**6. Sample 1D SSI problem as an example of different approaches to SSI analysis**

Let us consider 1D P-waves in a homogeneous massive rod, modeling soil. The only coordinate is *x*. Wave displacements *u(x,t)* are described by the wave equation

$$c^2 \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} \tag{11}$$

Soil-Structure Interaction 159

  (16)

0

*x*

are doubled:

"control motion" *U0*.

"problem A" are shown on Fig.7.

**Figure 7.** Layouts of "problem B" and "problem A"

Equation of motion for mass *m* in the frequency domain is

upcoming wave

2 0

0 20 2

0 0 00 1 2 (0) ( / )( ) 0 *<sup>u</sup> E Ei U U*

00 00 0 0 0 1 2 *U U U U u x U ix ix U x* / 2; / 2; ( ) [exp( / ) exp( / )] / 2 cos( / )

Two equations (14) and (15) give us the well-known "doubling rule": the upcoming wave reaching free surface reflects back. At the free surface displacements of the upcoming wave

So, (16) gives the whole solution of the "problem B" linking wave field in any point to the

Now let us move to the "problem A". Let structure be rigid and have mass *m* (this is a mass, related to the unit area of the cross-section of the rod). Schemes of "problem B" and

General goal of the SSI analysis is to obtain the response motion of the basement from the given control motion of the free surface. In this simple case we can do it directly (it is not the "direct approach" so far, just a simple solution!). In "problem A" wave field in the soil is still described by (12), but with coefficients *U<sup>А</sup>1* (upcoming wave) and *U<sup>А</sup>2* (wave coming down):

Problem *В* Problem *А*

1 2 ( ) exp( / ) exp( / ) *A A <sup>A</sup> u x U ix U ix* 

1 2 1 2 ( ) [( / ) ( / ) ] *<sup>A</sup> A AA*

Having (18) one can express the reflected wave amplitude from the amplitude of the

2 1 <sup>1</sup> [ / ] / [ / ] [1 ] / [1 ] *A A <sup>A</sup> m m U U iE m iE m U i i*

 

*mU U E i U i U*

 

(19)

 

(17)

(18)

(15)

Here *с* is wave velocity. In the frequency domain for certain frequency *ω* there exist two solutions of (11):

$$\mu\_1 = \mathcal{U}\_1 \exp(-i\mathbf{x} \text{ / } \mathcal{X}); \quad \mu\_2 = \mathcal{U}\_2 \exp(i\mathbf{x} \text{ / } \mathcal{X}); \quad \mathcal{X} = \mathbf{c} \text{ / } \mathcal{O} \tag{12}$$

The first wave *u1* described by (12) goes up along Ох, the second wave *u2* goes down. Each wave has own amplitude *U*. Wave velocity *c* is determined by mass density *ρ* and constrained elasticity module *Е0* as

$$c^2 = \mathbb{E}^0 \mid \rho \tag{13}$$

Let us now apply basic principles discussed above to this simple model: solve "problem B", "problem A", and then solve "problem A2" by direct and impedance methods. All problems will be solved in the frequency domain, using (12).

#### **6.1. The exact solution**

Let us start with "problem B" – wave solution without structure. Let *U0* be displacement at the free surface *x=0*. As we are going to obtain coefficient *U10* of the upcoming wave and coefficient *U20* of the wave coming down, now we have the first of the two equations for them

$$
\mu^0(\mathbf{0}) = \mathcal{U}\_1^0 + \mathcal{U}\_2^0 = \mathcal{U}^0 \tag{14}
$$

The second equation comes from the description of the "free" condition of the surface; total stress in the soil must be zero:

Soil-Structure Interaction 159

$$E^{0}\frac{\partial \boldsymbol{u}^{0}}{\partial \boldsymbol{\chi}}(\mathbf{0}) = E^{0}(\boldsymbol{i} \,/\ \mathcal{A})(-\boldsymbol{\mathcal{U}}^{0}\_{1} + \boldsymbol{\mathcal{U}}^{0}\_{2}) = \mathbf{0} \tag{15}$$

Two equations (14) and (15) give us the well-known "doubling rule": the upcoming wave reaching free surface reflects back. At the free surface displacements of the upcoming wave are doubled:

$$\text{LI}\_1^0 = \text{LI}^0 / 2; \quad \text{LI}\_2^0 = \text{LI}^0 / 2; \quad u^0(\mathbf{x}) = \text{LI}^0[\exp(-i\mathbf{x} \,/\, \lambda) + \exp(i\mathbf{x} \,/\, \lambda)] / 2 = \text{LI}^0 \cos(\mathbf{x} \,/\, \lambda) \tag{16}$$

So, (16) gives the whole solution of the "problem B" linking wave field in any point to the "control motion" *U0*.

Now let us move to the "problem A". Let structure be rigid and have mass *m* (this is a mass, related to the unit area of the cross-section of the rod). Schemes of "problem B" and "problem A" are shown on Fig.7.

**Figure 7.** Layouts of "problem B" and "problem A"

158 Earthquake Engineering

described above.

solutions of (11):

constrained elasticity module *Е0* as

**6.1. The exact solution** 

stress in the soil must be zero:

them

will be solved in the frequency domain, using (12).

**analysis** 

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)

**6. Sample 1D SSI problem as an example of different approaches to SSI** 

Let us consider 1D P-waves in a homogeneous massive rod, modeling soil. The only

2 2

*x t*

Here *с* is wave velocity. In the frequency domain for certain frequency *ω* there exist two

1 1 2 2 *u U ix u U ix c* exp( / ); exp( / ); /

2 0 *c E* /

Let us now apply basic principles discussed above to this simple model: solve "problem B", "problem A", and then solve "problem A2" by direct and impedance methods. All problems

Let us start with "problem B" – wave solution without structure. Let *U0* be displacement at the free surface *x=0*. As we are going to obtain coefficient *U10* of the upcoming wave and coefficient *U20* of the wave coming down, now we have the first of the two equations for

0 000

The second equation comes from the description of the "free" condition of the surface; total

The first wave *u1* described by (12) goes up along Ох, the second wave *u2* goes down. Each wave has own amplitude *U*. Wave velocity *c* is determined by mass density *ρ* and

2 2 *u u*

(11)

 

(13)

(12)

1 2 *u UUU* (0) (14)

coordinate is *x*. Wave displacements *u(x,t)* are described by the wave equation

2

*c*

> General goal of the SSI analysis is to obtain the response motion of the basement from the given control motion of the free surface. In this simple case we can do it directly (it is not the "direct approach" so far, just a simple solution!). In "problem A" wave field in the soil is still described by (12), but with coefficients *U<sup>А</sup>1* (upcoming wave) and *U<sup>А</sup>2* (wave coming down):

$$\mu^A(\mathbf{x}) = \mathcal{U}\_1^A \exp(-i\mathbf{x}/\lambda) + \mathcal{U}\_2^A \exp(i\mathbf{x}/\lambda) \tag{17}$$

Equation of motion for mass *m* in the frequency domain is

$$-\alpha^2 m \left( \mathcal{U}\_1^A + \mathcal{U}\_2^A \right) = -E^0 \left[ \left( -i \;/\; \lambda \right) \mathcal{U}\_1^A + \left( i \;/\; \lambda \right) \mathcal{U}\_2^A \right] \tag{18}$$

Having (18) one can express the reflected wave amplitude from the amplitude of the upcoming wave

$$\mathrm{L}\mathbf{U}\_{2}^{A} = \mathrm{L}\mathbf{U}\_{1}^{A}[\mathrm{i}\mathbb{E}^{0}\mid\mathbb{X}+o^{2}\mathrm{m}]/[\mathrm{i}\mathbb{E}^{0}\mid\mathbb{X}-o^{2}\mathrm{m}] = \mathrm{L}\mathbf{U}\_{1}^{A}[1-\mathrm{i}\frac{\mathrm{m}}{\rho\mathbb{X}}]/[1+\mathrm{i}\frac{\mathrm{m}}{\rho\mathbb{X}}] \tag{19}$$

