**4. Basic superposition**

148 Earthquake Engineering

structures.

soil samples proved to be considerably less than the damping measured in the dynamic

The nature of this effect was discovered in 1930-40s (Reissner in Germany [3], Schechter in the USSR) and proved to be in inertial properties of the soil. Inertia plus flexibility always mean wave propagation. It turned out that in the field tests actual energy dissipation in the soil was composed of two parts: conventional "material" damping (the same as in laboratory tests) and so-called "wave damping". In the latter case the moving stamp caused certain waves in the soil, and those waves took away energy from the stamp, contributing to the overall "damping" in the soil-structure system. This energy was not transferred from mechanical form into heat (like in material damping case), but was taken to the infinity in the original mechanical form. In reality waves did not go to the infinity, gradually dissipating due to the material damping in the soil, but huge volumes of the soil were involved in this wave propagation. Even without any material damping in the soil this "wave damping" contributed a lot to the response of the stamp. In practice it turned out that

the level of wave damping was usually greater than the level of material damping.

In parallel it turned out that when the base mat size is comparable to the wave length in the soil, not only damping, but stiffness also depends on the excitation frequency. This effect is invisible for comparatively small machine basements, but important for large and stiff

To study these wave effects new soil-structure models with infinite inertial soil foundations should be considered. That was the moment (1960-s) when the very term "SSI" appeared [2,4,5]. It happened so, that NPPs were actively designed at that time, including seismic regions (e.g., California), and intensive research was funded in the US to study the SSI effects controlling the NPP seismic response. Earthquake Engineering Research Center (EERC) in the University of Berkley, California became a leader with such outstanding

The main result of these investigations was the development of new powerful tools to analyze more or less realistic models. The earlier SSI models considered homogeneous halfspace with surface rigid stamp. They could be treated analytically or semi-analytically for

In practice soil is usually layered. Layering can lead to the appearance of the new wave types and change the whole wave picture. Important achievements of 1960-70s enabled to move from the homogeneous half-space to the horizontally–layered medium in soil modeling [7,8]. However, the infinite part of the foundation, excluding some limited soil volume around the basement, still remains a) linear, b) isotropic, c) horizontally layered. These limitations are due to the methods of the SSI analysis. The final masterpiece of Prof. Lysmer was SASSI code [9], further developed by F.Ostadan, M.Tabatabaie, D.Giocel et al. This code combined finite element modeling of the structure and limited volume of the soil with semi-analytical modeling of the infinite foundation (see below). Limitations on the

embedment depth and on the shape of the underground part have gone.

field tests with rigid stamps resting on the soil surface.

scientists as H.B.Seed and J.Lysmer on board [6].

simple stamp shapes (e.g., circle).

Today there exist three approaches to SSI problems, namely "direct", "impedance", and "combined" ones. To understand them all, let us start from common general approach based on the superposition of the wave fields.

Let us call the problem with seismic wave, soil and structure "problem A" and start with completely linear soil-structure model. Let Q be some surface surrounding the basement in the soil and dividing the soil-structure model into two parts: the "external" part Vext and internal part Vint. Let *(-F)* be additional external loads distributed over Vint and specially tuned so, to provide zero displacements in Vint. Then "problem A" can be split in the sum of two wave pictures: "problem A1", including seismic excitation and loads *(-F),* and "problem A2" including only loads *(F)* without seismic wave – see Fig.1.

This simple superposition leads to a number of important conclusions.


balanced by loads *(-F)*). Hence, Vint can be withdrawn or replaced by another medium (with zero displacements) without changing Vext, seismic excitation, or loads *(-F)*. In particular, Vint can be replaced by initial soil without structure.

Soil-Structure Interaction 151

**Figure 2.** Split of loads F into Fint and Fext

methods one by one.

**5.1. Direct approach** 

corresponding to the internal stress field.

determined by soil properties in Vext and initial seismic wave.

With given free-field wave field *U0* one can easily obtain *Fint* just as surface forces at Q

Theoretically we can change the content of the volume Vint in "problem B" (e.g., withdraw the medium completely, taking loads *Fint* to zero). This change will change operator *G0*, change wave field *U0*, but the left part of (1) will stay the same, as it is in fact fully

Even if there exists some physical non-linearity in the model, and this non-linearity is localized inside Vint, initial "problem A" can still be substituted by "problem A2" without changing the loads *F*, as compared to the linear case. This is a consequence of the logic of the

Current methods of SSI analysis can be classified according to the choice of surface Q. In "direct" method they try to put Q on some physical boundaries in the soil, usually apart from the basement. In "impedance" method they put Q right on the soil-structure contact surface, additionally presuming the rigidity of this surface. In "combined" method they put Q on the boundary of the "modified volume" (sometimes Q is a flexible soil-structure contact surface, but sometimes there exists some additional soil volume around the basement modified with the appearance of structure). Let us discuss some details of these

In direct method there always exists certain volume Vint. Usually the lower boundary (i.e., the bottom of Vint) is placed on the rock. In this case we presume that the additional waves radiating from the structure cannot change the motion of this boundary, as compared to the seismic field *U0*. As a consequence, we can substitute boundary conditions in

previous point: the loads are fully determined by soil in Vext and initial seismic wave.

