A Simple Model for Vertical Dynamic Interactions among a Group of Strip Footings Rested on Homogeneous Half-Space

A simple model for vertical dynamic interactions among a group of strip footings rested on half-space is presented in this paper. An analytical method is presented to obtain the contact pressures and the impedance matrix for a group of surface strip footings. In order to conveniently solve the unknown contact pressures between the soil and footings, the soil-footing interfaces are discretized into a series of strip elements. )e Green function for each element under uniform harmonic force is derived and calculated by the piecewise integration and Cauchy principal value integral. )e influences of footing and soil parameters on contact pressures and vertical dynamic impedances of footing groups are discussed in detail. )e SSSI effect between adjacent footings increases with the decrease of the distance ratio S/L. For three footings in a group, the middle footing experiences greater cross-interactional effect than the side ones. )e present method has high accuracy, which is not only simple but also suitable for the high-frequency analysis.


Introduction
e soil-structure interaction (SSI) has been paid comprehensive attention over the past few decades [1][2][3]. e substructure method has been widely applied in SSI research due to that the footing and the half-space can be analyzed separately by using the respectively suitable methods. In the substructure method, the reaction of the soil against the footing can be described by the frequency-dependent stiffness and damping coefficients which are commonly called as dynamic impedance. erefore, it is a key step to obtain the impedance function of the footing in the analysis of SSI.
Reissner [4] derived the first analytical solution for the vertical vibration of a circular plate subjected to a harmonic uniform force and marked the beginning of the elastic halfspace theory about SSI. Sung [5] presented three kinds of supposed contact pressure distributions (static rigidity, uniform, and parabolic) beneath the footing. e dynamic impedances for some representative cases [6,7] were obtained based on these assumed distributions. However, the assumed distribution could not yield a constant surface displacement of the supporting medium beneath the rigid footing as demanded from physical considerations; it is then necessary to find the mean value for the footing displacement through various averaging techniques. On the contrary, the footing impedances were analyzed more rigorously and treated as a mixed boundary-value problem. Luco and Westmann [8] presented the impedance for a massless rigid strip footing by letting the Cauchy singular dual integral equations reduce to the second kind of Fredholm integral equations. Ma et al. [9] transformed the dual integral equations into a set of linear equations using an infinite series of orthogonal Jacobi polynomials for the rocking impedance of the rigid strip footing. As the contact stresses cannot be expressed by the elementary functions, there are certain mathematical limitations to solve the mixed boundary-value problem. erefore, some semianalytical approaches were presented to obtain the impedance more conveniently. Jiang and Song [10] investigated the impedance of a massless rigid strip footing by the thin layered method [11], that is, the analytical solution in the horizontal direction and the finite element discretization in the vertical direction. Lin et al. [12] studied the similar problem based on the precise integration method [13].
In the aforementioned literature, dynamic interaction between single footing and elastic half-space was considered. However, the construction of the metros and highspeed trains is becoming prevalent with the acceleration of urbanization. In such a situation, the footings associate together through soil ground. is results in the structuresoil-structure interaction (SSSI) under the externally vertical exciting loads [14,15]. Taking the advantage in the SSI model proposed by Parmelee [16], Warburton et al. [17] derived governing equations for the response of two geometrically identical cylindrical bodies attached to the surface of an elastic half-space, which initiates the SSSI study. Liou [18] presented an analytical solution for the dynamic stiffness matrix of adjacent surface rigid footings, based on the assumed contact stress distribution which linearly varies in the radius direction in the cylindrical coordinate system. Currently, analyses of dynamic interactions between multiple footings are mainly through numerical methods [19][20][21][22] such as the finite element method and the boundary element method due to the rapid progress in computer technique. However, these methods are far too time-consuming and complicated for actual engineering and designers. e analytical method for the impedance matrix of footing group based on the elastic half-space theory is still very rare, while it is efficient and has great significance for solving the seismic response of structure groups and assessing structure safety.
In this paper, the Green function of uniform harmonic vertical force is derived and numerically calculated by the piecewise integrations and the Cauchy principal value integral. Contact pressures and the impedance matrix of multiple strip footings considering the SSSI effect are obtained by combining the Green function with the element discretization technique. In contrast with the mixed boundary-value method, the present method avoids the directly solving of contact pressures which cannot be expressed by the elementary functions. e validity and wide applicability of the present method has been verified by the comparative studies. e SSSI effect on the contact pressures and impedances for a group of surface strip footings are illustrated by the parametric studies.

