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Seismic behaviour of circular tunnels accounting for above ground structures interaction effects
Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
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Seismic behaviour of circular tunnels accounting for above ground structures interaction effects Kyriazis Pitilakis a,n, Grigorios Tsinidis a, Andrea Leanza b, Michele Maugeri b a Aristotle University of Thessaloniki, Research Unit of Soil Dynamics and Geotechnical Earthquake Engineering, Department of Civil Engineering, PO Box 424, GR-54124 Thessaloniki, Greece b University of Catania, Department of Civil and Environmental Engineering, Sicily, Italy
art ic l e i nf o
a b s t r a c t
Article history: Received 12 February 2014 Received in revised form 19 May 2014 Accepted 20 August 2014
Tunnels are commonly designed under seismic loading assuming “free field conditions”. However, in urban areas these structures pass beneath buildings, often high-rise ones, or are located close to them. During seismic excitation, above ground structures may cause complex interaction effects with the tunnel, altering its seismic response compared to the “free field conditions” case. The paper summarizes an attempt to identify and understand these interaction effects, focusing on the tunnel response. The problem is investigated in the transversal direction, by means of full dynamic time history analyses. Two structural configurations are studied and compared to the free field conditions case, consisting of one or two above ground structures, located over a circular tunnel. Above ground structures are modeled in a simplified way as equivalent single-degree of freedom oscillators, with proper mechanical properties. Several parameters that are significantly affecting the phenomenon are accounted for in this parametric study, namely the soil to tunnel relative flexibility, the tunnel dimensions, the tunnel burial depth and the soil properties and nonlinearities during shaking. Tunnels response characteristics are compared and discussed, in terms of acceleration, deformations and lining dynamic internal forces. Internal forces are also evaluated with analytical closed form solutions, commonly used in preliminary stages of design, and compared with the numerical predictions. The results indicate that the presence of the above ground structures may have a significant effect on the seismic response of the tunnel, especially when the latter is stiff and located in shallow depths. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Circular tunnels Seismic behavior Above ground structures–soil–tunnel dynamic interaction Dynamic analysis
1. Introduction Tunnels constitute crucial components of the transportation and utility networks in urban areas. The associated impact in case of earthquake induced damage denotes the importance of proper seismic design especially in seismic prone regions. It is generally believed that underground structures are less vulnerable to seismic shaking compared to above ground structures. However, several cases of severe damage or even collapse have been reported in the literature, mainly for shallow embedded structures in soft soils ([10,17,19,28,31,33,40,42,43] among others). During an earthquake, tunnels are subjected to shaking due to wave propagation and permanent ground displacements due to ground failure (lateral spreading, landslides and fault rupture). In both the cases, the kinematic loading imposed by the adjacent soils prevails, while the inertial loads are generally of secondary importance. Therefore, the seismic behavior of underground structures
n
Corresponding author. Tel.: þ 30 2310995693. E-mail address: [email protected] (K. Pitilakis).
http://dx.doi.org/10.1016/j.soildyn.2014.08.009 0267-7261/& 2014 Elsevier Ltd. All rights reserved.
and tunnels is quite distinct compared to the above ground structures [13,17]. Several methods are available in the literature for the evaluation of the seismic response of underground structures and tunnels [4,11,12,16,29,35,36,41]. The results of these methods may significantly deviate, even under the same design assumptions, due to both inherent epistemic uncertainties and knowledge shortfall regarding some crucial issues that considerably affect the seismic response [30]. In order to better understand the seismic behavior of these types of structures several experimental research studies have been recently carried out (e.g., [8,22,38,39] among others). Available design methods for shaking, usually assume free field conditions, precluding the existence of above ground structures in the adjacent area of the tunnel (e.g., above the tunnel). However, in urban areas, tunnels often pass beneath or close by high-rise buildings. During shaking, the vibration of these above ground structures may create complex interaction phenomena with the tunnel, passing often few meters below their foundation, which are expected to affect the seismic wave propagation field. In this sense, they may modify the dynamic response of the tunnel, while at the same time the existence of the tunnel close to the surface
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and the foundations of the buildings, may alter the response of the buildings themselves. Dynamic interaction effects in urban areas (e.g., “city effects”) have been mainly examined between above ground structures. A comprehensive review is made by Menglin et al. [27]. Regarding the dynamic interaction phenomena between above ground and embedded structures, most researchers focus on the effect of an underground structure, often a circular tunnel or a cavity, on the response of the above ground structures. The underground structure is commonly assumed to be embedded in an elastic halfspace, while the effects are usually expressed in terms of surface ground motion amplification [9,20,23–25,34,43]. On the other hand, the inverse problem, i.e., the effects of above ground structures on the response characteristics of embedded structures (e.g., shallow tunnels) have not been thoroughly studied. The present paper presents an attempt to identify, understand and quantify these potential effects. For this purpose, a numerical parametric study is conducted, assuming different structures, soil and tunnel configurations. The problem is investigated in the transversal direction, as this direction is related to the lining maximum developing stress states and affects directly the cross-sectional structural design of the tunnel. Parameters that significantly affect the phenomenon and considered herein are the soil to tunnel relative flexibility, the tunnel dimensions, the tunnel burial depth and the soil properties accounting also for their nonlinear behavior during strong shaking. The response of the examined cases is discussed in terms of acceleration, deformations and lining internal forces. Lining forces are also evaluated with existing closed form analytical solutions (e.g., [41]), commonly used in preliminary stages of design of circular tunnels in the absence of any above ground structure, and the results are compared to the numerical data.
2. Numerical simulation A series of dynamic time history analyses is performed on representative structural systems comprising of a circular tunnel and models of above ground buildings. The analysis is performed in the transversal direction, with the surface structures been simulated as equivalent single degree of freedom (SDOF) oscillators with rigid foundations. Inertial properties of the equivalent SDOFs correspond to usual buildings (e.g., 6–8 storey buildings). The case studies are summarized in Fig. 1. One of the buildings (e.g., Structure A) is located above the tunnel, assuming that the tunnel is constructed with an underground excavation method (e.g., using a tunnel boring machine). In cases of two surface structures, the second one (e.g., Structure B) is located just aside the first one. Table 1 tabulates the mechanical properties of the tunnel and structures along with the assumed fixed-based fundamental periods of the above ground structures (Tfix). Two different sand soil deposits are considered herein. One of them corresponds to a rather loose soil deposit representing a soil type C according to Eurocode 8 [7] with fundamental frequency equal to 1 Hz, while the second one corresponds to a stiffer deposit (i.e., soil type B according to Eurocode 8), with fundamental frequency equal to 2.5 Hz. The shear wave velocity gradient profiles are presented in Fig. 2. Mechanical properties of the assumed soil deposits are tabulated in Table 2. To study the effects of the tunnel size and burial depth, the diameter of the circular tunnel (d) is ranging between 5 and 10 m, while the tunnel burial depth (h) is also ranging between 5 and 10 m. Both dimensions and depths are common in practice. Moreover, the soil to tunnel relative flexibility is examined through the so called flexibility ratio [41], which is estimated using
the following analytical formulation: ES 1 ν2l r 3 F¼ 6El I l ð1 þ νS Þ
ð1Þ
where, Es is the soil elastic modulus, vs is the soil Poisson ratio, El is the lining elastic modulus, vl is the lining Poisson ratio, Il is the lining moment of inertia (per unit width) and r is the circular tunnel radius. The flexibility ratio of the investigated cases is ranging from almost zero (quite rigid tunnel) to 10 (quite flexible tunnel). Few very flexible tunnels (F430) are also evaluated, so as to study the effect of this crucial parameter to extreme ends. To achieve the desirable flexibility ratio, the tunnel lining thickness (tl) is adequately selected for each case. The analyses are performed, under plane strain conditions in total stresses, using the finite element code ADINA [3]. Despite the generic nature of the specific code, ADINA can efficiently reproduce the complex phenomena implicated in a dynamic time history analysis, including wave propagation through soil media and dynamic soil–structure interaction effects [2,18,26]. More specifically, the soil is meshed with plane strain elements, while the tunnel and the above ground structures (SDOFs) are modeled using beam elements (Fig. 3). The adopted element size is selected in a way that ensures the following criteria: (a) Efficient reproduction of all the waveforms of the whole frequency range under study (e.g., following the principle that the element size must be 8–10 times smaller than the minimum wavelength of interest), (b) converge criteria of the analysis (for elasto-plastic analysis) and (c) efficient simulation of the soil close to the tunnel. Therefore, a finer discretization near the tunnel is selected, allowing a low element aspect ratio (for the soil elements) and a low face corner angle (for the beam elements simulating the circular tunnels). The base boundary of the model is simulated as rigid bedrock, where the seismic input motion is applied in terms of displacement time history. For the vertical boundaries kinematic constrains are introduced, forcing the opposite vertical sides to move simultaneously, simulating the shear waves propagating upwards, e.g., tie constrains [1]. To simplify the analyses, a solid connection between the soil and structures is assumed. Although, interface characteristics are quite crucial for the dynamic response of embedded structures [15,21,30,32,37], this assumption is quite common in engineering practice, as it corresponds to an upper limit for the developed shear stresses around the tunnel. The tunnel and the above ground SDOF structures are assumed to behave within the linear elastic range. For the soil behavior two assumptions are made. In the first series of analyses a linear viscoelastic material is used, while for the final analyses an elastoplastic Mohr Coulomb material is implemented, in order to account for the permanent soil response due to yielding. In the latter case, the soil shear strength is assumed to increase with depth (Table 2), following the increase of the soil stiffness (Fig. 2). Viscous damping (5% for all the examined cases and elements, for sake of simplicity) is employed in the frequency depended Rayleigh type. For the elasto-plastic analyses additional energy dissipation is introduced by the hysteretic soil response. Investigated systems are subjected to a simplified Ricker wavelet of nominal frequency equal to 1 Hz and amplitude equal to 0.1 g, introduced at the model base (Fig. 4a). Input motion nominal frequency is selected equal to the fundamental frequency of one of the soil deposits (soil type C) in order to study the effects of soil resonance. In addition, input motion amplitude is selected in order to produce at the tunnel depth and the soil surface,
K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
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Fig. 1. Schematic representation of the examined cases.
Table 1 Structures mechanical properties.
