THREE-DIMENSION DYNAMIC SOIL-STRUCTURE INTERACTION ANALYSIS USING THE SUBSTRUCTURE METHOD IN THE TIME DOMAIN

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1 The 4 th October Bejng Chna THREE-DIENSION DYNAIC SOIL-STRUCTURE INTERACTION ANALYSIS USING THE SUBSTRUCTURE ETHOD IN THE TIE DOAIN X..Yang Y.Chen and B.P.Yang 2 Assocate Professor aculty of Constructon Engneerng Guangdong Unversty of Technology Guangzhou Chna 2 Professor Insttute of Engneerng echancs Chna Sesmologcal Bureau Harbn Chna Emal:xmyang_86@yahoo.com.cn ABSTRACT : Based on the wave propagaton theory and the lumped-mass explct fnte element procedure wth a local transmttng artfcal boundary the substructure method s formulated wth step-by-step ntegraton and central dfference approach. Snce the explct fnte element and step-by-step ntegraton scheme s adopted ths presented method leads to the ablty n computatonal feasblty and the applcablty n engneerng. In addton the local transmttng artfcal boundary results n the sgnfcant reducton n computatonal effort wth lttle loss of accuracy. To llustrate the effcacy of the procedure numercal examples are studed for a structure founded on a rgd foundaton whch s on the surface of sem-nfnte sol medum. The sol-structural dynamc predctons under the assumpton of rgd foundaton on a unform sm-nfnte medum usng ths substructure method are n good agreement to the reference solutons by drect method. Another numercal example s studed for the nfluence by the bg stffness of sol medum on the dynamc characters of the sol-structure system. Ths predcton can also be n agreement the reference by the typcal method. These predctons can dsplay the appled feasblty n engneerng. KEYWORDS: sol-structure nteracton substructure method transmttng boundary condton. INTRODUCTION SECTION HEADING Snce some structures such as nuclear power plants and hgh buldng have three-dmensonal sze the effect of sol and structure dynamc nteracton can be ratonally consdered only when three-dmenson analyss s performed because of structural ntegrty. There have been many studes on SSI theores and methods whch are generally classfed nto drect method and substructure method ether n tme doman or n frequency doman. In the drect method the upper structure the rgd foundaton and the bounded sol medum whch s adjacent to the foundaton are modeled as a whole wth the artfcal boundary. Snce all the sol-structure system s modeled the computatonal cost for the complex system s generally too hgh even to solve the three-dmenson lnear elastc system. In the substructure method the sol-structure system s dvded nto two or more substructures. Each substructure s modeled separately and s connected to the general structure through the nterface of adjacent to other substructures. The earlest substructure developments have been made for several effectve analytc solutons consderng the sm-nfnte sol zone as homogeneous sotropc and elastc medum as substructure [Luco & Westmann 97; Oen 973; Wong & Luco 978]. After solvng the mpedance functon of the sm-nfnte sol zone the nteracton force whch the sol apples to the actual structure s replaced by the mpedance functon [Veletsos & We 97; Wong & Luco 978]. However t s dffcult for the analyss method to solve the sol-structure nteracton problems when the sem-nfnte sol zone s heterogeneous or ansotropc or nonlnear. Latterly the researcher s nterests have been growng n developng the dscrete method to deal wth the sol-structure nteracton problems [Shah & Wong 982; Wolf 988; Zhang Wegner & Haddow 999]. Compared wth the drect method the substructure method can smplfy the computatonal model decrease the computatonal cost greatly and easly dealt wth the complex sol-structure system by dvdng the whole system nto necessary substructures. However snce most of substructure analyss methods are developed to use mplct equatons to solve ths nd of problems the

