INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, VOL. 18, 205-212 (1994)

SHORT COMMUNICATION

AMPLIFICATION OF VERTICALLY PROPAGATING SH WAVES BY MULTIPLE LAYERS OF GIBSON SOILS

ROB DAVIS AND BRUCE HUNT

Department of Civil Engineering, University of Canterbury, Private Bag 4800. Christchurch, New Zealand

SUMMARY

Using the Haskell-Thomson transfer matrix approach, an analytical solution is obtained for SH wave amplification by multiple layers of Gibson soils (i.e. viscoelastic layers with linearly varying shear moduli). Amplification spectra for typical soil and basement rock properties are calculated. A comparison of the Gibson soil response with that obtained for homogeneous soil models shows generally stronger amplifi- cations associated with the Gibson soil.

INTRODUCTION

The problem of determining the amplification of surface motion due to wave interaction within a layered elastic medium was initially solved by Thomson' and later refined by Haskell.' These solutions have been extensively used for the particular case of vertically propagating SH waves to estimate the site response characteristics of layered soils subjected to seismic loading. This paper considers the closely related problem of vertically propagating SH waves interacting with a layer or layers of Gibson soil. The term Gibson soil is used here to represent an elastic or viscoelastic material with constant density and with shear modulus which varies linearly with depth. For many practical situations, a Gibson soil will be a more realistic model of an actual soil than will a homogeneous linearly elastic or viscoelastic material.

The work presented here is modelled on the Haskell-Thomson analysis of multiple layers resting on a homogeneous half-space. A transfer matrix is derived giving the response at the top of a Gibson soil layer in terms of the response at the base. Multiplying the transfer matrices for each soil layer leads to relations between the free surface response and the basement rock incident wave form. This relation yields the free surface amplification function.

To our knowledge, the only work similar to this has recently been carried out by Towhata and T ~ y o t a . ~ They consider the case of an elastic layer with shear modulus which varies as a function of z", where z denotes depth and n is a parameter. They consider only a single layer excited by a prescribed base motion, and they obtain analytical solutions only for the zero damped, elastic case. The work presented here is less general in considering linear variation of shear modulus, but more general in considering multiple layers, the basement rock radiation condition and viscoelas- tic behaviour.

CCC 0363-9061/94/030205-08 0 1994 by John Wiley & Sons, Ltd.

Received 9 June 1993 Revised 25 August 1993

206 SHORT COMMUNICATION

ANALYSIS

Consider a layer of Gibson soil with uniform thickness h shown in Figure 1. Let z be the local depth co-ordinate within the layer and let u(z, t ) denote horizontal displacement. The governing equation for vertically propagating shear waves is

where p denotes mass density and G(z) is the complex shear modulus:

@z) = G(z)(l + ii(z)) (2)

Here G denotes elastic shear modulus and i is the hysteretic damping ratio. Solutions of (1) may be assumed to have the form

14 = @(z)eiof (3) where o denotes circular frequency and @ is as yet an undetermined function. Using (3) in (1) gives

Now suppose i; is a linear function of z, so that

i; -=CI++z P

where CI and p are complex constants. Then the solution to (4) is

4 = AJo(5) + SYO(5) (6) Here lo and Yo are zero-order Bessel functions, A and B are constants, and

The simplest variation of 6 would correspond to a constant damping ratio i and linearly varying elastic shear modulus G; thus

(8) Z G = G1 + (Gz - GI) - h

For this condition

(9) CI = -(I G1 + i i ) = ct(1 + i i ) P

Gibson soil layer h

Figure 1. Geometry of the Gibson soil layer

SHORT COMMUNICATION 207

and

Here ci = J ( G i / p ) represents the shear wave velocity at either the top (i = 1) or at the base (i = 2) of the layer. The amplification results given below are obtained with this constant damping model, but the general solution applies to the case where G and i have any form so long as 6 is linear in z.