Elastic modulus (GPa) SDOF mass (t) Storeys (n) Fixed-based period Tfix (s)
Tunnel
Structure A
Structure B
32 – – –
32 60 8 0.80
32 45 6 0.60
Vs (m/s) 0
0
200
400
600
800
Depth(m)
15
30
45 Soil Type C Soil Type B 60 Fig. 2. Shear wave velocity profiles of the soil deposits.
acceleration amplitudes of the order of 0.15 g and 0.30 g, (respectively for the two soil profiles), values that are often proposed in code prescriptions. The selection of this type of input motion instead of real accelerograms is usual in this type of analyses in order to avoid the dependence of the response to a specific time history. The selected Ricker wavelet characteristics match a broader band of input motions having at the same time high displacement amplitudes. Comparisons of displacement response spectra computed for the selected wavelet and for different real records (recorded in different earthquakes for soil class A of
Eurocode 8 and scaled at 0.1 g) are presented in Fig. 4b proving that the Ricker wavelet produces much higher displacements compared to the real records. Accounting for the dynamic response of embedded structures that is significantly affected by the soil kinematic response, this selection is considered to be conservative. At the same time the expected soil non-linear behavior is well achieved. All the analyses are performed in two steps. In the first step, the gravity loads are introduced (e.g., geostatic step), while in the second step the dynamic input motion is applied at the base of the numerical model (dynamic step). The boundaries are accordingly changed, between the two steps, using the restart option available in ADINA as shown in Fig. 3b and c. Construction sequence of the tunnel and the above ground structures, as well as construction stages of the circular tunnel, are not considered in this study. Accounting for the aim of this work that is to study the effect of above ground buildings on the tunnels seismic response, case specific stress and deformation variations related to the construction process would rather fade out the problem. These important aspects have actually no effect in case of elastic dynamic analyses, while it is recognized that in case of the elasto-plastic analyses, they may affect the soil yielding response and therefore the initial soil stress state around the tunnel. Nevertheless, since this study is focusing on the dynamic inelastic response and the main objective herein is to examine the differences assuming a reasonable and generally enough reference initial stress state, the conclusions drawn can be considered applicable and valid. Similar simplifications can be found in the literature (e.g., [14]). Time discretization is an important element for the accuracy of the dynamic analysis. ADINA offers an automatic time incrementation scheme, according to which the time step of the analysis is properly selected and changed in order to achieve a stable solution. This scheme was selected herein as adequate to achieve convergence for the solution.
3. Results Representative results are presented and discussed in this section in terms of horizontal acceleration, tunnel deformations and dynamic internal forces of the lining.
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Table 2 Mechanical properties of the soil deposits. Soil profile Soil type B Unit weight of volume, γ (kN/m3) Coefficient of earth pressure at rest, Ko Shear wave velocity, Vs (m/s) Damping, D (%) Poisson ratio, v Cohesion, c (kPa) Friction angle, φ (deg)
Soil type C
20 20 0.50 0.50 Variable with depth (see Fig. 1) 5% 5% 0.333 0.333 20 10 Variable with depth: Variable with depth: 0–30 (m): 35ο 0–30 (m): 28ο 30–60 (m): 45ο 30–60 (m): 35ο
3.2. Tunnel distortions Plain strain elements
Structure A
Beam elements
Structure B
Structure A Displacement constrains
u(t)
Fig. 3. Representative numerical models in ADINA, (a) 5 m diameter tunnel without any building at surface, (b) 10 m diameter tunnel with one building at surface, and (c) 10 m diameter tunnel with two buildings at surface.
3.1. Horizontal acceleration Fig. 5 presents the horizontal acceleration amplification along the vertical array crossing the center of a flexible shallow tunnel (5 m diameter, 5 m burial depth), computed for different soil profiles and assumptions regarding the soil response (visco-elastic and elasto-plastic approach). Visco-elastic analyses results for soil type C indicate an increase of the horizontal acceleration near the tunnel area (compared to the free field), while for the elastoplastic analysis, the behavior is more complex due to the soil yielding. The introduction of the above ground structures causes a slight increase of the horizontal acceleration near the tunnel. For soil type B (stiffer soil conditions) the amplification within the soil is in general reduced and it is not affected by the existence of the tunnel and the structures at the surface. This is mainly attributed to the significant difference between the soil deposit resonance frequency (2.5 Hz) compared to the signal frequency (1 Hz). Similar results are reported for the majority of the investigated systems.
Fig. 6 presents the effect of the above ground structures on the horizontal deformations of the tunnel. The results refer to elastoplastic analyses of a flexible and large tunnel (F4 9.6). Deformed shapes verify the ovaling deformation mode of tunnels during shear wave propagation. In addition, building masses alter the soil stress field around tunnel resulting in stress redistributions that can lead to a different soil permanent response. This complex response is clearer in Fig. 7 where the dynamic vertical displacement of the tunnel crown is presented for different tunnels (flexible and rigid tunnels of different dimensions embedded in different burial depths). The existence of above ground structures results in larger dynamic displacement increments. It is clear that for shallow burial depths and for larger dimension flexible tunnels, soil yielding increase, resulting in an increase of permanent deformations after shaking. This effect is less evident for rigid tunnels and for higher burial depths. Fig. 8 summarizes the numerical results of the variation of the ovaling ratio (R) with the flexibility ratio (F), for tunnels embedded in 5 m burial depth. The effect of the above ground structures on these relations is also presented. Similar results are presented in Fig. 9 for tunnels embedded in the 10 m burial depth. The computation of the ovaling ratio is performed according to the following formulation: R¼
Δstructure Δf f
ð2Þ
where Δstructure is the computed ovaling distortion of the tunnel and Δf f is the corresponding distortion of the soil at free field. In the absence of structures on the surface, numerical results are in good agreement with the F–R relation proposed by Penzien [29] for no slip conditions between the tunnels lining and the soil. Generally, the elastoplastic analyses predict slightly higher diametric deflection for the tunnel, due to the soil yielding. This observation is more significant for flexible structures in soil type C and for small burial depths (e.g., F48 in Fig. 8a). In the majority of the cases the introduction of above ground oscillating structures results in an increase of the tunnel's diametric deflection (larger ovaling ratio for the same flexibility ratio) compared to the case where no surface structures exist. This increase is more significant for shallow tunnels where ovaling ratio R is about 15–20% larger than the case where no above ground structures exist. As it has been expected, the increase of R for deeper tunnels is lower reaching 5–6% compared to the case where no above ground structures exist. Similar to the “free field conditions” case, elastoplastic analyses predict higher values for ovaling ratio as result of soil yielding around the tunnel.
K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
Input motion
5
Displacement response spectra
0.5
0.08 0.07 0.06
0
D (m)
a (m/s2)
0.05 0.04 0.03
−0.5
0.02 0.01 −1
0
1.5
3 t (s)
4.5
0
6
0
1
2 T(s)
3
4
Fig. 4. (a) Input motion acceleration time history (b) displacement response spectra computed for different real seismic records (recorded in soil type A conditions and scaled at 0.1 g) compared to the selected Ricker wavelet (black solid line).
A (m/s2) Visco−elastic, Type C 0
1
2
3
4
15
0 Depth(m)
Depth(m)
0
A (m/s2) Elasto−plastic, Type C
30 45 60
0
1
2
3
4
15 30 45 60
2
A (m/s ) Visco−elastic, Type B
Depth(m)
0
0
0.5
1
1.5
15 30
2
Tunnel Tunnel+Structure A Tunnel+Structures A+B Free field
45 60
Fig. 5. Horizontal acceleration amplification along the vertical array crossing the center of a shallow flexible tunnel (F=2.50); effect of the surface structures.
An increase of the diametric deflection leads to an increase of the lining dynamic internal forces, with the effect being more significant for stiffer tunnels, due to their relative inability to follow these increased deformations.
structures on the tunnel lining internal forces, is made, by plotting ratios of the computed maximum internal force accounting for the above ground structures to the maximum internal forces precluding them, called hereafter “free field conditions”.
3.3. Dynamic internal forces 3.3.1. Evaluation of dynamic internal forces Lining internal forces (e.g., bending moment and axial force), which are of great importance for the final detailing of the tunnel section, are discussed in this section accounting for the effects of the above ground structures. Representative results are presented in terms of time histories and distributions of the internal forces dynamic increments along the perimeter, the latter referring to the time step of the tunnel maximum ovaling distortion. In addition, we present the maximum envelope distributions of the lining forces (e.g., maximum values along the perimeter regardless the time step), as these are commonly used for the final lining detailing. The numerical dynamic increments distributions are compared with the results of closed form analytical solutions [41], which are commonly used during preliminary stages of design. A first attempt, to quantify the effects of the above ground
3.3.2. Axial force Fig. 10 presents time windows of representative time histories of the dynamic axial forces computed at a crucial section of the lining ο (θ ¼45 ) for different scenarios regarding the presence of above ground structures. The results refer to both flexible and rigid tunnels embedded in different burial depths. As expected, the consideration of the oscillating massive structures at the ground surface results in an increase of the axial forces, especially in the cases of shallow and stiffer tunnels. The elasto-plastic analyses reveal residual values for the axial forces after shaking, due to the soil yielding, with the phenomenon being amplified for flexible shallow tunnels. This observation comes in accordance with recent findings from a series of dynamic centrifuge tests that were performed on a flexible tunnel model embedded in dry sand [22].
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Fig. 6. Contour diagrams of horizontal displacements on models deformed shapes for elasto-plastic analyses; time step of maximum ovaling of a 10 m diameter tunnel in 5 m burial depth (a) no surface structures, (b) structure A and (c) structures A þ B.
h=5m, d=5m, F=0.04
u (cm)
h=5m, d=5m, F=2.50 0.8
0.8
0.4
0.4
0.4
0
0
0
−0.4
−0.4
Tunnel Structure A Structures A+B
−0.4 −0.8
0
1.5
3 t(s)
4.5
6
−0.8
0
1.5
3 t(s)
4.5
6
−0.8
0.8
0.8
0.4
0.4
0.4
0
0
0
−0.4
−0.4
−0.4
0
1.5
3 t(s)
4.5
6
−0.8
0
1.5
3 t(s)
4.5
1.5
3 t(s)
4.5
6
h=5m, d=10m, F=0.36
0.8
−0.8
0
h=5m, d=10m, F=9.60
h=10m, d=5m, F=0.04
u (cm)
h=10m, d=5m, F=2.50
0.8
6
−0.8
0
1.5
3 t(s)
4.5
6
Fig. 7. Dynamic vertical displacement of the tunnel crown for different tunnels computed by elastoplastic analysis (h: burial depth, d: tunnel diameter, and F: flexibility ratio).