General Formulation of SSSI Problem
For the strip footings such as track footings, dams, or building footings with high ratio of length to width, it is reasonable to consider the problem as a plane strain case with a coordinate system (x-z) where the z-axis is normal to the space surface. Consider a group of strip footings with different width L m (m � 1, . . . , M) and different separation distance S m , as shown in Figure 1. M footings rest on a linear elastic half-space, and the mth footing is excited by the vertical harmonic line excitations T m exp(iωt) (m � 1, . . . , M). e SSSI effect of the system consisting of the M footings can be conveniently expressed by where H m represents the vertical displacement of the mth footing. e impedance R mn in matrix [R] is a complex number in the matrix, which describes the coupling interaction effect between the mth footing and the nth footing.
. Here, the real part K mn acts as a spring which expresses the restraint from soil medium on the footing, whereas the imaginary part C mn acts as a dashpot which expresses the energy dissipation from the soil medium. Based on the above formulations, an equivalent simplified model as shown in Figure 2 visually describes the dynamic interaction among strip footings in a group through the soil.
A compact form of (1) can be written as e impedance matrix [R] should be determined with regard to the contact condition at the footing-soil interfaces. For the simplification in analysis, the interface between the supporting medium and each footing is divided into a number of surface strip elements. e coordinate information for each element is shown in Table 1.
e elements beneath the mth footing (m � 1, . . . , M) are numbered in succession from 1 to R m , starting from the left to the right. e width of the element is denoted by Δ m � L m /R m . It is assumed that the rth element (r � 1, . . . , R m ) of the mth footing is subjected to the uniform pressure q r m , as shown in Figure 3.

Green Function for Elastic Half-Space
Based on the Cartesian coordinate system (x-z) with z � 0 at the free surface, the wave equations in an elastic halfspace composed of homogeneous and isotropic solid are given by where ρ is the density of the elastic soil, G and λ are the elastic Lamé constants, and u and w are the displacement components of the soil in x and z directions. e steady-state vertical dynamic of the soil satis es u Ue iωt and w We iωt . e term exp(iωt) is hereafter omitted from all displacements and forces for brevity. ε (zu/zx) + (zw/zz) and ∇ 2 ((z 2 /zx 2 ) + (z 2 /zz 2 )) denote the volumetric strain and the Laplacian operator, respectively.
Introducing two potential functions Φ and Ψ, the displacement functions U and W are assumed to be satis ed by Substituting (4) in (3), the wave equations can be transformed into two Helmholtz equations: where h ω/V p , k ω/V s , V p (λ + 2G)/ρ is the dilatational wave (P wave) velocity, and V s G/ρ is the shear wave (S wave) velocity. e general solutions of (5) can be obtained by using the separation of variables method. ereafter, making use of (4), the general solutions of displacements U and W in a half-space can be expressed as

Shock and Vibration
where A and B are the integral coefficients dependent on the boundary conditions of the soil surface, α � Based on the relationships between stresses and displacements for plane problem in elasticity, σ zz � (2GzW/zz) + λε and τ zx � G((zU/zz) + (zW/zx)), the stress amplitudes at an arbitrary point in half-space can be derived from (6) and (7) as follows: Without loss of generality, we consider the vertical uniform pressure q r m applied at a strip element, as shown in Figure 4. e boundary conditions at the ground surface z � 0 have where [a, b] is the interval of strip element beneath the strip footings with its coordinate information shown in Table 1.
Making the Fourier transformation to (10) and expressing it in an integrated form with respect to the x-coordinate based on its inverse relationship, one has Comparing (11) with (8) and (9), the coefficients A and B can be uniquely determined by Substituting the above two coefficients back in (6) and (7), the vertical displacement field w r m (x, z) caused by the half-space caused by the uniform pressure can be obtained: Submitting the interval coordinates of each element into (13) and considering the symmetry of the integral, the vertical displacement field at the surface of the half-space caused by the uniform pressure q r m of the rth element beneath the mth footing can be obtained as

Impedance Matrix of a Strip Footing Group
Following the superposition approach, the vertical displacement W (x, 0) at any point of the soil surface caused by a series of strip elements beneath the footing group can be obtained from (14) as follows: Figure 4: e half-space acted by a vertical harmonic uniform excitation.

