consistency issues would be addressed, making sure 

that screening-level ground motions or assumed material properties were not used with state-ofthe-art nonlinear finite element analysis. 

34 
 

 

 

 

0.1 

Probability of Being Exceeded 

0.01 

0.001 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 

Peak Acceleration (g) 

5th 

15th 

50th 

85th 

95th 

Mean 

0.0001 

 

Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

Uncertainty comes in two forms: aleatory and epistemic. Aleatory uncertainty is attributed to 

inherent randomness, or natural variations. Aleatory uncertainty is what limits precision. Because 

there is random variability in a parameter, it cannot be assigned a specific value, but rather must 

be assigned a range of values. An example of aleatory uncertainty is shown in figure 6-1, 

showing a seismic hazard curve with uncertainty bands around the mean and how the bands get 

wider at longer return periods. The error bands show the magnitude of uncertainty that can be 

expected given the available data. Collecting more data might improve the estimated parameters 

and reduce the width of the error bands. However, expected error approaches a finite value as the 

sample size approaches infinity. Therefore, additional data can never eliminate aleatory 

uncertainty associated with random physical processes. 

Figure 6-1: Uncertainty bands around a seismic hazard curve (peak acceleration vs. exceedance 

probability). 

Epistemic uncertainty is attributed to lack of knowledge. The level of epistemic uncertainty 

cannot be quantified. It is unquantifiable because one does not know what one does not know. 

While aleatory uncertainty conditions precision, epistemic uncertainty affects accuracy. 

Epistemic uncertainty could, for example, be associated with the entire process of developing 

the curves being unknowingly flawed. Using the seismic hazard curve example, epistemic 

uncertainty would be that associated with the potential of omitting the governing seismogenic 

fault from the study, or epistemic uncertainty might be due to accelerometers in the field that are 

measuring the wrong parameter. As a result, statistical bounds drawn around data for the hazard 

curves are based on incorrect data. More seismic data to improve the aleatory estimation of 

uncertainty are useless based on the set of unknowingly bad data. 

35 
 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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When epistemic uncertainty is suspected, the appropriate response of the analyst is to request 

more information. Unlike aleatory uncertainty, which is quantifiable but irreducible, epistemic 

uncertainty can be reduced. For example, the analyst may be uncertain if a low strength clay 

seam runs under the dam, which could present a failure mechanism. This example of epistemic 

uncertainty could be eliminated by a foundation coring program. 

A systematic sensitivity and uncertainty analysis provides both the analyst and the 

decisionmakers the ability to judge the adequacy of the analysis and conclusions. Both Hartford 

and Baecher (2004) and Morgan and Hendron (1998) make a distinction between sensitivity 

studies and uncertainty analysis that is appropriate here. Sensitivity studies involve determining 

the change in response of a model to changes in either individual model parameters, more than 

one parameter, some key parameters across reasonable high and low ranges, or the model itself. 

They indicate how wide the spread of results can be if individual parameters vary within the 

bounds of their respective uncertainties. Sensitivity studies are a useful way of determining if 

some uncertainties significantly influence the results. For instance, the stability of a gravity dam 

could be computed for various uplift assumptions. The sensitivity of uplift on stability could then 

be determined. 

Finally, a certain level of uncertainty may be tolerated depending on the degree of conservatism 

of the assumptions and results, and on the level of demand on the structure. High levels of 

uncertainty may be acceptable knowing the level of conservatism introduced in the analyses. 

Factors of safety are still viable benchmarks for analyses’ results and decisionmaking given that 

there is an understanding of the parameters and procedures that produced those factors of safety. 

6.1.2 Confidence 

A structural analyst should perform analyses appropriate to the level of study, focus on capturing 

the desired structural response at an event node associated with a potential failure mode, and 

portray the uncertainties and assumptions to decisionmakers. Choosing the appropriate analysis 

method is key to answering a desired question about the dam. The problem being solved should 

dictate the analysis method selected. Table 4-1 provides some guidance on selecting an analysis 

method. Once the analysis method is chosen, the accuracy or usefulness of the solution depends 

on the understanding of the material properties, boundary conditions, and loads input into the 

program. The many choices in analyses cause variability in the computed results. These might 

include, but are definitely not limited to, choices between: 

• Average, generic material properties instead of site-specific material properties 

• Using a massless foundation instead of a foundation with mass 

• Viscous damping or Rayleigh damping 

• Westergaard’s added mass or compressible fluid elements for hydrodynamic interaction 

36 
 

 

 

 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

•	 Modeling the dam as a homogeneous structure instead of modeling geometric 

nonlinearities in the dam such as contraction joints or unbonded lift joints
 

•	 Linear or nonlinear material properties 

•	 A response spectrum analysis or a time-history analysis 

•	 Modal superposition or direct time integration 

•	 A coarse or a fine finite element mesh 

•	 Various types of solid elements 

•	 Ground motions generated at 0.02-second or at 0.005-second intervals 

•	 Modeling 3D or only 2D canyon effects 

•	 Applying ground motions in three orthogonal directions or in only one direction 

•	 Using appropriate ground motions for the site including spatial variations 

The analyst must be aware of the consequences of all of these modeling decisions. If known 

features are ignored or simplified, the analyst should include an explanation. 

Analysis methods do not introduce uncertainty. They do what they do, and we know what they 

do because we created them. An analysis can introduce error, but this should not be confused 

with uncertainty. All analysis tools make simplifying assumptions, and it is the responsibility of 

the analyst to understand the effects of these limitations. No uncertainty band can cover a faulty 

analysis. In such a situation, the analysis can be incorrect or inappropriate with 100-percent 

certainty. Epistemic uncertainty may be why critical features are ignored in the analysis, but that 

uncertainty is not the result of the analysis itself. Choosing to ignore known critical features for 

the sake of simplification is not a case of uncertainty; it is a case of error. 

Troubleshooting a numerical model, while important, is not an example of reducing uncertainty, 

but it does develop confidence in the analysis performed. Two effective techniques to 

troubleshoot models are to adjust parameters in the model until failure of the model can be 

reached or identify threshold levels for historic maximum loadings on the actual dam. For 

instance, in a nonlinear structural analysis of a potentially unstable foundation block, shear 

properties are reduced to the point where sliding commences for a historic maximum loading 

level. If the actual dam was stable for this historic loading but the finite element model shows 

movement with realistic shear properties, something is wrong with the analysis. If the analysis is 

run with zero cohesion and zero friction and the foundation blocks do not slide, something may 

be wrong with the analysis. The foundation block may be inadvertently held in place because of 

a modeling issue. Uncertainty analysis is an extension of sensitivity analysis, where probability 

37 
 

 

 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

distributions are associated with the various parameters or models being varied. Thus, the output 

is in the form of a probability distribution that specifies the likelihood of each possible result 

across a full range of possible results. 

Do the natural frequencies of the models match the field-measured frequencies? Do seasonal 

static deformations in the models match the measured deformations? Are static water pressures 

at locations on the upstream face of the dam in the finite element model the same before and 

after the earthquake? Do the friction forces developed along a contact surface match the 

computed normal force times the friction tangent? Do the velocities obtained by a potentially 

unstable rock block moving from a known constant force match the theoretical velocity? Are the 

results different with more finite element mesh density? Do the ground motions captured at the 

free-field surface of the model match the target motion provided by the seismologist? Are results 

of the progression of analyses, ranging from the simplest to the most complex, understandable? 

These checks of finite element models increase confidence in the models, help validate that the 

models are working correctly, and reduce doubt about how well the models are predicting reality. 

7.1 Conclusions 

The process this document suggests is to first consider potential failure modes and the events for 

each mode that would lead to failure of a concrete dam, then select appropriate structural 

analysis methods to help quantify the response of the dam at desired events to determine the 

probability of failure, quantify the uncertainty, or prove that the failure process stops at an event. 

Field investigations, instrumentation, and testing can then be focused on the requirements of the 

analyses. 

Potential failure modes start with some initiating event such as a flood, seismic event, or human 

operating error. These adverse changes can cause a sequence of interrelated and normally 

sequential events that must occur for failure to actually occur. 

Event trees graphically represent event sequences. Event trees are available to the engineer 

conducting a risk or reliability analysis as an efficient way to organize the chronological 

sequence of failure mode development. 

Structural analyses are then selected to determine the structural response at a given node. 

Structural analysis uses either limit equilibrium analysis or various types of finite element 

methods, namely 2D linear elastic, 2D nonlinear, 3D linear elastic, or 3D nonlinear. It must be 

realized that each of these analysis methods has a certain level of applicability and represents a 

certain level of reality. These methods also vary greatly in level of effort. The accuracy of the 

results depends on the accuracy of the material properties being used and the accuracy of the 

loading condition. However, more importantly, applicability of the analysis method being used 

depends on the question being asked. For instance, a linear analysis cannot be used to determine 

how far a crack progresses, because the linear elastic elements can carry tensile stresses higher 

38 
 

 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

than the material strength. Nonlinear methods, or the limit equilibrium method, answer this 

question better. Examples showing various structural analysis methods and results are provided 

in the body of the report and in appendix A. 

Evaluation of dam structures requires that the engineer make assumptions and use various 

parameters and structural analysis methods to develop a reasonably accurate representation of 

structural behavior. The impacts of particular parameters on analysis results vary based on the 

type of analysis. 

Though many efforts are made during structural analysis to achieve high quality results, some 

analysis methods just cannot model the structural behavior as accurately as others. Many times, 

the analyst makes simplifying assumptions that greatly limit the accuracy of the analysis, yet 

decisions about the safety of a structure are made assuming the results of an analysis are more 

accurate than they actually are. This is because structural analyses have levels of uncertainty and 

levels of accuracy throughout the process. 

There are uncertainties about the loads applied to the structure, the material properties used in 

the analysis, the analysis methods being used, and the interpretation of the results. Also, many 

times the level of uncertainty, level of accuracy, or level of reality may not be explained to or 

understood by decisionmakers as well as needed. It is important that decisionmakers understand 

the conservatisms, the accuracy, and limitations of the structural analyses so they can determine 

the safety of the dam and necessary actions to take. If they know the limitations, they might 

decide to update the loading, refine the structural analysis, or obtain actual material properties. 

Higher levels of effort may be needed in cases where risk is estimated to be near the agency’s 

guidelines. 

39 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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Bibliography 

Bureau of Reclamation, Best Practices for Dam Safety Risk Analysis, Technical Service Center, 

Denver, Colorado, 2012. 

Bureau of Reclamation, Public Protection Guidelines, January 3, 2003. 

Federal Energy Regulatory Commission (FERC), Engineering Guidelines for the Evaluation of 

Hydropower Projects, Chapter 14, “Dam Safety Performance Monitoring Program,” 2005. 

Georgia Institute of Technology, GT-STRUDL, Integrated CAE System for Structural 

Engineering Analysis and Design, School of Civil and Environmental Engineering, Atlanta, 

Georgia. 

Hartford, Desmond N.D. and Gregory B. Baecher, Risk and Uncertainty in Dam Safety, CEA 

Technologies Dam Safety Interest Group, 2004. 

Itasca, FLAC-3D, Minneapolis, Minnesota. 

Livermore Software Technology Corporation, LS-DYNA, Livermore, California. 

Morgan, M. Granger and Max Hendron, Uncertainty—A Guide to Dealing with Uncertainty in 

Quantitative Risk and Policy Analysis, Cambridge University Press, 1990 with reprint in 

1998. 

Parametric Technologies Corporation, MathCad 14, PTC Corporate Headquarters, Needham, 

MA. 

U.S. Army Corps of Engineers, Ice Engineering, 1110-2-1612, 2002.
 

University of Illinois at Urbana-Champaign, CAST, Version 0.9.11, 1998-2004.
 

Vose, David, Risk Analysis—A Quantitative Guide, second edition,” 2002.
 

Westergaard, H.M., Handbook of Dam Engineering, Chapter 8, Van Nostrand Reinhold 

Company Inc., 1977. 

Westergaard, H.M., “Water Pressures on Dams during Earthquakes,” Alfred R. Golze, ed., 

American Society of Civil Engineers Transactions, Paper No. 1835, November 1931 

Proceedings. 