The same remarks are made in Figs. 11 and 12, where the dynamic increments of the axial forces are presented along the perimeter of the tunnel, in terms of (i) maximum values regardless the time step (e.g. envelope values) and (ii) simultaneous
values computed for time step of maximum ovaling deformation of the tunnel. In both figures, the first line refers to visco-elastic analyses results, while the second line to elasto-plastic analyses results. The anomalies observed in the simultaneous axial force
3
3
2.25
2.25
2.25
1.5
R
3
R
R
K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
1.5
7
1.5
Penzien(2000) 0.75
Viscoelastic
0.75
0.75
Elastoplastic 0
0 1
2.5
5 F
7.5
0
10
3
3
2.25
2.25
0 1
2.5
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0
10
0 1
2.5
5 F
7.5
10
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R
R
0.5
1.5
5 F
.375
1.5
0.375
Penzien(2000) 0.25
0.75
0.75
.125 0
0
0.25
0 1
0
2.5
0.25
5 F
7.5
0
0.5
0
10
Tunnel
0.125
0 1
Structure A 0
2.5
0.25
5 F
Structures A+B
0.5
7.5
10
3
3
2.25
2.25
2.25
1.5
R
3
R
R
Fig. 8. Ovaling ratio (R) versus flexibility ratio (F) for tunnels embedded in 5 m burial depth. Computed results versus Penzien [29] solution: (a) tunnel without above ground structures, (b) tunnel and structure A, (c) tunnel and structures A þ B, (d) comparisons for visco-elastic analyses, and (e) comparisons for elasto-plastic analyses.
1.5
1.5
Penzien(2000) Viscoelastic
0.75
0.75
0.75
Elastoplastic 0
0 1
2.5
5 F
7.5
0
10
3
3
2.25
2.25
0 1
2.5
7.5
10
0
0 1
2.5
5 F
7.5
10
0.5
R
R
0.5
1.5
5 F
.375
1.5
0.375
Penzien(2000) 0.25
0.75
0.75
.125 0
0
0.25
0 1
2.5
0
0.25
5 F
7.5
0
0.5
10
0
Tunnel
0.125
0 1
2.5
Structure A 0
0.25
5 F
7.5
0.5
Structures A+B
10
Fig. 9. Ovaling ratio (R) versus flexibility ratio (F) for tunnels embedded in 10 m burial depth. Computed results versus Penzien [29] solution: (a) tunnel without above ground structures, (b) tunnel and structure A, (c) tunnel and structures A þB, (d) comparisons for visco-elastic analyses, and (e) comparisons for elasto-plastic analyses.
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K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
h=5m, d=5m, F=2.5
N (kN/m)
400 200
200
0
0
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−200
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N (kN/m)
3
4
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−400
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Structure A Structure B
θ
4
h=10m, d=5m, F=2.5 1
2 3 Tunnel t(s) Structure A Structures A+B
0
1
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3
4
h=10m, d=5m, F=0.04
400
200
1
h=5m, d=5m, F=0.04
400
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−400
N (kN/m)
2
h=5m, d=5m, F=0.04
400
N (kN/m)
1
h=5m, d=5m, F=2.5
400
0
1
2 t(s)
3
4
Fig. 10. Dynamic axial force time histories at θ ¼45ο. Left column: visco-elastic analysis results and right column: elasto-plastic analysis results.
N (kN/m)
800
800
h=5m, d=10m, F=0.28
800
h=10m, d=10m, F=7.50
800
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h=5m, d=10m, F=7.50
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h=5m, d=10m, F=0.28
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180 270 360
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180 270 360 θ(o) Tunnel
0
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180 270 360 θ(o)
Structure A
0
0
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180 270 360 θ(o)
0
θ
h=10m, d=10m, F=0.28
0
90
180 270 360
h=10m, d=10m, F=0.28
0
90
180 270 360 θ(o)
Structures A+B
Fig. 11. Maximum envelopes of dynamic axial force increments along the perimeter of the tunnel. First line: visco-elastic analysis results and second line: elasto-plastic analyses results.
K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
N (kN/m)
800
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h=5m, d=10m, F=0.28
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h=10m, d=10m, F=7.50
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θ
h=10m, d=10m, F=0.28
0
90
180 270 360
h=10m, d=10m, F=0.28
0
90
180 270 360 θ(o)
Structures A+B
Fig. 12. Dynamic axial force increments along the perimeter of the tunnel computed for the time step of maximum ovaling distortion. First line: visco-elastic analysis results and second line: elasto-plastic analyses results.
Viscoelastic
1.05
1.025 1.0125
0 1
2.5
5
7.5
1
10
F Viscoelastic
0 1
2.5
5
7.5
10
F Elastoplastic
1.35
Structure A Structure B
1.2625
rN
1.225
rN
1.08 1.04
1.3
1.15 1.075 1
Structure A
1.12
rN
rN
1.0375
1
Elastoplastic
1.16
1.175 1.0875
0 1
2.5
h=5m,d=5m
5 F
7.5
10
1
h=5m,d=10m
0 1
2.5
h=10m,d=5m
5 F
7.5
10
h=10m,d=10m
Fig. 13. rN response ratio as function of diameter, burial depth and flexibility ratio of the tunnel, computed from the maximum dynamic axial force increments regardless the time step (“envelope values”).
distributions (second line in Fig. 12) for flexible tunnels, are mainly attributed to stress redistributions within the adjacent soil, caused by soil yielding. Similar results are reported by Amorosi and Boldini [5]. Although, the oscillating massive
structures effect is more profound in the “envelope values” rather than the simultaneous values, it is observed in both cases, while it is amplified for rigid tunnels buried in small burial depths.
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K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
Viscoelastic
1.08
1.04 1.02
0 1
2.5
5 F
7.5
1
10
Viscoelastic
0 1
2.5
5 F
7.5
10
Elastoplastic
1.2
Structure A Structure B
1.15 rN
1.15 rN
1.04 1.02
1.2
1.1 1.05 1
Structure A
1.06 rN
rN
1.06
1
Elastoplastic
1.08
1.1 1.05
0 1
2.5
5 F
7.5
h=5m,d=5m
1
10
0 1
h=5m,d=10m
2.5
5 F h=10m,d=5m
7.5
10 h=10m,d=10m
Fig. 14. rN response ratio as function of diameter, burial depth and flexibility ratio of the tunnel, computed from simultaneous values of the dynamic axial force increments (time step of maximum ovaling deflection).
h=5m, d=5m, F=2.50
N (kN/m)
400
h=5m, d=5m, F=0.04
400
400
h=10m, d=5m, F=2.50
400
300
300
300
300
200
200
200
200
100
100
100
100
0
0
90
180 270 360
0
0
o
0
180 270 360 o
θ( ) Tunnel
90
90
180 270 360 o
θ( ) Structure A
0
θ( )
Structures A+B
0
h=10m, d=5m, F=0.04
0
90
θ
180 270 360 θ(o)
Wang(1993)
Fig. 15. Axial force dynamic increments along the perimeter of the tunnel for the time step of maximum ovaling distortion. Comparisons of visco-elastic analyses results with Wang's solution for no-slip conditions.
As a general remark, in the majority of the cases the consideration of the above ground massive structures results in an increase of the dynamic axial force. This effect seems to be related to the diameter, the burial depth and the flexibility of the tunnel. In a preliminary effort to quantify this increase for the investigated cases response ratios (rN) are computed according to the following expression and plotted against the aforementioned parameters: r N ðθjNTunnel ¼ max Þ ¼
N Tunnel þ Structures ðθjNTunnel ¼ max Þ N Tunnel ðθjNTunnel ¼ max Þ
ð3Þ
where N Tunnel ðθjNTunnel ¼ max Þ is the maximum dynamic axial force increment for the “free-field conditions” case and N Tunnel þ Structures ðθjNTunnel ¼ max Þ is the corresponding dynamic axial force increment at the same section accounting for the above ground structures.
The computed response ratios evaluated for the maximum dynamic axial force increments regardless the time step (envelope values) are presented in Fig. 13, as a function of the diameter, burial depth and flexibility ratio of the tunnel. Shallow and stiffer tunnels are affected more by the above ground structures. In addition, elasto-plastic analyses predict higher amplification of the axial force compare to the visco-elastic analyses, due to stress redistributions within the soil caused by the soil yielding. For the cases where only one structure is considered (e.g., Structure A), the dynamic axial force increase due to the above ground structure existence is as high as 16% for rigid shallow tunnels, assuming soil non-linear response. The introduction of the second structure (e.g., Structures A and B) results in even larger amplification, which reaches in this case, 30% of the envelope axial force in the absence of any structure at the surface. These results may be considered relatively conservative, as they
K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
h=5m, d=5m, F=2.5
M (kNm/m)
450
225
0
0
−225
−225
−450
0
1
M (kNm/m)
4
−450
225
0
0
−225
−225 0
1
2
3
4
−450
h=10m, d=5m, F=2.5
450
225
0
0
−225
−225
−450
−450
1
2
3
4
h=10m, d=5m, F=0.04
450
225
0
0
−225
−225
−450 0
1
2 t(s)
3
4
2
3
4
Structure A Structure B
0
1
2
3
4
θ
h=10m, d=5m, F=2.5
Tunnel Structure A Structures A+B
0
450
225
1
h=5m, d=5m, F=0.04
450
225
0
0
450
225
−450
M (kNm/m)
3
h=5m, d=5m, F=0.04
450
M (kNm/m)
2
h=5m, d=5m, F=2.5
450
225
11
1
2
3
4
h=10m, d=5m, F=0.04
−450 0
1
2 t(s)
3
4
Fig. 16. Dynamic bending moment time histories at θ¼ 45ο. Left column: visco-elastic analysis results and right column: elasto-plastic analysis results.
h=5m, d=5m, F=2.50
M (kNm/m)
450
450
h=10m, d=5m, F=2.50
450
337.5
337.5
337.5
337.5
225
225
225
225
112.5
112.5
112.5
112.5
0
0
90
180 270 360
h=5m, d=5m, F=2.50
450
M (kNm/m)
h=5m, d=5m, F=0.04
450
0
0
90
180 270 360
h=5m, d=5m, F=0.04
450
0
450
0
90
180 270 360
h=10m, d=5m, F=2.50
0
450
337.5
337.5
337.5
337.5
225
225
225
225
112.5
112.5
112.5
112.5
0
0
90
180 270 360 o
0
0
90
180 270 360
0
0
90
o
θ( )
o
θ( ) Tunnel
Structure A
180 270 360 θ( )
θ
0
h=10m, d=5m, F=0.04
0
90
180 270 360
h=10m, d=5m, F=0.04
0
90
180 270 360 θ(o)
Structures A+B
Fig. 17. Maximum envelopes of dynamic bending moment increments along the perimeter of the tunnel. First line: visco-elastic analysis results and second line: elastoplastic analyses results.
