Comparison of added mass method with sophisticated analytical BEM-FEM approach using ADAD-IZIIS software

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1 Comparson of added mass method wth sophstcated analytcal BEM-FEM approach usng ADAD-IZIIS software V. Mrcevska Insttute of Earthquake Engneerng and Engneerng Sesmology, uv. Ss. Cyrl and Methodus, Macedoa I. Bulajć Mng Insttute Belgrade, Serba K. Manova Insttute of Earthquake Engneerng and Engneerng Sesmology, uv. Ss. Cyrl and Methodus, Macedoa SUMMARY Ths paper presents a comparson of dynamc response of a flud-arch dam coupled system obtaned by use of the classcal Westergaard added mass formulaton and the more advanced analytcal FEM-BEM techque. Heren the nfluence of the appled techques on the dstrbuton and ntenstes of the mafested stresses s presented wthn the dam body. Dynamc response s acheved assumng that the arch dam has a lnear elastc materal behavor. For solvng the dynamc equaton of moton drect ntegraton techque s used, whereat the dam foundaton, as well as foundaton undergong the reservor, s assumed rgd. The FSI s performed followng the assumptons that the water n the reservor s nvscd wth rrotatonal moton of the water partcles lmted to small ampltudes of veloctes and acceleraton whle the gravty surface waves are neglected. The presented results are obtaned by use of FE BE orented software ADAD-IZIIS, especally wrtten for analyses of arch dams. The software also has the possblty for calculaton of hydrodynamc effects accordng to Westergaard added mass method. Keywords: flud-arch dam nteracton, BEM-FEM, added mass, ADAD-IZIIS software, hydrodynamc pressure 1. INTRODUCTION The phenomenon of FSI that occur durng the sesmc response of the coupled dam-reservor system has been for the frst tme physcally explaned and mathematcally solved by Westergaard (1933). The very frst dynamc analyses, consderng the hydrodynamc nertal forces, were based on applcaton of added mass method. Besdes ts smplcty n applcaton and popularty among the analysts, Kuo (198) and Kotsubo (1960) have emphaszed that evaluaton of hydrodynamc pressure usng the added mass approach s not accurate, snce the assumptons used for calculaton of the added mass are not n accordance wth random nature of the earthquake phenomenon. Ths type of analyss leads to a conservatve desgn. There s a common belef that n case of a rgd structure, the magtude of the hydrodynamc pressure becomes hgh, but ths s not always true. The magtude of hydrodynamc pressure may ncrease sgfcantly for flexble structures as well. If resonance effect and the energy release mechasm n the flud occur t can lead to the development of unsound desgn. Thus t s necessary to study the flud structure nteracton problem consderng the flexblty property of the structure that may alter the behavor of the flud doman sgfcantly. However, the added mass method s very effectve for calculaton of Egen values for coupled dam-reservor systems. Durng the last decade consequent and extensve research was carred out for development of more sophstcated approaches and analytcal methods amed towards overcomng of dsadvantages of the added mass method as well as lghteng the mpact of the characterstc features of ths phenomenon over the calculaton results. They are dvded nto soluton methods concerng FEM- FEM or BEM-FEM numercal techques based on tme or frequency doman approach, Seghr et al.

