17 ARCH DAM DESIGN AND ANALYSIS

HOWARD L. BOGGS

Consulting Civil Engineer Arvada, Colorado

GLENN S. TARBOX

Vice President and Chief Engineer Harza Engineering Company

Chicago, Illinois

ROBERT B. JANSEN

Consulting Civil Engineer Bellingham, Washington

INTRODUCTION

Background

The Handbook of Dam Engineering! presented the subject of concrete dam design from fundamental definitions of terms through planning, design, and construction consid­erations. Most of that information is still pertinent and us­able. Those portions directed toward arch dam design are still relevant for new dam design and the evaluation and rehabilitation of existing dams. With the Handbook and other well-accepted references as background, this chapter takes the opportunity to discuss areas of new and updated technology. The aspects of arch dam design that represent the most important changes in the last decade are pre­sented.

Scope

The subjects discussed in this chapter include: a historical perspective of the last decade, revised and new arch dam design criteria, evolution of arch shapes and optimum de­sign methods, static and dynamic nonlinear analysis, pre­and postprocessing of finite-element analysis data, suita­bility of roller-compacted concrete technology for arch dams, evaluation of existing arch dams, risk assessment applied to arch dams, case histories of Klang Gates Dam and Stewart Mountain Dam rehabilitation, and discussion of multiple-arch dams.

DESIGN CRITERIA

Background

A universally accepted fundamental of concrete arch dam design philosophy is the use of rational and consistent cri­teria to ensure safe, economical, functional, durable, and easily maintained structures. It follows naturally that de­sign criteria should be continually reviewed, evaluated, and modified in light of developing technology in all fields as­sociated with concrete dam design, to ensure that design practices remain up-to-date and consistent with currently accepted engineering practice.

The arch dam design criteria presented in the Handbook of Dam Engineering and in the comprehensive Design of Arch Dams, 2 both published in 1977, represented the stan­dards for design at that time. It is appropriate here to ad­dress changes and additions due to developments of the past decade. It is also important to remember that those criteria were developed principally to govern the design of new concrete arch dams, and this publication emphasizes as well the rehabilitation of existing structures.

With this understanding, consider the notion that, under special conditions, criteria used as the basis for rehabili­tation designs and analyses for existing dams need not nec­essarily be as stringent as those used for the design of new dams. For example, if a structure has provided satisfactory service for 40 or 50 years, and an evaluation of the foun­dation stability indicates that the safety factors against slid-

493

R. B. Jansen (ed.), Advanced Dam Engineering for Design, Construction, and Rehabilitation© Van Nostrand Reinhold 1988

494 ADVANCED DAM ENGINEERING FOR DESIGN. CONSTRUCTION. AND REHABILITATION

ing are somewhat less than those normally required for the construction of a new dam, the question arises of whether or not it is justified to require the dam owner to expend the money necessary to bring the factor of safety for sliding up to that required by criteria applied to new dams. The de­cision will depend on many factors requiring careful anal­ysis.

Historical Perspective 1977 to 1987. Before criteria revisions are discussed, it will be helpful to review the events that have taken place since 1977 that represent changes and advances in the field of concrete arch dam design. Most notable is the enormous level of effort that has been expended in the area of dam safety. Engineers and other experts have spent untold hours meeting in com­mittees and generating papers, reports, and other publica­tions on the subject of safety of dams. This body of liter­ature, taken collectively, has had a significant impact on the approaches that are now common practice within the profession for the evaluation of existing dams.

The singular event that provided the impetus to what has been a series of events related to dam safety in the United States was the signing of Public Law 92-367 on August 8, 1972, authorizing the secretary of the army, through the Corps of Engineers, to undertake a national program of inspection of dams. The Corps produced a report, "Na­tional Program of Inspection of Dams," dated May 1975.3

The report contained (1) an inventory compilation of fed­eral and nonfederal dams; (2) a survey of state capabilities, practices, and regulations in dam safety; (3) recommended guidelines for safety inspection of dams; and (4) recom­mendations for a dam safety program.

Following the failure of Teton Dam in 1976, a presiden­tial memorandum was issued on April 23, 1977, in which federal agencies involved in dam programs were instructed to initiate a review of federal dam safety procedures. From this impetus came a publication titled "Improving Federal Dam Safety," published by the Federal Coordinating Council for Science, Engineering and Technology (FCC SET) . 4 This report contained reports from each of the seven federal departments and agencies that had been in­structed to review and evaluate their in-house dam safety practices.

A second document that was produced as a result of the presidential initiative was titled "Federal Dam Safety," published on December 6, 1978.5 This report came from an independent review panel established by the Office of Science and Technology Policy (OSTP), the organization charged by the FCCSET with reviewing the federal agen­cies' dam safety practices and providing recommendations for improving federal dam safety. A final document, titled "Federal Guidelines for Dam Safety," was published by

OSTP on June 25, 1979.6 Each of the federal departments and agencies responsible for dam safety was directed to adopt and implement the federal guidelines in October 1979.

During this period, the U.S. government was also in the process of establishing a new agency responsible for the management of all emergency activities related to the fed­eral government. In 1979 the Federal Emergency Manage­ment Agency (FEMA) was created, and the responsibility for coordinating all efforts to promote dam safety was as­signed to the director of this newly formed agency.

The next event in the chronology was the formalization of an interagency committee of federal agencies known as ICODS (Inter-Agency Committee on Dam Safety). ICODS was created April 24, 1980 to encourage the establishment and maintenance of effective federal and state programs to ensure dam safety. The chief of federal dam safety within FEMA also serves as the chairman of ICODS.

One of the activities performed by FEMA in the early 1980s was to request that the National Research Council form a committee on the safety of existing dams. The re­sulting work was accomplished in two phases. In the first phase, FEMA asked the NRC to identify impediments to state-run programs for dam safety, to suggest federal ac­tions to remove or mitigate those impediments, and to de­fine how the U.S. government could help make nonfederal dams safer. In response, the NRC created the Committee on Safety of Nonfederal Dams, to review and discuss the issues involved. The efforts of that committee were com­pleted in February 1982 and reported in the publication "Safety of Nonfederal Dams, a Review of the Federal Role."

The second phase was concerned with technical consid­erations related to dam safety. The NRC created a new committee on the safety of existing dams in May 1982 to examine the technical issues of dam safety and to develop guidance on how to achieve improvements in the safety of dams. The work of that committee was completed in 1983 and published in the book "Safety of Existing Dams­Evaluation and Improvement.,,7 The National Research Council also received a request submitted jointly by the assistant secretary of the interior for water and science and the assistant secretary of the army for civil works. The re­quest was to establish a committee on safety criteria for dams to prepare an inventory of currently used criteria re­lated to safety from hazards of extreme floods and earth­quakes. The work of the committee and its findings and recommendations were published in 1985 in a book titled "Safety of Dams, Flood and Earthquake Criteria.,,8 Three other publications were also published during the early eighties by the United States Committee on Large Dams: "Current United States Practice for Numerical Analysis of

Dams, ,,9 "Guidelines for Selection of Seismic Parameters for Dam Projects, ,,\0 and' 'Dam Safety Practices and Con­cerns in the United States." 11

Areas of Change. All the findings and recommenda­tions from the work of various committees within the fed­eral government and professional organizations over the period from 1972 through 1985 form the foundation on which the revisions to design criteria discussed in this sec­tion are based. A review of all the recent publications con­cerning safety of dams reveals three areas in particular where developments in technology and new philosophy af­fect design criteria for existing dams most significantly. These areas are (1) criteria used to assess extreme flood hazards, (2) criteria for performing dam break analysis and (3) criteria used to assess earthquake hazards. A fourth area, which is gaining support, concerns developing criteria to evaluate dam safety using risk-based decision analysis. The interrelationship between flood and earthquake hazard as­sessment and risk analysis is included in the discussions that follow.

Extreme Flood Hazards

Background. Arch dams are designed and constructed to withstand the forces of nature that may be expected to occur over the life of the dam structure, based on past events and information. Essential to this process is the an­ticipation of future events that might result in floods that could cause dam failure. In spite of the use of currently available criteria to determine the capacity of dam struc­tures to withstand or pass floods, it is still possible for a flood event to exceed a dam's capacity to resist it. It is impossible to provide "absolute" safety against all haz­ards, especially from events produced by nature. The ob­jective is to balance the benefits of making dams safe against the increased cost of safety, and to reduce the risks to tolerable levels.

Although legally no statutes exist in the body of law that point to a specific basis for the design of a dam, the stan­dards against which engineers are judged in court decisions emphasize the need for designs to meet standards of rea­sonableness and prudence, that is, the exercise of good judgment and common sense. There appears to be no com­pletely satisfactory answer to the question of what consti­tutes reasonable care and prudence in selecting the mag­nitude of a flood for which a dam should be designed. Because the actual timing and magnitude of future flood events are indeterminate, such assessments cannot be based on rigorous analyses, but must be made by analyzing the probable behavior of a project during hypothetical design floods.

ARCH DAM DESIGN AND ANALYSIS 495

The most generally accepted hypothetical flood used for assessing the capability of a dam to withstand an extreme flood condition is called the spillway design flood (SDF). In general, there are no legal standards for the magnitude of a flood that a dam must safely pass, or the type of anal­yses to be used. Also, there are no procedures or criteria for such assessments that are universally accepted in the engineering profession.

Current Practices. As part of its work, sponsored by the National Research Council, the Committee on Safety Criteria for Dams submitted a questionnaire to the federal agencies most concerned with dams, the units of state gov­ernments associated with dam safety, and several private engineering companies in the dam design field. The re­sponses to those inquiries were published in the commit­tee's report in 1985. It is appropriate, when developing a recommended set of criteria on which to base a dam de­sign, to review carefully the current procedures and criteria used by the major dam-building organizations of the world.

The data presented in the committee's 1985 report rep­resent responses from 10 federal organizations, 35 state and local agencies, 9 private firms, and 4 professional engi­neering societies. Basic to all safety standards relating to hydrologic events are systems for classifying dams accord­ing to the probable damages caused by dam failure. There is considerable variety in the classification systems that have been adopted, which leads to difficulty in comparing the different criteria. The system for classifying the hazard potential of dams, as proposed by the U.S. Army Corps of Engineers in the National Dam Inspection Program, has been adopted here and is shown in Table 17-1.

U sing the hazard classifications given in Table 17-1, the responses from federal agencies, technical societies, a se­lection of U.S. consulting firms, and one foreign organi­zation have been compiled for comparison in Table 17-2, taken from the report by the Committee on Safety Criteria for Dams, published in 1985. This inventory of current practices shows considerable diversity in approach by var­ious agencies, professional societies, and privately owned firms. There is a reasonable consensus on the spillway re­quirements for large, high-hazard dams, but the inventory shows widespread uncertainty for the other classes of dams. In developing a set of criteria for spillway capacities for this chapter, an attempt has been made to differentiate be­tween criteria appropriate for new dams and criteria for the evaluation of existing dams. The practices used in the United States for computing spillway design floods (SDFs) are covered fully in Chapter 3.

Criteria for Spillway Capacity. The proposed criteria, contained herein, have been restricted to the definition of

496 ADVANCED DAM ENGINEERING FOR DESIGN, CONSTRUCTION, AND REHABILITATION

Table 17-1. Terms for Classifying Hazard Potentials.

Category

Size of Dam a

Small Intermediate Large

Category

Impoundment (ac-ft)

50 to 1.000 1,000 to 50,000

Over 50,000

Loss of Life (Extent of Development)

Height of Dam (ft )

25 to 40 40 to 100 Over 100

Economic Loss

Hazard Potential Classification Low None expected (no permanent structures

for human habitation) Minimal (undeveloped to occasional structures or agriculture) Appreciable (notable agriculture, industry, or structures)

Significant Few (no urban developments and no more than a small number of inhabitable structures)

High More than few

'Criterion that places project in largest category governs. Source: "Safety of Dams-Flood and Earthquake Criteria" (1985)8

what level of SDF should be incorporated in the design for a dam. Table 17-3 contains the proposed spillway capacity criteria for new dams, and Table 17-4 contains criteria for the evaluation of existing dams. The crest elevation of the dam should be detennined based on routing the SDF through the reservoir to detennine the maximum water sur­face elevation. For arch dams, freeboard requirements are generally satisfied by the upstream concrete parapet wall.

Exceptional Cases. Existing arch dams that are evalu­ated for safety, enlarged, or improved may have spillways smaller than required to safely pass some floods, as indi­cated in Table 17-4. There are cases when a smaller-ca­pacity spillway may be acceptable for a specific dam. This situation exists if it can be shown that the resulting dam break flood caused by an IDF that just exceeds the routing capacity of the reservoir would not cause additional loss of life or a significant increase in damage to properties over that which would have occurred without dam failure. If such conditions obtain, a risk assessment is a fonnal method that can be used to develop a basis for decisions for re­medial work.

Dam Break Analysis

Applications. Dam safety programs require the devel­opment, writing, and publication of emergency action plans. These plans are designed to infonn emergency ac­tion authorities and citizens living downstream of dams of the potential inundation areas that could occur in the event of a dam break, and to publish appropriate actions to be taken for warning and evacuation. To better assess the haz-

Excessive (extensive community, industry, or agriculture)

ard of a dam in a systematic and equitable manner, an an­alytical approach, called dam break analysis, should be used. Dam break analysis serves two primary goals. First, it provides infonnation to the engineer about classifying the potential hazard of a dam for detennining the recom­mended spillway capacity. Second, it predicts flood depths and wave arrival times and identifies areas that could be affected by floodwater, should a failure occur.

Types of Models. Many types of dam break models ex­ist. The objective of each is to simulate the failure of a dam by generating a dam failure hydrograph and routing the hy­drograph downstream. Some modeling procedures can be perfonned by hand, whereas others are complex and re­quire computer analysis. The decision as to which method to use for a given situation depends on the purpose for which the results of the dam break analysis would be ap­plied. Where approximate downstream impacts resulting from postulated dam failure are needed for preliminary planning, a simple handwork method can suffice. For sit­uations where detailed flood arrival times, depths, and ac­curate inundation mapping are required, a computer model approach should be used. A list of nine dam break models is discussed in "Safety of Existing Dams-Evaluation and Improvement. ,,7

Criteria for Dam Break Analysis. The criterion rec­ommended here is that anyone of the hand methods avail­able is suitable. Among the computerized dam break anal­ysis methods available, it is recommended that the National Weather Service or the U.S. Anny Corps of Engineers' HEC-1 programs be used. With the reminder that dam

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498 ADVANCED DAM ENGINEERING FOR DESIGN, CONSTRUCTION, AND REHABILITATION

break analyses are perfonned primarily to delineate the maximum downstream inundation areas and are not in­tended to be sophisticated structural analysis programs ca­pable of modeling the actual failure mechanics of an arch dam, the following guidelines are also recommended. These guidelines are intended for fonnulating emergency action plans and for use in risk-based decision analysis only. They should be carried out so that results in no way reflect on the structural integrity of the dam. Any report or inundation map produced as a result of the analysis should contain such a qualifying statement. For concrete arch dams, an instantaneous failure is assumed, thus maximiz­ing the downstream flooding. The shape of an instanta­neous failure breach is similar to the geometry of the chan­nel along the referenced plane of the dam. A time to failure equal to 0.15 hour should be used for the NWS and HEC-1 dam break models. The parabolic shape of the failure breach should be approximated by a trapezoidal section.

Earthquake Hazard

Background. Since 1977, there have been new devel­opments and a considerable body of literature written on earthquake hazards for dams. The National Research Council Committee on Safety Criteria for Dams, which sampled current practice related to dam safety evaluation against flood hazards, also solicited infonnation on current practices related to the evaluation of safety of dams against earthquake hazards. The responses provided to the com­mittee's questionnaire were incomplete and did not reflect the thoroughness of practices related to flood hazards; so they are not reproduced in this text. The following discus­sions are provided to describe the basis for defining various levels of earthquake events used in the evaluation of new and existing dams. The discussions are intentionally brief because it is not the purpose of this chapter to provide de­tailed discussions on the geological and seismological eval­uations required for the detennination of such events.

Ground shaking associated with an earthquake is due to stress waves that propagate outward from the earthquake's origin or focus. The effects that a particular earthquake would have on arch dam structures are a function of certain characteristic parameters. Therefore, seismological data have been collected on historical earthquakes including lo­cation, date of occurrence, magnitude, depth, duration, and so on. In addition, seismotectonic data have been collected on faults, including their identification, type of faulting, and estimates of most recent fault displacement. These data provide the basis for estimating the seismic hazard to arch dams. Various methods incorporate these data to estimate the intensity of ground shaking expected beneath a dam during an earthquake.

Deterministic-Statistical Method. The detenninistic­statistical method is appropriate in geographical regions that are seismically active, and where faults have been ob­served and mapped. This method is used to relate the earth­quake magnitude and the shortest distance from the caus­ative fault or the earthquake source zone to the dam site to peak ground motion parameters, for example, acceleration and duration of strong ground shaking.

