GSP 160 Dynamic Response and Soil Properties
Effect of underground structures in earthquake resistant design of
surface structures.
P. Yiouta-Mitra1, G. Kouretzis2, G. Bouckovalas2, A. Sofianos1
1
Tunnelling Laboratory, Division of Mining, National Technical University, Iroon
Polytechniou 9, 15780, Athens, Greece.
2
Geotechnical Division, National Technical University, Iroon Polytechniou 9, 15780,
Athens, Greece.
ABSTRACT
This research deals with the question of whether, and under what circumstances, the
presence of underground structures should be taken into account for the earthquake
resistant design of neighbouring surface structures. In order to investigate the effect
of underground structures on surface seismic motion, a series of dynamic planestrain numerical analyses were conducted, considering a circular tunnel embedded in
a viscoelastic half-space, and a harmonic SV-wave excitation. The numerical
methodology, based on the Finite Difference Method, aims at quantifying the effect
of the soil medium characteristics, excitation frequency, tunnel diameter, depth of
construction, and relative flexibility of the lining compared to that of the surrounding
soil. Conclusions contain preliminary criteria identifying the cases when the presence
of an underground structure should be considered in the design of a surface structure.
INTRODUCTION
The effect of the construction of underground structures, such as metro tunnels, on
aboveground buildings has been examined thoroughly in the past, focusing mainly
on the resulting surface settlements. However, the effect that the presence of a
massive circular tunnel has on the seismic design of nearby buildings is neglected in
current practice.
Nevertheless, many researchers (e.g Lee and Trifunac, 1979, Manoogian & Lee,
1996, Manoogian, 1998, Lee & Karl, 1992, Barros & Luco, 1993 and 1994, Lee et al,
2001) have examined analytically the effect that the presence of a circular
underground structure, embedded in an elastic half-space, has on the surface ground
motion. Summarizing their findings the following can be said:
Maximum amplification of the ground surface motion, consisting of SVwaves, due to the presence of an underground unlined circular structure
varies between 75% - 100%.
The presence of a cavity can cause, under certain conditions, intense and
selective de-amplification of the free-field motion. This phenomenon is
referred to as “shadow zone”.
The presence of an underground structure results also in a parasitic vertical
component of surface motion, with significant amplitude.
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GSP 160 Dynamic Response and Soil Properties
The most important parameters, rising from the analytical results of the
aforementioned researchers to be influencing the response at the ground surface, are:
The ratio of the depth of structure axis over the radius of structure, H/ . For
smaller depths, not only the amplification due to the underground cavity is
larger, but also seismic waves are “trapped” between the cavity and the
ground surface, thus producing more complex response patterns.
The dimensionless frequency n=2 / , where is the wavelength. Higher
values of n generally produce higher amplifications and more complex
response patterns.
The dimensionless distance from the structure axis on the ground surface,
x/ , where is the radius of the structure. The affected wave field, resulting
from the presence of the underground structure, attenuates at low rates and is
approximately analogous to the inverse square root of the distance from the
structure. However, analytical methods proposed in the literature have
produced results limited to x/a<4.
The relative stiffness of the structure, compared to the surrounding soil, as
stiffer structures appear to produce lower amplifications.
The analytical solutions proposed in the literature provide a clear overview of the
problem, and the basic parameters influencing the ground response. Nevertheless, as
they are based on complex mathematics, they cannot be used for parametric analyses
to examine the effect of a wide range of values of the aforementioned parameters.
For this reason, in the present work the problem is treated numerically, with the aid
of the commercially available Finite Difference Code FLAC (Itasca, 1999).
Following the verification of the numerical model, a series of parametric analyses are
presented, aiming at the definition of preliminary criteria identifying the cases when
the presence of an underground structure should be considered in the design of
surface structures.
NUMERICAL MODEL AND VERIFICATION
The numerical model is verified against the analytical results of Luco and De
Barros (1994), who used an indirect boundary integral method based on twodimensional Green’s functions, to obtain the harmonic motion on the ground surface.
Their model consisted of an infinitely long unlined cavity of circular cross-section
with radius , parallel to the free surface of the half-space and embedded at a depth
H. The soil was treated as a viscoelastic medium, characterized by density and
propagation velocities of P and SV waves equal to a and b respectively. The
simplifying assumption for hysteretic damping a= b= was also adopted.