2 The 4 th October Bejng Chna computatonal freedoms wll be relatve to the complexty condtons of sol medum and actual upper structure. Ths treatment often also leads to dffculty n dealng wth the engneerng problems. Snce the numercal procedure usng ether drect or substructure method can not drectly model the unbounded sol medum the fnte sol zone have to be consdered. The effect of the truncated sol medum s generally consdered approxmately by the artfcal boundary such as a vscous boundary [Lysmer & Kuhlemeyer 969] a superposton boundary [Smyh 974] and several others [Lysmer & Waas 972; Lao et al 984; E.Kausel 988]. In ths paper a new three-dmenson substructure numercal procedure s presented for analyss of dynamc sol-structure nteracton. Utlzng the advantage of local transmttng artfcal boundary [Lao et al 984] for wave propagaton explct numercal procedure the tme-doman explct fnte element method s formulated wth step-by-step ntegraton wth central dfference and lateral dfference approach. The ey to the computatonal effcency of the numercal procedure s the applcaton of the explct procedure for sol zone and local transmttng artfcal boundary as well as substructure treatment. Based on ths numercal procedure the correspondng tme-doman explct fnte element program T3DSSI s programmed to analyze the dynamc sol-structure nteracton problems. Program T3DSSI has such advantages as smple nput data fast computatonal speed and extensve engneerng applcablty. Under the assumpton of rgd foundaton the structure vbraton frequences can be obtaned usng ABAQUS mode-based analyss. The substructure method computed by program T3DSSI s tested by drect method and by a practcal example on the vbraton frequences of a structure on rgd medum. It shows that the sol-structural dynamc predctons by ths substructure method are n good agreement to the reference solutons. 2. THE THREE-DIENSION SUBSTRUCTURE ETHOD IN TIE DOAIN As shown n g. (a) a sol-structure dynamc nteracton model s composed of actual upper structure surface foundaton and fnte sol zone wth the artfcal boundary. Under the rectangular coordnate system (x- y- z-) the model s appled to the analyss of S wave propagaton through the sem-nfnte sol medum to the actual structure. n actual upper structure KZ KY Artfcal boundary J O Z Artfcal boundary zone J- J-2 foundaton 0 Internal sol zone Artfcal boundary zone S wave free surface Artfcal boundary zone X Artfcal boundary SZ SY KX SX (a) (b) gure (a) A typcal computatonal model for the sol-structure dynamc nteracton analyss; (b) The nteracton forces among sol medum rgd plate and upper structure 2.. Equaton of moton of sol doman As shown n g. (a) the fnte sol zone s dvded n two parts as the artfcal boundary doman and the nternal computatonal doman. Also as shown n g. (a) the nodes of the fnte element mesh n sol zone are

3 The 4 th October Bejng Chna dvded nto the artfcal boundary nodes the nternal nodes and the nodes connectng wth the foundaton. To nternal nodes of sol doman when the explct lumped-mass fnte element method n tme doman s adopted the equaton of moton can be expressed as: µ µ (2.) 2 { p + 2{ P { p U [ ] [ ( )[ ]{ p [ ]{ p { p = U U t + Kl Ul Kl Ul ] l t l t [] [K ] and { denote the dagonal lumped-mass matrx the stff coeffcents matrx and the external loadng vector {U {U & and {U && denote respectvely the dsplacement the velocty and the acceleraton of nternal node µ denotes the vscosty dampng coeffcent n proporton to the velocty of nternal node. The subscrpt or l denotes the nternal node and subscrpt l denotes the relatonshp between the nternal node and the node l adjoned wth node. The superscrpt p refers to the ncrement number and the relatonshp between tme t and no negatve ncrement number p s t=p t. The hstory of sesmc wave s assumed to be nputted from bottom the artfcal boundary nodes and the scatterng wave can be transmtted out through them. The moton of the artfcal boundary nodes can not be computed by equaton (2.) snce the moton of nodes whch connect the artfcal boundary but outsde computatonal zone s unnown. As shown n g. (a) assumed the node J on the artfcal boundary the node J- the node J-2 are denoted to