2 (009), Tsa and Lee (1987), Tsa (199), Tsa et al. (199). Ths paper deals wth coupled BE-FE analyss of the flud structure systems consderng the coupled effect of elastc structure and an ncompressble and nvscd flud. The soluton of the coupled system s accomplshed by solvng the moton of the two systems, the dam and the flud n the reservor, separately. The nteracton effects at the flud sold nterface are enforced by addng the matrx of hydrodynamc forces to the classc equaton of dynamc moton of dam, ADAD-IZIIS (008). Research was made for concrete arch dam wth structural heght of H=130m. The flud dam system was subjected n upstream drecton to El Centro N-S record of the Imperal Valley earthquake. The frst seven seconds of the record was used and scaled to the peak acceleraton of 0.3g. The tme step ncrement of 0.01s was chosen for the ntegraton process. In a general flud-sol-structure system, the structure and the reservor doman are supported by the elastc sol medum. However, n the analyss presented n ths paper the rock foundaton s assumed to be rgd whch means that the dam foundaton nteracton s neglected and the moton of all nodes at the flud dam-rock nterface are followng the nput ground acceleratons wthout any modfcaton. The dam propertes are as follows: Young s modulus E = 31.5 GPa; mass densty ρ = 450 kg/m 3 ; Posson s rato ν = 0.; the acoustc wave velocty n water c = 1440 m/s.. GOVERING EQUATIONS.1 Governg equatons for coupled BE-FE FSI soluton The ncremental form of dfferental equaton of moton for system subjected to the dynamc acton ncludng the effect of flud-dam nteracton s as follows: [ M] U(t) & + [ C] U(t) & + [ K] U(t) = [ M] [ R] a g (t) + HDF(t) & (.1) where [M] s the mass matrx, [R] s the acceleraton transformaton matrx, [C] s the structural dampng matrx, [K] s the structural stffness matrx, U(t) ; U & (t) ; U & (t) are vectors of nodal ncremental dsplacements, veloctes and acceleratons, relatve to the ground, a g (t) s the vector of ncremental ground acceleratons, HDF(t) s tme dependant vector of ncremental nodal forces at the dam-flud nterface caused by the propagaton of the acoustc waves generated n the reservor doman. Wlson-θ method s used for the soluton of the general equaton of moton Eqn..1 and for θ=1.38 uncondtonal stablty of the drect ntegraton process s acheved. Ths method requres the dampng matrx to be represented n explct form. Ths s accomplshed usng Raylegh dampng. Accordng to Raylegh, the structural dampng n the system can be ncluded by the Eqn... [ C] α [ M] + β [ K] = (.) R R The Raylegh dampng coeffcents α R and β R are determned usng two known dampng ratos ξ along wth the correspondng egenvalues ω Eqn..3 chosen to present energy dsspaton ablty of the flud-structure system n the best way. α ω1ωξ1 ωω1 ξ ωξ ω1ξ1 R = R = ω ω 1 ω ω1 β (.3) Durng the dynamc response, at any tme step, the vector of ncremental hydrodynamc forces s drectly added to the vector of sesmc force. The vector of HDF s defned applyng BEM numercal techque for soluton of Laplace s equaton of flud moton as well as a specally wrtten algorthm wthn the software ADAD-IZIIS. The governg equaton for solvng the small ampltude rrotatonal moton of the mpounded ncompressble and nvscd flud s governed by the three-dmensonal Laplace s equaton as follows:

3 W W W + + = 0 x y z (.4) where W(x, y, z) s the functon of the potental n the flud doman wth boundary surfaces Γ 1 and Γ where essental and natural type of boundary condtons exst. Applyng BE techque, the dscretzaton of boundary surfaces s by an assembly of eght nodded quadratc serendpty type of boundary elements as follows: 1 W + W p ( p W )dγ n n W p ( p W )dγ n n NEL NEL 1 + = nel= 1 Γ1 nel= 1 Γ 0 (.5) Where: =1, NBE; NBE - number of nodes n the boundary element model Father on, despte drect Czygan and Estorff (00), Yu et al. (001), Estorff and Antes (1991), Fuku (1987), Karabals and Beskos (1985), or teratve couplng methods Soares et al. (005), Soares and Estorff (004), Soares et al. (004), (005), the software utlze smple and effectve numercal techque of matrx of hydrodynamc nfluence that provdes a compatble lnk of both meda whch are n nteracton. It s accomplshed utlzng the concept of vrtual work of ut acceleraton.. Governg equatons of dynamc moton n case of applcaton of added mass method If added mass method s used for solvng FSI then t s necessary to calculate the amount of the vrtual mass fxed aganst the upstream face of the dam. The added mass should produce nertal forces equvalent to the expected hydrodynamc effects at any tme ncrement. The procedure for ths calculaton s gven n secton 3. If added mass method s used then the vector of ncremental nodal hydrodynamc forces n Eqn..1 s replaced by the modfed mass matrx. The structural mass matrx s modfed by the ncrement of the added mass that produces equvalent ncrement of hydrodynamc force durng the dynamc moton. Therefore, the dfferental equaton of dynamc moton Eqn..1, acqures the followng form: [ M + Ma] U(t) & + [ C] U(t) & + [ K] U(t) = [ M + Ma] [ R] a (t) & (.6) The added mass method mples rearrangng of the ncremental dfferental equaton. After some transformatons t acqures the followng form: 6 ˆ 6 3 k [ ] ˆ τ ( M + M ˆ ˆ A) U U& 3U&& + αm + βk U U&& 3U& + K U = P (.7) τk τk τk After arrangng known and unknown terms Eqn..6 becomes: where, K ˆ U = ˆ F (.8) 6 3 K = K + (M + MA ) + [ αm + βk ] τ τ (.9) k k g ˆ F = ˆ P + (M + M A 6 ) τk τk U& + 3U&& + [ αm + βk ] U&& + 3U& (.10)

