SPH simulation for seismic behavior of earth structures

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1 The 4 th World Conerence on Earthquake Engneerng October 2-7, 28, Beng, Chna ABSTRACT : SPH sulaton or sesc behavor o earth structures Y. Ono, S. shda 2 and J. Kyono 3 Assstant Proessor, Dept. o Urban Manageent, Kyoto Unversty, Kyoto, Japan 2 Graduate Student, Dept. o Urban Manageent, Kyoto Unversty, Kyoto, Japan 3 Assocate Proessor, Dept. o Urban Manageent, Kyoto Unversty, Kyoto, Japan Eal: ysk@box.kudpc.kyoto-u.ac.p, shngo@quake2.kucv.kyoto-u.ac.p, kyono@quake.kucv.kyoto-u.ac.p The earthquake response analyss o earth structures usng the Soothed Partcle Hydrodynacs (SPH) was ntroduced n ths paper. We showed that the correctve soothed partcle ethod (CSPM), whch s a oded verson o the conventonal SPH, could acheve hgher accuracy than the conventonal SPH or earthquake response analyss. To llustrate ths, three nuercal sulatons were carred out; the one densonal elastc wave propagaton proble wth relecton, the earthquake response analyss o the one densonal elastc sol and the two-densonal elasto-plastc sol. Fnally, we deonstrated the SPH sulaton o the collapse behavor o an earth structure. KEYWORDS: soothed partcle hydrodynacs, correctve soothed partcle ethod, earth structure and earthquake response analyss. ITRODUCTIO Strong ground oton causes large deoraton wth tenson cracks and slp alure to earth structures. The nte eleent analyss s wdely used to sulate the earthquake daage o earth structures. However, t s dcult to estate resdual dsplaceent precsely because the FE analyss hardly sulates behavor o earth structures ater dscontnuty such as sldng or tenson crack have taken place. In the present paper, the Soothed Partcle Hydrodynacs (SPH) s exaned to sulate dynac behavor o earth structures excted by strong ground oton. One advantage o the SPH ethod the FE analyss s that the SPH could handle dscontnuty destructon n a contnuu body. The SPH was nvented by Lucy (977) and Gngold and Monaghan (977) or dealng wth a copressve lud body. Snce 99s, the SPH has been appled to sold body dynacs(randles and Lbersky,996 and Gray et al.,2). The obectve o ths paper s to present the earthquake response analyss o earth structures. In the rst secton, the undaental equatons o the SPH are ntroduced. Also, the correctve Soothng Partcle Method (CSPM) whch acheves hgher accuracy than the SPH s descrbed. In the second secton, several nuercal tests are conducted to llustrate the capablty o the SPH and the CSPM. 2.SMOOTHED PARTICLE HYDRODYAMICS 2. Conventonal SPH The SPH s one knd o nterpolaton theory and t conssts o two nterpolaton technques,.e., kernel approxaton and partcle approxaton. The kernel approxaton begns wth the ollowng equaton. ( x ) ( x ) δ ( x x ) dx (2.) Ω where ( x) s a uncton o the three-densonal poston vector x and ( x x ) uncton. Ω s the doan where the uncton ( x) s dened. Replacng the delta uncton ( x x ) by a uncton W ( x x, we get δ s the Drac delta δ n (2.)

2 The 4 th World Conerence on Earthquake Engneerng October 2-7, 28, Beng, Chna ( ) ( x ) W ( x x h) dx x, (2.2) Ω x h s called a kernel uncton or soothng uncton. The angle bracket eans the kernel where W ( x, ) approxaton operator. h s a paraeter whch denes the nluence area o the kernel uncton W ( x x. Although any types o kernel unctons have been proposed, the cubc B-splne kernel uncton, whch s proposed by Monaghan and Lattanzo(985), s ost preerred n any SPH lteratures because o ts coputatonal ecency. The cubc B-splne kernel uncton s, R + R R < W ( R, h) α d ( 2 R) R < 2 (2.3) 6 R 2 x x' where R and or one-, two- and three-densonal, α d, 5 2 and 3 3, respectvely. In h h 7πh 2πh the present study we also use the cubc B-splne kernel uncton. In the SPH ethod, the ntegral representaton o the equaton (2,2), s converted to a suaton over a set o x at partcle s, dscrete partcles. The partcle approxaton or a uncton ( ) W ( x x where s the total nuber o partcles n the nluence doan o partcle as shown n Fgure. s the ass and densty assocated to partcle, respectvely. and (2.4) Fgure SPH partcles neghborng partcle In ost engneerng probles, the rst dervatve o a uncton ( x) ( x) tsel. Instead o drect applcaton o equaton (2.4) to the rst dervatve o a uncton ( x) ollowng or s coonly used. ( ) W ( x x s oten requred rather than the uncton, the The SPH ethod has been appled successully to varous contnuous echancs probles. However, two bg probles whch can cause naccuracy are known. Frst, the kernel approxaton, equaton (2.2) used cannot reproduce the uncton tsel copletely. In other words, the SPH does not satsy even the zero order consstency. Second, near or on a boundary, the ntegral (or suaton) s truncated due to partcle decency and the SPH approxaton becoes naccurate. Thereore, Chen et al. proposed the correctve soothed partcle ethod (CSPM) whch can elnate these probles successully. The CSPM orulaton s brely ntroduced n the next secton. 2.2 The Correctve Soothed Partcle Method(CSPM) (2.5)