locate the same normal lne of the artfcal boundary as the node J and arranged near the node J nsde the sol computatonal zone n turn. The dsplacements of the scatterng wave at the node J at moment (p+) t on the artfcal boundary can be determned by the dsplacements of the scatterng wave at the node J and the node J- the node J-2 at tme p t and ( p-) t the recurrence formulaton s [Lao et al 984] : N p n N { U + JR = ( ) + Cn { U n n= (2.2) { U p JR + denotes the dsplacement vector of scattng wave at the node J on the artfcal boundary at the moment (p+) t N s the transmttng orders vector{ U n s denoted as: { p n J N C n s the bnomal coeffcent and the dsplacement p n+ { { p n+ { { p n+ p n+ U n U J U J { U J 2 { U J n = L (2.3) U + denotes the dsplacement vector of scattng wave at the node J on the artfcal boundary at the moment (p-n+) t and { p U n + J denotes the dsplacement of scattng wave at the node J- near the artfcal boundary nsde the sol zone at the moment (p-n+) t. Therefore the dsplacements feld of the whole wave feld at the nodes { U p+ on the artfcal boundary can be determned as follow: J T { U = { U + { U p + p + p + J JR JI the dsplacement vector{ U p + JI denotes the free feld value when the node J s on the sde artfcal boundary or the nput wave value when the node J s on the bottom artfcal boundary. The moton of the nodes connected wth the foundaton s decded by the foundaton moton whch wll be presented n secton Equaton of moton of structure

4 The 4 th October Bejng Chna In ths secton the actual upper structure model s smplfed as a three-dmenson lumped-mass beam element model as shown n g. (a). All the nodes of beam elements are classfed as the nternal nodes and the nodes connected wth the foundaton. Wth the Raylegh dampng the equaton of moton to the nternal nodes of structure can be expressed as: { U = 2{ U { U p + p p 2 β α P β α p p t [ ] ( + )[ Kl ] + [ l ] { Ul ( [ Kl ] [ l ]){ Ul { t t + l t t (2.4) [] [K ] { {U {U & and {U && denote smlarly to the equaton (2.) and α β denote the Raylegh dampng coeffcents. The moton of the nodes connected wth the foundaton s decded by the foundaton moton whch wll be presented n next secton Equaton of moton foundaton The connecton part between the actual upper structure and sol medum structure s the foundaton by whch the nteracton forces from the unbounded sol medum are transferred. Snce the foundaton s often stff and thc t s often assumed to be a rgd plate n computatonal model. That s to say that the moton of the foundaton only ncludes the sx components whch are three concentrated force components ( X Y Z ) and three concentrated moment components ( x y z ) appled to the centre of the rgd plate. The total forces appled to the rgd plate whch gven by the actual upper structure and sol medum mae the rgd moton of the foundaton. The ey problem analyzng the dynamc sol-structure nteracton s to decde the total forces appled on the foundaton. The model used to descrbe the nteracton forces among sol medum rgd plate and upper structure s shown n g. (b). The node s assumed the one of the connectng ponts between the sol and the foundaton. Snce the deformaton of sol doman the force at node K appled by the sol doman s x y { { T x y z x y z = (2.5) z respectvely denote the concentrated force components at node along the x- y- and z- coordnate drectons and x y z respectvely denote the concentrated moment components at node along the x- y- and z- coordnate drectons and then { = µ µ + + p p ( )[ Kl ]{ Ul [ Kl ]{ Ul t l t (2.6) The total force at the foundaton appled by the sol doman s computed as: = (2.7) { [ ] T D A { { { T D DX DY DZ DX DY DZ m = s the total force vector at the foundaton appled by the sol doman m s the total number of the nodes connectng wth the foundaton and

