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Chapter 8Dynamic Analysis Methods andProcedures

8-1. Attributes of Dynamic Analysis Methods

A dynamic analysis method is identified by fourattributes: (1) material behavior, (2) design earth-quake definition, (3) dimensional representation ofproject conditions, and (4) model configuration. Thefirst two attributes have been discussed in precedingchapters. They are briefly summarized below, fol-lowed by a more detailed discussion of the latter twoattributes.

a. Material behavior. This attribute definesmaterial behavior as either (1) linear-elastic or(2) nonlinear. Associated with each of these twotypes of material behavior is a unique criterion forestablishing acceptable response. Refer to para-graphs 2-2d, 2-2e, and 3-10.

b. Design earthquake definition.This attributeestablishes which of two options will be used tospecify the free field ground motion for the designearthquakes. The options are (1) design responsespectra and (2) ground motion time-history records.Refer to Chapter 5 for details.

c. Dimensional representation of project condi-tions. This attribute defines whether project condi-tions will be represented in (1) two dimensions or(2) three dimensions. Project conditions refer to thegeometry of the dam, the foundation, and the reser-voir that have an affect on the seismic response.Examples of features governing which of these twooptions is appropriate include such things as layout ofthe dam axis, shape of the dam monoliths, foundationconditions, and orientation of potential fault slips ifapplicable.

(1) Two-dimensional (2-D) analysis. In theanalysis of most gravity dams, it is assumed that thedam is composed of individual transverse verticalelements or cantilevers each of which carry loads tothe foundation without transfer of load between adja-cent elements. This assumption also applies to mostRCC dams including dams with transverse joints thatseparate the dam into several monoliths, and damswith monolithic construction that contain no trans-verse joints. This assumption is usually valid, and

stress analyses including the dynamic stress analysisphase can be based on 2-D representation of the damcross-section. The design example provided inAppendix D presents a typical 2-D analysis. It dem-onstrates the most common procedure where a 2-Dcross section of the structure is analyzed. However,most principles and procedures applying to the 2-Danalysis also apply, or may be adapted to a 3-D anal-ysis discussed below.

(2) Three-dimensional (3-D) analysis. Occasion-ally there are exceptions to the assumption justifying2-D analysis. Dams in narrow canyons with a largeenough ratio of height of the dam to distance betweenabutments may cause significant two-way distributionof stresses. Dams which are aligned on a curved axismay also allow significant transfer of stress into theabutments by arch action. Unusual shaped monolithswhere there is substantial variation in the transversecross section across the width of the monolith alsomay not be analyzed satisfactorily by 2-D methods.Another exception occurs when the trace of a poten-tial fault slip is not parallel or nearly parallel to thedam axis. In this situation, a 2-D foundation faultdisplacement analysis will not adequately representproject conditions. All of these situations indicate theneed for 3-D analysis if the response is to be deter-mined to a reasonable degree of accuracy.

(a) Ground motion direction. The 3-D analysisintroduces additional variables into the dynamic anal-ysis. One important variable is determining the criti-cal direction of the horizontal ground motion. Thisintroduces a second horizontal component of groundmotion into the dynamic analysis. The critical direc-tion is defined by transforming the design earthquakeground motion into a pair of orthogonal components.Since no method exists to determine the critical direc-tion directly, it usually becomes necessary to makesome rough approximations.

(b) Simplified approach. This approach to deter-mining the critical horizontal direction of groundmotion is to select two orthogonal direction vectors(in the horizontal plane), and assume that the criticaltensile stress at various locations on the dam willoccur when the direction of ground motion is nearone or the other vector. Since the accompanyingorthogonal ground motion component is small, thestresses are assumed negligible and are neglected.Often the direction vectors are assumed to be theupstream-downstream direction, and the cross-stream

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direction. This approach requires performingseparate, independent dynamic analyses for the twoorthogonal ground motion directions.

(c) Conservative approach. Another more con-servative approach accounts for both orthogonal com-ponents of ground motion. It is necessary to performthe two dynamic analyses described above, but thefirst analysis includes the full magnitude design earth-quake ground motion component acting in anassumed direction with a fraction of the design earth-quake ground motion acting orthogonally. The sec-ond analysis includes the fractional part of the groundmotion acting in the assumed direction and the fullmagnitude ground motion acting orthogonally. Thefractional part of the design earthquake groundmotion is usually assumed to be 30 percent of thedesign earthquake ground motion. In a responsespectrum analysis, stresses produced by the two hori-zontal components of ground motion are addeddirectly to produce the resultant stress component forhorizontal ground motion. This resultant stress com-ponent is then combined with the stress componentproduced by the vertical component of ground motionusing SRSS.

