Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 47, 4, pp. 923-942, Warsaw 2009 LOCAL MODELLING OF BACKFILL EFFECTS FOR RIGID AXISYMMETRIC FOUNDATIONS UNDER DYNAMIC EXCITATION Zbigniew Sienkiewicz Koszalin University of Technology, Department of Civil and Environmental Engineering, Koszalin, Poland; e-mail: zbigniew.sienkiewicz@wbiis.tu.koszalin.pl The influence of backfill on the dynamic response of rigid axisymmetric foundations is described as reaction of an independent layer. It leads to the solution of the kinematic interaction problem by means of the com- plex stiffnessmatrix of a supportingmedium given by a specific increase in relation to the case of non-embedded foundations. The increases of stiffnesses are obtained in the closed-form from steady-state solutions to the equation of motion of an isotropic homogeneous medium under appropriate displacement boundary and radiation conditions. The ap- proximate modelling is compared with results of the rigorous boundary integral equation approach. Key words: kinematic interaction, embedded foundation, backfill, local modelling 1. Introduction In the dynamic analysis of foundations supported on a soil, the foundation block is massive and may be considered as a rigid body. The excitation is assumed to be harmonic being able to consider other excitations by means of Fourier transform techniques. The steady-state motion of a massive foun- dation can be analysed in two steps. In the first step, also called ”kinematic interaction”, the response of a massless rigid body on a supporting medium is computed due to the dynamic excitation under consideration. The dynamic response is determined by the stiffness matrix of the supporting medium for a given shape of the foundation. It is worth to say that dynamic forces and displacements related by the stiffness matrix are generally out of phase. It is convenient then to use complex notation to represent forces and displace- ments of the foundation and stiffnesses of the supporting medium. The real 924 Z. Sienkiewicz component of the stiffnesses reflects the stiffness and inertia of the soil. The imaginary component reflects the dampingof the system.Themainpart of the damping is due to the energy dissipated by the waves propagating away from the foundation (radiation damping). In addition to the radiation damping, a hystereticmaterial dampingmay exist. Once the dynamic stiffness of the sup- porting medium is known, the response of the foundation including its mass may be evaluated from the general equations of translational and rotational motion of the massive rigid body. Generally, the foundations are surrounded by a backfill, which modifies the dynamic stiffness of the supporting medium. The kinematic interaction problem is governed by amixed boundary value problem of three-dimensional elastodynamics if the soil medium is treated as a continuum half-space. The procedures utilised to solve the problem include the finite element method, the boundary elementmethod or hybrid approaches (Lysmer, 1980; Apsel and Luco, 1987; Mita and Luco, 1987; Emperador and Dominguez, 1987; Barros, 2006). Due to unbounded nature of the soil medium, the computational size of these methods is very large. Furthermore, the foundation embedment con- ditions are very complex practically due to uncertainties in the state of the soil. Then, approximatemodels are justified to supplementmore generally ap- plicable rigorousmethods. Approximate approaches to model the embedment effects of rigid cylindrical foundations in the dynamic soil-structure interaction analysis include: • a linear elastic weightless spring system (Prakash and Puri, 1988) • an empirical approach (Gazetas, 1991) • a lumped-parameter model (Wolf and Paronesso, 1992; Wu and Lee, 2002) • a cone model (Meek and Wolf, 1994; Jaya and Prasad, 2002; Takewaki et al., 2003; Wolf, 1994; Wolf and Preisig, 2003) • an independent continuum layer (Novak et al., 1977). Theobjective of this paper is topresent adynamic local approach tomodel the backfill effects on complex stiffnesses of a supportingmedium under following assumptions: (1) the backfill is modelled as an independent isotropic homogeneous me- dium in the plane and (or) anti-plane strain cases; (2) themediumis characterised by themassdensity ρB and complexLame’s constants µ∗B, λ ∗ B to includematerial damping. Local modelling of backfill effects... 