5 The 4 th October Bejng Chna [ A] Z Y 0 0 Z 0 X 0 0 Y X 0 = X Y Z s respectvely the relatvely coordnate values of node to the shape center of rgd plate. The node s assumed the one of the ponts of the upper structure but t s not connected wth the foundaton. Snce the moton of structure the nerta force at node wll brng about appled forces to the foundaton. The nerta force of node s gven: x y z- coordnate drectons and x { { T x y z x y z = (2.8) z respectvely denote the concentrated force components at node along the x- y- and y z respectvely denote the concentrated moment components at node along the x- y- and z- coordnate drectons and then from equaton (2.4) the nerta force of node s gven: { [( )[ ] [ ]]{ p p = + Kl + l Ul ( [ Kl ] + [ l ]){ Ul (2.9) l β β β α t t t t all the varables have been denoted n equaton (2.4). Smlarly the total forces on the foundaton appled by the structure can be expressed as: { [ ] T S A { { { T S SX SY SZ SX SY SZ n l = (2.0) = s the total force vector at the foundaton appled by the structure n s the total number of upper structure nodes except the nodes connected wth the foundaton and the defnton of matrx [A ] s the same as the equaton (2.7). rom the equaton (2.7) and equaton (2.0) the total forces appled by both sol doman and upper structure are: { { { [ ] T T = D + S = A { + [ A ] { (2.) the total force vector { { x y z x y z components have been denoted n the last two sectons. The equaton of moton of rgd foundaton s wrtten: m n = and the meanngs of other T [ ]{ U&& = { (2.2)

6 The 4 th October Bejng Chna [] s the dagonal lumped-mass matrx of rgd plate the nodes connectng wth the foundaton of sol medum and actual upper structure and { U && denotes the acceleraton vector of rgd plate system and { has been denoted n last equaton. Usng the explct lumped-mass fnte element method n tme doman the moton equaton (2.2) of rgd plate can be expressed as: U U U t p+ p p 2 { = 2{ { [ ] { (2.3) { U denotes the dsplacement vector of rgd plate at the moment p t and the other varables p have been denoted n ths and last two sectons. The moton of the nodes connected wth foundaton ether n the structure or n the sol doman s decded by the rgd moton of plate. The expresson of the moton of these nodes can be gven as follow: { U = [ A]{ U (2.4) p+ p+ { U p + [A] and U + have been denoted n ths and last two sectons. { p 3. NUERICAL EXAPLES Case one: dynamc response of rectangular rgd foundaton on a unform sm-nfnte medum As shown n g. (a) a structure wth a rgd thc plate foundaton s founded on the surface of the sem-nfnte sol layer. oreover the actual upper structure smplfed to a 9 three-dmenson beam element system s gnored to test the Program T3DSSI. The dmenson and the densty of rgd plate are L W H=78m 77m 6.5m and 2.5T/m 3 respectvely. The sm-nfnte sol medum s smplfed unform medum and the transmttng artfcal boundary s used to truncate the sol medum. The ncdent S wave s assumed as a unt pulse wth the wdth of 0.2s from the bottom of the transmttng boundary. The three-dmenson mesh s used to model the truncated sol medum. The drect method for ths nd module has been used n many papers [Lao et al 984; Yang et al 2007]. The horzontal accelerate hstory of rgd foundaton usng the drect method and substructure method n ths paper are shown n g.2. orm g.2 the results are agreement very well when shear wave velocty Vs n the rgd plate and unform medum are chosen 2500 m/s and 200 m/s respectvely. acceleraton tme hstory/gal DIRECT ETHOD SUBSTEUCTURE ETHOD acceleraton tme hstory/gal ABAQUS T3DSSI t/s t/s gure 2 The acceleraton tme hstory of the rgd foundaton gure 3 The horzontal acceleraton tme hstory of the hghest node of the structure Case two: The dynamc effect of the sol-structure nteracton under the bg stffness of sol medum Also as shown n g. (a) the actual upper structure smplfed to a 9 three-dmenson beam element system s

