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DEPARTMENT OF THE ARMY EP 1110-2-12U.S. Army Corps of
Engineers
CECW-ED Washington, DC 20314-1000
PamphletNo. 1110-2-12 30 September 1995
Engineering and DesignSEISMIC DESIGN PROVISIONS FOR ROLLER
COMPACTED CONCRETE DAMS
1. Purpose
The purpose of this engineer pamphlet (EP) is toprovide
preliminary guidance and direction for theearthquake-resistant
design of new roller compactedconcrete (RCC) dams, and for the
evaluation of safetyand serviceability of existing RCC dams
subjected toearthquake loading.
2. Applicability
This EP applies to all HQUSACE elements andUSACE commands having
responsibilities for thedesign of civil works projects.
3. Discussion
a. This EP presents preliminary guidanceconcerning the design of
new RCC dams and theevaluation of existing RCC dams located in
zones ofhigh seismic activity. References are included inAppendix
A.
b. Appendices B-D present examples ofapplying this guidance to
the design of a new RCCdam.
c. Both the preliminary guidance containedherein and the example
problems are based onEM 1110-2-2200 and ER 1110-2-1806. Both of
thesedocuments are under revision and the final guidancecontained
in these documents may vary somewhatfrom the provisions of this EP.
Draft copies of thesedocuments may be obtained from CECW-ED for
usein the design of RCC structures.
d. A dynamic stress analysis shall be performedas part of the
design procedure for all new RCCdams, or the evaluation of existing
RCC dams,located in areas of strong seismicity. Dams shall beshown
capable of satisfying general performancerequirements for design
earthquake seismic eventsdescribed herein. Linear-elastic analysis
methodsshall be used in performing dynamic stress analysis.
e. Consultation and approval of CECW-ED arerequired prior to
performing a nonlinear dynamicstress analysis based upon the theory
of fracturemechanics to qualify a new design or to evaluate
anexisting RCC dam with regard to dam safety.
FOR THE COMMANDER:
ROBERT H. GRIFFINColonel, Corps of EngineersChief of Staff
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DEPARTMENT OF THE ARMY EP 1110-2-12CECW-ED U.S. Army Corps of
Engineers
Washington, DC 20314-1000
PamphletNo. 1110-2-12 30 September 1995
Engineering and DesignSEISMIC DESIGN PROVISIONS FOR ROLLER
COMPACTED CONCRETE DAMS
Table of Contents
Subject Paragraph Page
Chapter 1IntroductionGeneral. . . . . . . . . . . . . . . . . .
. . . 1-1 1-1References . . . . . . . . . . . . . . . . . . 1-2
1-1Explanation of Terms. . . . . . . . . . . 1-3 1-1Background. . .
. . . . . . . . . . . . . . . 1-4 1-1Design Philosophy. . . . . . .
. . . . . . 1-5 1-1Design Earthquakes. . . . . . . . . . . . 1-6
1-1Acceptance Criteria. . . . . . . . . . . . 1-7 1-2Important
Factors. . . . . . . . . . . . . . 1-8 1-2Analysis Methods and
Procedure . . . 1-9 1-3Coordination . . . . . . . . . . . . . . . .
.1-10 1-3
Chapter 2Seismic Design CriteriaStability . . . . . . . . . . .
. . . . . . . . . 2-1 2-1Response to Ground Shaking. . . . . . 2-2
2-1Foundation Fault Displacement. . . . 2-3 2-2Refined Dynamic
Analyses Methods 2-4 2-6
Chapter 3Material Properties of RCCSimilarities of RCC and
Conventional Concrete. . . . . . . . . 3-1 3-1Compressive
Strength. . . . . . . . . . . 3-2 3-1Tensile Strength. . . . . . .
. . . . . . . . 3-3 3-1Shear Strength. . . . . . . . . . . . . . .
. 3-4 3-3Modulus of Elasticity . . . . . . . . . . . 3-5
3-3Poisson’s Ratio. . . . . . . . . . . . . . . 3-6 3-4Tensile
Stress/Strain Relationship . . . 3-7 3-5Dynamic Tensile Strength
(DTS) . . . 3-8 3-6Allowable Tensile Stresses. . . . . . . 3-9
3-7
Subject ParagraphPage
Chapter 4Design EarthquakesDefinition . . . . . . . . . . . . .
. . . . . . . 4-1 4-1Operating Basis Earthquake (OBE) . . 4-2
4-1Maximum Credible Earthquake
(MCE) . . . . . . . . . . . . . . . . . . . . . 4-3 4-1
Chapter 5Design Response Spectra andAcceleration Time
HistoriesDefining the Design Earthquake. . . . 5-1 5-1Developing
Design Response
Spectra. . . . . . . . . . . . . . . . . . . . . 5-2
5-1Developing Acceleration Time
Histories . . . . . . . . . . . . . . . . . . . . 5-3 5-1Dynamic
Analysis by Modal
Superposition . . . . . . . . . . . . . . . . 5-4 5-2Types of
Design Response Spectra . . . 5-5 5-2Horizontal and Vertical
Design
Response Spectra. . . . . . . . . . . . . . 5-6 5-3
Chapter 6Earthquake Load CasesLoad Combinations. . . . . . . . .
. . . . 6-1 6-1Dynamic Loads To Be Considered . . . 6-2 6-1Static
Loads To Be Considered. . . . . 6-3 6-1Static Loads Not To Be
Considered . . 6-4 6-2
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EP 1110-2-1230 Sep 95
Subject Paragraph Page
Chapter 7Factors Significantly AffectingDynamic
ResponseEvaluation Procedure and Objectives 7-1 7-1Design Response
Spectra. . . . . . . . 7-2 7-1Dam-Foundation Interaction,
Damping Effect. . . . . . . . . . . . . . 7-3 7-1Dam-Foundation
Interaction,
Foundation Modulus Effect. . . . . . 7-4 7-3Hydrodynamic Effect.
. . . . . . . . . . 7-5 7-5Reservoir Bottom Absorption. . . . . 7-6
7-7Method of Combining Modes. . . . . 7-7 7-8Vertical Component of
Ground
Motion . . . . . . . . . . . . . . . . . . . . 7-8 7-8
Chapter 8Dynamic Analysis Methods andProceduresAttributes of
Dynamic Analysis
Methods . . . . . . . . . . . . . . . . . . . 8-1 8-1Comparison
of Dynamic Analysis
Methods . . . . . . . . . . . . . . . . . . . 8-2 8-3Dynamic
Analysis Procedure. . . . . . 8-3 8-4Preliminary Design of New Dams
. . 8-4 8-5Final Design of New Dams. . . . . . . 8-5 8-6Evaluating
Existing Dams. . . . . . . . 8-6 8-6
Subject ParagraphPage
Appendix AReferences
Appendix BDesign Example Problem
Appendix CDesign Example - Chopra’s Simplified Method
Appendix DDesign Example - Finite Element Method
Appendix ETensile Strength of Roller CompactedConcrete
Appendix FGlossary
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EP 1110-2-1230 Sep 95
Chapter 1Introduction
1-1. General
Roller compacted concrete (RCC) dams are designedin accordance
with EM 1110-2-2200. Theproportions of the RCC dam are derived by
stabilityanalysis in a manner identical to that for aconventional
concrete gravity dam and are governedby the static forces to be
resisted and not by thedynamic forces generated during seismic
activity.After the geometric proportions are determined basedon the
static loads a dynamic analysis is conducted.Zones requiring
superior RCC mixes are established,and vibratory compaction methods
and jointpreparation methods which affect the RCC tensilestrength
are also established based on the criteriaprovided in this engineer
pamphlet (EP).
1-2. References
Required and related publications are listed inAppendix A.
1-3. Explanation of Terms
Abbreviations, symbols, and notations usedthroughout this EP are
explained in the glossary.
1-4. Background
Basic criteria and guidance for the design of RCCdams are
provided in EM 1110-2-2200. ER 1110-2-1806 provides guidance on
analysis methods andprocedures for new designs and an
investigativeprogram for existing dams. ETL 1110-2-301
givesadditional information on specifying earthquakeground motions
for a particular site. ETL 1110-2-303provides guidance on finite
element dynamic analysismethods and on evaluating the severity of
crackingbased on tensile stresses from the linear analysis.EM
1110-2-2006 provides guidance concerning RCCusage and mix
design.
1-5. Design Philosophy
a. Response spectrum analysis.The nonlinear-ities associated
with concrete behavior under seismicloading are difficult to assess
and beyond practicalanalyzing capabilities of most design
offices.Procedures which permit the use of a linear-elastictype of
dynamic analysis adjusted to provide areasonable but conservative
approximation of thenonlinear behavior are adequate in almost all
designsituations. The philosophy of design followed in thisEP will
be to establish the procedures applicable tothe majority of design
situations. This consists ofproviding in some detail the
requirements forperforming the linear-elastic response
spectrumanalysis and the criteria for evaluating the results.
b. Refined analyses.For the few occasionswhere this approach
does not produce a satisfactorydesign or where an existing dam does
not satisfycriteria, the designer is then advised to pursue themore
refined analysis methods. Should the evenmore complex nonlinear
analysis become necessary, itshould be performed under the guidance
of arecognized expert in this specialized field and shouldonly be
undertaken with approval of CECW-ED.
1-6. Design Earthquakes
The linear-elastic response spectrum method ofanalysis is the
simplest dynamic analysis method andprovides adequate results for
most designs. Theground motion is usually defined by design
responsespectra scaled to peak ground accelerations (PGA) forthe
two design earthquakes described below.
a. Operating basis earthquake.The operatingbasis earthquake
(OBE) is defined as the earthquakeproducing the greatest level of
ground motion that islikely to occur at the site during the
economic life ofthe dam.
b. Maximum credible earthquake.Themaximum credible earthquake
(MCE) is defined asthe earthquake which produces the greatest level
ofground motion at the site as a result of the largestmagnitude
earthquake that could reasonably occuralong the recognized faults
or within a particularseismic source.