4 Eqn..8 expresses the dynamc equlbrum of a lnear or nonlnear system at the ""-th tme ncrement and t has the same form as the statc equlbrum equaton. 3. WESTEGAARD SOLUTION OF HDP ADDED MASS METHOD When Westergaard formulaton s appled when solvng the FSI of arch dam, there s a need for some modfcaton of basc Westergaard assumpton. Frstly, t should be consdered that earthquake moton s not normal to the dam face. In general the upstream face of the dam s double curved and therefore, the orentaton of the faces relatve to the ground moton vares from pont to pont. Also recogton should be gven to the nfluence of dam flexblty over the magtudes of the mafested hydrodynamc effects. Namely the relatve response acceleraton of dam face vares from pont to pont and dffers from the earthquake acceleraton. Accordng the classc Westergaard soluton, the pressure caused by ut acceleraton at any pont s expressed by: f 8αwh πt n = cos π T 1,3,5 1 n c n e q n nπy sn h (3.1) where: q n nπcnx TS =, k = v s ρ, h c n 16h h sec. = 1 = 1 (3.) n l n 1000Tft. The parameters n the above expressons are well known Westergaard (1933). Calculaton s based on use of the frst terms of nfte row of Westergaard soluton. The perod of propagaton of the acoustc waves s selected from the Fourer spectrum of the exctaton and amounts to T=0.5sec. The normal acceleraton& r& due to Cartesan components s gven by the drecton cosnes between the coordnate axs and the normal. & r = L & r L = { λ λ λ } (3.3) The normal force at pont s obtaned as follows: P xn yn zn = F L & r (3.4) In order to use fte element analyss ths normal force must be resolved nto Cartesan components: P = L P = L F L & r (3.5) T T Hence the dagonal terms of the added mass matrx are as followng: m a T = L F L & r / ρ (3.6) Here ma s a dagonal (lumped) mass matrx for each upstream node, and s to be added approprately to the general mass matrx n order to obtan the full nertal effect. The added mass exhbts some propertes of ts own. Thus, n dstncton of the structural mass, ths value s ndependent of the drecton of the degrees of freedom, the value of the added mass depends on the drecton n whch the exctaton develops and the drecton of the degrees of freedom. Snce the earthquake s assumed to travel upon some drecton c and f the structural degrees of freedom are n drectons x, y, z, and then the components of the lumped mass have the followng expressons: m h = F / ρ (3.7)

