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Soil Dynamics and Earthquake Engineering 34 (2012) 62–68
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Soil Dynamics and Earthquake Engineering
0267-72
doi:10.1
n Corr
E-m
journal homepage: www.elsevier.com/locate/soildyn
Coupled gravity dam–foundation analysis using a simplified direct method ofsoil–structure interaction
A. Burman a, Parsuram Nayak b, P. Agrawal a, Damodar Maity b,n
a Department of Civil Engineering, Birla Institute of Technology, Mesra, Indiab Department of Civil Engineering, Indian Institute of Technology, Kharagpur, India
a r t i c l e i n f o
Article history:
Received 8 January 2010
Received in revised form
10 August 2011
Accepted 22 October 2011Available online 1 December 2011
Keywords:
Concrete gravity dam
Direct method of soil–structure interaction
Finite element method
Newmark algorithm
Koyna earthquake
Dynamic response analysis
61/$ - see front matter & 2011 Elsevier Ltd. A
016/j.soildyn.2011.10.008
esponding author. Tel.: þ91 3222283406; fa
ail address: [email protected] (D. M
a b s t r a c t
A time domain transient analysis of a concrete gravity dam and its foundation has been carried out in a
coupled manner using finite element technique and the effect of Soil–Structure Interaction (SSI) has
been incorporated using a simplified direct method. A two dimensional plane strain dam–foundation
model has been used for the time history analysis to compute the stresses and displacements against
earthquake loading considering the effect of soil–structure interaction. An effective boundary condition
has been implemented by attaching dashpots to the vertical boundaries. The material damping effects
have also been considered and the dam and foundation have both been modeled as linear, elastic
materials. To achieve a greater degree of accuracy, the displacements and stresses calculated in the
free-field analysis have also been added to those developed in the complete dam–foundation analysis.
The proposed algorithm has been simulated for the case of two published problems and in both the
cases the results have been found to be in close agreement. The proposed technique is quite simple and
easy to implement in the computer code. The outcomes of the results show the efficacy of the
developed method.
& 2011 Elsevier Ltd. All rights reserved.
1. Introduction
The response of a dam during an earthquake depends uponcharacteristics of the ground motion, the surrounding soil andreservoir and the dam itself. Damage sustained in recent earth-quakes, such as the 1995 Kobe Earthquake, have also highlightedthat the seismic behavior of a structure is highly influenced not onlyby the response of the superstructure, but also by the response ofthe foundation and the ground as well. Hence it becomes imperativeto consider the effect of soil–structure interaction for heavy struc-tures such as concrete gravity dams. Soil–Structure Interaction (SSI)problems have hitherto been dealt with by applying variousmethodologies, out of which the direct method is one of the mostaccurate and robust technique.
Wolf [19] first presented the direct method of soil–structureinteraction analysis. Using this method the soil region near thestructure along with the structure is modeled directly and theidealized soil–structure system was analyzed in a single step. Anexcellent amount of work on dam–reservoir–foundation interac-tion in the frequency domain has been carried out by Chopra andhis colleagues [1,6]. However, frequency domain based methodsare difficult to understand compared to the time domain based
ll rights reserved.
x: þ91 3222282254.
aity).
methods. Also, incorporation of nonlinear material behavior willbe a prohibitive task for frequency domain based methods. Thetime domain based methods do not suffer from such limitations.As the present method is fully defined in the time domain,incorporation of nonlinear material behavior for dam and founda-tion domain is quite straightforward. This is one of the advantagesof time domain based method suggested in this paper.
Felippa and Park [5] discussed, in detail, staggered solutionprocedure for solution of a variety of coupled field dynamicproblems. Direct solution methods of coupled field problemssuffer from the need of excessive storage requirements andcomputation time. The staggered solution schemes allow us touse reduced matrices for the particular subsystem leading to lesstime for solution. Rizos and Wang [15] developed a partitionedmethod for soil–structure interaction analysis in the time domainthrough a staggered solution method using both FEM and BEM.Maity and Bhattacharyya [12] suggested an iterative scheme inconjunction with the staggered solution procedure for the dam–reservoir interaction problems. However, the staggered approachshould be used with great care, since its stability is conditional toconvergence of solution at each time step. If a corrective iterationat each time step is employed, where the interface boundaryconditions are iteratively updated until convergence is achieved,one obtains an iterative coupling method. Jahromi et al. [8] usedpartitioned analysis technique involving iterative coupling proce-dure for solving different soil–structure interaction problems.
