Analytical solution for deep rectangular structures subjected to far-field shear stresses

Analytical solution for deep rectangular structures subjectedto far-field shear stressesH.Huo a ,A.Bobeta,*,G.Ferna´ndez b ,J.Ramı´rez aaSchool of Civil Engineering,Purdue University,550Stadium Mall Drive,West Lafayette,IN 47907-1284,United StatesbDepartment of Civil and Environmental Engineering,University of Illinois,Urbana,IL,United StatesReceived 9May 2005;received in revised form 15August 2005;accepted 10December 2005Available online 28February 2006AbstractUnderground structures located in seismic areas have to support the static loads transferred from the surrounding ground under nor-mal working conditions,as well as the loads imposed by any seismic event.Typically underground structures have cross section dimen-sions much smaller than the wave length of ground peak velocities,in which case inertial forces can be neglected and the structure can be designed using a pseudo-static analysis,where the seismic-induced loads or deformations can be approximated by a far-field shear stress or strain.Current close-form solutions for deep rectangular structures subjected to a far-field shear stress are approximations that do not consider all the relevant variables.An analytical solution is presented in this paper for deep rectangular structures with a far-field shear plex variable theory and conformal mapping have been used to develop the solution,which is applicable to deep rectangular structures in a homogeneous,isotropic,elastic medium.The solution shows that the deformations of the structure depend on the relative stiffness between the structure and the surrounding ground,and on the shape of the structure.The analytical solution has been verified by comparing its predictions with results from a finite element method and from previously published data.Ó2006Elsevier Ltd.All rights reserved.Keywords:Deep rectangular structure;Relative stiffness;Seismic design;Pseudo-static analysis;Analytical solution1.IntroductionThe final support system of underground facilities in seismic zones must be designed to support static overbur-den loads as well as to accommodate additional deformations imposed by earthquake-induced motions.Seismic-induced deformations of underground structures can be produced by direct shearing displacements of active faults intersect-ing the structure,by ground failure,or by ground shaking.Most of the analytical work done so far has concentrated on the evaluation of the effects of ground shaking.There are two basic approaches in present seismic design.One approach is to carry out dynamic,non-linear soil–structure interaction analysis using finite element orfinite difference methods,where inertia forces are included.The input motions in these analyses are time histories emulating design response spectra.Input motions are applied to the boundaries of a ‘‘soil island’’to represent vertically propagating shear waves.In the second approach,the pseudo-static approach,inertia forces are neglected.The earthquake loading is simulated as a static far-field shear stress or strain applied at the boundaries of the ground where the structure is embed-ded.In the pseudo-static approach soil–structure interac-tion may or may not be considered.When the interaction is included in the analysis,finite element or finite differ-ence methods are typically used;analytical solutions exist which provide relationships to evaluate the magnitude of seismic-induced displacements or strains in underground structures (Penzien and Wu,1998;Penzien,2000;a com-prehensive review is provided by Hashash et al.(2001)).If soil–structure interaction is neglected,it is assumed that0886-7798/$-see front matter Ó2006Elsevier Ltd.All rights reserved.doi:10.1016/j.tust.2005.12.135*Corresponding author.Tel.:+17654945033;fax:+17654961364.E-mail address:bobet@ (A.Bobet)./locate/tustTunnelling and Underground Space Technology 21(2006)613–625Tunnelling andUnderground Space Technologyincorporating Trenchless Technology Researchthe structure follows the deformations of the ground. This is the freefield approach(Hendron and Ferna´ndez, 1983;Merritt et al.,1985),where relationships are obtained based on the premise that the structure must accommodate the free-field deformations without loss of its integrity.This may not be entirely correct since the presence of the structure,if more rigid than the ground, would decrease the deformations of the surrounding ground.Although this effect may appear to bring predic-tions on the safe side,this is so only when the structure is stiffer than the surrounding ground.If the structure is moreflexible than the ground,predictions may be unsafe because the liner distortions are larger than the freefield deformations.To account for this effect Hendron and Ferna´ndez(1983),and Merritt et al.