"Problem A" and "problem B" have similar seismic excitation: in our case it means that the upcoming waves are similar:

$$\mathcal{U}\mathcal{U}\_1^A = \mathcal{U}\_1^0 = \mathcal{U}^0 / \mathcal{Z} \tag{20}$$

Soil-Structure Interaction 161

The same load *F* will impact Q in "problem В2" and in "problem А2". To complete the model we use the analogue between half-infinite rod and viscous dashpot, mentioned above

As a result, complete models for the direct approach look now like those shown on Fig.8

As previously mentioned it is worth solving "problem B2" first and compare the solution with free-field. Instead of that we can take "exact" displacement from exact wave field at Q, calculate total force impacting the upper part in the model on Fig.8 and check the equilibrium for Q. In our case the force impacting the upper part from dashpot is a product of exact velocity [*iωU0(-H)*] and viscous parameter of dashpot *(ρc)*; we see that it annihilates with *Fext* given by (25). The rest of the load *F* – the load *Fint* – annihilates with internal forces

Now let us check the equilibrium for Q in "problem A2". The exact solution is given by

<sup>0</sup> ( ) {exp( / ) [1 ] / [1 ]exp( / )} / 2 *<sup>A</sup> m m U x U ix i i ix*

So, Q is loaded by a) force from dashpot, b) internal force, acting from the upper part (not *Fint* any more, because in the upper part wave field has changed!) and c) the external load *F*

*<sup>A</sup> <sup>U</sup> i cU H E H F*

Let a reader obtain zero in (28) himself. General conclusion is that direct method provides

Now let us perform the substitution of the boundary conditions in the direct approach, shown on Fig.3, i.e. let us fix the motion of Q as *U0(-H)*. We see that "problem B2" without

<sup>0</sup> () () 0 *A*

*x*

 

(27)

 

*F -H*

*0*

*x*

(28)

in the upper part due to the origin (23). So, "problem B2" gives the exact solution.

(remember boundaries of Lysmer-Kuhlemeyer) and shown on Fig.4.

**Figure 8.** Direct approach models for "problem B2" and "problem A2"

*F -H*

*0*

*x*

(17,19,20):

given by (26):

exact results, if correctly applied.

Hence we get the final expression linking the displacement of the mass in "problem A" to the control motion at the free surface in "problem B":

$$\mathcal{U}\mathcal{U}^A = \mathcal{U}^0 \{ 1 + \{1 - i\frac{m}{\rho \lambda}\} / \{1 + i\frac{m}{\rho \lambda}\} \} / \mathcal{Z} = \mathcal{U}^0 \/ \{1 + i\frac{m\alpha}{\rho \mathcal{c}}\} \tag{21}$$

The transfer function in the frequency domain linking response displacements to the control displacements in (21) at the same time links response accelerations to the control accelerations. Having control accelerogram *a0(t)* one can get response accelerogram *aA(t)* easily using FFT technique.

#### **6.2. Direct approach**

Now we will show the direct approach for the same example, following p.4.1. Let surface Q be at depth *Н*. There the free-field wave according to (16) will be

$$
\mu^0(-H) = \text{l}\boldsymbol{\Gamma}^0 \cos(\boldsymbol{H} \,/\, \lambda); \quad \frac{\partial \boldsymbol{u}^0}{\partial \boldsymbol{\chi}}(-H) = \text{(l}\boldsymbol{\Gamma}^0 \,/\, \lambda)\sin(\boldsymbol{H} \,/\, \lambda) \tag{22}
$$

Loads *(-F)* may be obtained from "problem B1" as shown on Fig.2. In the upper part the resulting sum must be zero; so, the part *Fint* of the load, balancing the reflected wave in the upper part of the soil, is just "mirror" of the free field:

$$F\_{\rm int} = -E^0 \frac{\partial \mathcal{u}^0}{\partial \mathbf{x}} (-H) = -(E^0 \mathcal{U}^0 \,/\, \mathcal{\lambda}) \sin(H \,/\, \mathcal{\lambda}) = -\rho \rho \alpha \, \mathcal{U}^0 \sin(H \,/\, \mathcal{\lambda}) \tag{23}$$

The reflected wave *u1* in the lower part of the rod consists just of the single wave coming down. We know the displacement at Q; so, we can describe the additional wave in the lower part as

$$u^1(\mathbf{x}) = -\mathbf{U}^0 \cos(\mathbf{H} \,/\, \lambda) \exp[i(\mathbf{x} + \mathbf{H}) / \, \lambda]; \quad \frac{\partial u^1}{\partial \mathbf{x}}(\mathbf{x}) = -(i\mathbf{U}^0 \,/\, \lambda) \cos(\mathbf{H} \,/\, \lambda) \exp[i(\mathbf{x} + \mathbf{H}) / \, \lambda] \tag{24}$$

In this wave field at *x=-H* we get the following force *Fext*, impacting Q from the lower part of the soil

$$F\_{\rm ext} = \text{E}^0(\text{i}\text{L}^0/\text{ }\text{}\text{/}\text{\textmathcal})\cos(\text{H}/\text{\textdegree\text0}) = \text{i}\text{L}^0(\rho\,\text{c}^2\text{o}\text{/}\text{\textc})\cos(\text{H}/\text{\textdegree\text0}) = (\text{i}\,\rho\,\text{c})\text{L}^0\cos(\text{H}/\text{\textdegree\text0})\tag{25}$$

Thus, the total load *F* in "problem B1" is composed of two parts, given by (23) and (25):

$$F = F\_{\rm int} + F\_{\rm ext} = i\alpha\rho\sigma c\mathcal{U}^0 [\cos(H/\,\lambda) + i\sin(H/\,\lambda)] = i\alpha\rho\sigma c\mathcal{U}^0 \exp(iH/\,\lambda) \tag{26}$$

The same load *F* will impact Q in "problem В2" and in "problem А2". To complete the model we use the analogue between half-infinite rod and viscous dashpot, mentioned above (remember boundaries of Lysmer-Kuhlemeyer) and shown on Fig.4.

As a result, complete models for the direct approach look now like those shown on Fig.8

**Figure 8.** Direct approach models for "problem B2" and "problem A2"

160 Earthquake Engineering

upcoming waves are similar:

easily using FFT technique.

**6.2. Direct approach** 

part as

the soil

the control motion at the free surface in "problem B":

"Problem A" and "problem B" have similar seismic excitation: in our case it means that the

Hence we get the final expression linking the displacement of the mass in "problem A" to

0 0 {1 [1 ] / [1 ]} / 2 / [1 ] *<sup>A</sup> mm m UU i i U i*

be at depth *Н*. There the free-field wave according to (16) will be

upper part of the soil, is just "mirror" of the free field:

0

*x*

 0 0

 

The transfer function in the frequency domain linking response displacements to the control displacements in (21) at the same time links response accelerations to the control accelerations. Having control accelerogram *a0(t)* one can get response accelerogram *aA(t)*

Now we will show the direct approach for the same example, following p.4.1. Let surface Q

0 0 0 <sup>0</sup> ( ) cos( / ); ( ) ( / )sin( / ) *<sup>u</sup> uHU H H U H x*

Loads *(-F)* may be obtained from "problem B1" as shown on Fig.2. In the upper part the resulting sum must be zero; so, the part *Fint* of the load, balancing the reflected wave in the

> 0 0 0 0 int ( ) ( / )sin( / ) sin( / ) *<sup>u</sup> F E H E U H cU H*

The reflected wave *u1* in the lower part of the rod consists just of the single wave coming down. We know the displacement at Q; so, we can describe the additional wave in the lower

1 1 0 <sup>0</sup> ( ) cos( / )exp[ ( ) / ]; ( ) ( / )cos( / )exp[ ( ) / ] *<sup>u</sup> u x U H ix H x iU H i x H x*

In this wave field at *x=-H* we get the following force *Fext*, impacting Q from the lower part of

0 0 0 2 <sup>0</sup> ( / )cos( / ) ( / ) cos( / ) ( ) cos( / ) *ext F E iU H iU c c H i c U H*

int [cos( / ) sin( / )] exp( / ) *ext F F F i cU H i H i cU iH*

0 0

 

Thus, the total load *F* in "problem B1" is composed of two parts, given by (23) and (25):

 

 

1 1 / <sup>2</sup> *<sup>A</sup> U UU* (20)

(21)

(22)

(23)

 

 

 

*c* 

> 

 

(25)

(26)

  (24)

As previously mentioned it is worth solving "problem B2" first and compare the solution with free-field. Instead of that we can take "exact" displacement from exact wave field at Q, calculate total force impacting the upper part in the model on Fig.8 and check the equilibrium for Q. In our case the force impacting the upper part from dashpot is a product of exact velocity [*iωU0(-H)*] and viscous parameter of dashpot *(ρc)*; we see that it annihilates with *Fext* given by (25). The rest of the load *F* – the load *Fint* – annihilates with internal forces in the upper part due to the origin (23). So, "problem B2" gives the exact solution.

