A SIMPLIFIED SEISMIC DESIGN METHOD FOR UNDERGROUND STRUCTURES BASED ON THE SHEAR STRAIN TRANSMITTING CHARACTERISTICS

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1 A SMPLFED SESMC DESGN METHOD FOR UNDERGROUND STRUCTURES BASED ON THE SHEAR STRAN TRANSMTTNG CHARACTERSTCS Tutomu NSHOKA and Shieki UNJOH Public Work Reearch ntitute, -6, Minamihara, Tukuba, , JAPAN Public Work Reearch ntitute, -6, Minamihara, Tukuba, , JAPAN ABSTRACT: Thi paper preent a implified eimic dein method for underround tructure baed on the hear train tranmittin characteritic from urroundin round to the tructure. Since eimic deformation in the cro ection of underround tructure i mainly hear deformation, eimic performance i etimated by the hear deformation in implified eimic dein method. Thi paper clarifie that tructure-round hear train ratio i the hyperbolic function of round-tructure tiffne ratio and propoe an analytical method to etimate the eimic hear deformation uin the hear train tranmittin characteritic.. ntroduction Underround tructure were thouht to be relatively afe durin earthquake until ome of the ubway tunnel in Kobe uffered eriou damae from the 995 Hyooken-nanbu earthquake []. Seimic deformation method (SDM) i commonly applied to practical eimic dein of the cro ection of underround tructure [-3]. n thi method, free-field round diplacement i loaded on the tructure throuh Winkler-type oil prin. One of the problem in the SDM, however, i evaluatin the oil prin that i uppoed to imulate the oil-tructure kinematic interaction. The adequacy of the oil prin and it evaluation method are till under dicuion. Althouh traihtforward approache uch a finite element method (FEM) can olve the problem of the oil prin model, there till remain an important place for imple approache for the benefit of practical dein activitie. ntead of the SDM, implified eimic dein method to etimate eimic deformation of underround tructure are propoed thee day [3-4]. The eimic deformation in the cro ection of underround tructure i mainly hear deformation. Hence, eimic performance i etimated by the hear deformation baed on the round-tructure hear tiffne ratio in thee method. Thee current method, however, conider only horizontal deformation to evaluate the eimic hear deformation. The eimic deformation of underround tructure enerally include horizontal and vertical deformation, and the rotation of the whole tructure exit imultaneouly. n thi dicuion, coniderin the horizontal and vertical deformation and the rotation of the tructure, the eimic hear deformation i accurately evaluated. t i clarified that tructure-round hear train ratio depend only on round-tructure hear tiffne ratio and that the hyperbolic relation between the two parameter exit. Finally, a implified analytical method to etimate the eimic hear deformation of underround tructure i propoed baed on the hear train tranmittin characteritic.

2 . Analyi of Shear Strain Tranmittin Characteritic. Analyi Cae Common utility boxe with rectanular cro ection are adopted for the analyi of the hear train tranmittin characteritic. Seven cae are analyzed, a hown in Table. Parameter are apect ratio of the rectanular cro ection and tructure-round weiht ratio in order to invetiate the effect of the kinematic and the inertia interaction on the hear train tranmittin characteritic. Cae i the baic cae that correpond to RC common utility box with two hollow hown in Fi.. The apect ratio b/h vary from. to.79 in Cae ~4. The tructure-round weiht ratio W /W in Cae 5,6 are adjuted to be.,., repectively, by chanin RC weiht per unit volume. Cae Width b (m) Table Analyi cae Heiht h (m) Overburden D (m) Beam element 37 Surface layer thickne H (m) node Apect ratio b/h Riid zone Weiht ratio W /W Fi. Cro ection of the common utility box and the frame model in Cae 875 unit:mm Ground diplacement ditribution Ground level Underround tructure Lateral boundary (Free in the horizontal direction ) Quarter Bae layer (Fix) coine curve Surface layer Fi. Schematic diaram of the FEM analyi 55.7m 5.m Fi. 3 Finite element meh of Cae 76.m