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EP 1110-2-1230 Sep 95
c. Types of design spectra.Design responsespectra for the OBE
are usually developed using aprobabilistic approach, and design
response spectrafor the MCE are developed using a
deterministicapproach. Design response spectra are
furtherclassified into two types: (1) site-specific or(2) standard.
The seismic zone location of the site,the height of the dam, and
the proximity to activefaults are the factors used to determine if
it isnecessary to develop a site-specific design responsespectra or
if the standard spectra may be used in thedynamic analysis. When
standard design responsespectra are acceptable, Chapter 5 provides
theappropriate spectra along with the PGA values to beused for
scaling. These standard design spectra arebased on the mean level
of the ground motionparameters for the records selected in
thedevelopment of the standard spectra.
d. Ground motion time histories.The morerefined analysis methods
require a ground motiontime history representation of the design
earthquakes.These may be developed using actual past
earthquakeground motion records, synthetically, or by modifyingan
actual record. Ground motion time histories aredeveloped so their
response spectrum closely matchesthe site-specific design response
spectrum.
1-7. Acceptance Criteria
a. Cracking of RCC.The ground motion that isproduced during a
seismic event can cause cracks tooccur in an RCC dam. As cracking
progresses,serviceability is eventually impaired. If groundshaking
is extremely severe, or if strong groundshaking combines with a
foundation fault displace-ment, it is conceivable that continued
propagation ofthe system of cracks could eventually lead to a
failuremechanism where the dam is no longer capable ofcontaining
the pool. This EP establishes acceptancecriteria which maintain
serviceability during an OBE,and provide a reasonable safety factor
againstdeveloping a failure mechanism during a MCE.Because of the
complexity and the great number ofvariables involved in seismic
design, the EP criteriashould be supplemented with the judgment
ofstructural engineers experienced in seismic design.
b. Direct tensile strength.The direct tensilestrength of the RCC
is the design parameter used forestablishing the acceptance
criteria. Unlikeconventional concrete, tensile strength of RCC
depends on mix consistency and placement andcompaction methods
as well as mix proportions.Tensile strength of both the lift joint
and the parentconcrete shall be determined from cores taken
fromtest fill placements for new dam design and from thein-place
RCC for existing dams. Although splittingtensile tests may be used,
the test results shall beadjusted to reflect direct tensile
strength. From thedirect tensile strength, the allowable design
tensilestresses shall be established for both lift joints andparent
concrete by applying adjustment factors toaccount for high strain
rate associated with dynamicloading and certain nonlinear
characteristics of thestress/strain curve. Adjustment factors shall
beselected to maintain serviceability during an OBE andto produce a
reasonable safety factor for a MCE.
1-8. Important Factors
Discussed below are recommendations regardingfactors which are
important because they have asignificant impact on the dynamic
response.Recommendations that differ from those contained inETL
1110-2-303 and ER 1110-2-1806 are identified.
a. Effective damping.The material andradiation damping of the
foundation contributesignificantly to the damping of the
combineddam-foundation system, and must be considered inthe
analysis. This requires calculating an effectiveviscous damping
ratio to reflect the dampingcontribution of both the dam and the
foundation.This will result in a considerably higher dampingratio
for a foundation having a very low modulusthan the damping ratio
used previously.
b. Hydrodynamic effect. Added mass shall becalculated using
standard hydrodynamic pressurefunction curves which consider
compressibility of thewater, stiffness characteristics of the dam,
andreservoir bottom absorption (Fenves and Chopra1986). Appendix D
provides an example showing therequired procedure.
c. Mode combination methods.The completequadratic combination
method (CQC) of combiningmodes shall be used for final design of
dams undercritical seismic design conditions and for evaluationof
existing dams. Critical conditions are consideredto exist when
site-specific design response spectra arerequired by this EP.
Either the square root of thesum of the squares method (SRSS) or
the CQC
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EP 1110-2-1230 Sep 95
method is acceptable for all preliminary designs andfor final
designs under noncritical seismic conditions.Since the modal
frequencies are fairly well separatedin gravity dams, the simpler
SRSS method producesadequate results which are in balance with the
generallevel of precision required for preliminary ornoncritical
analyses.
d. Seismic zone map.The seismic zone map,Figure 5-1, shall be
used in the dynamic stressanalysis phase of the seismic design. The
peakground accelerations for use in scaling standarddesign response
spectra are contained in Table 5-2and are based on the zone map.
The seismic zonemaps and the seismic coefficients contained inER
1110-2-1806 shall be used only in the stabilityanalysis phase of
seismic design.
1-9. Analysis Methods and Procedure
In general a dynamic stress analysis shall beperformed, and the
results shall be evaluated todetermine if the response of the RCC
dam to thedesign earthquakes is acceptable. If the response isnot
acceptable, the design of a new dam may bemodified and reanalyzed
using the same analysismethod, or a more refined analysis method
may beemployed. For an existing dam, progressively morerefined
methods of analysis are employed.
a. Method attributes.There are four attributesthat characterize
a particular dynamic analysismethod.
(1) Material behavior. Options are (a) linear-elastic or (b)
nonlinear behavior.
(2) Design earthquake definition. Options are(a) design response
spectrum or (b) time historyground motion record input.
(3) Dimensional representation. Options are(a) two-dimensional
representation or (b) three-dimensional representation.
(4) Model configuration. Options are(a) Chopra’s “standardized”
model, (b) compositefinite element-equivalent mass system model,
or(c) finite element-substructure model.
b. Computer programs.Various computerprograms are available
which are identified withcertain analysis methods. Also, Chopra’s
Simplified
Method may be either hand-calculated or done by acomputer
program. Some computer programs, suchas the general purpose finite
element programs, allowthe attribute options to be changed so that
one ofseveral possible methods may be employed for thedynamic
analysis. This often allows a transition to amore refined method
without necessarily abandoningall the previous computer model input
effort. Othercomputer programs, such as the EAGD-84 program,and
Chopra’s Simplified Method are single methodprograms since they
have fixed attributes. Chapter 8discusses dynamic analysis methods
in more detail.
c. Preliminary and final design.The two-dimensional,
linear-elastic, response spectrum methodshall be used for the
preliminary design analysis.Either Chopra’s Simplified Method or a
general-purpose finite element program shall be employeddepending
on the design conditions. The simplestfinal design analysis
utilizes a composite finiteelement-equivalent mass system model and
general-purpose finite element program.
1-10. Coordination
A fully coordinated team of structural engineers,geotechnical
and materials engineers, geologists, andseismologists should ensure
that all factors relevant tothe dynamic analysis are correct and
that the resultsof the analysis are properly evaluated. Some of
thecritical analysis and design aspects requiring coordi-nation are
discussed below.
a. Design response spectra.Developing site-specific design
response spectra when required.
b. Tensile strength of RCC.Obtainingrepresentative cores from
test-fill placements for newdams or from the in-place concrete for
existing damsfor use in determining the direct tensile strength
anddynamic tensile strength of both the lift joints and theparent
RCC.
c. Foundation properties.Obtaining explora-tory corings and
evaluating tests to determine thefoundation deformation modulus and
other foundationproperties.
d. Foundation fault displacement.Evaluatinggeoseismic conditions
at the site to determine iffoundation fault displacement is
possible, and to mapthe location, strike, and dip of the potential
faults.
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EP 1110-2-1230 Sep 95
Chapter 2Seismic Design Criteria
2-1. Stability
a. Resultant location and sliding.RCC damsshall satisfy the
overturning and sliding stabilityrequirements for gravity dams
using inertia forcescalculated by the seismic coefficient method as
setforth in EM 1110-2-2200 and ETL 1110-2-256. Theseismic
coefficients shall be as shown on the seismiczone maps provided in
ER 1110-2-1806.
b. Extreme stability conditions.When intenseground shaking
causes serious tensile cracking at thedam-foundation interface, a
nonlinear time historyanalysis shall be performed to evaluate
cracking,potential permanent displacements, and the effectthese
have on sliding stability. Certain stipulationsregarding nonlinear
analyses are covered inparagraph 2-2g.
2-2. Response to Ground Shaking
RCC dams shall be capable of resisting the strongmotion ground
shaking associated with designearthquakes within the allowable
tensile stress designcriteria specified in Chapter 4. Dynamic
stressanalysis methods and procedures are described inChapter 8.