925 Thebackfill reduces the effect of embedment due to excavating and backfilling of the soil deposit in the place of building of the foundation. This fact can be taken into account by the appropriate choice of the mass density and dyna- micmaterial parameters. The parameters can be estimated by solution of the inverse problem if the dynamic response of the embedded foundation is given frommeasurements. 2. Statement of the problem Let Ω=ΩI∪ΩE ⊂R3 be thedomain in three-dimensional space R3 occupied by a rigid massless inclusion ΩI and an excavated elastic half-space ΩE, see Fig.1. Fig. 1. Description of the model The rigidmassless body ΩI is perfectly bonded to thehalf-space ΩE along the surface ∂ΩEC = ∂Ω E hor ∪∂ΩEver. The body Ω = ΩI ∪ΩE is in a state of motion relative to an inertial frame of reference. 2.1. Balance equations for the rigid massless inclusion ΩI The rigid massless inclusion ΩI is subjected to the external force ~P0(t) = ~P0e iωt and moment ~M0(t) = ~M0e iωt vectors acting at the referen- ce point x0 ∈ ΩI with harmonic time dependence of the type eiωt in which ω is the circular frequency and i = √ −1. Furthermore, due to deformation of the half-space ΩE, it is loaded by a field ~t(x,nI) of contact forces acting on ∂ΩI, where nI is the unit outer normal vector for ΩI at x∈ ∂ΩI. 926 Z. Sienkiewicz The momentum balance equation and balance of angular momentum for the massless rigid inclusion ΩI lead to ~P0e iωt+ ∫ ∂ΩI ~t(x,nI)eiωt dS =~0 x∈ ∂ΩI ~M0e iωt+ ∫ ∂ΩI (x−x0)×~t(x,nI)eiωt dS =~0 x∈ ∂ΩI (2.1) 2.2. Field equations for the excavated half-space ΩE The steady-state time-harmonic motion of the excavated half-space ΩE includes thefieldsofdisplacement ~u(x, t)= ~u(x)eiωt, strain ↔ ε(x, t)= ↔ ε(x)eiωt and stress ↔ σ(x, t)= ↔ σ(x)eiωt, x∈ΩE. The principle of linear momentum with the conservation of mass on the assumption of small deformations leads to the Cauchy equation of motion in the frequency domain (Achenbach, 1973) ∇·↔σ(x)+ρ~b(x)=−ω2ρ~u(x) x∈ΩE (2.2) where ~b(x) stands for the body force vector, ρ is themass density, ∇ denotes the del operator and ∇·(·) implies the divergence of (·). The principle of the angular momentum provides the symmetry of stress tensor ↔ σ(x). The generalizedHooke’s law in the frequency domain for a linear inelastic isotropic homogeneous medium is ↔ σ(x)=λ∗ tr ↔ ε(x) ↔ 1+2µ∗ ↔ ε(x) x∈ΩE (2.3) where tr ↔ ε means the trace of ↔ ε, ↔ 1 denotes the unit tensor and λ∗ and µ∗ are complex-valued Lame’ constants. The kinematical relation within the restrictions of the linearised theory is given by ↔ ε(x)= 1 2 (∇~u(x)+~u(x)∇) x∈ΩE (2.4) Substituting Eqs (2.3) and (2.4) into (2.2), leads to the Navier-Cauchy equ- ation of motion µ∗∇2~u(x)+(λ∗+µ∗)∇∇·~u(x)+ρ~b(x)=−ω2ρ~u(x) x∈ΩE (2.5) where ∇2 =∇·∇ is the Laplacian operator. Local modelling of backfill effects... 927 2.3. Elastodynamicproblem for the excavatedhalf-space in the frequency domain Theproblemcanbe statedas follows: tofindthe solution toNavier-Cauchy equation of motion (2.5) with the boundary conditions ~u(x)= ~u(x) x∈ ∂ΩEC (2.6) ~t(x,n)=n ·↔σ(x)=Tn(~u(x))=~0 x∈ ∂ΩEf = ∂ΩE −∂ΩEC where ~u(x), x∈ ∂ΩEC is the prescribed displacement on ∂ΩEC ,n denotes the unit outward normal vector to ∂ΩE and the operator Tn is defined by Tn(·)=λ∗n∇·(·)+2µ∗n ·∇(·) ↔ 1+µ∗n×∇× (·) (2.7) The considered elastodynamic problem includes the boundary extended to in- finity, then it is reasonable to require that the displacement at infinity must be bounded and that nowave can be reflected back from infinity. These condi- tions, called the radiation conditions are crucial in searching the unique solu- tion in unbounded domains, particularly in time-harmonic problems (Eringen and Suhubi, 1975). 