Now let us check the equilibrium for Q in "problem A2". The exact solution is given by (17,19,20):

$$\mathcal{U}^{A}(\mathbf{x}) = \mathcal{U}^{0}\{\exp(-i\mathbf{x}\,/\ \lambda) + [1 - i\frac{m}{\rho\lambda}] / [1 + i\frac{m}{\rho\lambda}] \exp(i\mathbf{x}\,/\ \lambda)\}/2\tag{27}$$

So, Q is loaded by a) force from dashpot, b) internal force, acting from the upper part (not *Fint* any more, because in the upper part wave field has changed!) and c) the external load *F* given by (26):

$$-i o \rho c \mathcal{U}^A (-H) + E^0 \frac{\partial \mathcal{U}^A}{\partial \mathbf{x}} (-H) + F = 0 \tag{28}$$

Let a reader obtain zero in (28) himself. General conclusion is that direct method provides exact results, if correctly applied.

Now let us perform the substitution of the boundary conditions in the direct approach, shown on Fig.3, i.e. let us fix the motion of Q as *U0(-H)*. We see that "problem B2" without

structure will be solved exactly, providing *U0(x)* in the whole upper part. So, this check is OK. However, in "problem A2" exact displacement *UA(-H)* given by (27) is different from *U0(-H)* given by (22). As a result, both upcoming wave and the wave coming down in Vint will be different from the exact solution. If this new solution is marked with upper index "3", then the displacement of the mass is given by

$$\text{L}\,\text{L}^3 = \text{L}\,\text{L}^3 + \text{L}\,\text{L}^3 = \text{L}\,\text{L}^0 / \left[\cos(H/\lambda) - \frac{m}{\rho\lambda}\sin(H/\lambda)\right] \tag{29}$$

Note that this solution depends on *H*, while the exact solution does not (see (21)). As our boundary is artificial, such dependence cannot be physical.

We can calculate the "error coefficient" relating approximate solution (29) to the exact solution (21). With dimensionless frequency

$$
\rho \varpi = \frac{\alpha \ m}{\rho \text{ c}} = \frac{m}{\rho \text{\AA}} \tag{30}
$$

Soil-Structure Interaction 163

h=1

h=2

h=3

**Figure 9.** "Error coefficient" for direct approach after changing the lower boundary condition

exact solution.

0

1

2

3

4

5

6

7

8

(dashpot in our case).

General conclusion is that the correct application of the impedance method provides the

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2 **Безразмерная частота**

The decisive factor in getting the exact solution both in direct and in impedance approaches for our 1D model was the ability to get exact model for substituting Vext

Unfortunately, this example is of methodology value only. While "problem B" (without structure) is often treated in practice with 1D models (though not homogeneous), "problem A" (with structure) is principally different. Great part of energy is taken from the structure

Historically, the first problems for impedances considering soil inertia were solved semianalytically for homogeneous half-space without the internal damping and for circular surface stamps. It turned out that horizontal impedances behaved more or less like pairs of springs and viscous dashpots (as mentioned above, in reality the dissipation of energy had completely different nature; mechanical energy was not converted in heat, as in dashpot, but taken away by elastic waves). One more "good news" was that the impedance matrix appeared to be almost diagonal. Though non-diagonal terms coupling horizontal translation with rocking in the same vertical plane were non-zero ones, their squares were considerably less in module than products of the corresponding diagonal terms of the impedance matrix.

by surface waves, which are not represented in 1D model.

**7. Impedances in the frequency domain** 

and dimensionless depth of Q

$$h = H \frac{\rho}{m} \tag{31}$$

this "error coefficient" makes

$$\mu = \frac{\mathcal{U}^3}{\mathcal{U}^A} = \frac{1 + i\sigma}{\cos(h\sigma) - \sigma\sin(h\sigma)}\tag{32}$$

Curves for different *µ(ω)* for different *h* are shown in Fig.9.

We see that the solution in not satisfactory. The increase of the boundary depth *h* does not improve the situation. General conclusion is that one must be very careful in placing lower boundary in the direct approach when there is no rock seen in the depth.

#### **6.3. Impedance approach**

Now let us apply the impedance approach to the same system. As our structure rests on the surface, we can use Fig.6. If the displacement of the mass is *Ub*, then the equation of motion (10) turns to

$$\mathbb{C}\left[\mathbb{C}(o) - o^{2}m\right] \mathbb{U}\_{b}(o) = \mathbb{C}(o) \wr \mathbb{U}^{0}(o) \tag{33}$$

Impedance is the same as in Fig.4: *C(ω)=iωρc*. So, from (33) we at once get the ultimate result which turns to be similar to the exact one (21):

$$\mathcal{U}\mathcal{U}\_b = \mathcal{U}^0 \;/\left[1 - \frac{\alpha \, m}{\mathrm{i}\rho \, \mathrm{c}}\right] = \mathcal{U}^0 \;/\left[1 + \mathrm{i}\, \frac{m}{\rho \, \mathrm{i}}\right] = \mathcal{U}^A \tag{34}$$

**Figure 9.** "Error coefficient" for direct approach after changing the lower boundary condition

General conclusion is that the correct application of the impedance method provides the exact solution.

The decisive factor in getting the exact solution both in direct and in impedance approaches for our 1D model was the ability to get exact model for substituting Vext (dashpot in our case).

Unfortunately, this example is of methodology value only. While "problem B" (without structure) is often treated in practice with 1D models (though not homogeneous), "problem A" (with structure) is principally different. Great part of energy is taken from the structure by surface waves, which are not represented in 1D model.

### **7. Impedances in the frequency domain**

162 Earthquake Engineering

"3", then the displacement of the mass is given by

solution (21). With dimensionless frequency

and dimensionless depth of Q

this "error coefficient" makes

**6.3. Impedance approach** 

(10) turns to

3330

boundary is artificial, such dependence cannot be physical.

boundary in the direct approach when there is no rock seen in the depth.

which turns to be similar to the exact one (21):

*b*

Curves for different *µ(ω)* for different *h* are shown in Fig.9.

structure will be solved exactly, providing *U0(x)* in the whole upper part. So, this check is OK. However, in "problem A2" exact displacement *UA(-H)* given by (27) is different from *U0(-H)* given by (22). As a result, both upcoming wave and the wave coming down in Vint will be different from the exact solution. If this new solution is marked with upper index

1 2 /[cos( / ) sin( / )] *<sup>m</sup> UUUU H H*

Note that this solution depends on *H*, while the exact solution does not (see (21)). As our

We can calculate the "error coefficient" relating approximate solution (29) to the exact

*h H m* 

<sup>3</sup> 1

*U i U h h*

*m m c* 

cos( ) sin( ) *<sup>A</sup>*

We see that the solution in not satisfactory. The increase of the boundary depth *h* does not improve the situation. General conclusion is that one must be very careful in placing lower

Now let us apply the impedance approach to the same system. As our structure rests on the surface, we can use Fig.6. If the displacement of the mass is *Ub*, then the equation of motion

2 0 [() ] () () () *C mU C U*

 

Impedance is the same as in Fig.4: *C(ω)=iωρc*. So, from (33) we at once get the ultimate result

*<sup>m</sup> <sup>m</sup> UU U i U i c* 

0 0 / [1 ] / [1 ] *<sup>A</sup>*

 

(34)

 

(32)

*<sup>b</sup>* (33)

(29)

(30)

(31)

Historically, the first problems for impedances considering soil inertia were solved semianalytically for homogeneous half-space without the internal damping and for circular surface stamps. It turned out that horizontal impedances behaved more or less like pairs of springs and viscous dashpots (as mentioned above, in reality the dissipation of energy had completely different nature; mechanical energy was not converted in heat, as in dashpot, but taken away by elastic waves). One more "good news" was that the impedance matrix appeared to be almost diagonal. Though non-diagonal terms coupling horizontal translation with rocking in the same vertical plane were non-zero ones, their squares were considerably less in module than products of the corresponding diagonal terms of the impedance matrix.