41 

 

 

 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

ICODS
 

Figure References for
 

Selecting Analytic Tools for Concrete Dams to Address Key
 

Events along Potential Failure Mode Paths
 

Sept 12, 2014
 

Cover 

Upper left Reclamation aerial photograph 

Upper right Reclamation finite element model 

Lower left Figure developed by ICODS Committee in PowerPoint 

Lower right Figure developed by ICODS Committee in PowerPoint 

Figure 2.1 Reclamation file photo of Grand Coulee Dam 

Figure 2.2 Reclamation file photo of Roosevelt Dam 

Figure 2.3 Reclamation file photo of Stony Gorge Dam 

Figure 2.4 Reclamation file photo of Pueblo Dam 

Figure 2.5 Reclamation file photo of Bartlett Dam 

Figure 2.6 Reclamation file photo of Coolidge Dam 

Figure 3.1 Reclamation file photo of Shasta Dam 

Figure 3.2 Figure developed by ICODS Committee in PowerPoint 

Figure 3.3 Figure developed by ICODS Committee in PowerPoint 

Figure 3.4 Figure developed by ICODS Committee in PowerPoint 

Figure 4.1 Figure developed by ICODS Committee in PowerPoint 

Figure 4.2 Figure developed by ICODS Committee in FLAC 

Figure 4.3 Figure developed by ICODS Committee in Spreadsheet 

Figure 4.4 Figure developed by ICODS Committee in Spreadsheet 

Figure 4.5 Figure developed by ICODS Committee in FLAC 

Figure 4.6 Figure developed by ICODS Committee in Spreadsheet 

Figure 4.7 Figure developed by ICODS Committee in PowerPoint 

Figure 5.1 Figure developed by ICODS Committee in PowerPoint and FLAC 

Figure 5.2 Figure developed by ICODS Committee in Spreadsheet 

Figure 5.3 Figure developed by ICODS Committee in Spreadsheet 

Figure 5.4 Figure developed by ICODS Committee in PowerPoint and FLAC 

Figure 5.5 Figure developed by ICODS Committee in PowerPoint and FLAC 

Figure 5.6 Figure developed by ICODS Committee in Spreadsheet and FLAC 

Figure 5.7 Figure developed by ICODS Committee in FLAC 

Figure 6.1 Figure developed by ICODS Committee in Spreadsheet 

Figure A1.1 Figure developed by ICODS Committee in PowerPoint 

Figure A1.2 Figure developed by ICODS Committee in PowerPoint 

Figure A1.3 Figure developed by ICODS Committee in PowerPoint 

Figure A1.4 Reclamation file photo of Shasta Dam 

43 
 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

Figure A1.5 Reclamation file photo of Kortes Dam 

Figure A1.5 Reclamation file photo of Kortes Dam looking downstream 

Figure A2.1 Figure developed by ICODS Committee in PowerPoint 

Figure A2.2 Figure developed by ICODS Committee in PowerPoint 

Figure A2.3 Figure developed by ICODS Committee in FLAC 

Figure A2.4 Figure developed by ICODS Committee in Spreadsheet 

Figure A2.5 Figure developed by ICODS Committee in Spreadsheet 

Figure A2.6 Figure developed by ICODS Committee in FLAC 

Figure A2.7 Figure developed by ICODS Committee in FLAC 

Figure A2.8 Figure developed by ICODS Committee in Spreadsheet 

Figure A2.9 Figure developed by ICODS Committee in PowerPoint 

Figure A2.10 Figure developed by ICODS Committee in FLAC 

Figure A2.11 Figure developed by ICODS Committee in Spreadsheet 

Figure A2.12 Figure developed by ICODS Committee in FLAC 

Figure A2.13 Figure developed by ICODS Committee in FLAC 

Figure A2.14 Figure developed by ICODS Committee in FLAC and Spreadsheet 

Figure A2.15 Figure developed by ICODS Committee in Spreadsheet 

Figure A2.16 Figure developed by ICODS Committee in Spreadsheet 

Figure A2.17 Figure developed by ICODS Committee in Spreadsheet 

Figure A2.18 Figure developed by ICODS Committee in PowerPoint 

Figure A2.19 Figure developed by ICODS Committee in PowerPoint 

Figure A3.1 Figure developed by ICODS Committee in PowerPoint 

Figure A3.2 Figure developed by ICODS Committee in PowerPoint 

Figure A3.3 Figure developed by ICODS Committee in PowerPoint 

Figure A3.4 Reclamation file photo of Hungry Horse Dam 

Figure A3.5 Reclamation file photo of Hungry Horse Dam foundation 

Figure A3.6 Figure developed by ICODS Committee in EACD3D96 

Figure A3.7 Figure developed by ICODS Committee in Spreadsheet 

Figure A3.8 Figure developed at Reclamation in LSDYNA 

Figure A3.9 Figure developed at Reclamation in LSDYNA 

Figure A4.1 Reclamation file photo of Stony Gorge Dam 

Figure A4.2 Reclamation file photo of Stony Gorge Dam 

Figure A4.3 Figure developed by ICODS Committee in PowerPoint 

Figure A4.4 Reprint of Corbel forces from ACI-318 

Figure A4.5 Reprint from Reclamation file drawing of Stony Gorge Dam corbel 

Page A-46 Reprint from Reclamation file drawing of Stony Gorge Dam buttress 

Page A-47 Same as Figure A4.5 

Page A-49 Reprint from Reclamation file drawing of Stony Gorge Dam buttress 

Page A-51 Same as Figure A4.5 

Page A-52 Figure developed by ICODS Committee in Computer Aided Strut and Tie 

Page A-53 Figure developed by ICODS Committee in Computer Aided Strut and Tie 

Page A-54 Figure developed by ICODS Committee in Computer Aided Strut and Tie 

44 
 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

Page A-55 Figure developed by ICODS Committee in Computer Aided Strut and Tie 

Figure A5.1 Upper figure developed by ICODS Committee in PowerPoint 

Bottom figures (verifying from Husein Hasan at TVA) 

Figure A5.2a Figure developed by ICODS Committee in PowerPoint 

Figure A5.2b Figure developed by ICODS Committee in PowerPoint 

Figure A5.3 Same as Figure A4.5 

Figure A5.4 Figure developed by ICODS Committee using GT-STRUDL 

Figure A5.5 Figure developed by ICODS Committee using GT-STRUDL 

Figure A5.6 Figure developed by ICODS Committee using GT-STRUDL 

Figure A5.7 Reprint from Reclamation file drawing of Stony Gorge Dam 

Figure A5.8 Figure developed by ICODS Committee using GT-STRUDL 

Figure A5.9 Figure developed by ICODS Committee using GT-STRUDL 

Figure A5.10 Figure developed by ICODS Committee using GT-STRUDL 

Figure B.1 Figure developed by ICODS Committee in PowerPoint 

Figure B.2 Figure developed by ICODS Committee in PowerPoint 

Note: Original drawings by Committee members are listed as PowerPoint. There may have been 

another drawing program used such as AutoCAD, but these figures are originally developed by 

the committee. 

45 

Appendix A1 

Increased Reservoir Level Causes 

Sliding in a Gravity Dam 

 

 

 

 

Appendix A1: Increased Reservoir Level Causes 

Sliding in a Gravity Dam 

This potential failure mode is a sliding failure at the foundation contact or along an existing 

plane of weakness within a gravity dam due to reservoir loading in excess of original design 

conditions. These changes may be a result of operational changes (storing more water), updated 

hydrologic analyses, or operational errors (failure to operate gates in accordance with established 

plans, failure of gate equipment, plugging of gates or spillway by debris, etc.). Figure A1-1 

shows the geometry of a typical gravity dam along with the free-body diagram of the portion of 

the dam above a plane of weakness. When the reservoir driving force exceeds the frictional 

resistance induced by the weight of the dam (reduced by the effect of uplift) the upper portion of 

the dam will slide in the downstream direction. Figure A1-2 is the event tree for this potential 

failure mode. 

A1.1 Node A: Reservoir Level Increases 

An increase in reservoir loading will increase driving forces and uplift pressures, which reduce 

resisting forces. The combination of these effects may result in initiation of this potential failure 

mode. 

Figure A1-1: Geometry of a typical gravity dam (left) and free-body diagram above plane of weakness (right). 

A1-1 
 

 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

Figure A1-2: Event tree for a failure mode where increased reservoir load causes sliding 

instability. 

If the dam has a drainage system that intersects the failure plane in question and the drainage 

system efficiency is reduced, the uplift forces will increase, and the portion of the failure surface 

upon which the resisting shear forces act may be reduced. 

A thorough evaluation of the hydraulic loadings and evaluation of the drainage system is 

required prior to moving on to node B of the event tree. 

A1.2 Node B: Tension Zone Exists 

To adequately address the sliding failure mode, an evaluation of the condition of the failure plane 

beginning at the location where the sliding plane intersects the upstream face is required. If the 

initiating event causes this location to enter into tension, or if the tensile stresses increase, a 

determination of the length of the resulting tension zone is required. If the upstream face is 

originally in tension, the initiating event will change the driving forces and/or the distribution of 

the uplift pressures causing the length of the tension zone to increase. 

Node B of the event tree is where an evaluation of the condition of the failure surface is 

performed to determine how the failure mode would progress. Various methods exist to 

determine the portion of the failure surface that is cracked, including classical limit state “crack 

based” computations founded on force and moment equilibrium, nonlinear finite element 

methods, and fracture mechanics, which is based on the fracture energy required to advance a 

crack. Refer to section 4 of this report for a discussion on the applicability of the various 

methods. 

A1-2 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Included in figure A1-3 is an example of the limit state computations for sliding on a plane 

within the dam. In this example, the length of the tension zone represents 15 percent of the width 

of the dam. The uplift diagram includes full reservoir head throughout the tension zone. In this 

example, drains are not present, so the uplift diagram is assumed to vary linearly from the point 

of zero total normal stress to the downstream face of the dam. 

A1.3 Node C: Overturning Stability Met 

This node is an informational node for this sliding failure mode. In this case, overturning is not 

an issue because, as shown in the previous analysis, the resultant stays within the base. It is 

included to point out that, based on the initiating event and node B having occurred, the 

overturning failure mode needs to be evaluated. 

In the example in figure A1-3, the location of the resultant based on the final iteration is outside 

the middle third but within the base of the dam. This indicates that the upstream face would be in 

tension and the dam would not overturn. 

A1.4 Node D: Sliding Initiates 

Sliding initiates when the driving forces exceed the resisting forces. The driving forces include 

the hydrostatic forces of the reservoir and any soil pressures (silt) acting on the upstream face 

of the dam. The resisting forces include soil conditions or tailwater that may exist on the 

downstream face, as well as frictional forces. The resisting forces are a combination of cohesion 

and friction. The frictional resistance is a function of the total forces normal to the sliding plane 

multiplied by the coefficient of friction (tan phi). Uplift pressures offset the weight of the 

structure; therefore, the determination of the pressure diagram is important and should be 

compatible with the evaluation of the sliding plane as determined in nodes B and C. If cohesion 

is to be considered, it can only be applied to the portion of the base that is bonded and should be 

verified if critical by a coring and laboratory program. 

Computations shown in figure A1-3 indicate that the factor of safety against sliding is less than 

1.0; therefore, sliding is judged to begin based on a 2-dimensional (2D) study. 

A1.5 Node E: Sliding Continues 

To this point, sliding stability has been evaluated based on the dam performing as a 2D structure. 

While stability computations can be applied to a 3-dimensional (3D) block, a 2D sliding stability 

analysis can be performed, and if 2D stability is met, failure will not occur. In the example, these 

computations suggest sliding will commence. Continuation of sliding is controlled by the 3D 

characteristics of the structure. 

A1-3
 
 

 

 

 

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Figure A1-3: Example dam showing computation of length of base in compression, corresponding uplift distribution, 

and sliding factor of safety. 

A1-4
 
 

 

 

 

 

 

 

 

 

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An evaluation of as-constructed conditions may suggest that there are 3D characteristics of the 

dam that would prevent continuation of the sliding failure. These may include, in the case of a 

curved gravity dam, individual monoliths that are narrower at the downstream face than the 

upstream face, thereby causing the blocks to wedge together if their 2D behavior would initiate 

downstream movement (figure A1-4). 

Figure A1-4: A curved gravity dam, Bureau of Reclamation. 

In the case of straight gravity dams, contraction joints may have been constructed with shear 

keys, which would require mobilization of adjacent monoliths prior to the monolith in question 

failing or shear failure of the concrete through the keys. 

A monolith on an abutment will have a different height on one side of the monolith than the 

other. A 2D stability computation of the tallest section may indicate initiation of sliding, but a 

similar analysis of the shorter side may indicate no sliding. In this case, an evaluation of the 

entire monolith in 3D is required. Figure A1-5 shows a gravity dam constructed in a steep 

canyon where the structural height of each monolith varies along the axis of the dam. 

The dam may be wedged in the canyon, which could add significant side resistance. 

If these conditions exist, an evaluation is required to determine if the 3D effects are sufficient to 

prevent continuation of the sliding failure mode. In the case of keyed contraction joints, the shear 

key geometry and concrete shear strength may provide significant shear resistance. 

A1-5
 
 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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Figure A1-5: Looking upstream and downstream from a gravity dam in a narrow canyon,
 

Bureau of Reclamation.
 

However, the total driving forces on the monolith (not those from a 2D slice) must be 

considered. A complicating factor in this situation is that the block may want to slide 

longitudinally (parallel to the axis of the dam), thereby providing a normal force across the 

contraction joint, which, in turn, would provide a frictional resistance to downstream sliding. 

Also in the situation of a narrow canyon, a 3D finite element analysis may indicate that 

longitudinal stresses develop due to bending of horizontal cross canyon planes (beams). This also 

could lead to a determination that overall 3D stability should be evaluated similar to an arch 

dam. These examples are included to demonstrate the importance of a thorough evaluation of the 

failure mode and associated event tree. 

