3-D Boundary element-finite element method for the dynamic analysis of piled buildings

3-D boundary element–finite element method for the dynamic analysis of piled buildings ∗ L.A. Padr´on, J.J. Azn´arez, O.Maeso Instituto Universitario de Sistemas Inteligentes y Aplicaciones Num´ericas en Ingenier´ıa (SIANI) Universidad de Las Palmas de Gran Canaria Edificio Central del Parque Cient´ıfico y Tecnol´ogico Campus Universitario de Tafira, 35017, Las Palmas de Gran Canaria, Spain {lpadron,jjaznarez,omaeso}@siani.es 14 September 2010 Abstract A boundary element – finite element model is presented for the three-dimensional dynamic analysis of piled buildings in the frequency domain. Piles are modelled as compressible EulerBernoulli beams founded on a linear, isotropic, viscoelastic, zoned-homogeneous, unbounded layered soil, while multi-storey buildings are assumed to be comprised of vertical compressible piers and rigid slabs. Soil-foundation-structure interaction is rigorously taken into account with an affordable number of degrees of freedom. The code allows the direct analysis of multiple piled buildings, so that the influence of other constructions can be taken into account in the analysis of a certain element. The formulation is outlined before presenting validation results and an application example. 1 Introduction The practical interest of understanding the dynamic behaviour of pile foundations becomes evident in view of the large number of related papers published in the last four decades. A highly significant proportion of these works use a boundary element approach to the problem due to the need to model the soil as an unbounded region (see, for example, reviews by Beskos [1, 2] and Padr´on [3]). Moreover, most studies focus on the computation of impedances [4, 5, 6], kinematic seismic response [7, 8, 9] or internal forces in pile foundations [10, 11], allowing the analysis of pile–supported structures by means of substructuring strategies. On the contrary, very few works have attempted to tackle the problem using a direct approach in which pile, soil and superstructures are modelled simultaneously [12, 13, 14]. Despite the efforts to analyse pile dynamic response, the influence of nearby pile–supported structures on the behaviour of pile foundations and piled buildings has received, up to the authors’ knowledge, no attention at all, except for some preliminary results presented in [15]. For this reason, a direct methodology to study simultaneously soil, piles and several superstructures has been formulated and implemented using the boundary element method to model the layered soil and finite elements for piles and superstructures. This numerical scheme represents an enhancement of a boundary element – finite element coupling model [16] formulated to study pile–soil interaction only. Such a formulation was motivated by the need to develop a flexible and accurate methodology to tackle the problem, and also by the necessity of reducing the meshing ∗This is the peer reviewed version of the following article: L.A. Padr´on, J.J. Azn´arez, O.Maeso, 3-D boundary element–finite element method for the dynamic analysis of piled buildings, Engineering Analysis with Boundary Elements 35 (2011) 465-477, which has been published in final form at doi:10.1016/j.enganabound.2010.09.006. This work is released with a Creative Commons Attribution Non-Commercial No Derivatives License. 1 efforts and the number of degrees of freedom of the model with respect to a multi-domain boundary element approach, as the one used in [17]. In order to meet these requirements, pilesoil interaction forces are modelled as internal loads, pile rigidity is introduced by means of mono-dimensional finite elements, and equilibrium and compatibility conditions are established along the pile axis. Following this strategy, pile-soil interfaces do not need to be discretized by boundary elements leading to a significant reduction in the number of degrees of freedom without appreciably compromising the accuracy of the results. The boundary element – finite element method formulation for the dynamic analysis of soil-pile-building systems is outlined in section 2. Then, section 3 presents some comparison results in order to check the method and validate its implementation. In order to illustrate the capabilities of the methodology, section 4 shows different types of analyses related to a group of piled buildings in a layered soil subjected to a harmonic excitation applied on the ground surface. Finally, some conclusions and future research directions are provided in the last section. Further details can be found in [3]. 