The dynamic analyses shall incorporatethe dynamic characteristics
of the dam, foundation,reservoir, and backfill or silt deposition
whenapplicable.
a. Defining ground motion.The free fieldground motions are used
to define the ground motionthat would be felt at the site due to
two designearthquakes. Free field ground motion associatedwith each
shall be represented by design responsespectra and, when required,
design acceleration timehistories. The design earthquakes are
operating basisearthquake (OBE), and maximum credible
earthquake(MCE). Both are discussed in detail in Chapter 4.
b. Propagation of cracks in RCC.Most damswith earthquake
resistant provisions will probablysurvive the most severe
earthquake shaking possibleat the site with little or no damage,
although highdams located near major faults have
experiencedextensive cracking during major earthquakes (Chopraand
Chakrabarti 1973). Concrete cracking due to
ground shaking combined with cracking due tofoundation fault
displacement could propagate to anextent where a failure mechanism
is formed thusimpairing the ability of the dam to contain the
pool.Criteria defining an acceptable response of the dam todesign
earthquakes are based on initiation andpropagation of tensile
cracking within the RCC.
c. Analyzing response to ground shaking.Theprocess of cracking
and the propagation of the cracksresult in nonlinear behavior of
the dam. There arealso nonlinearities associated with
dam-foundationinteraction and dam-reservoir interaction which
aredifficult to assess. Approximate linear relationshipsaccount for
some of the nonlinear dynamic behaviorand allow the response of the
dam to the designearthquake ground motion to be determined using
alinear-elastic analysis method. Tensile stresses canthen be
evaluated based on tensile strength parametersadjusted to be
compatible with linear-elastic analysismethods.
d. Analysis methods.The simplest of the linear-elastic methods
uses a response spectrum to definethe ground motion as outlined in
Chapter 5. MostRCC dams will be found adequate using this
method.For the few exceptions, the next level of refinementin
determining the dynamic response is the linear-elastic time history
method, and in rare cases anonlinear time history finite element
analysis may berequired.
e. Allowable tensile stress.The tensile strengthof the RCC is
the single concrete material propertyused to evaluate cracking, and
to establish acceptableresponse. Allowable tensile stresses are
defined inparagraph 4-2c and paragraph 4-3c for the OBE andMCE,
respectively.
f. Evaluating time-history response.Whendynamic response is
determined by the linear-elastictime-history method, the allowable
tensile stress is theprincipal criterion for evaluating acceptable
response,but additional criteria are also required to qualifyother
response characteristics such as the number ofstress cycles
approaching or exceeding the allowablestress, and the magnitude and
pattern of theseexcursions beyond the specified limits.
g. Evaluating nonlinear analyses.Whendynamic response is
determined by the nonlineartime-history method, criteria for
evaluating acceptableresponse are based on the theory of
fracture
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EP 1110-2-1230 Sep 95
mechanics. This type of analysis should only beundertaken in
consultation with and as approved byCECW-ED.
2-3. Foundation Fault Displacement
a. General. Most RCC dam sites are notsubject to any significant
differential displacement ofthe ground surface at the
dam-foundation interfaceduring a seismic event. Dam sites should
always beavoided when located near a major active faultsystem with
the potential to trigger sympatheticfoundation displacements at the
site. Occasionally itis not possible to avoid these sites, and it
becomesnecessary to evaluate the response of the dam shouldsuch a
foundation fault displacement occur.
(1) Considerable judgment is required in theevaluation process.
At best, analysis methods forfoundation fault displacement are
approximate and aregenerally unsupported by past observations of
theresponse of existing dams to fault displacementsoccurring at the
dam foundation. Furthermore,considerable judgment is required in
the prediction offuture fault movement and in the magnitude of
thefault displacement. For example, the estimate of themagnitude of
potential fault displacement provided bydifferent experts for a
specific site could vary from afew inches to several feet. This
necessitatesconsulting several geotechnical firms to provide
site-specific fault displacement estimates, and thencarefully
scrutinizing these estimates before finallyestablishing the design
fault displacement.
(2) Experts in plate tectonics, geology,seismology, and finite
element analysis techniquesshould be consulted to provide guidance
for any damlocated on a site subject to foundation
faultdisplacement. Because of the many uncertainties andthe risk
involved, approval by CECW-ED is requiredfor any RCC dam which is
located on a site subjectto foundation fault displacement.
b. Types of faults.Fault slip is the relativedisplacement of two
adjacent tectonic plates withrespect to each other. This refers to
large active faultsystems such as the San Andreas or Hayward
faultsin California. On a smaller scale, the foundation rockmass
beneath a dam contains various discontinuities,joint sets, and
shear and fault zones. Normally this isa system of historically
inactive discontinuities;however, there is a potential for fault
slippage
particularly when triggered by a great earthquake on anearby
large active fault. The three general types offault slips are
strike-slip, normal-slip (dip-slip), andreverse-slip (thrust-slip).
Refer to Figure 2-1 forillustrations of the various types of faults
and how themagnitude of slip is measured. The strike of the faultis
the trace the fault makes with respect to the groundsurface, and it
may be at any orientation with respectto the dam axis.
c. Design fault displacement.The design faultdisplacement (DFD)
is defined as the maximumpossible free field fault slip movement
that couldreasonably occur in the dam foundation as measuredat the
ground surface. The return period that wouldbe associated with the
DFD is similar to that of theMCE. Therefore, the DFD and the free
field groundmotion together specify the site-specific
seismicactivity associated with the MCE. To fully describethe DFD,
three factors must be specified: magnitude,type of slip, and strike
of the fault.
(1) The geology of the dam foundation iscomplex, and the
foundation may be crossed by anumber of discontinuities with fault
displacementpotential. Experts in the fields of geology
andseismology should be consulted to study thefoundation fault
system, determine which faults arecapable of surface displacement,
and finallyrecommend which faults are critical and specify theDFD
for each critical fault.
(2) Normally, foundation fault displacements arenot considered
to occur concurrently with strongmotion shaking associated with the
OBE. The activefault near the dam site that produces a seismic
eventof OBE magnitude is not likely to trigger sympatheticslippage
in the fault system in the dam foundation.The probability of
sympathetic foundation faultdisplacement is normally several orders
of magnitudeless than the recurrence rate for the strong
motionshaking associated with the OBE; therefore, theprobability of
the OBE being accompanied bysignificant foundation displacement is
usuallyconsidered negligible.
(3) On rare occasions, the probability logicdiscussed above may
not apply when considering if itis appropriate to combine
foundation faultdisplacement with ground shaking in specifying
theOBE. For example, unusual geology of thefoundation could make it
susceptible to a reservoir-induced foundation fault displacement or
to other
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EP 1110-2-1230 Sep 95
unusual causes of foundation fault displacement
Figure 2-1. Types of fault slips
discussed later in this chapter. In these situations thestrong
motion shaking accompanying the local faultslip may be nearly as
intense or even more intensethan the gound motion shaking
associated with anOBE produced by a major active fault slip
occurringsome distance from the site. When this is the case,
areduced value of the DFD would be included withfree field ground
motion to describe the OBE.
d. Combined DFD and ground shaking.Stresses associated with the
DFD result from highlycomplex nonlinear behavior; however,
simplified faultdisplacement analysis procedures, such as the
onedescribed below, are normally used to investigateconcrete
stresses that may occur due to fault displace-
ment. Stresses due to ground shaking are determinedby methods
discussed earlier in this chapter. Thus,stresses due to fault
displacement and stresses due toground shaking are obtained from
two separate, inde-pendent, and approximate analyses. The response
tothe design earthquake is then obtained by direct addi-tion of the
two sets of stresses without accounting forany interaction.
Actually, the fault displacement maycause inelastic behavior at the
dam-foundation inter-face, cracking within the RCC, or other
inelasticresponse which changes the dynamic characteristicsof the
dam, which in turn interacts with and effectsthe ground shaking
response. Because these simpli-fied and approximate procedures have
not been sup-ported by nonlinear finite element analyses
thatproperly combine the effects of fault displacement
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EP 1110-2-1230 Sep 95
and ground shaking, they should be used withcaution.
e. Simplified DFD analysis procedure.Thesimplified procedure
described below was used toinvestigate concrete stresses due to
fault displacementin the Auburn Dam in California (U.S. Department
ofthe Interior, Bureau of Reclamation 1980). The damand foundation
are modeled with finite elements withthe mesh geometry adjusted to
allow the fault to beproperly oriented. Refer to Figure 2-2.
Thefoundation model consists of a fixed block withconventional
boundary supports, and a movable blockwith special boundary
conditions that allow forces tobe applied at the boundary parallel
to the fault toproduce the DFD. The fixed and movable block
areseparated by elastic orthotropic elements which allowthe sharp
displacement discontinuity to take place asthe movable block
displaces upward.
(1) The finite element model is first loaded withthe gravity
loads followed by the hydrostatic loads,and finally the movable
block is forced to undergothe DFD. Each loading is applied
incrementally.After each loading increment, tensile stresses
areevaluated and elements are softened in areas wherethe tensile
strength is exceeded. Elements are soft-ened by reducing their
elastic modulus until thetensile stress is eliminated. Most
elements requiringsoftening are located in the foundation because
joint-ing and discontinuities in the rock prevent it fromsustaining
high tensile stress. When the DFD isreached, the extent of the
tensile failure areas isevaluated. The dam tends to bridge over the
fracturezone in the foundation. Resulting stresses induced inthe
RCC are obtained from the finite element analysisfor the final
increment of loading which produced theDFD.
(2) The method of incremental loading and soft-ening of element
properties allows the use of asimplified static, linear-elastic
finite element analysisapproach. Disadvantages of the procedure are
that itgives only an approximation of the complex nonlinearbehavior
associated with fault displacement, it is timeconsuming, and it
requires considerable judgment.
(3) The example shown in Figure 2-2 is typicalfor a normal or
reverse fault where the fault strike isapproximately parallel to
the dam axis so a two-dimensional analysis is adequate. If the
fault strike isnot close to parallel to the dam axis, or for a
strike-slip fault, a three-dimensional analysis is required.
The three-dimensional analysis is even more timeconsuming and
complex, but the principles andgeneral procedure are similar to the
two-dimensionalanalysis described.
f. Acceptable response to DFD.When the seis-mic activity
associated with the design earthquakeconsists of both fault
displacement and ground shak-ing, stresses for the combined
response described inparagraph 2-3d must satisfy the allowable
tensilestress criteria of paragraph 2-2e. Beyond thesetensile
stress requirements, additional consideration isrequired regarding
general performance requirementsof Chapter 4 related to dam safety
and operations inthe event of foundation fault displacement.