Seismotectonic (Semiprobabilistic) Method. In some geographical locations, such as the eastern United States, earthquakes usually cannot be associated with mapped faults. The rupture planes do not commonly ex­tend to the free surface; so the detennination of distance to the damsite, as required for detenninistic-statistical stud­ies, can only be approximated. Accordingly, in the seis­motectonic method, the country or a portion of the country is divided into regions with similar geological and seis­mological characteristics, and it is assumed that the spatial density of historical earthquakes is more or less unifonn in each region. Each region is called a seismotectonic prov­ince or region, and a maximum credible earthquake (MCE) is detennined for each seismotectonic province using a probabilistic procedure. Beyond this stage, the analysis proceeds as for the detenninistic-statistical method, except that the distance from the earthquake source to the damsite is considered to be the smallest distance from any point in the seismotectonic province to the site if the site lies out­side the province. When the damsite lies within the seis­motectonic province, it is considered sufficiently conserv­ative to assume that the epicentral distance to the damsite is some small portion of the province dimension. 8

Probabilistic Risk Analysis. This method of analysis differs from both the detenninistic-statistical and seismo­tectonic methods in that the frequency of occurrence of earthquakes is con~idered. One of the principal purposes of applying risk analysis methods is to take account of the frequency of earthquake occurrence in risk assessment.

Reservoir-Induced Earthquakes. Reservoir-induced earthquakes, although rare, have occurred following the building of a dam. The presence of active or potentially active faults within the regime of a deep reservoir can lead to the triggering of local earthquakes of potentially dam­aging intensity. This subject historically has been very con­troversial, and there is a significant difference of opinion throughout the profession as to the occurrence of reservoir­induced seismicity at damsites around the world. The six dams listed in Table 17-5 are generally agreed to have pre­cipitated or triggered earthquakes as a result of reservoir filling. The possibility of reservoir-induced earthquakes

Table 17-5. Dams at which apparent reservoir­induced earthquakes have been observed.

Dam Earthquake Dam Location height (m) magnitude

Koyna India 103 6.5 Kremasta Greece 165 6.3 Hsingfengkiang China 105 6.1 Kariba Zimbabwe/ 128 5.8

Zambia Hoover United States 221 5.0 Marathon Greece 63 5.0

Source: "Safety of Dams-Flood and Earthquake Criteria" (1985).8

should be given consideration in setting design criteria for new high dams, particularly if any active or potentially ac­tive faults are located within the reservoir regime. 9

Earthquake Design Criteria. Design Level Earth­quakes. The criteria contained in this section are intended to be used as general guides and not to be considered as standards. It is recognized that earthquake engineering is still in the developmental stage, and flexibility is desirable. Although these guidelines generally reflect a compilation of current practices, it will be necessary to make periodic revisions, additions, and deletions to maintain currency with state-of-the-art earthquake engineering.

The evaluation of new and existing dams should con­sider two levels of design earthquakes: the maximum cred­ible earthquake (MCE) and the operating basis earthquake (OBE).

• The MCE is defined as the earthquake associated with specific seismotectonic structures, source areas, or provinces that would cause the most severe vibratory ground motion or foundation dislocation capable of being produced at the site under the currently known tectonic framework. The deterministic-statistical method discussed previously should be used to deter­mine the MCE whenever possible. However, where earthquake sources are not well defined, the seismo­tectonic province (semiprobabilistic) approach should be adopted.

• The OBE is defined as the earthquake with the maxi­mum level of ground shaking expected to occur at the site during the economic life of the project. The OBE is usually determined on a probabilistic basis, consid­ering the regional and local geology and seismology, and reflects the level of earthquake protection desired for operational or economic reasons. (The economic life of a project is usually considered to be 100 years for dams.) See Chapter 5 for more detailed discus-

I

ARCH DAM DESIGN AND ANALYSIS 499

FACTORS TO CONSIDER IN SELECTION OF DESIGN EARTHQUAKES · REGIONAL TECTONIC SETTING · SEISMIC HISTORY · SEISMOTECTONIC STRUCTURES · LOCAL OR SITE GEOLOGY · SEISMIC ATTENUATION · RESERVOIR INDUCED SEISMICITY

~ SELECTION OF DESIGN EARTHQUAKES · MAXIMUM CREDIBLE EARTHQUAKES · OPERATING BASIS EARTHQUAKE , DETERMINATION OF GROUND MOTION

FOR THE DESIGN EARTHQUAKES · PEAK ACCELERATION, VELOCITY, ANO DISPLACEMENT · DURATION · ACCELERATION TIME- HISTORIES · RESPONSE SPECTRA ,

REQUIREMENT FOR EARTHQUAKE ANALYSIS · SEISMICITY AND GROUND MOTIONS · FOUNDATION CONDITIONS · TYPE OF DAM · CONSTRUCTION METHOOS · MATERIAL PROPERTI ES · PAST EXPERIENCE

NO A YES REQUIRED?

DOCUMENTATION

I METHODS OF ANALYS IS

OF · PSEUDOSTATIC METHOD

EVALUATION · DYNAMIC ANALYS IS METHODS RESPONSE SPECTRUM T1ME- HISTORY

EVALUATION OF STRUCTURAL ADEQUACY · EVALUATION OF ANALYSIS RESULTS · PAST EXPERIENCE OF DAMS · CONSEQUENCES OF FAt LURE

Figure 17-1. Flow chart depicting steps for earthquake analyses and de­sign of dams (modified). Source: ICODS draft of proposed guidelines.

sions concerning the source mechanisms of these earthquake events.

The entire process discussed in this section is depicted schematically by the flow chart shown in Fig. 17-1, which identifies the steps for earthquake analysis and design of dams. The types of mathematical analytical techniques available to perform dynamic analysis are discussed below in the section on "Analysis." The criteria presented in this section have been developed to be consistent with the an­alytical methods discussed. Additional discussions are con­tained in Chapter 15.

Response Requirements for Seismic Loading. The criteria published in the 1977 Handbook of Dam Engineering l re­quired a safety factor greater than 1 under factors of safety for the extreme loading condition, which included the max­imum credible earthquake. In evaluating stress results from a dynamic analysis where the structure's response remains elastic but stresses exceed the elastic limits of the mate-

500 ADVANCED DAM ENGINEERING FOR DESIGN, CONSTRUCTION, AND REHABILITATION

rials, the evaluation becomes a matter of judgment. It is easy to understand why design engineers look to the future and the capability of performing nonlinear dynamic anal­yses, whereby structural evaluation will be made easier and more accurate. For the present, however, the following principles are appropriate guidelines:

• Under loading from the MCE, the structures of a proj­ect vital to the retention or release of the reservior should be required to function (1) without permitting a sudden, uncontrolled release of the reservoir and (2) without compromising the ability to make a controlled release of the reservoir.

• Under loading from the OBE, the project facilities not critical to the retention or release of the reservoir should be designed to sustain the earthquake with re­pairable damage. The degree of damage that would be acceptable is based on an economic analysis or on an estimate of the cost of the repair versus the initial cost to control the damage.

Site Ground Motion. An intermediate step in comput­ing the dynamic loads for earthquake analysis of an arch dam is to determine site ground motion. The site or input ground motions are the links between the geology-seis­mology input and the dynamic loads applied to the dam. Ground motions are usually represented by time-history plots of ground acceleration called accelerograms. Unique to each accelerogram is its response spectrum curve.

A response spectrum is a plot of the maximum response of a single-degree-of-freedom system for a particular range of periods and reservoir damping. The response may be expressed in units of acceleration, velocity, or displace­ment. Usual practice is to use acceleration response spectra to perform a response spectrum analysis of a dam or as the basis for selecting accelerograms that are used as input for a time-history analysis.

The shape of a spectral curve is like the "fingerprint" for a particular earthquake ground motion because its shape uniquely defines the energy content of the ground motion, which in tum defines how a particular ground motion af­fects a dam. Spectra can be computed from actual earth­quake records or from empirical formulas based on the earthquake parameters of magnitude and distance between the damsite and the fault or earthquake epicenter.

The concept of idealized spectrum curves was intro­duced by Housner in 1959. 12 Examination of idealized, un­damped velocity spectra shows different characteristic shapes for earthquakes of different magnitudes at various distances. Figure 17-2 shows three distinct curves for three different magnitude/distance relationships. Various meth­ods have been developed for determining site response spectra (SRS) that provide these characteristic shapes.13 It

> !/)

,: ... u o ..J W >

25 MILES FROM CENTER OF LARGE EARTHQUAKE

PERIOD (sec.)

Figure 17-2. Idealized undamped velocity spectra. Source: Hausner (1959).12

is important to select a method appropriate to the actual magnitude/distance relationships in developing an SRS.

Earthquake ground motions usually are recorded as two orthogonally horizontal components and one vertical com­ponent. After site response spectra and the corresponding accelerograms have been developed for each earthquake to be applied to a dam, the motions are input to a dynamic analysis. General practice is to apply the larger horizontal component in the upstream/downstream direction, the smaller horizontal component in the cross-canyon direc­tion, and the vertical component vertically. It may be ar­gued that the source of motion will not necessarily match exactly the upstream/downstream and cross-canyon orien­tation of a dam. However, at this time it is considered pru­dent to follow this convention.

Once the SRSs are specified, compatible accelerograms must be selected. There are two acceptable methods. One is to use historically recorded natural accelerograms. An­other is to generate synthetic accelerograms mathemati­cally.13 Whichever method is used, the resulting motion must properly account for the required magnitude, dis­tance, and duration. The medium on which the records were recorded (in the case of natural accelerograms) must also be appropriate, for example, rock records as opposed to soil records for arch dams.

The techniques employed to determine the correct num­ber of modes, to account for damping effects, and to in­clude hydrodynamic and dam-foundation interaction ef­fects are described in more detail in the section on "Analysis" and in Chapter 15.

Dynamic Material Properties. Dynamic stress anal­yses of arch dams require the use of dynamic material prop­erties to account for short-term loading effects. The mate-

ARCH DAM DESIGN AND ANALYSIS 501

Table 17-6. Effect of speed of loading on strength.

Direct Tensile Strength.

Dam Slow

Crystal Springs 203 Big Tujunga A 139 Big Tujunga B 217 Santa Anita 224 Juncal Morris Average

Grand average all tensile factors = 1.56. Source: Raphael, ACI Journal (1984).17

psi

Fast

337 225 397 373

Factor Slow

1.66 490 1.62 440 1.83 1.53 440

462 474

1.66

rial properties most affected by. short-tenn loadings are concrete modulus and strength.

The increase in concrete modulus during dynamic load­ing is well documented by laboratory tests, and dynamic studies confinn the use of the instantaneous concrete mod­ulus. \3-16 A value of instantaneous concrete modulus ap­proximately 25 % larger than the sustained modulus of elas­ticity can be used for preliminary studies in the absence of actual laboratory test data. Laboratory tests should be per­fonned, however, to detennine instantaneous modulus of elasticity values for final design studies.

Splitting Tensile Strength, Compressive Strength,

psi psi

Fast Factor Slow Fast Factor

640 1.31 4,500 5,930 1.32 3,540 4,070 1.15 5,320 5,970 1.12

650 1.48 4,520 5,810 1.29 723 1.56 4,420 6,740 1.52 694 1.46 5,290 7,780 1.47

1.45 1.31

Dynamic strength values also are dependent on the rate of loading. 16-20 Tests perfonned in laboratories using rapid strain rates for tensile and compressive strength produce consistent results. The effects of rapid loading on com­pressive and tensile strength are demonstrated in Table 17-6. Results of rapid compression and flexual loading tests made in the laboratory by the Bureau of Reclamation are shown in Fig. 17-3. The results from all the tests indicate an approximately 30% increase for compressive strength and increases of slightly greater than 50% for the ultimate tensile strength based on modulus of rupture tests. 17

MASS CONCRETE COMPRESSION AND FLEXURE TESTS TEN 12 x 24 INCH CYLINDERS AND TEN 12 x 12 x 40 INCH BEAMS TESTED AT 90 DAYS FOR EACH TIME TO FAILURE

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Figure 17-3. Rapid strain rate laboratory test results. Courtesy Bureau of Reclamation.

LEGEND: t MAXIMUM

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FLEXURE TEST

T

* 1 COMPRESSIO N TEST

502 ADVANCED DAM ENGINEERING FOR DESIGN. CONSTRUCTION. AND REHABILITATION

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SEISMIC TENSILE STRENGTH I

o 2000 4000 6000 8000 10000

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Figure 17-4. Design chart for seismic tensile strength. Source: Raphael (1984).'7

A plot of tensile strength as a function of compressive strength is shown in Fig. 17-4. The equation of the lower curve, which represents actual tensile strength of mass con­crete under seismic load as a function of compressive strength, is:

it = 2.6f~/3

The upper curve represents the apparent tensile strength of mass concrete under seismic load. "Apparent tensile strength" is a term coined by Dungar (1981 )2l to account for the disparity between failure stresses predicted by linear analysis and actual nonlinear stresses. The equation of the upper curve, for apparent tensile strength as a function of compressive strength, is:

fr = 3.4f~/3

The apparent tensile strength should be used to evaluate tensile stresses under seismic loading with linearly elastic stress analyses of arch dams.

DESIGN

Evolution of Arch Shapes

The gradual evolution of arch dam design has been brought about by knowledge of the actual behavior of arch dams under service conditions and by stress analyses which re­flect that behavior. A third ingredient in the evolution has

been increased knowledge of the composition and proper­ties of mass concrete. As analyses have more closely rep­resented the actual behavior of arch dams, very subtle re­finements have been introduced into their design layout.

Until the mid-1920s, arch dams were designed as inde­pendent circular arches, each capable of independently supporting the water load corresponding to its depth in the reservoir. Analyses of stresses reflected the circular shapes, with no allowance for load transfer between arches. When Jurgensen advanced the idea of the economic central angle, more subtle designs with overhanging vertical sections re­sulted. Allowance for the higher stresses found at the abut­ments led to layouts with increasing arch thickness from the crown to abutment, but the basic horizontal section was circular.

The Stevenson Creek Experimental Arch Dam studies showed that arch dam behavior was not at all as predicted by cylinder theory. It was shown that interaction between arches occurred, and that the newly devised trial-load anal­ysis method predicted the behavior very closely y. 23

Experiments in Portugal with rubber membranes pro­vided a rational basis for defining arch dams geometrically with vertical and horizontal sections, with the vertical sec­tions bulging upstream to distribute loads more evenly in the structure. This, combined with the results of trial-load analyses, showed that the horizontal shape, rather than being based on constant-radius circles, should have greater curvature at the center of the arch, with flatter curves near the abutments. Arches thus evolved from circular curves, to compound circular curves, such as three-centered arches, to parabolas, and finally to ellipses. This evolution was aided greatly by the proliferation of electronic digital com­puters in the 1960s.

Optimum Shapes

Another development spawned in the 1960s with the ad­vent of commercially available computers was optimum structural design using linear programming. 24 Prior to this time, arch dam designs were done by making geometric layouts of the arches and cantilevers by hand. The design principles used to make an arch dam layout are the same whether done by hand or using a computer. Refer to the Handbook of Dam Engineering l and to Design of Arch Dami for discussions of basic design principles applicable to these dams.

From the field of operations research came several meth­ods of solving optimization problems. These were applied to a wide class of elastic structures in the aerospace indus­try and ultimately to arch dam design. It was then possible to generate arch dam layouts interactively using the com­puter in concert with plotters.

The arch dam design problem is a nonlinear mathemat­ical programming problem. One method for solving prob-

lems of this type is to transform the problem into a series of linear programming problems using the cutting-plane method to piecewise-linearize all the nonlinear equations. Solutions are then found using the simplex method, which is an efficient interaction algorithm developed for solving linear programming problems.2s More recently, methods have been developed to optimize arch shapes using pre­scribed shape functions. 26

The object of the optimization design approach is to maximize a merit function expressing cost and fundamental usefulness of the dam, subject to constraints on geometry, material properties, and behavior due to loads. Embedded within the optimum design computer program is an arch dam stress analysis program, which is used to compute de­flections, forces, moments, and stresses of each subopti­mum design during the iterative process.

Most recent developments allow a design engineer to de­velop optimum arch dam designs for a particular site using modem-day CADD systems. The result is an arch dam with optimum shape, to distribute the applied loads within al­lowable limits; optimum volume, to minimize costs; and optimum height, to maximize benefits. Results from optimi­zation designs confirm that polynomial arch formulations approximating ellipses are more efficient than circular arches.

It is only fair to point out that in spite of the capabilities that have been developed to automate and optimize the de­sign process, most arch dam designers continue to develop design layouts manually. A designer may choose to make a design layout totally by hand or partially by interactive graphics on a computer-a phenomenon that is not surpris­ing if the design of an arch dam is properly understood. An arch dam should not be thought of as just a structure, but as an architectural creation as well. In the field of ar­chitecture, line and form are critical elements among the fundamental principles followed to create a structure. The architect blends creative energy with scientific principles to sketch ideas on paper manually. When designing an arch dam, the designer is both engineer and architect. It is not surprising, then, that the designer chooses to be involved manually in the layout process; for at that point one gains a feel for the structure's behavior and can bring to bear creative talent and ideas to shape and mold the final design.