(1985),suggested to consider ground deformations compatible with the strain concentration imposed by the presence of the opening.Analytical relationships are presented in this paper fol-lowing the pseudo-static approach to estimate seismic-induced distortions(racking mode)in deep rectangular underground structures,and taking into account the inter-action between the soil and the structure.These relation-ships can be an effective tool for practitioners.They allow readily identification of the variables controlling the magnitude of the distortions and thus provide an insight into the behavior of the structure.The insight gained from the analysis can be used to optimize potential numerical remodeling efforts by identifying pertinent parameters for sensitivity analysis.Estimates obtained from analytical relationships can also be used in pre-feasibility and feasibility studies to obtain early and relatively accurate assessment of structural requirements.Finally,analytical relationships can also be used to check the validity of the results obtained from numerical modeling.2.Methodologies for seismic-induced structure deformationsThe subject of ground-support interaction has been trea-ted by numerous researchers(Savin,1961;Timoshenko and Goodier,1970;Peck et al.,1972;Einstein and Sch-wartz,1979;Hendron and Ferna´ndez,1983;Merritt et al.,1985;Penzien and Wu,1998;Penzien,2000;Hashash et al.,2001;Bobet,2003),and it has concentrated mostly on circular cross-sections.A circular cross section may be appropriate for lifelines and deep tunnels,but for shallow structures(e.g.,cut-and-cover)and some mines a rectangu-lar cross-section is more common.A close-form solution for deep rectangular tunnels has been proposed by Penzien (2000).Current seismic design approaches for circular tunnels proposed by Hendron and Ferna´ndez(1983),and Merritt et al.(1985),and for rectangular tunnels proposed by Penz-ien and Wu(1998),and Penzien(2000),suggest the use of simplified relationships to estimate both the seismic-induced longitudinal as well as the circumferential strains in tunnel liners.Three assumptions are usually made:(a)The dynamic amplification of stresses associated witha stress wave impinging on the opening is negligible.This assumption is correct if the wave length of peak velocities is at least eight times larger than the width of the opening(Hendron and Ferna´ndez,1983).Under these conditions the free-field stress gradient across the opening is relatively small and the seismic loading can be considered a pseudo-static load.The wave length can be estimated as V s/f;where V s is the shear wave velocity and f,the frequency of vibra-tion of peak ground motions.In most underground openings pseudo-static conditions are usually satisfied.(b)Plane strain conditions are assumed on any sectionperpendicular to the longitudinal axis of the tunnel.(c)Linear elastic deformations of ground and structure.In general racking of the structure(ovalization of the structure due to shear waves traveling perpendicular to the longitudinal axis of the structure)is the most critical deformation.In a pseudo-static analysis,the shear wave motions can be approximated by a farfield constant shear stress or shear strain.The magnitude of the farfield shear stress sffis equal to the shear modulus,G,of the soil times the shear strain of the soil,cff.That is:s ff¼G c ffð1ÞThe freefield shear strain cffcan be obtained as:c ff¼v s=V sð2Þwhere v s is the peak particle vibration velocity,and V s is the shear wave velocity of the soil.Wang(1993)ran dynamic parametric analyses on rect-angular structures with different dimensions and stiffnesses. He found that theflexibility ratio of the structure,F,corre-lates well with the structure’s deformations.Theflexibility ratio