Thepotential fault displacement and the effect it has onthe dam
must be evaluated on a case-by-case basis.The analysis procedures
described above forevaluating the effect of fault displacement are
roughapproximations, but they do provide an indication ofthe extent
of the fracture zones that could occur inthe foundation or lower
portions of the RCC dam.The analysis results must be coupled
withconsiderable judgment to determine if this damagecould lead to
the erosion of the foundation or RCCmaterials to the extent that
finally causes anuncontrolled release of the reservoir.
g. Dam failures caused by fault displacements.To help identify
some of the judgment factorsinvolved in evaluating sites with fault
displacementpotential, the following is a brief review of
historicalinformation on dams that failed directly or indirectlyas
a result of fault displacement. Differential dis-placements across
a fault have been recorded due to:triggering of the fault by a
seismic event; a differencein consolidation of materials on either
side of thefault; a reduction in resistance to fault
movementcreated by the lubricating effects of water, or theerosion
of fault materials by flowing water; andincrease in hydrostatic
pressures along the fault.
(1) Earth-fill dams, concrete gravity dams, andconcrete arch
dams have failed due to fault move-ments. Failures of the Baldwin
Hills earth-fill dam,the Malpasset concrete arch dam, and the St.
Francisconcrete gravity dam (James et al. 1988) can all
beattributed in part to forces and movements occurringalong fault
surfaces. Although these forces andmovements were not triggered by
seismic activity, itcan be surmised that if a seismic event had
occurred,it would have likely triggered similar failures.
Theseexamples show that fault movement can cause a
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failure mechanism to form in the dam structure whichresults in
dam failure; however, it is more likely thatthe fault movement
would create flow paths thatcould lead to a release of the
impounded reservoir.Seepage can erode dam or foundation materials
whicheventually results in failure because capability forcontrolled
release of the pool is lost.
(2) An earth-fill dam with a flexible core is nor-mally
considered less susceptible to failure due tofoundation fault
displacement because it would tendto conform to the displaced shape
of the foundation.Although this flexibility of the dam material
willreduce voids and flow paths in the dam and founda-tion it will
not completely eliminate them. Thus, anearth-fill dam is
susceptible to erosion of core orfoundation material from water
flowing through faultsor through voids in the dam or foundation
created byfault movements. For this reason, an earth-fill dam isnot
necessarily superior to a concrete gravity dam inresisting the
effects of fault movement.
h. Defensive design features.Defensive designfeatures which can
be employed in the design of anRCC dam susceptible to foundation
displacement arediscussed below.
(1) The arching action provided by laying out thedam axis on a
curve may better distribute the forceson a gravity dam due to
foundation fault displace-ment, and reduce the tensile stresses and
cracking ofthe RCC. This defensive feature is only effective ifthe
heave of the foundation block is generally in adownstream
direction, and providing the fault move-ment does not occur at
either abutment.
(2) Special sliding joints may also be used toreduce cracking of
the RCC due to fault displace-ment. For example, vertical joints
may be located inthe RCC to accommodate potential strike-slip
faultdisplacements where the strike is generally in
theupstream-downstream direction.
(3) A design feature for controlling the reservoirrelease is to
provide a buttress fill against theupstream face of the dam. This
requires the reservoirwater to pass through a succession of filters
andcrack stoppers in a manner analogous to the behaviorof the
transitions and filters in a zoned embankment
dam. This defensive measure would be effective forflood-control
projects where the reservoir pool eleva-tion is low enough that the
required height of thebuttress fill is economically feasible, and
does notimpair the stability of the dam.
2-4. Refined Dynamic Analyses Methods
a. Need for refinement.When the simplifiedlinear-elastic
analysis methods described above for anexisting RCC dam produce
tensile stresses in excessof the allowables discussed in paragraph
2-2e, morerefined analyses methods shall be pursued before thedam
is judged unsafe. Also, if all practical and eco-nomical
adjustments to the design of a new dam havebeen exhausted in the
attempt to satisfy the allowa-bles based on simplified
linear-elastic methods, themore refined analyses methods may be
pursued tobetter evaluate nonlinear structural behavior.
Refinedanalyses consist of linear or nonlinear time historyanalyses
as discussed in paragraph 2-2d, with someadditional details of the
nonlinear analysis providedbelow. The response produced by refined
analysesshall be evaluated in accordance with the stipulationsof
paragraphs 2-2f and 2-2g.
b. Fracture mechanics.Nonlinear dynamicanalysis is based on
fracture mechanics theory whichis presently in the research phase.
It is also difficultto determine just what level of structural
damage canbe sustained safely by the dam and still consider it
tosatisfy the performance requirements. The nonlinearattribute
requires this type of dynamic analysis beperformed in a time domain
(time history analysis)rather than a frequency domain (response
spectrumanalysis), and use a direct integration solution.
Theanalysis accounts for: energy dissipation by cracking,strength
of cracked concrete, changes in vibrationcharacteristics caused by
cracking, changes in damp-ing, and changes in strength due to
strain rate andloading history.
c. Nonlinear analysis requirements.Because itis very complex,
costly, and requires a considerableamount of judgment to interpret
the results, an expertin fracture mechanics and nonlinear analysis
tech-niques should be consulted to provide guidance whenpursuing a
nonlinear analysis.
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Chapter 3Material Properties of RCC
3-1. Similarities of RCC and ConventionalConcrete
The strength and elastic properties of RCC vary de-pending on
the mix components and mix proportionsin much the same manner as
that for conventionalmass concrete. Aggregate quality and
water-cementratio are the principal factors affecting strength
andelastic properties. Properties important to the seismicanalysis
of RCC dams include compressive strength,tensile strength, shear
strength, modulus of elasticity,Poisson’s ratio, and unit weight.
Except for unitweight, all these properties are strain rate
sensitive,and the strain rates that occur during major earth-quakes
are in the order of 1,000 times greater thanthose used in standard
laboratory testing. Guidanceconcerning the determination of RCC
material proper-ties is given in EM 1110-2-2006 and ETL
1110-2-343.
3-2. Compressive Strength
The relationship between water-cement ratio andcompressive
strength is the same for RCC as forconventional mass concrete.
Normally, for durabilityreasons, the RCC mix will be designed to
provide aminimum strength of 2,000 psi; however, for seismicreasons
higher compressive strengths are oftenrequired to achieve the
desired tensile and shearstrength. The compressive strength at
seismic strainrates will be 15 to 20 percent greater than that at
thequasi-static rates used during laboratory testing
(ACICommittee-439 1969); however, compressive strengthis never the
governing factor in seismic design.
3-3. Tensile Strength
The tensile strength of RCC shall be based on thedirect tensile
strength tests of core samples. For thefinal design of new dams,
cores shall be taken fromtest-fill placements made with the
proposed designmixes, and placed with the proposed consolidationand
joint treatment methods. When an existing damis evaluated for
compliance with the requirements ofthis EP, cores shall be taken
directly from the struc-ture. Cores should be taken vertically so
that testscan be made which reflect weaknesses inherent at lift
joint surfaces in addition to the tests to determine thetensile
strength of the parent concrete.
a. Location of critical tensile stress.Criticaltensile stresses
are located at the upstream and down-stream faces of the dam. The
tensile stress distribu-tion within the dam mass is of interest to
helpestablish zone boundaries for superior, higher strengthRCC
mixes that may be required to control crackingnear the faces.
(1) Usually the tensile stress in the lift joints inthe
direction normal to the joint surface is criticalnear the upstream
face of the dam. This is becausethe direction of the principal
tensile stress near theupstream face is very nearly normal to the
joint sur-face, thus there is little difference between the
jointstress and the maximum principal stress in the parentconcrete.
Since tensile strength of the lift joint isnotably less than the
parent RCC, it will control thedesign near the upstream face.
(2) Near the downstream face, the direction ofthe principal
tensile stress is nearly parallel to theface which results in
significantly higher principaltensile stresses in the parent
concrete compared to thetensile stresses in the lift joints normal
to the jointsurface. The ratio of the tensile strength of
parentconcrete to the tensile strength of the lift joints
variesaccording to several parameters including workabilityof the
mix, joint preparation, and maximum sizeaggregate. Thus, it usually
becomes necessary toinvestigate both the principal tensile stress
and thecomponent tensile stress normal to the lift joints
todetermine which is critical near the downstream face.
b. Preliminary design.For preliminary design,the tensile
strength of the RCC may be obtained fromFigures 3-1 through 3-6 for
the proposed concretecompressive strength (f’c). These figures show
boththe tensile strength of the parent material and thetensile
strength of the lift joint based on the proposedconsolidation and
joint treatment method. Thesefigures were developed from Tables E2
and E3,Appendix E.
c. Tensile strength tests.Splitting tensile testsare easier to
perform and provide more consistentresults than direct tensile
tests. However, splittingtensile test results tends to overpredict
actual tensilestrengths, and should be adjusted by a strength
reduc-tion factor to reflect results that would be obtainedfrom
direct tensile tests. When splitting tensile tests
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are used as the basis for determining the tensile
Figure 3-1. Tensile strength range, RCC, MSA ≤ 1.5 inches,
consistency < 30 seconds vibration, mortarbedding
strength of RCC, the test results shall be reduced by astrength
reduction factor of 75 percent as recom-mended in Appendix E.
d. Factors affecting tensile strength.The tensilestrength of
RCC, as well as of conventionally placedmass concrete, is dependent
on many variablesincluding paste and aggregate strength, aggregate
size,loading history, and load deformation rates. Seeparagraph 3-9
concerning strain rate sensitivity anddynamic tensile strength.
(1) RCC differs from conventionally placed massconcrete due to
the many horizontal planes of weak-ness (construction joints)
created during placement.RCC is placed and compacted in layers
ranging from6 to 24 inches with each layer creating a joint
withtensile strength less than that of the parent concrete.