3 The 4 th World Conerence on Earthquake Engneerng October 2-7, 28, Beng, Chna Chen et al.(999) proposed the ollowng or o the kernel approxaton nstead o equaton (2.2). ( x) Ω ( x ) W ( x x Ω W ( x x The denonator o equaton (2.6) s called the correctve ter and t s equal to, equaton (2.6) becoes dentcal to equaton (2.2). The partcle approxaton or o equaton (2.6) can be wrtten by W ( x x (2.7) W ( x x The rst dervatves or partcle can be obtaned by the ollowng sultaneous equatons. βα, α β A F where βα α α,β A ( x x ) W F β ( ) The ndces α and β denote spatal coponents and the superscrpt, α ndcates the rst dervatve or W W x x h. ote that the CSPM satsy the zero and rst order consstency spatal coponent α and ( ), condtons copletely. The eectveness o the CSPM or earthquake response analyss s deonstrated n the ollowng secton. W dx,β dx (2.6) (2.8) (2.9) (2.) 3. RESULTS AD DISCUSSIOS 3. Case : D Elastc Wave Propagaton To show the accuracy o the SPH ethod, elastc wave propagaton n one densonal bar (unaxal stran proble) s studed here. As shown n Fgure, we x the let end o the bar rgdly and stress σ ( t) s appled to the rght hand end. Young s odulus E and the densty s assued to be / 2 and kg/ 3, respectvely. The length o the bar s set to 4.. The governng equatons o ths proble are gven by v( x, t) σ ( x t), (3.) t x ( x, t) v σ& ( x, t) E (3.2) x n whch v s velocty at poston x. For the boundary treatent, a partcle s centered on a boundary and boundary values are assgned explctly. The explct Euler schee s eployed or te ntegraton. Although the SPH s a pure Lagrangan ethod, all partcles are spatally xed here. The partcles are arranged wth equal nterval.2. Thereore, total 2 partcles are used. The Fnte Derence Method (FDM) s also conducted or coparson. Fgure 2 D Elastc Wave Propagaton Test

4 The 4 th World Conerence on Earthquake Engneerng October 2-7, 28, Beng, Chna velocty(/s) t.(s) -. CSPM(kh3.2) CSPM(kh2.5) -.2 SPH(kh3.2) -.3 SPH(kh2.5) FDM poston() velocty(/s) t2.(s) CSPM(kh3.2) CSPM(kh2.5) SPH(kh3.2) SPH(kh2.5) FDM poston() Fgure 3 Velocty dstrbuton at t.s (LHS) and t2.s (RHS) Fgure 3 shows the snapshots o the velocty dstrbuton along the bar. In Fgure 3, kh ndcates the rato o the soothng length to the partcle spacng. At t.(s) the wave ront has not reached at the let end and the wave has relected once at the let end between.(s) and 2.(s). Two derent values o the soothng length are exaned or the conventonal SPH and the CSPM. The results ro the CSPM agree very well wth those ro the FDM ndependently on the soothng length whle the SPH results less accuracy and the wave speed s aected by the soothng length. Ths proble o the conventonal SPH s caused by the lack o the rst order consstency. rst dervatve (x)x kh SPH CSPM Exact velocty(/s) SPH(kh2.9) -.4 FDM poston() t.(s) t2.(s) Fgure 4 Varaton o the estated rst dervatve Fgure 5 Velocty dstrbuton wth the SPH (kh2.9dp) In Fgure 4 the rst dervatves o the uncton ( x) x calculated by the conventonal SPH and the CSPM are plotted. The horzontal axs corresponds to the soothng length rato to the partcle spacng. Here all partcles are arranged wth equal nterval. The soothng length does not aect the accuracy o the CSPM whle the accuracy o the conventonal SPH changes. Fgure 4 suggests that the SPH can gve the best approxaton when kh s equal to 2. or 2.9. It s conred n Fgure 5 where the conventonal SPH wth kh2.9 shows good agreeent wth the FDM. 3.2 Case 2: D Earthquake Response Analyss o Elastc Sol Model The response o one-densonal elastc sol colun odel to earthquake exctaton s analyzed here. The sol