(d) Complexity of analysis. A 3-D analysisrequires considerably greater effort to create the 3-Dmodel as compared to a 2-D model, and may requirea main frame computer and a substantial amount ofcomputer time to perform the analysis. It also pro-duces a large amount of output to evaluate and inter-pret. However, the general purpose structural finiteelement programs are continuously being improvedand are much more user oriented than they were inthe past. They have refined graphics capabilitieswhich help greatly in checking for errors in the com-puter model input, and in displaying the stress output.Also, specialized post-processors are being developedso that results can be evaluated much more effi-ciently. These advances greatly enhance the practi-cality of the 3-D analysis.

d. Model configuration.This attribute of thedynamic analysis method is dependent on the type ofmodel used to represent the dam-foundation-reservoirsystem. The three types of models used for dynamicanalysis of gravity dams are (1) the “standardized”model developed by Chopra and used in his Simpli-fied Method of Analysis, (2) the finite element-substructure model, and (3) the composite finiteelement-equivalent mass system model.

(1) Standardized model. This type of model isused in Chopra’s Simplified Method. It is based onstandardizing certain parameters that define the dam-foundation-reservoir system. It recognizes the factthat these parameters have little variation within therange of geometry common to gravity dams. Forexample, the normalized fundamental mode shapesfor six sample dam cross sections were studied andfound to be almost identical. A standardized modeshape was then developed for use in the calculationprocedure.

(a) Factors considered. In the latest version, thestandardized model considers dam-foundation rockinteraction, dam-reservoir effects, and reservoir bot-tom absorption. All of these factors are based onstandard curves and formulae.

(b) Model limitations. The standardized modelis the simplest of the three types of models. A com-puter is not required to formulate the model or evento perform the dynamic analysis. However, standard-izing the mode shape, frequency, and other parame-ters makes this an approximate method limited strictlyto the typical nonoverflow monolith shape.

(2) Finite element-substructure model. In thistype of model, different techniques are used to repre-sent the dam, foundation, and reservoir; however, byusing common node points at the interfaces, a com-puter model is formulated that can be analyzed byconventional matrix methods.

(a) Dam. The dam is modeled as an assemblyof discrete finite elements. Either solid quadrilateralplane stress or plane strain elements are used for a2-D model.

(b) Foundation. The foundation is idealized as aviscoelastic half-plane. The elastic properties of thefoundation are formulated into a substructure matrixusing the theory of elasticity. This matrix is com-bined with the structural stiffness matrix developedfrom the finite element representation of the dam.The substructure matrix introduces the foundationstiffness to the equations associated with the degrees-of-freedom of the node points at the dam-foundationinterface. There is no finite element model of thefoundation. The dimensions of the structural stiffnessmatrix are set by the finite element model of the dam.

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(c) Reservoir. The impounded water of thereservoir is idealized as a fluid domain of constantdepth and infinite length. This can be interpreted asa series of subchannels of infinite length discretizedto match the common upstream nodal points of thedam. The reservoir bottom absorption is modeled byadjusting the boundary condition at the reservoirbottom. This substructure representation of the reser-voir produces more accurate hydrodynamic responseto horizontal and vertical ground motion than does anequivalent mass system representation as described inparagraph 8-1d(3)(a).

(d) Specialized computer program. This type ofmodel requires a specialized computer program toallow the foundation and the reservoir effects to beformulated in the manner described above. Also, thesubstructure method requires the foundation to bemodeled as a uniform homogeneous material. Pres-ently, a computer program is available which devel-ops a 2-D finite element-substructure model forgravity dams. Refer to paragraph 8-2b.

(3) Composite finite element-equivalent masssystem model. This method models both the damand the foundation as an assembly of discrete finiteelements. Either solid quadrilateral plane stress orplane strain elements are used for 2-D models or 3-Disoparametric solid elements are used for 3-D models.The foundation consists of a rectangular block with awidth in the upstream-downstream direction about3 times the base width of the dam at the foundationplane, and with a height about 1.5 times the height ofthe dam.