Soil media
Structure
Common Nodes
U = v +uU = Absolute Displacementsv = Free Field Displacementsu = Added Displacements
Fig. 1. Soil–structure interaction model.
A. Burman et al. / Soil Dynamics and Earthquake Engineering 34 (2012) 62–68 63
But obtaining the convergence of solution in the iterative methodsis quite difficult and time-consuming process. The direct methodssuch as one suggested in the present paper do not suffer fromsuch limitations.
Also, coupled BEM–FEM [22,4] and coupled FEM and infiniteelement techniques [16,9] have been used tackle SSI problems.Clough and Penzien [2], Clough and Chopra [3] presented thedetailed SSI equations in the time domain. Also, Wolf and Song[20] developed the scaled boundary element method (also knownas consistent infinitesimal finite element cell method) by combin-ing the advantages of the boundary element method and thefinite element method. This method is exact in radial directionand converges to the exact solution in the finite element sense inthe circumferential direction, and is rigorous in space and time.However, these methods are not too simple from mathematicalpoint of view and wouldn’t be too attractive to a common user.From the user’s point of view, a method will be interesting only ifit is simple and easy to use. Such methods though mathematicallycorrect will not be too attractive to a common user apart fromscientist or researchers. The authors feel that the present methodin this paper is an attractive alternative to already existing morecomplex analysis procedures.
In this paper, the proposed algorithm is tested against theresults of the dam–foundation interaction problem analyzed byYazdchi et al. [21] for Koyna ground motion. In the present work,the free field response is obtained first for the foundation domainsubjected to ground motion. Later, the free field responses areused to determine the soil–structure interaction forces which areexerted on the coupled soil–structure system along with theexternally applied earthquake excitations. The responses obtainedfrom the latter are added to the free-field responses to calculatethe total displacements, velocities and accelerations of the systemconsidering the effects of SSI. In most of the soil–structureinteraction studies [7,19], special purpose programs involvingconvolution integrals and Fourier transforms are required tocalculate the soil–structure interaction forces. The evaluation ofconvolution integrals and Fourier transforms are complex andtime consuming processes. The present method is defined entirelyin time domain where there is no need to evaluate convolutionintegrals or Fourier transforms. Time domain based methods areeasier to envisage from engineer’s point of view compared to thefrequency domain based methods. The algorithm presented hereis robust in terms of runtime as well as memory allocationconsiderations.
2. Methodology
2.1. Mathematical model of coupled dam–foundation system
In dam–foundation interaction problems, the foundation and thestructure do not vibrate as separate systems under external excita-tions, rather they act together in a coupled way. Therefore, theseproblems have to be dealt in a coupled way. The most common SSIapproach used is based on the ‘‘added motion’’ formulation. Thisformulation is mathematically simple, theoretically correct, and iseasy to automate and is used within a general linear structuralanalysis program. In addition, the formulation is valid for free-fieldmotions caused by earthquake waves generated from all sources.The method requires that the free-field motions at the base of thestructure be calculated before the SSI analysis. To develop thefundamental SSI dynamic equilibrium equations, the soil–structuresystem, as shown in Fig. 1, is considered. The absolute displace-ments of the structure are considered to be the sum of two parts,viz. free field displacements and added part of the displacements.Free field displacement is found out by analyzing the foundation
domain with no structure present on it against the earthquakeforces. The added part of the displacement is found out by carryingout coupled soil–structure interaction model.