ANALYSIS

Introduction

Analysis is an integral part of the design procedure of an arch dam. Mathematical analyses are used to determine the stresses and deflections in the structure due to the applied loads. The design is accomplished through an iterative pro­cess of alternately modifying the structure geometrically

ARCH DAM DESIGN AND ANALYSIS 503

and checking the results of the analysis until the design objectives are satisfied within the allowable design criteria.

Arch dams are designed to withstand static and dynamic loads. The analytical methods that have been developed and evolved with time to evaluate dam response to these loads have steadily improved to produce results that give a good prediction of prototype behavior. Historically, both analytical and physical models have been used to predict arch dam behavior. An impressive body of literature de­scribes these methods, particularly as related to linear elas­tic behavior. Therefore, they will not be discussed here, but the reader is referred to the Handbook on Dam Engineeringl and to Design of Arch Dams2 for more de­tailed descriptions of analytical methods in particular.

Since 1977, when those books were published, advances have been made in nonlinear analytical methods for both static and dynamic loads. The focus of attention is directed toward these subjects next.

Nonlinear Static Analyses

Contraction Joints. When a linear-elastic numerical analysis method is used to analyze an arch dam, contrac­tion joints are not modeled, and the arches are assumed to be monolithic, homogeneous, and isotropic. Conse­quently, the results can indicate apparent large tensile stresses in the horizontal planes of the arches. If the loads are large enough, they can produce tensile stresses that ex­ceed the tensile strength of the concrete. Because arch dams are constructed in vertical blocks, however, the arches are divided into segments separated by contraction joints, which are later grouted. Grouting restores the dam to a monolithic structure in compression, but grouted joints can take little or no tension. Obviously, this represents a dis­crepancy between the dam modeled analytically and the prototype structure. Consider the following as an actual loading condition on a dam.

During the winter, the measured concrete temperature near the crest of the dam due to reservoir and ambient air temperatures is less than the temperature at which the dam was grouted to close the contraction joints. Such a negative temperature differential causes the arches to contract and shorten. The result is a "computed" arch tension that could suggest concrete cracking. This cannot happen physically because the contraction joints would simply open and pre­vent tension from occurring. This effect, of open or par­tially open contraction joints, is not introduced into the nu­merical analysis. Therefore, the indicated tension in the arches results in a misleading interpretation of the entire structure. Linear analyses cannot accurately describe this type of structural response.

Needless to say, engineers have been aware of this prob­lem for some time. It has led investigators to seek better

504 ADVANCED DAM ENGINEERING FOR DESIGN, CONSTRUCTION, AND REHABILITATION

and more accurate numerical analysis methods to deal with it.

A joint-type element developed specifically for arch dams to assess the effects of contraction joints having no tensile strength was reported by Boggs and Mays (1986).27 Anal­yses show that the load is redistributed from the portion of the arches with partially opened contraction joints to the intact portion. For example, the computed arch stresses from static loads on Morrow Point Dam are shown in Fig . 17 -5. The sequence ofloading used to simulate partial con­traction joint opening started with gravity and hydrostatic loads due to a half-full reservoir, followed by a tempera­ture increase causing an upstream expansion. Effects of the nonlinear joint are subtle, but it clearly is effective in re­ducing the calculated tensile arch stresses on the upstream face, as shown in Fig. 17-6.

Fracture Mechanics. During the life of an arch dam, conditions may occur that cause the concrete to crack. These cracks may occur during construction due to im­properly prepared lift surfaces, large thermal gradients, shrinkage, or other nonlinear volumetric changes . Cracks may also occur during operation of a dam due to differen­tial settlement, overloading, or large-magnitude earth­quakes. As such, they may represent weaknesses that threaten safety. Most present-day arch dam engineering methods incorporate state-of-the-art concepts and analyses, which, for the most part, produce structures that are almost entirely in compression so that cracking is not a problem. This is not necessarily true for existing dams, however, some of which are a century or more old.

It has not been possible to accurately assess the safety of a new or existing arch dam using numerical analyses limited to linear-elastic assumptions if the dam's response goes beyond the elastic range. In overloaded zones of an arch, when the computed stresses exceed the concrete strength, it is necessary to use judgment in assessing the dam's ability to redistribute the applied loads safely with­out either a partial or a total material failure.

Early efforts to address the phenomenon of cracking were made by A. A. Griffith, who proposed a general fracture theory in 1921.28 Simple models were attempted, which assumed that a crack would be initiated when the computed stress exceeded the concrete strength, and would propagate to a point of zero stress. Recent concrete research into frac­ture mechanics suggests this assumption is too conserva­tive. Over the 50 years following Griffith's work, fracture theory was investigated primarily in the academic com­munity, and fracture mechanics was more closely associ­ated with metals. 29 Not until recently, the mid-1970s, was fracture mechanics investigated in depth for concrete,30 specifically in the application of fracture mechanics to con­crete dams. 31

Two-dimensional finite-element programs have since been developed to incorporate fracture mechanics con­cepts. As shown in Fig. 17-7, cracking is described in three modes: (1) an opening mode, (2) a sliding mode, and (3) a tearing mode.

Numerical analytical simulation models of joints in con­crete have taken the form of finite-element representations of discrete arch joints, "smeared" cracks, and various other types of approximations. 27.32

GRAVITY LOAD ONLY GRAV I TY + WATER GRAV ITY + WATER + 24 0 F

o o ~

o o <D

o o Il)

I

o o ~

o o f")

o o N

o o o

Figure 17-5. Contours of arch stresses assuming no contraction joints.

o o

ARCH DAM DESIGN AND ANALYSIS 505

UPSTREA' M FACE UPSTREAM FACE

TENSION IN CONTRACTION JOINTS NO TEN S ION INC 0 N T R ACT I ON J 0 I NT S

DOWNSTREAM FACE DOWNSTREAM FACE

TENSION IN CONTRACTION JOINTS NO TENS ION IN CONTRACTION JOI NTS

Figure 17-6. Contours of arch stresses including effects of contraction joints.

Discrete Joint Models. The discrete joint includes gen­erally two types of elements, the contact element and the joint element. Both use two sets of nodes on each side of the joint so that in a no-compression condition each side moves more or less independently of the other. A realistic joint is simulated, with a good representation of strain de­formation and stress variations. Both types of elements in-

Modt- .n: SlidirHJ Mode

Figure 17-7. Three modes of cracking.

ModI!' m Tear1nFjJ Mode

troduce additional degrees of freedom requmng more equations than a linear system needs, and, consequently, they are more expensive. Discrete joint elements have a drawback because of difficulties in redefining the structural topology associated with partial joint openings, and diffi­culty in partial material property modifications because of the joint opening and closing.

Smeared Crack Models. Joints in concrete can be simu­lated as a fracture zone rather than a single discrete joint. 32

The joint is bounded by a fracture zone that can be de­scribed by judicious sizing of finite elements encompassing the zone. The basic principal is to "smear" out the local cracks that make up the fracture zone into the whole ele­ment. Because the finite-element solution is obtained by minimizing the minimum potential energy, the smeared crack method is acceptable if the strain energy released by the smeared cracks is equivalent to the energy released by the local cracks. Although this method may not exactly represent the physical phenomenon of joint behavior, it does provide for reduced analytical costs by not requiring

506 ADVANCED DAM ENGINEERING FOR DESIGN, CONSTRUCTION, AND REHABILITATION

topology restructuring, and it can approximate partial joint opening.

Other Methods. Other methods available to model joint­ing involve introducing specific crack properties into a model. Detailed discussions are available in the literature. At this time, however, especially because of the explosion of investigations relative to discrete and smeared methods, these other methods will not be discussed.

Application of these methods to concrete arch dams will not be commonplace until three-dimensional laboratory tests have been accomplished to corroborate three-dimen­sional numerical analyses. Prudence suggests that these precautions are necessary in going from two- to three-di­mensional nonlinear analysis.

Linear Dynamic Analysis

Dynamic analysis of arch dams is the area that has received the most attention in the last ten years. During its func­tional life, a dam may experience shaking from nearby earthquakes that excites its foundation, dam, and reservoir components. Each component, if examined separately, would produce an individual response that would differ from a composite interaction of the three. Analytical meth­ods are available to determine the steady-state harmonic response of arch dams, including hydrodynamic interaction effects of the reservoir and consideration of the foundation rock flexibility. 33

Hydrodynamic Interaction. Westergaard's attached mass concept assumes that the inertial water mass, which varies with depth of reservoir and ground acceleration, is added to the concrete inertia, and both are distributed throughout the dam: At the time of Westergaard's labora­tory investigations, arch dams were very massive, and the assumptions of no interaction and incompressible fluid ap­plied. Contemporary arch dams are thinner, and their re­sponse during earthquakes may be significantly influenced by water mass inertia. Previous analyses, which considered the reservoir water incompressible, have been reevaluated, and results suggest that compressibility should be included in the analyses. The reservoir shape, both in plan imme­diately upstream from the arch and in the cross-canyon di­rection, has significant influence on the arch response dur­ing an earthquake. Current analysis methods take into consideration the interactive response between the reser­voir and the arch. 33

Compressibility of Reservoir Water. Water is known to be compressible, but formerly in arch dam design the effect was considered to be small in comparison with other loads and the tolerances to which dams are designed, con­structed, and operated. A more recent development, how-

ever, accompanying the integration into the numerical analyses of the dam-reservoir-foundation interaction ef­fects, is the possibility of considering water compressible or incompressible. Analyses have shown significant differ­ences in crest acceleration on a simply shaped arch dam and reservoir due to water compressibility, 34 and studies on various heights of arch dams confirm the need to consider water compressibility. 35 Most recently, these effects have been applied to Morrow Point Dam, a 465-ft (142-meter)­high, thin, double-curvature arch dam, and the conclusion is that the response of dams may be significantly influenced by dam-water interaction, water compressibility, and ab­sorption of hydrodynamic waves at the reservoir bound­ary .36

Cavitation Effects. Theoretically, if during an earth­quake the negative hydrodynamic pressures of the reservoir at the upstream face of a dam exceeded the pre-earthquake hydrostatic plus atmospheric pressure, cavitation would occur. The occurrence of negative pressures or cavitation has been observed on laboratory models of gravity dams tested for dynamic loading using the shaking table. 37 Fur­ther studies have shown that cavitation may increase or de­crease tensile stresses on the face of a dam, depending on the direction of motion, and that the effects are greatest in the upper part of the dam. The effect observed for a gravity dam would be larger for a thin arch dam because the water mass is proportionately larger than the concrete mass.

Although cavitation effects undoubtedly exist, research­ers have concluded that these effects may be omitted from the dynamic analyses of arch dams, based on the fact that tensile stresses caused by upstream movement are reduced because cavitation effects reduce upstream motion. 38 Oth­ers have reached similar conclusions, citing evidence that cavitation alters the maximum deformation and stresses only under extreme seismic intensities, for very high struc­tures, so that neglect would not seriously alter the dam's safety. For example, in the design of 741-ft (226-meter)­high EI Cajon concrete arch dam, the effect of possible cavitation was considered unlikely to seriously alter the overall response of the dam. 39

Nonlinear Dynamic Analysis

As in static analysis, it is common practice to perform dy­namic analyses of arch dams using linear-elastic analytical methods. It is recognized that there are many shortcomings of the approach because many nonlinear mechanisms are involved in the dynamic behavior of an arch dam, partic­ularly when subjected to high-intensity ground motion. Al­though such an approach is theoretically possible, the cost associated with conducting a complete nonlinear dynamic analysis would be prohibitive at this time. However, re-

searchers have reported efforts to investigate the effects of some nonlinear mechanisms on the response of arch dams.

One study has presented a method for analyzing arches that uses a special joint element to efficiently represent a gradual joint opening and closing, using a single finite-ele­ment discretization in the thickness direction. 4o Nonlinear behavior in compression is not considered, nor is sliding. The element is represented by a rotational spring, a trans­lational spring, and a location variable for the translational spring. This method, while showing great computational savings, is still expensive, especially when applied to three­dimensional arch dams.

Others have performed shaking table scale model studies in two dimensions of a segmented arch ring and a cracked cantilever section.41 Tests of the arch model clearly indi­cate that the nonlinear joint-opening mechanism prevents the development of tensile stress in the arch direction and greatly influences the dynamic response behavior. Com­pressive stresses in the arch direction are substantially in­creased as a result of joint opening, because of the reduced contact area between the partially open arch segments. These experimental observations have helped to confirm designers' intuitions concerning arch dam behavior during earthquakes, providing a basis on which to interpret results from linear-elastic dynamic analyses and to make judg­ments concerning a dam's safety. From tests on the cracked cantilever model, it has been demonstrated that even for very high peak acceleration values (1.2 g), a cracked can­tilever can remain stable. Adding this observation and those from the arch model to the realization that the combined arches and cantilevers of an actual arch dam are even more resistant than these models to seismic loading, it is appar­ent that arch dams can undergo significant nonlinear be­havior without loss of the reservoir.

Effects of Damping. Designers generally agree that when tensile strain exceeds elastic limits, the load is redis­tributed to a stiffer structural element or transferred to a compressive element. Also, tension is generally associated with bending action within the arch or cantilever, and structural relaxation from tensile stress usually will be ac­commodated by a redistribution. 39

A technique for simulating these strain-softening effects into a linear-elastic analysis employs the assumed values of critical damping. The damping constant for an arch dam is dependent upon the magnitude of the displacements, opening of vertical contraction joints, and cracking that may occur. There is also an effect on the damping constant which suggests that damping increases with period. 15,42

Shaking tests of prototype structures have indicated damping constants of 2 to 7 % with most values in the 3 to 5 % range. From an experimental blast at Ambiesta Dam in Italy, which caused large displacements, a damping con­stant of 6.5 % was determined. 43 Hatano suggests a damp-

ARCH DAM DESIGN AND ANALYSIS 507

ing constant of 7 %,42 and Rouse6o suggests that damping for concrete dams is less than 5 % for small displacements and not over 10% for large displacements. The Bureau of Reclamation used 3 to 7 % damping for Auburn Dam earth­quake analyses, depending on the magnitude of the earth­quake. 13 Under conditions of high straining and joint open­ing, others have considered it appropriate to assume 10% critical damping for the MCE and 6% for the OBE.39 Un­published studies from the Bureau of Reclamation indicate a somewhat linear relationship in the maximum stresses for a dam using a linear-elastic analysis of a large-magnitude earthquake (M 7.5) at 13 km, and damping ratios of 5, 7.5, and 10% of critical.

This is an area where future research, such as large-force shaking of prototypes or structural scale models and struc­tural behavior measurements taken from dams excited by actual earthquakes, could provide valuable insights.

Pre- and Postprocessing Results

Present-day numerical analyses of concrete arch dams al­most always use finite-element methods to model the dam, a representative portion of the foundation, and the reser­voir. An important aspect in these analyses is sizing of the mesh or grid. A coarse mesh may produce results that are misleading or difficult to accurately interpret, whereas too fine a mesh requires excessive computer time and money. Assembly of the finite-element model by hand, which gen­era.lly results in a coarse mesh, is an arduous task to debug and to modify when necessary. Without the proper precau­tions, tremendous amounts of time can be wasted creating data files and interpreting results from finite-element pro­grams. Commercially available software can greatly in­crease productivity, save time, and improve efficiency.

Creation of data files for a finite-element analysis of an arch dam involves defining the geometry, material types, loads, boundary conditions, and temperatures of the struc­ture in numeric terms, which can be very cumbersome. First, the structure is divided into geometric shapes, called elements, such as quadrilaterals and triangles. Smaller ele­ments, although more accurate, drastically increase the volume of required input. Therefore, an optimum element size should be determined for all of the structure. Coordi­nates for each comer and midpoint nodes are calculated to position the elements exactly.

As an example, a model of Glen Canyon Dam and foun­dation, using finite elements, is shown in Fig. 17-8. The three-dimensional analysis for Glen Canyon required a 3583-line input file for the finite-element program SAPIV. 44 Each of the 1684 nodal points had three coor­dinate values (x, y, z), six boundary condition designa­tions, and a temperature value. Each of the 626 elements had 16 node designations, one material type, and an initial temperature.

508 ADVANCED DAM ENGINEERING FOR DESIGN, CONSTRUCTION, AND REHABILITATION

Figure 17-8. Finite-element model of an arch dam.

Output from a finite-element analysis can also get quite large with three displacements per node and six stresses per element face per load condition. An earthquake analysis, with stresses computed at time increments of 0.005 second, six stresses per element face, 200 element faces, and a 10-second earthquake duration results in 2,400,000 stress val­ues. Stress plots are vital so that an engineer can visualize all these stress interactions. Postprocessor/translator pro­grams are helpful because the computed results can be re­duced to multicolored graphics including tables or contours of stresses, deflections, and temperatures, on the faces of

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- 106

LOADING CONDITION:

the dam or sections, as shown in Fig. 17-9. Graphics plots are especially valuable for evaluation of dynamic analyses results, such as stress histories of selected elements, as shown in Fig. 17-10.