These two facts created a base for using the above mentioned "soil springs and dashpots". There are several variants of stiffness and dashpot parameters. You can see below a table 1 from ASCE4-98 [1] for circular base mats.

Soil-Structure Interaction 165

**Imaginary part of horizontal translational impedances**

0 5 10 15 20 25 30 **Frequency, Hz**

**Imaginary part of translational vertical impedances**

ImCz. SASSI\_8 ImCz. CLASSI\_8 ImCz. CLASSI\_16 ImCz. ASCE4-98

ImCz. SASSI\_16

ImCxx. SASSI\_8 ImCxx. CLASSI\_8 ImCxx. CLASSI\_16 ImCxx. ASCE4-98 ImCxx. SASSI\_16

ImCzz. SASSI\_8 ImCzz. CLASSI\_8 ImCzz. CLASSI\_16 ImCzz. ASCE4-98 ImCzz. SASSI\_16

0 5 10 15 20 25 30 **Frequency, Hz**

0 5 10 15 20 25 30 **Frequency, Hz**

0 5 10 15 20 25 30 **Frequency, Hz**

**Imaginary part of torsional impedances**

**Imaginary part of rocking impedance**

**Figure 10.** Impedances for a basement 30.6 x 30.6 meters on the surface of homogeneous half-space with mass density ρ=2 t/m3, wave velocities Vs=400 m/s, Vp=1100 m/s, internal damping γ=5%.

0,0E+00 5,0E+09 1,0E+10 1,5E+10 2,0E+10 2,5E+10 3,0E+10

**Real parts of horizontal translational impedances**

0 5 10 15 20 25 30 **Frequency, Hz**

0 5 10 15 20 25 30 **Frequency, Hz**

0 5 10 15 20 25 30 **Frequency, Hz**

0 5 10 15 20 25 30 **Frequency, Hz**

**Real part of torsional impedances**

ReCzz. SASSI\_8 ReCzz. CLASSI\_8 ReCzz. CLASSI\_16 ReCzz. ASCE4-98

ReCzz. SASSI\_16

**Real part of rocking impedances**

**Real part of translational vertical impedances**

0,0E+00 2,0E+07 4,0E+07 6,0E+07 8,0E+07 1,0E+08 1,2E+08 1,4E+08 1,6E+08 1,8E+08

0,0E+00 5,0E+07 1,0E+08 1,5E+08 2,0E+08 2,5E+08 3,0E+08 3,5E+08 4,0E+08 4,5E+08

0,0E+00 5,0E+09 1,0E+10 1,5E+10 2,0E+10 2,5E+10 3,0E+10 3,5E+10 4,0E+10 ImCx. SASSI\_8 ImCx. CLASSI\_8 ImCx. CLASSI\_16 ImCx. ASCE4-98 ImCx. SASSI\_16

0,0E+00 5,0E+06 1,0E+07 1,5E+07 2,0E+07 2,5E+07 3,0E+07 3,5E+07 4,0E+07



0,0E+00 2,0E+09 4,0E+09 6,0E+09 8,0E+09 1,0E+10 1,2E+10 ReCx. SASSI\_8 ReCx. CLASSI\_8 ReCx. CLASSI\_16 ReCx. ASCE4-98 ReCx. SASSI\_16

ReCz. SASSI\_8 ReCz. CLASSI\_8 ReCz. CLASSI\_16 ReCz. ASCE4-98 ReCz. SASSI\_16

ReCxx. SASSI\_8 ReCxx. CLASSI\_8 ReCxx. CLASSI\_16 ReCxx. ASCE4-98 ReCxx. SASSI\_16


Notes: = Poisson's ratio of foundation medium; *G* = shear modulus of foundation medium; *R* = radius of circular base mat; = mass density of foundation medium; *B = 3(1-)I0/8R5*; *I0*= total mass moment of inertia of structure and base mat about the rocking axis at the base; and *It*= polar mass moment of inertia of structure and base mat.

**Table 1.** Lumped Representation of Structure-Foundation Interaction at Surface for Circular Base

One more table of the same sort in the same standard [1] is given for rectangular base mats.

However, even for homogeneous half-space it turned out that vertical and angular impedances behaved differently from simple springs and dashpots. This can be seen in Fig.10, where the tabular impedances given by ASCE4-98 are compared to the wave solutions given by codes SASSI and CLASSI (number in the legend denotes number of finite elements along the side).

We understand now that the expressions given in the tables are just approximations for the real values. So, there cannot be "exact" or "true" expressions of this kind – different variants exist.

The expressions from the tables look particularly strange for angular damping, where parameters of the upper structure participate. It is real absurd from the physical point of view: impedances cannot depend on the upper structure; soil just "does not know" what is above rigid stamp. This is not an error, as some colleagues think. This is an attempt to take values from the frequency-dependent curves at certain frequencies. These frequencies estimate the first natural frequencies of rigid structure on flexible soil, so they depend on structural inertial parameters. Surely, such expressions are approximate.

The conclusion is that frequency dependence of the impedances exists even for the homogeneous half-space and spoils the spring/dashpot models in the impedance method.

mat; 

exist.

from ASCE4-98 [1] for circular base mats.

Horizontal 32(1 )

Rocking

Vertical

= mass density of foundation medium; *B*

elements along the side).

7 8 *<sup>x</sup> GR <sup>k</sup>* 

Torsion <sup>3</sup> 16 / 3 *<sup>t</sup> k GR* <sup>3</sup> 12/

**Table 1.** Lumped Representation of Structure-Foundation Interaction at Surface for Circular Base

Notes: = Poisson's ratio of foundation medium; *G* = shear modulus of foundation medium; *R* = radius of circular base

One more table of the same sort in the same standard [1] is given for rectangular base mats.

However, even for homogeneous half-space it turned out that vertical and angular impedances behaved differently from simple springs and dashpots. This can be seen in Fig.10, where the tabular impedances given by ASCE4-98 are compared to the wave solutions given by codes SASSI and CLASSI (number in the legend denotes number of finite

We understand now that the expressions given in the tables are just approximations for the real values. So, there cannot be "exact" or "true" expressions of this kind – different variants

The expressions from the tables look particularly strange for angular damping, where parameters of the upper structure participate. It is real absurd from the physical point of view: impedances cannot depend on the upper structure; soil just "does not know" what is above rigid stamp. This is not an error, as some colleagues think. This is an attempt to take values from the frequency-dependent curves at certain frequencies. These frequencies estimate the first natural frequencies of rigid structure on flexible soil, so they depend on

The conclusion is that frequency dependence of the impedances exists even for the homogeneous half-space and spoils the spring/dashpot models in the impedance method.

<sup>3</sup> 8 3 1 *GR <sup>k</sup>*

4 1 *<sup>z</sup> GR <sup>k</sup>*

> *= 3(1-)I0/8*

structural inertial parameters. Surely, such expressions are approximate.

mat about the rocking axis at the base; and *It*= polar mass moment of inertia of structure and base mat.

