1 s2.0 S2467967416300290 main
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Underground Space 2 (2017) 125–133
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Effect of frequency and flexibility ratio on the seismic response of
deep tunnels
Eimar Sandoval ⇑, Antonio Bobet
Lyles School of Civil Engineering, Purdue University, West Lafayette 47906, United States
School of Civil Engineering and Geomatics, Universidad del Valle, Cali, Colombia
Received 17 December 2016; received in revised form 6 April 2017; accepted 12 April 2017
Available online 1 June 2017
Abstract
Two-dimensional dynamic numerical analyses have been conducted, using FLAC 7.0, to evaluate the seismic response of underground structures located far from the seismic source, placed in either linear-elastic or nonlinear elastoplastic ground. The interaction
between the ground and deep circular tunnels with a tied interface is considered. For the simulations, it is assumed that the liner remains
in its elastic regime, and plane strain conditions apply to any cross section perpendicular to the tunnel axis. An elastoplastic constitutive
model is implemented in FLAC to simulate the nonlinear ground. The effect of input frequency and relative stiffness between the liner
and the ground, on the seismic response of tunnels, is evaluated. The response is studied in terms of distortions normalized with respect
to those of the free field, and load demand (axial forces and bending moments) in the liner. In all cases, i.e. for linear-elastic and nonlinear
ground models, the results show negligible effect of the input frequency on the distortions of the cross section, for input frequencies smaller than 5 Hz; that is for ratios between the wave length and the tunnel opening (k=D) larger than ten for linear-elastic and nine for nonlinear ground. Larger normalized distortions are obtained for the nonlinear than for the linear-elastic ground, for the same relative
stiffness, with differences increasing as the tunnel becomes more flexible, or when the amplitude of the dynamic input shear stress
increases. It has been found that normalized distortions for the nonlinear ground do not follow a unique relationship, as it happens
for the linear-elastic ground, but increase as the amplitude of the dynamic input increases. The loading in the liner decreases as the structure becomes more flexible with respect to the ground, and is smaller for a tunnel placed in a stiffer nonlinear ground than in a softer
nonlinear ground, for the same flexibility ratio.
Ó 2017 Tongji University and Tongji University Press. Production and hosting by Elsevier B.V. on behalf of Owner. This is an open access article
under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Deep circular tunnel; Dynamic numerical analysis; Flexibility ratio; Distortion; Seismic response
1. Introduction
Underground structures must be able to support static
overburden loads, as well as to accommodate additional
deformations imposed by seismic motions. Progress has
⇑ Corresponding author at: Lyles School of Civil Engineering, Purdue
University, West Lafayette 47906, United States.
E-mail addresses: esandov@purdue.edu, eimar.sandoval@correounivalle.edu.co (E. Sandoval), bobet@ecn.purdue.edu (A. Bobet).
Peer review under responsibility of Tongji University and Tongji
University Press.
been made in the last few years in understanding the soilstructure interaction mechanisms and the stress and displacement transfer from the ground to the structure during
a seismic event. For most tunnels, with some exceptions
including submerged tunnels, it seems well established that
the most critical demand to the structure is caused by shear
waves traveling perpendicular to the tunnel axis (Bobet,
2003; Hendron and Fernández, 1983; Merritt et al., 1985;
Wang, 1993). Those waves cause distortions of the cross
section (ovaling for a circular tunnel, and racking for a
http://dx.doi.org/10.1016/j.undsp.2017.04.003
2467-9674/Ó 2017 Tongji University and Tongji University Press. Production and hosting by Elsevier B.V. on behalf of Owner.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
126
E. Sandoval, A. Bobet / Underground Space 2 (2017) 125–133
rectangular tunnel) that result in axial forces (thrusts) and
bending moments.
Two approaches have been used to evaluate the seismic
response of underground structures. One is the free field
approach (Hendron and Fernández, 1983; Kuesel, 1969;
Merritt et al., 1985; Newmark, 1967), which assumes that
the structure follows the free field deformations of the
ground, and therefore accommodates them without loss
of its integrity. Authors supporting this approach have suggested computing deformations considering the perforated
ground, given that most tunnels in competent ground
would behave as perfectly flexible structures. This assumption however may result in extremely conservative designs,
especially for stiff structures in a soft medium. The second
approach is the soil-structure interaction approach (Bobet,
2003, 2010; Huo et al., 2006; Penzien, 2000; Wang, 1993).
This approach, which is discussed in this paper, states that
the underground structure modifies the free field deformation of the ground around it such that demand and
response depend on the relative stiffness between the
ground and the tunnel support. Two dimensionless coefficients have been proposed to consider soil-structure interaction, namely the flexibility ratio and the compressibility
ratio (Einstein and Schwartz, 1979; Peck et al., 1972).
The flexibility ratio (F), or flexural stiffness, is a measure
of the resistance of the system to change shape (i.e., distort)
under a state of pure shear. The compressibility ratio (C),
or extensional stiffness, is a measure of the all-around uniform pressure required to cause diametric strain without
change in shape. As the seismic shear waves distort the
cross section of the tunnel, the flexibility ratio (F) is
the main coefficient of interest in the evaluation of the seismic response of the cross-section of underground
structures.
Analytical studies by Paul (1963), Mow and Pao (1971),
Yoshihara (1963), Hendron and Fernández (1983), Merritt
et al. (1985) showed that the dynamic amplification of
stress waves impinging on a cavity is negligible when the
wave length (k) of the seismic peak velocities is at least
eight times larger than the width of the opening. This is
usually the case for most structures located far from the
seismic source. According to Dowding (1985) a far-field
motion can be considered for distances from the seismic
source between 10 and 100 km, with predominant frequencies between 0.1 and 10 Hz. With this assumption, pseudostatic numerical analyses have been carried out, and a number of closed-form solutions have been proposed for
seismically-induced motions (Bobet, 2003, 2010; Huo
et al., 2006; Penzien, 2000; Wang, 1993). These solutions
are based on elasticity (Timoshenko and Goodier, 1970),
on early work by Peck et al. (1972), and on the relative stiffness method of Einstein and Schwartz (1979).
The closed-form solutions are suitable for deep tunnels
placed in an infinite, linear-elastic, homogeneous, and isotropic medium. A tunnel is considered deep if the stress
gradient with depth has negligible or no effect on its behavior. A plane strain condition, which is reasonable for
sections located far from the face of the tunnel, is also
assumed. The previous work has shown that the most
important parameter determining the distortions of a cross
section of a tunnel is the relative stiffness between the medium and the liner (expressed by the flexibility ratio, F), and
that the depth and shape of the structure have secondorder effects (Bobet, 2010). This conclusion however has
been obtained considering linear-elastic ground. The more
realistic seismic response of tunnels when considering the
nonlinear stress-strain behavior of the ground under cyclic
loading has not been evaluated.
The paper provides results of 2D plane strain dynamic
numerical analyses, conducted in FLAC 7.0 (Itasca,
2011a), for deep circular underground structures subjected
to vertically traveling shear waves, produced by a sinusoidal input velocity. In the analyses, the liner is assumed
elastic with a tied interface, i.e. no relative displacement
between the structure and the ground. To evaluate the
effect of the flexibility ratio on the seismic response without
additional variables, the same interface is used in all the
analyses. A tied interface is selected, as it includes a combined normal-shear stress transmission at the interface.
The ground is considered dry (drained) i.e., no excess pore
pressures are generated, with either linear-elastic or nonlinear elastoplastic response. In both cases, the effect of the
input frequency on the distortions of the tunnel cross section is evaluated, as well as the effect of relative stiffness
on the distortions of the liner. For the nonlinear ground,
the loadings in the structure (thrusts and bending
moments) are also obtained. Comparisons in terms of distortions of the liner, for the linear-elastic and the nonlinear
elastoplastic ground, are provided.
