Soil-Structure Interaction

Synonyms

Dynamic soil-structure interaction; Seismic soil-structure interaction; SSI

Introduction

Seismic waves propagating through the soil impinge upon structures founded on the soil surface or embedded into it. Displacements are then produced both in the structure and in the soil. The mutual dependency of the displacements is called soil-structure interaction, abbreviated as SSI. Consequently, the motion occurring at the base of the structure is different compared to the free-field motion (motion in the absence of the structure). Soil-structure interaction characteristics depend on several factors:

• Intensity, wavelength, and angle of incidence of the seismic waves

• Soil stratigraphy

• Stiffness and hysteretic damping of the particular soil layers

• Geometry and rigidity of the foundation

• Embedment depth of the structure

• Inertia characteristics, slenderness, and natural vibration period (eigenperiod) of the superstructure

• Presence of nearby structures

Various effects are associated to this phenomenon:

• A building founded on compliant ground has different vibrational characteristics, for example, higher natural period compared to the same building on rigid base (solid rock). The softer the soil, the larger the difference.

• A part of the vibrational energy emanating from the compliant structure foundation is transmitted into the surrounding soil through wave radiation in the unbounded soil medium and hysteretic energy dissipation. Such effects do not occur in a rigidly supported structure.

• Due to the compliance of the foundation, the motion at the foundation base contains rocking and torsional components in addition to the translational components.

The mechanisms governing soil-structure interaction can be divided into two distinct interactions: inertial and kinematic interaction.

Kinematic interaction is the deviation of the soil response from the free-field motion due to the resistance of the stiffer foundation to conform to the distortions of the soil imposed by the traveling seismic waves. It is commonly expressed in terms of frequency-dependent transfer functions relating the disturbed motion at the interface foundation/soil to the free-field motion.

Inertial interaction arises as the structure responds to the soil motion induced by kinematic interaction at the foundation level. Inertial forces are developed in the structure being transmitted to the compliant soil. Frequency-dependent impedance functions are used to represent the stiffness of the foundation/soil system and the associated radiation damping.

The relative impact of each contribution is a function of the characteristics of the incoming waves, the foundation geometry and rigidity, and the soil conditions.

The analysis is particularly challenging due to the semi-infinite extent of the soil medium, the nonlinearity of the soil behavior, the inherent variability of the soil stratigraphy, and the dependency of the response on frequency. Several procedures of different degrees of complexity have been proposed during the past five decades. A historical overview is given by Kausel (2010). The book by Wolf (1985) provides a rigorous and comprehensive treatment of the topic including applications to seismic problems.

Even with the computational facilities available today, such analyses are associated with a major effort, both for modeling the soil-structure system and carrying out the calculation. In particular during the early design stage, parametric studies are necessary in order to assess the influence of the various parameters and optimize the system for purposes of cost estimation. This necessitates the application of simplified methods that capture the essential features of the system response. The next sections provide a brief overview with emphasis on such simplified methods.

Soil-Foundation-Structure Analysis Models

Two general approaches are commonly used for the analysis of soil-structure interaction problems.

Direct Approach

The soil and the structure are treated together in a combined analysis by modeling them using finite elements or finite differences in two or three dimensions. This offers the advantage that inelastic behavior, particularly for the soil, can be taken into account by the step-by-step numerical integration of the equations of motion within a time-domain algorithm. A drawback is the necessity to specify the input motion at the base of the model, where it is not known a priori. Since the design seismic motion is usually given at the free surface or at outcropping rock, a deconvolution is necessary to obtain the compatible bedrock motion. Often the bedrock is located at large depths thus prohibiting the modeling of the entire soil layer, and some artificial boundary is defined at a shallower depth. The deconvolution then involves an iterative procedure. For convenience, the deconvolution is often carried out using algorithms that are based on 1D vertical shear wave propagation, thus requiring an adjustment of the model parameters in order to achieve compatible solutions between 1D and 2D analyses. Attention is further required in the selection of appropriate boundary conditions at the side boundaries of the discretized domain to avoid spurious reflections that would contaminate the results. The composite soil-structure model is finally subjected to the previously determined base rock motion, and the evolution in time of displacement and stresses is computed.