The SSI model here is divided into three sets of node points, viz.the common nodes at the interface of the structure and soil areidentified with the subscript ‘‘c’’; the nodes within the structure arewith ‘‘s’’ and the nodes within the foundation are with ‘‘f’’. From thedirect stiffness approach in structural analysis, the dynamic forceequilibrium of the system is given in terms of the absolutedisplacements, U, by the following sub-matrix equation:
Mss Msc 0
Mcs Mcc Mcf
0 Mf c Mf f
264
375
€Us
€Uc
€Uf
8>><>>:
9>>=>>;þ
Css Csc 0
Ccs Ccc Ccf
0 Cf c Cf f
264
375
_Us
_U c
_Uf
8>><>>:
9>>=>>;
þ
Kss Ksc 0
Kcs Kcc Kcf
0 Kf c Kf f
264
375
Us
Uc
Uf
8><>:
9>=>;¼�
Mss Msc 0
Mcs Mcc Mcf
0 Mf c Mf f
264
375
€Ug
s
€Ug
c
€Ug
f
8>>><>>>:
9>>>=>>>;
ð1Þ
where the mass and stiffness at the contact nodes are the sum ofthe contributions from the structur (s) and foundation (f), and aregiven by
Mcc ¼MðsÞcc þMfcc Ccc ¼ CðsÞcc þCðf Þcc Kcc ¼ K ðsÞcc þK ðf Þcc ð2Þ
In order to solve the coupled soil–structure interaction problem,we would require to solve Eq. (1). Having solved Eq. (1) usingNewmark’s integration method, one would obtain the absolutedisplacements, velocities and accelerations of the coupled SSIproblem. To avoid solving the SSI problem directly, the dynamicresponse of the foundation without the structure is calculated. Thefree-field solution is designated by the free-field displacements v,velocities _v and accelerations €v. Here, €U
gis the ground acceleration
vector. By a simple change of variables, it becomes possible toexpress the absolute displacements U, velocities _U and accelerations€U in terms of displacements u, relative to the free-field displace-ments v. Or,
€Us
€U c
€U f
8>><>>:
9>>=>>;¼
€vs
€vc
€vf
8><>:
9>=>;þ
€us
€uc
€uf
8><>:
9>=>;
_U s
_Uc
_Uf
8>><>>:
9>>=>>;¼
_vs
_vc
_vf
8><>:
9>=>;þ
_us
_uc
_uf
8><>:
9>=>;
Us
Uc
Uf
8><>:
9>=>;¼
vs
vc
vf
8><>:
9>=>;þ
us
uc
uf
8><>:
9>=>; ð3Þ
After replacing the values of €U , _U and U from Eq. (3), Eq. (1) isexpressed as
Mss Msc 0
Mcs Mcc Mcf
0 Mf c Mf f
264
375
€us
€uc
€uf
8><>:
9>=>;þ
Css Csc 0
Ccs Ccc Ccf
0 Cf c Cf f
264
375
_us
_uc
_uf
8><>:
9>=>;
Fig. 2. Viscous dashpots connected to each degrees of freedom of a boundary
node.
A. Burman et al. / Soil Dynamics and Earthquake Engineering 34 (2012) 62–6864
þ
Kss Ksc 0
Kcs Kcc Kcf
0 Kf c Kf f
264
375
us
uc
uf
8><>:
9>=>;¼ RþF ð4Þ
where
R¼�
Mss Msc 0
Mcs Mcc Mcf
0 Mf c Mf f
264
375
€vs
€vc
€vf
8><>:
9>=>;�
Css Csc 0
Ccs Ccc Ccf
0 Cf c Cf f
264
375
_vs
_vc
_vf
8><>:
9>=>;
�
Kss Ksc 0
Kcs Kcc Kcf
0 Kf c Kf f
264
375
vs
vc
vf
8><>:
9>=>; ð5Þ
And, F ¼�
Mss Msc 0
Mcs Mcc Mcf
0 Mf c Mf f
264
375
€Ug
s
€Ug
c
€Ug
f
8>>><>>>:
9>>>=>>>;
ð6Þ
This is a numerically cumbersome approach; hence, an alter-native approach is necessary to formulate the solution directly interms of the absolute displacements of the structure. Since theanalysis is now for the foundation part only (free field analysis),hence the corresponding values of the displacement, velocity andacceleration for the structural part is taken as zero. This involvesthe introduction of the following change of variables:
€Us
€Uc
€Uf
8>><>>:
9>>=>>;¼
0
€vc
€vf
8><>:
9>=>;þ
€us
€uc
€uf
8><>:
9>=>;
_Us
_Uc
_Uf
8>><>>:
9>>=>>;¼
0
_vc
_vf
8><>:
9>=>;þ
_us
_uc
_uf
8><>:
9>=>;
Us
Uc
Uf
8><>:
9>=>;¼
0
vc
vf
8><>:
9>=>;þ
us
uc
uf
8><>:
9>=>; ð7Þ
In order to calculate the free field displacements v, only founda-tion domain is solved by considering no structure is present on it.The foundation domain is subjected to earthquake motion and thefree-field displacement for the common and other foundation nodesare obtained.