2 Boundary element – finite element method formulation In this formulation, the boundary element method is used to model the dynamic behaviour of the stratified soil region, while finite elements are used for piles and superstructures. From the boundary element (soil) point of view, the loads arising from the pile-soil interaction are modelled as distributions of interaction forces applied on an internal line defined by the pile axis, what in this work will be named ‘load-line’ (see figure 1). Piles rigidity is provided by monodimensional finite elements, bonded to the surrounding soil by equilibrium and compatibility conditions. } + Load lines in the soil region BEM discretization Interaction forces acting within the soil Interaction forces acting over the pile Pile foundation modelling BEM internal points FEM elements & nodes L d s x y z Figure 1: Pile foundation geometry and modelling through BEM-FEM coupling formulation 2.1 Pile foundation formulation Three-node Euler-Bernoulli beam finite elements with axial deformation are used to model the piles. Thirteen degrees of freedom are defined in the element: three orthogonal displacements at each node plus two rotations (torsion excluded) at each one of the end nodes. The displacement vector ue(ξ) at any point ξof an element eis expressed in terms of the vector of nodal displacements and rotations up eas ue(ξ) = Ψ(ξ)up e(1) 2 where Ψ(ξ) is the matrix of shape functions at point ξcontaining two different sets of functions: (i) a first set of five functions ϕused to build an interpolating polynomial of the lateral displacements through the three nodes with explicitly given rotations at both ends; and (ii) a second set of three functions φto approximate the axial deformation. This last set of functions, which correspond to the three Lagrangian polynomials of second order, are also used to approximate the pile-soil contact interaction forces qe(ξ) acting over the pile, within an element e, as qe(ξ) = Φ(ξ)qp e(2) where matrix Φis formed by shape functions φ(see [3, 16] for more details) and qp eis the vector of nodal interaction forces. The principle of virtual displacements allows to obtain the mass and stiffness matrices (Mand K) for the described element type [18], and also the matrix Qthat transforms nodal interaction forces components qpinto equivalent nodal forces Feq through the linear transformation Feq e= Qeqp e. Hence, after a process of discretization of the pile, computation of elemental matrices and assembly of global matrices, the dynamic behaviour of a certain pile can be represented, in the finite–element sense, by the matrix equation ¯ Kup=F+Qqp(3) where the vector of external forces Fcontains the equivalent nodal forces Feq from the pile-soil contact interaction, the forces Ftop acting at the top of the pile and the axial force Fpat the tip of the pile; where ¯ K=K−ω2M,ωbeing the frequency of excitation; and where all matrices must be understood as global to the whole pile. 2.2 Soil formulation In contrast, each stratum Γmof the soil is modelled by the BEM as a linear, homogeneous, isotropic, viscoelastic, unbounded region with complex valued shear modulus µof the type µ=Re[µ](1 + 2iβ), where βis the damping coefficient and i = √−1. The boundary integral equation for a time-harmonic elastodynamic state defined in a domain Ωmwith boundary Γm can be written in a condensed and general form as cιuι+ZΓm p∗udΓ = ZΓm u∗pdΓ + nm ll X j=1 "ZΓm pj u∗qsjdΓpj−δjΥj kFpj#(4) where cιis the local free term matrix at collocation point xι;uand pare the displacement and traction vectors; u∗and p∗are the elastodynamic fundamental solution tensors representing the response of an unbounded region to a harmonic concentrated unit load with a time variation eiωt applied at a point xι; Γm pjis the pile-soil interface along the load-line jwithin the domain Ωm;nm ll is the total number of load-lines in the domain Ωm;δjis equal to one if the load-line jcontains the tip of a floating pile and zero otherwise; Υj kis a three-component vector that represents the contribution of the axial force Fpjat the tip of the jth load-line; and qs=−qpis the vector of nodal interaction forces acting over the soil, as sketched in figure 1. The boundaries Γmare discretized into quadratic elements of triangular and quadrilateral shapes with six and nine nodes, respectively. Once all boundaries have been discretized, eq. (4) can