The finite-element method is a very powerful, state-of­the-art, necessary engineering tool; however, input and output are extremely cumbersome. Available software can significantly reduce the time and errors associated with FEM analysis.

CONSTRUCTION USING ROLLER-COMPACTED CONCRETE (RCC)

The most significant development in concrete dam design and construction in the decade from 1975 to 1985 was the use of roller-compacted concrete technology in construc­tion. Chapter 18 presents a detailed discussion of the sub­ject, particularly as it relates to gravity dams. To date, however, RCC technology has not been used to construct an arch dam.

This is so primarily because the damsites for which RCC dams have been designed were best suited for gravity-type structures (valleys with large width-to-height ratios). Sec­ondarily, there is a natural tendency for advances in dam engineering to evolve slowly, partly because of a long­standing tradition that advances are made in increments that deviate only slightly from established precedent. For ex­ample, the first RCC dam completed in the United States,

m '" "'. - 260 <:.-215

'? - 410 .... Ul '.'377

IS U'l

,+-320 ; ·296

.... ~+- 290

N m ':" .. -266

0 0 1..-380 .... m Ul

<:'.-361 cr'. , 183

DEAD WEIGHT NORMAL RES. WATER SURFACE UPSTREAM FACE OF DAM LOOKI NG DIS LOW TEMPERATURE

_ COM PRE SS ION

+ TENSION

Figure 17-9. Typical computer plot of horizontal and vertical static stresses on the face of an arch dam.

ARCH DAM DESIGN AND ANALYSIS 509

UPSTREAM CANTILEVER STRESS HISTORY 1500

1000

500

0

-500 MAX. TEN. STRESS 95 AT 4.810 SEC, 1-MAX. COM. STRESS -37 AT 3.170 SEC.

-1000 TEN. STRESS OF 400 REPEATED o TIMES -COM. STRESS OF -5000 REPEATED o TI M ES

-1500

DOWNSTREAM CANTILEVER STRESS HISTORY 1500

1000

500

0

(I) -500 MAX. TEN. STRESS 59 AT 3.380 SEC. t--a.. MAX. COM. STRESS -53 AT 3.620 SEC.

-1000 TEN. STRESS OF 400 REPEATED o TlMES-Z COM. STRESS OF -5000 REPEATED o TIMES

-1500

(I)

ILl 1000 UPSTREAM ARCH STRESS HISTORY

(I) If)

500 ILl 0::

l;; 0

-500

-1000

:" 1 1\ I r\I' 1\ I ~ I) 11\ ,. r\ rv !.t-v >.J - -~ IV II M AX. TEN. STRESS 777 AT 2.370 SEC. f--V MAX. COM. STRESS -1220 AT 3.280 SEC.

-1500 TEN. STRESS OF 400 REPEATED 3 TIMES-COM. STRESS OF -5000 REPEATED o TIMES

-2000

DOWNSTREAM ARCH STRESS HISTORY 1000

500

0

-500

1\ J' ilV"' ~\ rv .It hJl tv ~ V\ b,I' "..,.. ~ -I-

V IU

-1000 MAX. TEN. STRESS 240 AT 3.300 SEC.~

MAX .COM. STRESS -732 AT 2.370 SEC. -1500 TEN. STRESS OF 400 REPEATED o TIMES-

COM. STRESS OF -5000 REPEATED o TIMES -2000

o 2 3 4 5 6 7 8 9 10 II 12

TIME IN SECONDS

Figure 17-10. Typical computer plot of dynamic arch and cantilever stress histories at a point on the face of an arch dam.

Willow Creek Dam, was 169 ft high. Subsequent dams were higher but incrementally within reason.

Other factors present immediate difficulties in applying RCC technology to arch dam construction, such as com­plicated geometry, significant thermal effects to be consid­ered, equipment congestion on the dam, control of crack­ing, and so on. In spite of these and perhaps other considerations, there is no reason to believe that RCC tech­nology cannot eventually be used in constructing an arch dam.

Breakthroughs in forming technology, advances in in­tegrated delivery, spreading, and compaction equipment, and the application of already developed joint construction

techniques similar to Japanese practice will help pave the way.

REHABILITATION

Safety Evaluation of Existing Dams

The purpose of a safety evaluation of an arch dam is to assess its structural and operational safety. The evaluation is based on a review of past design, construction and O&M records, on-site examinations, and engineering analyses. Following the records review and on-site inspection, rec-

510 ADVANCED DAM ENGINEERING FOR DESIGN, CONSTRUCTION, AND REHABILITATION

ommendations can be made regarding required rehabilita­tion measures.45

Records Review

Records review begins with gathering data relative to the arch dam project. The project includes primarily the dam, foundation, appurtenances, and reservoir area. These data constitute the available historical record with regard to de­sign, construction, and operation of the dam. Records available from design include criteria, preconstruction field tests, laboratory tests for properties of foundation rock or concrete, geological records, consulting engineers' re­ports, correspondence, and specifications. Useful infor­mation recorded during construction may be obtained from daily, weekly, and monthly reports by the contractor or owners' representatives, photographs, geological interpre­tation related to the excavated surface or materials areas, laboratory tests of materials used in construction, subse­quent foundation treatment, correspondence, and the final construction report. Operational records may include ob­servations by the dam tender, periodic maintenance re­ports, safety inspection reports, photographs, structural be­havior measurements, and reports of considered or completed modifications or retrofits to the dam, appurte­nances, or reservoir area. Other available sources of infor­mation are professional journals, construction-related mag­azines, newspaper articles, or other published materials related to the project. Once assembled, these data should remain at a central point and be updated so that emergen­cies or normal operation requiring repair or rehabilitation can be quickly and economically evaluated, and alterna­tives can be identified.

On-Site Examination

Essential to the evaluation of the project is field inspection to corroborate and expand the mental image developed dur­ing review of the existing data.

The multidisciplined inspection team includes at least civil and mechanical engineers and a geologist, all with significant experience in design and evaluation of arch dam projects. Personnel contacts should include the original de­signer, the owner, the constructor, and previous care­takers/dam tenders/operators. An experienced civil engi­neer can detect incipient conditions, anticipate probable areas of future distress, and suggest actions to prevent se­rious accidents or distress and to minimize repair and re­habilitation. The geology and topography of the immediate site and the surrounding area and reservoir are examined for shears, slides, erosion, sinkholes, and so forth. On older arch dams, such observations as surficial deterioration due to shrinkage cracks, freeze-thaw damage, alkali-aggregate reaction, and structural cracks from foundation irregulari-

ties or excessive temperature differences are not unusual. These types of defects are not necessarily evidence of a safety problem. Thus, careful review of historical data and a thorough understanding of arch dams and their founda­tions are essential for correct interpretation of defects ob­served during the inspection.

The appurtenances, including the spillway, outlet works, and other hydraulic conduits and their accompanying me­chanical equipment and machinery, are examined for main­tenance and functional ability. Mechanical and electrical equipment should be operated during the inspection if fea­sible.

Checklists and inspection forms should be prepared be­fore the inspection, and they should contain items revealed in the data examination and reflect possible conditions dis­tressful to arch dams, based on the inspection team's ex­perience.

Photographs of all types should be taken because the cost of the inspection far exceeds the film cost. Photographs will continue to supplement the historical record and assist in the evaluation.

Evaluation

The assembly of data and the field inspection provide a solid base for an initial understanding and evaluation of the project condition, and for prediction of future events. Hy­drologic studies may indicate differences between the prob­able maximum flood (PMF) and spillway capacity. These differences can be significant for older arch dams. The spillway capacity may be insufficient to pass the flood, so that overtopping could occur. Similarly, earthquake effects determined according to modem standards may produce accelerations that are larger than those used for the original designs.

With these possibilities in mind, a state-of-the-art hy­draulic and structural analysis may be appropriate, which should include the dam structure, foundation stability, hy­drologic/hydraulic capacities, and downstream hazard as­sessment. Features of the dam or foundation shown to be in distress usually require another review of existing data and a field inspection. New data from field tests on the dam or foundation, laboratory tests of concrete and foundation cores, seismotectonic analyses, and so on, may be needed to confirm or alter antiquated or questionable values. If any discrepancies are determined, and, in the opinion of the analyst, they are important to the safety of the dam or re­habilitation considerations, then a reanalysis may be nec­essary.

Once all the analyses and evaluations are complete, suf­ficient information is available on the past and present con­dition of the project. The conclusions should reflect the best judgment of the examining team as to safety and es­sential areas of repair or rehabilitation.

Recommendations

Having completed the evaluation and made its judgments, the examining team should recommend-in sufficient de­tail-items or areas of the dam, foundation, appurtenances, and reservoir in need of repair or rehabilitation to ensure continued safety. These recommendations should be cate­gorized according to safety, maintenance, and discipline and priorities should be set according to urgency to mini­mize the owners' risk, costs, and down time.

Repair/Rehabilitation

The repair or rehabilitation of concrete arch dams for rea­sons of safety or continued uninterrupted use generally is infrequent. Tolla Dam in Corsica, a 295-ft (90-meter)-high experimental-type structure, was substantially reinforced following a close inspection of cracks.46 Wild Horse Dam in Utah was condemned and replaced entirely because of material deterioration from alkali-aggregate reaction. For the same reason, the spillway crest of Matilija Dam in Cal­ifornia was lowered by removal of concrete, significantly reducing the reservoir capacity. 47 Numerous dams have been rehabilitated or replaced because of increased need for storage capacity, such as East Canyon Dam in Utah.48

Some modifications may require additional concrete on the crest only for small raises, as proposed at Buffalo Bill Dam, Wyoming. Where structural conditions and shaping are satisfactory, substantial increases in height are possible, as at Ross Dam in Washington state.49

An adequate structural behavior instrumentation system is essential to arch dam safety monitoring. The existing system may need to be supplemented with additional in­struments, to have existing instruments reinstated, or to have a new system designed and installed. Structural be­havior systems not previously installed will require more frequent readings (weekly) for the first two years to estab­lish a reliable data base from which to judge future mea­surements.

Risk Assessment

Dealing with dam safety problems involves financial and institutional issues just as much as the more obvious tech­nical considerations. This is true because organizational decision makers charged with the responsible allocation of limited resources and the safety of dams are faced with the dilemma of how best to carry out their charge. As noted by the Committee on the Safety of Existing Dams: "In­creasingly, therefore, dam safety program managers, own­ers, and their technical staffs must be prepared to support and justify engineering decisions by the use of an analysis of the trade-off between cost and risk.' ,7 This especially applies where funding for remedial work is limited and ex-

ARCH DAM DESIGN AND ANALYSIS 511

penditures must be directed to achieve an optimum reduc­tion of risk.

The purpose of risk assessment is to provide a formal basis for evaluating the least-cost alternative for safety im­provements for new or existing dams. The risk assessment approach provides a framework within which significant risk factors can be quantified and the degree of safety jus­tified. The objective is to reach an optimum balance be­tween adverse outcomes and alternative actions. Optimi­zation can be done to minimize loss of life or property damage, or to maximize risk reduction benefits at mini­mum cost. The uncertainty connected with initiating events, system responses, and outcomes is the primary justification for using a formal probabilistic approach. Contrary to the arguments raised by its detractors, probabilistic risk as­sessment complements and strengthens engineering judg­ment rather than replacing it.

Levels of Risk Assessment. Different levels of detail in the risk assessment procedures used for dam safety eval­uations are appropriate at different stages in the life of a dam. As the data base for a dam grows, and as the issues to be addressed change from general questions of site se­lection to specific issues of the selection of design param­eters, the degree of detail that can be justified in the risk assessment grows correspondingly.50.51 For the purpose of this discussion, three levels of risk assessment applications to dams are distinguished. In order of increasing detail, they are the planning level, the screening level, and the design level.

At the planning level, it is desirable to introduce an es­timate of risk cost associated with dam failure into the ben­efit-cost analysis as a means of including societal risk in the process of deciding to build a dam. 52 At this level, the estimated probabilities and consequences of dam failure are only approximate and usually rely heavily on historical in­formation.

The screening problem is the ranking of "unsafe" dams in order of priority for expenditure of limited funds to pay for remedial action. In this context, the absolute values of probability and consequence estimates are less important than a consistent procedure for estimating them so that an accurate ranking will be achieved. At the screening level, site-specific conditions typically would be evaluated using reconnaissance-level investigations and only approximate engineering and economics. An example of screening-level analysis is the method developed for the Federal Emer­gency Management Agency by Stanford University. 53

A design-level risk assessment involves detailed ques­tions such as the selection of design standards and choices between design alternatives for the dam and its appurtenant structures. Carefully estimated probabilities and conse­quences must take into account site-specific conditions based on detailed site investigations and engineering and

512 ADVANCED DAM ENGINEERING FOR DESIGN, CONSTRUCTION, AND REHABILITATION

economic analyses. In addition, the sensitivity of conclu­sions must be investigated with respect to uncertainties in the estimates of both probabilities and consequences. Doc­umented procedures at the design level have been prepared by the U.S. Bureau of Reclamation. 54

Design-Level Risk Assessment. The successful de­sign and construction of dams requires the application of the judgment of highly experienced engineers, geologists, hydrologists, and others. Traditionally, the approach to dam design focuses deterministic analyses on extreme events, such as the probable maximum flood or the maxi­mum credible earthquake, and uses conservative estimates of such properties as concrete or soil strength. Safety fac­tors are used to evaluate the ratio of resisting to overturning moments for such failure modes as slope instability. As a result, through the practice of the traditional approach, which is based on the accumulation of many decades of dam engineering experience, an impressive safety record has been achieved.

However, the traditional approach does not attempt ex­plicitly to quantify all significant risk factors for a dam. Nor does it explicitly determine the degree of safety that can be justified for a particular structure, considering the potential consequences of a sudden release of the contents of a reservoir following dam failure. The risk assessment approach provides the framework for such a quantitative determination.

Perhaps the most critical step in the decision analysis process is to conceive alternatives for providing added pro­tection or for providing it more cheaply. For each alter­native, the engineer must evaluate the added cost in rela­tion to the "do nothing" alternative, as well as its

effectiveness in reducing the probability of failure or its consequences, or both. Based on this evaluation and guided by appropriate decision criteria, the engineer must select the most favorable alternative or, when acting in an advi­sory capacity, present all the facts on cost and risk to the decision maker.

In a formal risk assessment, estimates are made of oc­currence frequencies, relative likelihoods of different lev­els of response and damage, and the various components of cost and consequences. Although an actual value of risk cost is determined, this value often need not be considered in absolute terms but as a number suitable for comparison among alternative risk reduction measures. An integral part of a risk assessment is that assumptions should be varied about any of the study parameters to determine their effect on the risk cost. The approach used in applying risk as­sessment to an existing dam at a design level follows the identification, estimation, aversion, and acceptance steps as shown in Fig. 17-11.55 Each of the steps is described more fully in the following sections.

Risk Identification (Risk Model Development). A se­quence of events is identified, beginning with events that can initiate dam failure and ending with the consequences of the failure, as shown across the top row in Fig. 17-12. Initiating events can be classified as external or internal. External events include earthquakes, floods, and upstream dam failure. Internal events include chemical changes in soil or concrete properties, or latent construction defects. At low levels, initiating events would not lead to dam fail­ure. However, at high inflow rates, a rapid rise in pool level could lead to overtopping, or a severe earthquake could result in structural deformation or liquefaction. These

RISK ASSESSMENT

RI SK RISK RISK r---. ESTI MATION EVALUATION

IDENTIFICATION -DO NOTHING - ~

-ACCEPTABLE RISK?

NO

RISK .. ESTIMATION I--AVERSION MEASURES

l Figure 17-11. Risk assessment framework. Source: Bowles (1987).55

~ 1M PLEMENTATION

OF RISK AVERSION

MEASURES

..

ARCH DAM DESIGN AND ANAL VSIS 513

INITIATING SYSTEM RESPONSE OUTCOME EXPOSURE CONSEQUENCE EVENT (FAILURE MODE) (PARTIAL/COMPLETE

FAI LURE) EXTERNAL: OVERTOPPING TIME OF DAY ECONOMIC DAMAGE

IDENTIFICATION EARTHQUAKE CRACKING

FLOOD SEASON LOSS OF LIFE

FLOOD FOUNDATION WARNING TIME INTERNAL.