In this paper, the numerical model was constructed along the above lines in order to
provide the means of verification. The commercially available Finite Difference
Code FLAC (Itasca, 1999) was selected to perform the analyses. The boundary
conditions applied at the artificial boundaries were a) lateral dashpots to minimize
wave reflections and achieve free-field conditions, b) absorbing boundaries at the
bottom i.e. normal and shear dashpots of coefficient c= CsVs (Kramer, 1996) to
represent the effect of radiation damping and c) a stress boundary of amplitude
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GSP 160 Dynamic Response and Soil Properties
xy=2
CsVs at the bottom to simulate the incoming harmonic SV- wave. The factor of
2 in the above relation corresponds to the fact that half of the input energy is
absorbed by the viscous boundary, while in the above relations Cs is speed of S-wave
propagation through medium, is the mass density and Vs the seismic particle
velocity. Despite the existence of a vertical plane of symmetry, the full model was
used due to the limitations of the numerical code with respect to lateral dashpots.
Hysteretic damping is simulated in an approximate fashion as Rayleigh damping
according to the guidelines of Itasca (1999) and Sofianos (2003).
Figure 1 depicts a sample of the satisfactory comparison with the Luco & De
Barros analytical model. Displacement amplitudes are normalized against the
amplitude of the incident motion on the free-field ground surface Us.
Figure 1. Comparison of numerically computed normalized horizontal displacements
on the ground surface, with the results of Luco & De Barros (1994). The
case shown here applies to =0.001, n=0.5, =1/3.
PARAMETRIC ANALYSES
A series of parametric analyses were conducted to quantify the effects of
parameters H/
(dimensionless depth), x/
(dimensionless distance), n
(dimensionless frequency) and J (index of relative flexibility of the lining compared
to that of the surrounding soil), as discussed in the introduction.
The range of these parameters was based on common characteristics of
underground structures and seismic wave characteristics representing high, as well as
low frequency excitations. Table 1 contains the range of the structure, soil and
motion characteristics examined in the analyses. The combinations of these
parameters give a range of 0.05÷1 for the dimensionless frequency n, while the
dimensionless depth H/ was given values of 1÷7. Naturally, since the depth of one
radius would apply to a structure emerging at the surface, it can no longer be
considered as underground. Thus, it has been included only for reasons of
completeness.
The index of relative flexibility was calculated with the following equation (St.
John and Zahrah, 1987), considering a concrete structure lining with Elining = 30 GPa
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GSP 160 Dynamic Response and Soil Properties
and lining = 0.2. Combined to the soil characteristics, it yields two values, namely J=5
and J=150.
3
2
2 E soil (1 lining
) Rlining
J=
3
Elining (1 + soil )t lining
he extent of the retained strong ground motion was investigated for 18 structure
radii. Finally, a special parameter that was separately examined was the hysteretic
damping of the soil. Three values compatible with the elastic model were selected,
namely =1%, 2% and 5%, corresponding to soft rock-stiff soil formations. Results
have shown no practical difference between the three values, therefore =2% was
adopted in all analyses.
Table 1. Range of physical parameters employed in the numerical analyses
Description
Circular unlined tunnel
Circular lined tunnel
Figure
Dimensions (m)
Shear wave velocity,
Vs (m/sec)
Harmonic excitation
frequency, T (sec)
D = 5÷20
D = 5÷20, d = 4÷17
200÷1000
0.05 ÷1.0
RESULTS
A selection of diagrams is depicted from the results, where the normalized
amplitudes of ground surface displacements are plotted against the horizontal
distance from the structure axis of symmetry. Normalization has been achieved by
dividing the results with the free-field horizontal displacement amplitude even in the
case of vertical parasitic displacements, since there is no free-field vertical
component of motion.
1. Effect of dimensionless frequency n
Low frequency values such as n=0.05, 0.1, that infer high values of wavelength as
compared to the tunnel dimensions, result in wave transmission practically
unaffected by the structure. For n=0.2 and higher, the response changes dramatically.
De-amplifications in the shadow zone reach 55% and amplifications 30% (Fig. 2).
For n=0.4–1.0 the shadow zone becomes prominent (Fig. 3). Similar remarks can be
drawn for the vertical, parasitic, component of motion. Higher frequencies cause a
vertical component that cannot be ignored as in lower frequencies. Fig. 4 depicts the
highest of these values at 90% of the free-field horizontal component.
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GSP 160 Dynamic Response and Soil Properties
Figure 2. Normalized horizontal displacement amplitudes for n=0.2, H/ =2
Figure 3. Normalized horizontal displacement amplitudes for n=1.0, H/ =2
Figure 4. Normalized vertical displacement amplitudes for n=0.4, H/ =2
2. Effect of dimensionless depth H/
For depths larger than two radii, the horizontal component amplification is no more
than 25%. For depths larger than 7 radii it is further reduced to 15% (Fig. 6). It is
therefore confirmed that only for H/ 2 (Fig. 5) does any amplification arise, with
maximum amplifications occurring at H/ =1 and reaching 85%. Observations are
somewhat different in the case of the parasitic vertical component of motion. At
H/ =1.5, maximum amplifications for all frequencies are between 100-175 %.