**5. Classification of methods: direct and impedance approaches** 

**Figure 1.** Basic superposition: split of the "problem A"


$$-F(\mathbf{x}) = G\_0(\mathbf{x}, \mathbf{y})[-\mathcal{U}\_0(\mathbf{y})] \tag{1}$$

Formula (1) uses operator *G0* in the time domain. This operator is applied to the displacement field in the volume Vint and provides the loads, distributed over the volume (this formula can be applied to the whole volume Vint, but for the internal nodes the result will be zero). For linear initial soil this operator in the frequency domain will turn into complex frequency-dependent dynamic stiffness function. Note that for the given surface Q the loads *F* in (1) can be split in two parts: loads *Fint* acting from Vint and loads *Fext* acting from Vext. Physical meaning is illustrated in Fig.2.

**Figure 2.** Split of loads F into Fint and Fext

balanced by loads *(-F)*). Hence, Vint can be withdrawn or replaced by another medium (with zero displacements) without changing Vext, seismic excitation, or loads *(-F)*. In

4. We can withdraw structure from Fig.1 and call the problem with initial soil in Vint "problem B". This problem can be also split in "problem B1" and "problem B2" in the same manner as "problem A". Wave fields in Vext and loads *(-F)* are equal in "problem A1" and "problem B1". However, in "problem A2" and "problem B2" wave fields are different in spite of similar loads *F* and similar Vext in both problems. Generally, the motions of Q in "problem B2" and "problem A2" are different due to the waves,

5. Wave field in volume Vint in "problem B2" is the same as in "problem B" (see conclusion 1 above). Very often this field is known apriori or easily calculated. This creates a powerful tool to verify models suggested for "problem A2". Each of these models contains some description of the internal part Vint, external part Vext and the loads *F*. It is useful to take the same Vext and *F* and substitute the internal part Vint by the initial soil, thus coming from "problem A2" to "problem B2". The suggested Vext and *F* must provide adequate

6. Loads *(–F)* can be obtained from the wave field *U0* in "problem B" and dynamic

Formula (1) uses operator *G0* in the time domain. This operator is applied to the displacement field in the volume Vint and provides the loads, distributed over the volume (this formula can be applied to the whole volume Vint, but for the internal nodes the result will be zero). For linear initial soil this operator in the frequency domain will turn into complex frequency-dependent dynamic stiffness function. Note that for the given surface Q the loads *F* in (1) can be split in two parts: loads *Fint* acting from Vint and loads *Fext* acting

0 0 *Fx G xy U y* ( ) ( , )[ ( )] (1)

solution for "problem B2"; otherwise they cannot be applied to "problem A2".

stiffness operator *G0* for the initial unbounded soil as follows

particular, Vint can be replaced by initial soil without structure.

**Figure 1.** Basic superposition: split of the "problem A"

radiating from the structure in "problem A2".

from Vext. Physical meaning is illustrated in Fig.2.

With given free-field wave field *U0* one can easily obtain *Fint* just as surface forces at Q corresponding to the internal stress field.

Theoretically we can change the content of the volume Vint in "problem B" (e.g., withdraw the medium completely, taking loads *Fint* to zero). This change will change operator *G0*, change wave field *U0*, but the left part of (1) will stay the same, as it is in fact fully determined by soil properties in Vext and initial seismic wave.

Even if there exists some physical non-linearity in the model, and this non-linearity is localized inside Vint, initial "problem A" can still be substituted by "problem A2" without changing the loads *F*, as compared to the linear case. This is a consequence of the logic of the previous point: the loads are fully determined by soil in Vext and initial seismic wave.

#### **5. Classification of methods: direct and impedance approaches**

Current methods of SSI analysis can be classified according to the choice of surface Q. In "direct" method they try to put Q on some physical boundaries in the soil, usually apart from the basement. In "impedance" method they put Q right on the soil-structure contact surface, additionally presuming the rigidity of this surface. In "combined" method they put Q on the boundary of the "modified volume" (sometimes Q is a flexible soil-structure contact surface, but sometimes there exists some additional soil volume around the basement modified with the appearance of structure). Let us discuss some details of these methods one by one.