The joint strength can be improved by placing a layerof high
slump bedding mortar on each lift; however,the resulting joint
strength is always somewhat lessthan the parent concrete. The
consistency of RCCcan also affect tensile strength with lower
strengthvalues for harsh mixes with low paste contents.Refer to
Chapter 2 for additional discussion of thesefactors.
(2) Inherent in some RCC mixes are certainanisotropic material
properties. In the RCC compac-tion process, the flatter coarse
aggregate particles inthese mixes have a tendency to align
themselves inthe horizontal direction. When this occurs,
thestrength of vertical cores will be less, and the strengthof
horizontal cores greater than the average tensilestrength. The
variance from average could be as highas 20 percent, although in
general these effects will
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Figure 3-2. Tensile strength range, RCC, MSA > 1.5 inches,
consistency < 30 seconds vibration, mortarbedding
be small. If the coarse aggregate particle shape indi-cates the
possibility of significant anisotropy, bothvertical and horizontal
cores obtained from the labo-ratory test placement should be
tested.
3-4. Shear Strength
The shear strength along lift joint surfaces is alwaysless than
the parent concrete; therefore, final shearstrength determination
should be based on tests ofrepresentative samples from the dam or
test fill.Both the bond strength and the tangent of the angleof
internal friction can be increased by 10 percent toaccount for the
apparent higher strengths associatedwith seismic strain rates.
3-5. Modulus of Elasticity
RCC will usually provide a modulus of elasticityequal to, or
greater than, that of conventional massconcrete of equal
compressive strength. The modulusof RCC in tension is equal to that
in compression.The static modulus of elasticity, in the absence
oftesting, can be assumed equal to (ACI Committee-2071973):
E 57,000 fc
where E static modulus of elasticity
fc static compressive strength of RCC
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The relationship between strain rate and modulus of
Figure 3-3. Tensile strength range, RCC, MSA ≤ 1.5 inches,
consistency > 30 seconds vibration, mortarbedding
elasticity is as follows (Bruhwieler 1990):
E E(Er)0.020
where E static modulus of elasticity
E seismic modulus of elasticity at thequasi static rate
Erhigh seismic strain rate
quasi static rate
For a seismic strain rate equal to 1,000 times thequasi-static
rate the seismic modulus of elasticity is1.15 times the static
modulus. For long-term load-ings where creep effects are important,
the effectivemodulus of elasticity may be only 2/3 the static
mod-
ulus of elasticity calculated by the above formula(Dunstan
1978). The modulus of elasticity mayexhibit some anisotropic
behavior due to the coarseaggregate particle alignment as discussed
inparagraph 3-3d(2); however, the effects on themodulus will be
small and can be disregarded whenperforming a dynamic stress
analysis.
3-6. Poisson’s Ratio
Poisson’s ratio for RCC is the same as for conven-tional mass
concrete. For static loads, values rangebetween 0.17 and 0.22, with
0.20 recommended whentesting has not been performed. Poisson’s
ratio is alsostrain rate sensitive, and the static value should
bereduced by 30 percent when evaluating stresses dueto seismic
loads (Bruhwieler 1990).
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3-7. Tensile Stress/Strain Relationship
Figure 3-4. Tensile strength range, RCC, MSA > 1.5 inches,
consistency > 30 seconds vibration, mortarbedding
As mentioned in paragraph 2-2b, concrete cracking,crack
propagation, and the energy dissipated in theprocess are complex
and nonlinear in nature. For asimplified linear-elastic analysis, a
constant modulusof elasticity is required. Thus, a linear
stress/strainrelationship is used for the analysis with a
tensilemodulus equal to the modulus of elasticity for con-crete in
compression.
a. Compression and tension differences.Although a linear
relationship is assumed for theanalysis, in actuality the
stress/strain relationshipbecomes nonlinear after concrete stresses
reachapproximately 60 percent of the peak stress (Raphael1984). In
compression this does not cause a problembecause, in general,
concrete compressive stresseseven during a major earthquake are
quite low with
respect to the peak stress or ultimate capacity. Intension, it
is a different matter since tensile stress canapproach and exceed
the peak tensile stress capacityof the concrete and in some cases
cracking willoccur.
b. Tensile stress/strain curve.The actual non-linear
stress/strain relationship for RCC concrete isshown in Figure 3-7.
The assumed linear relationshipused for finite element analysis was
developed fromthe work done by Raphael (1984). The actual
nonlin-ear performance of concrete in tension consists of alinear
region from zero stress up to 60 percent of thepeak stress, a
nonlinear ascending region from60 percent of peak stress to peak
stress (this point onthe curve corresponds to the direct tensile
strengthtest value described in paragraph 3-3c), and a nonlin-ear
descending region from peak stress back to zero
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stress. The last region is termed the “tensile soften-
Figure 3-5. Tensile strength range, RCC, MSA ≤ 1.5 inches,
consistency > 30 seconds vibration, no mortarbedding
ing zone.” In this region, where deformationincreases with
decreasing stress, deformation con-trolled stable test procedures
are required to capturethe stress/strain behavior (Bruhwieler
1990), whereconventional test procedures will cause the strain
tofall off abruptly to zero strain at a point on the curvejust
beyond the peak stress point. The area under thetensile softening
region of the stress/strain curverepresents additional energy
absorbed by the RCCstructure during the crack formation process.
Assuch, this region is quite instrumental in dissipatingthe energy
imparted to the dam through seismicground motion. The transition
from linear to nonlin-ear in the ascending region of the
stress/strain curverepresents the development of microcracking
withinthe concrete. These microcracks eventually coalesceinto
macrocracks as the tensile softening zone isreached.
3-8. Dynamic Tensile Strength (DTS)
The tensile strength of concrete is strain rate sensi-tive.
During seismic events strain rates are related tothe fundamental
period of vibration of the dam withthe peak stress reached during a
quarter cycle ofvibration. The high strain rates associated with
damresponse to ground motion produce tensile strengths50 to 80
percent higher than those produced duringdirect tensile strength
testing where the strain rate isvery slow. For this reason, the
dynamic tensilestrength (DTS) of RCC shall be equivalent to
thedirect tensile strength multiplied by a factor of 1.50(Cannon
1991, Raphael 1984). This adjustment fac-tor applies to both the
tensile strength of the parentmaterial and to the tensile strength
at the lift joints.
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3-9. Allowable Tensile Stresses
Figure 3-6. Tensile strength range, RCC, MSA > 1.5 inches,
consistency > 30 seconds vibration, no mortarbedding
When the response to ground motion increasesbeyond the elastic
limit, energy is dissipated throughcrack development and crack
propagation in accor-dance with the stress/strain relationship
shown inFigure 3-7. To account for all nonlinear responseincluding
that in the tensile softening zone of thestress/strain curve
requires a complex nonlinear anal-ysis. The simpler linear-elastic
analysis may be uti-lized in a manner which accounts for response
in thelinear region, and the nonlinear pre-peak region.
a. Comparing linear and nonlinear curves.Since a linear-elastic
analysis converts strains tostress using a constant modulus of
elasticity, thestresses from the analysis will be higher than
actualstresses when in the nonlinear pre-peak and post-peakstrain
regions. This may be compensated for by
establishing an allowable tensile stress which isgreater than
the actual peak tensile stress as shown inFigure 3-7. In this
figure, the dashed line representsthe tensile stress/strain
relationship assuming linear-elastic behavior as opposed to the
actual nonlinearstress/strain relationship which is shown as a
heavysolid line. The amount the peak tensile stress isincreased in
establishing the allowable stress dependson the extent of tensile
cracking that can be tolerated,which in turn is based on the
performance require-ments for the design earthquake under
consideration.The economics of the design also becomes a factor
inthe higher seismic zones. In these zones, a somewhatgreater
amount of cracking can be justified economi-cally because there is
a point where the cost of pro-ducing RCC mixes with high tensile
strengths toresist cracking will exceed the cost of repairing
thecracks as long as the cracking is not too extensive.
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b. Key points on stress/strain curve.Several
Figure 3-7. Tensile stress/strain diagram for RCC
points on the stress/strain curve are of interest
whenestablishing the allowable tensile stresses that areused in
linear-elastic analyses (refer to para-graphs 4-2c and 4-3c). Based
onf ′t = actual peaktensile stress (tensile stress that corresponds
to thatwhich would be attained by a direct tensile strengthtest),
andft = the stress level based on linear-elasticbehavior (refer to
the dashed line in Figure 3-7), thefollowing key values offt are of
interest:
(1) ft = 0.60 f ′t -- the end of the elastic rangeand the
beginning of microcracking.
(2) ft = 0.90 f ′t -- this point was selected becausethe
stress/strain dashed line for linear-elastic behavioris just
beginning to significantly separate from theactual stress/strain
curve. If the tensile stresses for alinear-elastic analysis stay
within the stress level for
this point, the response can still be judged as primar-ily
linear.
(3) ft = 1.25 f ′t -- the area under the dashed linefor
linear-elastic behavior up to this stress level isapproximately
equal to the area under the solid linefor the actual stress/strain
curve up to the peak tensilestress point (this point is the end of
microcrackingand the beginning of macrocracking). Thus, theenergy
absorbed in a linear-elastic analysis to thispoint of stress is
equal to the actual energy absorbedthrough the microcracking
pre-peak region.
(4) ft = 1.33 f ′t -- the strain corresponding to thispoint of
stress based on linear-elastic behavior isequal to the strain
corresponding to the actual peaktensile stress. This strain point
signifies the end ofmicrocracking and the beginning of
macrocracking.This point also represents a practical limit for
the
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linear-elastic response spectrum analysis described inparagraph
2-2c. Beyond this point in the tensilesoftening zone, the
stress/strain relationship based onlinear-elastic behavior diverges
so rapidly from theactual stress/strain curve that a linear-elastic
analysis
will no longer provide an acceptable approximation ofeither the
energy absorbed by the dam-foundationsystem, or the strain
deformation of the system.Cracking could be extensive enough to
change thedynamic properties of the dam structure.