be written, for each region Ωm, at all nodes on Γmin order to obtain a matrix equation of the type Hssus−Gssps− nm ll X j=1 Gspjqsj+ nm ll X j=1 δjΥsjFpj= 0 (5) where usand psare the vectors of nodal displacements and tractions of boundary elements; Hss and Gss are coefficient matrices obtained by numerical integration over the boundary elements 3 of the fundamental solution times the corresponding shape functions; and Gspjis the coefficient matrix obtained by numerical integration over load-line jof the fundamental solution times the corresponding interpolation functions, when the unit load is applied on Γm. Furthermore, eq. (4) will be also applied at internal nodes belonging to load-line Γm pi, so that one can write C upi+Hpisus−Gpisps− nm ll X j=1 Gpipjqsj+ nm ll X j=1 δjΥpijFpj= 0 (6) where Hpisand Gpisare coefficient matrices obtained by numerical integration over the boundary elements of the fundamental solution times the corresponding shape functions; and Gpipj is the coefficient matrix obtained by numerical integration over load-line jof the fundamental solution times the corresponding interpolation functions, when the unit load is applied on loadline Γm pi. Here, upiis the vector of nodal displacements of the load-line i, which is multiplied by the diagonal matrix C, whose non-zero terms are valued 1/2 in positions corresponding to pile nodes placed on a smooth surface (as e.g. pile heads) and unity at the internal points. The situation of a pile crossing interfaces between adjacent regions (see figure 1) is worthy of attention. In this case, the distribution of the interaction forces qsalong the pile-soil interface is not continuous between layers, and different load-lines are considered in the upper and lower layers. This fact causes a kind of corner problem which can be overcome by applying a non-nodal collocation strategy. For nodes placed on the interface, eq. (6) is applied on an inner point of the pertinent element k, so that the integral equation becomes Φupi k+Hpisus−Gpisps− nm ll X j=1 Gpipjqsj+ nm ll X j=1 δjΥpijFpj= 0 (7) where upi kis the vector of nodal displacements of element Γkand Φis the matrix of shape functions specified at the collocation point. The axial punctual force Fpconsidered at the tip of a floating pile adds a new unknown per pile, in such a way that an extra equation needs to be written. The third row (corresponding to the application of the unit load in the vertical direction) of eq. (7) written for a non-nodal point placed between the two bottom nodes of the pile yields a suitable additional equation for this purpose. 2.3 Pile supported structures model This section describes the adopted formulation for the analysis of the dynamic behaviour of pile– supported multi-storey structures. For this purpose, piles are grouped and fixedly connected slab 1 slab j−1 slab j slab j−1 slab nT Figure 2: Two-dimensional view of considered pile–supported structures 4 to rigid pile caps on which the building is founded. The superstructures are assumed to be composed by any number of vertical extensible piers and horizontal rigid slabs (see figure 2). Piers are modelled as massless Euler-Bernoulli beams, with axial and lateral deformation, and with hysteretic damping through a complex valued stiffness of the type k=Re[k](1 + 2iζ). Torsional stiffness is not considered in the piers. The principal axes of inertia of rigid slabs may have any orientation with respect to the global coordinate axes and, at the same time, the position of their centre of gravity on the horizontal plane can change between storeys, which provides a great deal of generality and flexibility to the model. In order to write the equations directly in terms of slabs displacements and rotations (the most interesting parameters in this kind of study), all DoF at piers ends are condensated to the centre of gravity of slabs and pile caps. Let Uj= ({U1, U2, U3,Θ1,Θ2,Θ3}j)Tbe the vector defining displacements and rotations at the centre of gravity of the pile cap or slab j, and let Yj i= ({u1, u2, u3, θ1, θ2}j)Tbe the vector defining displacements and rotations at the end of pier or pile iconnected to slab or pile cap j. The kinematic relationship between the head of pile iand the corresponding pile cap j, or between a pier iand a certain slab or cap jcan be expressed as Yj i=Tj iUj(8) where Tj i=       1 0 0 0 0 −(xi2−xcgj 2) 0 1 0 0 0 xi1−xcgj 1 0 0 1 xi2−xcgj 2−(xi1−xcgj 1) 0 0 0 0 1 0 0 0 0 0 0 1 0        (9) where xi1and xi2are the x1and x2coordinates of pier or pile