#### **5.1. Direct approach**

In direct method there always exists certain volume Vint. Usually the lower boundary (i.e., the bottom of Vint) is placed on the rock. In this case we presume that the additional waves radiating from the structure cannot change the motion of this boundary, as compared to the seismic field *U0*. As a consequence, we can substitute boundary conditions in "problem A2", fixing the motion of the bottom (and obtaining it from "problem B") instead of modeling Vext and applying loads *F* at surface Q. This substitution of the boundary conditions is shown in Fig.3.

Soil-Structure Interaction 153

This boundary was far better than any elementary (fixed or free) boundary before. Horizontal body waves normal to the boundary did not have artificial reflection into Vint (that is why this boundary was called "non-reflecting boundary"). This boundary could work in the time domain. Besides, this boundary was "local": the response force acting to

This boundary was implemented in code LUSH [15], named after the authors (Lysmer, Udaka, Seed, Hwang). However, soon it turned out that the most important waves, radiating by structure in the soil underlain by rock are not horizontal body waves, but surface waves (see below). So, the artificial reflection at the boundary was not completely eliminated, and such a boundary could be placed only apart from the structure (3-4 plan

One should keep in mind the important limitation: when wave process in Vint is modeled by ordinary finite elements, the element size should be about 1/8 (1/5 at most [1]) of the wavelength. It means that one cannot increase the finite element size going away from the structure (as they often do in statics). So, the increase in Vint leads to the considerable

The next step was done by G.Waas [8,16,17]. Instead of replacing the infinite soil by number of horizontal rods he suggested to use "homogeneous solutions" – i.e. the waves in the horizontally-layered medium underlain by rigid rock. These waves can be evaluated in the frequency domain only. According to G.Waas, in 2D case each homogeneous displacement varies in the horizontal direction following complex exponential rule with a certain complex frequency-dependent "wave number"; in the vertical direction it varies according to the finite-element interpolation. Later it turned out that in 3D case cylindrical waves in Vext along horizontal radius followed not exponents, but certain special Hankel functions with

Any displacement of the lateral boundary (in the finite-element approximation) can be split into the sum of such waves in Vext. Then stresses are calculated at the boundary and finally nodal forces impacting Vint from Vext in response to boundary displacement can be evaluated. So, the final format of the Waas boundary was the dynamic stiffness matrix in the frequency domain, replacing Vext in the model. Unlike previous boundaries, this boundary was not local: response forces from Vext in the node depended not only on the motion of this very node, but on the motion of all the nodes. Thus, dynamic stiffness

New boundaries (they were called "transmitting boundaries") proved to be far more accurate than previous ones. They could be placed right near the basement, decreasing Vint and accelerating the analysis. So, LUSH was converted into FLUSH (=Fast LUSH) [18] for 2D problems. Then appeared code ALUSH (=Axisymmetric LUSH) [19] to solve 3D problems with axisymmetric geometry. However, there remained two important limitations: hard rock at the bottom and axisymmetric geometry of structure in 3D case. The next step forward was again done by J.Lysmer in EERC. But this new approach will be discussed later

the node depended on the velocities of this node only, not neighbors.

sizes from the basements) to get more or less reasonable results.

increase in the problem size and to the computational problems.

the same complex "wave numbers" as previously exponents in 2D case.

matrix became fully populated.

(see "combined method").

**Figure 3.** Substitution of the boundary conditions at the bottom of Vint in direct method

One should remember that such substitution is accurate for rigid rock only. Otherwise, an error occurs due to the reflection of the additional waves from such a boundary back into Vint. This error can be significant. The example will be shown below.

After the lower boundary is fixed, lateral boundaries (usually vertical) remain to be set. In 1970s, when direct method was popular, there was an evolution of these boundaries from "elementary" ones (fixed or free) towards more accurate ones. The first important step was so-called "acoustic boundaries" by Lysmer and Kuhlemeyer [13,14].

Physical base is as follows. 1D elastic (without material damping!) massive rod with shear or pressure waves can be cut in two, and one half of it can be accurately replaced by viscous dashpot. This is illustrated by Fig. 4.

Dashpot viscosity parameter is a product of mass density *ρ* and wave velocity *c*. So, Lysmer and Kuhlemeyer suggested to place distributed "shear" and "pressure" dashpots over vertical lateral boundary (three dashpots along three axes in each node). Practically they substituted the half-infinite layer in Fig.3 by number of horizontal 1D rods normal to the boundary.

**Figure 4.** Analogue between half-infinite rod of unit cross-section area and viscous dashpot with viscosity parameter b=ρc

This boundary was far better than any elementary (fixed or free) boundary before. Horizontal body waves normal to the boundary did not have artificial reflection into Vint (that is why this boundary was called "non-reflecting boundary"). This boundary could work in the time domain. Besides, this boundary was "local": the response force acting to the node depended on the velocities of this node only, not neighbors.