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Chapter 4Design Earthquakes
4-1. Definition
The term “design earthquake” refers to the specifica-tion of the
free field ground motion that would be feltat the dam site due to a
particular seismic event thatis used as the basis for earthquake
resistant design ofnew RCC dams, or to evaluate the response of
exist-ing RCC dams.
4-2. Operating Basis Earthquake (OBE)
The OBE is defined as the earthquake producing thegreatest level
of ground motion that is likely to occurat the site during the
service life of the dam. Theservice life shall be taken as 100
years for both newdams and existing dams. The seismic risk or
adverseconsequences of failure of an existing dam is notreduced as
long as the dam is in operation; therefore,the “remaining service
life” of an existing dam shallnot be substituted for the 100-year
service life speci-fied above. The OBE is determined using
probab-ilistic methods and, as such, is defined as the earth-quake
with a 50 percent chance of exceedance in theservice life of the
dam.
a. General performance requirements.Allstructural, mechanical,
and control equipment used toregulate the reservoir shall be
capable of remainingfully operational during and after an OBE.
NewRCC dams located in low seismic regions shall bedesigned to
prevent the initiation of cracking in theconcrete structure.
Tensile cracking in new RCCdams located in high seismic regions and
in existingdams in all seismic regions is allowed; however, itshall
be limited to only “minor cracking” that requireslittle or no
repair.
b. Structural criteria. The following generalstructural criteria
shall be the basis for satisfying theconcrete cracking performance
requirements statedabove.
(1) Initiation of cracking is prevented when thetensile stresses
are less than 0.60f ′t as shown inFigure 3-7.
(2) The level of cracking is considered to be“minor cracking”
when the tensile stresses are lessthan 1.25f ′t as shown in Figure
3-7.
c. Allowable tensile stress.The allowable ten-sile stressesft
(allowable) for the OBE are establishedbelow. The formulae apply to
the calculation of bothallowable tensile stress of the parent
material andallowable tensile stress of the lift joints. DTS
=Dynamic Tensile Strength, andf ′t = direct tensilestrength.
(1) Existing dams:
ft (allowable) = 1.25 × DTS = 1.875 ×f ′t
(2) New dams in seismic zones 0, 1, 2A,and 2B:
ft (allowable) = 0.60 × DTS = 0.90 ×f ′t
(3) New dams in seismic zones 3 and 4:
ft (allowable) = 0.90 × DTS = 1.35 ×f ′t
d. Damping. Studies on dams under severeground motion which
cause stresses in the upperreaches of the elastic range indicate a
dampenedresponse which corresponds to a damping factor ofabout 5
percent of critical. On this basis the OBEshall be analyzed using a
damping ratio equal to5.0 percent of critical damping for the
concrete damstructure only. This factor must be modified as
out-lined in paragraph 7-3 to account for foundationdamping.
4-3. Maximum Credible Earthquake (MCE)
The MCE is defined as the largest possible earthqu-ake that
could reasonably occur along the recognizedfaults or within a
particular seismic source. Oftenseveral fault sources must be
investigated to deter-mine which will produce the critical site
groundmotion. By definition the MCE has a very low prob-ability of
occurence. Ground motion associated withthe MCE is established
using the deterministicapproach.
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a. General performance requirements.Bothnew RCC dams and
existing dams shall be capable ofsurviving the MCE without a
failure of a type thatwould result in the loss of life or
significant damageto downstream property caused by an
uncontrolledrelease of the reservoir pool. Nonlinear behaviorwith
associated damage is permissible, but the postearthquake damaged
condition of the dam shall allowfor controlled lowering of the pool
to facilitate repair.
b. Structural criteria. The upper limit of linearelastic
analysis is considered to be that point on thestraight
stress/strain line corresponding to a linearstress level of 1.33f
′t (see Figure 4-7). When tensilestrains exceed the strain
associated with this linearstress limit, macrocracking occurs and
the RCC willbe subject to some degree of structural damage. Asthe
strain level increases well into the tensile soften-ing zone,
response becomes markedly nonlinear and itis clear that a
linear-elastic analysis no longer approx-imates the response.
Although crack damage
increases in this zone, performance requirements maystill be
satisfied. Thus, the structural criteria for theMCE, when using
linear-elastic analysis, are set bylimitations of the method of
analysis rather than oncriteria that relate to an acceptable level
of structuralconcrete damage.
c. Allowable tensile stresses.The allowabletensile stressft
(allowable) for the MCE is establishedbelow. DTS = Dynamic Tensile
Strength, andf ′t =the direct tensile strength.
ft (allowable) = 1.33 × DTS = 2.000 ×f ′t
d. Damping. The linear-elastic analysis for theMCE shall utilize
a damping ratio equal to 7.0 per-cent of critical damping for the
concrete dam struc-ture only. The increase in the damping ratio
from5 percent for the OBE to 7 percent for the MCEhelps account for
some additional nonlinear behaviorwhile using a linear-elastic
approach.
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Chapter 5Design Response Spectra andAcceleration Time
Histories
5-1. Defining the Design Earthquake
In a linear-elastic response spectrum analysis,response spectra
define the free field ground motionfor the design earthquake. A
response spectrum givesthe maximum damped response (expressed as
dis-placement, velocity, or acceleration) of all possiblelinear
single degree-of-freedom systems using thenatural frequency (or
period) to describe the system.Viscous damping expressed as a
percentage of criticaldamping is used to develop a response
spectra. Adesign earthquake is often defined by a set ofresponse
spectra for various damping ratios. Theresponse spectra produced by
recorded earthquakeevents are characterized by a jagged shape made
upof peaks and valleys of varying magnitude; however,design
response spectra are smoothed so that they arenot frequency
sensitive.
5-2. Developing Design Response Spectra
a. Deterministic and probabilistic approaches.Design response
spectra are developed by using eithera “deterministic approach” or
a “probabilisticapproach.” The probabilistic approach is based
onprobabilistic seismic hazard analysis methodologywhich in essence
uses the same elements as the deter-ministic approach, but adds an
assessment of thelikelihood that ground motion will occur during
aspecified time period.
b. Procedures.There are two basic proceduresfor developing
design response spectra using eitherthe deterministic or
probabilistic approach. They are:(1) anchoring the spectral shape
to the peak groundacceleration; and (2) estimating the spectrum
directly.Although procedure (1) is more often used, the use
ofprocedure (2) is increasing, and for some situations ispreferred
because it incorporates factors besides justthe local site
conditions.
c. Obtaining design response spectra.It isbeyond the scope of
this EP to present the detailedprocedures for developing design
response spectra, orfor forecasting PGA’s for design earthquakes.
Referto ETL 1110-2-301, ETL 1110-2-303, and “Tentative
Provisions for the Development of Seismic Regula-tions for
Buildings” (Applied Technology Council1984) for further information
on developing designresponse spectra to define the design
earthquakes.
5-3. Developing Acceleration TimeHistories
a. Matching design response spectrum.Themore refined methods of
analysis discussed in para-graph 2-2d are of the time-history type.
Time histo-ries usually express the ground motion as a record
ofacceleration with respect to time. Acceleration timehistories
should be developed so their response spec-trum is consistent with
the previously established site-specific design response spectrum
described inparagraph 5-5c. The time histories should also have
astrong motion duration appropriate to the particulardesign
earthquake.
b. Procedures.There are two basic proceduresfor developing
acceleration time histories: (1) select-ing a suite of past
recorded earthquake groundmotions, and (2) synthetically developing
or modify-ing one or more ground motions.
(1) When selecting a suite of time-historyrecords for the first
procedure, the intent is to coverthe valleys of the spectrum
produced by one record,which fall significantly below the
site-specific designresponse spectrum, with better matching
spectralvalues at these frequencies as produced by the otherrecords
in the suite. It is also necessary that thespectra produced by the
suite of records not signifi-cantly exceed the site-specific design
response spec-trum. Primary advantage of this procedure is that
thestructure is analyzed by real, natural ground motionsthat are
representative of what the structure couldexperience.
(2) When using the second procedure, it is possi-ble to either
completely synthesize an accelerogram,or modify an actual recorded
earthquake accelero-gram so that the response spectrum of the
resultantaccelerogram closely fits or matches the
site-specificdesign response spectrum. The primary advantage ofthis
procedure is that a good fit to the designresponse spectrum can be
achieved with a singleaccelerogram, thus only a single dynamic
analysis isrequired.