These two facts created a base for using the above mentioned "soil springs and dashpots". There are several variants of stiffness and dashpot parameters. You can see below a table 1

Motion Equivalent Spring Constant Equivalent Damping Coefficient

0.576 / *x x c kR G*

0.85 / *z z c kR G*

*t*

*R5*; *I0*= total mass moment of inertia of structure and base

*c*

*t t*

*I R*

*k I*

*t*

0.30 / <sup>1</sup> *c kR G B*

 

**Figure 10.** Impedances for a basement 30.6 x 30.6 meters on the surface of homogeneous half-space with mass density ρ=2 t/m3, wave velocities Vs=400 m/s, Vp=1100 m/s, internal damping γ=5%.

One more comment should be added here. If a package of horizontal layers is underlain by rigid rock, surface waves behave in a completely different manner than for homogeneous half-space. Instead of two surface waves (Love and Rayleigh ones) there exist an infinite number of surface waves. Each of these waves for low frequencies cannot take energy from the basement – they are "geometrically dissipating" (even without internal soil damping), or "locked". However, when frequency goes up, each of these waves one by one transforms from a "locked" wave into a "running" wave capable to take energy to the infinity. This behavior depends on soil geometry only, not on structure.

Soil-Structure Interaction 167

distributed). Of course we get some "platform" impedances *D(ω)*, different from "wave" impedances *C(ω)*, but the idea is to tune the platform excitation *Vb* so to account for the difference between "wave" impedances and "platform" impedances. Six components of the platform excitation may be tuned to reproduce six components of response – e.g., six components of the rigid base mat's accelerations. Such an approach combines the calculations in the frequency domain (platform seismic input) with calculations in the time domain (final dynamic analysis of the platform model), that is why this method is called "combined". Besides, this method is "exact" for rigid base mats only: the stiffer is a base mat, the more accurate are the results. That is why this method is also called "asymptotic" –

The last item to discuss in this part is practical tools to obtain impedances and seismic loads (or weightless base mats' motions) in the frequency domain. At the moment the author uses one of two computer codes. For rigid surface basements on a horizontally-layered soil code CLASSI is the most appropriate. For the embedded basements with possible local breaks in horizontal layering code SASSI is used (SASSI can be used for surface base mats also, but is

In both cases formula (3) is a basic equation for impedances, and the dynamic stiffness matrix *G0* linking set of nodes in the infinite soil is a key issue (the second matrix *Gint* is absent for surface basement in CLASSI and easily obtained by FEM for the embedded basement in SASSI). To get *G0*, they first obtain a dynamic flexibility matrix, describing displacements due to the unit forces (this is a Green's function). Here is a difference between

Professor J.Luco managed [7] to develop Green's function analytically for the case of surface load and surface response node in horizontally-layered soil in the frequency domain. Then contact surface Q was covered with number of rectangular elements of different shapes. Loads were applied in the centers of each element one by one; response displacements were obtained in the centers of each element for each load. The additional convenience was that for the horizontally layered soil one can shift the loaded node and the response node horizontally, and the link between them stays the same. So, in fact one needs Green's functions only for a single loaded node and for the response nodes placed at a distance from the loaded node within maximal size of the mat. Zero (for vertical load) and first (for horizontal load) Fourier terms describe the displacement field along angular cylindrical coordinate, enabling to store Green's functions only along 1D radial line. Thus, it is not very time-consuming to obtain Green's function (it is done in a separate module "Green") and to use it in order to obtain the full flexibility matrix (in separate module "Claff"). Then this full flexibility matrix is turned into a full stiffness matrix *G0*. Finally *G0* is condensed to the 6 x 6 impedance matrix *C*. The whole procedure is repeated for each frequency of the prescribed set. For a surface rigid basement and vertical waves there is no separate problem to find the load matrix *B* in equation (4): kinematical interaction does not change control motion, so *B* consists of the first three columns of *C*. This was a brief description of CLASSI ideology.

full name is "combined asymptotic method" (CAM) [22].

more sophisticated).

two codes.

It means that in the frequency domain there exists a certain low-frequency range, where the whole soil foundation is "locked". All the energy taken from the basement is only due to the internal damping. If there is no internal damping, complex impedances are completely real. In practice internal soil damping is several percent, so impedances are "almost real". After the first surface wave transforms into a running one, the soil foundation becomes "unlocked" – wave damping appears. Then one by one other surface waves turn into "running" ones, increasing the integral damping in the soil-structure system. The impedance functions for the same soil, but underlain by rigid rock at depth 26 m, are shown on Fig.11. One can see "locking phenomena" looking at the imaginary parts of the impedances.

The conclusion here is that the frequency dependence of the impedances may be rather sophisticated depending on soil layering. The attempts to "cover" the variety of soils by number of homogeneous half-spaces with different properties (usually this is an approach for "serial design" of structures) may lead to mistakes: there is always a possibility that real layered soil will not be "covered" by a set of half-spaces (e.g. no half-space can reproduce the "locking" effects described above).

The additional problem with springs and dashpots arises when the integral stiffness is distributed over the contact surface Q. Physically in every point of Q there are no distributed angular loads impacting basement from the soil. So, only translational springs and dashpots are usually distributed over Q, and angular impedances are the results of these distributed translational springs. For a surface basement vertical distributed springs are responsible for rocking impedances, and horizontal springs are responsible for torsional impedance. The problem here is that all attempts to find the distribution shape for vertical springs to represent rocking impedances simultaneously with vertical one have failed. If fact, integral rocking stiffness obtained from distributed vertical springs is always less than actual rocking stiffness; on the contrary, integral rocking damping obtained from distributed vertical dashpots is always greater than the actual one. Physical reason of this mismatch is the interaction between different points through soil. Spring/dashpot model is "local" in nature: the response is determined by motion of this very point, and not neighbors. This is not physically true.

The author found a way to treat both problems at once. The idea is to work in the time domain using a platform model of Fig.5 with conventional springs and dashpots (lumped or distributed). Of course we get some "platform" impedances *D(ω)*, different from "wave" impedances *C(ω)*, but the idea is to tune the platform excitation *Vb* so to account for the difference between "wave" impedances and "platform" impedances. Six components of the platform excitation may be tuned to reproduce six components of response – e.g., six components of the rigid base mat's accelerations. Such an approach combines the calculations in the frequency domain (platform seismic input) with calculations in the time domain (final dynamic analysis of the platform model), that is why this method is called "combined". Besides, this method is "exact" for rigid base mats only: the stiffer is a base mat, the more accurate are the results. That is why this method is also called "asymptotic" – full name is "combined asymptotic method" (CAM) [22].

166 Earthquake Engineering

impedances.

not physically true.

the "locking" effects described above).

One more comment should be added here. If a package of horizontal layers is underlain by rigid rock, surface waves behave in a completely different manner than for homogeneous half-space. Instead of two surface waves (Love and Rayleigh ones) there exist an infinite number of surface waves. Each of these waves for low frequencies cannot take energy from the basement – they are "geometrically dissipating" (even without internal soil damping), or "locked". However, when frequency goes up, each of these waves one by one transforms from a "locked" wave into a "running" wave capable to take energy to the infinity. This

It means that in the frequency domain there exists a certain low-frequency range, where the whole soil foundation is "locked". All the energy taken from the basement is only due to the internal damping. If there is no internal damping, complex impedances are completely real. In practice internal soil damping is several percent, so impedances are "almost real". After the first surface wave transforms into a running one, the soil foundation becomes "unlocked" – wave damping appears. Then one by one other surface waves turn into "running" ones, increasing the integral damping in the soil-structure system. The impedance functions for the same soil, but underlain by rigid rock at depth 26 m, are shown on Fig.11. One can see "locking phenomena" looking at the imaginary parts of the

The conclusion here is that the frequency dependence of the impedances may be rather sophisticated depending on soil layering. The attempts to "cover" the variety of soils by number of homogeneous half-spaces with different properties (usually this is an approach for "serial design" of structures) may lead to mistakes: there is always a possibility that real layered soil will not be "covered" by a set of half-spaces (e.g. no half-space can reproduce

The additional problem with springs and dashpots arises when the integral stiffness is distributed over the contact surface Q. Physically in every point of Q there are no distributed angular loads impacting basement from the soil. So, only translational springs and dashpots are usually distributed over Q, and angular impedances are the results of these distributed translational springs. For a surface basement vertical distributed springs are responsible for rocking impedances, and horizontal springs are responsible for torsional impedance. The problem here is that all attempts to find the distribution shape for vertical springs to represent rocking impedances simultaneously with vertical one have failed. If fact, integral rocking stiffness obtained from distributed vertical springs is always less than actual rocking stiffness; on the contrary, integral rocking damping obtained from distributed vertical dashpots is always greater than the actual one. Physical reason of this mismatch is the interaction between different points through soil. Spring/dashpot model is "local" in nature: the response is determined by motion of this very point, and not neighbors. This is

The author found a way to treat both problems at once. The idea is to work in the time domain using a platform model of Fig.5 with conventional springs and dashpots (lumped or

behavior depends on soil geometry only, not on structure.