The displacement field within the layer is now given by

u = [AJo(5) + BYo(<)]eiof (1 1)

The corresponding velocity u and shear stress z are easily found:

Let til and z1 denote the velocity and shear stress at the top of the layer ( z = 0). Equations (12) and (1 3) then give (ignoring the common eiot term)

Similarly, at the base of the layer, let ti2 and 72 denote the velocity and stress so that

IKI [::I = [ - P W J k + PhVl(52) - poJ(a + Ph) Yl(52)

iwJo(52) iw y o (52 1

In (14) and (15) the complex arguments of the Bessel functions are

2w C t = g J a

5 2 = p J(a + Bh)

and 2 0

Equations (14) and (15) can be combined to eliminate A and B. The result is

where

208 SHORT COMMUNICATION

The matrix T is the Haskell-Thomson transfer matrix for the layer with linear shear modulus variation. It can be used recursively to find the response of multiply layered sites. Consider the situation with (n - 1) layers illustrated in Figure 2. The free surface response, (ti1, zl), is related to the response at the basement rock interface, (ti", rn), by

where each of the matrices T'", T('), . . . is appropriate to each of the layers k = 1,2,. . . , and the matrix H is the product of all the Fk) matrices.

Assuming homogeneous elastic behaviour in the basement rock, the motion there can be specified by

u(z, r ) = Iceik' + De-ik']ei"' (20) where z is the depth measured from the interface between layer (n - 1) and the basement rock. The wave number k is given by k = w/co, where c,, denotes the basement shear wave velocity. The upward propagating wave has amplitude C. This is the incident wave, which will be denoted as u,:

u1 = Cei(kz+W (21)

The velocity and shear stress in the basement rock are easily found from (20):

where p o denotes mass density of the basement rock.

Figure 2. Multiple layers of Gibson soils above homogeneous basement rock

SHORT COMMUNICATION 209

Let tio and zo denote the velocity and shear stress at the top of the basement rock, found by setting z = 0 in (22) and (23). Combining the resulting equations, it can be seen that the following condition applies:

pocouo + zo = 2iwpocoCeiot = 2p0c0u, (24) where tiI denotes the particle velocity of the incident wave found by differentiating (21) and setting z = 0.

Finally, boundary conditions at the free surface and at the soil-rock interface can be applied. At the free surface, the shear stress must vanish

z1 = o (25) while continuity of velocity and stress at the interface between layer (n - 1) and the basement rock requires

u, = t io and z, = zo (26) Using (25) and (26) with (19) gives

tio = Hlllil and zo = Hzltil

and using these equations in (24) gives

P o c o f f l l ~ l + H 2 l h = 2POCO4 (27) Thus, the amplification function is

Here the absolute value signs imply the complex modulus. Equation (28) is the main result of this paper.

DISCUSSION

To illustrate the analysis given above, an example problem, shown in Figure 3, will be solved. A single layer of Gibson soil lies above a homogeneous elastic half-space. The layer thickness h is taken to be 50 m and the soil density is 1.90 x lo3 kg/m3. The elastic shear modulus G is assumed to vary linearly as in (8). Values of G I and G 2 are chosen so that cl, the shear wave velocity at the free surface, is 300 m/s while c2, the velocity at the base of the layer, is 800 mjs. The basement rock density po is taken to be 2.5 x lo3 kg/m3, and the shear wave velocity co is 2000 m/s.

The amplification function for the situation described was calculated using damping ratios [ of 0,2,5 and 10 per cent. The excitation frequency w was allowed to range between 0 and 160 rad/s. The results are illustrated in Figure 4. As one might expect, the soil layer causes strongly peaked amplifications at certain frequencies. Increasing the value of [ attenuates the amplification, and this effect is more pronounced at higher frequencies. All these basic features are found in the response of a homogeneous viscoelastic layer. There are subtle differences, however, between the Gibson soil response and that of a homogeneous layer. For the Gibson soil, the undamped amplification peaks grow slightly with increasing frequency. This is unlike the homogeneous layer response where the undamped peaks all have constant amplitude. A somewhat more interesting feature concerns the troughs rather than the peaks of the amplification function. For a homogene- ous layer, the smallest undamped amplification is exactly 1.0, which occurs between each peak.