2.4. Coupling equations The excavated half-space and the rigid inclusion substructures can be co- upled by enforcing the compatibility and the equilibrium conditions at their common interface ΩI ∩ΩE. The response of the rigid massless inclusion ΩI can be described by the displacement ~U0(t) = ~U0e iωt and the small rotation ~Φ0(t)= ~Φ0e iωt vectors at the point of reference x0 ∈ΩI. Then, the compatibi- lity of interaction displacements at the contact surface between the half-space and the inclusion requires that ~u(x)= ~U0+ ~Φ0× (x−x0) x∈ΩI ∩ΩE (2.8) The equilibrium of interaction forces at the medium-inclusion interface requ- ires that ~t(x,nI)+~t(x,n)=~0 x∈ΩI ∩ΩE (2.9) and nI =−n on ΩI ∩ΩE. Taking the equilibrium of interaction forces (2.9) in equations (2.1) into account and referring to an orthonormal basis {o; ê1, ê2, ê3} results in the matrix form {P0}= ∫ ∂ΩE C [g(x;x0)] ⊤{~t(x,n)} dS x∈ ∂ΩEC = ∂ΩEhor ∪∂ΩEver (2.10) 928 Z. Sienkiewicz where {P0}=(P01,P02,P03,M01,M02,M03)⊤, {~t}=(t1, t2, t3)⊤, and [g(x;x0)] =    1 0 0 0 (z−z0) −(y−y0) 0 1 0 −(z−z0) 0 (x−x0) 0 0 1 (y−y0) −(x−x0) 0    (2.11) Since the tractions ~t(x,n) on ∂ΩEC are linearly related to the displacement ~U0 and the rotation ~Φ0 of the rigid inclusion Ω I, the traction vector {~t(x,n)} can be written in the form {~t(x,n)}= [Hn(x)]{U0} (2.12) where {U0}=(U01,U02,U03,Φ01,Φ02,Φ03)⊤ and [Hn(x)] is the 3×6matrix of contact tractions on ∂ΩEC for unit rigid-body displacements of the foundation corresponding to each of the six degrees of freedom. Substitution from (2.12) to (2.10) gives {P0}= [K(x0)]{U0} (2.13) in which [K(x0)] = ∫ ∂ΩE C [g(x;x0)] ⊤[Hn(x)] dS x∈ ∂ΩEC (2.14) is the 6× 6 dynamic stiffness matrix of the supporting medium, referred to the point of reference x0. The Kpq component (p,q=1,2, . . . ,6) of the matrix is given by Kpq(x0)= ∫ ∂ΩE C {gp(x;x0)}⊤{Hnq (x)} dS x∈ ∂ΩEC (2.15) in which {gp(x;x0)} corresponds to the p-th column of thematrix [g(x;x0)] and {Hnq (x)} corresponds to the q-th column of the matrix [Hn(x)]. In the case of rigid foundations with a vertical axis of symmetry and x0 = (0,0,z0) the integration can be done in cylindrical coordinates (r,θ,z) and the azimuthal dependence can be factored out. The displacement vector ~u(x)= ~U0+ ~Φ0×(x−x0),x∈ΩI∩ΩE, on themedium-foundation interface can be written in the matrix form {u(x)}= [g(x;x0)]{U0} (2.16) Local modelling of backfill effects... 929 where {u(x)}= (ur(x),uθ(x),uz(x))⊤, {U0}= (U01,U02,U03,Φ01,Φ02,Φ03)⊤ and the rigid bodymotion influence matrix is given by [g(x;x0)] =    cosθ sinθ 0 −(z−z0)sinθ (z−z0)cosθ 0 −sinθ cosθ 0 −(z−z0)cosθ −(z−z0)sinθ r 0 0 1 r sinθ −rcosθ 0    (2.17) Considering the q-th component (q = 1,2, . . . ,6) of the generalized displa- cement vector U0q it is possible to write the corresponding displacement and traction components in the form {uq(x)}= {gq(x;x0)}U0q = [Aq(θ)]{gq(r,z)}U0q {~tq(x,n)}= {Hnq (x)}U0q = [Aq(θ)]{H n q (r,z)}U0q (2.18) x=(r,θ,z)∈ΩI ∩ΩE q=1,2, . . . ,6 Thediagonalmatrices [Aq(θ)] represent the azimuthal dependence of the q-th rigid-bodymotion and are given by [A1(θ)]= [A5(θ)]= diag(cosθ,sinθ,cosθ) [A2(θ)]= [A4(θ)]= diag(sinθ,−cosθ,sinθ) (2.19) [A3(θ)]= diag(0,0,1) [A6(θ)]= diag(0,−1,0) The vector {gq(r,z;z0)} corresponds to the q-th column of the matrix [g(r,z;z0)] [g(r,z;z0)] =    1 1 0 −(z−z0) (z−z0) 0 −1 −1 0 (z−z0) −(z−z0) −r 0 0 1 r −r 0    (2.20) Substitution of (2.18) into (2.15) and integration over θ from 0 to 2π leads to Kpq(z0)= 2πapq ∫ Lc {gp(r,z;z0)}⊤{H n q (r,z)}r dL(r,z) (2.21) inwhich Lc represents the line defined by intersection of the rz-plane (θ=0) and ∂ΩEC = ∂Ω E hor∪∂ΩEver and the matrix [a] = 1 2          1 0 0 0 1 0 0 1 0 1 0 0 0 0 2 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 