2. Linear elastic analysis
The ground-structure system is discretized in FLAC
with meshes with dimensions 600 m wide and 100 m or
50 m high, to evaluate the effect of input frequency and
the effect of flexibility ratio, respectively. The reasons to
select these two different heights are explained later. For
both meshes, square elements (1 m 1 m) are used away
from the tunnel, and square elements with a smaller size
(0.5 m 0.5 m), close to the tunnel, within a rectangular
section of 60 m 40 m. The center of the tunnel is located
25 m above the bottom of the model. Fig. 1 shows the
meshes used for the dynamic numerical analyses.
A sinusoidal input velocity is imposed at the bottom of
the discretization. Free-field boundaries on the sides and
quiet boundaries at the bottom of the model are used.
These are absorbing boundaries that use independent dashpots in the normal and shear directions, to prevent reflection of waves back into the model and avoid energy
radiation (these boundaries are a built-in option in FLAC).
By using absorbing boundaries, plane waves propagating
upward do not distort at the lateral boundaries and do
no reflect at the bottom boundary. (Itasca, 2011b;
Lysmer and Kuhlemeyer, 1969).
E. Sandoval, A. Bobet / Underground Space 2 (2017) 125–133
127
Fig. 1. Meshes configuration for evaluation of: (a) effect of input frequency; (b) effect of flexibility ratio.
The appropriate width of the model and good performance of the absorbing boundaries are verified by comparing shear strains at different points through the width of the
mesh, for specific depths. Although not included here, the
results show negligible differences at points located up to
80 m on each side of the center of the model. So the proposed discretization is considered adequate.
The tunnel support is modeled with elastic beam elements. Beam elements are frequently used in this type of
simulations because, similar to beams, liners must support
axial forces and bending moments (e.g., Bobet, 2010;
Wang, 1993).
2.1. Effect of input frequency on tunnel’s distortions
Previous research has found, when the ground’s
response is elastic, that the dynamic amplification of stress
waves impinging on a tunnel is negligible when the rise time
of the pulse is larger than about two times the transit time
of the pulse across the opening; that is, when the wave
length (k) of peak velocities is at least eight times larger
than the width (B) or diameter (D) of the opening
(Hendron and Fernández, 1983; Mow and Pao, 1971;
Merritt et al., 1985; Paul, 1963; Yoshihara, 1963). The following numerical analysis is intended to evaluate the previous finding and to use its results for comparison when the
response of the ground is nonlinear.
Distortions for circular tunnels with 4.0 m in diameter
and 0.40 m in thickness are obtained. The mesh has dimensions 600 m 100 m. The depth of the model is selected to
avoid the effect of waves reflection from the free surface,
especially for input frequencies equal or lower than
1.0 Hz. According to Itasca (2011b), the depth of the model
should be larger than one third to one fourth of the associated wave length, to avoid this effect. The amplitude of the
input velocity is 0.1 m/s, with frequencies ranging from
0.25 to 15 Hz. These frequencies correspond to k=D ratios
ranging from 200 to 3.3, given the dimensions chosen for
the tunnel. The wave length (k) is obtained as the ratio
between the shear wave velocity in the medium (C s ) and
the frequency of the dynamic input (f).
The elastic properties of the medium, shear modulus
(Gm ) and Poisson’s ratio (mm ), are assumed to be 80 MPa
and 0.25 respectively. For the liner, a value of 0.15 is used
for the Poisson’s ratio (ms ). A stiff tunnel with flexibility
ratio (F) equal to 0.125, and a flexible liner with F = 12.5
are used. The corresponding Young’s moduli for the liner’s
material (Es ) are 3.13 105 and 3.13 103 MPa, respectively. In this paper, the definition of flexibility ratio provided by Peck et al. (1972), shown in Eq. (1), is used.
F ¼
Em =ð1 þ mm Þ
6Es Is=ðR3 ð1 m2s ÞÞ
ð1Þ
where Em is the Young’s modulus of the medium, I s is the
moment of inertia of the liner per unit length, and R is the
radius of the tunnel.
Fig. 2 shows the results of both relative stiffnesses, when
the wave length or input frequency is changed. The inset in
the figure shows a schematic of the ovaling distortions for a
circular support. The figure plots the maximum distortions,
which occur at maximum amplitude of the sinusoidal input
velocity, normalized with respect to the ground distortions
in the free-field, i.e. distortions of the ground without the
liner. The normalized distortions are expressed as a function of the ratio k=D (bottom horizontal axis). For compar-
128
E. Sandoval, A. Bobet / Underground Space 2 (2017) 125–133
Fig. 3. Normalized distortions for circular tunnels placed in linear-elastic
medium.
Fig. 2. Effect of input frequency on distortions of circular tunnels placed
in linear-elastic medium.
ison purposes, the top horizontal axis presents the values of
the input frequency (f).
Fig. 2 shows that the more flexible liner (F = 12.5) has
larger normalized distortions than the stiffer liner
(F = 0.125). This behavior is expected as the flexible structure deforms more than the stiff structure and also more
than the soil it replaces. Small differences in response are
observed with the input frequency, depending on the flexibility ratio. For the stiff structure (F = 0.125), the normalized distortions are almost constant for k=D ratios larger
than ten (f < 5.0 Hz). For lower k=D ratios (higher f), a
small reduction in the normalized distortions is observed.
For the flexible structure on the other hand, negligible differences in the normalized distortions are observed for k=D
ratios larger than ten (f < 5.0 Hz). Afterwards, the normalized distortions increase up to a point, and then decreases
as the k=D ratios decrease (f increases). Overall, the effect
of frequency is negligible for k=D ratios larger than ten,
i.e., for input frequencies (f) lower than 5.0 Hz.
These results, obtained through fully dynamic numerical
analyses, confirm the assumption made for analytical solutions that a pseudo-static analysis is sufficient when the
ratio between the wave length and the opening of the liner
is larger than eight (ten in this paper), if a linear-elastic
medium is considered.
2.2. Effect of relative stiffness on tunnel’s distortions
Previous work, based on analytical solutions and
numerical analyses, has found that the main parameter
controlling the distortions of the cross section of the liner
is the relative stiffness between the structure and the ground
(e.g., Bobet, 2010; Penzien, 2000; Wang, 1993). To extend
the database available for linear-elastic ground, and to
have results that can be compared with those of the nonlinear ground in Section 3, dynamic numerical analyses for a
sinusoidal input velocity with 0.1 m/s of amplitude and
2.5 Hz of frequency are performed. Note that, in the preceding section, it was shown that at this frequency the
results are frequency-independent. The applied input velocity corresponds to an input shear stress equal to 40 kPa.
For the input frequency used (2.5 Hz), the depth of the
model can be taken as 50 m, since no unwanted reflections
from the surface are produced. This decreases the computation time of the simulations. Consequently, the effect of
the relative stiffness, for the linear-elastic ground and for
the nonlinear ground, is evaluated for a mesh
600 m 50 m. In this section, the same tunnel geometry
and elastic properties of the ground considered in the preceding section are used.
Fig. 3 shows the normalized maximum distortions, for
flexibility ratios ðF Þ ranging from 0.125 (a relatively very
stiff tunnel) to 15 (a relatively very flexible tunnel). It can
be seen in Fig. 3 that the normalized distortions increase
as the flexibility ratio increases. As stated in Section 2.1,
this trend is anticipated because as F increases, the tunnel
becomes much more flexible than the soil it replaces. For
an elastic analysis, the magnitude of the dynamic input
should not affect the results, due to the normalization.
Fig. 3 corroborates this assumption by including results
for three different flexibility ratios (F = 0.25, 1 and 12.5)
with dynamic input that changes by a factor of 3.
The flexibility ratio used in this paper, and proposed by
Peck et al. (1972), provides a good measure of the stiffness
of the structure relative to the ground. A value of F ¼ 1
represents a structure with stiffness equal to that of the
ground that it replaces, and so the distortions of the structure are those of the free field (normalized distortions equal
to 1.0). A value F > 1 represents a structure more flexible
than the ground, and the distortions of the structure are
larger than those of the free field. A value F < 1 represents
a structure stiffer than the ground, and the distortions are
smaller than those in the free field.