Shock and Vibration
where r � 1, . . . , R m and m � 1, . . . , M. R m is the number of strip element beneath the mth footing, and M is the footing number of the group. Substituting the central coordinate of each element into the above equation, in turn, the equilibrium equations for all strip elements can be written in a matrix form: where q m � q 1 m , . . . , q r m , . . . , q R m m T , e element A ij mn in matrix [A R m R n mn ] describes the relationship between the contact pressure applied on the jth element beneath the mth footing and the displacement of the i element beneath the nth footing, which is given as where Equation (18) is a multivalue improper integral, which can be precisely calculated by the piecewise integration and the Cauchy principal value integral, which can be expressed as in which, where ε is the root of F(p) and Ρ means the Cauchy principal value integral. For brevity, (16) can be written in a compact form: in which q � q T 1 , . . . , q T M and W � W in which, [X] where [X] is a matrix of dimension M m 1 R m × M and 0 m is a zero column vector of order R m .
Balance of excitation force and the surface contact forces for the mth footing yields T m ( I m ) T q m Δ m . erefore, one has in which, [Y] Submitting (22) and (23) in (25) leads to Finally, comparing the above equation with (1), these unknown parameters in Figure 2 can be obtained by the following impedance matrix of the footing group.

Convergence and Comparison Studies
Convergence and numerical stability of the proposed method are investigated with respect to the number of segmental strip elements. e vertical impedances of a single footing with di erent series terms are given in Table 2. It is seen from Table 2 that 100 elements for uniform pressures are enough to give results with at least three signi cant digits. e vertical displacement at the soil surface under unit harmonic force is compared with that obtained by the thin layered method (TLM) [11] to verify the numerical calculation of the multivalue improper integral in the Green function (13). e parameters used are soil density ρ 2000 kg/m 3 , shear wave velocity V s 500 m/s, Poisson's ratio v 0.4, the distance between the observation location and the force location d 40 m, and the interval width of the uniform pressure 1 m. e nondimensional excitation frequency is a 0 kd/2. e displacement response of the halfspace with respect to the force-frequency is given in Figure 5. It can be seen from Figure 5 that the present solutions agree with TLM's solutions. However, there are some minor di erences between the results from the two methods up to 30% in Figure 5. is is because the present model satis es the boundary condition of a semi-in nite half-space while the TLM model used the arti cial boundary. erefore, the present solution is more accurate than that of TLM. e exibility of a rigid strip footing was obtained by Luco and Westmann [8] based on the mixed boundary-value approach with Fredholm integral equations. As there are considerable di culties in solving Fredholm integral equations numerically or analytically, only a special case for Poisson's ratio v 0.5 was studied rigorously in Luco's solution.
e cases with Poisson's ratio v < 0.5 were approximately studied by using the dominant part of the singular integral equation to evaluate the impedances of the strip footing, which is only valid for the low frequency a 0 ≤ 1.  In order to compare with Luco's solutions, the impedance R is transformed into the exibility by F R −1 . e exibility of the footing with respect to dimensionless exciting frequency is plotted in Figure 6 for three di erent Poisson's ratios v 0.5, 1/3, and 0.25. In Figure 6, the maximum relative errors between two solutions are less than 10%. e agreement with Luco's solution approves the correctness and e ectiveness of the present method. In addition, Figures 6(b) and 6(c) show that the present method covers a wider range of frequency than Luco's method.

Vertical Dynamic Contact Pressures of Footing Group.
e contact pressures of footing groups, as well as that of a single footing, under the harmonic vertical excitations with dimensionless frequencies a 0 0.2 and 3 are plotted in Figure 7 for a group of two footings and in Figure 8 for a group of three footings. Horizontal coordinate x is a local coordinate with its origin at the center of each footing. Longitudinal coordinate q Lq/(K sig H) is a dimensionless contact pressure, in which K sig is the dynamic sti ness of a single footing and H is its displacement amplitude. It also can be seen from Figure 7 that the pressure distributions of the footings in the group are di erent from that of a single strip footing due to the SSSI e ect. Moreover, the comparison between Figures 7(a) and 7(b) indicates that the shape and magnitude of the contact pressures for both single footing and footing group change with the frequency of the excitation.
In addition, the di erences between the contact pressures of footing group and those of a single footing in  Figure 7 and two side footings in Figure 8, the pressure distribution around the outside edge is still in accord with that of a single footing, while the pressure distribution around the inside edge slopes downward because of the interference from the adjacent footing. Moreover, the middle footing in Figure 8 although presents a symmetrical distribution of pressure, its magnitude and shape are completely di erent from those of the single footing. erefore, those pressure distribution assumptions for single footings would cause a considerable error in SSSI analysis.