Substructure Approach

The underlying calculation method comprises three steps. At first the seismic motion acting at the foundation level is determined assuming a rigid but massless foundation. This is referred to as foundation input motion (FIM), and for an embedded structure, it will include both translational and rotational components. In the second step, the complex-valued frequency-dependent impedances for the foundation/soil system are determined. The real part of the impedance function represents a linear spring and the imaginary part, a viscous dashpot accounting for the energy radiation into the soil medium. Finally, the structure supported by the frequency-dependent springs and dashpots is subjected to the foundation input motion computed in the first analysis step.

While impedance functions are sufficient for rigid foundations, distributed springs and dashpots placed around the foundation are used for nonrigid embedded foundations when the distribution of sectional forces is sought. In this case, due to the vertical variation of ground motion, the imposed differential ground displacements vary over the height of the basement walls.

The validity of this approach – often called superposition theorem – is shown by Kausel and Roesset (1974). The main advantage of the method is that each step can be handled independently and with different algorithms. Further, it allows an insight into the contributions from each analysis step and is particularly suitable for parametric studies.

The application of the principle of superposition requires linear behavior. Inelastic behavior is implemented using equivalent linearization by selecting soil modulus and radiation damping to correspond to the likely effective strain level the soil will experience under the specific loading. This is achieved by means of an iterative procedure. Superposition is shown to be a reasonable approximation even when inertia forces produce large strains in the vicinity of the foundation, since shear strains due to kinematic interaction effects are usually significant in deeper soil regions.

Inertial Interaction

Shallow Foundations

The illustration of the concepts is made on the basis of a simple structure-soil system being composed by a linear structure of height h, mass m, lateral stiffness k, and damping ratio β_str that is connected to a rigid foundation of radius r resting on the surface of a homogeneous elastic half-space. The half-space is used to represent the unbounded soil medium and is characterized by its shear modulus G, Poisson’s ratio ν, and mass density ρ. Mass and moment of inertia of the foundation are neglected for simplification. The compliance of the soil is modeled by two frequency-dependent springs placed underneath the rigid foundation: a horizontal translation spring of stiffness K _x and a rotational spring of stiffness K _θ . Energy dissipation in the soil due to friction within the material (hysteretic damping) and wave radiation in the unbounded medium is modeled by a pair of frequency-dependent dashpots with coefficients C _x and C _θ attached parallel to the respective springs. This model may be viewed as a single- or multistory building after an appropriate reduction of the degrees of freedom.

Springs and dashpots for each degree of freedom j can be condensed to complex-valued impedances that are expressed in two equivalent forms:

$$ \tilde{K}={K}_j+i\omega {C}_j={K}_j\left(1+i2{\beta}_j\right) $$

(1)

where ω is the circular frequency of the excitation, i is the imaginary unit, and β _j is a damping coefficient that is related to the viscous dashpot coefficient of a simple oscillator by

$$ {\beta}_j\left(\omega \right)=\frac{\mathrm{Im}\left({\tilde{K}}_j\right)}{2\mathrm{R}\mathrm{e}\left({\tilde{K}}_j\right)}=\frac{\omega {C}_j}{2{K}_j} $$

(2)

The use of β _j has the advantage that at resonance of the compliant system β _j corresponds to the percentage of critical damping.

The undamped natural vibration period of the structure in its fixed-base condition is

$$ T=2\pi \sqrt{\frac{k}{m}} $$

(3)

For the case of a compliant base, it can be shown that the respective natural period is (Veletsos and Meek 1974)

$$ \tilde{T}=T\sqrt{1+\frac{k}{K_x}+\frac{k{h}^2}{K_{\theta }}} $$

(4)

Hence, the period of the flexibly supported structure is higher than that on rigid base. Since the spring stiffnesses are in general frequency dependent, an iterative procedure is necessary to evaluate the period $\tilde{T}$. A reasonable approximation consists in using the spring values corresponding to the fixed-base natural period, and even simpler is to use the static values of the springs.

The dimensionless parameters controlling the period lengthening are

$$ \mathrm{Stiffness}\ \mathrm{ratio}\ \mathrm{structure}-\mathrm{t}\mathrm{o}-\mathrm{soil} \overline{s}=\frac{h}{T\;{v}_s} $$

(5)

$$ \mathrm{Slenderness}\ \mathrm{ratio}\ \overline{h}=\frac{h}{r} $$

(6)

$$ \mathrm{Mass}\ \mathrm{ratio}\ \overline{m}=\frac{m}{\rho \pi {r}^2h} $$

(7)

with

$$ {v}_s=\sqrt{G/\rho } $$

(8)

denoting the shear wave velocity in the soil.