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E. Sandoval, A. Bobet / Underground Space 2 (2017) 125–133
as a function of the change in the effective confinement
(dr0m ), with r0m = 1/3(rkk ).
3. Nonlinear elastoplastic analysis
Soils and other geomaterials usually behave as elastic
only at very small strains. In general, those materials show
a nonlinear stress–strain behavior with strain, with a reduction of stiffness with increased strains. As a result, higher
strains in the free field are anticipated for nonlinear
ground, compared to a linear-elastic ground. Consequently, given that the response of a tunnel under seismic
loading is a soil-structure interaction problem, differences
in the seismic response are expected.
It must be noted that due to the stiffness degradation of
the ground, the flexibility ratio changes during the dynamic
loading. More precisely, if the liner remains in its elastic
regime F decreases, as the ground becomes more flexible
with deformation, i.e., the tunnel becomes stiffer with
respect to the ground as the ground deforms. The flexibility
ratios reported here are those computed using the degraded
stiffness of the medium at peak distortions of the tunnel.
The free field shear strains used for the normalization correspond to those of the degraded ground.
As with the elastic ground case, the effect of the input
frequency and the effect of the flexibility ratio on the distortions of the cross section are investigated. The effect of flexibility ratio on loadings of the liner is also evaluated. The
distortions obtained are then compared to those found
for a linear-elastic medium. A constitutive model to represent the nonlinear elastoplastic ground is presented and
implemented in FLAC.
3.1. Cyclic nonlinear elastoplastic model
The elastoplastic model proposed is based on that developed by Jung (2009) and modified by Khasawneh et al.
(2017), and is intended to simulate the nonlinear behavior
observed in geological materials, even for small shear
strains around 10 5 (Dobry et al., 1981; Vucetic, 1994).
The nonlinear behavior is simulated with a modified hyperbolic relationship. Jung (2009) proposed to obtain the
degraded shear stiffness of the ground (G) by means of
Eq. (2), as a function of the modified hyperbolic strain
(ch ) proposed by Hardin and Drnevich (1972), which is
given in Eq. (3).
ds
1
¼ Go
G¼
2
dc
ð1 þ ch Þ
i
c
ch
1 þ aexp bðcr Þ
ch ¼
cr
ð2Þ
ð3Þ
where cr ¼ sf =Go is the reference shear strain, sf is the
shear stress at failure, Go is the shear modulus at small
strains, c is the current shear strain, and a and b are soil
constants that determine the deviation of the stress–strain
relation from the hyperbolic function.
The model also includes the dependence of stiffness on
confinement. Jung (2009) proposed Eq. (4) to calculate
the change in the shear modulus for small strains (dGo ),
dGo ¼
2
1 Goðref Þ 1
dr0
2 r0mðref Þ Go m
ð4Þ
where r0mðref Þ is the reference effective mean stress (for geostatic conditions), and Goðref Þ is the corresponding reference
small strain shear modulus.
The hysteretic behavior during cyclic loading is based on
Pyke (1979), who proposed a modification to the Masing’s
rules. Jung (2009) provided Eq. (5) to calculate the stiffness
reduction in the hysteresis loop in the three dimensional
space. Eq. (6) is an extension hyperbolic strain (Eq. (3))
in 3D and is based on the octahedral shear strain (coct ),
defined in Eq. (8). This modification is required as the original hyperbolic strain proposed by Hardin and Drnevich
(1972) was obtained from 1D cyclic simple shear tests.
G ¼ Go
1
coct
coct;r
coct;rev h
1 þ aexp
Crev ¼
coct;r
rffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffi
2 1
eij eij
coct ¼ 2
3 2
1
kk dij
eij ¼ ij
3
1 @ui @uj
þ
ij ¼
2 @xj @xi
C¼
ð5Þ
2
1þ
Crev j
h
i
c
bð oct Þ
1 þ aexp coct;r
1
jC
n
bð
coct;rev
coct;r Þ
ð6Þ
i
ð7Þ
ð8Þ
ð9Þ
ð10Þ
where n is a scaling factor for the hysteresis loop (n = 2 to
satisfy Masing rules or n – 2 for non-symmetric cyclic
loading), coct;r is a reference octahedral shear strain, coct;rev
is the octahedral shear strain at reversal, eij is the deviatoric
strain tensor, ij is the Lagrangian strain tensor, and ui are
the displacements along the xi axis.
Small strain analysis is assumed, and thus the strain
increment (dij ) is decomposed into its elastic and plastic
components dij ¼ deij þ dpij , with the stress increment
computed from Eq. (11).
drkl ¼ C eklij deij
ð11Þ
where drkl is the incremental stress tensor, C klij is the elastic
modulus tensor, and dij is the incremental elastic strain
tensor.
After yielding, which is defined with the Drucker-Prager
(D-P) criterion, the plastic strains are determined by means
of a strain hardening law with a non-associated flow rule.
Eqs. (12) and (13) show the yield function and the plastic
potential, respectively. More details about the cyclic elastoplastic model, including the complete formulation, implementation and its verification can be found in
Khasawneh et al. (2017).
130
pffiffiffiffiffiffiffi 1
3J 2 þ I 1 tanðaÞ
3
pffiffiffiffiffi 1
I 1 tanðuÞ
g ¼ J2
3
f ¼
E. Sandoval, A. Bobet / Underground Space 2 (2017) 125–133
j
ð12Þ
ð13Þ
where J 2 = 1/2(S ij S ij ) is the second invariant of the deviatoric stress tensor, I 1 ¼ rkk is the first invariant of the stress
tensor, a is the D-P friction angle, j is the D-P cohesion,
and u is the dilation angle.
3.2. Effect of input frequency on tunnel’s distortions
A mesh 600 m 100 m with a circular tunnel placed
25 m above the bottom is used. The liner, assumed to
remain in its elastic regime, has again 4 m in diameter
and 0.4 m in thickness.
A ground with initial stiffness Goðref Þ ¼ 80 MPa, and
Poisson’s ratio mm ¼ 0:25 is used in the simulations. Similar
to the linear-elastic ground, a stiff and a flexible liner are
considered. The Poisson’s ratio for the liner’s material is
assumed ms ¼ 0:15, and the Young’s moduli (Es ) are
1.56 105 and 2.61 103 MPa, for the stiff and the flexible
structure, respectively. Such stiffnesses represent initial flexibility ratios (F) of 0.25 and 15 that are reduced during the
dynamic phase to values of 0.20 and 12.0, respectively (at
peak distortions). The input frequencies range from 0.50
to 15 Hz, which correspond to k=D ratios (considering
the degraded ground) ranging from 90 to 3, given the
dimensions chosen for the tunnels. In the results, displayed
in Fig. 4, it can be seen that negligible changes in the distortions of the liner are observed for k=D ratios larger than
nine (f < 5 Hz), for both the flexible and the stiff structure.
Similar to the linear-elastic ground, for k=D ratios smaller
than nine, a reduction in the normalized distortions is
observed in stiff liners. For flexible liners, an increase of
distortions with a later decrease is detected.
Fig. 4. Effect of input frequency on distortions of tunnels placed in
nonlinear ground.
For completeness, static numerical analyses for both the
flexible and the stiff tunnel are performed. A square model
50 m 50 m with the tunnel located at the center (25 m
from the bottom, as in the dynamic case) is used. In the
analyses, inertial forces are neglected and a simple shear
condition producing the same free field shear strain than
the free field induced by the dynamic loading is used. The
results are included as arrows on the right vertical axis of
Fig. 4. Negligible differences are observed in normalized
distortions obtained from the static analyses, or from
dynamic analyses for k=D ratios larger than nine
(f < 5 Hz). Consequently, for nonlinear ground, a
pseudo-static analysis is feasible for ratios between the
wave length and the tunnel opening larger than nine.
