Transverse response of underground cavities and pipes to incident SV waves

See discussions, stats, and author profiles for this publication at: https:///publication/227543624Transverse response of underground cavities and pipes to incident SV wavesARTICLE in EARTHQUAKE ENGINEERING & STRUCTURAL DYNAMICS · MARCH 2001Impact Factor: 2.31 · DOI: 10.1002/eqe.14CITATIONS 51READS1163 AUTHORS:Craig A. DavisLos Angeles Department of Water and P…48 PUBLICATIONS 264 CITATIONSSEE PROFILEVincent W LeeUniversity of Southern California215 PUBLICATIONS 2,534 CITATIONSSEE PROFILEJean-Pierre BardetUniversity of Texas at Arlington112 PUBLICATIONS 1,583 CITATIONSSEE PROFILEEARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct.Dyn.2001;30:383–410Transverse response of underground cavities and pipes toincident SV wavesC.A.Davis1,V.W.Lee2and J.P.Bardet2;∗1Los Angeles Department of Water and Power;Los Angeles;U.S.A.2Civil Engineering Department;University of Southern California;Los Angeles;U.S.A.SUMMARYThe transverse response of underground cylindrical cavities to incident SV waves is investigated.Analyt-ical solutions are derived for unlined cavities embedded within an elastic half-space using Fourier–Bessel series and a convex approximation of the half-space free surface.The computed displacements at the half-space free surface and the tangential stresses on the cavity are compared with the results of previ-ous investigations.The analytical solutions are extended to formulate approximate solutions for assessing hoop stresses within cavity liners impinged by low-frequency waves having wavelengths much longer than the cavity diameter.The approximate solutions are compared to existing numerical solutions,and used to evaluate the dynamic response of a exible buried pipe shaken by the1994Northridge earth-quake.The proposed approximate model for cavity liners is useful for the seismic analysis of underground pipes and small-diameter tunnels.Copyright?2001John Wiley&Sons,Ltd.KEY WORDS:underground cavity;buried pipes;tunnels;earthquake analysisINTRODUCTIONUnderground pipes and tunnel linings are rarely designed to withstand the transverse loads applied by earthquake ground motions[e.g.,References1,2],mainly because there has been very few well-documented case histories of transverse failures of underground structures during past earthquakes.However,the damage to shallow underground structures observed in near-ÿeld earthquakes indicates a need for improved analysis and design methods accounting for transverse shearing forces[e.g.,References3–6].The dynamic response of unlined cylindrical cavities in a half-space to transverse shear waves(i.e.SV waves)has previously been investigated by[7–9].Wong et al.[7]used a∗Correspondence to:J.P.Bardet,Civil Engineering Department,University of Southern California,KAP210, Los Angeles,CA90089-2531,U.S.A.Contract=grant sponsor:National Science Foundation;contract=grant number:CMS9714859Received3August1999Revised15February2000 Copyright?2001John Wiley&Sons,Ltd.Accepted20June2000384C.A.DAVIS,V.W.LEE AND J.P.BARDEThybrid method combining ÿnite elements and an expanded wave function expressed in terms of Green’s functions.Lee and Karl [8]derived a closed-form solution by approximating the at free surface of the half-space as a concave surface with a large radius of curvature.Luco and De Barros [9]used an indirect boundary integral method based on two-dimensional Green’s functions.El-Akily and Datta [10;11]investigated the response of cylindrical shells in a half-space using successive re ections and matched asymptotic expansions limited to long-period motions.Datta et al .and Chin et al .