5 mcx = m cos(n, c) cos(n, x) h h m m cos(n,c) cos(n, y) m h cy = h cz = m h h cos(n, c) cos(n, z) Fgure 1. Computaton dagram for added mass of water 3.1 Dervaton of approxmate Westergaard formulas For the purpose of approxmate computaton one mght replace the nfte row of Westergaard soluton Eqn. 3.1 by a parabola, even f ths parabola has a slopng, not vertcal, tangent at the bottom. p = Cα hy P C = (3.8) α hy The man source of varaton of C s the constant c 1 wth n=1, n Eqn As c 1 appears as denomnator n the frst term n each of the sums n eq. 3.1, t s reasonable to express C va expresson Eqn. 3.9, where K s a constant. K C = (3.9) c 1 Usng the fact that the coeffcent C depends on the rato of the mpounded water depth h and predomnate perod of exctaton T, and by nspecton of the numercal results obtaned by applcaton of the Westergaard equatons [1], t has been concluded that the most conveent value for the coeffcent K s K=0.055 ton/ft 3. Accordngly the approxmate formula for hydrodynamc pressure calculaton s: = Cα hy = 0.055ton / ft α hy (3.10) hsec Tft p 3 For the analyss presented heren the value of the coeffcent C s C=1.4 kn/m DISCUSSION OF THE RESULTS - WESTEGAARD VERSES BE-FE METHOD If flexble structure s consdered, lke an arch dam, added mass prncple, accordng to Westergaard does not gve accurate resultants, because Westergaard theory s based on straght rgd dam assumpton. Snce software ADAD-IZIIS have the possblty for calculaton of hydrodynamc pressure accordng to Westergaard added mass method and accordng to coupled FE-BE method, fallowng fgures present the dscrepances n obtaned resultants. Fgures, 3, and 4. present the tme hstory responses of relatve dsplacement, velocty and acceleraton for selected node n X, Y and Z drecton respectvely, at dam crest (crown cantlever) where the extremes of the response occurred. Obvously, the response acceleraton of the dam s modfed by 37% when BEM-FEM numercal method s used for solvng FSI. Westergaard added mass method gves 8% modfcaton of the dynamc response of the dam. The flexblty property of

6 the structures and the nfluence of the reservor doman alter the behavor of the flud sgfcantly and consequently the coupled system has a stronger response. Fgure 5. presents the tme hstores of the three prncpal stresses at the nodes where the extreme of each stress s acheved when FSI effect s omtted. It also presents the mpact of the FSI over the tme hstores of these stresses and ts extremes. Fgure 7. presents a snapshot of hydrodynamc pressure dstrbuton over the nterface, at tme T=4.95sec. It s obtaned under the assumptons that the topology of the canyon has a regular shape as ndcated n the drawng. However, the nsght n the developed hydrodynamc effects s not possble when added mass method s used. Accordng to presented results heren, t s evdent that added mass method s not applcable to flexble systems, because the dscrepances are sgfcant, whch s best shown n fgure 6. The fgure shows dagrams of stress dstrbuton wth and wthout ncluded hydrodynamc effects, whereat hydrodynamc effects are calculated accordng to added mass method and coupled BE-FE method. The stress extreme s ncreased by 15% f added mass method s used and 49% f coupled BE-FE method s used, whch gves error of 3% usng added mass method n arch dam desgn. There s a common belef that n case of a rgd structure, the magtude of the hydrodynamc effects becomes hgh, whch can not be generalzed. In order to confrm or deny ths statement, other analyses have been done for the 70m depth of mpounded water. As the dam s more rgd n the lower part and consequently the absolute acceleratons at the dam-flud nterface are lower t appears that the above statement s almost vald for ths partcular case, Fgure 8. Namely, despte the case of 100m water depth, here the msmatchng of both approaches s neglgble. Fgure. Modfcaton of the dam response n X drecton due to the FSI effect

7 Fgure 3. Modfcaton of the dam response n Y drecton due to the FSI effect Fgure 4. Modfcaton of the dam response n Z drecton due to the FSI effect

8 Fgure 5. Tme hstores of extreme prncpal stresses wth and wthout FSI effect a) Stress dstrbuton on dam extrados face G3, T=.4 s, wthout HDE b) Stress dstrbuton on dam extrados face G3, T=.4 s, wth HDE usng BEM-FEM c) Stress dstrbuton on dam extrados face G3, T=.4 s, wth HDE usng added mass method Fgure 6. Stress dstrbuton. Modfcaton of the dam response due to the FSI effect

9 Fgure 7. Snapshot of hydrodynamc pressure dstrbuton at tme T=4.95 sec Fgure 8. Modfcaton of the dam response n Y drecton due to the FSI effect