3.3. Effect of relative stiffness on tunnel’s distortions
As before, a mesh 600 m 50 m is used to assess the
effect of the relative stiffness on the tunnel response. The
tunnel is again placed 25 m above the bottom of the model,
and has 4 m in diameter and 0.40 m in thickness.
A parametric study for three different stiffnesses of the
ground (Goðref Þ = 40, 80, and 200 MPa) is performed. These
values represent medium stiff soil, very stiff soil, and weak
rock, respectively. An input velocity with 2.5 Hz frequency
is used. This frequency is selected, based on the results
obtained (shown in Fig. 4), to have the distortions independent of frequency. Given that the magnitude of the shear
stress induced by a dynamic input velocity or acceleration
is directly related to the stiffness of the ground, the amplitude of the input velocity is selected such that the same
input shear stress (s) at the bottom of the discretization is
obtained for the three stiffnesses of the ground investigated.
A shear stress with magnitude 40 kPa is applied, which corresponds to input velocities of 0.14, 0.10 and 0.06 m/s, for
Goðref Þ 40, 80 and 200 MPa, respectively. These velocities
are representative of strong to very strong earthquakes,
with magnitudes between VI and VII in the Modified Mer-
Fig. 5. Normalized distortions for circular tunnels placed in nonlinear
ground, for the same input shear stress amplitude and for different initial
ground stiffness.
E. Sandoval, A. Bobet / Underground Space 2 (2017) 125–133
calli Intensity Scale, according to the classification provided by Wald et al. (1999). The initial flexibility ratios
(F) range from 0.10 to 15, which result in final flexibility
ratios (F) ranging between 0.08 and 12, due to the stiffness
reduction in the ground with deformation.
The results of the simulations are shown in Fig. 5. For
comparison purposes, the plot also includes results using
a linear-elastic medium that has the same stiffness as that
of the degraded ground of the nonlinear analyses. It can
be seen in the figure that, for the same flexibility ratio,
higher normalized distortions occur with the nonlinear
ground, with differences increasing as the tunnel becomes
more flexible. This behavior is expected, as the nonlinear
ground surrounding the structure degrades, and so there
is a higher strain demand on the structure. For the nonlinear ground, it can also be seen that normalized distortions
are almost the same when the same input shear stress is
used, irrespective of the initial stiffness of the ground.
To verify whether the relationship between distortions
and flexibility ratio for a nonlinear ground can be
expressed by means of a single curve, as it is done for a
linear-elastic ground (see Fig. 3), analyses are performed
for different amplitudes of the input shear stress. The
amplitude of the input shear stress used in Fig. 5 is
increased by a factor of two and three, while the other
parameters are kept constant. Fig. 6 compares the normalized distortions for Goðref Þ = 80 MPa for the three cases.
Important differences in the normalized distortions are
observed when the amplitude of the input stress is changed,
for the same final (degraded) flexibility ratio. It can also be
seen in Fig. 6 that the differences with respect to the linearelastic case increase as the amplitude of the dynamic input
increases. Hence, there is not a unique relationship between
the flexibility ratio and the normalized distortions when a
nonlinear ground is assumed, in contrast to what happens
with a linear-elastic ground where the relationship is
unique. Indeed, the nonlinear ground, under a higher input
stress, will deform more than under a lower input stress,
Fig. 6. Normalized distortions for circular tunnels placed in nonlinear
ground, for different input shear stress amplitude and for the same initial
ground stiffness.
131
and so its stiffness will be smaller, which will result in larger
deformation demands to the structure.
3.4. Effect of relative stiffness on liner’s loading
The effect of the flexibility ratio on the thrusts and bending moments of the liner for a nonlinear ground is evaluated, by considering two different stiffnesses of the
ground, namely G0ðref Þ = 80 and 200 MPa.
Figs. 7 and 8 show the results for thrusts and bending
moments, respectively. In the figures, the thrusts are normalized by the product of the input shear stress and the
radius of the tunnel (sR); and the bending moments are
normalized by the product of the input shear stress and
the square of the radius of the tunnel (sR2 ), as proposed
by Einstein and Schwartz (1979).
It can be seen in the figures that larger thrust and bending moments occur for stiffer structures, i.e., for lower flexibility ratios (F), which decrease as the tunnel becomes
Fig. 7. Normalized axial forces (thrusts) for circular tunnels placed in
nonlinear ground.
Fig. 8. Normalized bending moments for circular tunnels placed in
nonlinear ground.
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E. Sandoval, A. Bobet / Underground Space 2 (2017) 125–133
more flexible (as F increases). In fact, almost negligible
bending moments are obtained for flexibility ratios larger
than 10, as the structure approaches a perfectly flexible tunnel (Peck et al., 1972) that cannot withstand bending
moments. These results support the statements made by
previous authors (e.g., Kuesel, 1969; Monsees & Merritt,
1991) in that increasing the stiffness of the structure is
not the best solution for seismic design, as stiffer structures
will take greater loads. For the seismic design, these
authors suggested increasing the ductility of the liner to
absorb the imposed seismic distortions, rather than increasing the stiffness or strength of the liner. It can also be seen
in Figs. 7 and 8 that for the same flexibility ratio, smaller
thrust and bending moments are observed for the stiffer
nonlinear ground than for the softer nonlinear ground, as
the first deforms less.
4. Summary and conclusions
The paper evaluates the dynamic response of deep circular tunnels placed in linear-elastic and nonlinear elastoplastic ground. Dynamic numerical analyses using FLAC 7.0
are performed. For the elastoplastic ground, a constitutive
model based on hyperbolic behavior is implemented in
FLAC. The effect of the input frequency on the distortions
of the cross section and the effect of the flexibility ratio on
the distortions of the liner are evaluated for linear-elastic
and nonlinear ground. For nonlinear ground, the effect of
the flexibility ratio on the loadings of the liner is also investigated. Plane strain conditions and deep tunnel with a tied
interface are assumed.
The results show that for both linear-elastic and nonlinear elastoplastic ground, there is no effect of the input frequency on the seismic response of the tunnel, when the
input frequency is smaller than 5 Hz (k=D ratios larger than
ten for linear-elastic and nine for nonlinear ground).
Pseudo-static analyses can therefore be conducted for tunnels located far from the seismic source (for distances
between 10 and 100 km from the epicenter, after
Dowding, 1985), irrespective of the soil model used in the
simulations.
As it has been found in previous work with the assumption of linear-elastic ground (e.g., Bobet, 2010; Wang,
1993), normalized distortions (tunnel distortions normalized by those of the ground far from the tunnel) increase
as the tunnel becomes more flexible with respect to the
ground i.e., when the flexibility ratio (F) increases. A
unique relationship between the flexibility ratio and the
normalized distortions of the tunnel is found for linearelastic ground, irrespective of the amplitude of the dynamic
input. Similarly, for nonlinear ground, normalized distortions increase when the flexibility ratio increases. The normalized distortions are higher than those considering a
linear-elastic medium with the same flexibility ratio; the differences increase as the tunnel becomes more flexible (as F
increases), and when the amplitude of the shear stress
imposed by the dynamic input increases. It must be noted
that, different to the linear-elastic medium, there is not a
unique relationship between the flexibility ratio and the
normalized distortions for nonlinear ground, as those
increase with the amplitude of the dynamic input, for the
same relative stiffness.
Higher axial forces and bending moments are observed
for stiffer structures (lower flexibility ratios) irrespective
of the ground model used. The loading of the liner
decreases as the tunnel becomes more flexible than the
ground (as the flexibility ratio increases). This finding is
informative, since providing ductility to the support to
absorb the deformations imposed by the earthquake may
be more effective than increasing its stiffness.
The results of the dynamic numerical analyses conducted provide clear support to the importance of the role
that the relative stiffness between the ground and the liner
has on the seismic response of underground structures. The
implication of this observation is significant: the free field
approach should be abandoned for design.
Acknowledgments
The financial support of the Colombia-Purdue Institute
for Advanced Scientific Research (CPI), Universidad del
Valle (Colombia) and Purdue University – United States
is gratefully acknowledged.