152 Earthquake Engineering

boundary conditions is shown in Fig.3.

dashpot. This is illustrated by Fig. 4.

viscosity parameter b=ρc

"problem A2", fixing the motion of the bottom (and obtaining it from "problem B") instead of modeling Vext and applying loads *F* at surface Q. This substitution of the

**Figure 3.** Substitution of the boundary conditions at the bottom of Vint in direct method

Vint. This error can be significant. The example will be shown below.

so-called "acoustic boundaries" by Lysmer and Kuhlemeyer [13,14].

One should remember that such substitution is accurate for rigid rock only. Otherwise, an error occurs due to the reflection of the additional waves from such a boundary back into

After the lower boundary is fixed, lateral boundaries (usually vertical) remain to be set. In 1970s, when direct method was popular, there was an evolution of these boundaries from "elementary" ones (fixed or free) towards more accurate ones. The first important step was

Physical base is as follows. 1D elastic (without material damping!) massive rod with shear or pressure waves can be cut in two, and one half of it can be accurately replaced by viscous

Dashpot viscosity parameter is a product of mass density *ρ* and wave velocity *c*. So, Lysmer and Kuhlemeyer suggested to place distributed "shear" and "pressure" dashpots over vertical lateral boundary (three dashpots along three axes in each node). Practically they substituted

the half-infinite layer in Fig.3 by number of horizontal 1D rods normal to the boundary.

**Figure 4.** Analogue between half-infinite rod of unit cross-section area and viscous dashpot with

This boundary was implemented in code LUSH [15], named after the authors (Lysmer, Udaka, Seed, Hwang). However, soon it turned out that the most important waves, radiating by structure in the soil underlain by rock are not horizontal body waves, but surface waves (see below). So, the artificial reflection at the boundary was not completely eliminated, and such a boundary could be placed only apart from the structure (3-4 plan sizes from the basements) to get more or less reasonable results.

One should keep in mind the important limitation: when wave process in Vint is modeled by ordinary finite elements, the element size should be about 1/8 (1/5 at most [1]) of the wavelength. It means that one cannot increase the finite element size going away from the structure (as they often do in statics). So, the increase in Vint leads to the considerable increase in the problem size and to the computational problems.

The next step was done by G.Waas [8,16,17]. Instead of replacing the infinite soil by number of horizontal rods he suggested to use "homogeneous solutions" – i.e. the waves in the horizontally-layered medium underlain by rigid rock. These waves can be evaluated in the frequency domain only. According to G.Waas, in 2D case each homogeneous displacement varies in the horizontal direction following complex exponential rule with a certain complex frequency-dependent "wave number"; in the vertical direction it varies according to the finite-element interpolation. Later it turned out that in 3D case cylindrical waves in Vext along horizontal radius followed not exponents, but certain special Hankel functions with the same complex "wave numbers" as previously exponents in 2D case.

Any displacement of the lateral boundary (in the finite-element approximation) can be split into the sum of such waves in Vext. Then stresses are calculated at the boundary and finally nodal forces impacting Vint from Vext in response to boundary displacement can be evaluated. So, the final format of the Waas boundary was the dynamic stiffness matrix in the frequency domain, replacing Vext in the model. Unlike previous boundaries, this boundary was not local: response forces from Vext in the node depended not only on the motion of this very node, but on the motion of all the nodes. Thus, dynamic stiffness matrix became fully populated.

New boundaries (they were called "transmitting boundaries") proved to be far more accurate than previous ones. They could be placed right near the basement, decreasing Vint and accelerating the analysis. So, LUSH was converted into FLUSH (=Fast LUSH) [18] for 2D problems. Then appeared code ALUSH (=Axisymmetric LUSH) [19] to solve 3D problems with axisymmetric geometry. However, there remained two important limitations: hard rock at the bottom and axisymmetric geometry of structure in 3D case. The next step forward was again done by J.Lysmer in EERC. But this new approach will be discussed later (see "combined method").

#### **5.2. Impedance approach**

If we presume the rigidity of the soil-structure contact surface and place surface Q in Fig.1 right at this surface, we get six degrees of freedom for Q. Corresponding forces *F* in "problem A2" are condensed to six–component integral forces, loading the immovable base mat during seismic excitation. In addition, when the mat moves, Vext impacts the moving base mat by response forces set by "dynamic stiffness matrix" 6 x 6. This matrix is a linear operators' matrix in the time domain for linear soil (soil in Vext was linear from the very beginning). In the frequency domain it becomes complex frequency dependent matrix called "impedance matrix". That is why the whole approach is called the "impedance" one. Impedances can be estimated in the field experiments with stamps excited by unbalanced rotors. They were the first parameters of the soil flexibility in the early times of SSI (when machinery base mats were analyzed).