The last item to discuss in this part is practical tools to obtain impedances and seismic loads (or weightless base mats' motions) in the frequency domain. At the moment the author uses one of two computer codes. For rigid surface basements on a horizontally-layered soil code CLASSI is the most appropriate. For the embedded basements with possible local breaks in horizontal layering code SASSI is used (SASSI can be used for surface base mats also, but is more sophisticated).

In both cases formula (3) is a basic equation for impedances, and the dynamic stiffness matrix *G0* linking set of nodes in the infinite soil is a key issue (the second matrix *Gint* is absent for surface basement in CLASSI and easily obtained by FEM for the embedded basement in SASSI). To get *G0*, they first obtain a dynamic flexibility matrix, describing displacements due to the unit forces (this is a Green's function). Here is a difference between two codes.

Professor J.Luco managed [7] to develop Green's function analytically for the case of surface load and surface response node in horizontally-layered soil in the frequency domain. Then contact surface Q was covered with number of rectangular elements of different shapes. Loads were applied in the centers of each element one by one; response displacements were obtained in the centers of each element for each load. The additional convenience was that for the horizontally layered soil one can shift the loaded node and the response node horizontally, and the link between them stays the same. So, in fact one needs Green's functions only for a single loaded node and for the response nodes placed at a distance from the loaded node within maximal size of the mat. Zero (for vertical load) and first (for horizontal load) Fourier terms describe the displacement field along angular cylindrical coordinate, enabling to store Green's functions only along 1D radial line. Thus, it is not very time-consuming to obtain Green's function (it is done in a separate module "Green") and to use it in order to obtain the full flexibility matrix (in separate module "Claff"). Then this full flexibility matrix is turned into a full stiffness matrix *G0*. Finally *G0* is condensed to the 6 x 6 impedance matrix *C*. The whole procedure is repeated for each frequency of the prescribed set. For a surface rigid basement and vertical waves there is no separate problem to find the load matrix *B* in equation (4): kinematical interaction does not change control motion, so *B* consists of the first three columns of *C*. This was a brief description of CLASSI ideology.

Soil-Structure Interaction 169

J.Lysmer for the same problem (but for the embedded loaded and response nodes) used direct approach described above. He made Vint a cylindrical column, with just a single element per radius. The bottom of this column was placed on a homogeneous half-space, so a dashpot was put there in each of three directions. Lateral boundaries were of Waas type. Loads were applied to the nodes at the vertical axis of cylinder. The response nodes were either at the same axis or out of the cylinder (for regular mesh the radius of cylinder was set equal to 0.9 of mesh size). In the first case, the response displacements were obtained from the FEM problem. In the second case the solution of the FEM problem gave the displacements of the nodes at the lateral boundary. Then the displacements in the whole infinite volume Vext were obtained using "homogeneous wave fields" in cylindrical coordinates described above (with Hankel's functions along radius). Like in CLASSI, horizontal shift of the loaded and response nodes is allowed in the horizontally-layered soil. So only a single cylinder has to be studied. Like in CLASSI, it is done in a separate module (in SASSI it is called POINT). After a flexibility matrix for a set of "interacting nodes" covering Vint is obtained, it is turned into a stiffness matrix *G0*. This matrix may be condensed into a part of

However, one can instead return to the "problem A2" without condensation (e.g., for flexible basement or locally modified soil around the basement). Bottom of surface Q is placed not on the rock, as it was in Fig.3, but where it is convenient (usually at the bottom of the modified soil volume, if any, or at the bottom of the basement). This approach may be called a "combined" one, because the direct approach is used to get Green's function only; further on, the impedance approach (extended for flexible underground volume Vint) is applied.

The problem of axial symmetry has gone both in CLASSI and in SASSI: though "basic problem" in Green or in POINT has an axisymmetric geometry and is solved in cylindrical

Another problem for SASSI is with flexible underlying half-space: Waas boundaries were developed for rigid half-space at the bottom only. Lysmer's team found a brilliant approximate solution. Energy for a cylinder in POINT is taken down in the half-space by vertical body waves (modeled by dashpots, see above) and by surface Rayleigh waves in the flexible half-space. These Rayleigh waves in the frequency domain have a certain depth where the wave displacements are close to zero. One can put "rigid" bottom at that very depth without spoiling the response in the upper part of the soil. The problem is that this depth is frequency-dependent: so, the model in POINT becomes non-physical. However, transmitting boundaries, as we remember, were developed by G.Waas in the frequency domain, so the problem is solved frequency by frequency. Hence, one can change the depth of "rigid boundary" at the bottom of the model according to the frequency change. In SASSI it is done automatically: soil model in POINT consists of the upper part with fixed layers,

For surface basements one can compare the impedances given by SASSI and CLASSI. In the frequency range below 15 Hz the results are very close to each other. Examples were shown

the impedance matrix like in CLASSI (another part comes from *Gint*).

coordinates, the "interaction nodes set" may be arbitrary in shape.

and of the lower part with frequency-dependent layers.

above (see Fig.10).

**Figure 11.** Impedances for a basement 30.6 x 30.6 meters on the surface of homogeneous layer with mass density ρ=2 t/m3, wave velocities Vs=400 m/s, Vp=1100 m/s, internal damping γ=5%, thickness h=26 m, underlain by rigid rock

J.Lysmer for the same problem (but for the embedded loaded and response nodes) used direct approach described above. He made Vint a cylindrical column, with just a single element per radius. The bottom of this column was placed on a homogeneous half-space, so a dashpot was put there in each of three directions. Lateral boundaries were of Waas type. Loads were applied to the nodes at the vertical axis of cylinder. The response nodes were either at the same axis or out of the cylinder (for regular mesh the radius of cylinder was set equal to 0.9 of mesh size). In the first case, the response displacements were obtained from the FEM problem. In the second case the solution of the FEM problem gave the displacements of the nodes at the lateral boundary. Then the displacements in the whole infinite volume Vext were obtained using "homogeneous wave fields" in cylindrical coordinates described above (with Hankel's functions along radius). Like in CLASSI, horizontal shift of the loaded and response nodes is allowed in the horizontally-layered soil. So only a single cylinder has to be studied. Like in CLASSI, it is done in a separate module (in SASSI it is called POINT). After a flexibility matrix for a set of "interacting nodes" covering Vint is obtained, it is turned into a stiffness matrix *G0*. This matrix may be condensed into a part of the impedance matrix like in CLASSI (another part comes from *Gint*).