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ScienceDirect
Underground Space 2 (2017) 125–133
www.elsevier.com/locate/undsp
Effect of frequency and flexibility ratio on the seismic response of
deep tunnels
Eimar Sandoval ⇑, Antonio Bobet
Lyles School of Civil Engineering, Purdue University, West Lafayette 47906, United States
School of Civil Engineering and Geomatics, Universidad del Valle, Cali, Colombia
Received 17 December 2016; received in revised form 6 April 2017; accepted 12 April 2017
Available online 1 June 2017
Abstract
Two-dimensional dynamic numerical analyses have been conducted, using FLAC 7.0, to evaluate the seismic response of underground structures located far from the seismic source, placed in either linear-elastic or nonlinear elastoplastic ground. The interaction
between the ground and deep circular tunnels with a tied interface is considered. For the simulations, it is assumed that the liner remains
in its elastic regime, and plane strain conditions apply to any cross section perpendicular to the tunnel axis. An elastoplastic constitutive
model is implemented in FLAC to simulate the nonlinear ground. The effect of input frequency and relative stiffness between the liner
and the ground, on the seismic response of tunnels, is evaluated. The response is studied in terms of distortions normalized with respect
to those of the free field, and load demand (axial forces and bending moments) in the liner. In all cases, i.e. for linear-elastic and nonlinear
ground models, the results show negligible effect of the input frequency on the distortions of the cross section, for input frequencies smaller than 5 Hz; that is for ratios between the wave length and the tunnel opening (k=D) larger than ten for linear-elastic and nine for nonlinear ground. Larger normalized distortions are obtained for the nonlinear than for the linear-elastic ground, for the same relative
stiffness, with differences increasing as the tunnel becomes more flexible, or when the amplitude of the dynamic input shear stress
increases. It has been found that normalized distortions for the nonlinear ground do not follow a unique relationship, as it happens
for the linear-elastic ground, but increase as the amplitude of the dynamic input increases. The loading in the liner decreases as the structure becomes more flexible with respect to the ground, and is smaller for a tunnel placed in a stiffer nonlinear ground than in a softer
nonlinear ground, for the same flexibility ratio.
Ó 2017 Tongji University and Tongji University Press. Production and hosting by Elsevier B.V. on behalf of Owner. This is an open access article
under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Deep circular tunnel; Dynamic numerical analysis; Flexibility ratio; Distortion; Seismic response
1. Introduction
Underground structures must be able to support static
overburden loads, as well as to accommodate additional
deformations imposed by seismic motions. Progress has
⇑ Corresponding author at: Lyles School of Civil Engineering, Purdue
University, West Lafayette 47906, United States.
E-mail addresses: esandov@purdue.edu, eimar.sandoval@correounivalle.edu.co (E. Sandoval), bobet@ecn.purdue.edu (A. Bobet).
Peer review under responsibility of Tongji University and Tongji
University Press.
been made in the last few years in understanding the soilstructure interaction mechanisms and the stress and displacement transfer from the ground to the structure during
a seismic event. For most tunnels, with some exceptions
including submerged tunnels, it seems well established that
the most critical demand to the structure is caused by shear
waves traveling perpendicular to the tunnel axis (Bobet,
2003; Hendron and Fernández, 1983; Merritt et al., 1985;
Wang, 1993). Those waves cause distortions of the cross
section (ovaling for a circular tunnel, and racking for a
http://dx.doi.org/10.1016/j.undsp.2017.04.003
2467-9674/Ó 2017 Tongji University and Tongji University Press. Production and hosting by Elsevier B.V. on behalf of Owner.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
126
E. Sandoval, A. Bobet / Underground Space 2 (2017) 125–133
rectangular tunnel) that result in axial forces (thrusts) and
bending moments.
Two approaches have been used to evaluate the seismic
response of underground structures. One is the free field
approach (Hendron and Fernández, 1983; Kuesel, 1969;
Merritt et al., 1985; Newmark, 1967), which assumes that
the structure follows the free field deformations of the
ground, and therefore accommodates them without loss
of its integrity. Authors supporting this approach have suggested computing deformations considering the perforated
ground, given that most tunnels in competent ground
would behave as perfectly flexible structures. This assumption however may result in extremely conservative designs,
especially for stiff structures in a soft medium. The second
approach is the soil-structure interaction approach (Bobet,
2003, 2010; Huo et al., 2006; Penzien, 2000; Wang, 1993).
This approach, which is discussed in this paper, states that
the underground structure modifies the free field deformation of the ground around it such that demand and
response depend on the relative stiffness between the
ground and the tunnel support. Two dimensionless coefficients have been proposed to consider soil-structure interaction, namely the flexibility ratio and the compressibility
ratio (Einstein and Schwartz, 1979; Peck et al., 1972).
The flexibility ratio (F), or flexural stiffness, is a measure
of the resistance of the system to change shape (i.e., distort)
under a state of pure shear. The compressibility ratio (C),
or extensional stiffness, is a measure of the all-around uniform pressure required to cause diametric strain without
change in shape. As the seismic shear waves distort the
cross section of the tunnel, the flexibility ratio (F) is
the main coefficient of interest in the evaluation of the seismic response of the cross-section of underground
structures.
Analytical studies by Paul (1963), Mow and Pao (1971),
Yoshihara (1963), Hendron and Fernández (1983), Merritt
et al. (1985) showed that the dynamic amplification of
stress waves impinging on a cavity is negligible when the
wave length (k) of the seismic peak velocities is at least
eight times larger than the width of the opening. This is
usually the case for most structures located far from the
seismic source. According to Dowding (1985) a far-field
motion can be considered for distances from the seismic
source between 10 and 100 km, with predominant frequencies between 0.1 and 10 Hz. With this assumption, pseudostatic numerical analyses have been carried out, and a number of closed-form solutions have been proposed for
seismically-induced motions (Bobet, 2003, 2010; Huo
et al., 2006; Penzien, 2000; Wang, 1993). These solutions
are based on elasticity (Timoshenko and Goodier, 1970),
on early work by Peck et al. (1972), and on the relative stiffness method of Einstein and Schwartz (1979).
The closed-form solutions are suitable for deep tunnels
placed in an infinite, linear-elastic, homogeneous, and isotropic medium. A tunnel is considered deep if the stress
gradient with depth has negligible or no effect on its behavior. A plane strain condition, which is reasonable for
sections located far from the face of the tunnel, is also
assumed. The previous work has shown that the most
important parameter determining the distortions of a cross
section of a tunnel is the relative stiffness between the medium and the liner (expressed by the flexibility ratio, F), and
that the depth and shape of the structure have secondorder effects (Bobet, 2010). This conclusion however has
been obtained considering linear-elastic ground. The more
realistic seismic response of tunnels when considering the
nonlinear stress-strain behavior of the ground under cyclic
loading has not been evaluated.
The paper provides results of 2D plane strain dynamic
numerical analyses, conducted in FLAC 7.0 (Itasca,
2011a), for deep circular underground structures subjected
to vertically traveling shear waves, produced by a sinusoidal input velocity. In the analyses, the liner is assumed
elastic with a tied interface, i.e. no relative displacement
between the structure and the ground. To evaluate the
effect of the flexibility ratio on the seismic response without
additional variables, the same interface is used in all the
analyses. A tied interface is selected, as it includes a combined normal-shear stress transmission at the interface.
The ground is considered dry (drained) i.e., no excess pore
pressures are generated, with either linear-elastic or nonlinear elastoplastic response. In both cases, the effect of the
input frequency on the distortions of the tunnel cross section is evaluated, as well as the effect of relative stiffness
on the distortions of the liner. For the nonlinear ground,
the loadings in the structure (thrusts and bending
moments) are also obtained. Comparisons in terms of distortions of the liner, for the linear-elastic and the nonlinear
elastoplastic ground, are provided.