[12;13]extended that type of analysis to investigate the response of pipes within a back-ÿlled trench subjected to P,SV and Rayleigh waves.The closed-form solution of Reference [8]is relatively simple,however it requires additional work to be applicable to the design of underground pipes.The other solutions [e.g.Reference 12]are more complicated and di cult to implement into design procedures.The method proposed herein for incident SV waves improves upon the solution of Reference [8]by eliminating undesirable re ections on the half-space free surface though a convex approximation.The proposed method is used to analyse the motions at the free surface and stress and displacements on unlined cavities over a broad range of frequencies.The derived solutions are extended to cavities reinforced with a exible liner,and used for determining approximate values of hoop stresses in underground pipes subjected to low-frequency seismic waves.ANALYTICAL SOLUTION FOR UNLINED CYLINDRICAL CAVITYModelFigure 1(a)shows the two-dimensional model used for this analysis.The half-space is made of an elastic and isotropic medium deÿned by the x –y co-ordinate system for y ¿0,and is uniform except for the circular cylindrical section of radius a that is removed to form an unlined cavity.The cylinder is centred at a depth h below the half-space surface.The medium properties are deÿned by the LamÃe constants and ,and mass density .The longitudinal and transverse wave speeds,denoted and ÿ,respectively,are related to Poisson’s ratio and given by=+2 ;ÿ=and ÿ= 2 1+ 1−2 (1)Figure 1(a)shows four co-ordinate systems used in the analysis.The cartesian and polarco-ordinate systems x –y and r –Âoriginate at point O ,whereas those referred to as x 1–y 1and r 1–Â1are centred at point O 1.Table I shows the co-ordinate relations.Linear elastic wave propagation theory is used [e.g.Reference 14].The cavity is subjected to incident plane SV waves having the potential i shown in Table I with an amplitude B 0,angle of incidence Âÿshown in Figure 1(a),angular frequency !,and transverse wave number k ÿ=!=ÿ.The potential satisÿes the equation of motion:∇2 +k 2ÿ =0(2)The harmonic nature of the input,which is implied hereafter,is characterized by e −i !t withi =√−1.TRANSVERSE RESPONSE OF UNDERGROUND CAVITIES AND PIPES385Figure 1.Dynamic model for an underground cavity:(a)co-ordinate systems,geometry,and materialproperties,(b)incident SV,re ected,and scattered wave potentials.The displacement vector ˜u i has in-plane horizontal u x and vertical u y components and am-plitude,respectively,given byu x =−@ i@y =B 0i k ÿcos Âÿe i k ÿ(x sin Âÿ−y cos Âÿ)(3a)u y =+@ i@x=B 0i k ÿsin Âÿe i k ÿ(x sin Âÿ−y cos Âÿ)(3b)|˜u i |=B 0k ÿ(3c)386 C.A.DAVIS,V.W.LEE AND J.P.BARDETTable I.Co-ordinate relations and potential functions for co-ordinate systemsoriginating at points O;O1,and O2.OriginO O1O2Co-ordinate x x=r sinÂx1=−r1sinÂ1x1=x x2=−r2sinÂ2 x2=x1y y=r cosÂy1=r1cosÂ1y1=h−y y2=r2cosÂ2 y2=D−y1r r=x2+y2r1=x21+y21r2=x22+y22ÂÂ=tan−1xyÂ1=tan−1−x1y1Â2=tan−1−x2y2Potential i i=B0e i kÿ(x sinÂÿ−y cosÂÿ)i=B0e−i kÿh cosÂÿe i kÿr1cos(Â1+Âÿ)r r=B1e i kÿ(x sinÂÿ+y cosÂÿ)r=B1e i kÿh cosÂÿe−i kÿr1cos(Â1−Âÿ)r r=A1e i k (x sin +y cos ) r=A1e i k h cos e−i k r1cos(Â1− )When there is no cavity,the incident plane SV wave is re ected from the free surface (y=0),generating