is expressed as:F¼Ga=S1b,where a and b are the dimensions of the structure(Fig.1),G is the shear modulus of the ground,and S1=1/D1with D1equal to the displace-ment produced on the structure by a unit lateral concen-trated force applied to the top of thestructure.614H.Huo et al./Tunnelling and Underground Space Technology21(2006)613–625Penzien(2000)proposed an approximate method to evaluate the racking deformation of deep rectangular tun-nels subjected to a farfield shear stress.It is an approxi-mate solution because of the assumption that the load transfer between the ground and the structure takes place only through shear stresses at the interface,and the defor-mations of a rectangular opening are approximated by those of a circular opening.Penzien showed that the defor-mations of the structure depend on the relative stiffness,or the stiffness ratio,between the ground and the structure. The relative stiffness is defined with the parameter k stru/ k soil,which is the ratio between k stru,the stiffness of the structure and k soil the stiffness of the soil.k stru is equal to the magnitude of a uniform shear stress applied to the perimeter of the structure that produces a unit displace-ment of the structure;and k soil=G/b,where G is the shear modulus of the soil and b is the height of the structure.The relative stiffness is the inverse of Wang’sflexibility ratio.The normalized deformation of the structure,or the ratio between the structure deformation(D stru)and the free-field ground deformation(Dff),D stru/Dff,can be obtained by:D stru D ff ¼4ð1ÀvÞ1þð3À4vÞK strusoilð3Þwhere v is the Poisson’s ratio of the ground.When the structure is much stiffer than the surrounding ground (i.e.,k stru)k soil),k stru/k soil is very large and the deforma-tion of the structure approaches zero.This corresponds to a very rigid ring embedded in a much softer material;the ring will keep its shape no matter the deformation that the surrounding material undergoes.On the contrary,if the structure is much moreflexible than the surrounding ground(i.e.,k stru(k soil),k stru/k soil is very small,and the deformation of the structure,D stru,is4(1Àv)Dff.This is as if the structure does not exist and the solution corre-sponds to the deformation of the opening.Note that the value4(1Àv)Dffis exact for circular openings subjected to a farfield constant shear stress,but it is an approxima-tion for rectangular openings.Another important assump-tion made in developing Eq.(3)is that ground displacements are imposed to the structure only through shear stresses at the interface,thus neglecting the contribu-tion of any normal stresses.A new theoretical solution is proposed in this paper. Two are the keynote factors:(1)the new solution considers the contribution of both normal and shear stresses at the interface;(2)it considers the actual deformations of a rect-angular opening.The work presented in this paper is part of an on-going research to develop practical solutions for seismic design of cut and cover rectangular structures. 3.Theoretical frameworkAs with other analytical methods,the following assump-tions are made in the solution proposed herein:1.Deep rectangular structure inside an infinite medium.2.Plane strain conditions in any section perpendicular tothe longitudinal axis of the structure.3.Elastic response of the structure and surroundingground.4.Pseudo-static analysis.The seismic deformations ofground and structure can be approximated by a con-stant far-field shear stress or strain.The solution of any problem in elasticity must satisfy force equilibrium,compatibility of deformations,and boundary conditions.This is equivalent tofind an Airy stress function U such that:r2ðr2UÞ¼0ð4Þwhere$2is the Laplacian operator.In2D Cartesian coordinates,the stress components can be expressed as:r x¼o2Uo y2;r y¼o2Uo x2;s xy¼Ào2Uo x o yð5Þand the strain components,in plane strain,are:e x¼o u xo x¼1E½ð1Àv2Þr xÀvð1þvÞr ye y¼o u yo y¼1E½ð1Àv2Þr yÀvð1þvÞr xð6Þwhere E is the Young’s modulus of the material and v is the Poisson’s ratio.Complex variable theory and conformal mapping tech-niques have been used for the solution of problems con-cerning rectangular openings in an infinite medium (Mindlin,1940,1948;Muskhelishvili,1954;Sokolnikoff, 1956).The fundamental theories of complex variable