In uence of Distance Ratio on SSSI E ect.
Two adjacent strip footings with the same width L and separated by a distance S are used to investigate the in uence of the distance ratio S/L on the SSSI e ect under vertical harmonic excitation. e variation of the vertical exibility with the dimensionless frequency is plotted in Figures 9-11 for . e maximum di erentia between the vertical exibility considering the SSSI e ect and that of a single footing is less than 10% in Figures 9-11 when the distance ratio between two adjacent footings reaches S/L 4. However, it can be seen from Figures 9-11 that the SSSI e ect between adjacent footings increases with the decrease of the distance ratio S/L. e vertical exibility for the case of a small distance ratio S/L 0.125 shows an obvious uctuation around the solution of an isolated footing which is induced by the SSSI e ect. In addition, the comparison among Figures 9-11 shows that Poisson's ratio of the soil produces little in uence on the e ective distance of the SSSI e ect.

Dynamic Interaction between Multiple Strip Footings.
is example shows the matrix impedance for a group of three footings in the case of a distance ratio S/L 0.  Shock and Vibration C ii (i 1, 2, and 3) are shown in Figure 12, and the corresponding coupling impedances K ij and C ij of paired footings are shown in Figure 13. e impedance of an isolated footing K sig and C sig without the SSSI e ect is also presented in Figures 12 and 13 for comparisons. It can be seen from Figure 12 that the vertical impedances of K ii and C ii in uenced by the SSSI e ect uctuate around that of the isolated one. It can be seen from Figure 13(a) that the coupling sti ness K 21 and K 23 are negative for low frequencies. erefore, although the sti ness of K 22 is higher than the sti ness of a single footing K sig in Figure 12 Figure 12, as well as C 11 and C 22 , indicates that the middle footing experiences a greater SSSI e ect than the side ones. e coupling impedances of paired footings given in Figure 13 show that the SSSI e ect between the middle footing and its adjacent ones is signi cantly greater than that between two side footings due to a smaller distance ratio.

Conclusions
An e cient semianalytical method has been presented for modelling the vertical dynamic interaction among a group of strip footings rested on the elastic half-space. e approach is of signi cance for dynamic analysis and seismic design of strip footings with close space, such as a series of parallel tracks or building foundations with a high ratio of length to width. e dynamic contact pressures and impedance matrix of a group of rigid strip footings considering the SSSI e ect have been studied in detail, and the following conclusions can be emphasized: (1) e present method overcomes the mathematical limitations in the mixed boundary-value method and can provide the dynamic impedances of multiple footings within a wide scope of excitation frequency. e computational stability and accuracy of the present method have been veri ed by the convergence studies and the comparison examples.
(2) e distributions of contact pressures for strip footing groups are in uenced by the frequency of excitation and the SSSI e ect. erefore, those symmetric distribution assumptions for a single footing would cause a considerable error in SSSI analysis.
(3) e SSSI e ect increases with the decrease of distance ratio S/L between paired strip footings. It is advisable to consider the SSSI e ect among footing groups when S/L reaches a small ratio less than 4.0, especially for the case of a low exciting frequency a 0 from 0.5 to 1.5. Poisson's ratio of the half-space has little e ect on this critical value of S/L. (4) e dynamic impedance of a strip footing in group uctuates around that of a single one. For strip footings located in the di erent positions of the group, the middle footing generally su ers a greater SSSI e ect than the side ones. (5) Although the results in this paper are limited to the homogeneous half-space, the presented discretized method can be extended to the layered half-space with the fundamental solution of the wave equation of layered half-space, which will be derived in our further research by a transferred matrix method.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no con icts of interest regarding the publication of this paper. Shock and Vibration 11