A1-6
 
Appendix A2 

Earthquake Causes Sliding in a Gravity Dam 

 

 

 

 

 

 

 

Appendix A2: Earthquake Causes Sliding in a 

Gravity Dam 

Consider the dam in figure A2-1. The potential failure mode being investigated in this example is 

a sliding failure only within the body of the dam on a potentially weak or unbonded lift joint at 

or about the location of the reentrant corner at elevation 720 feet. There are other failure modes 

that could be investigated, but it should be noted that potential failure modes have to be well 

described for analyses to be meaningful. That means that specific potential failure modes have to 

be considered independently. The event tree shown in figure A2-2 outlines the steps that must 

occur for an uncontrolled reservoir release as a result of a seismic induced sliding failure. 

Figure A2-1: Concrete gravity dam. 

A2-1
 
 

 

 

 

 

 

 

 

 

 

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Figure A2-2: Event tree for earthquake causes sliding in a gravity dam. 

Node key: 

A Preexisting crack or poorly bonded lift joint exists. 

B Earthquake sufficient to cause cracking. 

C Crack propagates through thickness of dam. 

D Removable block forms. Slide planes develop. 

E Duration and severity of shaking sufficient to cause failure. 

F Damage caused by earthquake results in the dam being unable to resist static loads. 

A2.1 Node A: Existing Crack or Poorly Bonded Lift Joint 

The reentrant corner on the upstream face at elevation 720 feet is an area of potential cracking, 

and it is wise to investigate. The first branch point of the failure event tree is whether a crack or 

poorly bonded lift joint already exists in the area of concern. Whether a crack in this area is 

already present may be determined by underwater survey by remotely operated vehicle. It might 

also be possible to ascertain the condition of the area through borings. As this example will 

show, in many cases, it is more prudent to assume a crack than try to prove the tensile strength 

will not be exceeded. 

If it is determined that the area of potential cracking is already cracked, or that the lift line is 

poorly bonded, the effect of the crack or line of weakness must be taken into account when 

evaluating the seismic response of the dam, and the analysis should proceed directly to node C. 

Linear elastic analyses are no longer appropriate if preexisting cracks have been identified. 

A2.2 Node B: Upstream Cracks 

If it is determined that the area of the dam in question is not pre-cracked and that the concrete 

is intact, a linear elastic dynamic analysis can be done to determine if the induced seismic 

stresses are likely to produce a crack. If the stresses exceed the dynamic tensile strength of 

intact concrete, cracking is likely. Figure A2-3 shows a 2D linear elastic model of the 

dam/foundation/reservoir. Uplift was not applied in the postulated failure plane because 

the assumption is that there was no crack prior to the earthquake. This model has a 

A2-2 
 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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wave-propagating foundation with wave-radiating boundaries. Hydrodynamic effects are 

modeled directly through use of elastic wave-propagating elements with zero shear stiffness, 

as shown in figure A2-3. 

The base of the foundation is excited with horizontal and vertical components of the 1999 

Chi-Chi CHY006 record. The response spectrum is shown in figure A2-4. The method of 

application is to impose dynamic shear and normal stresses to the bottom of the foundation and 

allow them to propagate upward. The stress magnitudes are set to produce the desired free-field 

ground motion at the free surface of the foundation without the reservoir and dam. 

Figure A2-3: Finite element model. 

Figure A2-4: Spectral acceleration (Chi-Chi CHY006) with 5-percent damping. 

A2-3 
 

 

 

 

 

 

 

 

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The results of the analysis are shown below in figure A2-5. For the sake of this exercise, 

dynamic tensile strength is assumed to be 400 pounds per square inch (lb/in2). Note that several 

excursions exceed this stress limit; the most significant is 1,256 lb/in2. 

Figure A2-5: Vertical stresses at reentrant corner, 2D model. 

It could be argued that the tensile strength is exceeded only a few times for short periods and, 

therefore, can be ignored. This argument is false. In a brittle material, cracking takes no time to 

initiate, and once initiated, cracks run through the material at sonic velocities. In addition to 

fast propagation, a stress concentration effect occurs at the crack tip, which allows the crack to 

propagate under a tensile stress that is much lower than the material’s original uncracked tensile 

strength. 

In reality, dams are not 2D structures. Gravity dams are capable of transferring load horizontally 

if the joints between monoliths are sufficiently keyed. If 3D behavior is factored into the 

analysis, induced dynamic stresses may be less than those indicated by the 2D model. 

Figures A2-6 through A2-8 show the setup and results of a rigorous 3D linearly elastic finite 

element model. Material parameters and earthquake input are identical to those used in the 2D 

model, so there are no cross-canyon ground motion components. 

A2-4
 
 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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Figure A2-6: 3D linear finite element model. 

Figure A2-7: 3D linear finite element model with the dam, foundation, reservoir, and free-field boundaries. 

A2-5 
 

 

 

 

 

 

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Figure A2-8: Vertical stress time history at reentrant corner, 3D model. 

Note that the aspect ratio (crest length to dam height) of the dam is 3 (see figure A2-6). Gravity 

dams in narrow valleys are likely to exhibit more 3D behavior than dams in wide valleys. In a 

wide valley, a 3D analysis would not be expected to differ much from a 2D analysis; therefore, a 

3D analysis would not be warranted. However in this case, it is not clear that 3D effects can be 

ignored. 

The full dam, reservoir, and foundation model is shown in figure A2-7. Looking at the relative 

complexity of the model above, one can see why 2D analyses are preferable from an ease-ofmodeling standpoint. Because of the desire to capture stress concentration effects at the upstream 

face near the reentrant corner at elevation 720 feet, the mesh of the dam must be sufficiently fine. 

This requirement for a fine mesh complicates the model and makes computation slower. 

Because this model is linearly elastic, the effects of vertical contraction joints between dam 

monoliths are ignored. In this model, the dead load of the dam is allowed to be horizontally, as 

well as vertically, distributed. One can compare the initial stress in the 2D model (figure A2-5) 

A2-6 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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with that of the 3D model (figure A2-8) and note that the stresses are essentially the same 

(25,000 pounds per square foot [lb/ft2], 170 lb/in2); therefore, this error was not significant. 

Figure A2-8 shows the vertical stress history at the sampling point shown in figure A2-6. Note 

that in the 3D analysis, unlike the 2D analysis, the tensile strength (400 lb/in2) is approached but 

not exceeded. The analyst now has somewhat of a dilemma. Which model is to be believed? It is 

likely that the 3D model shows lower dynamic stresses because there is another dimension 

available for energy loss. This wave radiation into the cross-canyon dimension is a real 

phenomenon; therefore, there is some cause to put more faith in the 3D model. 

However, while the 3D model accounts for more physical reality than the 2D model, it does not 

take into account that the dam is actually separated by vertical contraction joints, which tend to 

diminish 3D behavior. It would not be advisable to accept the results of the 3D model as 

conclusive. Only one earthquake record was evaluated, and in that case, while the tensile limit 

was not exceeded, it was approached. Would another earthquake record cause the strength to be 

exceeded? To conclusively rule out cracking at elevation 720 feet, several earthquake records 

would have to be evaluated. In addition, the effects of vertical contraction joints would have to 

be accounted for. 

The level of effort expended in evaluating this node on the event tree has been significant, and 

the results of the two analyses performed are contradictory. Rather than perform additional 

analyses to try to reconcile the conflicting results, it would be better to assume that a crack does 

form, but to try to rule out dam failure at another node on the event tree that may be evaluated 

more definitively. The question that then must be asked is: “If a crack does occur, how far will it 

run?” 

A2.3 Node C: Crack Propagates through Thickness 

Once it is determined that the upstream face would crack at the reentrant corner as expected, it is 

necessary to evaluate what the effect of the crack would be. It must be determined whether the 

crack propagates through the dam. 

How the crack propagation would be modeled is the first choice that must be made. Most 

analysis programs have what is called a gap friction element. This is a good choice for crack 

modeling because the element allows compression and frictional shear stress across the gap, but 

it disallows tension once a set tensile strength is exceeded. The problem with this approach is 

that gap elements have to be laid out before the analysis is performed. The analyst constrains the 

direction of crack propagation by the position of the gap friction elements. In the current 

problem, there are two probable directions for crack propagation. 

The first possibility is that the crack would be roughly horizontal. Dams are constructed with 

horizontal lift joints between concrete placements. These joints are almost always weaker than 

the parent concrete above and below them. It is usually the case that these horizontal joints direct 

cracking. 

A2-7 
 

 

 

 

 

 

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The second possibility is that the crack would dive downward (see figure A2-9). This is because 

the forces that produce tension in this zone of consideration also produce shear stress. This shear 

stress tilts the major principal axis of stress in the area downstream. Since the crack would 

propagate perpendicularly to the major principal axis of stress, the crack would turn down. If the 

concrete is of uniform strength and not affected by lift joints, this second possibility is the most 

likely. 

Figure A2-9: Crack trajectories. 

It is possible to allow the analysis itself to determine the direction of crack propagation. There 

are concrete constitutive models that check for cracking on any plane within each element. Once 

an element is determined to have cracked, the orientation of the crack is stored, and the stressstrain characteristics of the cracked element are modified accordingly as the analysis progresses. 

For the sake of this analysis, it was assumed that the lift joint at or near elevation 720 feet will 

control the direction of crack propagation. This assumption is reasonable and the more 

conservative of the two propagation assumptions. Horizontal propagation could eventually run 

all the way through the dam, enabling a sliding failure. A diving crack would not daylight, but 

rather turn down into the foundation. As a result, a sliding failure mechanism would not be 

enabled. Unless the analyst can be certain that lift joints would not provide planes of weakness, it 

should be assumed that they would control the direction of cracking. 

A2-8
 
 

 

 

 

 

 

 

 

 

 

 

 

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Also for the sake of this analysis, crack propagation will be modeled with gap-friction, limitedtension elements. While these elements ignore some of the finer points of fracture mechanics, it 

must be remembered that the analysis must prove that cracking cannot propagate through the 

dam. More rigor is not expected to change the conclusion. 

Figure A2-10 shows a 2D model with the addition of a gap-friction interface. For this example, 

this interface has been assigned the following characteristics: 

Tensile strength: 400 lb/in2 

Unbroken shear strength: No sliding allowed if bonded 

Broken shear strength: 55 degrees 

All other properties of the model are identical to those in linear elastic model used for evaluation 

of node B. Earthquake excitation is also identical. 

Figure A2-11 shows the results of the nonlinear analysis. The interface gap opening is plotted 

with respect to time. Note that the interface breaks at the upstream face at 16.35 seconds into the 

earthquake event. This corresponds to the first exceedance of tensile strength shown in 

figure A2-5. Note also that 0.45 second later, the crack has run all the way through the dam. This 

indicates that the crack has propagated at about 650 feet per second. 

As a result of the analysis above, it is clear that cracking will start and propagate through the 

body of the dam for the given set of modeling assumptions and material properties. It should 

be noted that if a through-going crack were simply assumed in node A of the failure tree, the 

analyses for nodes B and C could have been avoided. 

The question now is whether the dam can continue to resist static and dynamic loads in a cracked 

state. The potential for sliding on the plane induced by the crack will need to be evaluated. 

Figure A2-10: 2D nonlinear finite element model. 

A2-9
 
 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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Figure A2-11: Crack propagation. 

A2.4 Node D: Removable Blocks Form and Slide Planes Develop 

The dam is a 3D structure. The question that must now be considered is whether significant loads 

can be transmitted sideways, as well as directly downward to the foundation. If the dam can span 

from valley wall to valley wall above the crack, the cracking at elevation 720 feet will be less 

relevant. A nonlinear 3D model similar to the linear 3D model used in node B can be evaluated 

(see figure A2-12). 

It is important to note that a real dam is divided by vertical monolith joints that can inhibit lateral 

load transfer (see figure A2-13). A realistic 3D model must account for the effect of these joints. 

Monolith joints can be keyed. If they are, they may be able to transmit significant shear forces; 

but if they are not, lateral load cannot be transferred. In either case, no tension can be transmitted 

across a vertical monolith joint. The gap-friction interface element is an effective way to model 

monolith interaction across joints. In addition, it is typically assumed that the concrete/ 

foundation contact is a friction-only interface. Thus, the gap-friction interface is used to model 

the contact between concrete and rock above elevation 720 feet. It is the 3D analogue of the 

interface used in the 2D analysis performed for node C. 

A2-10
 
 

 

Selecting Analytic Tools for Concrete Dams to Address 

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Figure A2-12: Coarse 3D nonlinear finite element model. 

Figure A2-13: Interfaces. 

A2-11
 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Note the coarseness of the mesh used in this nonlinear model. In the linear elastic model used 

to evaluate node B, the dam alone consisted of 2,293 elements. In this nonlinear model, only 

233 elements are used in the dam. This is almost a tenfold reduction. It may be necessary to 

perform parametric studies to arrive at the appropriate fineness of mesh size when stresses do not 

significantly change. The reason for this mesh coarsening is that we are no longer trying to 

evaluate stresses in the area of the reentrant corner at elevation 720 feet. Accurate determination 

of stress is no longer a concern because, from previous analyses, it is clear that cracking would 

start and propagate through the dam. The failure mode would be dominated by the rigid body 

motion of dam monoliths slipping and rocking on interfaces. Nonlinear models typically require 

quite a bit of time to run. Run times correlate roughly to the number of elements squared. It 

therefore behooves the analyst to not have an overly fine mesh when performing nonlinear 

analysis. 