2          (2.22) 930 Z. Sienkiewicz Equation (2.21) indicates that generalized force-displacement relationship (2.13) for a rigid massless foundation bonded to the supporting medium sys- tem can be written in the form                  P01 P02 P03 M01 M02 M03                  =          KHH 0 0 0 KHM 0 0 KHH 0 −KHM 0 0 0 0 KVV 0 0 0 0 −KMH 0 KMM 0 0 KMH 0 0 0 KMM 0 0 0 0 0 0 KTT                           U01 U02 U03 Φ01 Φ02 Φ03                  (2.23) where the terms KHH,KMM,KHM =KMH,KVV , and KTT are the horizon- tal, rocking, coupling, vertical, and torsional stiffness functions, respectively. The functions are referred to the point of reference x0 =(0,0,z0) and can be written in the form KHH =GR(kHH +ia0cHH) KVV =GR(kVV +ia0cVV ) KHM =GR 2(kHM +ia0cHM) KTT =GR 3(kTT +ia0cTT) KMM =GR 3(kMM +ia0cMM) (2.24) inwhich G is a shearmodulus of reference, Rdenotes the radius of cylindrical foundation and a0 = ωR/VS is the dimensionless frequency defined on the basis of the S-wave velocity of reference VS. The terms kpq and cpq are the normalized stiffness and damping coefficients, respectively. Thekey step in the solution is the evaluation of the contact tractionmatrix [H n (r,z)], (r,z)∈Lc =Lhor∪Lver. 3. Local modelling Within the region ΩE one identifies a half-space subregion ΩN to represent the soil in its natural state and a layer ΩB to represent the disturbed soil (backfill): ΩE =ΩN ∪ΩB, see Fig.2. The perfect bonding exists only between the half-space ΩN and the rigid body ΩI along the contact surface ΩN ∩ΩI and between the layer ΩB and the body ΩI along the contact surface ΩB ∩ΩI. It is assumed however that at the horizontal interface ΩN∩ΩB between the half-space and the layer, the condition of continuity of displacements is not satisfied and that the surfaces ∂ΩN and ∂ΩB of the separated regions are free from tractions. Then, the tractions at the base of the rigid body are equal to those of the body placed Local modelling of backfill effects... 931 Fig. 2. Rigid body ΩI on half-space ΩN, surrounded by backfill layer ΩB on the soil surface, while the backfill reactions are to be evaluated indepen- dently by local modelling. These assumptions imply that the Kpq component (p,q=1,2, . . . ,6) of the dynamic stiffness matrix is given by Kpq(z0)=K 0 pq+∆K B pq(z0) (3.1) where K0pq denotes the component for surface foundation and ∆K B pq(z0) re- presents the increase due to local backfill reactions ∆KBpq(z0)= 2πRapq HB ∫ 0 {gp(R,z;z0)}⊤{H n q (R,z)} dz (3.2) where HB is the thickness of the backfill layer. The vector of contact tractions {Hnq} can be obtained in amathematically accurate form on the following assumptions: (1) the backfill ΩB is modelled as an inelastic isotropic homogeneous me- dium in the plane and (or) anti-plane strain cases (2) the medium is characterised by complex Lame’s constants µ∗B, λ ∗ B and mass density ρB. The governing Navier-Cauchy equation of backfill motion as two-dimensional approximation of the three-dimensional case is derived directly from (2.5) and solved in cylindrical coordinates under appropriate displacement boun- dary conditions ~u(x) = ~U0 + ~Φ0 × (x−x0), x ∈ ΩI ∩ΩB resulting from the rigid-body motion ~U0, ~Φ0 of the inclusion Ω I at the point of reference x0 =(0,0,z0). The vectors of contact tractions {H n q} are given explicitly for all the six cases of unit rigid-bodymotions U0q, q=1, . . . ,6 932 Z. Sienkiewicz (1) {U0}=(1,0,0,0,0,0)⊤ {Hn1}=      (λ∗B +2µ ∗ B)ŝ 2 1K1(ŝ1R)A −µ∗Bŝ2K1(ŝR)B 0      (3.3) where ŝ= iω √ ρB µ∗B ŝ1 = iω √ ρB λ∗B +2µ ∗ B A=−L(ŝ,R) M(R) B=−L(ŝ1,R) M(R) L(s,R)= sK0(sR)+2R −1K1(sR) M(R)= ŝ1ŝK0(ŝ1R)K0(ŝR)+ ŝ1R −1K0(ŝ1R)K1(ŝR)+ +ŝR−1K1(ŝ1R)K0(ŝR) and K0,K1 are the modified Bessel functions of the second kind, (2) {U0}=(0,1,0,0,0,0)⊤ {Hn2}= {H n 1} (3.4) (3) {U0}=(0,0,1,0,0,0)⊤ {Hn3}=          0 0 −µ∗Bŝ K1(ŝR) K0(ŝR)          (3.5) (4) {U0}=(0,0,0,1,0,0)⊤ {Hn4}=          (z0−z)H n 12 (z0−z)H n 22 −µ∗B [ŝRK0(ŝR) K1(ŝR) +1 ]          (3.6) (5) {U0}=(0,0,0,0,1,0)⊤ {Hn5}=          −(z0−z)H n 11 −(z0−z)H n 21 µ∗B [ ŝRK0(ŝR) K1(ŝR) +1 ]          (3.7) Local modelling of backfill effects... 