27(5A)-15(5868).fm

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1 ª Œª Œ 27ƒ 5A Á œ pp. 701 ~ 711 ª x 3 k w Radial 3-D Elastodynamic Infinite Elements Á Á½ Seo, Choon-GyoÁYun, Chung-BangÁKim, Jae-Min Abstract This paper presents new three dimensional radial infinite elements for soil-structure-interaction problem in a multi-layered half-space. Three kinds of radial infinite elements, including horizontal, vertical and corner infinite elements, are developed in Cartesian coordinates using approximate wave propagation functions with multiple wave components. umerical example analyses are presented for rigid arbitrarily shaped footings, and an embedded caisson on a homogeneous and layered half-space in frequency domain. The numerical results obtained show the effectiveness of the proposed infinite elements. Keywords : infinite element, soil-structure interaction, wave propagation, layered halfspace d - y w w 3 w w. w Cartesian t y, s, š w. w y w w x w w q w w w w ùk ü ww. w w w» w q ƒ w ww. ³ d x x» f» w, mw. w : w, - y, q, d 1. w w w x - y (SSI) y w w» w w q k q p x yw txw w., xy w x š w w» w (White 1977), (Tassoulas & Kausel, 1983; ½ 2000), w (Wong & Luco 1976, 1985; Liou, 1989; Chow 1986), (Chen & Penzien 1986; ½ 2000) w (Bettess 1977; Medina 1981; 1991, 1995, 2000, 2001, 2007; Chow & Smith 1981; Zhao & Valliappan 1993; Park & Watanabe 2003) w. - y w w v w» š, x y š (FLUSH, Lysmer et al., 1975; SASSI, Lysmer et z Á Áw w» y œw ( seosck@kaist.ac.kr) z Áw w» y œw ( ycb@kaist.ac.kr) z Á û w œw w y œw ( jm4kim@jnu.ac.kr) al., 1988; KIESSI, 2001). - y w xy w w w. w w xyw z ùkü» w š œ, y w w w ww» ƒ š. - y (, 2000), w y w (½», 2001; Park & Watanabe 2003). w ƒ w w ùký p w x w ƒ v w w w w v w. p d w k w w k q ( q, q t q) s ww q w txw w, e ( q, q Helmholtz ) w w y ùww š w. 27ƒ 5A œ 701

2 xy d w k w w» w w 2 e. 3 w ³ k w Zhao & Valliappan(1993) 3 w 3 mx w d w w Park & Watanabe(2003) 2.5 w w. (2007) w d w w w w. ù ¾ w 3 w x d xy w r. d - y w w w y w. w 3 w 1 x, x xyw yw. d p wš x w z w sd w ƒ. wš w 3 w ƒ w y xy x w {. w ƒ q w x w swwš, x w w q w q w w w w. w 3 w y» w, w ³ d x x q» f» fv v w wš, k ew e SSI qj w w w xy»w w y w w, q avier l xw. 1 - y 3 w xywš, w w xyw» w. w w» x-y-z t 2(a) x-y s w xy x x w. y ƒ j xy w yw r k. ù y w w»wxk w w w y w w w» w w x w w v. w» 2-(b) s d (Ω H ) s w (HIE, Horizontal IE), w w (Ω V, Ω C ) w (VIE, Vertical IE) w (CIE, Corner IE) xyw w. š ƒ w»w t 1 w. t 1 L j (Á): Lagrange w, w ùkü. w w ƒ 8- w, 2-(a) ùkù 4 s w š 2 w 2. w»w xy 702 ª Œª Œ