Interface parameters are reflective of the following assumptions: 

•	 Vertical interfaces between monoliths have some keying. This is modeled by assuming a 

high friction angle of 75 degrees. If monoliths are in contact at all, they are capable of 

transmitting some shear between them. Keys could have been modeled directly with a 

significant increase in model complication. 

•	 The interface between the abutments and the dam is assumed to be rough. Again,
 

75 degrees is used.
 

•	 The interface at elevation 720 feet is assumed to be broken concrete along a relatively 

smooth lift surface. A friction angle of 45 degrees is representative of this state. 

Figure A2-14 tracks displacement of three points during the earthquake: (1) the motion of the 

base concrete below elevation 720 feet, (2) the motion of the base of monolith 7, and (3) the 

motion of the base of monolith 8. 

At this history sampling point, the slip on plane 720 is about 0.1 foot (1.2 inches) at the end of 

strong shaking. Monoliths 7 and 8 are moving downstream with respect to the base. 

The induced offset can be captured and plotted around the perimeter of the potential failure 

surface (see figure A2-15). In general, offsets are on the order of 1 inch over a significant portion 

of the failure surface. 

The analysis performed above ignored the effects of uplift. If the crack at elevation 720 feet is 

not present until 15 seconds into the earthquake, it can be reasoned that there is insufficient time 

for the crack to be pressurized by the reservoir. This assumption may be reasonable, but it is 

somewhat unconservative. Figure A2-16 shows offsets produced when the effect of uplift is 

applied in the joint using concentrated forces at nodes. While offsets are several times greater 

than when uplift is ignored, they are still on the order of inches and not feet. 

A2-12
 
 

 

Selecting Analytic Tools for Concrete Dams to Address 

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Figure A2-14: Induced offsets. 

A2-13
 
 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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Figure A2-15: Without uplift, induced offsets at each contraction joint looking down at the 

model. 

Figure A2-16: With uplift, induced offsets at each contraction joint looking down at the 

model. 

A2-14
 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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One could argue that the 75-degree friction angle assumed on the right abutment interface is not 

appropriate considering the 6 inches of offset indicated in figure A2-16. It would be prudent to 

lower the friction angle to a residual value between 40 and 45 degrees and rerun the analysis. 

Certainly, larger offsets would result on the right abutment. However, this is technically a 

different failure mode and should be evaluated independently from the failure of blocks with a 

base plane at elevation 720 feet. 

It is also possible that if abutment sliding were inhibited, sliding of blocks with a base plane at 

elevation 720 feet could be affected. Numerous parameters could be investigated through 

sensitivity analysis. However, each time a parameter is varied, a new run would have to be done. 

Judgment must be applied considering that each run takes considerable time. 

A2.5 Node E: Duration and Severity of Shaking Sufficient to Cause 

Failure 

The analyses performed at nodes A through D have shown that cracking caused by the 

earthquake motions cannot be ruled out, and also that it is reasonable to expect that portions of 

the dam would slide downstream, but that the amount of displacement is on the order of inches. 

Is this induced offset cause for concern? The reservoir would not be released by blocks moving 

downstream a few inches. Was it necessary to progress through the entire failure tree, node by 

node, with ever more complicated analyses? It is probably not necessary. A simple Newmark 

style analysis could be done to estimate how big offsets would get. 

The equation below, an expression of F=MA in finite difference form, can be solved iteratively 

using a simple program, or even a spreadsheet. As opposed to the 3D nonlinear analysis 

associated with node D, this computation can be performed in seconds rather than hours. 

Ft = Ms * A 

D (t+∆t ) − 2 * D ( )t + D(t t −∆ ) A =
 ∆t

2
 

 D (t+∆t ) − 2 * D ( )t + D (t t −∆ )  Ft = Ms *   2  ∆t  

F * ∆t

2 

t = D (t+∆t ) − 2 * D ( )t + D(t−∆t ) Ms 

F * ∆t

2

D = t + 2*D − D eq. E1 ( ) t+∆t ( )t t( ) −∆t Ms

where, 

D = downstream displacement. 

t = time. 

Ft = total instantaneous force imbalance. This is the static and dynamic driving 

forces minus total frictional resistance. 

A2-15
 
 

 

 

 

 

 

 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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Ms =	 total system mass including the dam above the elevation 720 failure plane and 

the mass of participating water. 

Inherent in this analysis is the assumption that the failure plane is fully cracked and that a 

residual friction angle is approached due to slipping along the failure plane. In this example, 

45 degrees was assumed. Uplift was assumed to be acting on the failure plane. 

Results are shown in figure A2-17. Note that this simple rigid body sliding model results in 

8 inches of downstream offset, similar to the 6 inches of downstream offset indicated by the 3D 

nonlinear dynamic analysis done for node D. 

Figure A2-17: Results of Newmark analysis. 

Whether offset is 1 inch or 10 inches, this is a small amount compared to the 294 feet of contact 

length on the failure plane; however, drain efficiency may be compromised if offsets are greater 

than drain diameter or dilation on the plane allows for large water inflows. In addition, joint 

shear strength would be reduced as a result of sliding. 

A2.6 Node F: Damage Caused by Earthquake Results in the Dam Being 

Unable to Resist Static Loads 

The simple gravity analysis shown in figure A2-18 and accompanying results in table A2-1 

reveal that with a friction angle of 45 degrees, there is a post-earthquake factor of safety of 1.34. 

The block above elevation 720 feet is stable. 

A2-16 
 

 

Selecting Analytic Tools for Concrete Dams to Address 

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Figure A2-18: Post-earthquake analysis. 

Table A2-1: Results 

Forces Fx (kip) Y= (feet) Fy (kip) X= (feet) Moment (0,0) (kip-ft) 

Dam weight 0.00 0.00 -8,142.75 97.65 795,143.31 

Headwater 3,713.58 835.00 0.00 0.00 3,100,839.00 

Uplift 0.00 0.00 3,164.62 98.00 -310,132.38 

Totals 3,713.58 -4,978.13 3,585,850.00 

Resultant location at X= 183.216 ft, Y= 720 ft, sliding safety factor = 1.34 

Note: kip = 1,000 pounds, kip-ft = 1,000 pound-feet 

A2-17
 
 

 

 

 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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Input parameters: 

• Concrete unit weight = 150 pounds per cubic feet (lb/ft3) 

• Slide plane phi = 45 degrees 

• Head elevation = 1065 feet 

• Tailwater elevation = 720 feet 

One point that must be understood is that if the failure process is interrupted at any node, failure 

will not occur. It is not necessary to progress through the failure tree from the first node to the 

last. If failure can easily be ruled out at any node, it is most prudent to concentrate efforts on that 

node and ignore other nodes. Figure A2-19 illustrates this point. 

Figure A2-19: Path through event tree. 

A2-18
 
Appendix A3 

Earthquake Causes Sliding in Foundation Leading to 

Dam Failure 

 

 

 

 

 

 

 

Appendix A3: Earthquake Causes Sliding in 

Foundation Leading to Dam Failure 

This potential failure mode is due to a seismic event with sufficient energy to displace a 

removable rock block of sufficient size in an abutment causing loss of foundation support for a 

curved gravity dam (see figure A3-1). With loss of abutment support, sections of the dam may 

not be stable. The dam and foundation blocks may slide together and fail as a unit. 

Figure A3-1: Schematic of a dam built on a removable rock block in the 

foundation of sufficient size to affect dam stability if the block moves. 

The key point in the description of this potential failure mode is that the rock block is potentially 

removable and is of sufficient size to affect the stability of the dam. For this example, it is 

assumed that no potential failure mode is associated with rock blocks if there is no removable 

block in the foundation. A removable block has a base plane, side plane(s), and a release plane, 

as well as a free surface for the block to slide toward (see figure A3-1). The base plane of the 

block is a relatively horizontal surface forming the bottom of the block. The side planes of the 

block are steeply sloping, diverging surfaces oriented in the stream direction. The release plane is 

a steeply sloping surface oriented in the cross-canyon direction and positioned at the upstream 

extent of the block. Movement of the rock block would slide on both or either of the base and 

side planes and would pull away from the release plane. An open face is an unrestrained face. 

There is also no potential failure mode if the block is too small to affect the stability of the dam. 

In this case, the dam can bridge across the block. Considerable field work may be necessary to 

determine the likelihood that a block exists and that this potential failure mode is possible in the 

foundation. Movement of the rock block is a function of the driving forces on the block and the 

resisting forces as described below. Figure A3-2 shows graphically what might happen if the 

A3-1 
 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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rock block slides, the loading path changes, the shearing resistance changes, the concrete 

fractures, and the drainage system undergoes disruption. In addition, dilation of the foundation 

planes may result in increased seepage and a change in water pressures including the 

development of full hydrostatic pressure on the release plane. 

Figure A3-2: Potential failure mode of sliding of a removable block in the 

foundation induced by a seismic event and post-seismic considerations. 

The event tree for this potential failure mode is shown in figure A3-3 and has five nodes along 

the tree. In this case, it has been determined that there is a removable block in the foundation and 

the potential failure mode exists. The analyst could start at any point along the event tree to 

attempt to disprove failure, but in this example, the following sequence was used. 

A3.1 Node 1: Initiating Event: Earthquake Load with Normal 

Operating Loads 

Node 1 of the event tree considers if the earthquake has sufficient magnitude along with the 

corresponding static loads to potentially cause movement of the foundation block. When 

appropriate, static operating loads include gravity, reservoir, ice, tailwater, silt, temperature, 

uplift, and other loads during normal operations. The dynamic loads include inertial forces of the 

dam onto the foundation block, inertial forces of the foundation, and the hydrodynamic 

interaction of the reservoir. Some loads (typically temperature, ice, and silt loads) apply an initial 

state of stress in the dam and foundation but are relieved once movement occurs. 

A3-2
 
 

 

 

 

 

 

 

 

 

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Figure A3-3: Event tree for this potential failure mode. 

Conversely, follower loads (typically gravity, hydrostatic (reservoir, tailwater, and uplift), 

hydrodynamic, and seismic loads) continue to load the dam and foundation block even after 

movement occurs. These loads may change in magnitude as movement occurs. Static loads on 

the dam and foundation could be significantly different after the earthquake compared to the start 

of the earthquake. The dam may become unstable not from the seismic loads but from the new 

post-seismic static loads and material strengths. 

Following the earthquake, other potential failure modes not associated with the seismic event, 

such as increased reservoir elevation on the damaged structure, could develop and should be 

checked. 

A3.2 Node 2: Movement of Foundation Block Commences 

Node 2 of the event tree considers if the driving forces on the foundation blocks are sufficient 

to overcome the sliding resistance and sliding initiates. Movement of foundation blocks is a 

function of the magnitude and orientation of the applied loads, the cyclical nature and frequency 

content of the ground motion, the shear strength along sliding planes, and the orientation of the 

sliding planes. Shear strength along sliding planes is a function of joint roughness and infilling 

material. Depending on the orientation of the joints and the orientation of the loading, sliding can 

be along a single plane or along the intersection between two planes. 

A3-3 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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A3.3 Node 3: Sufficient Block Movement Affects Dam 

Node 3 of the event tree determines if the earthquake lasts long enough to cause sufficient sliding 

of the foundation blocks to affect the stability of the dam. The earthquake must have sufficient 

duration to induce sufficient movements in the foundation to have the potential to cause a dam 

failure. One large pulse of the earthquake may cause initial foundation block movement, but if 

the movement is too small and not sustained, redistributed loads within the dam may prevent 

failure. Longer duration earthquakes can cause larger movements in the foundation such that 

loads cannot be redistributed in the dam and thus might cause failure during the earthquake. For 

this reason, the event tree splits into two possible scenarios at node 3. If the earthquake has 

sufficient duration to move the foundation block enough to affect the stability of the dam, the 

tree progresses to node 5, where the stability of the dam is evaluated during the earthquake. If the 

earthquake does not have enough duration to fail the dam during the earthquake, the event tree 

progresses to node 4, where the post-earthquake stability of the foundation block is evaluated 

due to possible changes in static load and sliding resistance. 

A3.4 Node 4: Foundation Block Slides After the Earthquake (PostEarthquake) 

Node 4 of the event tree deals with post-seismic stability if the foundation blocks do not slide 

enough during the earthquake to adversely affect dam stability. Movements of the foundation 

block may change stability conditions, increasing uplift and waterflow around the block, 

crushing or fracturing rock blocks, or reducing shear strength from sliding. Post-seismic 

stability considerations might include aftershocks. Uplift might increase around the block from: 

(1) opening of the release plane, allowing full hydrostatic reservoir head to penetrate to the full 

depth of the block; (2) dilating of block planes, allowing more water to penetrate along the 

planes; or (3) severing or disruption of foundation drains, impairing their ability to relieve 

seepage pressures. 