both P and SV waves.The re ected SV wave is plane and the P wave may be either:(1)a plane body wave,if incidence is below the critical angleÂcr(Âÿ6Âcr), or(2)a surface wave ifÂÿ¿Âcr.The critical angle is[e.g.,Reference14]:Âcr=sin−1ÿ(4)Only plane body waves are considered in this analysis,i.e.the incidence angle of incoming waves is smaller thanÂcr.Figure1(b)shows the wave potentials generated at the model boundaries.At the free surface (y=0),the re ected longitudinal P and transverse SV plane waves have the potentials r and r,respectively,as shown in Table I.The P waves are characterized by the longitudinal wave number k =!= ,re ected angle ,and re ected coe cients A1and B1given by =sin−1ÿsinÂÿ(5a)A1=B0( =ÿ)2sin4Âÿsin2Âÿsin2 +( =ÿ)2cos22Âÿ(5b)B1=B0sin2Âÿsin2 −( =ÿ)2cos2Âÿsin2Âÿsin2 +( =ÿ)2cos22Âÿ(5c)TRANSVERSE RESPONSE OF UNDERGROUND CAVITIES AND PIPES387The incident SV and re ected SV and P wave potentials are expressed in terms of the r 1–Â1co-ordinate system as shown in Table I and expanded [15]in Fourier–Bessel series asi +r= i + r=∞ n =0J n (k ÿr 1)(C 0n sin nÂ1+D 0n cos nÂ1)(6a) r=∞ n =0J n (k r 1)(A 0n cos nÂ1+B 0n sin nÂ1)(6b)where J n (·)are Bessel functions of the ÿrst kind.The coe cients are determined byA 0n =A 1e i k h cos  n (−i)n cos n ;n ¿0(7a)B 0n =A 1e i k h cos  n (−i)n sin n ;n ¿0(7b)C 0n =[−B 0e −i k ÿh cos Âÿ(i)n +B 1e i k ÿh cos Âÿ(−i)n ] n sin nÂÿ;n ¿0(7c)D 0n =[B 0e −i k ÿh cos Âÿ(i)n +B 1e i k ÿh cos Âÿ(−i)n ] n cos nÂÿ;n ¿0(7d)where the coe cient 0=1and n =2for n ¿1.Analytical solutionAs shown in Figure 1(b),in the presence of the cylindrical cavity (r 1=a ),the incident waverepresented by i ,and the re ected waves represented by r and r are scattered o thecavity surface.The total SV and P waves generated at the cavity are represented by s1and s1.The cylinder vibrations are re ected o the half-space free surface generating new waves represented by s2and s 2.The potentials scattered at the cavity surface ares 1= s 1(r 1;Â1)=∞ n =0H (1)n (k r 1)(A 1n cos nÂ1+B 1n sin nÂ1)(8a) s 1= s 1(r 1;Â1)=∞n =0H (1)n (k ÿr 1)(C 1n sin nÂ1+D 1n cos nÂ1)(8b)The additional potentials scattered at the half-space free surface ares 2= s 2(r 1;Â1)=∞ n =0J n (k r 1)(A 2n cos nÂ1+B 2n sin nÂ1)(9a) s2= s 2(r 1;Â1)=∞n =0J n (k ÿr 1)(C 2n sin nÂ1+D 2n cos nÂ1)(9b)where H (1)n(·)is the Hankel function of the ÿrst kind.The resulting potentials are = r + s1+ s 2(10a) = i + r + s 1+ s2(10b)388C.A.DAVIS,V.W.LEE AND J.P.BARDETStresses and displacements in the medium can be evaluated in terms of potentials as follows [e.g.Reference 14]:rr = ∇2+2@2 @r 2+@@r 1r @ @ (11a) ÂÂ= ∇2 +2 r @ @r +1r @2 @Â2+1r @ @Â−@2@r @Â(11b) rÂ= 21r @2 @r @Â−1r 2@ @ + 1r 2@2@Â2−r @@r 1r @ @r(11c)u r =@ @r +1r @ @Âand u Â=1r @ @Â−@@r(12)The potentials in Equation (10)must satisfy the stress-free boundary conditions at the half-space and cavity surfaces:xy = yy =0at y =0andr 1Â1= r 1r 1=0at r 1=a(13)Since i , r ,and r already satisfy the stress free conditions at y =0,Equation (13)is only used to solve for the coe cients A 1n ;B 1n ;C 1n ;D 1n ;A 2n ;B 2n ;C 2n ,and D 2n for n ¿0in Equations (8)and (9).The boundary conditions are straightforward to apply at the cavity surface since the potentials are expressed in the r 1–Â1co-ordinate system.As shown in Figure 2,the stress-free conditions at y =0(i.e.Equation (13))are applied by approximating the plane half-space surface as a convex circular surface of large radius b a ,in contrast to References [15;8]who used a