933 (6) {U0}=(0,0,0,0,0,1)⊤ {Hn6}=          0 µ∗B [ ŝRK0(ŝR) K1(ŝR) +2 ] 0          (3.8) The components ∆KBpq, p,q=H,M,V,T of dynamic stiffnessmatrix of back- fill (3.2) are found to be ∆KBHH =πµ ∗ BHBâ 2 0 4K1(â0)K1(b̂0)+ â0K0(â0)K1(b̂0)+ b̂0K1(â0)K0(b̂0) â0b̂0K0(â0)K0(b̂0)+ â0K0(â0)K1(b̂0)+ b̂0K1(â0)K0(b̂0) ∆KBHM = (1 2 HB −z0 ) ∆KBHH ∆KBMM = (1 3 H2B −HBz0+z20 ) ∆KB11+πµ ∗ BHBR 2 [â0K0(â0) K1(â0) +1 ] (3.9) ∆KBVV =2πµ ∗ BHBâ0 K1(â0) K0(â0) ∆KBTT =2πµ ∗ BHBR 2 [â0K0(â0) K1(â0) +2 ] where â0 = iωR √ ρB µ∗ B b̂0 = â0 √ µ∗B λ∗ B +2µ∗ B (3.10) 4. Numerical results Application of the closed-form solution require themodel ofmaterial damping in the backfill to be specified. It is introduced by complex Lame’s constants µ∗B and λ ∗ B of the form µ∗B =µB(1+ i2ξ B S ) λ∗B +2µ ∗ B =(λB +2µB)(1+ i2ξ B P ) (4.1) where µB ≡ GB and λB are real Lame’s constants and ξBS and ξBP repre- sent the hysteretic damping ratios for the S- and P-waves, respectively. The corresponding complex S- and P-wave velocities are 934 Z. Sienkiewicz V̂BS =V B S √ 1+i2ξBS ≈VBS (1+ iξBS ) V̂BP =V B P √ 1+i2ξBP ≈VBP (1+ iξBP ) (4.2) in which VBS = √ µB/ρB and V B P = √ (λB +2µB)/ρB correspond to the real S- and P-waves velocities. The complex Poisson ratio ν̂B is ν̂B = 1−2 ( V̂B S V̂B P )2 2 [ 1− ( V̂B S V̂B P )2] ≈ νB − i(ξBS − ξBP ) ( VB S VB P )2 1− ( VB S VB P )2 (4.3) where νB = 1−2 ( VB S VB P )2 2 [ 1− ( VB S VB P )2] (4.4) is real Poisson’s ratio of the backfill. The stiffness functions ∆KBHH, ∆K B HM = ∆K B MH, ∆K B MM, ∆K B VV and ∆KBTT expressed by Eqs. (3.9) and referred to the point of reference x0 =(0,0,HB) can be written in the form analogous to (2.24) ∆KBHH =GBR(k B HH +ia B 0 c B HH) ∆KBHM =GBR 2(kBHM +ia B 0 c B HM) ∆KBMM =GBR 3(kBMM +ia B 0 c B MM) (4.5) ∆KBVV =GBR(k B VV +ia B 0 c B VV ) ∆KBTT =GBR 3(kBTT +ia B 0 c B TT) inwhich aB0 =ωR/V B S is the dimensionless frequency and the terms k B mn and cBmn denote the normalized stiffness and damping coefficients of the backfill, respectively. They depend on the dimensionless parameters: kBHH, c B HH, k B HM = k B MH, c B HM = c B MH, k B MM, c B MM : HB/R, VBP /V B S , ξ B P , ξ B S , a B 0 ; k B VV , c B VV , k B TT , c B TT : HB/R, ξ B S , a B 0 (4.6) The normalized stiffness and damping coefficients have been calculated for the case of the backfill characterised by the values HB/R = 1, VBP /V B S ∈ { √ 2.25, √ 3, √ 3.94, √ 101}, ξBP = 0.005, ξBS = 0.01 and aB0 in the range from 0.25 to 6.Tothegivenvalues of VBP /V B S therecorrespondthe follo- wing values of Poisson’s ratio νB ∈{0.1,0.25,033,0.495}. Note that the range Local modelling of backfill effects... 935 Table 1 VB P /VB S = √ 2.25 VB P /VB S = √ 3 VB P /VB S = √ 3.94 VB P /VB S = √ 101 (νB =0.1) (νB =0.25) (νB =0.33) (νB =0.495) aB 0 = ωR V B S kB HH cB HH kB HH cB HH kB HH cB HH kB HH cB HH 0.25 3.010 11.234 3.244 12.078 3.438 12.843 4.228 17.308 0.50 3.399 9.429 3.642 10.138 3.833 10.791 4.302 14.666 0.75 3.587 8.803 3.814 9.480 3.973 10.110 3.755 13.778 1.00 3.691 8.505 3.890 9.179 4.000 9.816 2.744 13.394 1.25 3.754 8.340 3.924 9.025 3.979 9.682 1.326 13.233 1.50 3.797 8.239 3.942 8.938 3.945 9.620 −0.473 13.199 1.75 3.831 8.170 3.957 8.883 3.916 9.593 −2.638 13.250 2.00 3.858 8.121 3.974 8.846 3.899 9.580 −5.155 13.367 2.25 3.880 8.084 3.992 8.818 3.895 9.572 −8.014 13.538 2.50 3.899 8.054 4.011 8.794 3.901 9.565 −11.202 13.759 2.75 3.913 8.030 4.029 8.775 3.914 9.558 −14.706 14.025 3.00 3.924 8.010 4.046 8.758 3.931 9.550 −18.506 14.335 3.25 3.933 7.994 4.061 8.743 3.950 9.542 −22.582 14.687 3.50 3.938 7.979 4.074 8.729 3.970 9.533 −26.910 15.078 3.75 3.942 7.967 4.085 8.717 3.988 9.525 −31.459 15.508 4.00 3.943 7.957 4.093 8.707 4.004 9.516 −36.199 15.974 4.25 3.942 7.948 4.099 8.697 4.019 9.508 −41.095 