and conformal mapping have been extensively described by Muskhelishvili(1954),and later on by Savin(1961)and Timoshenko and Goodier(1970).A number of researchers have successfully implemented this technique into different disciplines of engineering:(Theocaris and Petrou,1989; Theocaris,1991;Motok,1997;Gercek,1997;Exadaktylos and Stavropoulou,2002;Exadaktylos et al.,2003).For cir-cular tunnels the method has been very effectively used by Verruijt and coworkers(Verruijt,1997,1998;Strack and Verruijt,2002).According to complex function theory,any biharmonic function(for example,the Airy stress function)can be expressed as:U¼Re½ z uðzÞþvðzÞ ð7Þwhere U is the Airy stress function;z is a complex variable and z is the complex conjugate of z;u(z)and wðzÞ¼v0ðzÞ¼d vðzÞ=d z are two analytic complex functions,also known as‘‘complex potential functions’’.The stress compo-nents can be expressed in terms of complex potentials as: r xþr y¼2½u0ðzÞþr yÀr xþ2i s xy¼2½ z u00ðzÞþw0ðzÞð8ÞH.Huo et al./Tunnelling and Underground Space Technology21(2006)613–625615The displacements in plane strain are:2Gðu xþi u yÞ¼ð3À4vÞuðzÞÀz u0ðzÞÀwðzÞð9Þwhere u x and u y are horizontal and vertical displacements, respectively;G is the shear modulus of the material,G= E/[2(1+v)].Fig.1shows the problem to be solved:a rectangular structure in an infinite medium subjected to a far-field stress sff.Complex variable theory is used to determine stresses and deformations of the ground,while structural theory is used tofind stresses and deformations of the patibility of normal and shear stresses and displacements is invoked at the interface between the ground and the structure to solve the problem.Thus,the approach followed consists of the solution of two initially independent problems:(1)the ground with a rectangular opening;(2)the structure.For the ground the objective is tofind the two complex potentials,u(z)and w(z).The Airy stress function for the ground can be expressed as:U¼U0þUÃð10Þwhere U0is the Airy stress function corresponding to the infinite medium without the rectangular opening,and U* is the Airy stress function that includes the presence of the rectangular opening.The two complex potential functions are:uðzÞ¼u0ðzÞþuÃðzÞwðzÞ¼w0ðzÞþwÃðzÞð11Þwhere u0(z)and w0(z)are the complex potentials corre-sponding to the infinite medium without the opening,and u*(z)and w*(z)are the complex potentials that account for the presence of the opening.The complex potential functions depend on the boundary conditions,which are gi-ven by the known stressfield at infinity,sff,and by the stress field at the interface between the ground and the structure, which is unknown and depends on compatibility of defor-mations between the ground and the structure.This intro-duces an additional level of difficulty since the solution depends on the solution itself.However,numerical simula-tions show that the shape of the normal and shear stress dis-tribution at the interface is rather independent of the dimensions of the rectangular opening and of the elastic properties of the ground and structure.In fact the shear stress can be well approximated as uniformly distributed around the perimeter of the structure,and the normal stress as linearly distributed(see Fig.2).This stress distribution follows the symmetry of the problem;there are two anti-symmetric axes:one vertical and the other one horizontal through the center of the structure.Thus,a complete solu-tion can be obtained if the stress distribution at the interior boundary is known.Note that only the distribution of the stresses is required;the actual values are obtained by estab-lishing compatibility of stresses and deformations between the ground and the structure.Thefinal loads on the struc-ture will result from the addition of the seismic loads,which are the ones discussed in this paper,and the static loads from the overburden(not shown in Fig.2).The Airy stress function and the complex potentials for the infinite plate(no structure)subjected to the farfield stresses sffare:U0¼Às ff xyu0ðzÞ¼0wðzÞ¼i s ff zð12ÞConformal mapping allows the transformation of the rectangular opening to a unit circle(Churchill,1960).The transformation is performed with the holomorphic func-tion(Savin,1961):z¼xðfÞ¼a0fþa1fþa2f2þÁÁÁð13Þwhere a0,a1,a2,etc.,are complex constants.The