2. Linear elastic analysis
The ground-structure system is discretized in FLAC
with meshes with dimensions 600 m wide and 100 m or
50 m high, to evaluate the effect of input frequency and
the effect of flexibility ratio, respectively. The reasons to
select these two different heights are explained later. For
both meshes, square elements (1 m 1 m) are used away
from the tunnel, and square elements with a smaller size
(0.5 m 0.5 m), close to the tunnel, within a rectangular
section of 60 m 40 m. The center of the tunnel is located
25 m above the bottom of the model. Fig. 1 shows the
meshes used for the dynamic numerical analyses.
A sinusoidal input velocity is imposed at the bottom of
the discretization. Free-field boundaries on the sides and
quiet boundaries at the bottom of the model are used.
These are absorbing boundaries that use independent dashpots in the normal and shear directions, to prevent reflection of waves back into the model and avoid energy
radiation (these boundaries are a built-in option in FLAC).
By using absorbing boundaries, plane waves propagating
upward do not distort at the lateral boundaries and do
no reflect at the bottom boundary. (Itasca, 2011b;
Lysmer and Kuhlemeyer, 1969).
E. Sandoval, A. Bobet / Underground Space 2 (2017) 125–133
127
Fig. 1. Meshes configuration for evaluation of: (a) effect of input frequency; (b) effect of flexibility ratio.
The appropriate width of the model and good performance of the absorbing boundaries are verified by comparing shear strains at different points through the width of the
mesh, for specific depths. Although not included here, the
results show negligible differences at points located up to
80 m on each side of the center of the model. So the proposed discretization is considered adequate.
The tunnel support is modeled with elastic beam elements. Beam elements are frequently used in this type of
simulations because, similar to beams, liners must support
axial forces and bending moments (e.g., Bobet, 2010;
Wang, 1993).
2.1. Effect of input frequency on tunnel’s distortions
Previous research has found, when the ground’s
response is elastic, that the dynamic amplification of stress
waves impinging on a tunnel is negligible when the rise time
of the pulse is larger than about two times the transit time
of the pulse across the opening; that is, when the wave
length (k) of peak velocities is at least eight times larger
than the width (B) or diameter (D) of the opening
(Hendron and Fernández, 1983; Mow and Pao, 1971;
Merritt et al., 1985; Paul, 1963; Yoshihara, 1963). The following numerical analysis is intended to evaluate the previous finding and to use its results for comparison when the
response of the ground is nonlinear.
Distortions for circular tunnels with 4.0 m in diameter
and 0.40 m in thickness are obtained. The mesh has dimensions 600 m 100 m. The depth of the model is selected to
avoid the effect of waves reflection from the free surface,
especially for input frequencies equal or lower than
1.0 Hz. According to Itasca (2011b), the depth of the model
should be larger than one third to one fourth of the associated wave length, to avoid this effect. The amplitude of the
input velocity is 0.1 m/s, with frequencies ranging from
0.25 to 15 Hz. These frequencies correspond to k=D ratios
ranging from 200 to 3.3, given the dimensions chosen for
the tunnel. The wave length (k) is obtained as the ratio
between the shear wave velocity in the medium (C s ) and
the frequency of the dynamic input (f).
The elastic properties of the medium, shear modulus
(Gm ) and Poisson’s ratio (mm ), are assumed to be 80 MPa
and 0.25 respectively. For the liner, a value of 0.15 is used
for the Poisson’s ratio (ms ). A stiff tunnel with flexibility
ratio (F) equal to 0.125, and a flexible liner with F = 12.5
are used. The corresponding Young’s moduli for the liner’s
material (Es ) are 3.13 105 and 3.13 103 MPa, respectively. In this paper, the definition of flexibility ratio provided by Peck et al. (1972), shown in Eq. (1), is used.
F ¼
Em =ð1 þ mm Þ
6Es Is=ðR3 ð1 m2s ÞÞ
ð1Þ
where Em is the Young’s modulus of the medium, I s is the
moment of inertia of the liner per unit length, and R is the
radius of the tunnel.
Fig. 2 shows the results of both relative stiffnesses, when
the wave length or input frequency is changed. The inset in
the figure shows a schematic of the ovaling distortions for a
circular support. The figure plots the maximum distortions,
which occur at maximum amplitude of the sinusoidal input
velocity, normalized with respect to the ground distortions
in the free-field, i.e. distortions of the ground without the
liner. The normalized distortions are expressed as a function of the ratio k=D (bottom horizontal axis). For compar-
128
E. Sandoval, A. Bobet / Underground Space 2 (2017) 125–133
Fig. 3. Normalized distortions for circular tunnels placed in linear-elastic
medium.
Fig. 2. Effect of input frequency on distortions of circular tunnels placed
in linear-elastic medium.
ison purposes, the top horizontal axis presents the values of
the input frequency (f).
Fig. 2 shows that the more flexible liner (F = 12.5) has
larger normalized distortions than the stiffer liner
(F = 0.125). This behavior is expected as the flexible structure deforms more than the stiff structure and also more
than the soil it replaces. Small differences in response are
observed with the input frequency, depending on the flexibility ratio. For the stiff structure (F = 0.125), the normalized distortions are almost constant for k=D ratios larger
than ten (f < 5.0 Hz). For lower k=D ratios (higher f), a
small reduction in the normalized distortions is observed.
For the flexible structure on the other hand, negligible differences in the normalized distortions are observed for k=D
ratios larger than ten (f < 5.0 Hz). Afterwards, the normalized distortions increase up to a point, and then decreases
as the k=D ratios decrease (f increases). Overall, the effect
of frequency is negligible for k=D ratios larger than ten,
i.e., for input frequencies (f) lower than 5.0 Hz.
These results, obtained through fully dynamic numerical
analyses, confirm the assumption made for analytical solutions that a pseudo-static analysis is sufficient when the
ratio between the wave length and the opening of the liner
is larger than eight (ten in this paper), if a linear-elastic
medium is considered.
2.2. Effect of relative stiffness on tunnel’s distortions
Previous work, based on analytical solutions and
numerical analyses, has found that the main parameter
controlling the distortions of the cross section of the liner
is the relative stiffness between the structure and the ground
(e.g., Bobet, 2010; Penzien, 2000; Wang, 1993). To extend
the database available for linear-elastic ground, and to
have results that can be compared with those of the nonlinear ground in Section 3, dynamic numerical analyses for a
sinusoidal input velocity with 0.1 m/s of amplitude and
2.5 Hz of frequency are performed. Note that, in the preceding section, it was shown that at this frequency the
results are frequency-independent. The applied input velocity corresponds to an input shear stress equal to 40 kPa.
For the input frequency used (2.5 Hz), the depth of the
model can be taken as 50 m, since no unwanted reflections
from the surface are produced. This decreases the computation time of the simulations. Consequently, the effect of
the relative stiffness, for the linear-elastic ground and for
the nonlinear ground, is evaluated for a mesh
600 m 50 m. In this section, the same tunnel geometry
and elastic properties of the ground considered in the preceding section are used.
Fig. 3 shows the normalized maximum distortions, for
flexibility ratios ðF Þ ranging from 0.125 (a relatively very
stiff tunnel) to 15 (a relatively very flexible tunnel). It can
be seen in Fig. 3 that the normalized distortions increase
as the flexibility ratio increases. As stated in Section 2.1,
this trend is anticipated because as F increases, the tunnel
becomes much more flexible than the soil it replaces. For
an elastic analysis, the magnitude of the dynamic input
should not affect the results, due to the normalization.
Fig. 3 corroborates this assumption by including results
for three different flexibility ratios (F = 0.25, 1 and 12.5)
with dynamic input that changes by a factor of 3.
The flexibility ratio used in this paper, and proposed by
Peck et al. (1972), provides a good measure of the stiffness
of the structure relative to the ground. A value of F ¼ 1
represents a structure with stiffness equal to that of the
ground that it replaces, and so the distortions of the structure are those of the free field (normalized distortions equal
to 1.0). A value F > 1 represents a structure more flexible
than the ground, and the distortions of the structure are
larger than those of the free field. A value F < 1 represents
a structure stiffer than the ground, and the distortions are
smaller than those in the free field.