3 t 1. x w»w x y z HIE L( η, )x j 1 ( ξ) L( η, )y j 1 ( ξ) L j ( η, )z j VIE CIE L j ( ξη, )x j L( η)x j ( 1 ξ) L j ( ξη, )y j z j L( η)y j ( 1 ξ) z j xyw w. š x j, y j š z j t. 3. w x w w q w. u( xyz;ω,, ) p ( xyz;ω,, )p p [ 111 I,, I, I,, 11n n I I,, lm1 lmn I] T T T T T T p p 111, p 11n, p 211,, p 21n,, p l11, T p lmn (1) ƒƒ w w y w. H H u( xyz;,, ω) jlm xyz; (,, ω)p jlm HIE l 1m 1 V u( xyz;,, ω) jn xyz; (,, ω)p jn VIE n 1 H H V u( xyz;,, ω) jlmn xyz; (,, ω)p jlmn l 1m 1n 1 CIE» ùkü, H V ƒƒ sq q ùkü. š p jlm (w), p jn (w), p jlmn (w) ƒƒ x w jlm, jn, jlmn w s ùkü t l, x w t. jlm ;ω) L j f l ; ω)g m ;ω) : HIE jn ;ω) L j ( ξ, η)h n ( ; ω) : VIE jlmn ;ω) L j ( η)f l ; ω)g m ;ω)h n ( ; ω) : CIE f l g m ƒƒ t x y qw s w q w h n w q w w. q w yw l w ƒ w š, ƒ, d w q š w ( & 1991; 1995). (1) (2) (3) (4) e f l ;ω) e β x η, }ξ, }ξ, ) ik s x η, ) { β x η, ik p x η, { { e α x η, ik C a x }ξ s { } a 1 e g m ;ω) e β y η, }ξ, }ξ, ( C y η, ) ik s { β y η, ik p y η, ) { { e α y ik C a y }ξ { } ( h) s a 1» s t q ; ³, s 1 d s 2 w q H 2 S V 22Á S ; w q w (0, 0, 0)» ƒ, txw.» q w q k w. { h n ( ;ω) e β ik } { s e β ik } p e µ sa e µ,,, x α x α θ x α , ( h) β x η, pa x( η, ) C x ( η, ) x( η, ) θ x ( η, ) x( η, ) ( η, ) j 1 s a 1 (5) x ( ) β θ x β L j x j x θ x , L j )x j L j )y j θ y ( η, ) y( η, ) ( η, ) ( ) k SP, k SP, ( h) k SP, 1 up low 1 up low -- ( k 2 SP, k SP, ) -- ( k 2 SP, k SP, ) : HIE 2 ( ) µ sa k a ( ) 2 2 µ pa k a ( ) 2, k s h ( h) k p ( η, ) L j ( η, )x j L j ( η, )y j : VIE, CIE : HIE ( η, ) L j ( η)x j L j ( η)y j : CIE (6) 27ƒ 5A œ 703

4 k S (z) k P (z) s d q q q z-w ùkü. s d w s w, q q x y w. w, k w w (half-space) ( h ) S k ( h ) P q q q s ùkü ; { k a } a 1 d x ùkü t q q ; m sa m pa w w z- w t q w ùkü ; ( η, ) (h) 2 w j¾ s w tx ; a b ƒ»w e x mw ƒƒ ƒ w u (d) t 2. ƒ x w u (e) ( &, 1991;, 1995;, 2007), w. wr, (3) l p p w ƒ ƒ. w w w. 3 x w u d, u e, u f, ü u i w ùkü z w w., ux; u d ( x; ω) u e ( x; ω) u f ( x; ω) u i ( x; ω) u (f ) ü u (i) (7) HIE 2 H L j f 1 d j1 L φ e ( 2j 1) m ( 2j 1)m m 2 2 H m 2 L φ f ( 2j) m ( 2j)m VIE L j f 1 g j1 L 2j 1 2 V m 2 ψ e ( ) m ( 2j 1)m 2 V m 2 L ψ f ( 2j) m ( 2j)m CIE L j f 1 g 1 d j11 1 H 1 m 2 V n 2 L φ ( 2j 1) l g e 1 ( 2j 1)m1 L 2j 1 ( )1n f ψ e ( ) 1 n 2j 1 1 H V m 2n 2 H m 2 L φ ( 2j 1) m ψ f n ( 2j 1)mn L 2 φ m g 1 f 2m1 V n 2 L 2 f 1 ψ n f 21n H V m 2n 2 L 2 φ m ψ n i 2mn : φ m ( ξη;,, ω) f m ( ξη;,, ω) f 1 ( ξη;,, ω), ψ n g n ( ; ω) g 1 ( ; ω) 3. w x w ( ) 704 ª Œª Œ