10 CONCLUSION The Westergaard added mass method provdes acceptable results only n the range of restrcted hypothess. Snce t neglects the dam flexblty and water compressblty and does not requre any dscretzaton of the reservor doman wherever these features have an mpact on the magtude of hydrodynamc effects there wll be dscrepancy of the obtaned resultants. Recently develop BEM- FEM, FEM-FEM or hybrd methods are focused towards overcomng dsadvantages of added mass method. They gve far more realstc estmates of FSI effects. In ths partcular case whether added mass method would over or under estmate the hydrodynamc effects depends on the water depth n respect to the flexblty propertes of the dam. REFERENCES Westergaard H.M. (1933). Water pressure on dams durng earthquakes. Transactons of the Amercan Socety of Cvl Engneers, Vol. 98, Kotsubo, S. (1960). Dynamc water pressure on dams durng earthquake. nd World Conference on Earthquake Engneerng, Kuo J.S.H. (198). Flud-structure nteractons: Added mass computaton for ncompressble flud. UCB/EERC- 8/09 Report, Uversty of Calfora, Berkeley Seghr A., Tahakourt A. and Bonnet G. (009). Couplng FEM and symmetrc BEM for dynamc nteracton of dam reservor systems. Engneerng Analyss wth Boundary Elements. Vol. 33, Tsa C.S., Lee G.C. (1987). Arch dam flud nteractons by FEM-BEM and substructure concept. Internatonal Journal of Numercal Methods n Engneerng. Vol. 4, Tsa C.S. (199). Analyses of three-dmensonal dam-reservor nteracton based on BEM wth partcular ntegrals and sem-analytcal soluton. Computers and Structures. Vol. 43:5, Tsa C.S., Lee G.C. and Yeh C.S. (199). Tme-doman analyses of three-dmensonal dam-reservor nteractons by BEM and sem-analytcal method, Engneerng Analyss wth Boundary Elements. Vol. 10:, Czygan O. and Estorff von O. (00). Flud structure nteracton by couplng BEM and nonlnear FEM, Engneerng Analyss wth Boundary Elements. Vol 6, Yu G., Mansur W.J., Carrer J.A.M., and Le S.T. (001). A more stable scheme for BEM/FEM couplng appled to two-dmensonal elastodynamcs. Computers & Structures. Vol. 79, Estorff von O. and Antes H. (1991). On FEM BEM couplng for flud structure nteracton analyss n the tme doman. Internatonal Journal for Numercal Methods n Engneerng. Vol. 31, Fuku T. (1987). Tme marchng BE FE method n D elastodynamc problems. In: Proceedngs of the nternatonal conference on boundary element methods IX, Stuttgart. Karabals D.L and Beskos D.E. (1985). Dynamc response of 3D flexble foundatons by tme doman BEM and FEM. Sol Dyn Earthq Eng. Vol. 4, Soares D.J., Carrer J.A.M. and Mansur W.J. (005). Non-lnear elastodynamc analyss by the BEM: an approach based on the teratve couplng of the D-BEM and TD-BEM formulatons. Engneerng Analyss wth Boundary Elements. Vol. 9, Soares D.J. and Estorff von O. (004). Combnaton of FEM and BEM by an teratve couplng procedure, 4th European Congress on Computatonal Methods n Appled Scences and Engneerng, ECCOMAS, Jyva skyla, Fnland,. Soares D. J., Estorff von O. and Mansur W.J. (004). Iteratve couplng of BEM and FEM for nonlnear dynamc analyses. Computatonal Mechacs. Vol. 34, Soares D. J., Estorff von O. and Mansur W.J. (005). Effcent nonlnear sold flud nteracton analyss by an teratve BEM/FEM couplng, Internatonal Journal for Numercal Methods n Engneerng. Vol. 64, Mrcevska V. and Bckovsk (008). ADAD-IZIIS software: Analyss and Desgn of Arch Dams, IZIIS- uv. Ss. Cyrl and Methodus, User s Manual,