A3.5 Node 5: Dam Fails 

Node 5 of the event tree considers the stability of the dam given post-earthquake instability of 

the foundation block. The movement of the block has occurred either during the earthquake 

(node 3) or after the earthquake (node 4). Questions that could be asked are: How stable is the 

dam given a certain amount of movement in the foundation? How stable is the dam given a 

certain lack of foundation support? Are there other 3D mechanisms that might come into play 

that could stabilize the dam? Is the dam too deformed or damaged to be stable? Is the dam large 

enough to stabilize the foundation block and prevent large releases of the reservoir? 

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A3.6 Case Study of a Foundation Stability Study 

The following case study describes the process of progressing through an event tree multiple 

times. The potential failure mode is sliding of a large block in the foundation under a curved 

gravity concrete dam (see figure A3-4). Each assessment through the event tree represented more 

focused attention to a node in the tree, reducing uncertainty (associated with the seismic loading, 

geology, and material properties) and increasing understanding of the dam response by using 

more advanced analysis methods to remove known limitations of simpler methods, making a No 

failure decision at any of the nodes more defensible. 

The dam is generally in good condition and has performed well for over 50 years. Great attention 

was paid to the geology of the foundation during original construction, but little was understood 

at that time regarding potential failure modes related to sliding of foundation rock blocks. The 

primary dam safety issue had to do with potential sliding on weak bedding plane partings within 

the dolomitic limestone of the left abutment. The entire left abutment of the dam was founded on 

a 30-degree “dip slope” in the dolomitic limestone foundation rock (continuous bedding plane 

partings parallel the abutment contact at various depths (base planes) and daylight just 

downstream from the dam (open face)) (see figure A3-5). Faults and joints form additional 

discontinuities (side and release planes) within the rock mass potentially allowing movement of 

large foundation blocks. It was determined that the foundation blocks were removable and large 

enough to affect the stability of the dam. Block sliding in the foundation is definitely a potential 

failure mode. 

Figure A3-4: Aerial view of curved gravity dam,
 

Bureau of Reclamation.
 

A3-5
 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Figure A3-5: This photograph, taken during construction, shows dam-to-foundation contact and the 

continuous nature of slide planes on the left abutment; the side plane is the fault just below workers, the 

base plane is parallel to dam contact, and the release plane is vertical along the upstream face of the dam, 

Bureau of Reclamation. 

A3.6.1 Node 1: Seismic Load and Normal Loads 

Node 1 relates to the initiating event: an earthquake. The dam is in a relatively high seismic area, 

so a sufficiently large earthquake is of concern. Hydropower generation at the dam requires a 

constantly high reservoir level, making static driving forces and post-seismic stability a concern 

and generating hydrodynamic forces that would increase seismic loads. The dam is a massive 

curved gravity dam, so temperature loads, ice loads, and silt loads are of lesser concern. 

A3.6.2 Node 2 and Node 3: Block Movement Commences and Affects the Dam: 

Assessment 1 

Node 2 considers if movement of a foundation block occurs, and node 3 considers how much 

movement occurs. The first time through the event tree was based on simplified and preliminary 

assessments, based on existing knowledge about the foundation, screening-level seismic ground 

motions, assumed material properties, forces generated from the dam to foundation with linear 

elastic finite element models (see figure A3-6), assumed uplift pressures, approximated plane 

orientations, and simplified, linear, uncoupled rigid block Newmark analysis (see figure A3-7). 

Uncoupled analysis means that the dam analysis and the foundation stability analysis are 

separate computations. Forces from the dam are applied continuously on the foundation and do 

not change as the foundation moves. Also, any rotation of the foundation blocks is not 

considered. 

A3-6
 
 

 

 

 

 

 

 

 

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Figure A3-6: EACD3D96 finite element model of the dam computed forces from the dam 

into the foundation for an uncoupled rigid block Newmark foundation stability analysis. 

The computed maximum displacement of the foundation block was 180 feet (see figure A3-7). 

Because of the large magnitude of predicted displacements, the responses to nodes 2 and 3 were 

Yes. 

The alarming prediction of 180 feet of deformation needs to be tempered with the realization that 

because of 3D effects, the failure mode analyzed is probably impossible. If this analysis had 

concluded that the displacements were small, it could have been reasoned that failure could not 

occur. The fact that the simplified analysis predicted very large movements does not mean the 

converse, and further analysis is required. 

A3.6.3 Node 2 and Node 3: Block Movement Commences and Affects the Dam: 

Assessment 2 

The results of the first study had large uncertainties based on available information and 

simplified analysis. At this point, one could choose to refine the material properties and loads or 

change the analysis technique. The decision was made to reduce uncertainties, refining 

seismology with site-specific field investigations, performing site-specific geologic 

investigations, extracting core and performing laboratory testing of the block slide planes and 

concrete within the dam, including large scale roughness effects to increase sliding resistance, 

and conducting vibration testing of the dam. The cost of creating a nonlinear, coupled finite 

element model was significant, so this was not done during the second study. Uncertainty 

associated with the simpler, uncoupled analyses was known and accepted for this assessment. It 

was known at the start of studies 1 and 2 that the simplified, uncoupled rigid block Newmark 

method had limitations and would probably overpredict sliding displacements of the rock blocks. 

However, the analyses could be easily rerun, inputting new material properties. 

A3-7
 
 

 

 

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Figure A3-7: Results from the uncoupled Newmark rigid block sliding study. 

A3-8 
 

 

 

 

 

 

 

 

 

 

 

 

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No further studies would be necessary if the uncoupled analyses using the best available data 

showed acceptable displacements. Unfortunately, the displacements computed in this assessment 

were still significant enough to affect the stability of the dam, and the responses to nodes 2 and 3 

were still Yes. 

A3.6.4 Node 2 and Node 3: Block Movement Commences and Affects the Dam: 

Assessment 3 

It was decided that the third assessment through the event tree would be based on more advanced 

analyses. A nonlinear, coupled dam-foundation-reservoir, dynamic, structural analysis using 

LS-DYNA was the next step. A coupled analysis includes both the dam and a critical foundation 

block in the same finite element model so that the forces on the blocks change as movements 

occur and redistribution of loads and dam-foundation interaction takes place (see figures A3-8 

and A3-9). Nonlinear aspects of the analyses included contact surfaces to model the contraction 

joints in the dam and foundation sliding planes, and reservoir water modeled with finite 

elements. 

Figure A3-8: Closeup of LS-DYNA model showing dam, contraction joints, 

foundation, reservoir, and potentially unstable rock blocks, Bureau of 

Reclamation 

The LS-DYNA results indicated significantly smaller foundation block displacements of 

3.3 inches, depending on the magnitude of the earthquake (a few inches compared to tens of feet) 

for low Rayleigh damping levels (about 2 percent of critical), frictional damping, and radiation 

damping on the order of 0.4. This is, in large part, due to the 3D effects of the vertical 

contraction joints in the dam in controlling the mode of sliding. Contraction joints in the dam 

forced sliding of the blocks in an oblique, uphill direction along the theoretical intersection of the 

contraction joint and base plane, resulting in higher static safety factors and, thus, smaller 

seismic displacements. There also was a small clockwise rotation of the block not computed in 

the uncoupled analysis. The response for node 3 was now No. However, the response at node 4 

could not be a No because with 3.3 inches of movement, the foundation drains could be pinched 

shut, permitting increased uplift pressures, and the release plane was now open and joints dilated 

A3-9 
 

 

 

 

 

 

 

 

 

 

 

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Figure A3-9: Plan view of LS-DYNA model showing dam, contraction 

joints, foundation, reservoir, and potentially unstable rock block, Bureau of 

Reclamation. 

to permit direct inflow of reservoir water into the planes. Post-seismic stability would be in 

question. Although it is difficult to predict exactly how the foundation water pressures would 

change following earthquake movement, they are likely to increase as a result of the release 

plane opening up along the upstream dam contact. Post-earthquake foundation stability with 

increased uplift is, therefore, a concern. The results of the modeling indicated that the dam and 

foundation were likely to be stable during a large earthquake, but post-earthquake stability was 

still questionable. This prompted additional studies addressing nodes 4 and 5 and to further 

reduce some uncertainties. 

A3.6.5 Node 4: Post-earthquake Stability of the Foundation Blocks 

For the reasons just stated, a fourth time through the event tree for post-seismic stability was 

performed using an additional validation of the nonlinear finite element model and including 

post-seismic stability analyses. There is considerable uncertainty in the assumed uplift profiles. It 

was assumed an increase in uplift would be nearly certain if the dilation along the planes 

approached the maximum value of about 2.5 inches. For dilations less than an inch, increases in 

uplift were thought to be unlikely. If the horizontal displacement from LS-DYNA was calculated 

to be more than half the drain diameter of 3 inches, it was thought likely that the uplift would 

tend toward the linear case of full reservoir head on the upstream side to zero at the downstream 

point (point above tailwater). If the displacement was less than this, it was thought that drain 

effectiveness would likely remain, and the case of 50-percent drain effectiveness (drain factor 

K=0.5) was chosen to represent this condition. 

The post-seismic static sliding factors of safety were greater than one, showing stability even 

with inoperable foundation drains, varied uplift profiles, and reduced shear strengths, indicating 

that post-earthquake instability is unlikely in any case. So the response for node 4 was No. 

A3-10 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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A3.6.6 Node 5: Dam Fails 

Because the response at node 4 was No, the event tree sequence stopped and did not lead to a 

plausible failure of the dam. 

A3.7 Addressing Uncertainty 

This case study shows the process of approaching an assessment of a potential failure mode in 

phases from simple analyses with large uncertainties and known limitations to more complex 

analyses with reduced limitations and uncertainties. Because this process involved multiple 

analyses building on one another, it is important not to take conclusions from initial analyses that 

were superseded. The first phase assessment for nodes 2 and 3 resulted in significant sliding, 

using known limitations in the simplified Newmark rigid block analysis with accepted 

uncertainty in the available and assumed material properties and loading. The assessment was 

that the block could slide too far to maintain dam stability. The second phase assessment also 

focused on nodes 2 and 3 with the intent of reducing the uncertainties on the loads and material 

properties. Again, the assessment was that the block could slide too far to maintain dam stability. 

The third phase also focused on nodes 2 and 3 with the intent of reducing the limitations of the 

simplified analyses with nonlinear analyses. The nonlinear analyses eliminated some limitations 

of the uncoupled analysis because it modeled additional features. The assessment was that the 

block would slide but not far enough to fail the dam during the earthquake. This moved the 

assessment along the event tree to node 4 and, to a smaller degree, still in node 3. The additional 

uncertainties in the nonlinear analyses were addressed, and the post-seismic stability analyses 

determined that the dam would be stable after a large earthquake. 

Nonlinear analyses provide valuable insight into the nonlinear response of the dam and 

foundation block. The geometric nonlinearities in the model required numerous validation 

studies to gain confidence in the model and demonstrate it was correctly working. This would 

also be true if the material nonlinearities of the concrete or foundation rock were modeled. 

Nonlinear analyses require additional considerations for element size, load path, wave travel 

through the finite elements, time step, rate effects, effects of damage, and damping. There needs 

to be a demonstration of the model’s ability to replicate stress-strain paths and failure along 

strain paths not used in the fitting procedure. The evolution of damage needs to be represented. 

The rate effects attributable to inertial resistance to crack opening may also be an important 

parameter. 

Uncertainties were reduced by several means. Strength properties used in the Newmark and 

nonlinear models were based on a detailed, laboratory, direct shear testing program to determine 

the basic friction angles (with dilation removed) of the various bedding plane partings and 

detailed photogrammetric roughness profile evaluation to estimate the waviness component to 

shear strength. Concrete properties were determined from detailed laboratory testing and 

vibration testing. Foundation water pressures were determined from a series of permanent and 

temporary piezometers (placed in existing foundation drain holes). The approximate average 

displacements of the block from the finite element analyses were used to estimate the amount of 

probable dilation and resulting friction based on the roughness studies. 

A3-11 
 

 

 

 

 

 

 

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Some uncertainties were not addressed or still remained. First, a friction angle of 55 degrees was 

assigned to the contraction joints in the dam model. The contraction joints are keyed, providing 

higher shear strength than included in the finite element model. Second, full uplift was 

included on the back release surface in the analyses, which is unlikely to exist under current 

predisplacement conditions based on piezometric measurements and the tightness of the 

foundation. 

An understanding of the geometric interaction between the faults in the foundation and the 

contraction joints in the arch dam was vital when assessing results from the nonlinear coupled 

dam/foundation analysis. The contraction joints converge in the downstream direction, making 

their wedge shape inherently stable. The block planes diverge in the downstream direction, but 

the likely direction of sliding is controlled by a dam contraction joint at the downslope limit of 

the foundation block. This resulted in a more favorable condition and reduced sliding 

displacements. In essence, the dam helps stabilize the foundation. This may be acceptable for a 

massive dam such as a gravity dam. 