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K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
h=5m, d=5m, F=2.50
M (kNm/m)
450
450
h=10m, d=5m, F=2.50
450
337.5
337.5
337.5
337.5
225
225
225
225
112.5
112.5
112.5
112.5
0
0
90
0
180 270 360
h=5m, d=5m, F=2.50
450
M (kNm/m)
h=5m, d=5m, F=0.04
450
0
90
0
180 270 360
h=5m, d=5m, F=0.04
450
450
0
90
0
180 270 360
h=10m, d=5m, F=2.50
450
337.5
337.5
337.5
337.5
225
225
225
225
112.5
112.5
112.5
112.5
0
0
90
0
180 270 360
0
o
90
0
180 270 360
0
90
o
θ( )
o
θ( ) Tunnel
180 270 360 θ( )
Structure A
θ
0
h=10m, d=5m, F=0.04
0
90
180 270 360
h=10m, d=5m, F=0.04
0
90
180 270 360 θ(o)
Structures A+B
Fig. 18. Dynamic bending moment increments along the perimeter of the tunnel computed for the time step of maximum ovaling distortion. First line: visco-elastic analysis results and second line: elasto-plastic analyses results.
Viscoelastic
1.05
rM
rM
1.025
0 1
2.5
5 F
7.5
1
10
Viscoelastic
1.22
0 1
2.5
5 F
7.5
10
Elastoplastic
1.26
Structure A Structure B
1.195 rM
1.165 rM
1.08 1.04
1.0125
1.11
1.13 1.065
1.055 1
Structure A
1.12
1.0375
1
Elastoplastic
1.16
1 0 1
2.5
h=5m,d=5m
5 F
7.5
10
h=5m,d=10m
0 1
2.5
h=10m,d=5m
5 F
7.5
10
h=10m,d=10m
Fig. 19. rM response ratio as function of diameter, burial depth and flexibility ratio of the tunnel, computed from the maximum dynamic bending moment regardless the time step (“envelope values”).
computed based on “envelope values”, however one could easily imagine that the real case with several above ground structures will much strongly affect the tunnels response increasing even more the axial forces. Following the previous analysis, Fig. 14 summarizes the computed response ratios estimated by the simultaneous values of the
dynamic axial forces increments along the perimeter, referring to the time step of the tunnel maximum diametric deflection. The results reveal a similar tendency with the “envelope ratios” with a small reduction of the computed amplification though, which in this case do not exceed 15% for the case of two structures at the surface and for a rigid shallow tunnel.
K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
13
Structure A
Structure A Structure B
h=5m,d=5m
h=5m,d=10m
h=10m,d=5m
h=10m,d=10m
Fig. 20. rM response ratio as function of diameter, burial depth and flexibility ratio of the tunnel, computed from simultaneous values of the dynamic bending moment increments (time step of maximum ovaling).
h=5m, d=5m, F=2.50
M (kNm/m)
400
h=5m, d=5m, F=0.04
400
400
h=10m, d=5m, F=2.50
400
300
300
300
300
200
200
200
200
100
100
100
100
0
0
90
180 270 360
0
0
θ(o) Tunnel
90
0
180 270 360 θ(o)
Structure A
0
90
180 270 360 θ(o)
Structures A+B
Wang(1993)
0
h=10m, d=5m, F=0.04
0
90
θ
180 270 360 θ(o)
Fig. 21. Dynamic bending moment increments along the perimeter of the tunnel for the time step of maximum ovaling distortion. Comparisons of visco-elastic analyses results with Wang's solution for no-slip conditions.
Representative numerical results are compared with analytical results estimated for “free field conditions” using the Wang's closed form solutions (Fig. 15). The analytical solutions refer to the computation of the lining dynamic axial forces under S-waves propagation, assuming no-slip conditions for the soil-tunnel interface. For the implementation of the analytical solutions herein, the soil shear strain, which is a main input, is estimated through 1D equivalent linear response analyses of the soil deposits that are performed in the frequency domain using EERA [6]. The comparisons indicate a relatively good agreement between the numerical and analytical results. The observed deviations are more significant for rigid tunnels and are mainly attributed to the simplified assumptions of the closed form solutions (e.g. elastic homogeneous soil, sufficient burial depth etc). It is noteworthy that when the comparison is extended to the cases where above ground structures exist (also given in Fig. 15), closed form solutions
predict lower axial forces than the numerically derived accounting for these structures. Soil–tunnel interface characteristics and more specifically the interface friction properties are affecting significantly the shear stresses distribution around the tunnel perimeter and therefore the lining axial forces [21,37]. In this parametric study we assumed perfect bonding between the soil medium and the tunnel. To this respect the computed axial forces may be considered as an upper bound for the lining. The introduction of an interface, allowing a certain slip between the two media, will result in a reduction of the lining axial forces and will probably affect the presented results.
3.3.3. Bending moment Fig. 16 summarizes representative time histories of the dynamic ο bending moment computed at a crucial section of the lining (θ ¼45 )
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K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
for different scenarios regarding the existence of above ground structures. The results indicate a similar trend to that of the axial forces. The effects of the above ground oscillating massive structures are again more evident for shallow and rigid tunnels (Figs. 17–18). Similar to the axial forces, response ratios (rM) are derived, between the maximum dynamic bending moment increment assuming “free-field conditions” M Tunnel ðθjMTunnel ¼ max Þ and the corresponded dynamic bending moment increment accounting for the above ground structures M Tunnel þ Structures ðθjMTunnel ¼ max Þ, and plotted against the diameter, burial depth and flexibility ratio of the tunnel. r M ðθjMTunnel ¼ max Þ ¼
M Tunnel þ Structures ðθjMTunnel ¼ max Þ M Tunnel ðθjMTunnel ¼ max Þ
ð4Þ
The computed response ratios are presented in Figs. 19 and 20, for the “envelope” and simultaneous dynamic bending moment increments, respectively. The interpretation of the results reveals similar observations with the axial forces. Shallow and stiffer tunnels are affected more by the presence of the surface structures. The dynamic bending moments can be amplified up to 25% in terms of envelope values compared to the case where the above ground structures are disregarded. This amplification reaches 16% in terms of simultaneous values. Finally, representative comparisons are given in Fig. 21 between numerically derived dynamic bending moment increments and results of Wang's analytical solution [41]. Analytical solutions overestimate the dynamic bending moment of flexible tunnels, compared to the numerical results, while the opposite is observed for rigid tunnels. Generally, the effect of the interface properties (e.g., interface friction characteristics) is less important for the lining bending moment amplitude, compared to the axial force. However, these properties may affect the soil non-linear response near the tunnel (e.g., effect on soil stiffness and strength degradation under severe shaking), resulting in a different soil yielding response [15], [37]. The latter can lead to stress redistributions within the soil (e.g., stress concentrations at crucial locations) and therefore to different distributions of the bending moment along the lining.
4. Conclusions The paper presents and discusses a series of numerical analyses that performed to investigate the dynamic response of shallow circular tunnels, accounting for the adjacent above ground oscillating buildings interaction effects. The analysis is focused to the more critical transversal response that affects directly the cross-sectional structural design. Parameters that significantly affect the phenomenon, namely the soil–tunnel relative flexibility, the tunnel dimensions, the tunnel burial depth and the soil properties and non-linearity during shaking, are accounted for within this parametric analysis. The main conclusions are summarized as follows:
The existence of above ground structures affects the shear wave
field altering the input motion compare to the “free field conditions” case. The effect is more evident for shallow burial depths. Numerical results prove the tunnel ovaling deformation mode during shear shaking. The existence of oscillating massive above ground structures may affect the amplitude of lining ovaling distortions with this effect being more pronounced for the shallow tunnels. For the herein studied cases, the ovaling ratio R (structural to soil diametric deflection) is increased compared to the “free field conditions” case to about 20% for shallow burial depths. For deeper tunnels the increase is of the
order of 6% compared to the case where no above ground structures exist. Existent above ground structures alter the soil stress field around the tunnel, therefore affecting the soil yielding response. The effects are more significant in the case of soil type C due to the reduced strength compared to soil type B. For flexible tunnels, the consideration of the above ground structures results in larger permanent deformations at crucial sections of the lining (e.g., tunnel crown), with the effect being more significant for reduced burial depths. Soil non-linear behavior in case of severe earthquakes, capable to produce very large shear strains, may further affect the present results. However, the range of ground acceleration amplitudes reached in this paper are within the most commonly expected ground accelerations and consequently the present results and conclusions should be considered reliable for these levels of moderate to strong ground shaking. The existence of above ground structures generally leads to an increase of the lining dynamic internal forces (compared to the “free field conditions” case) with that being more significant for shallow and stiffer tunnels. Elastoplastic analyses predict an increased response (larger lining forces) for the majority of the cases compared to the visco-elastic analyses. In terms of maximum dynamic increment distributions regardless the time step (e.g., envelope values) the increase may reach 30% for the axial force (for shallow stiff tunnels) and 25% for the bending moment. In terms of simultaneous dynamic increment distributions the amplification is of the order of 15% for both axial forces and bending moments. Analytical solutions proposed by Wang for no-slip conditions give comparable results for the dynamic axial forces with the numerical predictions. On the contrary, for the bending moments the differences are more significant. For flexible tunnels the analytical solutions overestimate the lining bending moment, while the opposite is observed for rigid tunnels. As a general conclusion, for the majority of the investigated cases the consideration of the above ground structures results in an increase of the tunnel dynamic response both in terms of deformations and lining forces with that being more significant for shallower tunnels. Considering the large number of the parameters involved, and the associated uncertainties, it is quite difficult to come out with a strict quantification of the response modification. However, this quantification is of great importance for engineering practice. To this respect, accounting for the most conservative results of this preliminary study (e.g., envelope results for shallow rigid tunnels considering two above ground structures), we may conclude that the consideration of above ground structures may result in an increase up to 25% for the lining bending moment and 30% for the lining axial force compared to the “free field conditions” case.
Further research deemed to be necessary to further understand and rigorously quantify in engineering terms the effects of above ground structures on the dynamic response of shallow tunnels, passing below these structures in urban areas. Issues, such as (i) the input motion characteristics (e.g., frequency content, amplitude, and duration), (ii) the soil–tunnel and soil–structures interface properties and response during severe shaking, (iii) the number, position, and geometrical and mechanical characteristics of the structures (e.g., existence of basements, mass and stiffness properties of the equivalent SDOFs, structures non linear response), and (iv) the complex wave propagation phenomena related within the longitudinal direction (e.g. input motion incoherence), may significantly affect the structures– soil–tunnel interaction effects. The present study should be considered as a first step to acknowledge the problem and quantify it in a simple, yet engineering meaningful way.