i, and xcgj 1and xcgj 2are the x1 and x2coordinates of the centre of gravity of slab or pile cap j. On the other hand, the equilibrium at any slab jcan be expressed as Fj ext + nj X i=1 (Tj i)Tfj itop + nj+1 X i=1 (Tj i)Tfj+1 ibottom +ω2MjUj= 0 (10) where Fj ext is the vector of external forces and moments acting on the slab j,njis the number of piers in the inter-storey immediately below slab j,nj+1 is the number of piers in the interstorey immediately above slab j,fj itop = ({f1, f2, f3, m1, m2}j)Tis the vector of forces and moments in the connection between slab jand pier ifrom the inter-storey immediately below, fj+1 ibottom = ({f1, f2, f3, m1, m2}j+1)Tis the vector of forces and moments in the connection between slab jand pier ifrom the inter-storey immediately above, and Mj=MjI3×3∅ ∅I(11) where Mjis the mass of the slab, I3×3is the identity matrix of order 3, and Iis the 3×3 tensor expressing the slab moments of inertia. As mentioned above, piers are modelled as massless Euler-Bernoulli beams with axial deformation. In this way, and in the absence of external forces applied along the pier, the relationship between forces and displacements at both ends of the pier is given by a 10 ×10 stiffness matrix of the form fib fitj =Kbb iKbt i Ktb iKtt ijYib Yitj (12) where band tstand for bottom and top respectively. 5 By substitution of (12) into (10), and taking (8) into account, the equilibrium equation for slab jcan be written as Fj ext +Kj j−1Uj−1+ (ω2Mj+Kj j)Uj+Kj j+1Uj+1 = 0 (13) where Kj j = nj X i=1 (Tj i)TKttj iTj i+ nj+1 X i=1 (Tj i)TKbbj+1 iTj i(14) Kj j−1= nj X i=1 (Tj i)TKtbj iTj−1 i(15) Kj j+1 = nj X i=1 (Tj i)TKbtj+1 iTj+1 i(16) These submatrices represent the three terms of the superstructure stiffness matrix that appear in the row corresponding to slab j, and where the contribution of all piers in each storey above and below such a slab is taken into account. On the other hand, an equation equivalent to (10) can be written for each of the pile caps on which the superstructure is founded. To express such an equilibrium equation in the format of eq. (13), eq. (3) must be rewritten for the top element of pile iin terms of submatrices affecting either the pile head node or the other two underground nodes of the element as "5×5¯ Khh 5×8¯ Khu 8×5¯ Kuh 8×8¯ Kuu #iuh uui +5×9Qh 8×9Qui qsi=fh fui (17) where hand ustand for head and underground respectively, and where the dimensions of the submatrices are written on its top left side. Then, the forces at the top of the pile, equivalent to the term fj itop of eq. (10), can be obtained as fh=¯ Khh iuh i+¯ Khu iuu i+Qh iqsi(18) which, taking eq. (8) into account, allows us to express the equilibrium equation at a certain pile cap cas Fc ext +Kc1U1+ (ω2Mc+Kc c)Uc+ nc piles X i=1 ˜ Kc iuu i+˜ Qc iqsi= 0 (19) where Kc1= n1 X i=1 (Tc i)TKbt1 iT1 i(20) Kc c = nc piles X i=1 (Tc i)T¯ Khh iTc i+ n1 X i=1 (Tc i)TKbb1 iTc i(21) ˜ Kc i =(Tc i)T¯ Khu i(22) ˜ Qc i =(Tc i)TQh i(23) 2.4 Seismic excitation modelling The system described in the previous sections can be excited either by forced displacements prescribed at certain points, by applied external forces or by a far-source earthquake. This section discusses how this last kind of action is modelled. 6 When seismic waves impinge on the site under study, reflection and refraction phenomena take place, and the arising wave field modifies the incident wave train. The original wave field, which is assumed to come from a far away source, is the incident field uI, while the one produced by the reflection and refraction phenomena receives the name of scattered field uS. The resulting displacement and traction fields (total fields) can be obtained by superposition as u=uI+uS and p=pI+pS. Let us consider that a pile foundation embedded in a layered soil is subjected to timeharmonic incident waves. In this case, eq. (5) represents the discretized boundary integral equation for each region Γmin terms of the total field. On the other hand, as the incident, scattered and total fields satisfy the governing equations, such an equation can also be written in terms of the incident field as Hssus I−Gssps I= 0 (24) where interaction forces along the pile-soil interface qsjand forces at piles tip Fpjare not present because they only exist in the scattered fields. The subtraction of eq. (24) from eq. (5) yields Hssus−Gssps− nm ll X j=1 Gspjqsj+ nm ll X j=1 δjΥsjFpj=Hssus I−Gssps I(25) where the right–hand vector is known because the analytic expressions of us Iand ps Ican be easily obtained. The same procedure can be repeated to obtain the BE equations (6) and (7) for load-lines. In contrast, the pile FE eq. (3) holds true for the total fields and, therefore, does not need to be rewritten. Finally, note that the magnitude of the scattered field decreases with distance due to the existing material and radiation damping in the soil, and consequently, boundaries far from the pile foundation do not need to be discretized. 