(a) Reservoir effects. The reservoir effects aremodeled by developing an equivalent mass systemwhich consists of adding mass to the finite elementmodel to correctly alter the dynamic properties. Theadded mass is active in the direction normal to thevertical upstream face of the dam. This method alsoallows the reservoir bottom absorption characteristicsto be incorporated into the analysis by using Chopra’sstandard hydrodynamic pressure function curves todetermine the added mass. Although use of thesecurves in developing the equivalent mass system isonly approximate, it has been shown to be reasonablyaccurate. Refer to paragraphs 7-5c and 7-5d andAppendix D for details.

(b) Boundary conditions. With this type ofmodel, the earthquake ground motion is introduced atthe rigid boundary. This boundary is along the sides

and bottom of the rectangular foundation block ratherthan at the ground surface (dam-foundation interface)where the design earthquake ground motion is speci-fied. To account for this, the foundation is assumedmassless. Therefore, no wave propagation takesplace in the massless foundation so the groundmotion is transmitted to the dam-foundation interfacewithout modification.

(c) Flexibility in modeling. The composite finiteelement model may be formulated to represent avariety of design conditions for both 2-D and 3-Dmodels. For example, most any geometric shape maybe accommodated, various zones of superior RCCmix may be incorporated in the dam model, anddiscontinuities such as fault zones or changes ofdeformation modulus in the foundation may also beincluded.

8-2. Comparison of Dynamic AnalysisMethods

This section will describe the attributes associatedwith the most commonly used dynamic analysismethods, and the methods will be evaluated andcompared.

a. Chopra’s simplified method.This methoduses the standardized model described in para-graph 8-1d(1). Other attributes include 2-D repre-sentation, linear-elastic material behavior, andresponse spectrum definition of the design earth-quake. This method is not flexible because all ofthese attributes are fixed.

(1) Equivalent lateral force. The simplifiedmethod develops the maximum response to the firstmode as a set of equivalent lateral forces. It alsoapproximates the equivalent lateral forces associatedwith the higher vibration modes using a “static cor-rection” method. The two sets of equivalent lateralforces are treated as statically applied distributedlateral loads. At present, response to a vertical com-ponent of ground motion is not possible with thistype of model. Stresses may be hand calculated bybeam theory treating the dam as a simple cantileverbeam, or the static load may be applied to a finiteelement model of the dam to gain a more realisticstress distribution pattern.

(2) Advantages and limitations. The simplifiedmethod is easy to use and can be done without a

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computer. However, it takes less time and effort toperform a simple 2-D analysis using a general pur-pose finite element program on a personal computer(PC) and the results of the finite element analysis willbe more accurate. Also, comparative studies haveindicated that as the flexibility of the foundationincreases, the response calculated by the simplifiedmethod tends to diverge from the response deter-mined by more refined methods, and the simplifiedmethod is not always conservative.

(3) Recommended use. Because of the limita-tions of the simplified method, it should be used onlyfor preliminary design work as described in para-graph 8-4a. However, appropriate equations anddesign figures used in this method are helpful inchecking the results from other more refined analysesand to prepare the computer input for these methods.

b. EAGD-84 Analysis Method.EAGD-84, AComputer Program for Earthquake Analysis of Con-crete Gravity Dams (Fenves and Chopra 1984), is aspecialized computer program that allows the founda-tion and the reservoir effects to be characterized bythe substructure model described inparagraph 8-1d(2).

(1) Other attributes. Other attributes that definethe EAGD-84 analysis method include 2-D represen-tation, linear-elastic material behavior, and time-history ground motion definition of the designearthquake. All attributes of EAGD-84 are fixed andcannot be changed.

(2) Advantages and limitations. When comparedto either a standardized model or a finite element-equivalent mass system model, the EAGD-84 sub-structure model is a better representation of thefoundation and reservoir, as long as the project condi-tions properly fit the program requirements. Also,the time-history definition of ground motion is a levelof refinement beyond response spectrum definition.Therefore, the EAGD-84 method is capable of pro-ducing the most accurate response, and the time-history response output provides additionalinformation often needed to evaluate acceptable per-formance. The biggest disadvantage of EAGD-84 isthe lack of attribute flexibility.

c. General purpose finite element programanalysis methods.This comprises a number ofmethods each with a different combination ofattributes, but all having the composite finite element-

equivalent mass system model as a common attribute.These methods use any one of several proven generalpurpose finite element computer programs to performthe dynamic analysis. Examples are ANSYS, SAP6,GT-STRUDL, and STAAD III. The material behav-ior attribute for most of the general purpose programsis linear-elastic; however, some programs such asANSYS and ADINA have nonlinear capability.