Let *Rj(x)* be real "rigid" vector displacements field of the contact surface Q, when one of six general coordinates of the surface Q (coordinate number *j*) gets a unit displacement. If some contact vector forces *F(x)* act over Q, the condensed integral general force along coordinate *j* is given by

$$P\_j = \int\_Q R\_j^T(\mathbf{x}) \, F(\mathbf{x}) \, d\mathbf{Q} \tag{2}$$

Soil-Structure Interaction 155

(5)

*<sup>s</sup>* (6)

(8)

If *U0* is a control motion of a certain point in "problem B" (generally multi-component one, though usually only three-component one), then the condensed total forces (not just *F*!)

<sup>0</sup> () () () () () *<sup>b</sup> P BU CU*

Here *Ub(ω)* is a six-component motion of the rigid base mat (i.e., soil-structure contact surface Q), *C(ω)* is 6 x 6 impedance matrix, *B(ω)* is usually 6 x 3 seismic loading matrix.

The important particular case is a surface basement – then Vint goes and *Gint* in right-hand part of (3) goes as well. If in addition we presume *Uk(y)=Rk(y)=*1, *k*=1,2,3 (it means that the whole future contact surface Q in "problem B" has no seismic rotations and has the same translations as the control point; this is the case for vertically propagating seismic body waves in the horizontally-layered media), then matrix *B* is composed of the first three

Let *K(ω)* and *M* be stiffness and mass matrices of the structure without soil foundation (*K* may be complex to account for the structural internal damping), and let *Us(ω)* be absolute displacements of structure in the frequency domain. Then seismic motion in the soil-

<sup>2</sup> [() ] () () *K MU P*

Matrix [*K-ω2M*] is huge in size. Loads *P(ω)* are non-zero only for the rigid basement's six degrees of freedom. As displacements *Ub* take part in (5), and they are at the same time part

<sup>0</sup> [ ( ) ( )] ( ) ( ) ( ) *K MC U B U*

In (7) *C(ω)* has to be huge matrix of the same size as *K* and *M*, but only 6 x 6 block of it

Now let us imagine the same rigid contact surface on the same soil with the same seismic excitation, with the only difference: structure with rigid basement is weightless. The righthand part of (7) will stay in place; in the left-hand part only *C(ω)* represents dynamic stiffness, because weightless structure is moving rigidly together with the basement, and no forces occur due to *K*. Let *Vb(ω)* be the response of the rigid weightless basement (instead of

<sup>0</sup> () () () () *CV BU*

<sup>2</sup> [ ( ) ( )] ( ) ( ) ( ) *K MC U C V*

This equation is of great importance because it describes the so-called "platform" model, shown in Fig.5. Rigid platform is excited by a prescribed motion *Vb(ω)*. Structural model is resting on a "soil support" with prescribed stiffness described by the impedance matrix *C(ω).*

   

> 

 *s b* (9)

   

 

 *<sup>s</sup>* (7)

of *Us*, they should better join the left-hand part of (6), making (6) look like

2

 *b* 

Using (8) one can replace the right-hand part of (7) and come to the equation

  

acting from the soil to the basement are given by

structure system is described by the equation

columns of matrix *C*.

referring to *Ub* is populated.

*Ub(ω)* for massive structure:

Scalar product of two 3D vectors is used under the integral (2). If *G0* is the dynamic stiffness operator in the initial soil, and Q moves by unit along the general coordinate *k*, then the displacement field at Q is described by *Rk(x)*, and response forces impacting Q in the initial soil are given by *G0 Rk*. However, for the embedded Q these total forces consist of the part *Fk(x)* impacting Q from Vext and another part impacting Q from Vint (see Fig.2). If *Gint* is the analogue of *G0* for the finite volume Vint, then *Fk=(G0-Gint) Rk*. In the frequency domain all linear operators become matrices. The impedance *Cjk*, meaning the condensed force acting from Vext to Q along coordinate *j* in response to the unit rigid displacement of Q along coordinate *k* is then given by

$$\mathbf{C}\_{jk}(o\rho) = \int\_{\mathcal{Q}\_x} \int\_{\mathcal{Q}\_y} R\_j^T(\mathbf{x}) \left[ \mathbf{G}\_0(o\rho) - \mathbf{G}\_{\text{int}}(o\rho) \right] R\_k(y) \, d\mathbf{Q}\_x d\mathbf{Q}\_y \tag{3}$$

Double integration in (3) reflects the fact that *G* provides the response forces in the node *x* due to the displacements in the node y.