2.5 Assembly of the global system matrix of equations Equations (3), (5), (6), (7), (13) and (19) (or the corresponding equations for incident seismic fields, if it were the case) can be arranged to form a linear system of equations representing a general problem consisting on a certain number of superstructures, each founded on one or more caps, each one of which groups two or more different piles embedded in a viscoelastic soil of generic stratigraphy and topography. Equilibrium and compatibility fully-bonded contact conditions over the different interfaces of the problem are imposed. The most general situation is that of a problem with multiple superstructures founded on different pile caps on a layered soil, the system being subject to external forces and/or incident seismic waves. In such a general case, the system of equations is of the form Aus,ps,qs,Fp,up,UjT=B(26) where A, whose structure is sketched in figure 3, is the square matrix of coefficients and B is the known vector, both computed by rearranging the equations and prescribing the known boundary conditions. The vector of unknowns includes the displacements usand/or tractions psat boundary element nodes, the interaction forces at pile-soil interface qs, the forces at pile tips Fp, the nodal translations and rotations at pile nodes up, and the degrees of freedom defined at the structures Uj. Note that the displacements at the pile head nodes have been condensed and that the first five equations of expression (3) have been included in (19) in such a way that only the equations corresponding to underground nodes have to be written in the form of eq. (3). 7 BEM eqs. on boundaries BEM eqs. on load-lines FEM eqs. on piles FEM eqs. on structure us|psqsFp upUj Figure 3: Structure of the system matrix of coefficients A        Slab Inertial properties Mass ms= 105kg Inertia Is xx =Is yy = 1728 m4        Pier properties ζ= 5% Epier = 3 ·1010 N/m2                Pile properties ξ= 5% Ep= 3 ·1010 N/m2 ρp= 2500 kg/m3 L= 30 m Figure 4: Two-dimensional sketch of considered pile–supported structures 3 Validation results As mentioned in the introduction, the formulation proposed in this work represents an enhancement of a work presented a few years ago for the dynamic analysis of pile foundations exclusively [16]. This part of the formulation, related only to the response of the foundation, has been thoroughly validated [16, 3]. This section intends to validate the whole soil-foundation-building formulation by comparing results obtained from the direct methodology presented herein against results from a substructuring approach. The example used for this purpose is depicted in figure 4, where a small five-storey RC building founded on a group of 16 piles is sketched. The system is modelled through five rigid slabs, nine 0.4×0.4 m vertical piers per storey, and a rigid pile cap for the 4×4 pile group embedded in a homogeneous half-space. The main structural properties are included in the figure, while the soil properties are: soil internal damping coefficient β= 0.05, soil density ρs= 1750 kg/m3, soil Poisson’s ratio νs= 0.4 and shear wave velocity cs= 239 m/s. These data yield a pile-soil modulus ratio Ep/Es= 107.1. 8 3.1 Substructuring model Regarding the substructuring approach, a two-dimensional model of the system is used, with 2 degrees of freedom (horizontal displacement and rotation) per slab and also on the pile cap. The stiffness matrices, damping properties, inertial features and input motions characterizing the system must be defined for the specific case at hand. In this way, for each storey of the building described above, and modelling piers as compressible Euler-Bernoulli beams, the stiffness submatrix Ks ij between slabs i(upper slab, first two DoF) and j(lower slab, second two DoF) is defined as Ks ij =    k11 −k12 −k11 −k12 −k12 k22 +kak12 k22/2−ka −k11 k12 k11 k12 −k12 k22/2−kak12 k22 +ka    ij (27) where k11 = 9 ·12(EI)pier/H3,k12 = 9 ·6(EI)pier/H2,k22 = 9 ·4(EI)pier/H and ka= 6 · (EA)piersc/L,Apier and Ipier being the area and inertia of the pier section, Epier