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5-4. Dynamic Analysis by ModalSuperposition
a. Frequencies and mode shapes.The linear-elastic response
spectrum method utilizes modalsuperposition dynamic analysis to
determine the struc-tural response.
b. Time-history analysis.Once the modes arederived, the response
of the complex multiple degree-of-freedom system is reduced to the
solution of thesimple, single basic equation of motion for a
singledegree-of-freedom (SDOF) system. For time-historyanalysis,
the response is easily obtained using step-by-step integration of
the equation of motion for theSDOF system for each significant mode
based on thefrequency (eigenvalue) of the mode. In essence
theresponse contribution of each mode is determined fora series of
time steps using a prescribed time-stepinterval, and the response
at each time step is simplythe superposition, or addition, of
characteristic modeshapes adjusted by coefficients obtained from
theintegration procedure. Normally, only a few modeshapes are found
to contribute significantly to theresponse, so that the modal
superposition methodproduces a precise response with minimum
computa-tional effort.
c. Response spectrum analysis.In a responsespectrum analysis,
the step-by-step integration part ofthe dynamic analysis, described
above for time-history analysis, is performed in the process of
devel-oping the response spectrum. The response spectrummay be
envisioned as a display of the results of thispart of the modal
analysis, and it is presented in theform of “maximum” response
versus frequency (orperiod). In the response spectrum modal
analysis,eigenvalues, eigenvectors, and modal participationfactors
are computed and used in the analysis proce-dure just as they are
in a time-history modal analysis.Precise “maximum” modal responses
are easily calcu-lated from a simple equation that relates these
param-eters and the appropriate spectral value thatcorresponds to
the modal frequency.
d. Combining modal responses.The final stepin a response
spectrum analysis consists of correctsuperpositioning of the
“maximum” modal responses;however, there is not a unique solution
to this finalstep in the response spectrum method. This isbecause
the exact mode contributions at the criticalpoint in time when the
response peaks are not avail-able from a response spectrum
representation of a
particular ground motion. One advantage of asmooth design
response spectrum is that it is a statis-tical representation, or
an envelope, of the manypossible ground motions that could occur at
the siterather than only a single ground motion. The super-position
of the maximum modal responses is accom-plished by use of one of
several statistical methodsdescribed in Chapter 7.
5-5. Types of Design Response Spectra
a. Probability level. Design response spectraare usually based
statistically either on the mean,median (50th percentile
probability level), or themedian plus one standard deviation (84th
percentileprobability level), of the ground motion parametersfor
the records chosen. Design response spectra usedfor design of new
RCC dams or for evaluation of thesafety and serviceability of
existing dams shall bebased on the mean level of the ground
motionparameters.
b. Type of spectrum required.Either a “site-specific” or a
“standard” design response spectra shallbe used to describe the
design earthquakes. The typerequired shall be based on the seismic
zone, the prox-imity of the seismic source, and the maximum
heightof the dam.
c. Site-specific design response spectra.Thesite-specific design
response spectra should bedeveloped based on earthquake source
conditions,propagation path properties, and local
foundationcharacteristics associated with the specific site.
Thistype of design spectra may be established by anchor-ing a
selected response spectral shape for the site tothe estimated peak
ground acceleration, or by estimat-ing the design spectra directly
using response spectralattenuation relationships, performing
statistical analy-sis of strong-motion records, or applying
theoretical(numerical) ground motion modeling. In the require-ments
that follow, a site is classified as a “high seis-mic risk site”
when it is located within 20 kilometersof an active fault or area
source in the western UnitedStates (WUS), or within a tectonic
province in theeastern United States (EUS) where the source
orprovince has a maximum local magnitude of 6.0 orgreater. The
boundary between the WUS and theEUS is defined as the eastern
boundary of the RockyMountains. Site-specific design response
spectra arerequired for:
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(1) Dams greater than 100 feet in height locatedat a site
classified as a “high seismic risk site.”
(2) Dams greater than 100 feet in height locatedin Seismic Zone
2B, 3, or 4 even though the site isnot classified as a “high
seismic risk site.”
(3) Dams not greater than 100 feet in heightlocated in Seismic
Zone 2B, 3, or 4 when the site isclassified as a “high seismic risk
site.”
d. Standard design response spectra.Standarddesign response
spectra are based on fixed spectralshapes established for very
general site classificationssuch as rock or soil site. They ignore
the effects ofearthquake magnitude and distance, and the
specificfoundation characteristics at the site. The standarddesign
spectra are usually “anchored” to the estimatedpeak ground
acceleration (PGA) established for thedesign earthquake. The fixed
spectral shape is usu-ally presented such that it is normalized to
a 1.0 gvalue of maximum ground acceleration. This normal-ized value
can be easily checked by observing thespectral acceleration value
from the spectrum plot forfrequencies above about 50 cps where the
responseand the maximum ground acceleration coincide.Standard
design response spectra are adapted to theseverity of ground motion
associated with the OBE orMCE by using the PGA as a scaling factor.
Thestandard design response spectra can be used for:
(1) Dams greater than 100 feet in height locatedin Seismic Zone
0, 1, or 2A when the site is notclassified as a “high seismic risk
site.”
(2) Dams not greater than 100 feet in heightlocated in Seismic
Zone 0, 1, or 2A.
(3) Dams not greater than 100 feet in heightlocated in Seismic
Zone 2B, 3, or 4 when the site isnot classified as a “high seismic
risk site.”
e. Required design spectrum.When it isacceptable to use a
standard design response spectrumto define the design earthquakes,
the standard designspectrum shown in Figure 5-2 shall be used
(AppliedTechnology Council 1984). This spectrum is consid-ered
conservative but reasonable for essential struc-tures such as dams.
It is fully described by only five
control points on a tripartite plot. Table 5-1 presentsthe
spectrum in equation format so it is easily devel-oped for any
damping value. The standard designspectrum shown in Figure 5-2 and
defined in equationformat in Table 5-1 is normalized to 1.0 g PGA.
Thestandard spectrum shall be anchored to the PGA forthe OBE and
the MCE by using the appropriate scal-ing factors provided in Table
5-2. The correct scal-ing factors are selected based on the seismic
zonelocation of the site using the seismic zone map shownin Figure
5-1.
5-6. Horizontal and Vertical DesignResponse Spectra
a. Site-specific design response spectra.Whensite-specific
design response spectra are required inaccordance with paragraph
5-5c, two independentdesign response spectra shall be developed,
one todefine the horizontal component of ground motion,and the
second to define the vertical component. Thevertical component of
ground motion usually containsmuch higher frequency content than
the horizontalcomponent, therefore the spectral shape is quite
dif-ferent than that of the horizontal component. ThePGA associated
with the vertical component will alsobe different than the PGA of
the horizontal compo-nent. Both values of PGA are dependent on the
dis-tance from the source, but for short distances, thePGA of the
vertical component may actually exceedthe PGA of the horizontal
component.
b. Standard design response spectra.When itis acceptable to use
standard design response spectrato define the design earthquakes,
the horizontal com-ponent of ground motion shall be defined by
anchor-ing the standard design response spectra for theappropriate
damping factor developed from Table 5-1with the scaling factor
provided in Table 5-2. Thevertical component of ground motion shall
utilize thesame standard design response spectrum used for
thehorizontal component, but it shall be scaled using
theappropriate ratio of the PGA for the vertical compo-nent to the
PGA for the horizontal component asprovided in Figure 5-3. This
ratio is based on thesite to source distance (R) and the
fundamental natu-ral period of vibration of the structure.
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Figure 5-1. Seismic zone map of the United States. (Uniform
Building Code, 1988 Edition)
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Figure 5-2. Standard design response spectra for horizontal
component of ground motion - normalized toPGA = 1.0 g. (Applied
Technology Council ATC-3-06 Tentative Provisions, 1984)
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Table 5-2Peak Ground Accelerations (PGA’s) for Use in Scaling
the Standard Design ResponseSpectra
PGA
Seismic ZoneOperating BasisEarthquake (OBE)
Maximum CredibleEarthquake (MCE)
0 0.030 0.130
1 0.050 0.210
2A 0.095 0.360
2B 0.115 0.430
3 0.210 0.550
4 0.270 0.610
NOTES:
1. Refer to Figure 5-1 for the seismic zone maps.2. PGA’s are
expressed as the decimal ratio of the acceleration due to gravity
(g).3. PGA’s are obtained from curves of “Annual Risk of Exceedance
vs. PGA” in Figure C1-7 of ATC-3 Tentative Provisions,
April 1984.4. The PGA for the OBE is based on a 50 percent
chance of exceedance in 100 years.5. The MCE is considered to be
the event with a 5,000-year return period (annual risk of
exceedance = 0.0002
chance/year).
Figure 5-3. Ratio of PGA for the horizontal component to the PGA
for the vertical component as a functionof source to site distance
(R) and the fundamental period of vibration of the structure
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Chapter 6Earthquake Load Cases
6-1. Load Combinations
The cyclic and oscillatory nature of vibratoryresponse can cause
critical tensile stresses to occur ineither the upstream or the
downstream face of thedam. Therefore, the earthquake load cases
mustconsider combinations of the design earthquake load-ing with
other loads which lead to critical tension inboth the upstream and
downstream faces. Usuallytwo or more OBE load cases and two or more
MCEload cases must be evaluated. The discussion ofearthquake load
cases that follows refers to seismiccriteria regarding ground
shaking and foundation faultdisplacement as discussed in paragraphs
2-2 and 2-3,respectively, and not stability criteria described
inparagraph 2-1. Load case requirements for stabilityare covered in
EM 1110-2-2200.
6-2. Dynamic Loads To Be Considered
The design earthquake imposes several types of dy-namic loads on
the dam. The greatest dynamic loadis the inertia load caused by the
response of the con-crete mass to ground motion accelerations. Next
isthe hydrodynamic load created by a high reservoirand tailwater
condition. Hydrodynamic forces areimposed on the dam due to motions
of the dam react-ing with the surrounding water, and motions of
thereservoir bottom. Finally, backfill or silt depositsagainst the
faces of the dam will interact with thestructural mass of the dam
in a manner similar to thehydrodynamic load.
6-3. Static Loads To Be Considered
The effects on the dam structure due to static loads,as
discussed below, are determined by conventionalstatic analysis
methods. The results of the dynamicand static analyses are combined
by superposition todetermine the total stresses for the earthquake
loadcase.
a. Reservoir and tailwater loads.Load casesshall be included to
cover both the highest and thelowest reservoir pool elevations that
can be judged on
a statistical basis to have a reasonable chance ofoccurrence at
the time of the design earthquake.
(1) Flood frequency data from project flood flowand flood
routing studies provide a basis for estab-lishing reasonable high
pool elevations. Each dammust be evaluated based on its own set of
uniqueconditions.