The stiffness ratio will be larger for stiff structural systems such as shear walls and smaller for flexible systems such as moment frames. For soil and weathered rock sites, this term is typically smaller than 0.1 for flexible systems such as moment frames and between approximately 0.1 and 0.5 for stiff systems such as shear wall and braced frame structures. The period lengthening variation with the stiffness ratio is shown in Fig. 1a for typical values of the parameters involved.

Soil-Structure Interaction, Fig. 1

Effects of soil-structure interaction: (a) elongation of natural period in dependency on the ratio of structure-to-soil stiffness $\overline{s}$; (b) increase in effective damping in dependency on the natural period ratio $\tilde{T}/T$. Curves are for mass ratio $\overline{m}$ = 0.15 and different values of the slenderness ratio $\overline{h}$. Poisson’s ratio ν = 0.45 (Adapted from Veletsos (1977). In: Hall WJ (ed) Structural and geotechnical mechanics, 1st edn, © 1977. Reprinted by permission of Pearson, Inc., Upper Saddle River, NJ)

For the overall effective damping ratio of the system, several approaches have been proposed in the literature differing in the degree of approximation involved. Usually products of damping ratios are neglected as higher-order terms. The most widespread among these solutions – that also entered design codes – is that derived by Veletsos and Meek (1974). Assuming structural damping of viscous nature, the overall effective damping becomes

$$ \tilde{\beta}={\beta}_0+\frac{\beta_{\mathrm{str}}}{{\left(\tilde{T}/T\right)}^3} $$

(9)

where β₀ represents the contribution from the soil-structure interaction – being referred to as foundation damping – that includes both material and radiation damping (Veletsos 1977). The respective expression is written here in the more general form

$$ {\beta}_0={\left(\frac{T}{\tilde{T}}\right)}^3\left|k\frac{\tilde{T}}{T}\left(\frac{\beta_x}{{\tilde{K}}_x}+\frac{\beta_{\theta }}{{\tilde{K}}_{\theta }}{h}^2\right)\right| $$

(10)

From Eq. 9 it is evident that the effectiveness of the structural damping is reduced by soil-structure interaction as the period ratio $\tilde{T}/T$ increases. This may lead to a decrease in overall damping unless this reduction is compensated by the increase in the foundation damping. In practice, effective damping is taken higher than the structural damping, the value 5 % used in the development of design provisions being considered as a lower bound. Figure 1b shows the significant increase of the foundation damping with decreasing slenderness ratio h/r: rocking motion that is characterized by small radiation damping dominates the response of slender structures, whereas for squat structures the prevailing motion is horizontal translation that radiates energy into the soil more efficiently.

Observations based on data from instrumented buildings confirmed the analytical findings. For the majority of structures, the stiffness ratio h/Tv _s will be less than 0.5 and the mass ratio will range between 0.1 and 0.2 with a typical average of 0.15 (Stewart et al. 2003). The case studies analyzed revealed that the governing parameter for inertial interaction is the stiffness ratio and that these effects can be neglected for values less than 0.1.

Impedances for Shallow Foundations

Frequency-dependent springs and dashpots for shallow foundations have been determined in the last decades by several authors for different geometries and soil stratigraphies. In most cases radiation damping is expressed in terms of the dashpot coefficient C _j , as given in Eq. 1. The stiffness K _j at zero frequency is referred to as the static foundation stiffness and is denoted by K ⁰ _j . The effects of frequency on the spring values for the particular vibrational mode j are then given by stiffness modifiers such that

$$ {K}_j={K}_j^0 {k}_j $$

(11)

Exact closed-form solutions are available only for perfectly rigid circular foundations and relaxed boundary conditions at the soil-foundation interface, i.e., normal stresses are neglected for swaying and shear stresses for rocking. These solutions are

Horizontal translation

$$ {K}_x^0=\frac{8}{2-\nu }Gr $$

(12)

Rocking

$$ {K}_{\theta}^0=\frac{8}{3\left(1-\nu \right)}G{r}^3 $$

(13)

These expressions may be used for square foundations – and also for rectangular foundations with aspect ratio less than 3 – by replacing the radius by an equivalent value that yields the same footprint area for swaying and equal moments of inertia for rocking.