The next step is to condense load *F* over rigid surface Q. If *Uk(x)* is a transfer function from the control motion along coordinate *k* to the free-field wave in "problem B", then in the frequency domain the transfer function *Bjk* to the condensed load along coordinate *j* according to (1) is given by

$$B\_{jk}(oo) = \int\limits\_{V\_x \ V\_y} R\_j^T(\infty) \left[ G\_0(oo) \right] \mathcal{U}\_k(y) \, dV\_x dV\_y \tag{4}$$

If *U0* is a control motion of a certain point in "problem B" (generally multi-component one, though usually only three-component one), then the condensed total forces (not just *F*!) acting from the soil to the basement are given by

154 Earthquake Engineering

is given by

**5.2. Impedance approach** 

machinery base mats were analyzed).

coordinate *k* is then given by

according to (1) is given by

If we presume the rigidity of the soil-structure contact surface and place surface Q in Fig.1 right at this surface, we get six degrees of freedom for Q. Corresponding forces *F* in "problem A2" are condensed to six–component integral forces, loading the immovable base mat during seismic excitation. In addition, when the mat moves, Vext impacts the moving base mat by response forces set by "dynamic stiffness matrix" 6 x 6. This matrix is a linear operators' matrix in the time domain for linear soil (soil in Vext was linear from the very beginning). In the frequency domain it becomes complex frequency dependent matrix called "impedance matrix". That is why the whole approach is called the "impedance" one. Impedances can be estimated in the field experiments with stamps excited by unbalanced rotors. They were the first parameters of the soil flexibility in the early times of SSI (when

Let *Rj(x)* be real "rigid" vector displacements field of the contact surface Q, when one of six general coordinates of the surface Q (coordinate number *j*) gets a unit displacement. If some contact vector forces *F(x)* act over Q, the condensed integral general force along coordinate *j*

() () *<sup>T</sup>*

Scalar product of two 3D vectors is used under the integral (2). If *G0* is the dynamic stiffness operator in the initial soil, and Q moves by unit along the general coordinate *k*, then the displacement field at Q is described by *Rk(x)*, and response forces impacting Q in the initial soil are given by *G0 Rk*. However, for the embedded Q these total forces consist of the part *Fk(x)* impacting Q from Vext and another part impacting Q from Vint (see Fig.2). If *Gint* is the analogue of *G0* for the finite volume Vint, then *Fk=(G0-Gint) Rk*. In the frequency domain all linear operators become matrices. The impedance *Cjk*, meaning the condensed force acting from Vext to Q along coordinate *j* in response to the unit rigid displacement of Q along

0 int ( ) ( ) [ ( ) ( )] ( )

 

*C R xG G Ry*

*jk j k xy*

Double integration in (3) reflects the fact that *G* provides the response forces in the node *x*

The next step is to condense load *F* over rigid surface Q. If *Uk(x)* is a transfer function from the control motion along coordinate *k* to the free-field wave in "problem B", then in the frequency domain the transfer function *Bjk* to the condensed load along coordinate *j*

<sup>0</sup> ( ) ( ) [ ( )] ( )

*jk j k xy*

 

*R x G U y dV dV* (4)

*T*

*x y*

*Q Q*

*B* 

due to the displacements in the node y.

*T*

*x y*

*V V*

*P R x F x dQ* (2)

*dQ dQ* (3)

*j j Q*

$$P(o) = B(o) \cup L\_0(o) - C(o) \cup L\_b(o) \tag{5}$$

Here *Ub(ω)* is a six-component motion of the rigid base mat (i.e., soil-structure contact surface Q), *C(ω)* is 6 x 6 impedance matrix, *B(ω)* is usually 6 x 3 seismic loading matrix.

The important particular case is a surface basement – then Vint goes and *Gint* in right-hand part of (3) goes as well. If in addition we presume *Uk(y)=Rk(y)=*1, *k*=1,2,3 (it means that the whole future contact surface Q in "problem B" has no seismic rotations and has the same translations as the control point; this is the case for vertically propagating seismic body waves in the horizontally-layered media), then matrix *B* is composed of the first three columns of matrix *C*.

Let *K(ω)* and *M* be stiffness and mass matrices of the structure without soil foundation (*K* may be complex to account for the structural internal damping), and let *Us(ω)* be absolute displacements of structure in the frequency domain. Then seismic motion in the soilstructure system is described by the equation

$$\left[\left(K(\phi) - \phi^2 M\right)\mathcal{U}\_s(\phi) = P(\phi)\right] \tag{6}$$

Matrix [*K-ω2M*] is huge in size. Loads *P(ω)* are non-zero only for the rigid basement's six degrees of freedom. As displacements *Ub* take part in (5), and they are at the same time part of *Us*, they should better join the left-hand part of (6), making (6) look like

$$\left[\left[K(\alpha) - \alpha^2 M + \mathbb{C}(\alpha)\right] \mathbb{U}\_s(\alpha) = \mathbb{B}(\alpha) \,\mathbb{U}\_0(\alpha)\right] \tag{7}$$

In (7) *C(ω)* has to be huge matrix of the same size as *K* and *M*, but only 6 x 6 block of it referring to *Ub* is populated.