concave surface.The curved surface is centred about O 2at distance D (D a )above O 1.The curved half-space surface is deÿned by introducing the x 2–y 2and r 2–Â2co-ordinate systems centred at point O 2(Figure 2),which are related to each other and the x 1–y 1coordinates as shown in Table I.The cylindrical wave potentials originate from the r 2–Â2co-ordinate system and are de-scribed ass 1= s 1(r 2;Â2)=∞m =0J m (k r 2)(A ∗1m cos mÂ2+B ∗1msin mÂ2)(14a) s 1= s 1(r 2;Â2)=∞m =0J m (k ÿr 2)(C ∗1m sin mÂ2+D ∗1mcos mÂ2)(14b) s 2= s 2(r 2;Â2)=∞m =0H (1)m (k r 2)(A ∗2m cos mÂ2+B ∗2m sin mÂ2)(14c) s 2= s 2(r 2;Â2)=∞m =0H (1)m (k ÿr 2)(C ∗2m sin mÂ2+D ∗2mcos mÂ2)(14d)The wave potentials are transformed from the r 2–Â2to the r 1–Â1co-ordinate system.Equations (14)are related to the potentials in Equations (8)and (9)through the followingTRANSVERSE RESPONSE OF UNDERGROUND CAVITIES AND PIPES389Figure2.Approximation of the half-space free surface with a large curved surface.transformation[16]:Z n(k r1)cos nÂ1sin nÂ1=∞n=−∞Z m+n(k D)J m(k r2)cos mÂ2sin mÂ2for r2¡D(15)where Z n(·)stands for the Bessel J n or Hankel H(1)n functions,and k is either k or kÿdepending on whether Equation(15)is applied to P or SV waves.Applying Equation(15)leads to the following relationships between coe cients of the r1–Â1and r2–Â2co-ordinate systems:A∗1m C∗1m= m2∞n=0T+mn(k D)00T−mn(kÿD)A1nC1n(16a)B∗1m D∗1m= m2∞n=0T−mn(k D)00T+mn(kÿD)B1nD1n(16b)390C.A.DAVIS,V.W.LEE AND J.P.BARDETA 2n C 2n=n 2∞ m =0T +nm (k D )00T −nm (k ÿD )A ∗2mC ∗2m(16c)B 2n D 2n=n2∞m =0T −nm (k D )T +nm(k ÿD )B ∗2mD ∗2m(16d)where 0=1,and n =2for n¿0,andT +mn (k D )=H (1)m +n (k D )+(−1)n H (1)m −n (k D )(17a)T −mn (k D )=H (1)m +n (k D )−(−1)n H (1)m −n (k D )(17b)In Equation (17),k =k or k ÿ.Hereafter,the following matrices are introduced:[T +m p ]=T +m p (k D )00T −m p (k ÿD )and[T −m p ]=T −m p (k D )00T +m p (k ÿD )(18)The transformation formulas (Equations (16)–(18))are di erent than those in References[8;15],due to the change from a concave to a convex free-surface approximation and the new range of D values used.The change in free-surface curvature was required to eliminate the incorrect cancellation of coe cients A 1n ;B 1n ;C 1n ,and D 1n ,which resulted from using the concave surface that represented both the re ected potentials r and r and scattered potentials s2and s 2with Bessel J functions in the r 2–Â2co-ordinate system.The convex free surfacecorrects this problem by allowing s2and s 2to be represented in terms of an outgoing Hankelfunction.The stress-free boundary conditions at the half-space surface can now be written asr 2Â2= r 2r 2=0at r 2=b(19)for the potentials s 1; s 1; s 2,and s 2in Equation (14).Introducing the potentials in Equation (10)into the stress Equations (11a)and (11c)for the r 1–Â1and r 2–Â2co-ordinate systems gives the following relations:r 1r 1 r 1Â1=2r 21∞n =0[E (3)(r 1;n )]A 1nC 1n+[E (1)(r 1;n )]A 0n +A 2n C 0n +C 2nTcos nÂ100sin nÂ1+2 r 21∞ n =0 [E (3)−(r 1;n )] B 1n D 1n +[E (1)−(r 1;n )] B 0n +B 2n D 0n +D 2n T sin nÂ10cos nÂ1(20a)TRANSVERSE RESPONSE OF UNDERGROUND CAVITIES AND PIPES391r 2r 2 r 2Â2=2r 22∞n =0[E (3)(r 2;n )]A ∗1nC∗1n+[E (1)(r 2;n )]A 0n +A ∗2nC 0n+C∗2nTcos nÂ200sin nÂ2+2 r 22∞ n =0 [E (3)−(r 2;n )] B ∗1n D ∗1n +[E (1)−(r 2;n )] B 0n +B ∗2n D 0n +D ∗2nTsin nÂ200cos nÂ2(20b)where the superscript T denotes transpose.The matrices[E (l )(r;n )]=E (l )11(r;n )E (l )12(r;n )E (l )21(r;n )E (l )22(r;n )(21a)[E (l )−(r;n )]=E (l )11(r;n )−E (l )12(r;n )−E (l )21(r;n )E (l )22(r;n )(21b)have the following coe cients for r =r 1and r 2:E (l )11=(n 2+n −12(k ÿr )2)Z (l )n (k r )−k rZ (l )n −1(k r )(22a)E (l )12=n [−(n +1)Z (l )n (k ÿr )+k ÿrZ (l )n −1(k ÿr )](22b)E (l )21=n [(n +1)Z (l )n (k r )−k rZ (l )n −1(k r )](22c)E (l )22=−(n 2+n −12(k ÿr )2)Z (l )n (k ÿr )+k ÿrZ (l )n −1(k ÿr )(22d)The superscript l (l =1or 3)stands for the J n or H (1)nfunctions,such that Z (1)n(·)=J n (·)and Z (3)n (·)=H (1)n (·)(23)The equations needed to solve for the coe cients of Equation (20)have now been for-mulated.Equation (20a),which is set equal to zero by applying the boundary conditions ofEquation (13),can only have a non-trivial solution by invoking the orthogonality condition and simultaneously equating to zero the functions containing sin nÂ1and cos nÂ1terms.This decouples the A 1n ;C 1n ;A 2n ,and C 2n coe cients from the B 1n ;D 1n ;B 2n ,and D 2n coe cients with the former found from:[E (3)(a;n )]A 1n C 1n +[E (1)(a;n )] A 0n C 0n +[E (1)(a;n )] A 2nC 2n = 00(24)Similarly,the A ∗1m ;C ∗1m ;A ∗2m ,and C ∗2mcoe cients are found by applying the boundary condi-tions of Equation (19)to Equation (20b):[E (3)(b;m )] A ∗1m C ∗1m +[E (1)(b;m ] A ∗2m C ∗2m= 00(25)Solving for (A ∗2mC ∗2m)in Equation (25)and substituting into Equation (16c)gives A 2n C 2n =− n 2∞ m =0[T +nm ][E (1)(b;m )]−1[E (3)(b;m )] A ∗1m C ∗1m(26)Substituting Equation (16a)into Equation (26)givesA 2n C 2n =− n 2∞ m =0 m 2∞ p =0[T +nm ][E (1)(b;m )]−1[E (3)(b;m )][T +m p] A 1p C 1p(27)Combining Equation (27)with Equation (24)gives[E (1)(a;n )] n 2∞ m =0 m 2∞ p =0[T +nm ][E (1)(b;m )]−1[E (3)(b;m )][T +m p ]A 1pC 1p−[E (3)(a;n )]A 1n C 1n−[E (1)(a;n )]A 0n C 0n=(28)A similar matrix system of equations can be solved for in terms of (B 1nD 1n)giving [E (1)−(a;n )] n 2∞ m =0 m 2∞p =0[T −nm ][E (1)−(b;m )]−1[E (3)−(b;m )][T −m p ]B 1pD 1p−[E (3)−(a;n )] B 1n D 1n −[E (1)−(a;n )] B 0n D 0n = 00(29)Equations (28)and (29)are similar to those of Reference [8],but have di erent co-ordinate transformations.The remaining coe cients (A 2n ;B 2n ;C 2n ,and D 2n )can be solved using Equa-tion (24),and the stresses and displacements in the medium can be determined.Equations (28)and (29)form a system of equations of inÿnite dimensions,which can be approximated by truncating the matrices at order N (where p =m =n =N −1).N can be determined by truncating the series terms that do not contribute signiÿcantly to stresses and displacements computed at speciÿc locations.The selection of N depends on the prescribed error tolerance,r 1; ;Âi ;h=a ,and the dimensionless frequency Á:Á=2a ÿ=!a ÿ=k ÿa=2a ÿT p (30)where ÿis the SV wavelength and T p =2 =!is the wave period.Figure 3(a)shows theexponential increase of N with Áfor calculations performed at the cavity and half-space surfaces using h =2a; =0:3;Âÿ=0and an error tolerance of 0.005per cent.The increase in N is almost linear for 5¡N .As shown in Figure 3(a),the increases of N with Áat the cavity and half-space surfaces can be approximated for N ¿5with the following linear relation N =5Á+5and N =18Á+5,respectively.Figure 3(b),shows the linear increase of N with r 1for displacement calculations performed along the x -axis at the half-space free surface y =0,in the absence of a buried cavity.Figure3.Number of coe cients required for accurate calculations(Âÿ=0; =0:3;error=0:005%)for:(a)a given dimensionless frequencyÁ(h=2a),and(b)a given distance r1.Lower and upper bounds of free-surface radius bAs shown in Figure2,D is