A3-12
 
Appendix A4 

Increased Silt Load Causes Failure of Reinforced 

Concrete Corbels of a Buttress Dam 

 

 

 

 

 

 

 

 

Appendix A4: Increased Silt Load Causes Failure of 

Reinforced Concrete Corbels of a Buttress Dam 

This potential failure mode considers increased loads due to silt buildup on the slabs of the 

buttress dam. Increased silt loading takes place when the elevation of silt buildup is higher than 

the anticipated height for design. The increased elevation of the silt would cause an increase in 

lateral and vertical earth pressures that act on the slabs of the buttress dam which are supported 

by the corbels. Most dams are designed to resist the forces of silt up to a certain elevation, at 

which time the silt would be removed to a lower level and the silt buildup process would start 

over again. Silt buildup can occur more rapidly than expected due to occurrences upstream of 

the dam such as heavy construction or poor land management techniques. The situation is 

compounded by a lack of maintenance or a disregard for the level of silt. Figures A4-1 and A4-2 

show a typical slab and buttress dam. 

The event tree for this potential failure mode is presented in figure A4-3. The resulting 

overstressing and failure of the corbels could lead to loss of support of the buttress slabs 

which could lead to loss of storage and instability of the buttress dam system. Several failure 

mechanisms of the corbel must be investigated to determine the mechanism with the least 

strength associated with it. Figure A4-4 presents the possible failure mechanisms that must 

be investigated to determine the strength of the corbel system. The four potential failure 

mechanisms are: (1) direct shear failure at the interface between bracket or corbel and supporting 

member; (2) yielding of the tension tie due to moment and/or direct tension; (3) crushing of the 

internal compression “strut;” and (4) bearing or shear failure under the loaded area. 

The analysis addresses all four mechanisms of failure of the corbel that could lead to the failure 

of the slab that could release the reservoir, constituting failure of the dam. This example assumes 

failure of the corbel results in failure of the dam. Analysis was accomplished without load factors 

because actual failure states were of interest. The concrete compressive strength, originally at 

3,000 lb/in2

, was assumed to deteriorate to 2,500 lb/in2, and the yield strength of the steel was 

assumed to be 40,000 lb/in2. 

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Figure A4-1: Downstream side of a slab and buttress dam, Bureau of Reclamation. 

Figure A4-2.—Closer photograph of the downstream side of the dam 

showing buttresses, struts, corbels, and upstream slab. This example looks at 

silt accumulation on the upstream side of the dam and the effect on the 

corbels, Bureau of Reclamation. 

A4-2
 
 

 

 

 

 

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Figure A4-3: Event tree. 

Figure A4-4: Structural action of corbel. 

A4.1 Node 1: Initiating Event: Silt Buildup with Normal Loads 

This node investigates whether the design elevation for silt buildup has been exceeded in the 

presence of normal loads. The original design documents should be reviewed to determine the 

level of silt buildup accounted for in the original design, if any. Obviously, if the level of silt 

A4-3 
 

 

 

 

 

 

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buildup is less than allowed in the original design and the strength level of the concrete is 

adequate to support the design silt buildup level, failure of the corbel is unlikely due to silt 

buildup. Figure A4-5 shows the area of the corbel/buttress system being investigated. 

Figure A4-5: Corbel/buttress section. 

A4.2 Node 2: Crack Initiation at Corbel/Buttress Interface 

Destructive and/or nondestructive tests on the concrete and reinforcing steel of the structural 

elements of the buttress dam should also be performed, depending on the age of the structure, to 

determine compressive strength levels of the concrete and condition of the reinforcing steel. 

Obviously, destructive testing needs to be done at a structurally inconsequential area. 

Information can be gleaned from the current conditions and construction records. Cracking at the 

interface between the corbel and buttress would indicate that service level loading has increased 

beyond the uncracked flexural capacity of the corbel and can also mean that the strut and tie 

phenomenon of the corbel system has been engaged. Further evidence of the strut and tie 

phenomenon would be diagonally oriented cracking within the corbel and buttress. The tension 

reinforcement near the top of the corbel will be engaged to balance the strut action of the strut 

and tie system being developed. 

Analysis computations based on the silt and water levels, as well as the compressive strength of 

the in situ concrete, can predict the strength level of the various failure mechanisms. All failure 

mechanisms would involve various levels of cracking of the corbel and buttress. Because 

cracking would commence at the upstream side of the corbel, the presence of cracking could not 

be seen and could only be determined by nondestructive testing. Nondestructive testing would 

have to be capable of detecting interior delaminations as this would be the type of cracking 

involved. Access to the top side of the corbel is not possible due to the presence of the slabs, silt, 

and water from the reservoir. 

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A4.3 Node 3: Yielding of Tension Steel at Corbel/Buttress Interface 

Yielding of the tension steel at the top of the corbel would occur in conjunction with the 

compressive action of the inclined compression struts. Initially, the tension in the steel is due to 

flexural actions of the corbel just after first cracking at the interface between the corbel and 

buttress. Additional cracking, along with rotations of the formed struts, would cause additional 

tensioning of the steel until it reaches its yield strength. Actually, it is most desirable for the steel 

to reach its yield strength before the compressive struts crush since a brittle failure situation 

would exist if the concrete crushes before the steel yields. Analysis computations can determine 

the strength level of the corbel/buttress system at yield of the steel. An example of a corbel 

analysis is provided after section A4.6. 

A4.4 Node 4: Wedge Displacement of Corbel 

This node describes the final stages of failure of the corbel. Signs of failure would be visible 

from the downstream face of the corbel/buttress. The wedge displacement would be 

characterized by a pair of inclined to nearly vertical cracks, with a horizontal crack running 

between the two cracks that is located near the intersection of the corbel and buttress face. Since 

the corbel is integral to the top of a sloping buttress wall, the loading on the corbel is variable, 

and the loading is reduced as the elevation of the corbel increases. The failure wedge could be 

less than the width of a slab panel; thus, there would not be a loss of support for the entire width 

of the slab. 

A4.5 Node 5: Entire Slab Panel Width of Corbel in Wedge Displacement 

This node deals with a length of failed corbel that is equal to the width of slab. Without corbel 

support, the slab would displace downward, allowing water to leak from the reservoir. Damage 

to the bearing ends of the slab panels could eventually occur that would allow the slab to be 

completely dislodged. Overall instability of the slab and buttress system could result. 

A4.6 Node 6: Corbel Fails Leading to Dam Failure 

This node deals with the loss of corbel support for the slab for the complete width of a slab 

panel. Loss of storage through the resulting opening could lead to instabilities of the adjacent 

panels. Progressive collapse of the system of slabs and buttresses could result. 

A4-5
 
 

 

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A4-6
 
 

 

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A4-7
 
 

 

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A4-8
 
 

 

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A4-9
 
 

 

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A4-10
 
 

 

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A4-11
 
 

 

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A4-12
 
 

 

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A4-13
 
 

 

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A4-14
 
 

 

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A4-15
 
 

 

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A4-16
 
Appendix A5 

Freeze-Thaw Deterioration Removes Concrete Cover 

and Causes Upstream Slab to Fail 

 

 

 

 

 

Appendix A5: Freeze-Thaw Deterioration Removes 

Concrete Cover and Causes Upstream Slab to Fail 

Buttress dam concrete slabs subjected to freezing and thawing cycles/action may experience 

deterioration (figure A5-1). This deterioration could lead to significant loss of concrete surface, 

as well as corrosion of reinforcing steel. Deterioration of concrete by freeze thaw action needs to 

be evaluated to determine the buttress dam slab adequacy. The following example shows the 

process by which the slab may be evaluated (see the event tree in figure A5-2a and the analysis 

tree in figure A5-2b). While codes are cited in this example, we are evaluating failure limits; 

therefore, load factors are not applied. 

Figure A5-1: Depiction of failure mode and example of freeze-thaw in a buttress dam,
 

Bureau of Reclamation.
 

A5-1
 
 

 

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Figure A5-2a: Event tree. 

Figure A5-2b: Analysis tree. 

A5-2 
 

 

 

 

 

 

 

 

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A5.1 Node 1: Freezing and Thawing 

Hydraulic structures are especially susceptible to freeze-thaw damage because they are more 

likely to be critically saturated. Older structures are also more susceptible to freeze-thaw damage 

since the concrete was not air entrained. 

Deterioration of concrete from freeze-thaw actions may occur when the concrete is critically 

saturated, which is when approximately 91 percent of its pores are filled with water. When water 

freezes to ice, it occupies 9 percent more volume than when it is water. If there is no space for 

this volume expansion in a porous, water containing material like concrete, freezing may cause 

distress in the concrete. 

A5.2 Node 2: Inspection for Freeze-Thaw Damage 

High water content and non-air-entrained concrete were commonly used in older buttress type 

dams, which contributed significantly to their potential for freeze-thaw damage. Particular 

attention must be paid to the concrete in face slabs. Because of their relative thinness, they 

cannot withstand excessive scaling or spalling, which decreases the strength of the slab structure. 

Typical signs of freeze-thaw damage are: 

• Spalling and scaling of the surface 

• Large chunks (inches size) are coming off 

• Exposing of aggregate and/or reinforcing 

• Surface parallel cracking 

A5.3 Node 3: Extent of Deterioration 

Distress to critically saturated concrete from freezing and thawing would commence with the 

first freeze-thaw cycle and continue throughout successive winter seasons, resulting in 

repeated loss of concrete surface. An inspection for freeze-thaw deterioration should include 

determination of extent of spalling, scaling, and cracking. The visual inspection should serve to 

examine and record the condition of the slabs and buttress structure. 

A5.4 Node 4: Steel Reinforcement 

Corrosion of reinforcing steel produces iron oxides and hydroxides, which have a volume much 

greater than the volume of the original metallic iron. This increase in volume causes local radial 

cracks, formation of longitudinal cracks (parallel to the bars), and spalling with loss of cover 

A5-3 
 

 

 

 

 

 

 

 

 

 

 

 

 

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concrete. This cracking and/or spalling, if significant enough, can result in delamination of the 

surface (a well-known problem in bridge decks). Corrosion of reinforcing steel also results in 

loss of tension steel area, which reduces the strength of the slab (structure). Corrosion also 

reduces the ability of reinforcing steel to develop a bond between bars at splice locations. 

A5.5 Node 5: Slab Evaluation for Structural Adequacy 

Based on an inspection/testing program of the buttress slab(s), determine if a strength 

evaluation of the slab(s) is required. This determination should consider the amount of concrete 

deterioration and loss of reinforcing steel area that have occurred. To illustrate this approach, a 

typical buttress slab will be analyzed for strength as shown in the example below. 

A5.6 Introduction 

The buttress dam face slab analyzed in this exercise was constructed in the 1920s. The face slab 

thickness varies from approximately 3 feet 8 inches at elevation 738 to approximately 1 foot 

5 inches at elevation 847. The face slab is reinforced with 1-inch-diameter bars in the 

downstream face of the slab with a 3-inch clear cover. The bar spacing varies from 4 inches on 

center to 12 inches on center. The normal water surface is at elevation 841, and for this exercise, 

sediment up to elevation 758 will be included. The 14-foot-long face slab was poured in place 

and has shear keys at the buttress walls as shown in figure A5-3. 

Figure A5-3: Typical section at buttress wall. 

A5.6.1 Event Tree Node 1 

For the purposes of this exercise, the buttress dam is assumed to be in an area susceptible to 

subfreezing temperatures. Since air-entrained concrete was not used at the time of construction, 

the buttress dam face slab could potentially experience freeze-thaw damage. 

A5-4
 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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A5.6.2 Event Tree Node 2 

The following inspection scenario was developed for this exercise: An inspection of the buttress 

dam face slab revealed a concrete spall at approximately the center of the face slab span, 

extending from elevation 738 to elevation 742. The concrete spall is approximately 6 inches 

wide and 2 inches deep. Additional cracking was noted in the damaged area. 

A5.6.3 Event Tree Node 3 

Due to the depth of the concrete spall (only 1 inch of reinforcement cover remaining) and the 

location of the damage (near the center of the slab span), the concrete deterioration is determined 

to be significant and could potentially impact the safety of the structure. 

A5.6.4 Event Tree Node 4 

The additional cracking noted in the concrete spall area indicates that the reinforcement has 

likely been exposed to moisture and has been degraded to some extent. The degrading 

reinforcement would have a larger volume, which could have caused the cracking documented 

in the inspection. 

In the event that the reinforcing development length does not comply with current American 

Concrete Institute (ACI) code requirements, assume the section is unreinforced (because 

inspections have revealed that the steel is exposed and unbonded) and evaluate for structural 

integrity as a plane concrete section. Boundary conditions at the end of the slab should be 

carefully considered in modeling the slab to take advantage of any provided restraint. The 

conservative assumption of no effective steel will not result in slab failure, as will be seen. 

However, the horizontal thrust that will appear at the supports must be resisted. 

A5.6.5 Event Tree Node 5 

Analysis is required at node 5 to determine if a slab failure is imminent. Under the provisions of 

ACI 318-05 section 10.7, the face slab section analyzed in this exercise is classified as a deep 

beam because the loads are applied to the face opposite the face that is supported, and the slab 

depth to span ratio is greater than 0.25. Since the slab is classified as a deep beam, the analysis 

must account for the nonlinear strain distribution in the slab. Therefore, a 3D finite element 

analysis of the face slab was performed using GTStrudl Version 28. Since the reinforcement has 

potentially been impacted and the deep beam behavior would likely result in arching action, the 

slab was modeled as plain concrete. The concrete stresses will be compared to the plain concrete 

tensile and compressive strength in ACI 318-05. ACI 350-06 was not used because it addresses 

reinforced concrete design of environmental engineering structures and does not address the 

tensile and compressive strength of plain concrete. 