A review of available solutions for foundation impedances is presented by Pais and Kausel (1988) and Gazetas (1991) and the update by Mylonakis et al. (2006). Approximate expressions and graphs are compiled for various configurations and for all six modes of vibration. They include static values for rectangular foundations, stiffness modifiers, and expressions for the radiation damping.

We restrict here the presentation of results to swaying and rocking motion for rectangular foundations with footprint area 2a × 2b with $a\ge b$ with the x-axis running parallel to the longer foundation side. The subscripts θ _x and θ _y in the impedances indicate rotation around the x- and y-axis, respectively. The weak coupling between translational and rocking mode is neglected. The frequency dependency is captured by the dimensionless parameter

$$ {a}_0=\frac{\omega b}{v_s} $$

(14)

and the foundation aspect ratio is denoted by

$$ \ell =\frac{a}{b}\ge 1 $$

(15)

The approximate expressions obtained by Pais and Kausel (1988) are displayed in the following.

Surface Foundations

The static solutions are:

$$ \mathrm{Swaying}\ {K}_x^0=\frac{Gb}{2-\nu}\left[6.8{\ell}^{0.65}+2.4\right] $$

(16)

$$ \mathrm{Swaying}\ {K}_y^0=\frac{Gb}{2-\nu}\left[6.8{\ell}^{0.65}+0.8\ell +1.6\right] $$

(17)

$$ \mathrm{Rocking}\ {K}_{\theta_x}^0=\frac{G{b}^3}{1-\nu}\left[3.2\ell +0.8\right] $$

(18)

$$ \mathrm{Rocking}\ {K}_{\theta_y}^0=\frac{G{b}^3}{1-\nu}\left[3.73{\ell}^{2.4}+0.27\right] $$

(19)

The frequency-dependent stiffness modifiers are:

$$ \mathrm{Swaying}\ {k}_x=1 $$

(20)

$$ \mathrm{Swaying}\ {k}_y=1 $$

(21)

$$ \mathrm{Rocking}\ {k}_{\theta_x}=1-\left[\frac{0.55{a}_0^2}{0.6+\frac{1.4}{\ell^3}+{a}_0^2}\right] $$

(22)

$$ \mathrm{Rocking}\ {k}_{\theta_y}=1-\left[\frac{\left(0.55+0.01\sqrt{\ell}\right){a}_0^2}{2.4-\frac{0.4}{\ell^3}+{a}_0^2}\right] $$

(23)

The viscous damping coefficients accounting for radiation damping as determined from the dashpot coefficients using Eq. 2 are:

$$ \mathrm{Swaying}\ {\beta}_x=\left[\frac{4\ell }{K_x^0/Gb}\right]\left[\frac{a_0}{2{k}_x}\right] $$

(24)

$$ \mathrm{Swaying}\ {\beta}_y=\left[\frac{4\ell }{K_y^0/Gb}\right]\left[\frac{a_0}{2{k}_y}\right] $$

(25)

$$ \mathrm{Rocking}\ {\beta}_{\theta_x}=\left[\frac{\left(4/3\right)\overline{v}\;\ell\;{a}_0^2}{\left({K}_{\theta x}^0/G{b}^3\right)\left[\left(2.2-\frac{0.4}{\ell^3}\right)+{a}_0^2\right]}\right]\left[\frac{a_0}{2{k}_{\theta x}}\right] $$

(26)

$$ \mathrm{Rocking}\ {\beta}_{\theta_y}=\left[\frac{\left(4/3\right)\overline{v}{\ell}^3{a}_0^2}{\left({K}_{\theta y}^0/G{b}^3\right)\left[\left(\frac{1.8}{1+1.75\left(\ell -1\right)}\right)+{a}_0^2\right]}\right]\left[\frac{a_0}{2{k}_{\theta y}}\right] $$

(27)

where

$$ \overline{v}=\sqrt{2\left(1-\nu \right)/\left(1-2\nu \right)} \overline{v}\le 2.5 $$

(28)

is the ratio of compressional wave velocity to shear wave velocity in the soil.