complex potentials u*(f)and w*(f)can be found from the following equation(Muskhelishvili,1954;Savin, 1961):2p i uÃðfÞþZcxðrÞx0ðrÞuÃ0ðrÞd rrÀf¼Zcf01þi f02rÀfd r2p i wÃðfÞþZcx0ðrÞuÃ0ðrÞd rrÀf¼Zcf01Ài f02rÀfd r8>>><>>>:ð14Þwhere u*(f)and w*(f)are the two complex potentials in terms of f;r denotes any point on the unit circle;c is theunit circle;and f01and f02are two boundary functions de-fined as(Muskhelishvili,1954;Savin,1961):f01þi f02¼f1þi f2Àu0ðrÞþxðrÞu0ðrÞþw0ðrÞ"#ð15Þwhere f1and f2are:f1¼ÀIÀr yd xd sþs xyd yd s!d sf2¼Ir xd yd sÀs xyd xd s!d sð16Þwhere r x,r y,and s xy are the stresses at the perimeter of the opening,and the integrals are performed along the contour of theopening.616H.Huo et al./Tunnelling and Underground Space Technology21(2006)613–6254.Analytical solutionThe conformal mapping of a rectangle can be approxi-mated with the following terms (Savin,1961):z ¼x ðf Þ¼aR 1f þða þ a Þ2f þða À a Þ224f 3þða 2À a 2Þða À a Þ80f 5!ð17Þwhere a =e 2k p i =cos(2k p )+i sin(2k p );k is a mathematical parameter related to the shape of the opening k ,k =a /b ,the ratio of the length ‘‘a ’’and the height ‘‘b ’’of the rect-angular opening.The following equation relates the size of the opening,given by a and b ,the aspect ratio k ,and the parameter k .k ¼a =b ¼1þcos 2k p À16sin 22k p À120sin 2k p sin 4k p 1Àcos 2k p À16sin 22k p þ120sin 2k p sin 4k p ð18ÞFor k <1/4,the length is larger than the height (a >b );for k >1/4,the length is smaller than the height (a <b ).For k =1/4,a =b ,which is a square structure,k =0.25,0.2and 0.172correspond to k =a /b =1,2and 3,respectively.Parameter R in Eq.(17)is defined as:R ¼121þcos 2k p À1sin 22k p À1sin 2k p sin 4k p ÀÁð19ÞThe rectangle ABCD in the ‘‘z ’’plane,shown in Fig.3,is transformed into a unit circle A 0B 0C 0D 0in the ‘‘f ’’plane.Point A 0in the unit circle corresponds to point A in the rect-angle,and so on.Note that ABCD is in a clockwise direction,whereas A 0B 0C 0D 0is counter-clockwise.This is so because the selected conformal mapping function transforms infinity in the ‘‘z ’’plane into the origin in the ‘‘f ’’plane,and vice versa.In other words,the area of interest (the ground)in the ‘‘z ’’plane is outside the rectangle,which corresponds to the area enclosed by the unit circle in the ‘‘f ’’plane.The distributions of normal and shear stresses at the perimeter of the rectangular opening are also shown in Fig.3.These are the stresses imposed on the ground by the structure.Identical stresses,but with opposite sign are applied to the structure (see Fig.2).The magnitudeof the shear stress s i ,which is unknown,is constant along the perimeter of the rectangle.The normal stress has a linear distribution on the four sides of the rectan-gle with a maximum magnitude p i 1along the side of length ‘‘a ’’and p i 2along the height ‘‘b ’’;both magnitudes are unknown.The positive sign of the normal stress denotes compression whereas the negative sign denotes tension (decompression given the initial static loading due to the overburden).According to the Cauchy theorem,the first integral of the first equation in (14)is zero because the function inside the integral is analytic along the unit circle c (Churchill,1960;Timoshenko and Goodier,1970).The integral along the unit circle on the right-hand side of (14)is composed of four integrals:one for each segment A 0B 0,B 0C 0,C 0D 0and D 0A 0on the unit circle (see Fig.3).The integrals are solved analytically using Taylor series expanion of the functions inside the integrals (Huo,2005).The functions u (f )and w (f )depend on p i 1;p i 2and s i .Compatibility of deformations between the ground and the structure are needed to solve for p i 1;p i 2and s i .The dis-placements of any point of the ground at the interface,given by Eq.