(1) Primary advantage. Attribute flexibility isthe primary advantage of the general purpose finiteelement methods. Except for the common attributementioned above, design methods are possible whichfeature most of the other possible combinations of theremaining attributes. This allows the dynamic analy-sis phase to start with a simple method such as the2-D, linear-elastic, response spectrum method. If theresults of the simple analysis or the project conditionsindicate the need of a more refined analysis, theprocedure may transition conveniently into a morerefined analysis by modifying or adding to the inputto the same general purpose program.

(2) Other advantages. The general purpose finiteelement programs discussed above are large, compre-hensive programs developed for main frame com-puters. In addition to these programs are severalsmaller general purpose finite element programsspecifically developed for PC’s. Since these desk-topPC’s are now a standard item in most design offices,a considerable amount of the dynamic analysis phasemay be completed without the need or expense of alarge main frame computer.

8-3. Dynamic Analysis Procedure

The dynamic analysis procedure described hereafter isderived with the objective of arriving at a reasonableand economic design of a new dam, and evaluatingthe seismic resistance of existing dams using an anal-ysis method with the simplest attributes possible. Ingeneral the procedure is to perform a dynamic stressanalysis and evaluate the results to determine if theRCC dam response to the design earthquakes isacceptable. If not acceptable, the design of a newdam may be modified and reanalyzed, or a morerefined analysis method may be employed whenanalyzing either new dams or existing dams.

a. Evaluating acceptable response.Theresponse is judged acceptable for a linear-elasticanalysis when the tensile stresses are within the

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established allowables and the analysis method pro-vides a reasonably accurate or conservative represen-tation of project conditions. Should the analysismethod utilize an extremely simplified representationof project conditions, the response may not neces-sarily be conservative and will likely be of relativelylow order of accuracy. However, the response maystill be judged acceptable without pursuing morerefined analyses on the basis that the tensile stressesare far enough below the established allowables toclearly infer that the response satisfies the require-ments and criteria described above. Refer to para-graphs 2-2e, 2-2f, and 2-2g for information on allow-able tensile stress criteria for various methods ofanalysis.

b. Modifying the design of a new dam.Whenthe response from a dynamic stress analysis for a newdam is judged not acceptable, consideration shall begiven to modifying the design, adjusting the computermodel to reflect the modifications, and reanalyzing.Modifications include:

(1) Modify geometric configuration.

(2) Superior mixes. Use richer, higher strengthsuperior RCC mixes in overstressed areas.

(3) Reducing aggregate size. Increase tensilestrength by reducing the maximum size aggregate.

(4) Mortar bedding. Provide mortar bedding toincrease tensile strength at lift joints.

(5) Zone boundaries. Adjust the zone boundariesof the superior RCC mixes to better fit the tensilestress pattern.

c. Refining the dynamic analysis methods.When the response from a dynamic stress analysis ofan existing dam is judged not acceptable, the nextstep in the procedure shall be to reanalyze using ananalysis method with more refined attributes. Incontrast to this, there is no clearly defined point inthe design procedure for new dams that indicateswhen the analysis method should be refined. Thedesign conditions and results of the design procedurealready completed must be evaluated to determinewhen it is appropriate to suspend the design modifica-tion process, and pursue a more refined analysis ofthe latest modified design. When the attributes of thedynamic analysis method are to be refined, it is

recommended that the refinements be considered inthe following order:

(1) 3-D representation. Consider refining theanalysis from two to three dimensions when the accu-racy of the response from a 2-D analysis cannot leadto a confident judgment that the response isacceptable.

(2) Time-history analysis. Consider defining thedesign earthquakes with appropriate ground motiontime-history records, and performing a time-historyanalysis when additional insight into the structuralbehavior beyond that provided by the response spec-trum analysis is needed. A time-history analysisyields additional information regarding the excursionsof tensile stress cycles beyond the allowables andprovides a better understanding of the response. Thisapplies both to existing dams or to the design of anew dam when all practical and economical modifi-cations to the design of a new dam have beenexhausted.