The deep beam analysis of the face slab described above is necessary due to the ratio of the slab 

depth to the clear span. However, the face slab thickness decreases at higher elevations of the 

dam from 48 inches thick at the base to 16.75 inches at the crest. Since the loads on the slab are 

not concentrated, once the ratio of the face slab thickness to the clear span is less than 0.25, the 

face slab is no longer classified as a deep beam and the finite element analysis is no longer 

necessary. The face slab must be analyzed as a reinforced concrete beam in accordance with 

chapter 10 of ACI 318-05 and meet the minimum reinforcement requirements. 

A5-5 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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A5.7 Finite Element Model Description 

A 12-inch-wide section of the face slab extending from elevation 738 to elevation 738.71 was 

modeled. However, to simplify the model geometry and load application, the slab was not 

inclined as it is actually constructed. Because both the hydrostatic pressure and the sediment 

pressure vary with the depth, they were decreased across the width of the model. This 

simplification is conservative because the full dead weight of the slab in the model is supported 

by the buttress walls, which results in higher bending stresses in the slab. The actual inclined 

orientation of the slab would result in lower bending stresses due to dead load because a portion 

of the dead load would be transmitted to the foundation at the base of the face slab. The face slab 

in the finite element model has a thickness of 41.75 inches after removing the concrete cover on 

the reinforcement. The face slab was loaded with three independent load cases and one 

dependent load case. Table A5-1 shows a summary of the loads applied to the model. 

Table A5-1: Loads applied to the model 

Load name Load description 

Case 1 

Case 2 

Case 3 

Case 4 

Dead load of the concrete slab. 

Hydrostatic pressure on the slab with the water surface at elevation 841. 

Pressure due to the buoyant weight of sediment up to elevation 758. It was 

assumed that the saturated unit weight of the sediment is 120 lb/ft3 . 

Dependent load case including independent load cases 1, 2, and 3. The load factors 

for the independent load cases were taken as 1.0. 

The portion of the face slab modeled is 14 feet long and has a clear span of 12 feet 7 inches. 

The face slab supports were modeled by restraining the translational degrees-of-freedom and 

releasing the rotational degrees-of-freedom at the model joints located where the slab rests on the 

support corbels. Compression only nonlinear springs oriented perpendicular to the ends of the 

slab to model the slab interaction with the buttress wall. A second support configuration that 

locates the supports only at the middle of the slab thickness was used to model a theoretical 

simple beam for model calibration using flexural analysis. 

Based on the age of the buttress dam, the concrete is assumed to be a normal weight concrete 

with a compressive strength of 2,500 lb/in2

. The modulus of elasticity of the concrete is 2.85 x 

106 lb/in2 (57,000 (fc ′) 0.5). 

A5.7.1 Finite Element Load Calculation 

Calculate hydrostatic pressure: 

w = 62.4 pcf 841 ft − 738 ft = 6, 427 psf hydro_ 738 ( )

w = 62.4 pcf 841 ft − 738.71 ft = 6, 383 psf hydro_ 738.71 ( )

A5-6 
 

 

 

 

 

 

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Calculate buoyant sediment pressure: 

wse diment _ 738 = (758 ft − 738 ft) (120 pcf − 62.4 pcf) = 1,152 psf
 

wse diment _ 738.71 = (758 ft − 738.71 ft) (120 pcf − 62.4 pcf) = 1,111 psf
 

A5.7.2 Finite Element Model Calibration 

To calibrate the model, the stress in the extreme tension and compression fibers of the face slab 

was calculated based on a 14-foot simple span beam as shown below. 

Calculate Slab Section Modulus
 

bh2 12"(41.75" )2 

3 S = = = 3486.in 

6 6
 

Calculate Hydrostatic Pressure
 

whydro = 62.4pcf(1ft)(841ft − 738ft) = 6427plf
 

Calculate Bouyant Sediment Pressure
 

w = (758ft − 738ft)(120pcf − 62.4pcf) = 1152plf
 sediment 

Calculate DistributedLoad on Slab
 

 1ft 
 w = 41.75" (1ft) * 150pcf + 6427plf + 1152plf 12in 
 

w = 8,101plf
 

Calculate Maximum Moment
 

wl 2 8101plf * (14.0 ft)

2
 

M = = =198,475lb − ft max 8 8
 

M max = 2,381,964lb − in
 

Calculate Stress
 

M max 2,381,964lb − in F b = = 3 S 3486in
 

Fb = 683.2 psi
 

The maximum tensile and compressive stresses in the calibration model were 683 lb/in2 and 

693 lb/in2, respectively. The maximum variance between the calculated stress and model results 

was only 1.4 percent. This variance shows that the model geometry and load applications are 

accurate. 

A5-7
 
 

 

 

 

 

 

 

 

 

 

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A5.7.3 Finite Element Analysis Results 

The supports were relocated to the bottom edge of the face slab, and the compression only 

nonlinear springs were added to model the actual support configuration. Since the slab was 

analyzed as an unreinforced elevated slab, the tensile stress was compared to the reduced 

modulus of rupture of the concrete in accordance with ACI 318-05. The reduced modulus of 

rupture for the concrete (φ7.5 (fc ′) 0.5, where φ= 0.55) was calculated to be 206 lb/in2. However, 

an elevated plain concrete slab is only permitted by ACI 318-05 if the slab exhibits arch action. 

Therefore, an iterative process was used to run the model and remove any elements where the 

tensile stress exceeded that calculated tensile strength of 206 lb/in2

. Figure A5-4 shows the final 

model geometry once the elements exceeding the tensile strength were removed. 

Figure A5-4: Final model geometry once the elements exceeding the tensile strength were 

removed. 

The stress contours in the face slab under the load case 4, shown in figure A5-5, demonstrate the 

arch action in the slab. The maximum tensile stress in the model was 20.4 lb/in2, which is well 

below the tensile strength of 206 lb/in2

. Since the slab develops arch action, the compressive 

stress in the concrete was compared to the compression limit for plain concrete in accordance 

with chapter 22 of ACI 318-05. The compression limit (φPn = 0.6f’c[1-(lc/32h)2], where 

φ = 0.55) was calculated to be 815 lb/in2

. The maximum compressive stress at mid-span was 

571 lb/in2, and the maximum compressive stress at the edge of the corbel was 1,037 lb/in2. 

However, the stress at the edge of the corbel diminishes to less than 800 lb/in2 within 1.5 inches 

of the joint. The corbel was modeled as a perfectly rigid support and, as shown in figure A5-6, a 

sharp change in the deformed shape occurs at this location. In reality, the corbel would have 

some flexibility and deformation, which would relieve some of the stress at this location. 

Therefore, the localized stress concentrations at the edge of the corbel were discounted. 

A5-8
 
 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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Figure A5-5: Face slab stress (SXX) contours. 

Figure A5-6: Deformed shape. 

Based on the results of the finite element analysis, the determination can be made that the 

deteriorated face slab in this exercise does not fail. Therefore, the slab does not fail in its current 

state. If deterioration is allowed to continue, its effects would have to be considered. 

In this example, phi factors were used to address material uncertainty and brittleness of the 

failure mechanism. If the conclusions of the analysis had been that the slab had failed, one would 

have to question whether or not failure was falsely indicated by excessive phi factors. In this 

case however, the analysis results without factors would indicate the same conclusion (see 

figure A5-7). 

A5-9
 
 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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Figure A5-7: Slabs: values for 12-inch 

strip. Reinforced for tension only. 

A5-10
 
 

 

Selecting Analytic Tools for Concrete Dams to Address 

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Calculate Hydrostatic Pressure at Center of 12" Wide Strip 

w hydro = 62.4 pcf (1 ft)(841 ft − 738.35 ft) = 6,405 plf 

Calculate Maximum Moment 

wl 2 6,405 plf * (12.583 ft)

2 

M max = = = 126,767lb − ft 8 8 

M max = 1,521,206lb − in 

π (1") 2 

As = 3× = 2.356in 2

4 

b = 12" 

d = 41.75" 

As 2.356in2 

ρ = = = 0.005

bd (12")(41.75") 

Ec = 57,000 2500 psi = 2,850,000 psi 

Es = 29,000,000 psi 

Es 29,000,000 psi n = = = 10.18

Ec 2,850,000 psi 

nρ = 10.18(0.005) = 0.0509 

k = (nρ)

2 + 2nρ − nρ

k = 0.05092 + 2(0.0509) − 0.0509 = 0.272
 

k 0.272 j = 1− = 1− = 0.909 

3 3 

Calculate steel stress : 

M max 1,521,206lb − in f s = = As jd 2.356in2 (0.909)(41.75") 

fs = 17,057 psi ≈ 17,000 psi 

Calculate concrete stress : 

M max 1,521,206lb − in f c = = 6 jkd 2 6(0.909)(0.272)(41.75") 2

f c = 588.3psi ≈ 513 

A5-11
 
 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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A5.8 Uncertainty 

The boundary conditions of the mathematical model are typically a major uncertainty in the 

development of the model. For example, three different boundary conditions were applied to the 

buttress dam face slab finite element model to demonstrate the effect on the stress results under 

the same load application. The support conditions included the following: restraint of the 

translational and rotational degrees-of-freedom at the center of the slab thickness, generating 

traditional beam behavior; restraint of the translational degrees-of-freedom where the face slab 

bears on the corbel, generating deep-beam arching behavior; and restraint of the translational 

degrees-of-freedom where the face slab bears on the corbel and along the end of the face slab. 

Figures A5-8 through A5-10 show the stress contours for each boundary condition, respectively, 

and table A5-2 shows a summary of the stress results at center span. 

The second boundary condition most closely models the actual boundary conditions of the 

buttress dam face slab being modeled. As shown in table A5-2, modeling the support conditions 

as a traditional fixed beam overstates the tensile stresses by 194 percent and the compressive 

stresses by 73 percent. While completely restraining the ends of the fixed beam only overstates 

the tensile stress by 2.5 percent, the compressive stress is underestimated by 40 percent. 

Figure A5-8: Stress contours for restraint of the translational and rotational degrees-offreedom at the center of the slab thickness. 

A5-12
 
 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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Figure A5-9: Stress contours for restraint of the translational degrees-of-freedom at the 

support corbel. 

Figure A5-10: Stress contours for restraint of the translational degrees-of-freedom at the 

support corbel and along the vertical edge. 

A5-13
 
 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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Table A5-2: Summary of model results 

Support condition 

Stress at center span (lb/in2) 

Tensile Compressive 

Beam action—fixed 

(see figure A5-8) 

688 693 

Deep beam—pinned 

(see figure A5-9) 

234 400 

Deep beam—fixed 

(see figure A5-10) 

240 240 

If the analysis is being performed using numerical methods, the options for boundary conditions 

are typically limited to fixing or releasing the translational and the rotational degrees-of-freedom 

at the supports. In addition to merely restraining or releasing the translational or rotational 

degrees-of-freedom, the engineer may utilize spring constants for any or all degrees-of-freedom. 

The spring constants are determined based on the stiffness of the foundation material and may be 

linear or nonlinear. The friction angle between the structure and foundation material will have an 

impact on the analysis results. The engineer will need to evaluate the effects of various 

foundation conditions on the structure if accurate material properties are not available and 

determine if it is necessary to perform testing to determine the foundation properties. 

A5-14
 
Appendix B 

Loads and Loading Requirements 

 

 

 

 

 

 

 

 

 

 

Appendix B: Loads and Loading Requirements 

There are several types of loads to consider in the complete analysis of concrete dams. Usually, 

no single load acting alone will cause stability problems, but the combinations of loads under 

certain conditions can lead to instability of a concrete dam. Although there are several types of 

loads, each load will fall into one of two categories: follower forces capable of doing work and 

high-frequency dynamic forces, or forces relieved by slight motion. Only follower forces can 

cause a dam to fail; however, other forces can cause damage, making it easier for follower forces 

to cause failure. Follower forces continue to load the structure as movement occurs (such as 

during foundation block sliding), such as loading from reservoir, gravity, or seismic activity. 

Loads that do not continue to apply force to the dam as movements occur are from temperature, 

ice, or silt. 

B.1 Gravity/Dead Load 

The dead load is the weight of the structure plus any appurtenant structures, such as gates, piers, 

and bridges. For computational purposes, the dead load is the cross-sectional area of the dam 

section being analyzed times the unit weight of the material. The unit weight of mass concrete 

is largely dependent on the specific gravity of the aggregates. Unit weights can range from 

130 pounds per cubic feet (lb/ft3) to 160 lb/ft3. A unit weight of 150 lb/ft3 is usually assumed 

in analyses. Mass in the foundation is usually not included in static analyses because the 

foundation has settled from self-weight long before the construction of the dam and filling of 

the reservoir. The dam and reservoir add additional weight that the foundation has not had to 

previously support, causing it to deflect further. The method of building the dam in stages and 

relatively small concrete placements compared to the size of the dam are considered when 

applying static weight to the dam. Including the stage construction process in the application of 

static weight provides a more realistic stress distribution in the dam. This eliminates the effect of 

the dam hanging from the upper abutments, causing high stress concentration. Preferably, mass 

in the foundation is included in dynamic analysis. This way, inertial effects and radiation 

damping effects are included in the dynamic response of the dam-reservoir-foundation system. 