It should be mentioned that the exact curves for the stiffness modifiers and the damping factors have in general a smooth wavy form, and the expressions given above consist approximations to these curves.

Key features of the system behavior are:

Dynamic modifiers for translational stiffness are almost unity independent of the foundation aspect ratio, whereas rocking modifiers are significantly reduced with frequency in a very weak dependency on the foundation aspect ratio.

Radiation damping for the horizontal translational mode is only modestly influenced by the direction of vibration or the foundation aspect ratio. For rocking on the other hand, the damping is strongly affected by the aspect ratio and the direction of vibration, increasing with the foundation aspect ratio. At low frequencies damping in rocking motion is smaller compared to that in horizontal translation due to interference phenomena; it only overweighs translational damping at higher frequencies and for elongated foundations when excited in the direction of the longer foundation side. Hence, translational foundation movement may often be predominant with respect to radiation damping.

Embedded Foundations

The references cited above contain also information for embedded foundations. Embedment increases static foundation stiffness. According to the review by Pais and Kausel (1988), dynamic stiffness modifiers remain largely unaffected. The dynamic analyses for obtaining such impedances usually assume a perfect contact between the soil and the basement walls, a situation that seldom occurs in reality. This yields higher damping values as observed from the actual response of buildings. A practical, conservative approach consists in considering the embedment effects only for the static stiffness and applying the dynamic modifiers of surface foundations. Alternatively, one may use the formulae given by Gazetas (1991) and by Mylonakis et al. (2006) that consider an effective height of the contact zone along the perimeter of the embedded foundation.

Soil Layering

Impedance functions for multilayered soils can only be determined with specialized software that is not easily accessible to practicing engineers. Available algorithms are mostly based on finite element procedures incorporating efficient consistent boundaries for the proper energy radiation at the domain boundaries.

A particular case constitutes a soil layer of finite depth on rock where a cutoff frequency exists, below which there is no radiation damping. The respective formulae given in the above references may be used for a two-layer system when the shear wave velocity in the top layer is less than half of that of the underlying stratum. Impedances for square foundations on uniform or nonuniform soil layer overlying a half-space are tabulated by Wong and Luco (1985).

Parameters for Soil Behavior

The expressions given above assume linear elastic or viscoelastic soil behavior. However, for moderate or strong seismic excitations, the nonlinearity of the soil must be taken into account. Hence, the values of the shear modulus entering the equations for the SSI effects must be adjusted to reflect the strain level in the ground associated with the stipulated design ground motion. In critical projects seismic site response analyses are carried out with the soil properties being determined from special dynamic laboratory tests on undisturbed samples. First-order estimates for the strain-compatible values are given in some code provisions. Typical values as recommended by Eurocode 8, Part 5 (CEN 2004), are tabulated below in terms of their small-strain amplitude values G₀ and v_s0 in dependency on the effective ground acceleration defined as the spectral acceleration at the plateau of the response spectrum divided by 2.5. Guide values for the hysteretic soil damping are also given.

The small-strain values of the soil shear modulus or the shear wave velocity may be determined by a variety of methods, the choice depending on the variability of the soil conditions, available knowledge on the material behavior, and the importance of the structure. These methods include: (i) empirical relationships in terms of the SPT blow count or of the tip resistance of the CPT tests, (ii) geophysical field methods based on wave propagation, and (iii) dynamic laboratory tests. An overview of the testing procedures and available design equations is summarized by Kramer (1996).

	Effective spectral ground acceleration [g]
	0.10	0.20	0.30
G/G0	0.80 (±0.10)	0.50 (±0.20)	0.36 (±0.20)
vs/vs0	0.90 (±0.07)	0.70 (±0.15)	0.60 (±0.15)
Damping ratio	0.03	0.06	0.10

Adaption in Design Codes and Implication for the Design

The implications of inertial SSI for design are illustrated in Fig. 2 with reference to the acceleration response spectrum used for evaluating seismic base shear forces in buildings. Idealized envelope spectra in modern codes initially increase with period, attaining a plateau value, and start decreasing monotonically after a certain period that is in the order of 0.4 to 1.0 s. For buildings with periods larger than about 0.5 s, consideration of period elongation and flexible base damping will lead to a reduction of the base shear demand. Hence, in most cases, SSI effects are neglected in the frame of conservative design.