K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
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Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn
Seismic behaviour of circular tunnels accounting for above ground structures interaction effects Kyriazis Pitilakis a,n, Grigorios Tsinidis a, Andrea Leanza b, Michele Maugeri b a Aristotle University of Thessaloniki, Research Unit of Soil Dynamics and Geotechnical Earthquake Engineering, Department of Civil Engineering, PO Box 424, GR-54124 Thessaloniki, Greece b University of Catania, Department of Civil and Environmental Engineering, Sicily, Italy
art ic l e i nf o
a b s t r a c t
Article history: Received 12 February 2014 Received in revised form 19 May 2014 Accepted 20 August 2014
Tunnels are commonly designed under seismic loading assuming “free field conditions”. However, in urban areas these structures pass beneath buildings, often high-rise ones, or are located close to them. During seismic excitation, above ground structures may cause complex interaction effects with the tunnel, altering its seismic response compared to the “free field conditions” case. The paper summarizes an attempt to identify and understand these interaction effects, focusing on the tunnel response. The problem is investigated in the transversal direction, by means of full dynamic time history analyses. Two structural configurations are studied and compared to the free field conditions case, consisting of one or two above ground structures, located over a circular tunnel. Above ground structures are modeled in a simplified way as equivalent single-degree of freedom oscillators, with proper mechanical properties. Several parameters that are significantly affecting the phenomenon are accounted for in this parametric study, namely the soil to tunnel relative flexibility, the tunnel dimensions, the tunnel burial depth and the soil properties and nonlinearities during shaking. Tunnels response characteristics are compared and discussed, in terms of acceleration, deformations and lining dynamic internal forces. Internal forces are also evaluated with analytical closed form solutions, commonly used in preliminary stages of design, and compared with the numerical predictions. The results indicate that the presence of the above ground structures may have a significant effect on the seismic response of the tunnel, especially when the latter is stiff and located in shallow depths. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Circular tunnels Seismic behavior Above ground structures–soil–tunnel dynamic interaction Dynamic analysis
1. Introduction Tunnels constitute crucial components of the transportation and utility networks in urban areas. The associated impact in case of earthquake induced damage denotes the importance of proper seismic design especially in seismic prone regions. It is generally believed that underground structures are less vulnerable to seismic shaking compared to above ground structures. However, several cases of severe damage or even collapse have been reported in the literature, mainly for shallow embedded structures in soft soils ([10,17,19,28,31,33,40,42,43] among others). During an earthquake, tunnels are subjected to shaking due to wave propagation and permanent ground displacements due to ground failure (lateral spreading, landslides and fault rupture). In both the cases, the kinematic loading imposed by the adjacent soils prevails, while the inertial loads are generally of secondary importance. Therefore, the seismic behavior of underground structures
n
Corresponding author. Tel.: þ 30 2310995693. E-mail address: [email protected] (K. Pitilakis).
http://dx.doi.org/10.1016/j.soildyn.2014.08.009 0267-7261/& 2014 Elsevier Ltd. All rights reserved.
and tunnels is quite distinct compared to the above ground structures [13,17]. Several methods are available in the literature for the evaluation of the seismic response of underground structures and tunnels [4,11,12,16,29,35,36,41]. The results of these methods may significantly deviate, even under the same design assumptions, due to both inherent epistemic uncertainties and knowledge shortfall regarding some crucial issues that considerably affect the seismic response [30]. In order to better understand the seismic behavior of these types of structures several experimental research studies have been recently carried out (e.g., [8,22,38,39] among others). Available design methods for shaking, usually assume free field conditions, precluding the existence of above ground structures in the adjacent area of the tunnel (e.g., above the tunnel). However, in urban areas, tunnels often pass beneath or close by high-rise buildings. During shaking, the vibration of these above ground structures may create complex interaction phenomena with the tunnel, passing often few meters below their foundation, which are expected to affect the seismic wave propagation field. In this sense, they may modify the dynamic response of the tunnel, while at the same time the existence of the tunnel close to the surface
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and the foundations of the buildings, may alter the response of the buildings themselves. Dynamic interaction effects in urban areas (e.g., “city effects”) have been mainly examined between above ground structures. A comprehensive review is made by Menglin et al. [27]. Regarding the dynamic interaction phenomena between above ground and embedded structures, most researchers focus on the effect of an underground structure, often a circular tunnel or a cavity, on the response of the above ground structures. The underground structure is commonly assumed to be embedded in an elastic halfspace, while the effects are usually expressed in terms of surface ground motion amplification [9,20,23–25,34,43]. On the other hand, the inverse problem, i.e., the effects of above ground structures on the response characteristics of embedded structures (e.g., shallow tunnels) have not been thoroughly studied. The present paper presents an attempt to identify, understand and quantify these potential effects. For this purpose, a numerical parametric study is conducted, assuming different structures, soil and tunnel configurations. The problem is investigated in the transversal direction, as this direction is related to the lining maximum developing stress states and affects directly the cross-sectional structural design of the tunnel. Parameters that significantly affect the phenomenon and considered herein are the soil to tunnel relative flexibility, the tunnel dimensions, the tunnel burial depth and the soil properties accounting also for their nonlinear behavior during strong shaking. The response of the examined cases is discussed in terms of acceleration, deformations and lining internal forces. Lining forces are also evaluated with existing closed form analytical solutions (e.g., [41]), commonly used in preliminary stages of design of circular tunnels in the absence of any above ground structure, and the results are compared to the numerical data.
2. Numerical simulation A series of dynamic time history analyses is performed on representative structural systems comprising of a circular tunnel and models of above ground buildings. The analysis is performed in the transversal direction, with the surface structures been simulated as equivalent single degree of freedom (SDOF) oscillators with rigid foundations. Inertial properties of the equivalent SDOFs correspond to usual buildings (e.g., 6–8 storey buildings). The case studies are summarized in Fig. 1. One of the buildings (e.g., Structure A) is located above the tunnel, assuming that the tunnel is constructed with an underground excavation method (e.g., using a tunnel boring machine). In cases of two surface structures, the second one (e.g., Structure B) is located just aside the first one. Table 1 tabulates the mechanical properties of the tunnel and structures along with the assumed fixed-based fundamental periods of the above ground structures (Tfix). Two different sand soil deposits are considered herein. One of them corresponds to a rather loose soil deposit representing a soil type C according to Eurocode 8 [7] with fundamental frequency equal to 1 Hz, while the second one corresponds to a stiffer deposit (i.e., soil type B according to Eurocode 8), with fundamental frequency equal to 2.5 Hz. The shear wave velocity gradient profiles are presented in Fig. 2. Mechanical properties of the assumed soil deposits are tabulated in Table 2. To study the effects of the tunnel size and burial depth, the diameter of the circular tunnel (d) is ranging between 5 and 10 m, while the tunnel burial depth (h) is also ranging between 5 and 10 m. Both dimensions and depths are common in practice. Moreover, the soil to tunnel relative flexibility is examined through the so called flexibility ratio [41], which is estimated using
the following analytical formulation: ES 1 ν2l r 3 F¼ 6El I l ð1 þ νS Þ
ð1Þ
where, Es is the soil elastic modulus, vs is the soil Poisson ratio, El is the lining elastic modulus, vl is the lining Poisson ratio, Il is the lining moment of inertia (per unit width) and r is the circular tunnel radius. The flexibility ratio of the investigated cases is ranging from almost zero (quite rigid tunnel) to 10 (quite flexible tunnel). Few very flexible tunnels (F430) are also evaluated, so as to study the effect of this crucial parameter to extreme ends. To achieve the desirable flexibility ratio, the tunnel lining thickness (tl) is adequately selected for each case. The analyses are performed, under plane strain conditions in total stresses, using the finite element code ADINA [3]. Despite the generic nature of the specific code, ADINA can efficiently reproduce the complex phenomena implicated in a dynamic time history analysis, including wave propagation through soil media and dynamic soil–structure interaction effects [2,18,26]. More specifically, the soil is meshed with plane strain elements, while the tunnel and the above ground structures (SDOFs) are modeled using beam elements (Fig. 3). The adopted element size is selected in a way that ensures the following criteria: (a) Efficient reproduction of all the waveforms of the whole frequency range under study (e.g., following the principle that the element size must be 8–10 times smaller than the minimum wavelength of interest), (b) converge criteria of the analysis (for elasto-plastic analysis) and (c) efficient simulation of the soil close to the tunnel. Therefore, a finer discretization near the tunnel is selected, allowing a low element aspect ratio (for the soil elements) and a low face corner angle (for the beam elements simulating the circular tunnels). The base boundary of the model is simulated as rigid bedrock, where the seismic input motion is applied in terms of displacement time history. For the vertical boundaries kinematic constrains are introduced, forcing the opposite vertical sides to move simultaneously, simulating the shear waves propagating upwards, e.g., tie constrains [1]. To simplify the analyses, a solid connection between the soil and structures is assumed. Although, interface characteristics are quite crucial for the dynamic response of embedded structures [15,21,30,32,37], this assumption is quite common in engineering practice, as it corresponds to an upper limit for the developed shear stresses around the tunnel. The tunnel and the above ground SDOF structures are assumed to behave within the linear elastic range. For the soil behavior two assumptions are made. In the first series of analyses a linear viscoelastic material is used, while for the final analyses an elastoplastic Mohr Coulomb material is implemented, in order to account for the permanent soil response due to yielding. In the latter case, the soil shear strength is assumed to increase with depth (Table 2), following the increase of the soil stiffness (Fig. 2). Viscous damping (5% for all the examined cases and elements, for sake of simplicity) is employed in the frequency depended Rayleigh type. For the elasto-plastic analyses additional energy dissipation is introduced by the hysteretic soil response. Investigated systems are subjected to a simplified Ricker wavelet of nominal frequency equal to 1 Hz and amplitude equal to 0.1 g, introduced at the model base (Fig. 4a). Input motion nominal frequency is selected equal to the fundamental frequency of one of the soil deposits (soil type C) in order to study the effects of soil resonance. In addition, input motion amplitude is selected in order to produce at the tunnel depth and the soil surface,
K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
3
Fig. 1. Schematic representation of the examined cases.
Table 1 Structures mechanical properties.
Elastic modulus (GPa) SDOF mass (t) Storeys (n) Fixed-based period Tfix (s)
Tunnel
Structure A
Structure B
32 – – –
32 60 8 0.80
32 45 6 0.60
Vs (m/s) 0
0
200
400
600
800
Depth(m)
15
30
45 Soil Type C Soil Type B 60 Fig. 2. Shear wave velocity profiles of the soil deposits.
acceleration amplitudes of the order of 0.15 g and 0.30 g, (respectively for the two soil profiles), values that are often proposed in code prescriptions. The selection of this type of input motion instead of real accelerograms is usual in this type of analyses in order to avoid the dependence of the response to a specific time history. The selected Ricker wavelet characteristics match a broader band of input motions having at the same time high displacement amplitudes. Comparisons of displacement response spectra computed for the selected wavelet and for different real records (recorded in different earthquakes for soil class A of
Eurocode 8 and scaled at 0.1 g) are presented in Fig. 4b proving that the Ricker wavelet produces much higher displacements compared to the real records. Accounting for the dynamic response of embedded structures that is significantly affected by the soil kinematic response, this selection is considered to be conservative. At the same time the expected soil non-linear behavior is well achieved. All the analyses are performed in two steps. In the first step, the gravity loads are introduced (e.g., geostatic step), while in the second step the dynamic input motion is applied at the base of the numerical model (dynamic step). The boundaries are accordingly changed, between the two steps, using the restart option available in ADINA as shown in Fig. 3b and c. Construction sequence of the tunnel and the above ground structures, as well as construction stages of the circular tunnel, are not considered in this study. Accounting for the aim of this work that is to study the effect of above ground buildings on the tunnels seismic response, case specific stress and deformation variations related to the construction process would rather fade out the problem. These important aspects have actually no effect in case of elastic dynamic analyses, while it is recognized that in case of the elasto-plastic analyses, they may affect the soil yielding response and therefore the initial soil stress state around the tunnel. Nevertheless, since this study is focusing on the dynamic inelastic response and the main objective herein is to examine the differences assuming a reasonable and generally enough reference initial stress state, the conclusions drawn can be considered applicable and valid. Similar simplifications can be found in the literature (e.g., [14]). Time discretization is an important element for the accuracy of the dynamic analysis. ADINA offers an automatic time incrementation scheme, according to which the time step of the analysis is properly selected and changed in order to achieve a stable solution. This scheme was selected herein as adequate to achieve convergence for the solution.