16.473 4.50 3.940 7.940 4.103 8.689 4.032 9.500 −46.110 17.004 4.75 3.937 7.933 4.106 8.681 4.043 9.493 −51.204 17.561 5.00 3.933 7.927 4.106 8.674 4.051 9.486 −56.337 18.143 5.25 3.927 7.922 4.106 8.668 4.058 9.480 −61.472 18.744 5.50 3.921 7.917 4.104 8.662 4.063 9.474 −66.568 19.362 5.75 3.914 7.913 4.101 8.657 4.066 9.469 −71.589 19.990 6.00 3.906 7.909 4.096 8.653 4.068 9.464 −76.502 20.626 ofPoisson’s ratio cover fullydrained (νB =0.1-0.2) toundrained (νB =0.495) conditions. The numerical results are presented in Tables 1, 2 and 3. To validate the proposed approach, the impedance functions Kpq(HE), p,q = H,M,V,T for the rigid massless cylindrical foundation embedded to a depth HE in a uniform inelastic half-space are considered, where the po- int of reference is the centre of the bottom of the foundation (z0 = HE). The uniform half-space is characterised by complex-valued Lame’s constants µ∗ =µ(1+i2ξS), λ ∗+2µ∗ =(λ+2µ)(1+i2ξP), where µ≡G and λ are real Lame’s constants and ξS and ξP represent the hysteretic damping ratios for the S- and P-waves, respectively. The corresponding complex S- and P-wave velocities are V̂S =VS(1+ iξS), V̂P =VP(1+ iξP), in which VS = √ µ/ρ and VP = √ (λ+2µ)/ρ represent the real S- and P-waves velocities, respectively, and ρ is the mass density. 936 Z. Sienkiewicz Table 2 VB P /VB S = √ 2.25 VB P /VB S = √ 3 VB P /VB S = √ 3.94 VB P /VB S = √ 101 (νB =0.1) (νB =0.25) (νB =0.33) (νB =0.495) aB 0 = ωR V B S kB MM cB MM kB MM cB MM kB MM cB MM kB MM cB MM 0.25 3.861 5.080 3.939 5.361 4.004 5.616 4.267 7.105 0.50 3.648 5.054 3.730 5.291 3.793 5.508 3.950 6.800 0.75 3.455 5.228 3.531 5.454 3.584 5.664 3.511 6.887 1.00 3.309 5.371 3.375 5.595 3.412 5.808 2.993 7.000 1.25 3.201 5.473 3.258 5.701 3.276 5.920 2.392 7.104 1.50 3.122 5.545 3.170 5.778 3.171 6.005 1.698 7.198 1.75 3.063 5.596 3.105 5.834 3.091 6.070 0.906 7.289 2.00 3.017 5.633 3.055 5.875 3.030 6.119 0.013 7.382 2.25 2.980 5.660 3.018 5.905 2.985 6.156 −0.984 7.478 2.50 2.951 5.680 2.988 5.927 2.951 6.184 −2.083 7.582 2.75 2.925 5.696 2.964 5.944 2.926 6.205 −3.281 7.694 3.00 2.903 5.707 2.944 5.956 2.906 6.220 −4.573 7.816 3.25 2.884 5.716 2.927 5.966 2.890 6.232 −5.955 7.947 3.50 2.866 5.724 2.911 5.973 2.876 6.241 −7.417 8.090 3.75 2.849 5.729 2.897 5.979 2.864 6.248 −8.951 8.243 4.00 2.833 5.734 2.883 5.984 2.853 6.254 −10.548 8.406 4.25 2.818 5.738 2.870 5.988 2.843 6.258 −12.195 8.580 4.50 2.803 5.741 2.857 5.990 2.834 6.261 −13.880 8.762 4.75 2.789 5.744 2.845 5.993 2.824 6.264 −15.591 8.953 5.00 2.775 5.746 2.833 5.995 2.814 6.266 −17.315 9.151 5.25 2.761 5.748 2.821 5.996 2.805 6.267 −19.039 9.355 5.50 2.748 5.749 2.808 5.998 2.795 6.268 −20.749 9.564 5.75 2.734 5.751 2.796 5.999 2.785 6.269 −22.434 9.777 6.00 2.721 5.752 2.784 6.000 2.775 6.270 −24.082 9.991 In accordance with the local modelling of embedment effects, the impe- dance functions Kpq are given by (3.1) Kpq(HE)≈K0pq+∆KBpq(HE) (4.7) where K0pq denotes the component for surface foundation and ∆K B pq(HE) represents the increase due to local backfill reactions at the point of refe- rence x0 = (0,0,HE). Assuming the parameters of backfill µB = µ ≡ G, λB =λ, ρB = ρ, ξ B P = ξP , ξ B S = ξS and HB =HE, the functions ∆K B pq(HE), p,q=H,M,V,T canbecalculated fromequations (3.9) and (3.10) and expres- sed in form (4.5). A test of all five impedance functions KHH,KHM =KMH, KMM, KVV , and KTT is realised by comparison of the results of Apsel and Luco (1987) Local modelling of backfill effects... 