Moreover, at a depth of three radii there is still a 70% amplification for n=0.6 (Fig. 7)
and even at four radii there is a 50% amplification for n=0.4. The effect attenuates
faster and drops at 15% at a depth of seven radii.
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GSP 160 Dynamic Response and Soil Properties
Figure 5. Normalized horizontal displacement amplitudes for n=0.2, H/ =1.5.
Figure 6. Normalized horizontal displacement amplitudes for n=0.8, H/ =7.
Figure 7. Normalized horizontal and vertical displacement amplitudes for n=0.6,
H/ =3.
3. Effect of relative flexibility of structure J
At small depths, it can be observed from Figs, 8, 9 and 12 that decreasing values of
J tend to eradicate the effect of the presence of the tunnel. At larger depths, the effect
of the tunnel lining is practically nullified. The last remark can be only cautiously
extended to all frequencies for the case of horizontal component of motion. As can
be seen from Figures 10 and 11, despite the large depth of six radii, increasing
frequencies result in horizontal amplifications larger even than those of the unlined
tunnel. More specifically, for J=5, when n=1.0 and H/ =6.0, ANx fluctuates from
0.45 to 1.38, while the same analyses for an unlined tunnel resulted in amplifications
of 0.54 to 1.17. With respect to the vertical components however, the increase of the
structure stiffness always results in a decrease of amplifications and more
prominently so at small depths.
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GSP 160 Dynamic Response and Soil Properties
Figure 8. Normalized horizontal displacement amplitudes for n=0.2, H/ =2 for an
unlined cavity (J=inf), very flexible lining (J=150) and stiff lining (J=5).
Figure 9. Normalized horizontal displacement amplitudes for n=0.2, H/ =4 for an
unlined cavity (J=inf), very flexible lining (J=150) and stiff lining (J=5).
Figure 10. Normalized horizontal displacement amplitudes for H/ =6 n=0.4 for an
unlined cavity (J=inf), very flexible lining (J=150) and stiff lining (J=5).
Figure 11. Normalized horizontal displacement amplitudes for H/ =6 n=1.0 for an
unlined cavity (J=inf), very flexible lining (J=150) and stiff lining (J=5).
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GSP 160 Dynamic Response and Soil Properties
Figure 12. Normalized vertical displacement amplitudes for n=0.4, H/ =2 and 5 for
an unlined cavity (J=inf), very flexible lining (J=150) and stiff lining (J=5).
CONCLUSIONS
In order to extract preliminary criteria identifying the cases when the presence of an
underground structure can or cannot be neglected in the design of a surface structure,
all results were evaluated via combined plots of the maximum amplification values
and of the location of that maxima on the ground surface. Figures 13 and 14 are only
samples of such plots, as the complete results cannot be shown due to space
limitations. From Figure 13 it is drawn that maximum amplification of the horizontal
motion reaches about 1.5 for a shallow (H/ <2) unlined tunnel. In Figure 14 it may
be seen that for small depths (H/ <2), only the area close to the tunnel is affected
(0<x/ <1.0). For 2<H/ <4 the affected area extends to 5 radii and for larger depths it
starts at 2 and extends to 7 radii.
Summarizing observations from all relevant diagrams, the following can be said for
an unlined tunnel:
Ground response for excitations with wavelengths larger than the tunnel
diameter is not affected by the presence of the tunnel.
The presence of a tunnel results in amplifications that should be considered
in the design of a surface structure for ratios H/ <3.0.
The horizontal component of motion can be amplified by 20% to 85% within
a distance of eleven radii from the tunnel axis (0<x/ <11).
The ground response is further complicated by the appearance of a parasitic
vertical component of motion. Maximum vertical displacement amplitudes
occur within four radii from the tunnel axis (0<x/ <4), reaching values of 1/3
up to 3 times the value of horizontal free-field component.
From the analyses for a lined tunnel, the following observations can be made:
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GSP 160 Dynamic Response and Soil Properties
Stiffer linings generally attenuate the effect of the tunnel presence on the
ground response.
For n>0.8 and H/ >6.0 stiffer structures cause unexpected for such depths
amplifications of about 30%.
The horizontal component of motion can be amplified from 20% to 40%.
The vertical component of motion reaches values from 1/3 to 1.5 times the
value of the horizontal free-field component.
The value of the relative stiffness does not alter the affected surface area,
which therefore coincides with that of the unlined tunnel.
Figure 13. Combined plot of maximum normalized horizontal displacement
amplitudes due to the presence of an unlined tunnel, for all depths and frequencies.
Figure 14. Combined plot of the maxima amplification location, due to the presence
of a flexible tunnel (J=150), for all depths and frequencies.
ACKNOWLEDGEMENTS
The research was funded by Earthquake Planning and Protection Organization, Greece.
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GSP 160 Dynamic Response and Soil Properties
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