Soil-Structure Interaction, Fig. 2

Effect of natural period elongation and foundation damping on a typical acceleration design spectrum (Adapted from Stewart et al. (2003) by permission of the Earthquake Engineering Research Institute)

However, there are various seismic environments with recorded response spectra exhibiting their peak at periods greater than 1.0 s. Spectra from some prominent records are contrasted to a typical design spectrum for soil in Fig. 3. SSI phenomena in these earthquakes had detrimental effects as revealed by analyses linking site conditions and building natural periods to observed damage. In the 1985 Mexico City earthquake, for example, due to SSI effects, the natural period of 10–12 story buildings founded on soft clay was altered from about 1.0–1.5 s to nearly 2.0 s, thus coinciding with the peak of the response spectrum at the particular site. The associated phenomena are elucidated among others by Gazetas and Mylonakis (1998).

Soil-Structure Interaction, Fig. 3

Ratio of spectral acceleration to peak ground acceleration for 5 % structural damping for some severe earthquakes with long-period components compared to that of a typical code for soft soil (Adapted from Gazetas and Mylonakis (1998) by permission of the American Society of Civil Engineers)

Hence, proper assessment of both the anticipated seismic input and the prevailing soil conditions is an indispensable prerequisite in any SSI analysis. In modern seismic codes the site characterization for deep soil deposits is based almost exclusively on the near-surface region of the soil (often the top 30 m), disregarding the depth of the underlying rock. The representative average shear wave velocity to this depth in this deposit is used as parameter for the classification; cf. Dobry et al. (2000).

Pile Foundations

Single Pile

Consider a pile horizontally loaded at its head at the ground surface. The deformed shape of the pile extends down to a so-called active (or effective) length below which it becomes negligible. This length depends on the pile diameter, the elastic modulus of the soil, the ratio of pile modulus to soil modulus, and the fixity conditions. Expressions for static and dynamic loading are given by Gazetas (1991). For static loads this length is of the order of 10–20 pile diameters, while for dynamic loading this length will be greater due to the wave propagation. With respect to flexural response, the pile can be modeled without significant error as an infinite-long beam when its length is greater than the active length. Two models are commonly used for the analysis: elastic continuum theory or Winkler spring models (Pender 1993).

Following the same principles as for shallow foundations, the horizontally loaded soil-pile system may be represented by three impedances corresponding to swaying, rocking, and cross-swaying-rocking. Consideration of the latter is necessary since the reference level is located at the pile head and the resultant of the reactions acts at a specific depth thus inducing a bending moment at the pile head. Expressions synthesized from results by various authors are summarized by Gazetas (1991). The static stiffnesses are expressed in terms of the pile diameter d and Young’s moduli of the soil and the pile E and E _p , respectively:

$$ \mathrm{Swaying}\ {K}_x^0=dE {\left(\frac{E_p}{E}\right)}^{0.21} $$

(29)

$$ \mathrm{Rocking}\ {K}_{\theta}^0=0.15{d}^3E{\left(\frac{E_p}{E}\right)}^{0.75} $$

(30)

$$ \mathrm{Cross}-\mathrm{swaying}-\mathrm{rocking}\ {K}_{x\theta}^0=-0.22{d}^2E{\left(\frac{E_p}{E}\right)}^{0.50} $$

(31)

The dynamic modifiers are approximately equal to unity:

$$ {k}_x={k}_{\theta }={k}_{x\theta}\approx 1 $$

(32)

The expressions for the radiation damping β _j as defined by Eq. 2 are given in dependency on the dimensionless frequency

$$ {a}_0=\frac{\omega d/2}{v_s} $$

(33)

as follows:

$$ \mathrm{Swaying}\ {\beta}_x=0.35 {a}_0{\left(\frac{E_p}{E}\right)}^{0.17} $$

(34)

$$ \mathrm{Rocking}\ {\beta}_{\theta }=0.11 {a}_0{\left(\frac{E_p}{E}\right)}^{0.20} $$

(35)

$$ \mathrm{Cross}-\mathrm{swaying}-\mathrm{rocking}\ {\beta}_{x\theta }=0.27 {a}_0{\left(\frac{E_p}{E}\right)}^{0.18} $$