5 The 4 th World Conerence on Earthquake Engneerng October 2-7, 28, Beng, Chna odel conducted s shown n Fgure 4. The governng equaton s wrtten by v( x, t) σ ( x t), + a( t) t x ( x, t) v σ& ( x, t) G (3.4) x n whch a(t) s acceleraton o the nput ground oton and G s shear odulus. Fgure 5 shows the nput ground oton used here. The densty and Posson s rato ν s set to be 2kg/ 3 and.4 or all layers respectvely. The nte eleent ethod (FEM) s also conducted or coparson. (3.3) A Vs/s Vs2/s Vs3/s Vs5/s acceleraton(/s/s) te(s) Fgure 6 D Sol colun odel Fgure 7 The nput ground oton (JMA Kobe 995) Relatve Error SPH(kh2.) SPH(kh2.5). SPH(kh2.9) CSPM(kh2.) CSPM(kh2.5) CSPM(kh2.9) Partcle Spacng() velocty(/s) FEM CSPM(kh2.2dp, dp.) te(s) Fgure 8 Varaton o the relatve error η (Case 2) Fgure 9 The te hstory at Pont A by CSPM We ran 8 sulatons usng ths sol odel wth the partcle spacng dp.,.5,2.,5.() and the soothng length kh2.dp, 2.5dp, 2.9dp or the conventonal SPH and the CSPM. To dscuss the accuracy o each odel we dene the relatve error η by n whch v FEM ( t) s the velocty gven by the FEM and v SPH ( t) T FEM SPH v v dt η (3.5) T 2 FEM v dt s by the conventonal SPH or the CSPM. T s the duraton o the analyss. The result s suarzed n Fgure 8. One can conclude ro results depcted n Fgure 8 that the CSPM acheves hgher accuracy than the conventonal SPH when the partcle spacng s sall. The hghest accuracy s obtaned by the CSPM wth kh2.dp and dp.(). The te hstory at the ground 2

The 4 th World Conerence on Earthquake Engneerng October 2-7, 28, Beng, Chna surace (Pont A n Fgure 6) s plotted n Fgure 9 and copared to the result ro the FEM. 3.

6 The 4 th World Conerence on Earthquake Engneerng October 2-7, 28, Beng, Chna surace (Pont A n Fgure 6) s plotted n Fgure 9 and copared to the result ro the FEM. 3.3 Case 3: 2D Earthquake Response Analyss o Elasto-Plastc Sol Model In ths exaple the two densonal sol odel s analyzed by the conventonal SPH and the CSPM. The sol odel whch s 2 wde and heght (see Fgure ) s excted by the acceleraton shown n Fgure 9. The plan stran s assued. The botto boundary partcles are xed. For the boundary partcles on the both sde, the vertcal dsplaceent s xed to be zero whle the horzontal one s ree. Stress ree condton s explctly gven to the ree-surace boundary partcles. The vscous dapng ter cv s added to the RHS o equaton (3.3) where c s the dapng coecent and set to be.2 here. The elasto-plastc odel wth the Mohr-Coulob yeld surace and the assocated low rule s eployed. The nternal rcton angle and the coheson coecent are set to be 3(deg.) and 2(k/ 2 ), respectvely. The shear wave velocty, the densty and Posson s rato are (/s), 9(kg/ 3 ) and.4. B Vs /s 9kg/ 3 ν.4 Relatve Error SPH(kh.25dp) SPH(kh.5dp) CSPM(kh.25dp) CSPM(kh.5dp) Partcle Spacng() Fgure Two densonal ground odel Fgure Varaton o the relatve error η (Case 3) velocty(/s) FEM CSPM(kh.25dp,dp.5) te(s) Fgure 2 Te hstory o response at Pont B Fgure 3 Earth structure odel Fgure shows the varaton o the relatve error η coputed by equaton (3.5) wth the partcle spacng dp and the soothng length kh. It s clear that the CSPM gves better results than the conventonal SPH. Fgure 2 copares the te hstores o the response at Pont B shown n Fgure coputed by the CSPM (dp.5, kh.25dp) wth that o the FEM. 3.4 Case 4: 2D Collapse Sulaton o Earth structure The earth structure odel depcted n Fgure 3 s excted by the acceleraton shown n Fgure 9. The shear wave velocty s 2/s or the doan and 4/s or the doan 2. The densty and Posson s rato are gven to be 9kg/ 3 and.35 respectvely or the entre odel. The boundary partcles on the both sdes o the doan 2 and the rght-hand-sde o the doan are constraned as well as the partcles on the botto edge