129
E. Sandoval, A. Bobet / Underground Space 2 (2017) 125–133
as a function of the change in the effective confinement
(dr0m ), with r0m = 1/3(rkk ).
3. Nonlinear elastoplastic analysis
Soils and other geomaterials usually behave as elastic
only at very small strains. In general, those materials show
a nonlinear stress–strain behavior with strain, with a reduction of stiffness with increased strains. As a result, higher
strains in the free field are anticipated for nonlinear
ground, compared to a linear-elastic ground. Consequently, given that the response of a tunnel under seismic
loading is a soil-structure interaction problem, differences
in the seismic response are expected.
It must be noted that due to the stiffness degradation of
the ground, the flexibility ratio changes during the dynamic
loading. More precisely, if the liner remains in its elastic
regime F decreases, as the ground becomes more flexible
with deformation, i.e., the tunnel becomes stiffer with
respect to the ground as the ground deforms. The flexibility
ratios reported here are those computed using the degraded
stiffness of the medium at peak distortions of the tunnel.
The free field shear strains used for the normalization correspond to those of the degraded ground.
As with the elastic ground case, the effect of the input
frequency and the effect of the flexibility ratio on the distortions of the cross section are investigated. The effect of flexibility ratio on loadings of the liner is also evaluated. The
distortions obtained are then compared to those found
for a linear-elastic medium. A constitutive model to represent the nonlinear elastoplastic ground is presented and
implemented in FLAC.
3.1. Cyclic nonlinear elastoplastic model
The elastoplastic model proposed is based on that developed by Jung (2009) and modified by Khasawneh et al.
(2017), and is intended to simulate the nonlinear behavior
observed in geological materials, even for small shear
strains around 10 5 (Dobry et al., 1981; Vucetic, 1994).
The nonlinear behavior is simulated with a modified hyperbolic relationship. Jung (2009) proposed to obtain the
degraded shear stiffness of the ground (G) by means of
Eq. (2), as a function of the modified hyperbolic strain
(ch ) proposed by Hardin and Drnevich (1972), which is
given in Eq. (3).
ds
1
¼ Go
G¼
2
dc
ð1 þ ch Þ
i
c
ch
1 þ aexp bðcr Þ
ch ¼
cr
ð2Þ
ð3Þ
where cr ¼ sf =Go is the reference shear strain, sf is the
shear stress at failure, Go is the shear modulus at small
strains, c is the current shear strain, and a and b are soil
constants that determine the deviation of the stress–strain
relation from the hyperbolic function.
The model also includes the dependence of stiffness on
confinement. Jung (2009) proposed Eq. (4) to calculate
the change in the shear modulus for small strains (dGo ),
dGo ¼
2
1 Goðref Þ 1
dr0
2 r0mðref Þ Go m
ð4Þ
where r0mðref Þ is the reference effective mean stress (for geostatic conditions), and Goðref Þ is the corresponding reference
small strain shear modulus.
The hysteretic behavior during cyclic loading is based on
Pyke (1979), who proposed a modification to the Masing’s
rules. Jung (2009) provided Eq. (5) to calculate the stiffness
reduction in the hysteresis loop in the three dimensional
space. Eq. (6) is an extension hyperbolic strain (Eq. (3))
in 3D and is based on the octahedral shear strain (coct ),
defined in Eq. (8). This modification is required as the original hyperbolic strain proposed by Hardin and Drnevich
(1972) was obtained from 1D cyclic simple shear tests.
G ¼ Go
1
coct
coct;r
coct;rev h
1 þ aexp
Crev ¼
coct;r
rffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffi
2 1
eij eij
coct ¼ 2
3 2
1
kk dij
eij ¼ ij
3
1 @ui @uj
þ
ij ¼
2 @xj @xi
C¼
ð5Þ
2
1þ
Crev j
h
i
c
bð oct Þ
1 þ aexp coct;r
1
jC
n
bð
coct;rev
coct;r Þ
ð6Þ
i
ð7Þ
ð8Þ
ð9Þ
ð10Þ
where n is a scaling factor for the hysteresis loop (n = 2 to
satisfy Masing rules or n – 2 for non-symmetric cyclic
loading), coct;r is a reference octahedral shear strain, coct;rev
is the octahedral shear strain at reversal, eij is the deviatoric
strain tensor, ij is the Lagrangian strain tensor, and ui are
the displacements along the xi axis.
Small strain analysis is assumed, and thus the strain
increment (dij ) is decomposed into its elastic and plastic
components dij ¼ deij þ dpij , with the stress increment
computed from Eq. (11).
drkl ¼ C eklij deij
ð11Þ
where drkl is the incremental stress tensor, C klij is the elastic
modulus tensor, and dij is the incremental elastic strain
tensor.
After yielding, which is defined with the Drucker-Prager
(D-P) criterion, the plastic strains are determined by means
of a strain hardening law with a non-associated flow rule.
Eqs. (12) and (13) show the yield function and the plastic
potential, respectively. More details about the cyclic elastoplastic model, including the complete formulation, implementation and its verification can be found in
Khasawneh et al. (2017).
130
pffiffiffiffiffiffiffi 1
3J 2 þ I 1 tanðaÞ
3
pffiffiffiffiffi 1
I 1 tanðuÞ
g ¼ J2
3
f ¼
E. Sandoval, A. Bobet / Underground Space 2 (2017) 125–133
j
ð12Þ
ð13Þ
where J 2 = 1/2(S ij S ij ) is the second invariant of the deviatoric stress tensor, I 1 ¼ rkk is the first invariant of the stress
tensor, a is the D-P friction angle, j is the D-P cohesion,
and u is the dilation angle.
3.2. Effect of input frequency on tunnel’s distortions
A mesh 600 m 100 m with a circular tunnel placed
25 m above the bottom is used. The liner, assumed to
remain in its elastic regime, has again 4 m in diameter
and 0.4 m in thickness.
A ground with initial stiffness Goðref Þ ¼ 80 MPa, and
Poisson’s ratio mm ¼ 0:25 is used in the simulations. Similar
to the linear-elastic ground, a stiff and a flexible liner are
considered. The Poisson’s ratio for the liner’s material is
assumed ms ¼ 0:15, and the Young’s moduli (Es ) are
1.56 105 and 2.61 103 MPa, for the stiff and the flexible
structure, respectively. Such stiffnesses represent initial flexibility ratios (F) of 0.25 and 15 that are reduced during the
dynamic phase to values of 0.20 and 12.0, respectively (at
peak distortions). The input frequencies range from 0.50
to 15 Hz, which correspond to k=D ratios (considering
the degraded ground) ranging from 90 to 3, given the
dimensions chosen for the tunnels. In the results, displayed
in Fig. 4, it can be seen that negligible changes in the distortions of the liner are observed for k=D ratios larger than
nine (f < 5 Hz), for both the flexible and the stiff structure.
Similar to the linear-elastic ground, for k=D ratios smaller
than nine, a reduction in the normalized distortions is
observed in stiff liners. For flexible liners, an increase of
distortions with a later decrease is detected.
Fig. 4. Effect of input frequency on distortions of tunnels placed in
nonlinear ground.
For completeness, static numerical analyses for both the
flexible and the stiff tunnel are performed. A square model
50 m 50 m with the tunnel located at the center (25 m
from the bottom, as in the dynamic case) is used. In the
analyses, inertial forces are neglected and a simple shear
condition producing the same free field shear strain than
the free field induced by the dynamic loading is used. The
results are included as arrows on the right vertical axis of
Fig. 4. Negligible differences are observed in normalized
distortions obtained from the static analyses, or from
dynamic analyses for k=D ratios larger than nine
(f < 5 Hz). Consequently, for nonlinear ground, a
pseudo-static analysis is feasible for ratios between the
wave length and the tunnel opening larger than nine.