3 . FEM Analyi Structure-round hear train ratio are analyzed by two-dimenional FEM. The tructure i modeled a the beam element illutrated in Fi.. The urface layer i modeled a the plain train element hown in Fi.. Static horizontal inertia force are applied to both the round and the tructure element o that round diplacement ditribution would be the quarter coine curve. Fi. i the chematic diaram of the FEM analyi. FEM boundary condition are fixed on the bae layer and free in the horizontal direction at the lateral boundarie of the urface layer. Fi. 3 how finite element meh of Cae. Horizontal ditance between the tructure and the lateral boundarie i et to be approximately three time a loner a the thickne of the urface layer. The urface layer i aumed to be elatic and homoeneou in order to implify the round hear tiffne. Two cae of hear wave velocity V = 5, (m/) are taken into account. Weiht per unit volume and Poion' ratio of the oil are 8 (kn/m 3 ) and.45, repectively. The tructure i modeled a two type of frame to invetiate the effect of tructural linearity and non-linearity on the hear train tranmittin characteritic. One i an equivalent linear frame model and the other i a non-linear frame model. The non-linear frame model ha the tri-linear moment-curvature relationhip that have the concrete crack point, the reinforcement yield point, and the ultimate point. The ultimate point i defined a the point where concrete compreion train reache.35 [5]. Weiht per unit volume, modulu of elaticity, and Poion' ratio of the RC are aumed a 4.5 (kn/m 3 ),.35 4 (N/mm ), and /6, repectively. Δ P = b G (Δ /h) Fi. 4 Etimation of the equivalent hear tiffne G of the whole tructure γ δ h δ r δ γ d b d.3 Computation of Shear Stiffne and Shear Strain of Ground and Structure Ground hear tiffne G i iven by G = ρv () / δ (a) n cae of G > G δ where V = hear wave velocity of the urface layer, ρ = weiht per unit volume of the oil, = acceleration of ravity. The equivalent hear tiffne G of the whole tructure, imple-upported at the lower lab hown in Fi. 4, i etimated by G = ( P / b) /( / h) () γ δ r γ (b) n cae of G < G Fi. 5 Ground hear train γ and the equivalent hear train γ of the whole tructure b h

4 where P = horizontal force applied to the upper lab, = horizontal diplacement of the upper lab, h = heiht of the cro ection, b = width of the cro ection. The round hear train γ on the tructure' underround level i obtained by γ = δ h (3) / γ/γ.5 5% of Flexural % of Riidity Flexural.5 Riidity Nonlinear, V=5m/ 5% of Nonlinear, V=m/ Flexural Equivalent Linear, Vm/ Riidity G/G where δ = relative horizontal diplacement of free-field round on the tructure' underround level hown in Fi. 5. The equivalent hear train γ of the whole tructure i computed, baed on the eometric property of the paralleloram in Fi. 5, by γ = δ / h = ( d d ) /(4bh) (4) where δ = diplacement of the upper lab due to the hear deformation of the tructure, d, d = horter and loner diaonal of the cro ection, repectively. t hould be noted that the yrotatic rotation of the whole tructure exit when G > G and that the counterclockwie rotation of the lower lab exit becaue of the larer hear deformation of the tructure than that of the round when G < G. Fi. 6 Comparion of the non-linear and the equivalent linear tructure frame model (Cae ) b/h=..5 b/h=.47 b/h=.6 b/h=.9 b/h= G/G.4 Effect of Linearity and Non-linearity of the Structure Frame Model Fi. 6 how the comparion of tructure-round hear train ratio γ /γ between the equivalent linear and the nonlinear tructure frame model in Cae. n the equivalent linear model, 5,, and 5 % of the initial flexural riidity of the beam element are analyzed. n the nonlinear model, G decreae a γ increae. The more round-tructure hear tiffne ratio G /G increae, The more γ /γ increae. γ /γ of the equivalent linear model are very cloe to thoe of the nonlinear frame model. t i alo found that, in the nonlinear model, γ /γ of V = 5, (m/) form a nearly identical curved line..5 Effect of Apect Ratio of the Cro Section γ /γ are analyzed in Cae ~4 where apect ratio b/h of the rectanular cro ection vary from. to.79. t i noted that the equivalent linear tructure frame model are applied to Cae ~4 in order to implify the analytical procedure. Fi. 7 how the relationhip between G /G and (δ +δ r )/δ, where δ r i the horizontal diplacement due to rotation of the lower lab, obtained by ubtractin δ from the relative horizontal diplacement (δ +δ r ) of the upper lab. Where G /G <, the maller b/h are, the larer (δ +δ r )/δ are. Where G /G >, on the other hand, the advere tendency i oberved. (δ+δr)/δ.5 Fi. 7 Relationhip between G /G and (δ +δ r )/δ (Cae ~4)