5 4. x w w t q q (x-y s ) ux; q ( x; ω)qx;», q ( x; ω) [ d, e, f, i ], q d T, e T, f T, i T T š, lq (d), (e), s (f) š ü (i). (3) (8) l t l p(w) q(w) (½ &, 1995). p T pq q (8) (9) (10)», T pq yw t x w. q ( xyz;,, ω) p ( xyz;,, ω)t pq (11) x w t 2 3 ƒ ùkü, q p»w txw š. 4 5 x w w, w t q q q w., 4-(a) 4 w, w ü qx s qx ùkû ù, 4-(b) yw x q wš. wr 5, q x q x-z s n w.» w q r, s qx š ùkû. ƒ x t sx». w s w q w w x wù s q ùkú. 4. w w w w w, w. ( e) K qq Ω T e B qdbq d Ω, M qq ( ) T qρq Ω dω (12)», dω dxdydz;d k w š B q x w. w w w m Gauss-Legendre, w w Gauss-Laguerre w. e t w z w. š yw w w w. ( e) K qq T e T pqkpp ( )T, M e pq qq ( ) T ( e) T pqmpp T pq», T pq (10) yw. K pp M pp p(ω) w w, w w wù q sww ƒ w. q ùký. S ( e) ( e) ( 1 i2β d )K e ( ) ω ( ) ω 2 M ( e) 5. x w w q q (x-z s ) S nn S ni U n F n f S in S ii S ii U i 0 (14) 27ƒ 5A œ 705

6 », U(ω) F(ω) w l ; S(ω) w wš, β d l p v ; f w ; n š i ùkü. 5. w w w mw» w w ww. ³ d w x q» fv w, ³ ü f» x q v w. 6 e x» ¾ w w j w. w» w ¾ (D/B) 2.0~2.5 w ( &, 1991; ½ &, 1995;, 2007). 5.1 x t» fv (14) - w ww, m w w w w (fv v ) yw y» w. fv v w ƒ, a o (ω B /c s1 ) w.» c s1 6. s d» ƒ 1/4 e x 706 ª Œª Œ

7 d s d q. x q» fv w w (Karasudhi, 1991). G 1 B x G C HH ( a 0 ) B z, C H VV ( a 0 ) , V G 1 B 3 θ y G 1 B 2 x C MM ( a 0 ) , C M HM ( a 0 ) M (15)» C HH C VV ƒƒ s fv w, C MM C HM ƒƒ k fv fv. š B x q» s w. (16) H, M, V 5-(a) w ƒƒ s, z, w. š G 1 2 d d k ( ρc s1 ), θ ƒ w z ƒ. 7 ³ x q w fv w e x e ù ùküš, ew w.» ƒx 6-(b) 6-(c) w w fv ùkü. 7- (a)~(c) y w, k wš š, Chow w w š., 8 d x q» fv w ùkü. 6-(a) ùkü, q s d ¾ (h/b)ƒ 2.0, d (ρ 2 /ρ 1 ) q (c s2 /c s1 )ƒ ƒƒ 1.13, w w Wong & Luco 3 SSI w v SASSI e w (Wong & Luco, 1985; Lysmer et al., 1988). 8 s,, fv fv w w w, k Luco j w. Luco w t» w (relaxed condition)w w š, SASSI w š (bonded condition) w» q. š Luco k fv ƒ š sƒ, 7-(c) ³ x q k w C MM (0) l w j ƒ w ƒ (Dobry & Gazetas, 1985; Chow, 1986). 7. ³ x q» fv w (β d 0.02) 27ƒ 5A œ 707

8 8. d x q» fv w 5.2 f» v 9 ³ w mx f x w v w wwš Chen & Penzien(1986) w w.» t»» p fv z ƒ j w. sta f» x q v w K ij K ij sta ( k ij ia o c ij ) txw.», K ij x» š, k ij, c ij. f» e p š w š p (torsion) w, 3Ü3 v w w xk - y x» w - w. H K HH K HM 0 x M K HM K MM 0 θ y V 0 0 K VV z 9. f» e x (17)» K HH K W ƒƒ s v, K MM K HM k fv v w. 10 f» x q» w v w ùkü, v s R v s R 3 w y., SASSI e» w Chen & Penzien w w š. w w 3 w w (, 2006) SASSI(Lysmer et al. 1988) t 3 w. t 3 SASSI, w wš w w x ¾ w w y». ù SASSI v w w fv v w wš w w w. š w, w»w ƒw 3 w ùkù. wr ww, v w SASSI w 708 ª Œª Œ