(36)

Pile Groups

Building foundations are always constructed as groups of piles. In evaluating the dynamic stiffness of a pile group, the interactions between the piles must be taken into consideration, just like in the case of static loading. However, the cross-interaction of individual piles is strongly dependent on frequency, thus precluding description by simple explicit formulae. The rigorous solution methods available are based on the thin-layer method (Kaynia and Kausel 1982; Waas and Hartmann 1984). Fortunately, a remarkable simple solution procedure was discovered by Dobry and Gazetas (1988) that is straightforward to implement, thus facilitating the assessment of the associated SSI effects with a very good accuracy. The respective interaction coefficients between the piles are given in terms of pile spacing, excitation frequency, and the wave velocity through the soil between the piles. The values for the overall stiffness and damping of the pile group are then assembled using the respective values of the single piles and these interaction factors.

Kinematic Interaction

Shallow Foundations

Kinematic interaction is induced by the presence of a stiff foundation that forces the foundation motions to deviate from the free-field motions. The associated phenomena are due to (i) base-slab averaging of inclined or incoherent seismic waves, and (ii) embedment of the foundation.

Base-Slab Averaging

Seismic waves impinging at directions other than vertical arrive at different points along the foundation at different times giving rise to the so-called wave passage effects. The apparent propagation velocity of the waves is in the order of 1.5–3.5 km/s and is controlled by the wave propagation in the underlying rock. In addition to this, ground motion is in most cases inherently incoherent resulting from inhomogeneities along the travel path from the source to the site.

Studies conducted hitherto mainly address the wave passage problem that is amenable to analytical treatment. They show that the slab due to its stiffness and flexural rigidity averages the free-field displacement pattern by reducing the translational motions and at the same time introducing rotational motions. The latter include rocking in the presence of inclined SV-, P-, or Rayleigh waves, and torsion in the presence of SH- or Love waves. The torsion of symmetrical buildings observed in earthquakes is a consequence of obliquely incident seismic waves. Further, the modification of the seismic motion depends on the frequency content of the seismic motion with high-frequency components being filtered out by the slab when the respective apparent wavelength is shorter than an effective length of the foundation slab (the diameter for circular foundations).

Kinematic interaction effects are expressed in terms of transfer functions relating the amplitude of the foundation input motion to that of the free-field motion. The system considered consists of a rectangular foundation with area 2a × 2b, $a\ge b$ that is subjected to harmonic SH waves of circular frequency ω with particle motion in the direction of the x-axis impinging on the foundation at an angle α _v with the vertical and propagating along the positive y-axis. The transfer functions derived by Veletsos et al. (1997) include both coherent and incoherent seismic motions. They are given in dependency on the dimensionless parameter

$$ {\tilde{a}}_0=\frac{\omega {b}_e}{v_s}\sqrt{\kappa^2+{ \sin}^2{\alpha}_v{\left(\frac{b}{b_e}\right)}^2} $$

(37)

where ${b}_e=\sqrt{ab}$ is the half-side length of an equivalent square foundation, v _s is the shear wave velocity, and κ is a ground motion incoherence parameter. The curves shown in Fig. 4 represent the two limiting cases with κ = 0 and α _v = 0, respectively. The transfer functions for torsional motions are referred to the foundation edge being the product of foundation half-width b and rotational angular distortion.

Soil-Structure Interaction, Fig. 4

Amplitude of transfer functions between free-field and foundation input motion for rectangular foundations subjected to obliquely incident shear waves: (a) vertically incident, incoherent waves; (b) non-vertically incident, coherent waves. The solid lines are for the horizontal motion and the dashed lines for the induced torsional component. Curves computed from expressions in Veletsos et al. (1997)

Lateral transfer functions are for both types of wave motion only very weak dependent on the aspect ratio a/b suggesting that the governing parameter is the foundation area. The induced torsional component, however, is very sensitive both to the aspect ratio and the type of wave motion.

Recent observations on buildings indicate that the apparent value of κ (denoted by κ _a ) is nearly proportional to the small-strain shear wave velocity v _s yielding roughly κ _a = 0.2 at a typical value v _s = 250 m/s, Kim and Stewart (2003).