168 Earthquake Engineering

ReCx. SASSI\_8 ReCx. ASCE4-98 ReCx. SASSI\_16





**Imaginary part of horizontal translational impedances**

0 5 10 15 20 25 30 **Frequency, Hz**

**Real part of vertical translational impedances**

0 5 10 15 20 25 30 **Frequency, Hz**

0 5 10 15 20 25 30 **Frequency, Hz**

0 5 10 15 20 25 30 **Frequency, Hz**

**Imaginary part of torsional impedances**

**Imaginary part of rocking impedances**

0 5 10 15 20 25 30 **Frequency, Hz**

0 5 10 15 20 25 30 **Frequency, Hz**

0 5 10 15 20 25 30 **Frequency, Hz**

0 5 10 15 20 25 30 **Frequency, Hz**

**Real part of torsional impedances**

**Real part of rocking impedances**

**Real part of vertical translational impedances**

ReCz. SASSI\_8 ReCz. ASCE4-98 ReCz. SASSI\_16 0,0E+00 5,0E+07 1,0E+08 1,5E+08 2,0E+08 2,5E+08 3,0E+08 3,5E+08

0,0E+00 1,0E+08 2,0E+08 3,0E+08 4,0E+08 5,0E+08 6,0E+08 7,0E+08

0,0E+00 5,0E+09 1,0E+10 1,5E+10 2,0E+10 2,5E+10 3,0E+10 3,5E+10 4,0E+10 4,5E+10 5,0E+10 ImCx. SASSI\_8

ImCx. ASCE4-98

ImCx. SASSI\_16

ImCz. SASSI\_8 ImCz. ASCE4-98 ImCz. SASSI\_16

ImCxx. SASSI\_8 ImCxx. ASCE4-98 ImCxx. SASSI\_16

ImCzz. SASSI\_8 ImCzz. ASCE4-98

ImCzz. SASSI\_16

m, underlain by rigid rock

ReCzz. SASSI\_8 ReCzz. ASCE4-98 ReCzz. SASSI\_16

**Figure 11.** Impedances for a basement 30.6 x 30.6 meters on the surface of homogeneous layer with mass density ρ=2 t/m3, wave velocities Vs=400 m/s, Vp=1100 m/s, internal damping γ=5%, thickness h=26

0,0E+00 5,0E+09 1,0E+10 1,5E+10 2,0E+10 2,5E+10 3,0E+10 3,5E+10 4,0E+10 However, one can instead return to the "problem A2" without condensation (e.g., for flexible basement or locally modified soil around the basement). Bottom of surface Q is placed not on the rock, as it was in Fig.3, but where it is convenient (usually at the bottom of the modified soil volume, if any, or at the bottom of the basement). This approach may be called a "combined" one, because the direct approach is used to get Green's function only; further on, the impedance approach (extended for flexible underground volume Vint) is applied.

The problem of axial symmetry has gone both in CLASSI and in SASSI: though "basic problem" in Green or in POINT has an axisymmetric geometry and is solved in cylindrical coordinates, the "interaction nodes set" may be arbitrary in shape.

Another problem for SASSI is with flexible underlying half-space: Waas boundaries were developed for rigid half-space at the bottom only. Lysmer's team found a brilliant approximate solution. Energy for a cylinder in POINT is taken down in the half-space by vertical body waves (modeled by dashpots, see above) and by surface Rayleigh waves in the flexible half-space. These Rayleigh waves in the frequency domain have a certain depth where the wave displacements are close to zero. One can put "rigid" bottom at that very depth without spoiling the response in the upper part of the soil. The problem is that this depth is frequency-dependent: so, the model in POINT becomes non-physical. However, transmitting boundaries, as we remember, were developed by G.Waas in the frequency domain, so the problem is solved frequency by frequency. Hence, one can change the depth of "rigid boundary" at the bottom of the model according to the frequency change. In SASSI it is done automatically: soil model in POINT consists of the upper part with fixed layers, and of the lower part with frequency-dependent layers.

For surface basements one can compare the impedances given by SASSI and CLASSI. In the frequency range below 15 Hz the results are very close to each other. Examples were shown above (see Fig.10).

## **8. Soil non-linearity – SHAKE ideology – Contact non-linearity**

Soil in reality is considerably non-linear. That is why H.B.Seed [23] suggested special method to describe seismic response of horizontally-layered soil to the verticallypropagating free-field seismic waves by equivalent linear model. Parameters of this model are obtained in iterative calculations, as modules and damping in each layer are straindependent. This method is used in the famous SHAKE code [24]. Usually 7-8 iterations are enough to get the error less than 1%.

Soil-Structure Interaction 171

OBE SSE 1,4SSE

**Figure 12.** Typical strain-dependent characteristics of soils


0

**Shear wave velocity Vs, m/s**

0,2

0,4

1 - Clays 2 - Sands 3 - Gravel sands 4 - Mergel

0,6

**G/Gmax**

0,8

1

**Figure 13.** Typical results of the SHAKE calculations

0 5 10 15 20 25 30 35 40 **Depth from the level +28,35 m**

(usually in two planes they are close to each other for every layer).

Typical results of SHAKE calculations for three levels of seismic intensity are given on Fig.13. Abbreviations of the level names mean: OBE – operational basis earthquake; SSE – safe shutdown earthquake (operations and shutdown refer to nuclear reactors activity). The results are calculated separately in two vertical planes (OXZ and OYZ) and then averaged

0


0 5 10 15 20 25 30 35 40 **Depth from the level +28,35 m**

6,67

13,34

**Коэффициент демпфирования, %** 20,01

1 - Clays 2 - Sands 3 - Gravel sands 4 - Mergel

26,68

We see that near the surface the equivalent soil characteristics are close to the low-strain

1,4SSE <sup>0</sup>

0,05

0,1

0,15

**Material damping coefficient**

0,2

0,25

Usually the greatest relative shift in velocities and damping is at the bottom of soft soil layers. If the intensity is very high, the iterations during deconvolution may not converge. Physically it means that the given surface motion is no longer compatible with given soil

values. This is because of zero strains at the free surface for every intensity level.

Low strains OBE SSE

properties; such motion cannot be transferred by given soil to the surface.

SHAKE calculations may be started from the surface (i.e. surface motion is given; for each other layer the motion is to be calculated together with equivalent soil properties); this case is called "deconvolution". On the contrary, SHAKE calculations may be started from some depth where the motion is given; this case is called "convolution". Motion for one and the same point in depth may be given in one of two variants. The first variant is just a motion of the point inside soil composed of the upcoming wave and wave coming down (see the 1D example in p.5 above). The second variant is to set just the upcoming wave amplitude. Sometimes this motion is called "outcropped" motion, because they double the upcoming wave amplitude, like it is at the free surface (see the example in p. 6.1 above). However, such a "doubled" motion will be similar to the motion of the really outcropped surface only at the surface of the underlying homogeneous half-space, and not in the middle of the layered soil.

The typical case consists of two sequential SHAKE calculations. First, deconvolution is performed using the initial soil model and initial seismic motion of the free surface of the initial soil (the author usually calls it "foundation no.1"). Note, that the underlying halfspace is not modified in the process of SHAKE iterations. That is why one should include into the modified part at least one upper layer of half-space. After the deconvolution is through, one should change the half-space manually making parameters equal to the lowest modified layer. The main result of deconvolution is an "outcropped" motion of the halfspace surface. The second stage is a convolution. Half-space stays the same as in deconvolution (after manual modification), but the upper layers may change due to various reasons starting from the direct changes in the construction process (very often the upper several meters of the initial soil are withdrawn). Sometimes the weight of the future structure changes the properties of the soil. Anyhow, we generally have "foundation no.2" instead of the first soil foundation. In the convolution procedure the outcropped motion of the half-space surface is given, and the whole wave field with equivalent soil properties is obtained as a result together with motion of the new surface.

Typical strain-dependent characteristics of different soils are shown on Fig.12. Sometimes they call it "degradation curves".

One should remember that degradation curves for shear modulus G are given in relative terms G/G0, where G0 is a low-strains value. On the contrary, strain-dependent characteristics of the internal damping are given for dimensionless coefficients without using low-strain values.

**Figure 12.** Typical strain-dependent characteristics of soils

enough to get the error less than 1%.