937 Table 3 aB 0 = ωR V B S kB VV cB VV kB TT cB TT 0.25 2.231 8.873 11.999 3.173 0.50 2.524 7.513 11.314 4.074 0.75 2.674 7.033 10.801 4.755 1.00 2.762 6.795 10.440 5.197 1.25 2.815 6.658 10.183 5.486 1.50 2.848 6.570 9.996 5.681 1.75 2.869 6.511 9.854 5.817 2.00 2.880 6.468 9.745 5.915 2.25 2.885 6.437 9.657 5.987 2.50 2.886 6.414 9.585 6.041 2.75 2.883 6.395 9.525 6.083 3.00 2.878 6.380 9.474 6.116 3.25 2.871 6.369 9.429 6.142 3.50 2.863 6.359 9.389 6.163 3.75 2.854 6.351 9.353 6.181 4.00 2.843 6.344 9.321 6.195 4.25 2.832 6.338 9.290 6.207 4.50 2.820 6.334 9.262 6.216 4.75 2.808 6.329 9.236 6.225 5.00 2.795 6.326 9.211 6.232 5.25 2.782 6.322 9.187 6.238 5.50 2.768 6.320 9.164 6.243 5.75 2.755 6.317 9.142 6.248 6.00 2.741 6.315 9.121 6.252 based on the rigorous non-singular integral equation approach to the dyna- mic response of embedded foundations with the values estimated from formu- la (4.7), where the functions K0pq for the surface foundation were calculated by the approach of Wong and Luco (1976). The dimensionless normalized stiffness kpq and damping cpq coefficients defining the form of generalized force-displacement relationship given by (2.23) and (2.24) were calculated for embedment ratios HE/R = 0.25, 0.5, 1, 2 at fixed values of VP/VS = √ 3, ξP =0.005, ξS =0.01. The calculations were performed for a number of valu- es of the dimensionless frequency a0 =ωR/VS in the range from 0.25 to 6.00. Comparisons are presented in Figs.3-7. Inspection of the Figures indicates that the stiffness coefficients for embedded foundations, obtained by the pre- sent approach, generally underestimate the values from the integral equation approach, with the exception of kVV for HE/R = 1, 2 where a small overe- stimation can be observed in limited ranges of the dimensionless frequency. 938 Z. Sienkiewicz On the contrary, the damping coefficients determined by the present approach generally overestimate the values from the integral equation approach at low dimensionless frequencies and tend to theApsel andLuco solutionas frequency increases. Fig. 3. Comparision of normalized horizontal stiffness and damping coefficients for cylindrical foundations with embedment ratios HE/R=0.25, 0.5, 1, 2: rigorous integral equation approach of Apsel and Luco (1987) – solid lines, local modelling of embedment – dotted lines Fig. 4. Comparision of normalized coupling stiffness and damping coefficients for cylindrical foundations with embedment ratios HE/R=0.25, 0.5, 1, 2: rigorous integral equation approach of Apsel and Luco (1987) – solid lines, local modelling of embedment – dotted lines Local modelling of backfill effects... 939 Fig. 5. Comparision of normalized rocking stiffness and damping coefficients for cylindrical foundations with embedment ratios HE/R=0.25, 0.5, 1, 2: rigorous integral equation approach of Apsel and Luco (1987) – solid lines, local modelling of embedment – dotted lines Fig. 6. Comparision of normalized vertical stiffness and damping coefficients for cylindrical foundations with embedment ratios HE/R=0.25, 0.5, 1, 2: rigorous integral equation approach of Apsel and Luco (1987) – solid lines, local modelling of embedment – dotted lines 5. Conclusions The dynamic generalized force-displacement relationship for a rigid massless foundation bonded to a supporting medium which represents the kinematic interaction can be expressed in the form of stiffness functions which depend on the soil properties, frequency of excitation and geometry of the foundation. On the assumption of local modelling of the backfill, the complex stiffnesses of the supportingmedium are decomposed into the sumof two parts: the first one corresponding to thekinematic interaction of the rigidmassless foundation placed on the surface of the supportingmedium and the second one represen- 940 Z. Sienkiewicz Fig. 7. Comparision of normalized torsional stiffness and damping coefficients for cylindrical foundations with embedment ratios HE/R=0.25, 0.5, 1, 2: rigorous integral equation approach of Apsel and Luco (1987) – solid lines, local modelling of embedment – dotted lines ting the increases generated by the backfill. For rigid cylindrical foundations, the second part of the stiffness functions can be expressed in the closed-form. The accuracy of the simplified dynamic model is contained within the limits of the strength-of-materials approach to the foundation