3.3. Effect of relative stiffness on tunnel’s distortions
As before, a mesh 600 m 50 m is used to assess the
effect of the relative stiffness on the tunnel response. The
tunnel is again placed 25 m above the bottom of the model,
and has 4 m in diameter and 0.40 m in thickness.
A parametric study for three different stiffnesses of the
ground (Goðref Þ = 40, 80, and 200 MPa) is performed. These
values represent medium stiff soil, very stiff soil, and weak
rock, respectively. An input velocity with 2.5 Hz frequency
is used. This frequency is selected, based on the results
obtained (shown in Fig. 4), to have the distortions independent of frequency. Given that the magnitude of the shear
stress induced by a dynamic input velocity or acceleration
is directly related to the stiffness of the ground, the amplitude of the input velocity is selected such that the same
input shear stress (s) at the bottom of the discretization is
obtained for the three stiffnesses of the ground investigated.
A shear stress with magnitude 40 kPa is applied, which corresponds to input velocities of 0.14, 0.10 and 0.06 m/s, for
Goðref Þ 40, 80 and 200 MPa, respectively. These velocities
are representative of strong to very strong earthquakes,
with magnitudes between VI and VII in the Modified Mer-
Fig. 5. Normalized distortions for circular tunnels placed in nonlinear
ground, for the same input shear stress amplitude and for different initial
ground stiffness.
E. Sandoval, A. Bobet / Underground Space 2 (2017) 125–133
calli Intensity Scale, according to the classification provided by Wald et al. (1999). The initial flexibility ratios
(F) range from 0.10 to 15, which result in final flexibility
ratios (F) ranging between 0.08 and 12, due to the stiffness
reduction in the ground with deformation.
The results of the simulations are shown in Fig. 5. For
comparison purposes, the plot also includes results using
a linear-elastic medium that has the same stiffness as that
of the degraded ground of the nonlinear analyses. It can
be seen in the figure that, for the same flexibility ratio,
higher normalized distortions occur with the nonlinear
ground, with differences increasing as the tunnel becomes
more flexible. This behavior is expected, as the nonlinear
ground surrounding the structure degrades, and so there
is a higher strain demand on the structure. For the nonlinear ground, it can also be seen that normalized distortions
are almost the same when the same input shear stress is
used, irrespective of the initial stiffness of the ground.
To verify whether the relationship between distortions
and flexibility ratio for a nonlinear ground can be
expressed by means of a single curve, as it is done for a
linear-elastic ground (see Fig. 3), analyses are performed
for different amplitudes of the input shear stress. The
amplitude of the input shear stress used in Fig. 5 is
increased by a factor of two and three, while the other
parameters are kept constant. Fig. 6 compares the normalized distortions for Goðref Þ = 80 MPa for the three cases.
Important differences in the normalized distortions are
observed when the amplitude of the input stress is changed,
for the same final (degraded) flexibility ratio. It can also be
seen in Fig. 6 that the differences with respect to the linearelastic case increase as the amplitude of the dynamic input
increases. Hence, there is not a unique relationship between
the flexibility ratio and the normalized distortions when a
nonlinear ground is assumed, in contrast to what happens
with a linear-elastic ground where the relationship is
unique. Indeed, the nonlinear ground, under a higher input
stress, will deform more than under a lower input stress,
Fig. 6. Normalized distortions for circular tunnels placed in nonlinear
ground, for different input shear stress amplitude and for the same initial
ground stiffness.
131
and so its stiffness will be smaller, which will result in larger
deformation demands to the structure.
3.4. Effect of relative stiffness on liner’s loading
The effect of the flexibility ratio on the thrusts and bending moments of the liner for a nonlinear ground is evaluated, by considering two different stiffnesses of the
ground, namely G0ðref Þ = 80 and 200 MPa.
Figs. 7 and 8 show the results for thrusts and bending
moments, respectively. In the figures, the thrusts are normalized by the product of the input shear stress and the
radius of the tunnel (sR); and the bending moments are
normalized by the product of the input shear stress and
the square of the radius of the tunnel (sR2 ), as proposed
by Einstein and Schwartz (1979).
It can be seen in the figures that larger thrust and bending moments occur for stiffer structures, i.e., for lower flexibility ratios (F), which decrease as the tunnel becomes
Fig. 7. Normalized axial forces (thrusts) for circular tunnels placed in
nonlinear ground.
Fig. 8. Normalized bending moments for circular tunnels placed in
nonlinear ground.
132
E. Sandoval, A. Bobet / Underground Space 2 (2017) 125–133
more flexible (as F increases). In fact, almost negligible
bending moments are obtained for flexibility ratios larger
than 10, as the structure approaches a perfectly flexible tunnel (Peck et al., 1972) that cannot withstand bending
moments. These results support the statements made by
previous authors (e.g., Kuesel, 1969; Monsees & Merritt,
1991) in that increasing the stiffness of the structure is
not the best solution for seismic design, as stiffer structures
will take greater loads. For the seismic design, these
authors suggested increasing the ductility of the liner to
absorb the imposed seismic distortions, rather than increasing the stiffness or strength of the liner. It can also be seen
in Figs. 7 and 8 that for the same flexibility ratio, smaller
thrust and bending moments are observed for the stiffer
nonlinear ground than for the softer nonlinear ground, as
the first deforms less.
4. Summary and conclusions
The paper evaluates the dynamic response of deep circular tunnels placed in linear-elastic and nonlinear elastoplastic ground. Dynamic numerical analyses using FLAC 7.0
are performed. For the elastoplastic ground, a constitutive
model based on hyperbolic behavior is implemented in
FLAC. The effect of the input frequency on the distortions
of the cross section and the effect of the flexibility ratio on
the distortions of the liner are evaluated for linear-elastic
and nonlinear ground. For nonlinear ground, the effect of
the flexibility ratio on the loadings of the liner is also investigated. Plane strain conditions and deep tunnel with a tied
interface are assumed.
The results show that for both linear-elastic and nonlinear elastoplastic ground, there is no effect of the input frequency on the seismic response of the tunnel, when the
input frequency is smaller than 5 Hz (k=D ratios larger than
ten for linear-elastic and nine for nonlinear ground).
Pseudo-static analyses can therefore be conducted for tunnels located far from the seismic source (for distances
between 10 and 100 km from the epicenter, after
Dowding, 1985), irrespective of the soil model used in the
simulations.
As it has been found in previous work with the assumption of linear-elastic ground (e.g., Bobet, 2010; Wang,
1993), normalized distortions (tunnel distortions normalized by those of the ground far from the tunnel) increase
as the tunnel becomes more flexible with respect to the
ground i.e., when the flexibility ratio (F) increases. A
unique relationship between the flexibility ratio and the
normalized distortions of the tunnel is found for linearelastic ground, irrespective of the amplitude of the dynamic
input. Similarly, for nonlinear ground, normalized distortions increase when the flexibility ratio increases. The normalized distortions are higher than those considering a
linear-elastic medium with the same flexibility ratio; the differences increase as the tunnel becomes more flexible (as F
increases), and when the amplitude of the shear stress
imposed by the dynamic input increases. It must be noted
that, different to the linear-elastic medium, there is not a
unique relationship between the flexibility ratio and the
normalized distortions for nonlinear ground, as those
increase with the amplitude of the dynamic input, for the
same relative stiffness.
Higher axial forces and bending moments are observed
for stiffer structures (lower flexibility ratios) irrespective
of the ground model used. The loading of the liner
decreases as the tunnel becomes more flexible than the
ground (as the flexibility ratio increases). This finding is
informative, since providing ductility to the support to
absorb the deformations imposed by the earthquake may
be more effective than increasing its stiffness.
The results of the dynamic numerical analyses conducted provide clear support to the importance of the role
that the relative stiffness between the ground and the liner
has on the seismic response of underground structures. The
implication of this observation is significant: the free field
approach should be abandoned for design.
Acknowledgments
The financial support of the Colombia-Purdue Institute
for Advanced Scientific Research (CPI), Universidad del
Valle (Colombia) and Purdue University – United States
is gratefully acknowledged.
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