5 Fi. 8 how the relationhip between G /G and δ r /δ. t i found that δ r /δ > where G /G < and that δ r /δ < where G /G >. The lant of δ r /δ with maller b/h are teeper. Thi can be explained by the yrotatic rotation of the whole tructure in G /G < and the counterclockwie rotation of the lower lab in G /G > a decribed above. f G /G =, the hear deformation of the tructure i a much a that of the round. Fi. 9 how the relationhip between G /G and the tructural deformation. Fi. how the relationhip between G /G and γ /γ. γ /γ are turned to be δ /δ by multiplyin the numerator and the denominator by the heiht of the cro ection. γ /γ in Cae ~4 have the imilar curved line reardle of b/h. t i made clear that b/h have little effect on γ /γ..6 Effect of Structure-round Weiht Ratio γ /γ are analyzed in Cae 5, 6 in which tructure-round weiht ratio W /W are.,., repectively. Fi. how the relationhip between G /G and γ /γ. W i the equivalent weiht per unit volume of the whole tructure that i computed by dividin all the RC member' weiht by the volume of the whole ection includin hollow. No inertia force are applied to the tructure when W /W =., wherea the whole inertia force of the tructure are equal to thoe of the round when W /W =.. γ /γ in Cae 5, 6 come into the approximately ame curved line. t i found that the inertia force of the tructure have little effect on γ /γ. 3. Etimation of Seimic Shear Deformation 3. Formulation of the Shear Strain Tranmittin Characteritic The phyical bai of the SDM i explained by the tatic ubtructure δr/δ b/h=. b/h=.47 b/h=.6 b/h=.9 b/h= G/G Fi. 8 Relationhip between G /G and δ r /δ (Cae ~4) (a) G /G < (b) G /G = (c) G /G > Fi. 9 Relationhip between G /G and the tructural deformation γ/γ (δ/δ).5 b/h=..5 b/h=.47 b/h=.6 b/h=.9 b/h= G/G Fi. Relationhip between G /G and γ /γ (Cae ~4) γ/γ.5.5 W/W=. W/W= G/G Fi. Relationhip between G /G and γ /γ (Cae 5, 6)

6 method derived from the dynamic ubtructure method [6]. Equation of motion of the SDM are iven by K SS K S rs M SS r S = + + G G F F K S K + K r M K r r q (5) where K = tiffne matrix of the tructure, M = ma matrix of the tructure, r = diplacement vector. The ubcript and S denote the node on the oil-tructure interface, and the remainin node of the tructure, repectively. The upercript dot denote time derivation. K G = round impedance matrix, r F = free-field round diplacement vector, q F = free-field round internal force on the oil-tructure interface. The econd row of Equation (5), the equilibrium on the oil-tructure interface, i expreed a ( K S S G F F r + K r ) = M r + K ( r r ) + q (6) Since only the hear deformation of the tructure and the round are dicued herein, the equilibrium of one-dimenional hear tre in Fi. i applied to Equation (6). The equilibrium i iven by Ground diplacement ditribution G (γ -γ ) G γ Ground level G γ = G ( γ γ ) + G γ (7) γ γ G γ where G γ = tructure reaction hear tre, G (γ -γ ) = round hear tre due to the relative hear train between the free-field round and the tructure, G γ = free-field round hear tre on the tructure' underround level. The firt term of the riht ide of Equation (6), the inertia force of the tructure are inored becaue the effect of the inertia interaction are very mall, hown in Fi.. The tructure-round hear train ratio γ /γ i expreed a γ / γ (G / G ) /( G / G + ) (8) = Fi. 3 how the comparion between γ /γ in Cae ~4 and Equation (8). t i found that Equation (8) approximately coincide with γ /γ in Cae ~4. γ/γ Bae layer Fi. Equilibrium of one-dimenional hear tre.5.5 Cae~4 Equation (8) G/G Fi. 3 Comparion of Equation (8) and γ /γ (Cae ~4)