PIPING EROSION

ESTIMATION LOADING j.l FAI LURE ~ BREACH ~ EXPOSURE ~ EXPECTED

PROB (E) PROB (FIE) PROB (OlF) PROB (LIO) LOSSES

UPSTREAM WATERSHED STRUCTURAL MODIFIC- STRUCTURAL WARNING SYSTEMS RELOCATIONS CHANGES TIONS, SAFETY INSPEC- MODIFICATIONS

TIONS, INSTRUMENTA-FLOOD PROOFING LAND USE ZONING EMERGENCY PREPARED-

TION, OPERATIONAL NESS AVERSION RESTRICTION S

'------I SELECTION OF RISK AVERSION MEASURES

ACCEPTANCE

I. P ( E) = ANNUAL PROBABILITY OF OCCURRENCE OF LOADING EVENT IN A RANGE OF MAGN ITUDES. 2. P (F/E)= CONDITIONAL RESPONSE PROBABILITY OF DAM FAILURE BY A SPECIFIC FAILURE MODE,F, GIVEN THE LOADING EVENT OCCURS. 3. P(O/F)=CONDITIONALOUTCOME PROBABILlTY,O,GIVEN THAT FAILURE MODE, F, OCCURS. 4. P(L/O)= CONDITIONAL LIFE LOSS PROBABILITY, L, FOR A POPULATION AT RISK GIVEN THAT OUTCOME, 0, OCCURS.

Figure l7-12. Risk-based method for dam safety improvements. Adapted from Bowles (1987).55

and other dam-reservoir system responses are failure modes that can lead to the outcome of sudden release of the res­ervoir contents. The magnitude of the resulting property damage and life loss will depend on the various exposure factors, which include flood routing to determine the path of the flood wave and the area of inundation, the time of day and season of the year, and the effectiveness of any warning systems and evacuation plans. Consequences are classified as life loss and economic loss, including property damage, cost of dislocations, and loss of project benefits.

During the identification step, professional judgment and experience, review of available information, and site visits are used to list types of initiating events, system responses, outcomes, exposure factors, and consequences applicable to a particular dam-reservoir system. A diagram such as that shown in Fig. 17-13 is then constructed to describe the event-consequence sequences or initiating event-system response-outcome-exposure-consequence pathways.

Risk Estimation. The second step in the design-level risk assessment procedure is estimation of the probability and

consequence components of risk. The types of probabilities to be estimated, shown in Fig. 17-12, are as follows:

• Annual probability of occurrence of loading (e.g., flood) in a range of magnitudes, E-Prob (E).

• Conditional response probability of dam failure by a specified failure mode, F, given that the loading oc­curs in the range, E-Prob (F I E).

• Conditional probability of the outcome, 0, release of reservoir contents, given that failure mode, F, oc­curs-Prob (0 I F).

• Conditional probability of life loss, L, for a popUla­tion at risk (PAR) given that the outcome, 0, occurs­Prob (L I 0).

A historical/empirical approach combined with engi­neering judgment is used to estimate probabilities. The larger the available sample size used, the better the esti­mates are expected to be. Thus, a long record of flows or seismicity can be expected to yield better (i.e., less uncer­tain) estimates of extreme floods or earthquakes than a short

514 ADVANCED DAM ENGINEERING FOR DESIGN, CONSTRUCTION, AND REHABILITATION

EVENT SYSTEM RESPONSE OUTCOME EXPOSURE CONSEQUENCE

FLOO 0

SPILLWAY CREST FAILURE

EROSION OF SPILLWAY TAILRACE CHANNEL

MALFUNCTION OF GATES

OVERTOPPING OF DAM LOSS OF LI FE

PARTIAL AND NO FA I LU R E

LAN 0 SLID E I N TO RESERVOIR

FOUNDATION LIQUEFACTION

PROPERTY DAMAGE RES I DENTIAL COMMERCIAL PERSONAL PUBLIC

TOTAL FAILURE

EXPOSURE FACTORS TIME OF DAY TIME OF YEAR TYPE OF FAILURE FLOOD WARN I NG EARTHQUAKE

(GROUND SHAKI NG AT DAMSITE)

S LO PE INSTABI LITY

PIPING AROUND OUTLET TUNNEL

LANDSLIDE INTO RESERVOIR

FOUNDATION FAILURE

EMBANKMENT (DIKES) DEFORMATION

CRACKING OF DAM

OUTLET TUNNEL RUPTURE

SEICHE

TIME LOSS OF REVENUE WAGES BUSINESS INCOME UTI LI TI ES

DAMAGE TO DAM

Figure 17-13. Event-system response-outcome-exposure-consequence diagram.

record. Similarly, a large data base of dam failures, cate­gorized according to such factors as the type, size, age, and location of dam, can be expected to provide better es­timates of the probability of failure due to a specified fail­ure mode than would a small data base. However, the available data base for dam failures is usually small for a particular category of dam, especially large dams, and in any case will provide only an estimate of probability of failure for the "average" condition rather than for the spe­cific condition of the dam that is under evaluation.

In order to incorporate additional information about a specific dam obtained from field work, laboratory testing, engineering analyses, and expert judgment, the historical! empirical probability estimates are treated as initial esti­mates, which are updated judgmentally using the addi­tional information. The Bayesian approach to estimation provides a formal method for updating historical probabil­ity estimates using "judgmental" information, as follows:

Prob" (F I X)

Prob ' (F) L(X I F) Prob ' (F) L(X I F) + Prob ' (F) L(X I F)

in which:

Prob" (F I X) = updated estimate of probability of fail­ure, given additional information X about the dam.

Prob ' (F) = prior estimate of probability of fail­ure.

Prob I (F) = prior estimate of probability of no fail­ure.

L(X I F) = likelihood of observing information X if the dam were to fail.

L(X I F) likelihood of observing information X if the dam were not to fail.

Probability of Hydrologic Events. The probability of a particular hydrologic event occurring can be described using discharge-probability relationships. For drainage basins with a systematic record of annual peak flow of 30 years or more, a flood probability relationship may be de­veloped using statistical procedures that can reasonably be extrapolated to return periods from 100 to 200 years, pro­vided the flood data used are based on the same meteoro­logical causative factors as the floods that formed the an­nual peak (snowmelt, thunderstorm, etc.). For drainage basins without an adequate historical record of stream­flows, discharge-probability relationships can be devel­oped by using recorded flood data from adjacent or similar basins. If a drainage basin has no streamflow records but does have a sufficient number of precipitation records, a runoff-precipitation relationship may be developed. Precip­itation records usually are available, and often are for longer return periods than runoff records.

Two general approaches may be employed in the devel­opment of these relationships. The first and preferred method is the statistical analysis of recorded, historical, and paleohydrologic streamflow data. The detailed meth­odology used in the statistical analyses of streamflow rec­ords is well defined in Water Resources Council (WRC) Bulletin No. l7B, "Guidelines for Determining Flood Flow Frequency. ,,61 The second approach uses rainfall runoff models in which precipitation amounts of specified proba­bility are converted into runoff of like probability. The lat­ter approach is used only when data are not available for the statistical approach.

Given the flood-frequency relationship to the 100-year return period based on historical events and the magnitude of the PMF (probable maximum flood), it is possible to extrapolate to about the 200-year return period and to es­tablish a relationship that provides an estimate of the an­nual probability of occurrence of floods, from the magni­tude representing approximately the 200-year flood through the magnitude representing the PMF.

An approach suggested by the National Research Councils is to construct two separate curves. The first por­tions of the curves, out to the 200-year event or four times the length of record if longer, are the 5% and 95% confi­dence limit curves, determined using the procedures con­tained in Bulletin No. l7B. The second portions of the curves, beyond the 200-year events, are constructed as straight-line extensions of the confidence limit curves on lognormal probability paper. The straight-line extensions are from the 5 % and 95 % confidence limit curves at the 1-in-200-year event to the PMF level at exceedance proba­bilitIes of 10-4 and 10-6, respectively. The curves con­structed in this manner are not consistent with an upper bound flood concept, but are constructed only to facilitate an assessment of various alternative spillway capacities un-

ARCH DAM DESIGN AND ANALYSIS 515

der consideration in a risk assessment. The constructed curves result in an apparent discrete probability for the PMF (10-4 and 10-6) that involves a significant level of uncer­tainty. Risk costs based on extreme hydrologic events must be developed for both 5 % and 95 % confidence limit curves to adequately depict the uncertainty involved in the esti­mates.

Probability of Seismic Events. Earthquake frequency of occurrence has been related empirically to magnitude ac­cording to the following equation (Richter 1958):62

lOglO N = a - bM

where N is the number of events of magnitude M or greater per year for a particular seismic source, and a and bare constants. The distribution is cumulative, so that the num­ber of events occurring per year within the magnitude in­terval MI to M2, denoted by I::..N, is given by:

The recurrence interval T for some earthquakes having a magnitude within the interval MI to M2 is the reciprocal of I::..N, or:

The cumulative recurrence gives a reasonable descrip­tion of occurrence for small to moderate-magnitude events. It should be modified, however, so that the frequency of events goes to zero for some M = Mmax. Two simple mod­ifications are generally considered for this purpose: at M = Mmax one may either (a) truncate the probability density function (number of events per unit magnitude range), or (b) truncate the cumulative distribution (number of events of magnitude M or larger).

For earthquake occurrence on single faults, the truncated cumulative distribution may be most appropriate. Geolog­ical observations in the early 1980s along the Wasatch fault in Utah suggest that fault segments rupture with a charac­teristic earthquake of magnitude near Mmax. The character­istic event is not really a periodic event of a particular mag­nitude, but is rather an event that repeatedly occurs over an interval near Mmax. This is the type of occurrence that is described by the truncated cumulative distribution model. This concept also lends itself very well to risk assessment models because conditional system response probabilities are developed for ranges of earthquake magnitudes as op­posed to singular earthquake events having a specific mag­nitude.

In spite of the apparent advantages of the truncated cu­mulative distribution for describing the frequency of oc-

516 ADVANCED DAM ENGINEERING FOR DESIGN, CONSTRUCTION, AND REHABILITATION

currence of earthquakes, some limitations must be taken into consideration in using the model for very large-mag­nitude events near M = Mmax. The model indicates an ab­rupt drop to zero in the cumulative occurrence of events at M = Mmax where an actual distribution tails off at some magnitude less than Mmax. Therefore, care must be taken in using this recurrence relation for events at or near Mmax.

When generating a magnitude frequency of occurrence relation, ideally one would have data with which to accu­rately define the "tail" of the distribution. Unfortunately, such data are rarely available, so a model simple enough to be accurately constrained by what data there are, such as the truncated cumulative distribution model, must be used. Suffice it to say, suitable precautions should be taken when using this approach, and the experience and judg­ment of a skilled seismologist should be consulted.

After the recurrence relationship is completed, the ap­propriate probabilities can be determined to satisfy the earthquake magnitude ranges defined in the risk assessment model.

Probability of Static Failures. The conditional system re­sponse of a dam due to static loadings can be estimated in the following manner. Static loading on a dam is due to the hydrostatic pressure of the reservoir. The magnitude of this load varies with reservoir stage, which is determined by the relationship between reservoir inflow and outflow. Al­though rarely, concrete dams have failed under static load­ing conditions independently of the occurrence of floods or earthquakes. Such failures typically can be attributed to either a design or latent construction defect, or to a dete­rioration in the properties of materials in the structure. Where evidence exists indicating that a particular dam may be endangered by a static failure mode, then this failure mode should be explicitly included in the risk model. Also,

its system response and outcome probabilities should be assessed by experienced dam engineers, using the results of investigations at the dam and subsequent analyses. When no evidence of potential static failure modes is known to exist at a particular dam, estimates can be made using the historical frequencies of static failures of similar dams.

The historical frequency method of dealing with static failure modes was proposed by Bowles et al. (1978)50 and subsequently pursued by the Bureau of Reclamation (" Ap­plication of Statistical Data from Dam Failures and Acci­dents to Risk Based Decision Analysis on Existing Dams," USBR Internal Report, 1987).63 The Bureau assembled an extensive data base of U.S. dams and their failures. This data base was added to by Stanford University (1985) un­der funding from the Federal Emergency Management Agency (FEMA), and expanded by Hatem (1985) in a the­sis written at Stanford University. 64

The annual failure probabilities estimated using the his­torical frequency method are entered into the risk model the same as the system responses for hydrologic and seis­mic events. They become a part of each partial risk cost computation, and the proportion of the overall risk costs due to static events is accounted for in the ultimate total annual risk costs.

Risk Costs. So that all event sequences will be consid­ered, an event tree such as the one shown in Fig. 17-14 is constructed. The partial risk cost for each pathway is ob­tained by taking the product of the four probabilities shown in Fig. 17-12 and the economic consequences, as follows:

Prob (E) Prob (F I E) Prob (0 I F) Prob (L I 0) LE

The total risk cost is obtained by summing the partial risk costs over all, N, mutually exclusive pathways. To facili-

EVENT RESERVOIR LEVEL

SYSTEM RESPONSE OUTCOME

--E RESERVOIR LOSS-HIGH FLOOD

SPILLWAY GATE FAILURE RESERVOIR RELEASE-LOW FLOOD

NO RESERVOIR LOSS

-E RESERVOIR LOSS-HIGH FLOOD

FLOOD lEVEL __ -LEVEL FACTOR -+-- TOE/ABUTMENT EROSION RESERVOIR RELEASE-LOW FLOOD

20-40% PMF NO RESERVOIR LOSS

{RESERVOIR LOSS- HIGH FLOOD

SPI LLWAY CHUTE EROSIO N

RESERVOIR RELEASE- LOW FLOOD

NO FAILURE------- NO RESERVOIR LOSS

Figure 17-14. Example of part of an event tree for a dam. Source: Bowles (1987).55

tate the computations, the risk model (see Fig. 17-14) de­veloped for a particular project can be reproduced in the computer using a spreadsheet program. An example spreadsheet representation of a risk model is shown in Fig. 17-15.

Risk Aversion. The product of the second step is an esti­mate of the probability of failure, risk cost, and life loss that would be associated with each failure mode, or com­bination of failure modes, for the baseline or "do nothing" alternative. If these risks are unacceptable, the analyst moves to the third step. This involves formulation and evaluation of alternatives for risk aversion. Risk aversion can be achieved by reducing the probabilities associated with a pathway or by reducing the consequences. In both cases, structural or non structural measures may be consid­ered. Figure 17-12 lists examples of aversion measures and shows the probability or consequence that would be ex­pected to be reduced by their implementation. An impor­tant and sometimes difficult part of the evaluation of risk aversion measures is estimation of the reduction in the probabilities or consequences that would be expected as a result of implementing a measure. For example, what could be the reduction in Prob (F IE) for foundation failure dur­ing an earthquake if rock anchors were placed in the foun­dation? Or, what would be the reduction in Prob (L I 0) if a flood warning system and an evacuation plan were im­plemented?

The product of the aversion step is an estimate of the reduction in probability of failure, risk cost, and life loss that could be attributed to the implementation of a risk aversion measure. Such reductions are used as an estimate of the benefits of the measure, and hence a benefit-cost analysis can be performed.

Risk Acceptance. The final step in the risk assessment process is to decide what degree of safety should be achieved, or equivalently, what residual risk will be ac­cepted. Although the analyst can supply information and recommendations for this decision, it is usually made by a decision maker such as the dam owner, operator, or regu­lator. The decision is especially sensitive and difficult where lives are at risk and where large investments will be required to improve safety with little or no effect on the project benefits, except, of course, on their longevity con­sidering the reduced likelihood of dam failure.

CASE HISTORY-KLANG GATES DAM ENLARGEMENT

History

Klang Gates Dam is located about 7! miles (12 km) north of Kuala Lumpur, the capital city of Malaysia, which is

ARCH DAM DESIGN AND ANALYSIS 517

located in the western central area of west Malaysia. In the early 1950s, the government of Malaysia wanted to de­velop a new water supply for the city of Kuala Lumpur by building a dam and reservoir on the Klang River. The Bu­reau of Reclamation was asked to provide technical assist­ance in designing Klang Gates Dam. Construction began in 1955 and was completed a few years later. The reservoir was allocated exclusively to municipal and industrial pur­poses, so any flood flows into the reservoir had to be passed over the uncontrolled spill way. In January 1971, the city of Kuala Lumpur experienced serious flooding from the Klang, Batu, and Gombak rivers, causing extensive dam­age and loss of life. This prompted the government of Ma­laysia to create the Kuala Lumpur Flood Mitigation Proj­ect, consisting of the enlargement of Klang Gates Dam and the construction of two new dams on the Batu and Gombak rivers.

The existing Klang Gates Dam was a concrete arched gravity dam with an uncontrolled overflow spillway. The major dimensions of the dam were maximum height 111 ft (33.8 meters), crest length 365 ft (111 meters), crest el­evation 311 ft (94.79 meters), crest thickness 8 ft (2.44 meters), spillway crest elevation 300 ft (91.44 meters), and spillway length 84 ft (25.6 meters). Reservoir storage capacity at elevation 300 was 16,481 acre-feet (20.3 mil­lion m3 ), and reservoir capacity at elevation 311 was 22,707 acre-feet (28.0 million m3 ). The domestic water is passed through the dam through two 45-in. outlet pipes, which connect downstream to two pipelines that carry the water to a treatment plant located near Kuala Lumpur. The Malaysian government's original proposal for the enlarge­ment of Klang Gates Dam was to raise the dam crest 10 to 15 ft (3 to 4.6 meters) and through the use of gated spill­ways, the reservoir would be raised 21 to 26 ft (6.4 to 7.9 meters) above elevation 300. A general plan, elevation, and sections of the dam are shown in Fig. 17-16.