The 4 th World Conerence on Earthquake Engneerng October 2-7, 28, Beng, Chna o the doan 2. The ree stress condton s drectly assgned to the other boundary partcles.

7 The 4 th World Conerence on Earthquake Engneerng October 2-7, 28, Beng, Chna o the doan 2. The ree stress condton s drectly assgned to the other boundary partcles. In ths case the Lagrangan approach s eployed, naely the partcle arrangeent s updated at every te step and the kernel unctons are rebuld. The Jauann stress rate tensor s used or the nte rotaton eect. The sae consttutve relatonshp as the prevous exaple s used except that the coheson coecent c and the nternal rcton angle φ are dropped ater yeldng. Here we use c6(k/ 2 ), φ2.(deg.) beore yeldng and c3(k/ 2 ), φ.(deg.) or post-yeldng. Total 5 partcles are used and they are regularly dstrbuted wth.25 nterval at the ntal te step. Fgure 3 shows the snapshots o the deoraton o the earth structure. At t4.s, one can see that the shear alure band was developed and the slp began. At t8.s, the slp alure took place and a ew gaps appeared on the top o the earth structure. (a).s (b) 4.s (c) 8.s Fgure 3 The deoraton o the earth structure odel excted by the earthquake ground oton (coputed by the CSPM) (a).s (b) 4.s (c) 8.s Fgure 4 The captured boundary partcles When the deoraton o the odel becoes large under the sesc exctaton, new boundary partcles ay appear due to cracks or collapse. These new boundary partcles ust be captured at every te step. In order to denty the boundary partcles we used the crtera whch s dened by Ψ ε (3.6) where Ψ W ote that Ψ corresponds to the outward noral vector at the partcle when the partcle locates on the boundary. I the partcle s ar ro the boundary and surrounded by other partcles enough, Ψ becoes zero. We deterne ε.5 by tral and error or ths case. Fgure 4 shows that the boundary partcles are captured properly. (3.7)

8 The 4 th World Conerence on Earthquake Engneerng October 2-7, 28, Beng, Chna 4. COCLUSIOS We dscussed the capablty o the SPH and the CSPM, whch s one odcaton o the SPH, or earthquake response analyss o sol wth a seres o the nuercal sulatons. We showed that the CSPM could gve better results or earthquake response analyss o sol than the conventonal SPH. Also we showed that the SPH can be used to sulate collapse behavor o earth structures. REFERECES Lucy, L.B. (977). A nuercal approach to the testng o the sson hypothess. Astroncal Journal 82, 3-2. Gngold, R.A. and Monaghan J.J. (977) Soothed partcle hydrodynacs: Theory and applcaton to non-sphercal stars. Monthly otces or the Royal Astronocal Socety 8, Randles, P.W. and Lbersky, L.D. (996). Soothed Partcle Hydrodynacs: Soe recent proveents and applcatons. Coputer Methods n Appled Mechancs and Engneerng 39, Gray, J.P., Monaghan J.J. and Swt, R.P. (2) SPH elastc dynacs. Coputer Methods n Appled Mechancs and Engneerng 9, Monaghan, J.J. and Lattanzo, J.C. (985) A rened partcle ethod or astrophyscal probles, Astronoy and Astrophyscs 49, Chen, J.K., Beraun, J.E. and Jh, C.J. (999) Copleteness o correctve soothed partcle ethod or lnear elastodynacs. Coputatonal Mechancs 24,

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