dynamics. Once the responseof amassless rigid foundation is computed, thedynamics ofmassive foundations can be studied by themethods of structural dynamics. Then, the presented local modelling is, among others, a simple, yet rational way to include the effect of backfill in the analysis of the dynamic response of embedded foundations, preferred in the preliminary stage of practical design. References 1. Achenbach J.D., 1973, Wave Propagation in Elastic Solids, North-Holland, Amsterdam 2. ApselR.J., Luco J.E., 1987, Impedance functions for foundations embedded in a layered medium: an integral equation approach, Earthquake Engineering and Structural Dynamics, 15, 213-231 3. BarrosP.L.A.,2006, Impedancesof rigidcylindrical foundationsembedded in transversely isotropic soils, International Journal of Numerical and Analytical Methods in Geomechanics, 30, 683-702 4. Emperador J.M., Dominguez J., 1987, Dynamic response of axisymmetric embedded foundations,Earthquake Engineering and Structural Dynamics, 18, 1105-1117 Local modelling of backfill effects... 941 5. Eringen A.C., Suhubi E.S., 1975,Elastodynamics, 2, Academic Press, New York 6. Gazetas G., 1991, Formulas and charts for impedances of surface and em- bedded foundations, Journal of Geotechnical Engineering (ASCE), 117, 9, 1363-1381 7. Jaya K.P., Prasad A.M., 2002, Embedded foundation in layered soil under dynamic excitations, Soil Dynamics and Earthquake Engineering, 22, 485-496 8. Lysmer J., 1980, Foundation vibrations with soil damping, Civil Engineering and Nuclear Power, ASCE, II, paper 10-4, 1-18 9. Meek J.W., Wolf J.P., 1994, Cone models for an embedded foundation, Journal of Geotechnical Engineering (ASCE), 120, 1, 60-80 10. Mita A., Luco J.E., 1987, Dynamic response of embedded foundations: a hybrid approach, Computer Methods in Applied Mechanics and Engineering, 63, 233-259 11. Novak M., Nogami T., Aboul-Ella F., 1977, Dynamic soil reactions for plane strain case, Journal of The Engineering Mechanics Division (ASCE), 104, 953-959 12. Prakash S., Puri V.K., 1988,Foundations for Machines: Analysis and De- sign, Wiley, NewYork 13. Takewaki I., TakedaN., Uetani K., 2003, Fast practical evaluation of soil- structure interaction of embedded structures, Soil Dynamics and Earthquake Engineering, 23, 195-202 14. Wolf J.P., 1994, Foundation Vibration Analysis Using Simple Physical Mo- dels, EnglewoodCliffs, NJ, Prentice-Hall 15. Wolf J.P., Paronesso A., 1992, Lumped-parametermodel for a rigid cylin- drical foundations embedded in a soil layer on rigid rock,Earthquake Engine- ering and Structural Dynamics, 21, 1021-38 16. Wolf J.P., Preisig M., 2003, Dynamic stiffness of foundation embedded in layered halfspace based onwave propagation in cones,Earthquake Engineering and Structural Dynamics, 32, 1075-1098 17. Wong H.L., Luco J.E., 1976, Dynamic response of rigid foundations of ar- bitrary shape,Earthquake Engineering and Structural Dynamics, 4, 579-587 18. WuW.-H., LeeW.-H., 2002, Systematic lumped-parametermodels for foun- dations based on polynomial-fraction approximation, Earthquake Engineering and Structural Dynamics, 31, 1383-1412 942 Z. Sienkiewicz Lokalne modelowanie wpływu zasypki na dynamicznie obciążane sztywne osiowo-symetryczne fundamenty Streszczenie Wpływ zasypki na dynamiczną odpowiedź sztywnych osiowo-symetrycznych fun- damentów opisano jako reakcję niezależnej warstwy.Rozwiązanie problemu interakcji kinematycznej dane jest w postaci przyrostu zespolonej macierzy sztywności podło- ża względem przypadku fundamentów niezagłębionych. Przyrosty sztywności otrzy- mano z rozwiązań w dziedzinie częstości równań ruchu izotropowego jednorodnego ośrodka z odpowiednimi przemieszczeniowymi warunkami brzegowymi i warunkami wypromieniowania. Modelowanie przybliżone porównano z wynikami numerycznego rozwiązania problemuw postaci brzegowego równania całkowego. Manuscript received June 20, 2008; accepted for print February 19, 2009