The concrete in the dam was in excellent condition after 20 years of service, and the spillway had operated regularly with no problems. Although the dam is not drastically curved in plan, there is sufficient arch action to compensate for the thinner cross sections that result from the 0.5: 1 downstream slope. A gravity analysis of a typical cross section of the dam would yield results that do not satisfy the usual gravity dam criteria. Therefore, the dam was de­signed and always considered to act as an arch gravity dam or thick arch dam.

Alternative Designs

The basic concept underlying the revised design studies was to enlarge the dam without significantly altering the state of stress in the dam and its foundation under the increased loading conditions of the higher reservoir water surface. To accomplish this, the existing dam was analyzed to establish

518 ADVANCED DAM ENGINEERING FOR DESIGN, CONSTRUCTION, AND REHABILITATION

AlTERNATIVE A, EXISTING DAM

LOADING SYSTEM RESPONSE OUTC[)I.1E PROB •••••••••• • •••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• (1 ) (2J (3) (4) (5) (6) (7) (B) 19) EVENT EVENT RESPONSE RESPONSE OUTcmE RESERVOIR OUTCOME BREACH PATII'IAY TYPE PROB TYPE PROS TYPE STAGE PROB PEAK FLIlf PROII.

(FT HSL) (K CFS)

25-30 B.OOE-04 NO FAIL (& PF) 0.45 3431 0.00036 (20-24) SPILL CREST F 0.3000

F - SPILLWAY 1.0000 710 2.40E-04

TAILRACE EROS. 0.2500 F STILL B. 1.0000 710 2.00E-04

PIPING ClINKER 0.0000 F SPILLWAY 1.0000 710 0.00E+00

OVERTIFPING 0.0000 F - BREACH 1.0000 71 0 O. OOE+OO

31l-50 B.OOE-04 NO FAIL (& PF) 0.15 3433.2 0.00012 (24-42) SPILL CREST F 0.4000

F - SPILLWAY 1.0000 B60 3.20E-04

TAILRACE EROS. 0.4000 F STILL B. 1.0000 860 3.20E-04

PIPING CLINKER 0.0600 F SPILLWAY 1.0000 860 4.00E-06

OVERTIFPING 0.0000 F - BREACH 1.0000 BBO 0.00E+00

5O-S0 3.00E-04 NO FAIL (& PF) 0 3437.3 0 (42-6B) SPILL CREST F 0.4500

F - SPILLWAY 1.0000 1130 1.35E-04

TAILRACE EROS. 0.4500 F STILL B. 1.0000 11301.35E-D4

PIPING ClINKER 0.1000 F SPILLWAY 1.0000 1130 3.00E-D5

OVERTIFPING 0.0000 F - BREACH 1.0000 1130 0.00E+00

?ga!183) loBOE-04 NO FAIL (& PF) 0 3441.6 0

SP ILL CREST F 0.4500 F - SPILLWAY 1.0000 1390 B.10E-05

TAILRACE EROS. 0.4500 F STILL B. 1.0000 1390 B.10E-05

PIPING CLINKER 0.1000 F SPILLWAY 1.0000 1390 1.00E-05

OVERTIFPING 0.0000 F - BREACH 1.0000 1390 O.OOE+OO

Figure 17-15. Example spreadsheet reproduction of risk model.

a point of reference against which all other stress results could be compared for various design alternatives. Two alternative designs were investigated. The first had the dam crest raised to elevation 321, with the 8-ft crest thickness of the existing dam maintained. The second alternative was to raise the dam to elevation 321, but with crest thickness increased to 12 ft (3.66 meters). The existing dam and the two alternatives were analyzed for hydrostatic and dead load, using the computer program ADSAS (arch dam stress analysis system). Seismic loads were not considered be­cause of the large distance to the nearest seismic area, which resulted in ground accelerations at the site of less

than 0.1 g. Temperature effects were included in the anal­yses by assuming that concrete temperatures varied uni­formly throughout each arch section. Temperature differ­ences were computed based on the difference between the concrete temperature at the time of loading and the as­sumed closure temperature. Although there were no rec­ords of the dam's having been grouted, it was assumed that the structure acted monolithically with continuous arch ac­tion whenever the temperature of the concrete exceeded 78°P. The selection of the 78°P closure temperature was based on a review of construction records made during the 1950s, when the dam was built. Neither silt nor tailwater

ARCH DAM DESIGN AND ANALYSIS 519

Spring FIDod Event, Sheet 2 Dr 3

ECIJNOI.IIC DAMAGE THREAT TO LIFE ..................................................... ------------------(10) (11) (12) (13) (14) (15)

PER ANHUAl RISK COST 116)

PIlPULA TlON EVENT ----------------- AT RISK ('·10**6) (TOTAL!

('*10**6) ,,*10"6) [.·10·*6) (.*10.*6) EVENT)

0.124 DAM - POINT A 282.5 0.088 BELOW POINT A

DAM - POINT A 282.5 0.057 BELOW POINT A

(17) (18) PIlPULATlON EXPOSURE AT RISK PROB

114 0.075000 10032 0.000200

114 0.075000 10032 0.000200

(19) LOSS OF LIFE PER EVENT

8.5 2.0

8.5 2.0

DAM - POINT A 114 0.075000 8.5 282.5 0.000 BELOW POINT A

DAM - POINT A 282.5 0.000 BELIIII POINT A

0.197 DAM - POINT A 289.5 0.093 BELIIII POINT A

DAM - POINT A 289.5 0.093 BELOW POINT A

DAH - POINT A 289.5 0.012 BELOW POINT A

DAM - POINT A 289.5 0.000 BELIIII PO INT A

0.089 DAH - POINT A 287 0.040 BELOW POINT A

DAH - POINT A 297 0.040 8ELIIII POINT A

DAM - POINT A 297 0.009 BELOW POINT A

DAM - POINT A 297 O.DOO 9ELIIII POINT A

0.055 DAM - POINT A 303 0.025 BELIJ'I POINT A

OAM - POINT A 303 0.025 BEL(lrI POINT A

OAM - POIUT A 303 0.005 BELIJ'I POINT A

DAM - POINT A 303 0.000 9ELIJI POINT A

Figure 17-15. (Continued)

loads were considered in the analyses. For comparison of the existing dam and alternatives 1 and 2, the results for only one loading combination are discussed.

Analytical Results. The analyses were performed for the following loading condition: normal reservoir water surface elevation, the effects of minimum concrete tem­peratures, and dead weight. In the existing dam, no arch action was assumed above the spillway crest elevation of 300. Arch action was assumed to the top of the dam, ele­vation 321, however, for alternatives 1 and 2. An exami­nation of the stress results revealed that the dam acts mon-

10032 0.000200 2.0

114 0.075000 8.5 10032 0.000200 2.0

114 0.075000 8.5 10032 0.000200 2.0

114 0.075000 8.5 10032 0.000200 2.0

114 0.075000 8.5 10032 0.000200 2.0

114 0.075000 8.5 10032 0.000200 2.0

114 0.075000 8.5 10032 0.000200 2.0

114 0.075000 8.5 10032 0.000200 2.0

114 0.075000 9.5 10032 0.000200 2.0

114 0.075000 9.5 10032 0.000200 2.0

114 0.075000 9.5 10032 0.000200 2.0

114 0.075000 9.5 10032 0.000200 2.0

114 0.075000 8.5 10032 0.000200 2.0

114 0.075000 8.5 10032 0.000200 2.0

olithically as an arch at all times below elevation 270. Above elevation 270, loads on the dam are taken by can­tilever action when concrete temperatures fall below 78 OF and the contraction joints are at incipient opening. The thicker cross section of alternative 2 helped to reduce the magnitude of tensile principal stresses that occur near the abutments at the upstream face, particularly for low-tem­perature conditions. The additional thickness provides greater stiffness in both the vertical and horizontal direc­tions, which improves the cantilever stresses directly and allows the arches to carry a larger proportion of the loads in a horizontal direction in the upper portion of the dam. It

320

Cr e s t £1 30.0.0.0.

520

s/o

t ,/Axis of dam f' JZ IOO

----''''-+1-.111.81"=

O".n;nfjS OM mechanlcol _qu iptnilnt no' .how/I

SECTION THRU SPILLWAY

PLAN

UPSTREAM ELEVATION (OEVE LOPED ALONG AXI S )

SECTION THRU DOMESTIC WATER SUPPLY OUTLETS

Figure 17-16. Plan, elevation, and sections of Klang Gates Dam.

was concluded, therefore, that alternative 2, with a crest thickness of 12 ft, was the preferred alternative. It was de­termined as well that the spillway also should be designed for orifice flow, so beams were constructed between the piers to carry horizontal loads across the spillway open­ings.

Selected Alternative. Based on the analytical stress re­sults and economic and geological considerations, it was decided to enlarge Klang Gates Dam by raising the top 10 ft (3.05 meters) and increasing the width of the crest from 8 to 12 ft (2.44 to 3.66 meters). The length of the dam also had to be increased to ensure good contact between the dam and the canyon walls. Thrust blocks were designed and constructed at each abutment. Because the enlarged dam had a crest thickness of 12 ft compared with the orig­inal crest thickness of 8 ft, a portion of the new concrete was extended down the downstream face of the dam and bonded to the original concrete. This resulted in construc­tion joints between the new and old concrete, along nearly vertical planes.

Bonding of Concrete at Downstream Face

Because the concrete industry at that time had little or no experience in the structural integrity of vertical and near­vertical construction joints, the Bureau performed labora­tory tests to determine the strength characteristics of the bonded surface between old and new concrete along ver­tical and near-vertical planes.

Laboratory Tests. Four specimens were cast and tested in the laboratory program. The test procedures used were as follows:

Specimen No.1. Four concrete slabs, 2.5 ft (0.76 meter) wide,S ft (1.52 meter) long, and 6 in. (15 cm) thick, were cast in an upright position to represent the old concrete in Klang Gates Dam. The slabs were moist-cured for 14 days by wrapping the slabs, with forms intact, with plastic and keeping the concrete wet. The forms were stripped after 14 days, and the slabs were allowed to dry in the laboratory air. The construction joint was cast in a vertical position when the slab, representing the old concrete, was 50 days old. The face of the concrete slab that formed the construc­tion joint was dry-sandblasted a few days prior to the plac­ing of the fresh concrete against the joint. The sandblasted face was brought to a saturated surface dry condition be­fore the fresh concrete was placed against it. The new con­crete was placed against the slab using three 20-in. lifts and standard placing procedures. The concrete was consoli­dated using a 2-in.-diameter immersion-type vibrator. The dimensions ofthe new concrete were: 2.66 ft (0.81 meter) wide,S ft (1.52 meter) long, and 2 ft (0.61 meter) thick.

ARCH DAM DESIGN AND ANALYSIS 521

Curing of the new concrete and construction joint was ac­complished by wrapping the specimen, with the form in­tact, with plastic and keeping the concrete and form wet for 14 days.

The form was then stripped, and the specimen was al­lowed to dry in the laboratory air. Cores 4 in. in diameter and 12 in. deep were drilled through the old concrete into the new concrete. These cores were not broken from the specimen but were left in place so that testing in tension could be done by bonding only one tension plate to the core. The core drilling for the 28-day testing was done with a specimen in a vertical position, with the drill entering the specimen horizontally. Cores drilled for testing at 60 and 90 days were drilled vertically with the specimen lying down. No attempt was made to remove drill water or drill fines that enveloped the core. These wet drill fines kept the construction joints moist through the 90-day test period. A 4-in.-diameter tension plate was bonded to the concrete core, using epoxy bonding agent.

It was theorized that the hydraulic head caused by the depth of plastic concrete would improve the bond at the construction joint. In an effort to resolve that theory, cores were tested from near the top, middle, and bottom of the specimen. Data shown in Tables 17-7 through 17-9, rep­resenting Specimen No.1, appear to support the theory. The average of the three tests at each level shows an in­crease in tensile strength at the lower levels.

Specimen No.2. This specimen was cast in the same manner as Specimen No.1. The 4-in.-diameter cores were drilled with the construction joint in the vertical position, with the core drill entering the specimen horizontally. This permitted the drill water to rinse the drill fines from the specimen and allowed the specimen to dry more rapidly. As in Specimen No.1, the tensile strength was greater at lower depths in the specimen, and tensile strength in­creased with age through 60 days. The 90-day tensile strengths in Specimen No.2 were lower than the 60-day strength. The lower 90-day tensile strength in Specimen No.2 may have been caused by drying shrinkage at the construction joint.

Specimen No.3. This specimen was cast in a vertical po­sition and cured for 14 days in the same manner as Speci­mens No.1 and No.2. Because it was theorized that drying shrinkage at the construction joint in Specimen No.2 had caused low 90-day tensile strengths, an environment sim­ilar to the Kuala Lumpur area of Malaysia was established for storage of the test specimens. After 14 days of moist cure, the specimen was moved into a calorimeter room where the temperature was maintained at 79°F ± and a rel­ative humidity of 70% ±.

Specimen No.4. This specimen was cast in a position that was thought to represent a less severe joint condition. The

522 ADVANCED DAM ENGINEERING FOR DESIGN. CONSTRUCTION. AND REHABILITATION

Table 17-7. Tensile Strength of Vertical Construction Joint at about 30 Days Age, Klang Gates Dam.

Tensile Strength

Location Ib/in. 2

Joint Age in Feet Specimen Specimen Break Location and Test on Date from Top No.1 No.2 Moisture Condition No. Tested of Lift Moist Joint Dry Joint of Joint

Near the Top of the Specimen

1-1 28 days 0.79 160 Moist 4-1 32 days 0.79 130 Moist-Broke in concrete 7-1 34 days 0.74 160 Moist-Broke in concrete

Average 150 1-2 29 days 0.50 160 Moist 4-2 31 days 0.47 205 Moist 7-2 32 days 0.99 150 Moist

Average 170 Near the Middle of the Specimen

2-1 28 days 2.05 235 Moist 5-1 32 days 2.00 160 Moist 8-1 34 days 2.10 160 Moist

Average 185 2-2 29 days 2.10 170 Moist-Some concrete broken 5-2 32 days 2.07 175 Moist 8-2 33 days 2.06 195 Moist

Average 180 Near the Bottom of the Specimen

3-1 28 days 3.61 6-1 32 days 3.65 9-1 34 days 3.63

Average 3-2 29 days 3.90 6-2 31 days 3.88 9-2 32 days 4.49

Average

Courtesy Bureau of Reclamation.

concrete slab representing the old in-place concrete was placed in a fonn tilted at 26 0 off vertical. As for the three other specimens, the construction joint was cleaned using conventional methods, and then brought to a saturated sur­face dry condition prior to placing of the new concrete. The new concrete, representing concrete required to increase the height of the dam, was placed against the joint in three equal lifts using an immersion-type vibrator for consoli­dation. After a l4-day moist cure, the specimen was stored in the calorimeter with controlled temperature and humid­ity. The average tensile strengths for cores drilled through the construction joints in all test specimens are shown in Table 17-10. They indicate that if the construction joint is allowed to dry as in Specimen No.2, the tensile strength is decreased. The following observations were made, based on the test results:

• At any age the construction joint or concrete, which­ever failed, was generally stronger near the bottom of the 5-ft block than at the top.

315 270 280 290

Moist-Broke in concrete Moist-Broke in concrete Moist-Broke in concrete

160 Moist 145 Moist 155 Moist 155

• When it was exposed to a humid environment, the strength of the construction joint or concrete generally increased with age.

• Tensile strength at the construction joint tilted at 26 0

off vertical was greater than the tensile strength of the vertical construction joints.

• The tensile strength of the four construction joints tested averaged 5 % of the compressive strength of the surrounding concrete.

Construction Considerations. To enhance a long-last­ing bond between the existing concrete and the new con­crete, a system of 4-in. split clay pipes was called for in the construction at Klang Gates Dam. The pipes were placed on the downstream edge of each lift line and con­traction joint, and covered by the new concrete. The split clay pipes were intended to intercept any water that might seep through the dam, thus preventing hydrostatic pres­sures from developing along the interface between the new and the old concrete. Details showing the arrangement of

ARCH DAM DESIGN AND ANALYSIS 523

Table 17-8. Tensile Strength of Vertical Construction Joint at about 62 Days Age, Klang Gates Dam.