Mcc Mcf
Mf c Mf f
" #€vc
€vf
( )þ
Ccc Ccf
Cf c Cf f
" #_vc
_vf
( )þ
Kcc Kcf
Kf c Kf f
" #vc
vf
( )
¼�Mcc Mcf
Mf c Mf f
" # €Ug
c
€Ug
f
8<:
9=; ð8Þ
After obtaining the free field response (i.e. v, _v and €v) theinteraction force R is calculated using Eq. (9) in the followingsimplified manner:
R¼�
Mss Msc 0
Mcs Mscc 0
0 0 0
264
375
0
€vc
0
8><>:
9>=>;�
Css Csc 0
Ccs Cscc 0
0 0 0
264
375
0
_vc
0
8><>:
9>=>;
�
Kss Ksc 0
Kcs Kscc 0
0 0 0
264
375
0
vc
0
8><>:
9>=>; ð9Þ
After obtaining the interaction forces R, the added responses ofthe dam and foundation domain are calculated using Eq. (10). Andthen the added responses (i.e. u, _u and €u) are added to the freefield responses to get the absolute responses of the coupled soiland structure domain, following Eq. (7):
Mss Msc 0
Mcs Mcc Mcf
0 Mf c Mf f
264
375
€us
€uc
€uf
8><>:
9>=>;þ
Css Csc 0
Ccs Ccc Ccf
0 Cf c Cf f
264
375
_us
_uc
_uf
8><>:
9>=>;
þ
Kss Ksc 0
Kcs Kcc Kcf
0 Kf c Kf f
264
375
us
uc
uf
8><>:
9>=>;¼ RþF ð10Þ
The main assumptions used in this model are that the inputmotions at the level of the base rock are not considered to beaffected by the presence of the dam and that all interface nodeswill be subjected to the same free-field accelerogram [10]. Intheory any desired spatial variation of the free-field componentscould be considered at the interface, but there is seldom sufficientinformation to specify such variation. In this case, the mass of thefoundation is taken into account in the analysis such that it willrepresent the dam–foundation interaction in a relatively morerealistic manner.
2.2. Effect of hydrodynamic pressure
The effect of hydrodynamic pressure is considered according toadded mass technique originally proposed by Westergaard [17].Assuming the reservoir water to be inviscid and incompressible andits motion to be of small amplitude, the governing equation forhydrodynamic pressure is expressed as
r2p¼ 0 ð11Þ
The solution of this equation is proposed by Westergaard [17]and is used in the present work to calculate the hydrodynamicpressure imposed on the upstream face of the dam body duringany earthquake.
3. Absorbing boundary
A way to eliminate waves propagating outward from thestructure is to use Lysmer and Kuhlemeyer [11] boundaries. Thismethod consists of simply connecting dashpots to all degrees offreedom of the boundary nodes and fixing them on the other end(Fig. 2). Lysmer and Kuhlemeyer [11] boundaries are derived foran elastic wave propagation problem in one-dimensional semi-infinite bar. The damping coefficient C of the dash pot equals
C ¼ Arc ð12Þ
where A is the cross section of the bar, r is the mass density and c
is the wave velocity that has to be selected according to the type
A. Burman et al. / Soil Dynamics and Earthquake Engineering 34 (2012) 62–68 65
of wave that has to be absorbed (shear wave velocity cs orcompressional wave velocity cp). In two dimensions Eq. (12) takesthe following form, which results in damping coefficient Cn and Ct
in the normal and tangential direction, respectively.