210 SHORT COMMUNICATION

Free surface Shear wave velocity

t Depth

Figure 3. Example problem

5 . 0 - - -

4 . s - -

4 . 0 - z - 0 + 3 .5 - 4 - -

0 4 3 . 0 - 1 a- = 2 . 5 : 4

- 2 . 0

l.5[

0 20 40 60 80 1 0 0 1 2 0 140 160 180 200

FREQUENCY (rad/sec)

Figure 4. Amplification spectra for different values of damping ratio

The troughs shown in Figure 4 are all significantly stronger than 1.0; thus the Gibson soil causes greater amplification at non-resonant frequencies.

The response of the Gibson soil layer is compared with that for a homogeneous layer in Figure 5. Figure 5(a) shows the Gibson soil response together with the amplification response of a homogeneous layer whose constant shear wave velocity is 550 m/s, the average shear wave velocity in the Gibson soil layer. Both soils have 5 per cent damping. The resonant frequencies for

z I-- < 0

0 G 2 CL I <

5 . 0

4 . 5

SHEAR MODULUS VAR I A T I ON

\ I .-, - , 1 . 0 ' ' - 4 - <

I I I I I I I I I I l I I I I L I I I I I

0 20 40 60 80 100 120 140 160 180 200

FREQUENCY (rad/sec)

6 . 5

6 . 0

5 . 5

5 . 0

4 . 5

4 . 0

s . 5

3 . 0

2 . 5

2 . 0

1 . 5

I . o

.s

211

Figure 5. Comparison of the Gibson soil response with that for homogeneous soils: (a) comparison with a homogeneous layer having c = 550 m/s. (b) comparison with a homogeneous layer having c = 300 mjs. Damping equals 5 per cent in all

the cases shown

212 SHORT COMMUNICATION

the two situations are clearly similar, but the Gibson soil exhibits significantly high amplification at all excitation frequencies higher than the first resonant frequency. Figure 5(b) compares the Gibson soil response with that for a homogeneous layer having shear wave velocity equal to 300m/s, the smallest shear wave velocity found in the Gibson layer. Five per cent damping applies for both soils. In this case the peak amplifications for the homogeneous layer are significantly stronger at low excitation frequencies due to the greater impedance mismatch at the basement rock interface. The resonant frequencies are quite different for the two cases, and the Gibson soil response becomes stronger at higher frequencies.

CONCLUDING REMARKS

The analysis of multiply layered Gibson soil site response of vertically propagating SH waves presented above is fundamentally no more difficult than the common homogeneous layer analysis. These calculations normally require the use of a digital computer, and the only significant difference between the Gibson and homogeneous soil models lies in the calculation of Bessel functions with complex arguments, a task readily performed by nearly all mathematical analysis packages. In many situations the Gibson soil may be a more realistic representation of true soil properties than is a homogeneous soil model. The use of a homogeneous model with average soil properties may produce correct estimates for resonant frequencies but will underesti- mate the amplification at most excitation frequencies.

REFERENCES

1. W. T. Thomson, ‘Transmission of elastic waves through a stratified solid medium’, J . Appl. Phys., 21, 89-93 (1950). 2. N. A. Haskell, ‘Crustal reflection of plane SH waves’, J . Geophys. Rex, 65, 414774150 (1960). 3. I. Towhata and H. Toyota, ‘Wave propagation on ground in which shear modulus varies continuously with depth’ (in

Japanese), to be published in Proc. Annual Meeting of the Japanese Society of Soil Mechanics and Foundation Engineering, 1993.