Tensile Strength

Location Ib/in. 2

Joint Age in Feet Specimen Specimen Break Location and Test on Date from Top No.1 No.2 Moisture Condition No. Tested of Lift Moist Joint Dry Joint of Joint

Near the Top of the Specimen

10-1 60 days 1.13 190 Dry 13-1 62 days 1.40 245 Dry-Broke in concrete 16-1 67 days 1.48 245 Moist

Average 225 10-2 62 days 0.66 265 Dry 13-2 63 days 0.55 265 Dry 16-2 64 days 0.57 335 Dry

Average 290 Near the Middle of the Specimen

11-1 60 days 2.52 280 Moist-Some concrete broken 14-1 62 days 2.69 325 Moist 17-1 67 days 2.48 290 Moist

Average 300 11-2 62 days 2.40 365 Dry 14-2 63 days 2.52 275 Dry 17-2 64 days 2.51 300 Dry-Some concrete broken

Average 315 Near the Bottom of the Specimen

12-1 60 days 4.14 15-1 62 days 4.02 18-1 67 days 4.09

Average 12-2 62 days 3.83 15-2 63 days 4.08 18-2 64 days 4.02

Average

Courtesy Bureau of Reclamation.

the split clay pipes are given in Fig. 17-17. The installation of the split clay pipes in the field is shown in Fig. 17-18.

Spillway Revisions

In addition to raising the nonoverflow section of the dam by 10ft, the Klang Gates Dam enlargement also provided for the addition of four 16.5 X 9-ft radial gates with head­walls allowing the maximum water surface to rise to ele­vation 321. At that elevation the capacity of the reservoir was increased approximately 6600 acre-feet (8.14 million m3 ) to 29,300 acre-feet (36.1 million m3 ). Approximately 4500 acre-feet (5.55 million m3 ) of this additional storage was allocated to exclusive flood control. Before the radial gates could be installed onto the existing spillway, the cen­ter spillway pier and bridge deck had to be removed, and three interior spillway piers, the headwall, and the hoist deck had to be constructed. The spillway modifications are shown in Fig. 17-16.

Because of the higher operating heads on the existing

340 290 330 320

Moist-Broke in concrete Moist-Broke in concrete Moist-Broke in concrete

415 Dry-Broke in concrete 330 Dry 330 Dry 360

spillway and with the addition of spillway radial gates, it was determined that laboratory model tests should be con­ducted on the revised spillway. The laboratory studies in­dicated that the hydraulic performance of the spillway modification was satisfactory. Some channelization down­stream of the spillway stilling basin was included in the construction modifications, based on results from the lab­oratory tests. The completed project is shown in Fig. 17-19.

Conclusion

Klang Gates Dam enlargement is an impressive example of how an existing dam project with inadequate spillway ca­pacity can be modified to provide the required level of downstream safety to people and property with the appli­cation of creative thinking and sound engineering practice. The modification was accomplished within the established budgets at a cost of approximately 3,400,000 U.S. dollars in 1980. The project continues to operate satisfactorily at this time.

524 ADVANCED DAM ENGINEERING FOR DESIGN, CONSTRUCTION, AND REHABILITATION

Table 17-9. Tensile Strength of Vertical Construction Joint at about 91 Days Age, Klang Gates Dam.

Tensile Strength

Location Ib/in. 2

Joint Age in Feet Specimen Specimen Break Location and Test on Date from Top No.1 No.2 Moisture Condition No. Tested of Lift Moist Joint Dry Joint of Joint

Near the Top of the Specimen

19-1 89 days 1.71 385 Moist-Broke in concrete 22-1 91 days 1.67 325 Moist-Broke in concrete 25-1 94 days 1.60 305 Dry

Average 340 19-2 94 days 1.36 115 Dry 22-2 96 days 1.75 120 Dry 25-2 97 days 1.33 130 Dry

Average 120 Near the Middle of the Specimen

20-1 89 days 2.99 330 Moist 23-1 91 days 2.95 230 Moist 26-1 94 days 3.03 290 Moist

Average 285 20-2 94 days 2.53 165 Dry 23-2 96 days 2.91 230 Dry to moist 26-2 97 days 2.65 180 Dry

Average 190 Near the Bottom of the Specimen

21-1 89 days 4.60 24-1 91 days 4.50 27-1 94 days 4.54

Average 21-2 94 days 3.50 24-2 96 days 3.51 27-2 97 days 3.53

Average

Courtesy Bureau of Reclamation.

Table 17-10. Summary of Tensile Strengths of Vertical Construction Joints for Specimens 1

Through 4 at 28, 60, and 90 Days Age, Klang Gates Dam.

Specimen No.

1 2 3 4

2 3 4

2 3 4

Tensile Strength Ib/in. 2

28 Days 60 Days 90 Days

Near the Top of the Specimen

150 225 340 170 290 120 185 215 335 225 415 370

Near the Middle of the Specimen

185 300 285 180 315 190 180 200 295 270 415 465

Near the Bottom of the Specimen

290 320 360 155 360 145 235 365 370 265 350 410

Courtesy Bureau of Reclamation.

Number of Cores That Broke at

Construction Joint

40f9 All

60f9 None

All All

6 of 8 None

None 80f9 20f9 40f9

390 Moist-Broke in concrete 370 Moist-Broke in concrete 325 Moist-Broke in concrete 360

185 Dry 135 Dry 110 Dry 145

CASE HISTORY-STEWART MOUNTAIN DAM

Stewart Mountain Dam, shown in Fig. 17-20, was con­structed in 1928-30. It is located 32 miles (51 kIn) north­east of Phoenix, Arizona, and creates a reservoir with a capacity of about 70,000 acre-feet (86.3 million m3 ). The project is a composite of several individual structures. The central structure is a thin arch dam 207 ft (63.1 meters) high with a 583-ft (177.7 -meter) crest length, flanked by two thrust blocks 90 and 105 ft (27.4 and 32.0 meters) high supporting the left and right arch abutments, respec­tively. Attached to each thrust block is a gravity concrete wing dam. The right gravity section connects to the right abutment, while the left gravity section connects to the main spillway of nine gates. A general plan, elevation, and sec­tion are shown in Fig. 17-21.

Field reports in 1937 discussed excessive movements and cracking on the right half of the arch's downstream face, top of the thrust blocks, deck above the gravity sections, and parapet walls. Deflections were sufficient to cause the field staff to cut the reinforcement between the dam and powerhouse. Laboratory tests on extracted cores confirmed in 1943 that the concrete was experiencing a phenomenon

ASSVAlCD

DRA I NACE OuT/.. (T _

rop OF 01..0 OA W

DOWN STR EA loll ELEVATION OEVELOPEO AlO~G AXIS

ARCH DAM DESIGN AND ANALYSIS 525

.. - SPLI r CL AY PIPE LOCATED ON EXISTING LI~T LI ~ Es .d~D CONTRACTiON JOIN rs

J-~ ___ .Q5SUItttCO f'XCAVATlOM

4 ~ SPl..lr CLAY PIPE ro Bf LOCAT(,O AT LOW

POINTS ALON G liFT LI NES ro F'ACIUTATC ORAI NA GE

DRA I NAGE OUr L(T AT LOW POINT

ENT/lANCE

.. spur CLAY IIlPE

S£CUFlf' PIPE ~ND CA I. K TO PRE VENT

CLOGGING OR DISP<AC"AI"N r DURING CO~CRETE PLACEME~r

TYPICAL SPLIT P I PE I NSTil. LLATIO N

SECTION A-A

Figure 17-17. Arrangement of split clay pipes. Source: Bureau of Reclamation (1977).

described as alkali-aggregate reaction. The concrete ma­terials contain aggregate types that, when mixed with high­alkali cement in the mass concrete, produce a silica gel that expands the concrete. A plot of crest deflections, in Fig. 17-22, shows dramatic upstream movements of the dam due to expansion from the alkali-aggregate reaction.

Figure 17-18. Split clay pipe installation. Figure 17-19. Modified Klang Gates Dam.

526 ADVANCED DAM ENGINEERING FOR DESIGN, CONSTRUCTION, AND REHABILITATION

Figure 17-20. Stewart Mountain Dam.

Over the years, the slow, progressive material degrada­tion and accompanying structural cracking were continu­ously monitored with a variety of field tests and evaluations by various engineers. In general, bond in the lift joints var­ied from mostly poor to excellent. These conclusions were based on laboratory tests of extracted cores, some up to 10 in. in diameter. In spite of apparent distress, a 24-in.-di­ameter calyx hole drilled in 1948 failed to show the con­tinuity of a crack very evident on the downstream face. Soniscope tests made by the Portland Concrete Association in 1949 indicated satisfactory concrete. About the same time, movement of the arch was observed to have slowed. Simultaneously with the field and laboratory tests, studies were made to determine various remedies to prolong the life of the structure. The downstream face of the arch was whitewashed to reduce the temperature effects of solar ra­diation. A 6-ft (1.83-meter) layer of concrete was added to the downstream face of the left thrust block for increased stability. For the next 25 years, the Stewart Mountain Dam complex was operating apparently as intended while being carefully monitored. There was renewed interest in the concrete materials and structural integrity in the late 1970s when additional 6-in. cores were tested. Results indicated strength and modulus increases, compared with core tests made in 1964. In 1981, an engineering consulting team examining the arch concluded that the structure was safe for static loads. The Bureau began further studies, tested more cores, and concluded in 1983 that modification was needed for stability during flood and earthquake events. Various plans considered ranged from complete replace-

ment with a new concrete dam downstream to enhancing stability with massive concrete placements against the ex­isting components. Another plan considered and aban­doned was to remove the top 40 ft ( 12 meters), which ap­peared to bound the upper material degradation and structural distress, and restore that part of the arch with fresh concrete. In 1985, a consulting board consisting of materials engineers, in concert with core tests at the Con­crete Technology Laboratory, a division of PCA, con­cluded that material degradation was caused by alkali-ag­gregate reaction, that minor amounts of reaction would continue but would cause relatively small movements, that the lift surfaces may never have had uniformly good bond, that epoxy injection was not feasible, and that the project could expect another 50 years of serviceable life from the existing concrete.

This evaluation, coupled with the Bureau's own find­ings, resurrected the concept of using posttension cables. At this time (1988), modifications by posttensioning and concrete overlay are in progress.

Arch Dam

Structural analyses of normal loadings indicated that stress and stability met Bureau criteria. 56 A reevaluation of the hydrologic/hydraulic conditions demonstrated that the river system could not satisfactorily control the new floods, and overtopping of Stewart Mountain Dam was very possible. Subsequent analyses for this load combination cast doubt on the stability of the upper 5-ft-high lift blocks. Concur-

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rently, results from dynamic analyses using the most recent seismotectonic evaluation of the site also suggested insta­bility of the upper lifts. Based on these studies, posttension cables were determined as the most viable method of res­toring structural integrity to the arch dam. Parameters that influenced the cable design are horizontal and vertical earthquake components, interaction of the lifts and blocks, lift surface friction, arch geometry, open vertical contrac­tionjoints, cable orientation and anchorage, and static loads of reservoir pressure, temperature, gravity, and uplift.

Foundation considerations included geological units of varying quality and orientation, such as the high-angle dip of the fault zone along the stream channel, the rock types consisting of hard medium-grain quartz diorite cut by dikes of hard medium-grain granite, and near the spillway chute a rock type identified as granite mylonite. The right gravity section, right thrust block, and right portion of the arch are mostly founded on slightly weathered to fresh diorite and granite. Rock in the left abutment is more fractured and sheared than that in the right abutment. Foundation stabil­ity is based upon accepted procedures for estimating rock properties from in-situ laboratory tests and geology logs. Based on similar accepted procedures for estimating rock properties, foundation moduli were determined and in­cluded in the finite-element model of the dam and foun­dation and in the design of the cables.

The current concept is to place on the existing arch crest a 3-ft-thick cap of reinforced concrete, in which a recess is

formed for the upper anchorage head. Cables will be low­ered through an 8-in.-diameter hole drilled vertically through each block at least 30 ft (9 meters) into the foun­dation rock. Pullout tests on the left abutment, to confirm anchorage length, are planned prior to installation in the dam. Axis spacing of 10 ft (3.05 meters) will require that each group of tendons be tensioned in the proper sequence and percent of final pull to avoid eccentric or abnormal loading. The object of the posttension cables is to restore the structural continuity of each vertical block by mechan­ical means.

An elaborate, redundant, structural behavior instrumen­tation network has been established to monitor the arch dis­placement, especially from the posttension cables. Period­ically, for the first 5 years, each cable will be tested for relaxation, which will be primarily due to the plastic be­havior of the concrete. The unrestrained vertical growth that has occurred over 60 years will require a period of time to recover. The exact time period will be a function of the rate of change in deflection measurements and pull of each cable set.

The new structural improvements of Stewart Mountain Dam include;65

• Drains installed through the downstream face of the arch and into the dam foundation.

• Collector drain systems for the thrust blocks and grav­ity sections.

• Grouting of the foundation under the left thrust block. • Foundation drains under the left thrust block and left

gravity section. • Concrete overlays on the thrust blocks and gravity

sections. • A 3-ft-thick concrete overlay on top of the dam. • Posttensioned cables through the arch dam from the

crest into the foundation. • Posttensioned cables through the left thrust block from

the downstream face into the foundation.

In the modification of the arch dam, the cable design for a lO-ft length of the crest provides for a 750-kip design load, 896-kip lock-off load, 1024-kip test load, and 1280-kip GUTS (Guaranteed Ultimate Tensile Strength). Final cable spacing ranged from 8.5 ft to 10.0 ft.

Hydrologic/hydraulic concerns will be satisfied with re­habilitation of the existing spillway to the left of the left gravity wing and a new auxiliary spillway to the right of the right gravity wing.

MULTIPLE-ARCH DAMS

Although few multiple-arch designs are being considered for new projects, this type of dam is important because of continuing interest in reevaluation and modification of ex­isting structures, particularly in regions with cold climates or seismic activity.

History

In the 1920s and 1930s, when multiple-arch dams had many advocates, their economic advantages could be well dem­onstrated in certain cases. Their merits were most evident in remotely located projects where requirements for labor could be weighed favorably against costs of transporting materials.

The typical multiple-arch dam consists of inclined arches that span between buttresses. Stability depends primarily on the water pressure against the sloping upstream face. In most of these dams, the arch barrel slope is between 45° and 50° with the horizontal. Buttress spacing generally is 50 ft to 60 ft (15 to 18 meters) in the most common design. The typical dam is a continuous structure in which the sta­bility of each element may depend on the stability of ad­jacent elements.

In the mid-1920s, there was concern about the stability of slender buttresses that had been used in some of the ear­lier designs. These buttresses were typically only a few feet thick, even in some dams higher than 100 ft (30 meters). In common practice, therefore, bracing was placed be­tween buttresses to improve lateral stability. In many cases, horizontal struts spanned the bays. In some of the higher dams of this type, flanges or pilasters are integral parts of

ARCH DAM DESIGN AND ANALYSIS 529

the buttresses, either in lieu of or in addition to struts. Most multiple-arch dams are of reinforced-concrete construc­tion. The reinforcing steel in the struts is usually continu­ous through at least three bays. In some dams it is contin­uous throughout the structure.

In dams with single-wall buttresses, corbels were com­monly used to transfer the load from the adjoining arches. In 1924, Noetzli57 proposed a double-wall or hollow but­tress, which was readily accepted. Load transfer in this de­sign was simplified by making the outer face of the buttress a continuation ofthe intrados ofthe arch (Fig. 17-23). The face slab joining the two buttress walls was designed to carry the water load and the moments introduced in the end of the buttress by the arch. Each buttress was to be stable by itself, thus eliminating the need for struts and/or pilas­ters. Cross walls and braces between the buttress walls en­sured unity of action. The buttress could be designed for practically any height, by placing the two walls far enough apart. Heavy reinforcing bars were placed at the junctions of the members to tie them together, and reinforcement was also placed in the walls near the outside faces. The com­mon practice was to design the buttresses so that there would be no tension.

The individual arches were designed by methods for sin­gle-arch dams, and typically they were reinforced if the analysis indicated tensile stresses. The slope of the up­stream face, the seasonal temperature range, and the arch

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530 ADVANCED DAM ENGINEERING FOR DESIGN, CONSTRUCTION, AND REHABILITATION

central angle governed design of the multiple-arch system. The economical central angle was determined on the basis of specific site conditions , and generally ranged from 150 0

to 180 0 , depending on temperature variations and the up­stream slope. Angles close to the lower limit were econom­ical where the temperature range would be small,and 180 0

was the most economical for extreme temperature varia­tions. Central angles as small as 100 0 were adopted in ear­lier dams. In later designs, there was a trend toward central angles approaching 180 0 to reduce lateral forces on the buttresses.

As the concept of a dam composed of multiple arches evolved, designs became more sophisticated. An outstand­ing representative of advanced engineering of this type is Daniel Johnson (Manicouagan No.5) Dam, 703 ft (214 meters) high and 4311 ft (1314 meters) long, located in northern Quebec, Canada. This monumental structure, completed in 1968, consists of 13 arches supported by 12 buttresses, with the highest arch spanning 530 ft (161.5 meters) . Daniel Johnson Dam recently has been the subject of study with a view toward enhancement of its perfor­mance under extreme temperature variation.