Cn ¼ A1rcp ð13Þ
Ct ¼ A2rcs ð14Þ
The shear wave velocity cs and compression wave velocity cp isgiven by
cs ¼
ffiffiffiffiG
r
sð15Þ
cp ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEð1�nÞ
ð1þnÞð1�2nÞr
sð16Þ
where G is the shear modulus of the medium and is expressed as
G¼E
2ð1þnÞ ð17Þ
here E is the Young’s modulus and n is Poisson’s ratio.However, in general, the directions of the incident waves are
not known in advance. In these cases, it’s advantageous to use a‘diffused’ version as suggested by White et al. [18].
Assuming that the wave energy arrives at the boundary withequal probability from all directions, effective factors A1 and A2
are evaluated by minimizing the ratio between the reflectedenergy and the incident energy over the range of incident angles.For an isotropic medium this results in
A1 ¼8
15pð5þ2S�2S2
Þ ð18Þ
A2 ¼8
15pð3þ2S2
Þ ð19Þ
where
S2¼ð1�2nÞ2ð1�nÞ
ð20Þ
This gives slightly better overall efficiency than the originalLysmer–Kuhlemeyer [11] approach.
4. Numerical results
4.1. Selection of an optimum mesh size
A dam (Fig. 3) of height 15.0 m, crest-width 2.0 m and basewidth 10.0 m discretized with isoparametric linear quadrilateral
Fig. 3. The geometry of dam–foundation prototype [21].
elements. This particular dam–foundation system was originallysolved by Yazdchi et al. [21] using coupled FEM–BEM techniqueconsidering soil–structure interaction effects. While solving thedam–foundation interaction problem, the side nodes of thediscretized finite elements in the foundation portions wereconsidered to be connected to dashpots allowing only the hor-izontal movements and the bottom nodes were considered to berollers. The middle node at the base of the foundation is kept fixedin order to prevent rigid body translation in the x direction. A2�2 Gauss Integration rule is adopted for the calculation of boththe stiffness matrix and the mass matrix. The dam and thefoundation are assumed to be linear elastic with the followingmaterial properties. Poisson’s ratio 0.2; modulus of elasticityEd¼3�107 kN/m2 and mass density as 2600 kg/m3. The Poisson’sratio and the mass density of the foundation were assumed to bethe same as those of the dam.
In order to arrive at a suitable mesh grading for the dam–foundation prototype, convergence studies are carried out tocheck the convergence of the time periods obtained from eigenvalue analysis. Also, a horizontal static load of 1000 kN is appliedon the rightmost crest point of the dam shown in Fig. 3. Thevalues of displacements obtained for the application of the staticload are also noticed. It is observed that the time periods and thehorizontal displacements at the rightmost crest point convergessufficiently for a mesh division of 4�6 (horizontal� vertical) inthe dam portion as well as for a mesh division of 12�4 for thefoundation region. However, a mesh division of 4�10 (horizon-tal� vertical) in the dam portion has been chosen for the valida-tion purpose.
4.2. Validation of proposed algorithm
The results of the present model are compared with the resultsof dam model (Fig. 3) analyzed by Yazdchi et al. [21] for Koynaground motion. The foundation size of 100 m�50 m has beenconsidered for validating the algorithm with the dam prototypesolved by Yazdchi et al. [21] considering the effect of soil–structure interaction. An attempt is made to compare the presentresults with the result obtained by Yazdchi et al. [21]. All thevertical side nodes are fitted with dashpots while the bottomnodes are considered to be rollers. The Koyna earthquake accel-eration (Fig. 4) is applied to the dam prototype with a scalingfactor of 2.5. Yazdchi et al. [21] solved this problem by thecoupled FEM–BEM method. They included the effect of viscousdamping with a damping ratio of 0.05. Moreover, the effect ofhydrodynamic pressure was incorporated in the analysis by theadded mass concept proposed by Westergaard [17]. The effect ofwave scattering and reflection was tackled by coupled FEM–BEMmethod. When the same dam–foundation model is solved by theproposed scheme, the effect of viscous damping and the hydro-dynamic pressure is also considered in similar way. Also, initially thedam has been analyzed considering the effects of its self weight andthe hydrostatic pressure which produced initial acceleration in thedam body.
Fig. 4. Koyna longitudinal earthquake motion 1967 [21].