3. Results Representative results are presented and discussed in this section in terms of horizontal acceleration, tunnel deformations and dynamic internal forces of the lining.
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Table 2 Mechanical properties of the soil deposits. Soil profile Soil type B Unit weight of volume, γ (kN/m3) Coefficient of earth pressure at rest, Ko Shear wave velocity, Vs (m/s) Damping, D (%) Poisson ratio, v Cohesion, c (kPa) Friction angle, φ (deg)
Soil type C
20 20 0.50 0.50 Variable with depth (see Fig. 1) 5% 5% 0.333 0.333 20 10 Variable with depth: Variable with depth: 0–30 (m): 35ο 0–30 (m): 28ο 30–60 (m): 45ο 30–60 (m): 35ο
3.2. Tunnel distortions Plain strain elements
Structure A
Beam elements
Structure B
Structure A Displacement constrains
u(t)
Fig. 3. Representative numerical models in ADINA, (a) 5 m diameter tunnel without any building at surface, (b) 10 m diameter tunnel with one building at surface, and (c) 10 m diameter tunnel with two buildings at surface.
3.1. Horizontal acceleration Fig. 5 presents the horizontal acceleration amplification along the vertical array crossing the center of a flexible shallow tunnel (5 m diameter, 5 m burial depth), computed for different soil profiles and assumptions regarding the soil response (visco-elastic and elasto-plastic approach). Visco-elastic analyses results for soil type C indicate an increase of the horizontal acceleration near the tunnel area (compared to the free field), while for the elastoplastic analysis, the behavior is more complex due to the soil yielding. The introduction of the above ground structures causes a slight increase of the horizontal acceleration near the tunnel. For soil type B (stiffer soil conditions) the amplification within the soil is in general reduced and it is not affected by the existence of the tunnel and the structures at the surface. This is mainly attributed to the significant difference between the soil deposit resonance frequency (2.5 Hz) compared to the signal frequency (1 Hz). Similar results are reported for the majority of the investigated systems.
Fig. 6 presents the effect of the above ground structures on the horizontal deformations of the tunnel. The results refer to elastoplastic analyses of a flexible and large tunnel (F4 9.6). Deformed shapes verify the ovaling deformation mode of tunnels during shear wave propagation. In addition, building masses alter the soil stress field around tunnel resulting in stress redistributions that can lead to a different soil permanent response. This complex response is clearer in Fig. 7 where the dynamic vertical displacement of the tunnel crown is presented for different tunnels (flexible and rigid tunnels of different dimensions embedded in different burial depths). The existence of above ground structures results in larger dynamic displacement increments. It is clear that for shallow burial depths and for larger dimension flexible tunnels, soil yielding increase, resulting in an increase of permanent deformations after shaking. This effect is less evident for rigid tunnels and for higher burial depths. Fig. 8 summarizes the numerical results of the variation of the ovaling ratio (R) with the flexibility ratio (F), for tunnels embedded in 5 m burial depth. The effect of the above ground structures on these relations is also presented. Similar results are presented in Fig. 9 for tunnels embedded in the 10 m burial depth. The computation of the ovaling ratio is performed according to the following formulation: R¼
Δstructure Δf f
ð2Þ
where Δstructure is the computed ovaling distortion of the tunnel and Δf f is the corresponding distortion of the soil at free field. In the absence of structures on the surface, numerical results are in good agreement with the F–R relation proposed by Penzien [29] for no slip conditions between the tunnels lining and the soil. Generally, the elastoplastic analyses predict slightly higher diametric deflection for the tunnel, due to the soil yielding. This observation is more significant for flexible structures in soil type C and for small burial depths (e.g., F48 in Fig. 8a). In the majority of the cases the introduction of above ground oscillating structures results in an increase of the tunnel's diametric deflection (larger ovaling ratio for the same flexibility ratio) compared to the case where no surface structures exist. This increase is more significant for shallow tunnels where ovaling ratio R is about 15–20% larger than the case where no above ground structures exist. As it has been expected, the increase of R for deeper tunnels is lower reaching 5–6% compared to the case where no above ground structures exist. Similar to the “free field conditions” case, elastoplastic analyses predict higher values for ovaling ratio as result of soil yielding around the tunnel.
K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
Input motion
5
Displacement response spectra
0.5
0.08 0.07 0.06
0
D (m)
a (m/s2)
0.05 0.04 0.03
−0.5
0.02 0.01 −1
0
1.5
3 t (s)
4.5
0
6
0
1
2 T(s)
3
4
Fig. 4. (a) Input motion acceleration time history (b) displacement response spectra computed for different real seismic records (recorded in soil type A conditions and scaled at 0.1 g) compared to the selected Ricker wavelet (black solid line).
A (m/s2) Visco−elastic, Type C 0
1
2
3
4
15
0 Depth(m)
Depth(m)
0
A (m/s2) Elasto−plastic, Type C
30 45 60
0
1
2
3
4
15 30 45 60
2
A (m/s ) Visco−elastic, Type B
Depth(m)
0
0
0.5
1
1.5
15 30
2
Tunnel Tunnel+Structure A Tunnel+Structures A+B Free field
45 60
Fig. 5. Horizontal acceleration amplification along the vertical array crossing the center of a shallow flexible tunnel (F=2.50); effect of the surface structures.
An increase of the diametric deflection leads to an increase of the lining dynamic internal forces, with the effect being more significant for stiffer tunnels, due to their relative inability to follow these increased deformations.
structures on the tunnel lining internal forces, is made, by plotting ratios of the computed maximum internal force accounting for the above ground structures to the maximum internal forces precluding them, called hereafter “free field conditions”.
3.3. Dynamic internal forces 3.3.1. Evaluation of dynamic internal forces Lining internal forces (e.g., bending moment and axial force), which are of great importance for the final detailing of the tunnel section, are discussed in this section accounting for the effects of the above ground structures. Representative results are presented in terms of time histories and distributions of the internal forces dynamic increments along the perimeter, the latter referring to the time step of the tunnel maximum ovaling distortion. In addition, we present the maximum envelope distributions of the lining forces (e.g., maximum values along the perimeter regardless the time step), as these are commonly used for the final lining detailing. The numerical dynamic increments distributions are compared with the results of closed form analytical solutions [41], which are commonly used during preliminary stages of design. A first attempt, to quantify the effects of the above ground
3.3.2. Axial force Fig. 10 presents time windows of representative time histories of the dynamic axial forces computed at a crucial section of the lining ο (θ ¼45 ) for different scenarios regarding the presence of above ground structures. The results refer to both flexible and rigid tunnels embedded in different burial depths. As expected, the consideration of the oscillating massive structures at the ground surface results in an increase of the axial forces, especially in the cases of shallow and stiffer tunnels. The elasto-plastic analyses reveal residual values for the axial forces after shaking, due to the soil yielding, with the phenomenon being amplified for flexible shallow tunnels. This observation comes in accordance with recent findings from a series of dynamic centrifuge tests that were performed on a flexible tunnel model embedded in dry sand [22].
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K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
Fig. 6. Contour diagrams of horizontal displacements on models deformed shapes for elasto-plastic analyses; time step of maximum ovaling of a 10 m diameter tunnel in 5 m burial depth (a) no surface structures, (b) structure A and (c) structures A þ B.
h=5m, d=5m, F=0.04
u (cm)
h=5m, d=5m, F=2.50 0.8
0.8
0.4
0.4
0.4
0
0
0
−0.4
−0.4
Tunnel Structure A Structures A+B
−0.4 −0.8
0
1.5
3 t(s)
4.5
6
−0.8
0
1.5
3 t(s)
4.5
6
−0.8
0.8
0.8
0.4
0.4
0.4
0
0
0
−0.4
−0.4
−0.4
0
1.5
3 t(s)
4.5
6
−0.8
0
1.5
3 t(s)
4.5
1.5
3 t(s)
4.5
6
h=5m, d=10m, F=0.36
0.8
−0.8
0
h=5m, d=10m, F=9.60
h=10m, d=5m, F=0.04
u (cm)
h=10m, d=5m, F=2.50
0.8
6
−0.8
0
1.5
3 t(s)
4.5
6
Fig. 7. Dynamic vertical displacement of the tunnel crown for different tunnels computed by elastoplastic analysis (h: burial depth, d: tunnel diameter, and F: flexibility ratio).
The same remarks are made in Figs. 11 and 12, where the dynamic increments of the axial forces are presented along the perimeter of the tunnel, in terms of (i) maximum values regardless the time step (e.g. envelope values) and (ii) simultaneous
values computed for time step of maximum ovaling deformation of the tunnel. In both figures, the first line refers to visco-elastic analyses results, while the second line to elasto-plastic analyses results. The anomalies observed in the simultaneous axial force
3
3
2.25
2.25
2.25
1.5
R
3
R
R
K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
1.5
7
1.5
Penzien(2000) 0.75
Viscoelastic
0.75
0.75
Elastoplastic 0
0 1
2.5
5 F
7.5
0
10
3
3
2.25
2.25
0 1
2.5
7.5
0
10
0 1
2.5
5 F
7.5
10
0.5
R
R
0.5
1.5
5 F
.375
1.5
0.375
Penzien(2000) 0.25
0.75
0.75
.125 0
0
0.25
0 1
0
2.5
0.25
5 F
7.5
0
0.5
0
10
Tunnel
0.125
0 1
Structure A 0
2.5
0.25
5 F
Structures A+B
0.5
7.5
10
3
3
2.25
2.25
2.25
1.5
R
3
R
R
Fig. 8. Ovaling ratio (R) versus flexibility ratio (F) for tunnels embedded in 5 m burial depth. Computed results versus Penzien [29] solution: (a) tunnel without above ground structures, (b) tunnel and structure A, (c) tunnel and structures A þ B, (d) comparisons for visco-elastic analyses, and (e) comparisons for elasto-plastic analyses.