Florence Lake Dam

A notable example of the multiple-arch type, which serves to illustrate structural performance and current analytical approaches, is Florence Lake Dam, on the South Fork of the San Joaquin River, on the western slope of the Sierra Nevada in Fresno County, California. The structure is 3156 ft (962 meters) long, and is composed of 58 arches and 59 buttresses (Figs. 17-24 through 17-27). The top is at El. 7329.0 ft above sea level. The maximum arch height is approximately 150 ft (46 meters). The dam was completed in 1926 by Southern California Edison Company.

As shown in Fig. 17-26, the alignment of Florence Lake Dam comprises five tangents joined at various angles by four gravity sections, one of which incorporates a spillway with two drum gates. The dam was founded almost entirely on a ridge composed of sound granodiorite.

In adoption of the multiple-arch design, full considera­tion was given to the potential effect of cold weather on the concrete. Florence Lake was planned to be emptied by about December 1 of each year, and the most severe tem­peratures do not prevail until after that date.

Buttress stiffness was secured by three counterforts on each side and a T-section at the downstream edge. The head of the buttress was enlarged by corbels, further en­hancing lateral stability. An upstream face slope of 9: 10 was adopted; downstream buttress slope is 2: 10. No struts were placed between buttresses, and the spacing of the but­tresses is 50 ft (15.24 meters).

The heavy gravity buttresses at three angle points were designed to take the thrust from a fully loaded arch on each side and the water pressure on the face between arches, without any tension being produced.

The designers of the dam reasoned that dependence on cross ties between buttresses for their lateral stability was objectionable because in case of failure of anyone or more of the ties in anyone span, side thrusts or pulls of consid­erable magnitude would be set up on the remaining but­tresses from changes in temperature or shrinkage. This would become particularly pronounced in dams of great length, where unrestrained shortening or lengthening of the struts under widely varying temperatures would be large in comparison with permissible side deflections in the but­tresses . Therefore, it seemep advisable to design the but­tresses so that each one was stable in itself, which could be done by using either the hollow or the counterfort type. The advantages of the hollow type did not appear to offset

Figure 17-24. Florence Lake Dam viewed from right abutment. Courtesy : R. A. Burks and T. A. Kelly , Southern California Edison Company.

Figure 17-25. Florence Lake Dam viewed from downstream. Courtesy: R. A. Burks and T. A. Kelly, Southern California Edison Company.

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531

532 ADVANCED DAM ENGINEERING FOR DESIGN, CONSTRUCTION, AND REHABILITATION

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its disadvantages in comparison with the counterfort type for heights up to 150 ft (45 meters). The designers judged the main disadvantage of the double-wall type to be the development of large bending stresses in the buttress head under varying conditions, which they saw as requiring a rather massive concrete section at that point. The quantity

of concrete, the form area, and the amount of excavation were also seen as greater for a hollow buttress of moderate height than for the counterfort type.

The arches were designed with five centers in a normal plane. Their compressive stress was not to exceed 650 psi (4.48 MPa) with full water load, weight of arch, rib short-

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FLORENCE LAKE DAM

Figure 17-27. (Continued)

ening, and temperature variation from a drop of 50°F to a rise of 25 OF. Under the same loading, but without temper­ature change, the design provided that the maximum com­pressive stress would be limited to 500 psi (3.45 MPa). The arches were analyzed by the elastic theory as fixed

ARCH DAM DESIGN AND ANALYSIS 533

arches. Under partial load and empty condition, a temper­ature drop of 80°F was assumed, and the reinforcing steel was calculated to carry all tension with a maximum stress of 16,000 psi (110 MPa) , and the concrete to carry a compression of 650 psi (4.48 MPa). Extra reinforcement was used in the top ofthe arch. The maximum compressive stress in a buttress was limited to 400 psi (2.76 MPa), and the maximum shear stress to 125 psi (0.86 MPa). The arches were founded in trenches in the granodiorite. Heavy rail steel was grouted into the foundation at the arch bases.

Florence Lake Dam thus was designed conservatively, using well-accepted practices of that time. However, it showed surficial freeze-thaw effects in its early years. Cores were first taken in 1940. Subsequently, there was some apparent decline in strength, but this was slowed by diligent protective work. In 1940 the average of all com­pressive test strengths was about 3474 psi (23.95 MPa). The records indicate that the average strength of cores was about 2700 psi (18.62 MPa) in 1945 and 2900 psi (20.00 MPa) in 1949. Cores tested in 1961-62 had compressive strengths averaging 4330 psi (29.86 MPa), with a range from 2539 psi (17.51 MPa) to 7128 psi (49.15 MPa). The strength assumed in the original design was 2000 psi (13.79 MPa).

The program for protection of the concrete has been comprehensive. Following tests of many coatings, Asbes­tile compound together with asphaltic felt sheathing was applied to the upstream face of Arch 8 in 1940 and to other arches in succeeding years. The surface was prepared in advance by removing deteriorated concrete and restoring the section with gunite. Reflecting paint was applied to the finished coating to reduce temperature fluctuations. These measures and more recent poly sulfide applications over gunite-restored surfaces evidently have been so successful that the deterioration of early years has been effectively arrested. This has been checked by regular ultrasonic test­ing.

Arch 21 was instrumented for strain and deflection mea­surements in 1950, this arch being selected as representa­tive of all the arches. The principal result of the measure­ments in that year was that no departure from normal arch action was indicated. The arch was judged to be function­ing well, with no serious weakness. Additional measure­ments of that arch in 1962, 1973, and 1979 verified this judgment.

A factor of primary importance at multiple-arch dams of appreciable height, such as Florence Lake Dam, is stability against seismic forces parallel to the axis. Although this dam does not have lateral bracing between buttresses, it does have substantial stabilizing elements, including coun­terforts, angles in the alignment ofthe arch series, and the considerable mass at the spillway section. In the periodic evaluations conducted over the years prior to the 1980s,

534 ADVANCED DAM ENGINEERING FOR DESIGN, CONSTRUCTION, AND REHABILITATION

the dam was consistently judged to be structurally capable. However, more recent assessments of the potential for seis­mic activity in the region led to the decision to make an analysis by advanced methods. The dam is about 19 miles (30 km) southwest of the Hilton Creek-Round Valley Fault Complex and 31 miles (50 km) west of the Owens Valley Fault Zone.

Seismic Analysis. The new study, made by Dynamic Analysis Corporation (DAC) of Saratoga, California in 1982, was based upon a three-dimensional finite-element model of the River Section (Arches 50 to 58, inclusive), assuming the high arches in that section to be most critical in dynamic response. It was further assumed that the River Section could be uncoupled from the rest of the dam, be­cause of the rigid comer buttress, No. 49. The effects of two maximum credible earthquakes, alternatively, were considered: near-field Earthquake A (M 6.0 at 20-30 km) with maximum acceleration of 0.25 g, predominant fre­quency range of 10 to 15 Hz, and duration of 10 seconds; and far-field Earthquake B (M7.5-8.0 at 50 km) with max­imum acceleration of 0.20 g, predominant frequency range of 5 to 10 Hz, and duration of 30 seconds.

This was the second multiple-arch dam seismic stability analysis conducted by DAC for Southern California Edison Company, the first being for Gem Lake Dam, an 80-ft (24.4 meter)-high dam on the eastern slope of the Sierra Nevada (Fig. 17-28). The analytical procedures and the scope nec­essary to describe the stress state and related seismic sta­bility of Florence Lake Dam evolved from the Gem Lake study.

The analysis of Florence Lake Dam was made by using the SPAR computational system developed by Engineering Information Services Inc. of Saratoga, California. SPAR is

designed primarily for stress, buckling, and vibrational analysis of large-scale linear finite-element systems. It can also be applied in an iterative mode in obtaining certain classes of nonlinear solutions.

Two River Section finite-element models were con­structed. The first, a complete system model, was used in all static load analyses. To minimize computer cost, a modal synthesis approach to solving the system modes and frequencies was used. A second model, with five subsys­tems, was generated (Fig. 17-29).

A side view of the finite-element grid for Arch 53 is shown in Fig. 17-30. The grids for Buttress 54 and Buttress 54 Trailing Edge stiffener are illustrated in Figs. 17-31 and 17-32. For both static and dynamic analyses, "pinned" boundary conditions were applied at the base of each arch and buttress. The massive granodiorite foundation was considered to be rigid relative to the dam. The complete model contains 3453 nodes, 458 beam elements, 192 three­mode shell elements, and 2770 four-mode shell elements.

The "effective" modulus ofthe reinforced concrete was difficult to estimate. Compressive tests of core specimens were considered as indicative; however, it was recognized that the overall field performance could be substantially dif­ferent from test results. A set of static measurements was used in deducing the "effective" modulus of the arches, drawing from the data obtained in the deflection and strain measurement program on Arch 21. A comparison was made between measured and computed Arch 21 deflections due to temperature and hydrostatic loading on a radial plane located at approximately midheight of that arch. A maxi­mum measured deflection of 0.076 in. with full reservoir corresponds to a modulus of 1.7 X 106 psi (11.72 GPa). This "effective" modulus was assigned to all arches and buttresses.

Figure 17-28. Gem Lake Dam. Courtesy: R. A. Burks and T. A. Kelly, Southern California Edison Company.

ARCH DAM DESIGN AND ANALYSIS 535

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In the Arch 21 study, both "pinned" and "fixed" boundary conditions at the springlines were examined. Comparison of measured and computed radial deflections showed close agreement for "pinned" boundary condi­tions. Therefore, "piano hinge" connections of arches to buttresses were assumed. This assumption is further sup­ported by the existence of a "cold joint" near the spring­line.

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536 ADVANCED DAM ENGINEERING FOR DESIGN, CONSTRUCTION, AND REHABILITATION

For multiple-arch dams, structural damping depends on the material and the joints and cracks. Damping due to hy­drodynamic pressure on the arches also must be consid­ered. Both material and joint or crack damping effects usu­ally are nonlinear, in that increased damping is associated with increased strain levels. For complex multicomponent structures, the state of the art has not been developed ad­equately to enable accurate analytical prediction of damp­ing effects. To ensure an appropriate degree of conserva­tism in the Florence Lake Dam dynamic analysis, the composite, effective, equivalent-linear, modal damping was estimated at 3 % of critical for the strain levels in the case with full reservoir and maximum credible earthquake. For half-full reservoir, a modal damping of 2.75% was esti­mated on the basis that damping would be predominantly of material type rather than hydrodynamic.

Hydrodynamic loading was applied according to the in­compressible fluid, "added mass" concept. The limita­tions of this approach are: (1) the fluid pressure response is independent of the elastic response of the arch; (2) the in-phase component of pressure is frequency-independent (incompressible flui.d); (3) the out-of-phase component of pressure is included as external damping in the modal equations-of-motion; and (4) surface waves and cavitation effects are not included. The major difficulty in applying the "added mass" approach is in defining the in-phase component of "rigid arch" pressure. Both longitudinal and lateral earthquake directions are of interest. Because no re­lated data had been developed for mUltiple arches, use was made of the pressure distributions for single arches re­ported by Zienkiewicz and Nath.58 Because of the rigid sidewalls that bound the fluid laterally, the pressure is asymmetric about the single-arch centerline. Application of this pressure distribution to the multiple-arch configu­ration would result in pressure discontinuities at the inter­faces between interior arches. Because data on applicable lateral pressure distributions were not available, and the in­phase component of pressure in that direction was consid­ered to be relatively minor, the Florence Lake Dam anal­ysis was made with "added mass" included in the longi­tudinal direction only.

To avoid the computer cost of calculating the maximum principal stress at each time point, only a short segment including the peak value was used in making spatial plots of maximum peak tensile stress. For critical elements within the arches and buttresses, complete time histories of principal stress were computed.

Stress response in the Florence Lake Dam arches and buttresses was obtained for the four combinations of Earth­quake A, Earthquake B, reservoir full, and reservoir half­full. For maximum tension in the arches, the case with Earthquake B (lateral direction) and reservoir full was found to be critical. High tensile stresses were indicated near the tops of Arches 52, 53, 54, and 55, as shown in Fig. 17-

33. A maximum tension of 581 psi (4.01 MPa) was indi­cated in the intrados at the top of Arch 52. The maximum tension in the buttresses and in the buttress stiffeners was about half of that developed in the arches. Maximum but­tress tension was 222 psi (1.53 MPa), and maximum stif­fener tension was 265 psi (1.83 MPa).

To assess potential damage, an ultimate-load criterion was applied to maximum shear stress, and a repeated-load criterion was applied to combined static and dynamic ten­sion. It was recognized that either the ultimate-load or re­peated-load criterion would have limitations in the evalu­ation of complex and redundant structures such as multiple­arch dams.

Large margins of safety were indicated when the maxi­mum shear stress was compared with the shear strength of the concrete. All compressive stresses were well within ac­ceptable limits.

The repeated-load criterion developed for Florence Lake Dam was based on the cyclic compressive failure data of Hatano and Watanabe.59 A 6.0-Hz cyclic rate was selected as representative of this multiple-arch structure's dynamic response. A static compressive strength of 3000 psi (20.7 MPa) was assumed for both arches and buttresses. The re­sulting compression damage criterion was extrapolated to the range of 1 to 100 cycles. A corresponding tension dam­age graph (Fig. 17-34) was drawn by scaling the compres­sion data by a 15 % strength ratio. Tensile strength varied from 490 psi (3.38 MPa) at a single peak to 434 psi (2.99 MPa) at 100 cycles. The significant feature of the damage criterion is that repeated cycles of loading less than the dy­namic strength are taken into account.

Using this criterion, potential damage was assessed. Only tension damage was considered because the maxi­mum compressive stresses were well below the damage level. Of the combinations of static and dynamic loading studied, only the case with reservoir full and far-field Earthquake B resulted in potential damage. This was in localized areas near the top of the arches. No exceedances of the damage criterion were indicated for the buttresses or buttress stiffeners. For the loading with reservoir full and Earthquake B, potential damage surfaces, defined as any surface where one or more tensile peaks would exceed the criterion of Fig. 17-34, are shown crosshatched in Fig. 17-33. The maximum number of exceedances (Arch 52 intra­dos) was three, with a maximum overstress of 20%, as indicated in Fig. 17-34. For other combinations of static and dynamic loadings, no exceedances were indicated.

The repeated-load damage criterion was derived from available data on cyclic compressive loading failure of plain concrete cylinders. The ideal analysis would entail moni­toring and prediction of crack initiation and crack propa­gation throughout the earthquake time history, accom­panied by redefinition of structural properties and redistribution of stresses. Such techniques are beyond the

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Figure 17-33. Florence Lake Dam, maximum combined static and dynamic tension, Arches 52, 53, 54 , and 55 (intrados) with full reservoir and far-field

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Figure 17-34. Florence Lake Dam, maximum number of tensile stress exceedances, Arch 52 (intrados) with full reservoir and far-field Earthquake B.

537

538 ADVANCED DAM ENGINEERING FOR DESIGN, CONSTRUCTION, AND REHABILITATION

present state of the art of finite-element analysis of concrete dams; so the damage criterion applied in the Florence Lake Dam evaluation can be regarded as a reasonable attempt to draw an envelope of possible extreme response.

As the critical tension is caused by bending about the axis of the arch barrel, the cracking would be expected to be essentially vertical. The cracks would tend to close as the concrete surface is subjected to the opposite or com­pressive phase of the loading cycle. As can be seen in Fig. 17-33, the high tensile stresses near the top of Arch 52 attenuate rapidly in the downward direction to values be­low the damage threshold. The momentary cracking that might occur would be localized, and downward crack prop­agation would be unlikely. No tension was indicated under static loading in the areas that might be cracked during dy­namic loading. No horizontal cracking was indicated by the analysis; so shear failure would be unlikely.

Conclusion. In evaluating the structural analyses that have been made of Florence Lake Dam, it must be ac­knowledged that there are unknowns that are not amenable to numerical resolution, even by techniques such as those employed in the study by Dynamic Analysis Corporation. Although the basic assumptions in that study appear rea­sonable, the tensile strength criterion is necessarily approx­imate. Tests of specimens taken from the dam have indi­cated compressive strengths ranging from about 2500 psi (17 MPa) to 7000 psi (48 MPa). On that basis, tensile strengths also could range widely. On the other hand, fa­vorable elements not reflected in the finite-element analysis are the resistances offered by the reinforcing steel in the arches and buttresses and by the concrete walkway on top of the dam. These could be important in limiting cracking that the analysis showed to be possible in the tops of the high arches.

The dam presently is believed to be essentially free of cracking that would limit its structural capability. Calcu­lated maximum static compressive stresses in the arches and buttresses in the DAC study were 481 psi (3.32 MPa) and 712 psi (4.91 MPa), respectively. These are amply lower than the allowable static compressive stress of 1000 psi (6.90 MPa), if taken as one-third of the assumed static compressive strength of 3000 psi. Maximum static tensile stress, in the intrados at the base of Arch 53, was computed to be 184 psi (1.27 MPa). This also is evidently within acceptable limits.

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