A. Burman et al. / Soil Dynamics and Earthquake Engineering 34 (2012) 62–6866
The nodes at the truncated boundary as well as at the bottom areprovided with dashpots in both the horizontal and tangentialdirection. In spite of the differences between the solution procedureadopted with one used by Yazdchi et al. [21], the proposed modelyields a similar displacement pattern of the horizontal displacementat the crest node though the magnitudes differed. In the presentwork, the hydrodynamic pressures at the upstream face of the damare calculated from the solution of Eq. (11). These hydrodynamicforces are applied on the upstream face of the dam. Table 1 showsthe comparison between the results of Yazdchi et al. [21] and that ofthe proposed method for different Ef / Ed (impedance ratio) ratios. Themaximum crest displacement of the dam under seismic excitation byboth the method has been tabulated in the Table 1 for a comparisonpurpose. The present results are also compared with the resultsobtained by Reddy et al. [14]. The obtained displacements by theproposed interaction scheme are in very close agreement with theresults obtained both by Yazdchi et al. [21] and Reddy et al. [14].Table 2 shows the comparison between the values of horizontal crestaccelerations (a/g) for different values of impedance ratios (Ef/ Ed)with those obtained by Yazdchi et al. [21]. It is a common notionthat as the impedance ratio (Ef/ Ed) increases, the responses (bothdisplacements and normalized acceleration values i.e. a/g values)shall decrease. The results obtained in the present paper arecommensurate with that interpretation. The slight discrepancybetween the two results obtained in both cases might be due to:
i.
TabCom
H
d
Im
N/A
TabCom
H
Im
Use of different methods and numerical tools for the solutionof the coupled system.
ii.
Different mesh sizes considered for the problem.In spite of differences in the methodologies used for solvingthis particular problem, appreciable agreement in the obtained
le 1parison of maximum horizontal crest displacements.
orizontal crest
isplacements (mm)
Coupled FE–BE solution
Yazdchi et al. [21]
Reddy
et al. [14]
Proposed
method
pedance ratio (Ef/Ed)
0.5 6.89 8.60 6.75
�7.53 N/A �8.62
1.0 4.38 4.60 5.03
�4.41 N/A �6.08
2.0 4.27 4.50 4.48
�3.85 N/A �5.22
4.0 4.11 3.90 4.09
�3.70 N/A �4.70
—Data Not Available in aforementioned paper.
le 2parison of maximum horizontal crest accelerations (a/g).
orizontal crest
accelerations (a/g)
Coupled FE–BE
solution [21]
Proposed
method
pedance ratio (Ef/Ed)
0.5 3.99 4.87
�3.41 �4.28
1.0 3.48 3.59
�3.18 �2.91
2.0 3.89 3.18
�3.48 �2.60
4.0 5.08 3.15
�4.19 �2.49
values of horizontal crest displacements is achieved by thepresent method. In the present analysis, the side nodes at theboundary of the foundation domain are attached to viscousdashpots and the nodes at the base are fixed apart from themiddle node. Therefore, in subsequent calculations, the bottomnodes of the foundation domain are provided with rollers exceptthe center node which will remain to be fully fixed to prevent anytype of rigid body translation.
4.3. Response of Koyna dam
The seismic response of Koyna dam has been investigatedconsidering the interaction behavior of a linear concrete dam andan elasto-plastic foundation subjected to Koyna earthquake(1967) acceleration. The foundation material is assumed to be ofhard rock. The width and the depth of the foundation are assumedto be 350.0 m and 100.0 m, respectively. The geometry of thedam–foundation system chosen for the analysis purpose is shownin Fig. 5. The material properties of the dam are as follows:
The Young’s modulus, Poisson’s ratio and mass density of dambody are considered as 3.15eþ10 N/m2, 0.235 and 2415.816 kg/m3,respectively. Similarly the Young’s modulus, Poisson’s ratio andmass density of foundation are considered as 1.75eþ10 N/m2,0.2 and 1800.0 kg/m3, respectively.
In order to arrive at an optimum mesh grading for thisparticular problem, convergence study has been carried out underexternal loadings. It is observed that the results convergedsufficiently for a mesh grading of 8�5 for the dam domain andfor a mesh grading of 12�5 for the foundation domain withsufficient degree of accuracy. The results for this analysis are notprovided because they are similar in nature with the results ofconvergence analysis carried out in Section 4.1 for the solution ofdam–foundation prototype.