(9)must be the same as the displacements of the same point of the structure.Points at the corners and centers of the sides of the rectangle are chosen to impose compatibility of deformations.Also,the structure must be in self-equilibrium.Based on moment equilibrium,the relation between p i 1and p i 2is:p i 1a 2¼p i 2b 2ð20ÞThe structure is assumed to deform only due to bending (deformations of structural elements due to axial forces are neglected).This is a common assumption in structural mechanics.The structure deformations,in plane strain,are given by:D stru ¼ð1Àm 2S Þðs i D s i þp i2D p i 2Þð21Þwhere m s is the Poisson’s ratio of the structure,D s i is the deformation of the structure due to a shear stress s i ,and D p i 2is the deformation of the structure due to a linear nor-mal stress distribution given by p i 1;p i 2,with the condition p i 1a 2¼p i 2b 2.D stru is the structure’s distortion,and is equal to the difference between the displacements of the top and bottom of the structure (see Fig.4).The actual values of D s i and D p i 2,can be obtained analytically or numerically from structural analysis.For the case of a rectangular structure with width ‘‘a ’’and height ‘‘b ’’with an interior central column,with members’stiffness:lateral walls,(EI)w ;top and bottom slabs,(EI)s ;central column,(EI)c ,the deformations are:D s i ¼124k b 42ðEI Þc k ðEI Þs þ1ðEI Þwh i þk 2ðEI Þs 3k 2ðEI Þs þ2ðEI Þw h i 2ðEI Þcþ1ðEI Þw þ3kðEI Þs D p 2i ¼14b 41c k s þ1w h i þk s 7k s þ5wh i 1cþ1wþk sð22Þz planep 1ip 2iaiiA BCDA‘B‘C‘D‘1(a)(b)Fig.3.Conformal mapping H.Huo et al./Tunnelling and Underground Space Technology 21(2006)613–625617There is a mismatch between the dimensions of the struc-ture and the opening,with a difference equal to the thick-ness of the structure members.This introduces errors in the formulation,which may be important because the thick-ness of structural members may not be small compared to the size of the opening.This is generally the case for rectan-gular structures because its resisting elements must with-stand significant bending moments.This issue is discussed later with the verification of the analytical solution.The D s i=D p i2,ratio depends on the shape of the structureand not on its absolute dimensions.Taking as reference the stiffness of one of the structure members(E s I s),and express-ing all the other element stiffnesses as a function of the refer-ence member,Eq.(22)can be written,in a general form,as:D s i¼K sðkÞE s I sk b4D p i2¼K pðkÞE s I sb4D s i=D p i2¼K sðkÞK pðkÞkð23Þwhere the parameters K s(k)and K p(k)depend on the shape of the structure and on the Young’s modulus and dimen-sions of the members of the structure.For a rectangular structure with no central column and members of equal stiffness(E s I s),the above equations reduce to:D s i¼ð1þkÞ24E s I sk b4D p i2¼ð1þkÞ60E s I sb4D s i=D p i2¼52kð24ÞImposing compatibility of deformations between the structure and the ground at the interface,one gets:p i1¼p i2kp i2¼M s iþN s ffs i¼L s ffð25Þwherewhere j=(3À4m)for plane strain.The functions f1(k), f2(k),etc.are included in Appendix1.Note that the coeffi-cients M and N depend only on the shape of the structure, k,and on the Poisson’s ratio of the ground,m.The param-eter L also depends on the stiffness of structure and bining Eqs.(23)and(26),one gets:Hence the parameters M,N,and L depend only on:the Poisson’s ratios of the structure,m s,and ground,m;on the shape of the structure,k;and on the factor X¼E s I s=Gb3, which is a measure of the relative stiffness between the ground and the structure.Both factors,k and X are non-dimensional.Fig.5is a plot of the parameters M and N as a function of the aspect ratio of the structure,k and Fig.6is a plot of L with k,and with the relative stiffness,X,computed for a rect-angular structure with uniform thickness.The values of the Poisson’s ratios are taken as m=0.35and m s=0.15.As it will be discussed later the effect of the Poisson’s ratios is small.The freefield ground distortion,Dff,is defined as the dis-placement difference between a point on the top slab of the structure and a point on the bottom slab,due to the free field ground deformation cff.Thus,D ff¼c ff b¼s ffbGð28ÞNormalization of the structure’s deformation with respect to the freefield ground deformation results in:D struD ff¼G D strus ff b¼ð1Àm2sÞ½N D p i2þðM D p i2þD s iÞLGbð29ÞM¼j f1ðkÞþf2ðkÞj f3ðkÞþf4ðkÞN¼f5ðkÞÀ0:076j ½j f3ðkÞþf4ðkÞ f6ðkÞL¼Àð1Àm2sÞN D p i2ÀaG1þ1kÀÁj½Nf7ðkÞþf8ðkÞ þN1À1kÀÁf9ðkÞþf10ðkÞÈÉð1Àm2sÞM D p i2þD s iÀa1þ1ÀÁj½Mf7ðkÞþf11ðkÞ þM1À1ÀÁf9ðkÞþf12ðkÞÈÉð26ÞM¼j f1ðkÞþf2ðkÞj f3ðkÞþf4ðkÞN¼f5ðkÞÀ0:076j ½j f3ðkÞþf4ðkÞ f6ðkÞL¼Àð1Àm2sÞNÀE s I sGb3kp1þ1ÀÁj½Nf7ðkÞþf8ðkÞ þN1À1ÀÁf9ðkÞþf10ðkÞÈÉð1Àm2sÞMþK sðkÞpkÀE s I sGb3kp1þ1ÀÁj½Mf7ðkÞþf11ðkÞ þM1À1ÀÁf9ðkÞþf12ðkÞÈÉð27Þ618H.Huo et al./Tunnelling and Underground Space Technology21(2006)613–625。