Now let us imagine the same rigid contact surface on the same soil with the same seismic excitation, with the only difference: structure with rigid basement is weightless. The righthand part of (7) will stay in place; in the left-hand part only *C(ω)* represents dynamic stiffness, because weightless structure is moving rigidly together with the basement, and no forces occur due to *K*. Let *Vb(ω)* be the response of the rigid weightless basement (instead of *Ub(ω)* for massive structure:

$$\mathcal{C}(o)\,\,V\_{\flat}(o) = \mathcal{B}(o)\,\,\mathcal{U}\_0(o)\tag{8}$$

Using (8) one can replace the right-hand part of (7) and come to the equation

$$\left[\left[K(\rho) - \alpha^2 M + \mathbb{C}(\rho)\right] \mathbb{U}\_s(\rho) = \mathbb{C}(\rho) \right] V\_b(\rho) \tag{9}$$

This equation is of great importance because it describes the so-called "platform" model, shown in Fig.5. Rigid platform is excited by a prescribed motion *Vb(ω)*. Structural model is resting on a "soil support" with prescribed stiffness described by the impedance matrix *C(ω).*

Soil-Structure Interaction 157

However, even for the surface basements the second assumption is not always correct. If the free-field motion *U0* of the soil surface is not "rigid" over Q, the rigid basement's response is somehow averaging it over the mat; in this case the translational response *Vb* may be less than control motion *U0* in a single node. On the other hand, the variability of horizontal *U0* under the mat may cause torsion in *Vb*, and the variability of vertical *U0* under the mat may

The second part of the problem (i.e. the dynamic analysis of the platform model from Fig.5) is called "inertial soil-structure interaction problem". Often they solve it putting the coordinate system on the platform (thus, coming to the fixed platform, but instead

One can easily see that conventional structural seismic response without SSI is obtained from the platform model as well. The difference with Fig.5 is only in the soil support and in the platform excitation. If we make soil stiffer and stiffer, *Vb* comes closer and closer to *U0*; "soil support" becomes stiff, and this difference also goes. So, we get a model without SSI as

Platform model can be extended to the case of several buildings having common soil foundation. The main limitations are the same – rigid basements and linear foundation/contact. Note that generally (with "kinematic interaction" on Fig.5) no requirements are made for soil layering, embedment depth, shape of the underground

The alternative form of the platform model uses fixed platform, and dynamic loads are directly applied to the rigid basement. These loads correspond to the right-hand parts of (9),

Platform model is convenient to work with in the frequency domain. Huge matrix [*K-ω2M*] can be condensed to the rigid base mat using natural modes and natural frequencies of the fixed-base structure [20]. As a result, we get *M(ω)* – the "dynamic inertia" complex

<sup>0</sup> [ ( ) ( )] ( ) ( ) ( ) *C M U BU*

It is easily solved, because maximal size is 6. The response in the time domain is further obtained using Fast Fourier Transform (FFT). For multiple-base structure the dynamic stiffness is condensed in a somewhat different way [21], though the size of the resulting

However, usually engineers prefer the time domain for the dynamic problems. Platform model of Fig.5 and equation (9) can be transferred to the time domain. Structural part is transferred easily. Kinematic excitation *Vb* can be also transferred to the time domain using

The most popular variant is just a set of six springs and six viscous dashpots (one spring and one dashpot along each of six coordinates). In the frequency domain an ordinary spring

   

 *<sup>b</sup>* (10)

frequency-dependent matrix 6 x 6 – similar in size to *C(ω)*. Equation (7) turns to

FFT. The only problem is to transfer to the time domain impedance matrix *C(ω)*.

2

matrix is still 6 x (number of base mats).

introducing inertial forces impacting all the masses of structure).

a limit case of the platform model with SSI with extremely hard soil.

cause rocking in *Vb*.

part, etc.

(8) or (7).

**Figure 5.** Scheme of platform model for SSI analysis

One should clearly understand that this platform is not physical: in the real world there is no rigid platform having such a motion. Often they try to imagine some hard rock somewhere – there is a mistake! This model is applicable, for example, for homogeneous half-space – no rock is present anywhere. In fact, platform model is just a mechanical analogue of the soil-structure model in terms of the structural response.

Let us work a little with this platform model. In the popular particular case with two assumptions mentioned above (i.e. surface rigid basement and vertical seismic waves in the horizontally layered medium) weightless rigid base mat will move exactly with "control" accelerations from free field. In other words *Vb=U0*. This case is shown in Fig.6.

**Figure 6.** Scheme of platform model for SSI analysis with two assumptions: rigid surface basement and "rigid" free-field motion under it

If any of the two assumptions is not valid, one has to solve a special problem to obtain *Vb* – it is called "kinematic soil-structure interaction problem". For embedded basements it is allowed to neglect the embedment depth if it is less then 30% of the effective radius of the mat. In this case upper layers of the soil foundation are just withdrawn from the soil model. This may sometimes cause some changes in the free-field motion.