related to the free surface radius b and cavity depth h throughD=b+h(31) As h tends to inÿnity,so does D and the transformation matrices[T+m p]and[T−m p]in Equation (18)go to zero independently of b.The solutions of Equations(28)and(29)reduce to the full-space solution[17]when the re ected input waves are zero(i.e.A1=B1=0).As b approaches inÿnity in the case of a at free surface,D approaches inÿnity independently of h.The solutions of Equations(28)and(29)behave like those of an inÿnite space,with the incident and re ected potentials acting as input waves,without any e ects from scattered wave potentials s2and s2.Thus,b cannot be selected too large for a correct half-space solution. Meanwhile,b cannot be taken too small,otherwise the problem becomes that of two cavities in a full space.Therefore,b has upper and lower bounds.References[8;15;18–20]presented P,SV,and Rayleigh wave solutions using a curved half-space approximation.Scheidl and Zieglar[18]envisioned that convex and concave sur-faces would provide upper and lower bound solutions for any value of b or D.References [8;15;19;20]describe that D=a must be large and recommend values of506D=a6100. However,no earlier works provide guidance on the determination of an appropriate value forD.Consequently,criteria had to be established for determining D and b.Figure4.Dimensionless frequencyÁat which the large curved surface behaves as a half space(u x=B0kÿ=2)for an incident SV wave:(a)variation with angleÂ2for =1=4,and(b)variation with Poisson’s ratio forÂ2=0.An analysis was performed to determine the minimum radius b for which displacements on the curved free surface match those of a at surface in the freeÿeld.Free-ÿeld surface displacements for a vertically incident SV wave are:u x=2and u y=0.The curved free surface was modelled as a large circular cavity embedded in a full elastic space free of any other cavity.The present model was reduced to a full-space solution,by setting the transformation matrix in Equation(18)to zero,so that the displacements at y2=b can be evaluated.The radius b was increased with respect to the input SV wavelength ÿuntil free-ÿeld displacements were obtained.Figure4shows u x evaluated at angles ofÂ2=0;±10;±20;±30,and±40◦for various Poisson ratios and diameter to wavelength ratiosÁb=2b= ÿ.As shown in Figure4(a), for =1=4the curved surface behaves as a at surface for−30◦6Â26+30◦andÁb¿70. The curved surface approximation does not apply outside this range,even whenÁb is increased to500orÂ2decreased to zero.Figure4(b)compares u x atÂ2=0◦for =1=4;1=3;0:3;and0.4.As shown in Figure4(b), the maximum radius b at which the curved free surface behaves as a at surface varies with .Áb ranges from approximately60for =1=3to approximately90for =0:3and0.4.At Áb=100,the curved surface behaves as a at surface for a wide range of .Reference[17]showed similar e ects for P waves,but at much lower values ofÁb.The value b=50 ÿis the minimum radius for the curved surface to behave as a at surface for all variables,and is appropriate for long-period waves.As b increases,the e ects of s2 and s2decrease,and the minimum value should be used in the analysis.Deÿning the surface curvature in terms of ÿrequires b to decrease with smaller wavelengths.For very high-frequency waves and large cavities,b can become smaller than a.Without further analysis, it is recommended to follow the criteria similar to References[8;19]and select b=50a to 100a as a lower bound.Results were found to change very little