In the following section results of the comparison of the dam–foundation interaction analysis with that of the rigid analysis ispresented. Rigid analysis is carried out by considering the Koynaearthquake (1967) excitation forces for the dam portion only. Inthese analyses, to propound the effect of soil–structure interac-tion upon the rigid analysis, the self weights of the dam andfoundation have not been considered. In rigid analysis the bottomnodes of the dam have been considered to be fixed, while in theinteraction analysis the sides have been fixed with dashpots,the middle node at the base of the foundation has been fixed, andthe remaining nodes at the base of the foundation have been
Fig. 5. The geometry and boundary condition of the Koyna dam–foundation
system.
A. Burman et al. / Soil Dynamics and Earthquake Engineering 34 (2012) 62–68 67
fitted with rollers allowing only horizontal translation. Fig. 6shows the comparison between the variation of crest displace-ments, obtained with and without dam–foundation interactionanalysis. The maximum and minimum values of the horizontalcrest (node ‘B’ as shown in Fig. 5) displacements for the interac-tion analysis are found to be 7.76 cm and �8.08 cm respectively,while those for rigid analysis are 6.06 cm and �7.22 cm. Fig. 7shows the comparison between the variations of major principalstresses at the neck of the dam (Node A) for rigid and flexible baseof the dam. A maximum value of 15.80 MPa is obtained withflexible base while the maximum value for rigid base is found tobe 14.40 MPa. Fig. 8 shows the comparison between the varia-tions of minor principal stresses at the heel node of the dam(Node A), obtained for the dam–foundation interaction analysisand rigid analysis. A minimum value of �14.0 MPa is obtained forthe dam–foundation coupled system while the minimum valuefor rigid base is found to be �12.60 MPa. It is evident from theabove observations that the displacements have increased for
-0.1-0.08-0.06-0.04-0.02
00.020.040.060.08
0.1
0Tim
Hor
izon
tal c
rest
dis
plac
emen
t (m
)
1 2 3
Fig. 6. Variation of crest displac
-4.00E+00-2.00E+000.00E+002.00E+004.00E+006.00E+008.00E+001.00E+011.20E+011.40E+011.60E+011.80E+01
0
Maj
or p
rinci
pal s
tres
s (M
Pa)
1 2 3
Fig. 7. Major principal stre
-16-14-12-10
-8-6-4-2024
0Tim
Min
or p
rinci
pal s
tres
s (M
Pa)
1 2 3
Fig. 8. Minor principal stre
interaction analysis compared to the rigid case. Also, the stressesat the neck have increased when the flexibility effect of founda-tion is taken care in the coupled analysis. Since the neck region ofthe dam experiences more stresses compared to the stressesobserved at the heel, therefore the stresses obtained from dam–foundation interaction analysis should govern design criteria ofthe dam.
5. Conclusion
The paper presents a methodology for the analysis of concretegravity dam subjected to seismic excitations considering the soil–structure interaction effect. The proposed method is validated fromthe literature which shows the accuracy of the developed algorithm.The dam like structure, having the coupling effect due to the under-lying foundation material during earthquake excitations is analyzed.The numerical results presented here prove the efficiency of the
e (sec)
Rigid analysisDam-foundation interaction analysis
4 5 6 7 8
ement for Koyna Excitation.
Time (sec)
Rigid analysisDam-foundation interaction analysis
4 5 6 7 8
ss variation at node A.
e (sec)
Rigid analysis
Dam-foundation interaction analysis
4 5 6 7 8
ss variation at node A.
A. Burman et al. / Soil Dynamics and Earthquake Engineering 34 (2012) 62–6868
present algorithm to solve a soil–structure coupled problem ofmassive structures such as concrete gravity dams. The advantageof using the present direct method is that it requires lesscomputational effort, in terms of both time and memory. Theresponses of the soil–structure system considering an absorbingboundary indicate that the incident energy is effectively absorbedat the truncation boundary. Another advantage of this method isthat it requires less computational effort since it avoids evalua-tion of convolution integrals and Fourier transforms to calculatesoil–structure interaction forces. The algorithm presented here issimple so that it may be programmed easily. The results showthat the displacements and stresses have increased for the elasticas compared to the rigid base. Hence it is advisable to carry outthe interaction analysis for massive structures like dams underflexible base. It is also observed that the neck is the most severelystressed zone; hence one may expect the appearance of cracksaround the neck region of the dam.