the modulus of elasticity, and Hthe pier height. In order to include hysteretic damping in the model, all the elements of matrix Ks ij are complex numbers of the type k=Re[k](1 + 2iζ). The dynamic stiffness of the foundation (where two DoF are also considered) is assumed to be defined by swaying (Kxx = kxx + iaocxx), rocking (Kθθ = kθθ + iaocθθ) and the crossed horizontal-rocking (Kxθ = kxθ +iaocxθ) impedance functions, where kij and cij are the frequency dependent dynamic stiffness and damping coefficients, respectively, ao=ωd/csis the dimensionless frequency and csis the soil shear-wave velocity. Therefore, the stiffness matrix of the pile foundation can be written as Kf=Kxx Kxθ Kxθ Kθθ (28) Such impedance functions have been computed for our particular case as specified in [16], and are shown in figure 5. 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0 0.05 0.1 0.15 0.2 0.25 k × 109 ao Kxx (N⁄m) Kxθ/10 (N) Kθθ/100 (N⋅m) 0.0 5.0 1.0 1.5 2.0 2.5 3.0 0 0.05 0.1 0.15 0.2 0.25 c × 1010 ao −2.0 0.0 2.0 4.0 6.0 8.0 1.0 1.2 0 0.05 0.1 0.15 0.2 0.25 ub = u/uff ao Real part Imag part −2.0 −1.0 0.0 1.0 2.0 3.0 0 0.05 0.1 0.15 0.2 0.25 θb = θ⋅d/uff ao Figure 5: Dynamic stiffness (top) and kinematic interaction (bottom) functions of 4 ×4 pile group 9 Figure 8: Representation of the mesh (modelling a quarter of the geometry) used for computations. The area where excitation is applied is marked in red in the upper illustration. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 0.05 0.1 0.15 0.2 0.25 abs(ux/uf) ao 1st floor 2nd floor 3rd floor 4th floor 5th floor Figure 9: Absolute horizontal displacements in floor slabs for case A. 16 −30 −25 −20 −15 −10 −5 0 00.5⋅1071⋅107 depth (m) Bending moment / uf (N⋅m/m) case A, p1 p2 p3 p4 case B, p1 p2 p3 p4 0.1 0.2 0.3 ao ✲ ∼1.8·107 ✲ ∼1.4·107 Figure 10: Comparison of envelopes of pile bending moments −30 −25 −20 −15 −10 −5 0 00.5⋅1071⋅107 depth (m) Shear force / uf (N/m) case A, p1 p2 p3 p4 case B, p1 p2 p3 p4 0.1 0.2 0.3 ao ✲ ∼1.9·107 ✲ ∼1.3·107 Figure 11: Comparison of envelopes of pile shear forces 17 −30 −25 −20 −15 −10 −5 0 00.5⋅1081⋅1081.5⋅1082⋅108 depth (m) Axial force / uf (N/m) case A, p1 p2 p3 p4 case B, p1 p2 p3 p4 0.1 0.2 0.3 ao Figure 12: Comparison of envelopes of pile axial forces 18 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Re(ux/uf), x=0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Im(ux/uf), x=0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Re(ux/uf), x=1.5b -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Im(ux/uf), x=1.5b -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Re(ux/uf), x=3b -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Im(ux/uf), x=3b -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Re(ux/uf), x=4.5b -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Im(ux/uf), x=4.5b -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Re(ux/uf), x=6b Without buildings case A case B -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Im(ux/uf), x=6b ao ao Figure 13: Comparison of horizontal displacements on different points in the ground surface. 19 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Re(uz/uf), x=0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Im(uz/uf), x=0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Re(uz/uf), x=1.5b -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Im(uz/uf), x=1.5b -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Re(uz/uf), x=3b -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Im(uz/uf), x=3b -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Re(uz/uf), x=4.5b -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Im(uz/uf), x=4.5b -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Re(uz/uf), x=6b Without buildings case A case B -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 Im(uz/uf), x=6b ao ao Figure 14: Comparison of vertical displacements on different points in the ground surface. 20 -0.08 -0.04 0 0.04 0.08 0 0.05 0.1 0.15 0.2 0.25 Re(ux/uf) -0.08 -0.04 0 0.04 0.08 0 0.05 0.1 0.15 0.2 0.25 Im(ux/uf) case A case B -0.08 -0.04 0 0.04 0.08 0 0.05 0.1 0.15 0.2 0.25 Re(θ⋅b/uf) ao -0.08 -0.04 0 0.04 0.08 0 0.05 0.1 0.15 0.2 0.25 Im(θ⋅b/uf) ao Figure 15: Horizontal displacements and rotations at pile cap of central building. Case A Case B Figure 16: Inter-storey drift in each storey of central building. 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 case A case B inter-storey drift 1 2 3 4 5 storey Figure 17: Maximum relative inter-storey drift in central building. 21