within this range.The lower bound value of b=100a was used for all calculations presented herein.Stress and displacements in half-spaceThe radial rr and shear rÂstress can be evaluated in the medium using Equation(20). Substitution of the SV and P wave potentials of Equation(10)into Equation(11b)gives the tangential stress:Â1Â1=2r21∞n=0[E(3)31E(3)32]A1nC1n+[E(1)31E(1)32]A0n+A2nC0n+C2ncos nÂ1+2r21∞n=0[E(3)31−E(3)32]B1nD1n+[E(1)31−E(1)32]B0n+B2nD0n+D2nsin nÂ1(32)where the stress coe cient terms for the r1–Â1co-ordinate system areE(l)31=−(n2+n+12(kÿr1)2−(k r1)2)Z(l)n(k r1)+k r1Z(l)n−1(k r1)(33a)E(l)32=E(l)12=n[(n+1)Z(l)n(kÿr1)−kÿr1Z(l)n−1(kÿr1)](33b) The medium displacements are determined in the r1–Â1co-ordinate system by introducing the potentials of Equations(10)into Equations(12)givingu r1 uÂ1=1r1∞n=0[D(3)(r1;n)]A1nC1n+[D(1)(r1;n)]A0n+A2nC0n+C2nTcos nÂ100sin nÂ1+1r1∞n=0[D(3)−(r1;n)]B1nD1n+[D(1)−(r1;n)]B0n+B2nD0n+D2nTsin nÂ100cos nÂ1(34)The displacement coe cient matrices are[D(l)(r1;n)]=D(l)11(r1;n)D(l)12(r1;n)D(l)21(r1;n)D(l)22(r1;n)(35a)[D(l)−(r1;n)]=D(l)11(r1;n)−D(l)12(r1;n)−D(l)21(r1;n)D(l)22(r1;n)(35b)and have the following coe cients:D(l)11=k r1Z(l)n−1(k r1)−nZ(l)n(k r1);D(l)12=nZ(l)n(kÿr1)(36a)D(l)21=−nZ(l)n(k r1);D(l)22=nZ(l)n(kÿr1)−kÿr1Z(l)n−1(kÿr1)(36b) where Z(l)n(·)is deÿned by Equation(23).Results and comparison with previous modelsFigure5shows normalized ÂÂ,u r,and uÂresults on the cavity surface r1=a using the present unlined cavity model.The results are shown for0:0016Á¡10atÂ1=15,45,75,90, 120,and150◦using the particular values =0:25,h=a=2,andÂÿ=0◦. ÂÂis normalized by the incident shear stress amplitude B0 k2ÿand u r and uÂare normalized by the incident displacement amplitude B0kÿ.As shown in Figure5(a),at high frequencies(i.e.Á¿0:3) ÂÂ, u r,and uÂvary widely withÂ1andÁ.At low frequencies the variation decreases asÁdecreases and u r and uÂbecome uniform atÁ¡0:01.The results of the present solution are now compared to those of previous analyses in terms of Poisson’s ratio ,embedment ratio h=a,incident angleÂÿ,and dimensionless frequencyÁ. Figure6compares the total horizontal surface displacement u x normalized by the magnitude of the incident displacement B0kÿfor the present model and that of Lee and Karl[8],who used a concave free surface.The results are presented for =0:25,h=a=2,Á=2,andÂÿ=0 and30◦.As shown in Figure6,the two solutions have similar shapes but di erent magnitude, which results from di erent coordinate transformations.Figure7compares the tangential stress ÂÂon the cavity surface with that found by Luco and De Barros[9],who used a boundary integral method based on two-dimensional Green’s functions in a viscoelastic half space for =1=3,h=a=5,Á=0:5,andÂÿ=0◦. ÂÂis normalized by the magnitude of the incident shear stress B0 k2ÿ.The present elastic model cannot be compared exactly to the viscoelastic model of Reference[9].The small amount of viscous damping in Reference[9]attenuates stress with distance and changes the coupling between scattered P and SV waves through the ratio =ÿ.However,as shown in Figure7,the present results compare favourably with those of Reference[9].Figure8compares the normalized vertical displacements on the half-space free surface with those found by Wong et al.[7]and Luco and De Barros[9]for incident SV waves.As shown in Figure8(a),using the same properties as in Figure7,the maximum surface dis-placements for the present model are smaller than that of Reference[9].Figure8(b)compares。