However, even for the surface basements the second assumption is not always correct. If the free-field motion *U0* of the soil surface is not "rigid" over Q, the rigid basement's response is somehow averaging it over the mat; in this case the translational response *Vb* may be less than control motion *U0* in a single node. On the other hand, the variability of horizontal *U0* under the mat may cause torsion in *Vb*, and the variability of vertical *U0* under the mat may cause rocking in *Vb*.

156 Earthquake Engineering

**Figure 5.** Scheme of platform model for SSI analysis

"rigid" free-field motion under it

One should clearly understand that this platform is not physical: in the real world there is no rigid platform having such a motion. Often they try to imagine some hard rock somewhere – there is a mistake! This model is applicable, for example, for homogeneous half-space – no rock is present anywhere. In fact, platform model is just a mechanical

*Weightless rigid basement with* 

*seismic response Vb*

*C(ω)* 

*C(ω)* 

*u0*

*Vb*

Let us work a little with this platform model. In the popular particular case with two assumptions mentioned above (i.e. surface rigid basement and vertical seismic waves in the horizontally layered medium) weightless rigid base mat will move exactly with "control"

**Figure 6.** Scheme of platform model for SSI analysis with two assumptions: rigid surface basement and

If any of the two assumptions is not valid, one has to solve a special problem to obtain *Vb* – it is called "kinematic soil-structure interaction problem". For embedded basements it is allowed to neglect the embedment depth if it is less then 30% of the effective radius of the mat. In this case upper layers of the soil foundation are just withdrawn from the soil model.

This may sometimes cause some changes in the free-field motion.

analogue of the soil-structure model in terms of the structural response.

accelerations from free field. In other words *Vb=U0*. This case is shown in Fig.6.

*u0*

The second part of the problem (i.e. the dynamic analysis of the platform model from Fig.5) is called "inertial soil-structure interaction problem". Often they solve it putting the coordinate system on the platform (thus, coming to the fixed platform, but instead introducing inertial forces impacting all the masses of structure).

One can easily see that conventional structural seismic response without SSI is obtained from the platform model as well. The difference with Fig.5 is only in the soil support and in the platform excitation. If we make soil stiffer and stiffer, *Vb* comes closer and closer to *U0*; "soil support" becomes stiff, and this difference also goes. So, we get a model without SSI as a limit case of the platform model with SSI with extremely hard soil.

Platform model can be extended to the case of several buildings having common soil foundation. The main limitations are the same – rigid basements and linear foundation/contact. Note that generally (with "kinematic interaction" on Fig.5) no requirements are made for soil layering, embedment depth, shape of the underground part, etc.

The alternative form of the platform model uses fixed platform, and dynamic loads are directly applied to the rigid basement. These loads correspond to the right-hand parts of (9), (8) or (7).

Platform model is convenient to work with in the frequency domain. Huge matrix [*K-ω2M*] can be condensed to the rigid base mat using natural modes and natural frequencies of the fixed-base structure [20]. As a result, we get *M(ω)* – the "dynamic inertia" complex frequency-dependent matrix 6 x 6 – similar in size to *C(ω)*. Equation (7) turns to

$$\left[\left[\mathbf{C}(o) - \alpha^2 M(o)\right] \mathbf{U}\_b(o) = \mathbf{B}(o) \,\mathbf{U}\_0(o) \right.\tag{10}$$

It is easily solved, because maximal size is 6. The response in the time domain is further obtained using Fast Fourier Transform (FFT). For multiple-base structure the dynamic stiffness is condensed in a somewhat different way [21], though the size of the resulting matrix is still 6 x (number of base mats).

However, usually engineers prefer the time domain for the dynamic problems. Platform model of Fig.5 and equation (9) can be transferred to the time domain. Structural part is transferred easily. Kinematic excitation *Vb* can be also transferred to the time domain using FFT. The only problem is to transfer to the time domain impedance matrix *C(ω)*.

The most popular variant is just a set of six springs and six viscous dashpots (one spring and one dashpot along each of six coordinates). In the frequency domain an ordinary spring corresponds to the real frequency-independent impedance, and viscous dashpot corresponds to the frequency-linear purely imaginary impedance. So, matrix *C(ω)* corresponding to such a set of springs/dashpots will be diagonal complex matrix 6 x 6 with frequency-independent real parts and frequency-linear imaginary parts. Is it realistic for real-world soil foundations? This question deserves special discussion. But before we enter it, let us consider a very simple example, illustrating methodology of SSI problem as a whole and two basic different approaches to this problem (i.e., direct and impedance ones) described above.