7 The 4 th October Bejng Chna added to the last example. The parameters such as the beam sze area mass rotary nerta and moment of nerta are gven n Table. In addton the parameters of elastc modulus shearng modulus of beams s gven respectvely as: E= and G= The dmenson of rgd plate the transmttng artfcal boundary and the ncdent S wave are assumed the same as last case. The three-dmenson mesh s used to model the truncated sol medum. As we nown when the stffness of sol zone below the upper structure s bg enough the structure appled force by the sol-structure nteracton wll have a lttle nfluence to the structure nature vbraton frequences. Ths character s often used to prove the effectve of the adopted methods. The bg stffness sol medum parameters nclude shear wave velocty Vs=5000m/s cut frequency f=25hz the denstes of sol=2.5t/m 3 Posson raton ν=0.25 and the dampng rato of beam ξ=0.00. Number Length Area of across Table the characters of beam oment of nerta ass ( 0 3 g m 2 ) Rotary nerta (m 4 ) of beam (m) secton (m 2 ) ( 0 3 g) x- y- z- x- y- z The structure nature frequences computng ether by transfer functon method for T3DSSI or mode-based analyss for ABAQUS can gve smlar results as shown n Table 2. These results proved the three dmenson sol-structure nteracton analyss procedure and program T3DSSI to be reasonable under the bg stffness sol medum. Table 2 The horzontal nature frequences of structure wth dampng rato ξ= 0.0 for the structure Nature vbraton frequences 2 3 Adopted method ABAQUS(mode-based analyss) T3DSSI (transfer functon method) The hghest node horzontal acceleraton tme hstory of substructure module gven by T3DSSI s shown n g3. To compared the upper structure wth a rgd foundaton s modeled usng drect method of the commercal

8 The 4 th October Bejng Chna software ABAQUAS. Wth the same unt pulse wth the wdth of 0.05s ncdent S wave nputted from the rgd plate the acceleraton tme hstores of the hghest node of the structure s also shown n g.3. rom g.3 the dynamc acceleraton tme hstory by T3DSSI s n good agreement to the reference solutons usng ABAQUS. 4. CLOSURE A fully numercal procedure s presented to solve the problems of thee-dmenson sol-structure nteracton analyss. The procedure starts wth tme-doman substructure dea. The explct fnte element and correspondng transmttng model are taen to model the nfnte sol medum. These procedures can solve the hgh cost n three dmenson sol structure nteracton computaton usng drect method. After studyng problems on foundaton-soft sol medum and bg stff of sol medum-structure systems the proposed procedures and programs are found to possess reasonable results and hgh computatonal cost. 5. ACKNOWLEDGENT The fnancal support of the Guangdong Provnce Natural Scence oundaton (Project Number: ) and Guangdong Unversty of Technology n the form of a Natural Scence oundaton are gratefully acnowledged. REERENCES ABAQUS (200). ABAQUS Theory and User s anuals Verson 6.2 Hbbtt Karlsson & Sorensen Inc. Pawtucet Rhode Island USA. Kausel E. (988). Local transmttng boundares. J.Eng. ech Lao Z.P. Wong H.L. Yang B.P. and Yuan Y.. (984). A transmttng boundary for transent wave analyss. Scenta Snca (Seres A) XXVII Lu S.C. and agel L.W. (97). Earthquae nteracton by ast ourer Transform. J. Eng. ech. Dv. ASCE Luco J.E. and Westmann R.A (97). Dynamc response to crcular footngs. J. Eng. ech.dv. ASCE Lysmer J. Kuhlemeyer R.L. (969). nte dynamc model for nfnte meda. J. Eng. ech. Dv. ASCE Lysmer J. Waas G. (972). Shear waves n plane nfnte structures. J. Eng. ech. Dv. ASCE Smth W.D. (974). A nonreflectng plane boundary for wave propagaton problems. J. Comput. Phys Oen.A. (973) Steady moton of a plate on an elastc half space. Appl. ech. Trans. ASE Shah A.H. Wong K.C. Datta S.K. (982). Dffracton of plane SH waves n half-space. Earthq. Eng. Struct Dynam Veletsos A.S. and We Y.T. (97). Lateral and rocng vbraton of footngs. J. Sol. ech. ound. Dv. ASCE Wolf J.P. (988). Sol structure n the tme doman Englewood Clffs NJ: Prentce-Hall. Wong H.L. and Luco J.E. (978). Dynamc response of rectangular foundatons to oblquely ncdent sesmc waves. Earthq.Eng. Struct. Dyn Yang X.. Wang H.T. Yang B.P (2008). The nfluence of three-dmenson fault stes on the ground moton. J. Earthq. Eng. Eng. Vb (n Chnese) Zhang X. Wegner J.L. Haddow J.B. (999). Three dmensonal sol-structure-wave nteracton analyss n tme doman. E Earthq. Eng. Struct Dynam