References
[1] Chopra AK, Chakrabarty P. Earthquake analysis of concrete gravity damsincluding dam-fluid-foundation rock interaction. Earthquake Engineeringand Structural Dynamics 1981;9:363–83.
[2] Clough RW, Penzien J. Dynamics of structures. first ed. McGraw-Hill Com-pany; 1975. p. 584–8.
[3] Clough RW, Chopra AK. Earthquake response analysis of concrete dams. In:Hall WJ, editor. Structural and geotechnical mechanics, Chap-19. Prentice-Hall; 1977. p. 384–6. 1977.
[4] Estorff OVon, Kausel E. Coupling of boundary and finite elements for soil-structure interaction problems. Earthquake Engineering and StructuralDynamics 1989;18:1065–75.
[5] Felippa CA, Park KC. Staggered transient analysis procedures for coupledmechanical systems: formulation. Computer Methods in Appl. Mech. Engg1980;24:61–111.
[6] Fenves G, Chopra AK. Simplified earthquake analysis of concrete gravitydams: Separate hydrodynamic and foundation effects. Journal of EngineeringMechanics, ASCE 1985;111(6):715–35.
[7] Guan F, Moore ID. New techniques for modeling reservoir–dam and foundation–dam interaction. Soil Dynamics and Earthquake Engineering 1997;16:285–93.
[8] Jahromi HZ, Izzuddin BA, Zdravkovic L. Partitioned analysis of nonlinear soil–structure interaction using iterative coupling. Interaction and MultiscaleMechanics 2007;1(1):33–51.
[9] Khalili N, Valiappan S, Tabatabaie Yazdi. J, Yazdchi M. 1D infinite elementfor dynamic problems in saturated media. Communications in NumericalMethods in Engineering 1997;13:727–38.
[10] Leger P, Boughoufalah M. Earthquake input mechanisms for time-domainanalysis of dam-foundation systems. Engineering Structures 1989;11(1):33–46.
[11] Lysmer J, Kuhlemeyer RL. Finite dynamic model for infinite media. Journal ofEngineering Mechanics Division, ASCE 1969;95(EM4):859–77.
[12] Maity D, Bhattacharyya SK. A parametric study on fluid structure interactionproblems. Journal of Sound and Vibration 2003;263:917–35.
[14] Reddy BV, Burman A, Maity D. Seismic response of concrete gravity damsconsidering foundation flexibility. Indian Geotechnical Journal 2008;38:187–203.
[15] Rizos DC, Wang Z. Coupled BEM–FEM solutions for direct time domain soil–structure interaction analysis. Engineering Analysis with Boundary Elements2002;26:877–88.
[16] Valiappan S, Zhao C. Dynamic response of concrete gravity dams includingdam-water-foundation interaction. International Journal Numerical Methodsfor Engineering 1992;16:79–99.
[17] Westergaard HM. Water pressures on dams during earthquakes. Transactionsof the ASCE 1933;98:418–72.
[18] White W, Valliappan S, Lee IK. Unified boundary for finite dynamic models.Journal of Engineering Mechanics Division, ASCE 1977;103:949–64.
[19] Wolf JP. Dynamic soil–structure interaction. Englewood Cliffs, NJ: PrenticeHall; 1985.
[20] Wolf JP, Song C. Finite element modeling of unbounded media. New York:Wiley; 1996.
[21] Yazdchi M, Khalili N, Valliappan S. Dynamic soil-structure interactionanalysis via coupled finite element-boundary element method. SoilDynamics and Earthquake Engineering 1999;18:499–517.
[22] Zienkiewicz OC, Kelly DW, Bettes P. The coupling of the finite elementmethods and boundary solution procedures. International Journal of Numer-ical Methods in Engineering 1977;11:355–77.