B.2 Water Loading Conditions Resulting from Reservoir Water Surface 

Elevation 

All water loading conditions resulting from reservoir water surface and tailwater elevations 

(figure B-1) are based on hydrologic information, which gives water elevations in terms of return 

periods. 

B-1 
 

 

Figure B-1: Loads on typical gravity section —normal operating condition. 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

Key Events Along Potential Failure Mode Paths 

B.2.1 Coincident Pool 

Coincident pools represent the water elevations used in combination with seismic events. 

B.2.2 Normal Operation Pool 

For hydropower dams, the pool will be fairly high for normal operation, while for some floodcontrol dams, the pool will be low for normal operation. For navigation projects, the maximum 

loading for normal operation might correspond to the normal water level, combined with the 

tailwater expected. Water loads defined by the normal operation loading condition are sometimes 

combined with other types of events. 

B.2.3 Nappe Forces and Tailwater 

The forces acting on an overflow dam or spillway section are complicated by steady state 

hydrodynamic effects. Hydrodynamic forces result from water changing speed and direction as it 

flows over a spillway. At small discharges, nappe forces may be discounted in a stability 

analysis; however, when the discharge over an overflow spillway approaches the design 

discharge, nappe forces can become significant and should be taken into account in the analysis 

of dam stability. Some gravity dam guidance recommends ignoring the weight of the nappe on 

top of the structure and requiring that the tailwater be assumed to be 60 percent of its expected 

height, and some do not. This method does not sufficiently account for subatmospheric crest 

pressures and high bucket pressures, and in some cases, it can yield unconservative results. 

While this practice is still acceptable, it may be desirable to determine forces due to the nappe 

B-2 
 

 

 

 

 

 

 

 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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and tailwater more rigorously. A procedure for computing the nappe and tailwater forces is 

presented in the Federal Energy Regulatory Commission Engineering Guidelines (2005). 

B.3 Uplift Loads 

Uplift loads have significant impact on stability and should be consistent with the failure mode 

being considered. Measured uplift pressures may not be indicative of pressures occurring during 

failure. For instance, uplift pressures along a slide plane are very different if sliding has occurred. 

The joint might be dilated from riding up over aggregate, and the drain efficiency may be 

reduced or eliminated from interruption of the drain holes. Sliding stability, resultant location, 

and flotation are all aspects of a stability analysis where safety can be improved by reducing 

uplift pressures. Since uplift pressures are directly related to flow paths beneath the structure, 

uplift pressure distribution may be determined from a seepage analysis. Such an analysis must 

consider the types of foundation and backfill materials, their possible range of horizontal and 

vertical permeabilities, and the effectiveness of cutoffs and drains. Seepage analysis techniques 

to determine uplift pressures on structures may include flow nets, finite element methods, the 

line-of-creep method, and the method of fragments. Uplift pressures resulting from flow through 

fractures and jointed rock, however, are poorly understood and can only be accurately known by 

taking measurements at the point of interest. Joint asperities, changes in joint aperture, and the 

degree to which joints interconnect with tailwater influence uplift pressures and pressure 

distribution. Uplift pressures are site-specific and may vary at a given site due to changes in 

geology. Uplift pressures can be reduced through foundation drainage or by various cutoff 

measures such as grout curtains, cutoff walls, and impervious blankets. Uplift pressures should 

be based on relatively long-term water elevations. Short duration fluctuations, such as from 

waves or from vibrations due to high-velocity flows, may be safely assumed to have no effect on 

uplift pressures. 

B.4 Silt 

Silt can accumulate upstream of dams. Not all dams will be susceptible to silt accumulation, and 

the structural engineer should consult with hydraulic engineers to determine if silt buildup was 

possible, and to what extent it might have accumulated over time. Measuring the silt level by 

hydrosurvey would be prudent. 

For an approximation, horizontal silt pressure is assumed to be equivalent to that of a fluid 

weighing 85 lb/ft3

, and vertical silt pressure is determined as if silt were a soil having a density 

of 120 lb/ft3. These values include the effects of water within the silt. If significant, the sediment 

depth for an existing dam can be based on hydrographic surveys. 

Reservoir silt can reduce uplift under a dam, similar to an upstream apron, by increasing the 

seepage path or increasing the head loss through the silt layer. However, uplift reduction should 

be justified by instrumentation because potential liquefaction of the silt during a seismic event 

may negate uplift reduction. If liquefaction occurs, pore pressure in the silt will increase. This 

B-3 
 

 

 

 

 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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condition of elevated pore pressure may persist for some time after the seismic event. For this 

reason, uplift reduction due to silt may not be relied upon when considering post-earthquake 

stability. 

B.5 Thermal and Alkali/Aggregate Reactivity 

Volumetric changes caused by thermal expansion and contraction, or by alkali/aggregate 

reactivity, affect the cross-valley stresses in the dam. These stresses are important when threedimensional behavior is being considered. Expansion will cause a straight gravity dam to wedge 

itself into the valley walls more tightly, increasing its stability. Contraction has the opposite 

effect. While these effects are acknowledged, the beneficial effect of expansion is difficult to 

quantify even with elaborate finite element models because it is contingent on the modulus of 

deformation of the abutments, which is highly variable. For this reason, the beneficial effects of 

expansion should not be relied upon in three-dimensional stability analyses. In the case of an 

arch dam, volumetric expansion causes upstream deflection of the arch and associated stresses. 

Volumetric contraction would have the opposite effect. The consequences of these stresses 

should be evaluated. If it appears that contraction will cause monolith joints to open and, thus, 

compromise force transfer from monolith to monolith, this effect should be considered. 

B.6 Earthquake Loads 

Earthquake loads are used to represent the inertial effects attributable to the structure mass, the 

surrounding foundation, and the surrounding water (hydrodynamic pressures). Earthquake 

loadings should be selected after consideration of the accelerations that may be expected at each 

project site as determined by the geology of the site, proximity to major faults, and earthquake 

history of the region as indicated by available seismic records. Seismic zone maps can be used to 

establish the probability zone for projects that do not have detailed seismicity studies, but 

typically do not provide enough information for structures such as dams that can have large 

consequences associated with failure. 

Site-specific seismic studies may be required for more critical structures. A set of seismic zone 

maps are available from the United States Geological Survey at: 

http://earthquake.usgs.gov 

Other widely accepted seismic zone maps can also be used as a starting point for determining 

seismic loading. General seismic hazard maps, such as that cited above, may not sufficiently 

account for local seismicity. 

Several analytical methods are available to evaluate the dynamic response of structures during 

earthquakes: seismic coefficient, response spectrum, and time history. The current state-of-theart method uses linear elastic and nonlinear finite element time history analysis procedures, 

which account for the dynamic interaction between the structure, foundation, soil, and water. The 

B-4 
 

 

 

 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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seismic coefficient method fails to account for the true dynamic characteristics of the structurewater-foundation system, but it can give a preliminary indication for stiff structures. 

The horizontal component of the dam inertial force is assumed to act at the structure’s center of 

mass and that the structure is a rigid body. In actuality, almost all structures have some 

flexibility, and the use of the rigid body concept often underestimates the magnitude of the 

inertial force. The location of the horizontal inertial force is also related to the flexibility of the 

structure and usually acts at a location higher than the center of mass. However, because of the 

cyclic nature of earthquake loads, there is little probability of a rotational-stability related failure. 

Backfill or silt material adjacent to a structure will induce inertial forces on the structure during 

an earthquake. The pseudo-static inertial force of the backfill or silt is added algebraically to 

the static lateral earth pressures in the seismic coefficient method. In a dynamic finite element 

analysis that incorporates reservoir interaction, the silt can be modeled as a saturated soil or more 

dense fluid. 

Water that is above the ground surface and adjacent to or surrounding a structure will increase 

the inertial forces acting on the structure during an earthquake. The displaced structure moves 

through the surrounding water, thereby causing hydrodynamic forces to act on the structure. The 

water inside and surrounding the structure alters the dynamic characteristics of the structural 

system, increasing the periods of the fundamental modes of vibration and modifying the mode 

shapes. In seismic coefficient methods, the hydrodynamic effects are approximated by using the 

Westergaard method (equation B1) (Westergaard, 1931). The hydrodynamic force can either 

increase or decrease the water force, depending on direction of seismic acceleration. Figure B-2 

illustrates hydrodynamic pressures based on the Westergaard method. 

Figure B-2: Hydrodynamic forces for freestanding water on a gravity dam. 

B-5 
 

 

 

 

 

 

 

 

 

 

 

 

 

Selecting Analytic Tools for Concrete Dams to Address 

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pE =(7/8) αh γw h eq. B1 

where, 

pE = hydrodynamic pressure at base of dam 

αh = horizontal seismic coefficient 

γw = unit weight of water 

h = water depth 

The hydrodynamic pressure is added algebraically to the static water pressure to get the total 

water pressure on the structure. The pressure distribution is parabolic, and the line of action for 

the resulting force, PWeq, is approximately 0.4 h above the ground surface. The Westergaard 

method assumes the structure is rigid and the water is incompressible. In 1952, Zangar studied 

dynamic water pressures on sloping faces and provided adjustments to Westergaard’s added 

mass. In the 1980s and 1990s, there was considerable work on formulating incompressible and 

compressible hydrodynamic approaches in finite element modeling. The incompressible 

formulation is more accurate than Westergaard’s added mass because the reservoir topography 

and shape of the upstream face of the dam are modeled. Compressible material formulations 

have been observed to transmit pressure waves and damping mechanisms in the reservoir better 

than incompressible material formulations. Current-day finite element methods can model the 

reservoir with fluid elements. Analysts must use care to verify the dynamic behavior of the fluid 

elements and ensure static water pressures remain on the structure after the earthquake as the 

structure may displace. 

Engineers using the seismic coefficient approach for stability analyses should be aware of the 

limitations and the simplifying assumptions made with respect to hydrodynamic pressures and 

their distribution on the structure. 

When time history analysis is done, a suite of time earthquake records should be used. 

Peculiarities of one particular record may produce results that are different from the general 

behavior from all other time histories. The results should be used as an aggregate to predict 

potential behavior of the structure. 

B.7 Ice Loads 

Ice pressure is created by thermal expansion of the ice and by wind drag (U.S. Army Corps of 

Engineers, 2002). Pressures caused by thermal expansion are dependent on the temperature rise 

of the ice, the thickness of the ice sheet, the coefficient of expansion, the elastic modulus, and 

the strength of the ice. Wind drag is dependent on the size and shape of the exposed area, the 

roughness of the surface, and the direction and velocity of the wind. Ice loads are usually 

transitory. Not all dams will be subject to ice pressure, and the engineer should decide whether 

an ice load is appropriate after considering all relevant factors. An example of the conditions 

conducive to the development of potentially high ice pressure would be a reservoir with hard 

rock reservoir walls that totally restrain the ice sheet. In addition, the site meteorological 

conditions would have to be such that an extremely rigid ice sheet develops. For the purpose of 

B-6 
 

 

 

 

 

 

 

 

 

 

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Key Events Along Potential Failure Mode Paths 

the analysis of structures for which ice loading occurs, it is recommended that a pressure of 

5,000 pounds per square feet (lb/ft2) be applied to the contact surface of the structure in contact 

with the ice, based on the expected ice thickness. The existence of a formal system for the 

prevention of ice formation, such as an air bubble system, may reduce or eliminate ice loading. 

Information showing the design and maintenance of such a system must be provided in support 

of this assumption. Ice pressure should be applied at the normal pool elevation. If the dam is 

topped with flashboards, the strength of the flashboards may limit the ice load. 

B.8 Ice/Debris Impact 

Some rivers are subject to ice and debris flow. Current-borne ice sheets weighing several tons 

and/or debris can impact dams and cause local damage to piers, gates, or machinery. Several 

dams have experienced significant reservoir surcharges under moderate flood events due to 

debris or floating ice plugging spillway bays. When the ability of a spillway to pass floods is 

evaluated, the effect of ice and debris should be considered. 

B.9 Wind Loads 

Wind loads are usually small compared to other forces acting on dams. A wind pressure of 50 

lb/ft2 has been applied to the exposed surfaces of dams to address stability, but it has usually 

been shown not to be a critical load. If it is critical, site-specific wind data are used. 

References 

Federal Energy Regulatory Commission, Engineering Guidelines for the Evaluation of 

Hydropower Projects, Chapter 14, “Dam Safety Performance Monitoring Program,” 2005. 

U.S. Army Corps of Engineers, Ice Engineering, 1110-2-1612, 2002. 

Westergaard, H.M., “Water Pressures on Dams during Earthquakes,” Alfred R. Golze, ed., 

American Society of Civil Engineers Transactions, Paper No. 1835, November 1931 

Proceedings. 

B-7 

U. S. DEPARTMENT OF

HOMELAND SECURITY

FEMA P-1016 

Catalog No. 14177-1 

FEMA