**8. Soil non-linearity – SHAKE ideology – Contact non-linearity** 

underlying homogeneous half-space, and not in the middle of the layered soil.

obtained as a result together with motion of the new surface.

they call it "degradation curves".

using low-strain values.

Soil in reality is considerably non-linear. That is why H.B.Seed [23] suggested special method to describe seismic response of horizontally-layered soil to the verticallypropagating free-field seismic waves by equivalent linear model. Parameters of this model are obtained in iterative calculations, as modules and damping in each layer are straindependent. This method is used in the famous SHAKE code [24]. Usually 7-8 iterations are

SHAKE calculations may be started from the surface (i.e. surface motion is given; for each other layer the motion is to be calculated together with equivalent soil properties); this case is called "deconvolution". On the contrary, SHAKE calculations may be started from some depth where the motion is given; this case is called "convolution". Motion for one and the same point in depth may be given in one of two variants. The first variant is just a motion of the point inside soil composed of the upcoming wave and wave coming down (see the 1D example in p.5 above). The second variant is to set just the upcoming wave amplitude. Sometimes this motion is called "outcropped" motion, because they double the upcoming wave amplitude, like it is at the free surface (see the example in p. 6.1 above). However, such a "doubled" motion will be similar to the motion of the really outcropped surface only at the surface of the

The typical case consists of two sequential SHAKE calculations. First, deconvolution is performed using the initial soil model and initial seismic motion of the free surface of the initial soil (the author usually calls it "foundation no.1"). Note, that the underlying halfspace is not modified in the process of SHAKE iterations. That is why one should include into the modified part at least one upper layer of half-space. After the deconvolution is through, one should change the half-space manually making parameters equal to the lowest modified layer. The main result of deconvolution is an "outcropped" motion of the halfspace surface. The second stage is a convolution. Half-space stays the same as in deconvolution (after manual modification), but the upper layers may change due to various reasons starting from the direct changes in the construction process (very often the upper several meters of the initial soil are withdrawn). Sometimes the weight of the future structure changes the properties of the soil. Anyhow, we generally have "foundation no.2" instead of the first soil foundation. In the convolution procedure the outcropped motion of the half-space surface is given, and the whole wave field with equivalent soil properties is

Typical strain-dependent characteristics of different soils are shown on Fig.12. Sometimes

One should remember that degradation curves for shear modulus G are given in relative terms G/G0, where G0 is a low-strains value. On the contrary, strain-dependent characteristics of the internal damping are given for dimensionless coefficients without Typical results of SHAKE calculations for three levels of seismic intensity are given on Fig.13. Abbreviations of the level names mean: OBE – operational basis earthquake; SSE – safe shutdown earthquake (operations and shutdown refer to nuclear reactors activity). The results are calculated separately in two vertical planes (OXZ and OYZ) and then averaged (usually in two planes they are close to each other for every layer).

**Figure 13.** Typical results of the SHAKE calculations

We see that near the surface the equivalent soil characteristics are close to the low-strain values. This is because of zero strains at the free surface for every intensity level.

Usually the greatest relative shift in velocities and damping is at the bottom of soft soil layers. If the intensity is very high, the iterations during deconvolution may not converge. Physically it means that the given surface motion is no longer compatible with given soil properties; such motion cannot be transferred by given soil to the surface.

Both SHAKE results – profiles of wave velocities/damping and surface motion – are further on used in SSI analysis.

Soil-Structure Interaction 173

**9. Non-mandatory assumptions** 

limited volume near the structure).

author's reports in SMiRT-21 [25,26].

moves "rigidly".

accepted for the whole SSI problem.

**10. Some examples of the SSI effects in practice** 

acceleration response spectra on the elevation +21.5 m.

So, basic assumptions currently used in the SSI analysis are a) linearity of the soil, of the structure and of the soil-structure contact; b) horizontal layering of the soil (except some

There are two other assumptions - not mandatory, but usually used in the SSI analysis. The first assumption is the rigidity of the soil-structure contact surface. Usually base mats are not extremely rigid, but they are considerably enforced by rather dense and thick shear walls, so in fact their behavior is almost rigid. Standards ASCE4-98 allow the treatment of base mats of the NPP structures as rigid ones. However, SASSI can treat flexible base mats as well. Different parameters of structural seismic response show different sensitivity to the flexibility of the base mat. Some examples are presented in the

The second assumption is about seismic wave field in the soil without structure. Usually, one starts from the three-component acceleration recorded on the surface of the soil (in some "control point", as they say). As we saw, in SSI problem one needs to know the motion of a certain soil volume (at least the soil motion in the nodes of the future contact surface). So some additional assumptions are introduced. The most common assumption "is vertically propagating body seismic waves in horizontally-layered medium". This assumption means, that three components of the acceleration in the control point are produced by three separate vertically propagating waves: vertical acceleration is a result of the P-wave, two horizontal accelerations are the result of two S-waves in main coordinate vertical planes. Each wave can be analyzed separately by SHAKE, providing seismic motion of any point in depth. Another consequence of this assumption is that seismic motion depends on vertical coordinate, but not on the horizontal coordinates: every horizontal plane in the free field

Again, this assumption is not mandatory. In SASSI one can set up other assumptions linking the whole wave field to the control point motion. However, as SHAKE (usually used as a preprocessor for SASSI) implements that very assumption, most often this assumption is

Concluding the chapter, the author should like to give some practical examples. One of recently built NPPs was analyzed for different soil and excitation models. Let us look at the

The first comparison is for rigid soil and flexible soil (without embedment). As we remember, rigid soil means the absence of the SSI effects. Flexible soil in this case was of medium type. Structure was one of the NPP buildings. One and the same three-component seismic excitation (corresponding to the standard spectra described in RG1.60) was applied

Theoretically a wide range of nonlinearities can be described by strain-dependent properties. However, these properties should be transient, i.e. they should vary from one time point to another during one seismic event. In the Seed's approach these properties in each soil layer are established once for the whole event duration (changing not from time to time, but from one linear run to another one). So, in fact soil properties depend not on the instant transient strains, but on some "effective" strains (in practice - on some portion of the maximal strain over the duration). In fact, Seed provided a tool to extrapolate the results of the lab test with harmonic excitation of the soil sample to another situation with nonharmonic wave in the same sample.

Nowadays seismologists are ready to provide more sophisticated approaches to the soil description. They can model accelerations in the soil more accurately with truly transient soil properties. But in the SSI problems (e.g., in SASSI or CLASSI) one will need linear soil with "effective" soil properties, so SHAKE still remains the best "pre-SSI" processor.

The result of SHAKE is obtained for horizontally-layered soil without structure. So, they say, that it contains "primary" non-linearity. The same equivalent strain-dependent properties can be further applied in the SSI problems to account for the "secondary" nonlinearity caused by structure, but the result will spoil horizontally-layered geometry of the soil. As we know, that will spoil CLASSI model and create additional problems in SASSI model.

That is why modern standards (e.g., ASCE4-98) require the consideration of the primary non-linearity, but do not require consideration of the secondary non-linearity of the soil.

Looking at soil-structure system, one can find nonlinearities not only in soil, but at the contact surface. Even if both soil and structure are modeled by linear systems, they may have various contact terms. Most often full contact is assumed – it does not spoil linearity. However, soil tension (unlike soil compression) is very limited, and that may cause a) uplift of the base mat from the soil, b) separation of the embedded basement walls from the soil.

Base mat uplift may be estimated if linear "full contact" vertical forces over contact surface Q are compared to the static vertical forces caused by structural weight. In practice the full uplift is seldom met, but rocking of the structure can cause dynamic tension near the edges of the base mat. This "partial" dynamic uplift usually occurs for stiff soils and sizable structures. Does it change the response motion considerably? Today they believe that if the area of partial dynamic uplift is less than 1/3 of the total contact area, one can neglect this uplift and still use SSI linear model with full contact.

Separation of vertical embedded walls is treated as follows. In the upper half of the embedment depth (but only up to 6 meters from the surface) they break soil-structure contact completely. Below this level they assume full contact.