1.5
1.5
Penzien(2000) Viscoelastic
0.75
0.75
0.75
Elastoplastic 0
0 1
2.5
5 F
7.5
0
10
3
3
2.25
2.25
0 1
2.5
7.5
10
0
0 1
2.5
5 F
7.5
10
0.5
R
R
0.5
1.5
5 F
.375
1.5
0.375
Penzien(2000) 0.25
0.75
0.75
.125 0
0
0.25
0 1
2.5
0
0.25
5 F
7.5
0
0.5
10
0
Tunnel
0.125
0 1
2.5
Structure A 0
0.25
5 F
7.5
0.5
Structures A+B
10
Fig. 9. Ovaling ratio (R) versus flexibility ratio (F) for tunnels embedded in 10 m burial depth. Computed results versus Penzien [29] solution: (a) tunnel without above ground structures, (b) tunnel and structure A, (c) tunnel and structures A þB, (d) comparisons for visco-elastic analyses, and (e) comparisons for elasto-plastic analyses.
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h=5m, d=5m, F=2.5
N (kN/m)
400 200
200
0
0
−200
−200
−400
0
N (kN/m)
3
4
−400
200
0
0
−200
−200 0
1
2
3
4
h=10m, d=5m, F=2.5
400
−400
200
0
0
−200
−200 0
1
2
3
4
h=10m, d=5m, F=0.04
400
−400
200
0
0
−200
−200
−400
0
1
2 t(s)
0
3
4
−400
2
3
4
1
2
3
Structure A Structure B
θ
4
h=10m, d=5m, F=2.5 1
2 3 Tunnel t(s) Structure A Structures A+B
0
1
2
3
4
h=10m, d=5m, F=0.04
400
200
1
h=5m, d=5m, F=0.04
400
200
−400
0
400
200
−400
N (kN/m)
2
h=5m, d=5m, F=0.04
400
N (kN/m)
1
h=5m, d=5m, F=2.5
400
0
1
2 t(s)
3
4
Fig. 10. Dynamic axial force time histories at θ ¼45ο. Left column: visco-elastic analysis results and right column: elasto-plastic analysis results.
N (kN/m)
800
800
h=5m, d=10m, F=0.28
800
h=10m, d=10m, F=7.50
800
600
600
600
600
400
400
400
400
200
200
200
200
0
800 N (kN/m)
h=5m, d=10m, F=7.50
0
90
180 270 360
h=5m, d=10m, F=7.50
0
800
0
90
180 270 360
h=5m, d=10m, F=0.28
0
800
0
90
180 270 360
h=10m, d=10m, F=7.50
0
800
600
600
600
600
400
400
400
400
200
200
200
200
0
0
90
180 270 360 θ(o) Tunnel
0
0
90
180 270 360 θ(o)
Structure A
0
0
90
180 270 360 θ(o)
0
θ
h=10m, d=10m, F=0.28
0
90
180 270 360
h=10m, d=10m, F=0.28
0
90
180 270 360 θ(o)
Structures A+B
Fig. 11. Maximum envelopes of dynamic axial force increments along the perimeter of the tunnel. First line: visco-elastic analysis results and second line: elasto-plastic analyses results.
K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
N (kN/m)
800
800
h=5m, d=10m, F=0.28
800
h=10m, d=10m, F=7.50
800
600
600
600
600
400
400
400
400
200
200
200
200
0
800 N (kN/m)
h=5m, d=10m, F=7.50
9
0
90
0
180 270 360
h=5m, d=10m, F=7.50
800
0
90
0
180 270 360
h=5m, d=10m, F=0.28
800
0
90
0
180 270 360
h=10m, d=10m, F=7.50
800
600
600
600
600
400
400
400
400
200
200
200
200
0
0
90
0
180 270 360
0
o
90
0
180 270 360
0
90
o
θ( )
o
θ( ) Tunnel
θ( )
Structure A
0
180 270 360
θ
h=10m, d=10m, F=0.28
0
90
180 270 360
h=10m, d=10m, F=0.28
0
90
180 270 360 θ(o)
Structures A+B
Fig. 12. Dynamic axial force increments along the perimeter of the tunnel computed for the time step of maximum ovaling distortion. First line: visco-elastic analysis results and second line: elasto-plastic analyses results.
Viscoelastic
1.05
1.025 1.0125
0 1
2.5
5
7.5
1
10
F Viscoelastic
0 1
2.5
5
7.5
10
F Elastoplastic
1.35
Structure A Structure B
1.2625
rN
1.225
rN
1.08 1.04
1.3
1.15 1.075 1
Structure A
1.12
rN
rN
1.0375
1
Elastoplastic
1.16
1.175 1.0875
0 1
2.5
h=5m,d=5m
5 F
7.5
10
1
h=5m,d=10m
0 1
2.5
h=10m,d=5m
5 F
7.5
10
h=10m,d=10m
Fig. 13. rN response ratio as function of diameter, burial depth and flexibility ratio of the tunnel, computed from the maximum dynamic axial force increments regardless the time step (“envelope values”).
distributions (second line in Fig. 12) for flexible tunnels, are mainly attributed to stress redistributions within the adjacent soil, caused by soil yielding. Similar results are reported by Amorosi and Boldini [5]. Although, the oscillating massive
structures effect is more profound in the “envelope values” rather than the simultaneous values, it is observed in both cases, while it is amplified for rigid tunnels buried in small burial depths.
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K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
Viscoelastic
1.08
1.04 1.02
0 1
2.5
5 F
7.5
1
10
Viscoelastic
0 1
2.5
5 F
7.5
10
Elastoplastic
1.2
Structure A Structure B
1.15 rN
1.15 rN
1.04 1.02
1.2
1.1 1.05 1
Structure A
1.06 rN
rN
1.06
1
Elastoplastic
1.08
1.1 1.05
0 1
2.5
5 F
7.5
h=5m,d=5m
1
10
0 1
h=5m,d=10m
2.5
5 F h=10m,d=5m
7.5
10 h=10m,d=10m
Fig. 14. rN response ratio as function of diameter, burial depth and flexibility ratio of the tunnel, computed from simultaneous values of the dynamic axial force increments (time step of maximum ovaling deflection).
h=5m, d=5m, F=2.50
N (kN/m)
400
h=5m, d=5m, F=0.04
400
400
h=10m, d=5m, F=2.50
400
300
300
300
300
200
200
200
200
100
100
100
100
0
0
90
180 270 360
0
0
o
0
180 270 360 o
θ( ) Tunnel
90
90
180 270 360 o
θ( ) Structure A
0
θ( )
Structures A+B
0
h=10m, d=5m, F=0.04
0
90
θ
180 270 360 θ(o)
Wang(1993)
Fig. 15. Axial force dynamic increments along the perimeter of the tunnel for the time step of maximum ovaling distortion. Comparisons of visco-elastic analyses results with Wang's solution for no-slip conditions.
As a general remark, in the majority of the cases the consideration of the above ground massive structures results in an increase of the dynamic axial force. This effect seems to be related to the diameter, the burial depth and the flexibility of the tunnel. In a preliminary effort to quantify this increase for the investigated cases response ratios (rN) are computed according to the following expression and plotted against the aforementioned parameters: r N ðθjNTunnel ¼ max Þ ¼
N Tunnel þ Structures ðθjNTunnel ¼ max Þ N Tunnel ðθjNTunnel ¼ max Þ
ð3Þ
where N Tunnel ðθjNTunnel ¼ max Þ is the maximum dynamic axial force increment for the “free-field conditions” case and N Tunnel þ Structures ðθjNTunnel ¼ max Þ is the corresponding dynamic axial force increment at the same section accounting for the above ground structures.
The computed response ratios evaluated for the maximum dynamic axial force increments regardless the time step (envelope values) are presented in Fig. 13, as a function of the diameter, burial depth and flexibility ratio of the tunnel. Shallow and stiffer tunnels are affected more by the above ground structures. In addition, elasto-plastic analyses predict higher amplification of the axial force compare to the visco-elastic analyses, due to stress redistributions within the soil caused by the soil yielding. For the cases where only one structure is considered (e.g., Structure A), the dynamic axial force increase due to the above ground structure existence is as high as 16% for rigid shallow tunnels, assuming soil non-linear response. The introduction of the second structure (e.g., Structures A and B) results in even larger amplification, which reaches in this case, 30% of the envelope axial force in the absence of any structure at the surface. These results may be considered relatively conservative, as they
K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
h=5m, d=5m, F=2.5
M (kNm/m)
450
225
0
0
−225
−225
−450
0
1
M (kNm/m)
4
−450
225
0
0
−225
−225 0
1
2
3
4
−450
h=10m, d=5m, F=2.5
450
225
0
0
−225
−225
−450
−450
1
2
3
4
h=10m, d=5m, F=0.04
450
225
0
0
−225
−225
−450 0
1
2 t(s)
3
4
2
3
4
Structure A Structure B
0
1
2
3
4
θ
h=10m, d=5m, F=2.5
Tunnel Structure A Structures A+B
0
450
225
1
h=5m, d=5m, F=0.04
450
225
0
0
450
225
−450
M (kNm/m)
3
h=5m, d=5m, F=0.04
450
M (kNm/m)
2
h=5m, d=5m, F=2.5
450
225
11
1
2
3
4
h=10m, d=5m, F=0.04
−450 0
1
2 t(s)
3
4
Fig. 16. Dynamic bending moment time histories at θ¼ 45ο. Left column: visco-elastic analysis results and right column: elasto-plastic analysis results.
h=5m, d=5m, F=2.50
M (kNm/m)
450
450
h=10m, d=5m, F=2.50
450
337.5
337.5
337.5
337.5
225
225
225
225
112.5
112.5
112.5
112.5
0
0
90
180 270 360
h=5m, d=5m, F=2.50
450
M (kNm/m)
h=5m, d=5m, F=0.04
450
0
0
90
180 270 360
h=5m, d=5m, F=0.04
450
0
450
0
90
180 270 360
h=10m, d=5m, F=2.50
0
450
337.5
337.5
337.5
337.5
225
225
225
225
112.5
112.5
112.5
112.5
0
0
90
180 270 360 o
0
0
90
180 270 360
0
0
90
o
θ( )
o
θ( ) Tunnel
Structure A
180 270 360 θ( )
θ
0
h=10m, d=5m, F=0.04
0
90
180 270 360
h=10m, d=5m, F=0.04
0
90
180 270 360 θ(o)
Structures A+B
Fig. 17. Maximum envelopes of dynamic bending moment increments along the perimeter of the tunnel. First line: visco-elastic analysis results and second line: elastoplastic analyses results.