7 3. Etimation of the Seimic Shear Deformation of Underround Structure Seimic repone analyi of free-field round ive the round hear train γ on the tructure' underround level and the round hear tiffne G. Subtitutin γ and G into Equation (8) yield one relation of the equivalent hear train γ and the equivalent hear tiffne G of the whole tructure. Puhover analyi of the tructure, modeled by the imple-upported non-linear frame in Fi. 4, provide another relation of γ and G. Durin earthquake, γ i determined by the interectin point of the two relation. Fi. 4 how the flow diaram of etimation of the eimic hear deformation baed on the hear train tranmittin characteritic. 3.3 Practical Example A practical example of etimation of the eimic hear deformation i provided a follow. Fi. 5 how the multi-layered round condition and the common utility box located in Kobe. The one-dimenional eimic repone analyi (SHAKE) of the urface layer i conducted. The eimic motion oberved underround (G.L.-83m) in the Kobe reclaimed iland durin the 995 Hyooken-nanbu earthquake i applied to SHAKE. From the reult of SHAKE, the peak round hear train γ,max on the tructure' underround level i n cae that the thickne of the urface layer i one fourth of the hear wavelenth, the equivalent hear wave velocity V,eq of the round i iven by V /, eq = 4H T (9) where H = total thickne of the urface layer, T = fundamental natural period of the urface layer analyzed by SHAKE. The equivalent round hear tiffne G,eq i evaluated by V,eq and ρ ave that i the averae weiht per unit volume weihted by each layer thickne. Fi. 6 how the two relation of γ and G. The interectin point of the two relation ive γ =.. At the ame time, the FEM tatic Total thickne H = 38. (m) Evaluatin γ and G by eimic repone analyi of free-field round Equation (8), Shear train tranmittin characteritic Relation of γ and G baed on Equation (8) Etimation of γ Puhover analyi of the tructure Structural nonlinear relation of Δ and P (Fi. 4) Structural nonlinear relation of γ and G Solution of the two relation Fi. 4 Flow diaram of etimation of the eimic hear deformation baed on the hear train tranmittin characteritic Each layer thickne (m) ρ(kn/m 3 ) Ground level V (m/) h = 5.4 (m) D =.95 (m) b = 4. (m) γ,max =.98-3 Fi. 5 The multi-layered round condition and the common utility box in Kobe

8 analyi, in which the horizontal inertia force at the time of γ,max =.98-3 in SHAKE are applied to both the tructure and the round, i conducted on the identical condition. The traihtforward approach ive the more precie olution of γ =.4. f the capacity hear deformation γ u i aumed a the minimum hear deformation for the non-linear repone of one tructural member to reach the ultimate curvature, γ u reult in.4 by the puhover analyi of the tructure. t i found that γ i much maller than γ u, which can explain the mall damae of the common utility boxe in Kobe durin the Hyooken-nanbu earthquake. The etimation of γ baed on the hear train tranmittin characteritic propoed herein i applicable enouh for the practical eimic dein method of underround tructure. 4. Concluion G(kN/m ).E+5 8.E+4 6.E+4 4.E+4.E+4 γ =. Puhover analyi Equation (8).E γ Fi. 6 Etimation of the eimic hear deformation baed on the hear train tranmittin characteritic The hear train tranmittin characteritic from the urroundin round to the underround tructure have been intenively dicued in thi paper. t i clarified that the apect ratio of the cro ection of the tructure and the tructure-round weiht ratio have little influence on the tructure-round hear train ratio. The hyperbolic relationhip between the round-tructure hear tiffne ratio and the tructure-round hear train ratio i developed by the equilibrium of one-dimenional hear tre between the tructure and the round. Finally, a implified eimic dein method for underround tructure baed on the hear train tranmittin characteritic i propoed. Reference [] SAMATA S., "Underround Subway Damae durin the Hyooken-nanbu Earthquake and Recontruction Technoloy", JSCE Journal of Contruction Manaement and Enineerin, No. 534, V-3, 996.3, pp. -7. [] JAPAN ROAD ASSOCATON, "Specification of Dein and Contruction for Parkin Structure", Maruzen nc., Tokyo, 99., pp [3] RALWAY TECHNCAL RESEARCH NSTUTUTE, "Specification of Seimic Dein for Railway Structure", Maruzen nc., Tokyo, 999., pp [4] WANG J. N., "Seimic Dein of Tunnel, A State-of-the-Art Approach", Monoraph 7, Paron Brickerhoff Quade & Doula, nc., New York, 993. [5] JAPAN SOCETY OF CVL ENGNEERS, "Standard Specification for Concrete Structure, Dein Edition", Maruzen nc., Tokyo, 996.3, pp [6] TATESH A., "A Study on Loadin Method of Seimic Deformation Method", JSCE Journal of Structural Mechanic and Earthquake Enineerin, No. 44, -8, 99., pp