9 10. f q» v w t 3. w SASSI w (f» w ) w SASSI ( ) w ( ) 252( : 56) 128( : 56) w ( ) 76 - z ( ) q ( ) ƒw. ù, w w- w ƒ ww w w, ƒ w ƒw. w w š w» ( : v k w ) w SASSI w w w û. 5.3 d x t» fv, 11 x x t» w. šwƒ, 12 ƒ - x w ƒƒ w w š w w., w ƒ w ƒ š w, 12-(c) (d) w xy kw w r. d ƒx» w w, x t» q s w 11. d x t» y w ƒw ƒƒ s fv ùkü. s fv, x-1 wƒ w ƒ ƒ ùš. ƒ s³ e 3% ( ea ( o ) 100 )ƒ ùkù, q w ¾ ƒ ƒà» w - ƒ w w š q. ù x-2 w w š z. wr fv z ƒ ùkù, r j. 27ƒ 5A œ 709

10 12. 3 x w w 4ƒ x 1/4 13. s fv 14. fv q x k w w, e mw w k w m 3 - y w w w. 3 - w» w ª Œª Œ

11 ü w 3 w wš w w 3 w w. w x w d q š w x w w q w w w w w. w w» w ³ d x t» fv w ³ f» v w w,» w. w 3 w w w ƒ, SSI v w e y w z. w - x w, d x t» fv w w w mw k w. wz 3» ³ew»wx» ƒ m / x w - y š w w w y» w. š x, (1991) - y w w w, wm wz, wm wz, 11«3y, pp ½, (1995) d w - y w w w, wm wz, wm wz, 15«1y, pp ,, ½, (2000) x w y - y, wm wz, wm wz, 20«5-Ay, pp ½»,, ½ z(2001) w w w 2 - w, wm wz, wm wz, 21«4y, pp ,, ½,, ½,, ½», x, (2006) 2, e y w w mw KIESSI, w w» y œw w.,, ½ (2007) 3 - y w w x w, w œwz, w œwz, 20«, 1y, pp Zhao,GC. and Valliappan, S. (1993) A Dynamic infinite element for three-dimensional infinite domain wave problem, Int. J. umer. Methods Eng., Vol. 36, pp White, W., Valliapan, S., and Lee, I.K. (1977) Unified boundary for finite dynamic model, J. Eng. Mech. Div., ASCE, Vol Tassoulas, J.L. and Kausel, E. (1983) Elements for the numerical analysis of wave motion in layered media, Int. J. umer. Methods Eng., Vol. 19, pp Kim, J.K., Koh, H.M., Kwon, K.J., and Yi, J.S. (2000) A threedimensional transmitting boundary formulated in Cartesian coordinate system for the dynamics of non- axisymmetric foundations, Earthquake Eng. Struct. Dyn., Vol. 29, pp Wong, H.L. and Luco, J.E. (1985) Tables of impedance functions for square foundations on layered media, Soil Dynamics and Earthquake Engineering, Vol. 4, pp Liou, G.S. (1989) Analytical solutions for soil-structure interaction in layered media, Earthquake Eng. Struct. Dyn., Vol. 18, pp Chen, C.H. and Penzien, J. (1986) Dynamic modeling of axisymmetric foundation, Earthquake Eng. Struct. Dyn. Vol. 14, pp Kim, M.K., Lim, Y.M., and Rhee, J.W. (2000) Dynamic analysis of layered half planes by coupled finite and boundary elements, Engineering Structures, Vol. 22, pp Bettess, P. (1977) Infinite element, Int. J. umer. Methods Eng., Vol. 11, pp Medina, F. (1981) An axisymmetric infinite element, Int. J. umer. Methods Eng., Vol. 17, pp Chow, Y.K. and Smith, I.M. (1981) Static and Periodic infinite solid elements, Int. J. umer. Methods Eng., Vol. 17, pp Park, K.R. and Watanabe, E. (2003) Development of 3 dimensional dynamic infinite elements in layered media, Proceeding of 16 th KKC Symposium, Gyeongju, Korea, pp Karasudhi, P. (1991) Foundations of solid mechanics, Kluwer Academic Publisher. Wong, H.L. and Luco, J.E. (1976) Dynamic response of a rigid foundation of arbitrary shape, Earthquake Engineering and Structural Dynamics, V 4, pp Chow, Y.K. (1986) Simplified analysis of dynamic response of rigid foundations with arbitrary geometries, Earthquake Eng. Struct. Dyn., Vol. 14, pp Dobry, R. and Gazetas, G. (1985), Dynamic response of arbitrarily shaped foundations, technical notes, ASCE, pp Lysmer, J. et al. (1988) SASSI: A System for Analysis of Soil- Structure Interaction; User s manual, University of California, Berkeley. ( : / : / : ) 27ƒ 5A œ 711