Foundation Embedment

Embedment effects result from the scattering of incoming waves. Rocking motions develop due to nonuniformly distributed tractions against the side walls. Assessment is made by means of transfer functions relating the base-slab translational and rocking motions to the free-field motions. An accurate numerical solution for cylindrical foundations subjected to coherent shear wave motions is provided by Day (1977). The embedment depth e is normalized with respect to the radius of the foundation r, and the frequency dependency is captured by the dimensionless parameter ${a}_0=\omega r/{v}_s$. Figure 5 shows typical patterns for vertically propagating waves.

Soil-Structure Interaction, Fig. 5

Amplitude of transfer functions between free-field and foundation input motion for cylindrical, embedded foundation subjected to vertically incident coherent shear waves for different normalized embedment depths. (a) horizontal translation; (b) rocking component (Redrawn from Day (1977))

Piles

Piles embedded in a soil stratum respond to incident vertical shear waves in dependence on their flexural rigidity in relation to the stiffness of the surrounding soil. The incoming wave field is modified, the displacement at the pile head differs from that of the free field, and pile-head rotation is induced. The displacement reduction depends on the ratio of pile modulus to soil modulus, the slenderness of the pile, and the frequency of excitation with high-frequency components being filtered out especially by relatively short, rigid piles.

Analyses of kinematic interaction that include variable stratigraphy are carried out mostly by using the Winkler spring model. The ends of the springs and dashpots that capture SSI effects are connected to the free field where the soil response is imposed. The latter is computed independently. The accuracy of this simplified approach depends on the selection of the springs and dashpots that are obtained adopting physically justified approximations (Pender 1993).

Attention deserves the bending moment induced in the pile during the passage of seismic waves. The maximum value occurs, as expected, at soil layer interfaces, strongly increasing with the contrast in shear wave velocity between the bottom and top layer (Nikolaou et al. 2001).

The numerical study by Fan et al. (1991) using the continuum model by Kaynia and Kausel (1982) for pile groups excited by vertically propagating shear waves provides graphs showing the effects of pile rigidity relative to that of the soil, pile slenderness, pile spacing, number of piles, and pile-head fixity conditions. An idealized general shape of the frequency dependence of the kinematic response is defined in terms of a displacement factor relating the pile-head displacement to that of the free field. This factor is approximately unity at low frequencies with the pile closely following the ground movement; in the medium frequency range, it decreases with frequency and beyond a distinct frequency fluctuates around a constant value of 0.2–0.4. The difficulty consists in defining the transition frequencies for the particular system layout.

It must be realized though that there is no simple means for evaluating kinematic interaction for pile groups. In noncritical situations, however, this difficulty may be circumvented by neglecting kinematic interaction. This is justified by findings that kinematic effects for pile groups are similar to those for individual piles, in particular for horizontal translation and to a lesser extent for torsional and rocking vibration modes.

Concluding Remarks

Despite its inherent complexity, the theory of linear soil-structure interaction and the implications in structural performance are now well understood. Refinements, optimization, and validation studies are subjects of ongoing research. The variability in the stratigraphy of soil deposits, the nonlinearity of the soil behavior, the frequency dependency of the response, and the limited availability of specialized software for the analysis make the proper assessment of the SSI effects still a difficult task, requiring physical insight when applying such concepts. It should be self-evident that the effective implementation in an integrated structural design asks for a close collaboration between structural and geotechnical engineers.

Summary

The main effects of soil-structure interaction on the seismic response of structures founded on compliant ground are presented. The modeling concepts to capture the associated modification of the building natural period and the energy dissipation due to radiation damping are highlighted by reference to relatively simple structures. Both kinematic and inertial actions are treated. Available expressions for the dynamic impedance functions are summarized both for shallow foundations and piles. A brief account is given of the implications in seismic design provisions for buildings.

Cross-References

• Building Codes and Standards

• Earthquake Response Spectra and Design Spectra

• Engineering Characterization of Earthquake Ground Motions

• Response Spectrum Analysis of Structures Subjected to Seismic Actions

• Seismic Analysis of Masonry Buildings: Numerical Modeling

• Selection of Ground Motions for Response History Analysis

• Site Response: 1-D Time Domain Analyses

• Substructuring Methods for Finite Element Analysis