(2) The conservation pool elevation for the proj-ect shall be
used for earthquake load cases involvinglow pool conditions. If
there is no established con-servation pool, use the lowest average
pool elevationthat can best be judged to exist for a 30-day period
ina normal yearly flow cycle.
(3) Where tailwater is applicable for an earth-quake load case,
the elevation shall be selected whichincreases the response while
being consistent with thereservoir conditions.
b. Backfill load. Earth or rock fill placedagainst either face
of the dam has both a static anddynamic load effect during an
earthquake. Theseloads shall be included in all earthquake load
cases.Static loading shall be based on at-rest pressures.Dynamic
loading may be approximated by theMononobe and Okabe method
utilizing the inertiaforce acting on the Coulomb sliding wedge in
theappropriate direction as discussed in EM 1110-2-2502. For finite
element analyses the dynamic effectmay be approximated by added
mass based on theCoulomb sliding wedge.
c. Siltation load. During the life of the dam,silt may build up
against the upstream face to a depthwhich may cause a moderate
increase in the tensilestresses in load cases where tension in the
upstreamface is critical. For these load cases, siltation
loadingshall be considered based on the full depth expectedduring
the life of the dam. In load cases where ten-sion in the downstream
face is critical, the siltationload will decrease the tensile
stresses. For these loadcases a zero depth of silt shall be
assumed. Whensilt is included, both static and dynamic
loadingeffects should be incorporated using the same meth-ods as
discussed for backfill loads.
d. Gravity loads. Gravity loads shall includethe weight of the
RCC, weight of backfill or silt onbattered faces of the dam, and
weight of equipment ifsignificant.
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6-4. Static Loads Not To Be Considered
There are several types of loads where the magnitudeof the load
and the load pattern that would exist atthe time of the design
earthquake event cannot bedefined on a logical basis or to any
degree of accur-acy. However, based on the general nature and
rangeof magnitude normally associated with loads of thistype, and
in comparing these loads with the dynamicand static loads already
discussed, these loads nor-mally do not contribute significantly to
the results ofthe analyses for earthquake load cases. However,
thedesigner should at least make a cursory evaluation ofthese loads
to be sure that no unusual site conditionsexist that would warrant
including one or more ofthem in the earthquake load cases. For this
reason, abrief discussion of these loads is included.
a. Pore pressure.When evaluating dam stabil-ity using the
seismic coefficient method described inparagraph 2-1, uplift is
considered to act over the
entire interface area. Under the MCE, any crackingin the
concrete would only extend just beyond themicrocracking level.
These fine cracks are open andsubject to buildup of internal water
pressure for ashort period of time due to the oscillatory nature
ofthe dynamic response. Therefore, uplift or internalwater pressure
within concrete cracks would be quitesmall and may be ignored in
the dynamic analysisphase of design.
b. Temperature stresses.Except under extremeclimatic conditions,
temperature stresses need not beincluded as part of the earthquake
load cases.
c. Wind load. Wind load on an RCC dam is sosmall it can be
considered insignificant.
d. Ice load. Ice loading need not be included aspart of an
earthquake load case except for unusualclimatic conditions which
would cause a great depthof ice to exist over an extended period of
time.
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Chapter 7Factors Significantly AffectingDynamic Response
7-1. Evaluation Procedure and Objectives
There are many important factors in a dynamic stressanalysis
that can greatly affect the response of a dam.The influence which
the various material and strengthparameters and loads have on the
final results mustbe evaluated. This can be done by executing
themodel using a typical dam cross-section and typicalmaterial
properties, then modifying the loads andparameters one-by-one to
give an indication of theinfluence each factor has on the dynamic
response.Once the important factors have been identified, thedesign
effort should concentrate on the more criticalfactors that form the
input to the dynamic analysis.Following is a discussion of the
impact some of theparameters have on the response of a dam.
7-2. Design Response Spectra
a. Spectral shape.Both the shape of the spec-trum and the PGA
used to anchor the spectrum affectthe dam response and should be
established carefully.The dynamic response in a linear-elastic
analysis isdirectly proportional to the PGA, but minor changesin
the shape of the spectra may not result in propor-tional changes in
the response.
b. Comparison of standard spectra.For com-parison purposes,
three widely accepted standarddesign response spectra will be
considered, eachrepresenting the same site conditions. The
designspectra are: (1) Applied Technology Council spec-trum for
rock of any characteristic whether shalelikeor crystalline in
nature (ATC 1984), (2) H. B. Seedspectrum for rock based on 28
records (Seed 1974),and (3) Newmark-Hall spectrum using
recommendedvalues for maximum ground velocity and displace-ment for
competent crystalline rock (Newmark andHall 1987). Figure 7-1 shows
all three spectra nor-malized to 1.0 g PGA for the same rock
foundationsite conditions. The Newmark-Hall spectrum is basedon the
median or 50th percentile cumulative probabil-ity, where the other
two spectra are based on themean of the records used in their
development. Thisdifference in probability level is reflected in
the spec-tral shape. The primary cause for the difference in
shape of these three spectra can be attributed to theassumptions
and techniques used in smoothing thejagged spectra produced from
the statistical combina-tion of real earthquake records.
c. Spectral accelerations.Referring toFigure 7-1, the range of
interest of natural periodwould be for periods of less than 1.0
second. Thisrange would cover the mode shapes that
producesignificant response. In this range the spectral
accel-eration values for a given period vary between spectraup to
as much as 65 percent. The ATC spectrumenvelopes the other two
design spectra, and is rec-ommended for use as the standard design
responsespectrum. In linear-elastic response spectrum analy-ses,
dynamic response of a particular system eval-uated by two different
response spectra is directlyproportional to the spectral ordinates
taken from thetwo spectra at the natural period of the system.
Thusthe shape of the design response spectrum greatlyinfluences the
results of the dynamic analysis.
7-3. Dam-Foundation Interaction, DampingEffect
a. Properties of the foundation.The two prop-erties of the
foundation rock that have a significantinfluence on the dynamic
response are the dampingratio and the deformation modulus. The
dampingcharacteristics of the foundation contribute signifi-cantly
to the damping of the combined dam-foundation system and must be
considered in theanalysis. When the foundation deformation
modulusis low, the damping ratio of the combined system
isconsiderably higher than the damping ratio of theRCC dam
structure alone.
b. Effective damping ratio.There are twosources of damping for
the foundation rock:(1) material (hysteretic) and (2) radiation. In
contrastto this type of damping is the viscous type of damp-ing
(directly proportional to velocity) used in pro-ducing design
response spectra. Therefore, it isnecessary to develop an effective
viscous dampingratio to represent the combined dam-foundation
sys-tem in a response spectrum analysis. This isaccomplished by
using the curves provided in Fig-ure D-6 of Appendix D, and the
following equation isfor an empty reservoir condition which allows
theeffects of foundation damping to be isolated. Thismethod,
developed by A. K. Chopra, is based on the
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Figure 7-1. Comparison of design response spectra for rock
foundations
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fundamental mode of vibration, and has been shownto be
reasonably close for the significant highervibration modes (Fenves
and Chopra 1986). In Fig-ure D-6, damping for the foundation rock
is expres-sed by the constant hysteretic damping factor.
ξ11
(Rf)3ξ1 ξf
where
ξ1 = the effective viscous damping ratio for theempty reservoir
condition
ξ1 = the viscous damping ratio for the RCC damstructure only
ξ1 = 5.0 percent for the OBEξ1 = 7.0 percent for the MCE
Rf = ratio of the fundamental period of the dam on arigid
foundation to the fundamental period ofthe dam on a foundation with
a deformationmodulus =Ef
ξf = added damping ratio due to dam-foundationrock interaction
taken from Figure D-6
c. Effect of damping on response.To determinethe effect that the
damping ratio has on the responseof a dam, the fundamental
frequency of the compositefinite element dam-foundation model must
be deter-mined. It is noted that for the response spectrummethod,
the effects of damping are contained only inthe response spectrum
itself. Thus, the ratio of theresponse of a dam/foundation system
responding atone damping factor to the same system responding ata
second damping factor is equal to the ratio of thespectral
ordinates taken from the two spectra eval-uated at the fundamental
frequency of the system.
d. Conclusion. The damping characteristics ofthe foundation can
have a great influence on thedynamic response. This indicates the
need to care-fully determine the value of the constant
hystereticdamping factor for the foundation rock. This can
bedetermined from experimental tests of appropriaterock samples
subject to harmonically varying stressand strain. From such tests,
the inelastic energy lostand the strain energy stored per cycle are
determinedand the hysteretic damping factor is calculated.
7-4. Dam-Foundation Interaction, Founda-tion Modulus Effect
a. Modulus of deformation.The flexibility ofthe jointed rock
foundation is characterized by themodulus of deformation which
represents the relation-ship between applied load and the resulting
elasticplus inelastic deformation. It is best determined byin-situ
testing, but may be estimated from the elasticmodulus of the rock
by applying an appropriatereduction factor. In a linear-elastic
analysis, themodulus of deformation is synonymous with
Young’smodulus of elasticity (Ef).
b. Dynamic characteristics affected.The elasticmodulus of the
foundation influences the responsebecause it directly affects the
following dynamiccharacteristics of the dam-foundation system:
(1) Modal frequencies. As the modulus of defor-mation decreases,
the modal frequencies of the com-posite dam/foundation system also
decrease.
(2) Mode shapes. As the modulus of deforma-tion decreases, the
mode shapes are affected byincreased rigid body translations and
rotation of thedam on the elastic foundation.
(3) Effective damping ratio. As the modulus ofdeformation
decreases, the effective damping ratio ofthe dam/foundation system
increases.
c. Effect of foundatio