(3) Nonlinear analysis. The analysis based onnonlinear material behavior represents the greatestpossible refinement and it produces the most accurateresults. However, it is also the most complex and themost costly. It requires time-history ground motioninput, direct integration solution, a large main framecomputer, specialized computer programs, and aconsiderable amount of computer time. As such, it isthe last recourse in the attribute refining process. Thenonlinear analysis should only be undertaken underthe guidance of an expert in the field of fracturemechanics and finite element methods.

8-4. Preliminary Design of New Dams

Preliminary design includes engineering and designthrough the Feasibility Phase, or through the GeneralDesign Memorandum (GDM) phase if a GDM isprepared for the project.

a. Initial dynamic analysis.The initial dynamicstress analysis shall use the simplest analysis methodwhich is identified by the following attributes:(1) linear-elastic material behavior, (2) 2-D represen-tation, and (3) design response spectrum definition ofthe design earthquake. The analysis shall be per-formed using the cross-section of the critical trans-verse element of the dam which usually consists of a

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section of the nonoverflow monolith with the greatestheight. The dam-foundation-reservoir system shall berepresented by a composite finite element-equivalentmass system model for RCC dams subject to criticalseismic design conditions. For other conditions, thedam-foundation-reservoir system may be character-ized by either the standardized model using Chopra’ssimplified method or the composite finite elementmodel described above.

b. Seismic and foundation investigations.Appropriate investigations of the regional tectonicsand site seismicity shall be conducted at the prelimi-nary design stage. When required, the site-specificdesign response spectra shall be developed in accor-dance with paragraph 5-5c. Preliminary dam site andreservoir geology investigations shall be conductedincluding exploratory corings and load testing todetermine foundation conditions and deformationmodulii.

c. Tensile strength.For preliminary design, thetensile strength may be taken from Figures 3-1through 3-6 for the proposed basic RCC mix and forsuperior RCC mixes in the critical zones.

d. Satisfying criteria. The preliminary designprocedure shall progress to the point where itbecomes evident that the preliminary design will leadto a final design that fully satisfies established perfor-mance requirements and criteria.

8-5. Final Design of New Dams

The final design of an RCC dam shall result in adesign that satisfies the provisions of this EP. Thedynamic analysis phase for RCC dams under criticalseismic design conditions shall be presented in anappropriate feature design memorandum.

a. Final design analysis method.The dynamicanalysis method for the final design shall evolve fromthe simple initial method described in paragraph 8-4ato more refined methods of design conditions as war-ranted. RCC dams analyzed by Chopra’s simplifiedmethod during the preliminary design phase shall

be reanalyzed using a composite finite element-equivalent mass system model and general purposefinite element program in the final design.

b. Foundation and material investigations.Thefoundation conditions for the final design shall reflectthe latest exploratory coring and other foundation andgeology investigations. The final design shall bebased on the RCC material properties obtained fromtests on core samples taken from test fill placementsmade with the proposed design mixes.

8-6. Evaluating Existing Dams

The dynamic analysis procedure for evaluating exist-ing dams is essentially the same as the combinedpreliminary design and final design procedures for anew dam, except modification of the design discussedin paragraph 8-3b does not apply to existing dams.As with the design of new dams, the dynamic analy-sis procedure shall utilize an analysis method with thesimplest attributes possible to determine if the exist-ing dam is capable of responding to the design earth-quakes in an acceptable manner.

a. Material properties. Material properties ofthe RCC for an existing dam, including tensilestrength, shall be obtained from tests on core samplestaken directly from the dam.

b. Using available records.Exploratory coringlogs, laboratory test data, and field geologic testresults conducted during design and constructionshould be used for an existing dam and to provideinformation needed to model the foundation. Reser-voir data should be used to determine the reservoirand tailwater elevations for earthquake load cases.

c. Special requirements and analysis methods.The regional tectonics and site geology and seismicityshall be investigated as required to develop a site-specific design response spectra in accordance withparagraph 5-5c. The initial analysis of an existingdam shall utilize a composite finite element-equivalent mass system model. Existing dams shallnot be analyzed by Chopra’s simplified method.

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