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K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
h=5m, d=5m, F=2.50
M (kNm/m)
450
450
h=10m, d=5m, F=2.50
450
337.5
337.5
337.5
337.5
225
225
225
225
112.5
112.5
112.5
112.5
0
0
90
0
180 270 360
h=5m, d=5m, F=2.50
450
M (kNm/m)
h=5m, d=5m, F=0.04
450
0
90
0
180 270 360
h=5m, d=5m, F=0.04
450
450
0
90
0
180 270 360
h=10m, d=5m, F=2.50
450
337.5
337.5
337.5
337.5
225
225
225
225
112.5
112.5
112.5
112.5
0
0
90
0
180 270 360
0
o
90
0
180 270 360
0
90
o
θ( )
o
θ( ) Tunnel
180 270 360 θ( )
Structure A
θ
0
h=10m, d=5m, F=0.04
0
90
180 270 360
h=10m, d=5m, F=0.04
0
90
180 270 360 θ(o)
Structures A+B
Fig. 18. Dynamic bending moment increments along the perimeter of the tunnel computed for the time step of maximum ovaling distortion. First line: visco-elastic analysis results and second line: elasto-plastic analyses results.
Viscoelastic
1.05
rM
rM
1.025
0 1
2.5
5 F
7.5
1
10
Viscoelastic
1.22
0 1
2.5
5 F
7.5
10
Elastoplastic
1.26
Structure A Structure B
1.195 rM
1.165 rM
1.08 1.04
1.0125
1.11
1.13 1.065
1.055 1
Structure A
1.12
1.0375
1
Elastoplastic
1.16
1 0 1
2.5
h=5m,d=5m
5 F
7.5
10
h=5m,d=10m
0 1
2.5
h=10m,d=5m
5 F
7.5
10
h=10m,d=10m
Fig. 19. rM response ratio as function of diameter, burial depth and flexibility ratio of the tunnel, computed from the maximum dynamic bending moment regardless the time step (“envelope values”).
computed based on “envelope values”, however one could easily imagine that the real case with several above ground structures will much strongly affect the tunnels response increasing even more the axial forces. Following the previous analysis, Fig. 14 summarizes the computed response ratios estimated by the simultaneous values of the
dynamic axial forces increments along the perimeter, referring to the time step of the tunnel maximum diametric deflection. The results reveal a similar tendency with the “envelope ratios” with a small reduction of the computed amplification though, which in this case do not exceed 15% for the case of two structures at the surface and for a rigid shallow tunnel.
K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
13
Structure A
Structure A Structure B
h=5m,d=5m
h=5m,d=10m
h=10m,d=5m
h=10m,d=10m
Fig. 20. rM response ratio as function of diameter, burial depth and flexibility ratio of the tunnel, computed from simultaneous values of the dynamic bending moment increments (time step of maximum ovaling).
h=5m, d=5m, F=2.50
M (kNm/m)
400
h=5m, d=5m, F=0.04
400
400
h=10m, d=5m, F=2.50
400
300
300
300
300
200
200
200
200
100
100
100
100
0
0
90
180 270 360
0
0
θ(o) Tunnel
90
0
180 270 360 θ(o)
Structure A
0
90
180 270 360 θ(o)
Structures A+B
Wang(1993)
0
h=10m, d=5m, F=0.04
0
90
θ
180 270 360 θ(o)
Fig. 21. Dynamic bending moment increments along the perimeter of the tunnel for the time step of maximum ovaling distortion. Comparisons of visco-elastic analyses results with Wang's solution for no-slip conditions.
Representative numerical results are compared with analytical results estimated for “free field conditions” using the Wang's closed form solutions (Fig. 15). The analytical solutions refer to the computation of the lining dynamic axial forces under S-waves propagation, assuming no-slip conditions for the soil-tunnel interface. For the implementation of the analytical solutions herein, the soil shear strain, which is a main input, is estimated through 1D equivalent linear response analyses of the soil deposits that are performed in the frequency domain using EERA [6]. The comparisons indicate a relatively good agreement between the numerical and analytical results. The observed deviations are more significant for rigid tunnels and are mainly attributed to the simplified assumptions of the closed form solutions (e.g. elastic homogeneous soil, sufficient burial depth etc). It is noteworthy that when the comparison is extended to the cases where above ground structures exist (also given in Fig. 15), closed form solutions
predict lower axial forces than the numerically derived accounting for these structures. Soil–tunnel interface characteristics and more specifically the interface friction properties are affecting significantly the shear stresses distribution around the tunnel perimeter and therefore the lining axial forces [21,37]. In this parametric study we assumed perfect bonding between the soil medium and the tunnel. To this respect the computed axial forces may be considered as an upper bound for the lining. The introduction of an interface, allowing a certain slip between the two media, will result in a reduction of the lining axial forces and will probably affect the presented results.
3.3.3. Bending moment Fig. 16 summarizes representative time histories of the dynamic ο bending moment computed at a crucial section of the lining (θ ¼45 )
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K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
for different scenarios regarding the existence of above ground structures. The results indicate a similar trend to that of the axial forces. The effects of the above ground oscillating massive structures are again more evident for shallow and rigid tunnels (Figs. 17–18). Similar to the axial forces, response ratios (rM) are derived, between the maximum dynamic bending moment increment assuming “free-field conditions” M Tunnel ðθjMTunnel ¼ max Þ and the corresponded dynamic bending moment increment accounting for the above ground structures M Tunnel þ Structures ðθjMTunnel ¼ max Þ, and plotted against the diameter, burial depth and flexibility ratio of the tunnel. r M ðθjMTunnel ¼ max Þ ¼
M Tunnel þ Structures ðθjMTunnel ¼ max Þ M Tunnel ðθjMTunnel ¼ max Þ
ð4Þ
The computed response ratios are presented in Figs. 19 and 20, for the “envelope” and simultaneous dynamic bending moment increments, respectively. The interpretation of the results reveals similar observations with the axial forces. Shallow and stiffer tunnels are affected more by the presence of the surface structures. The dynamic bending moments can be amplified up to 25% in terms of envelope values compared to the case where the above ground structures are disregarded. This amplification reaches 16% in terms of simultaneous values. Finally, representative comparisons are given in Fig. 21 between numerically derived dynamic bending moment increments and results of Wang's analytical solution [41]. Analytical solutions overestimate the dynamic bending moment of flexible tunnels, compared to the numerical results, while the opposite is observed for rigid tunnels. Generally, the effect of the interface properties (e.g., interface friction characteristics) is less important for the lining bending moment amplitude, compared to the axial force. However, these properties may affect the soil non-linear response near the tunnel (e.g., effect on soil stiffness and strength degradation under severe shaking), resulting in a different soil yielding response [15], [37]. The latter can lead to stress redistributions within the soil (e.g., stress concentrations at crucial locations) and therefore to different distributions of the bending moment along the lining.
4. Conclusions The paper presents and discusses a series of numerical analyses that performed to investigate the dynamic response of shallow circular tunnels, accounting for the adjacent above ground oscillating buildings interaction effects. The analysis is focused to the more critical transversal response that affects directly the cross-sectional structural design. Parameters that significantly affect the phenomenon, namely the soil–tunnel relative flexibility, the tunnel dimensions, the tunnel burial depth and the soil properties and non-linearity during shaking, are accounted for within this parametric analysis. The main conclusions are summarized as follows:
The existence of above ground structures affects the shear wave
field altering the input motion compare to the “free field conditions” case. The effect is more evident for shallow burial depths. Numerical results prove the tunnel ovaling deformation mode during shear shaking. The existence of oscillating massive above ground structures may affect the amplitude of lining ovaling distortions with this effect being more pronounced for the shallow tunnels. For the herein studied cases, the ovaling ratio R (structural to soil diametric deflection) is increased compared to the “free field conditions” case to about 20% for shallow burial depths. For deeper tunnels the increase is of the
order of 6% compared to the case where no above ground structures exist. Existent above ground structures alter the soil stress field around the tunnel, therefore affecting the soil yielding response. The effects are more significant in the case of soil type C due to the reduced strength compared to soil type B. For flexible tunnels, the consideration of the above ground structures results in larger permanent deformations at crucial sections of the lining (e.g., tunnel crown), with the effect being more significant for reduced burial depths. Soil non-linear behavior in case of severe earthquakes, capable to produce very large shear strains, may further affect the present results. However, the range of ground acceleration amplitudes reached in this paper are within the most commonly expected ground accelerations and consequently the present results and conclusions should be considered reliable for these levels of moderate to strong ground shaking. The existence of above ground structures generally leads to an increase of the lining dynamic internal forces (compared to the “free field conditions” case) with that being more significant for shallow and stiffer tunnels. Elastoplastic analyses predict an increased response (larger lining forces) for the majority of the cases compared to the visco-elastic analyses. In terms of maximum dynamic increment distributions regardless the time step (e.g., envelope values) the increase may reach 30% for the axial force (for shallow stiff tunnels) and 25% for the bending moment. In terms of simultaneous dynamic increment distributions the amplification is of the order of 15% for both axial forces and bending moments. Analytical solutions proposed by Wang for no-slip conditions give comparable results for the dynamic axial forces with the numerical predictions. On the contrary, for the bending moments the differences are more significant. For flexible tunnels the analytical solutions overestimate the lining bending moment, while the opposite is observed for rigid tunnels. As a general conclusion, for the majority of the investigated cases the consideration of the above ground structures results in an increase of the tunnel dynamic response both in terms of deformations and lining forces with that being more significant for shallower tunnels. Considering the large number of the parameters involved, and the associated uncertainties, it is quite difficult to come out with a strict quantification of the response modification. However, this quantification is of great importance for engineering practice. To this respect, accounting for the most conservative results of this preliminary study (e.g., envelope results for shallow rigid tunnels considering two above ground structures), we may conclude that the consideration of above ground structures may result in an increase up to 25% for the lining bending moment and 30% for the lining axial force compared to the “free field conditions” case.
Further research deemed to be necessary to further understand and rigorously quantify in engineering terms the effects of above ground structures on the dynamic response of shallow tunnels, passing below these structures in urban areas. Issues, such as (i) the input motion characteristics (e.g., frequency content, amplitude, and duration), (ii) the soil–tunnel and soil–structures interface properties and response during severe shaking, (iii) the number, position, and geometrical and mechanical characteristics of the structures (e.g., existence of basements, mass and stiffness properties of the equivalent SDOFs, structures non linear response), and (iv) the complex wave propagation phenomena related within the longitudinal direction (e.g. input motion incoherence), may significantly affect the structures– soil–tunnel interaction effects. The present study should be considered as a first step to acknowledge the problem and quantify it in a simple, yet engineering meaningful way.
K. Pitilakis et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 1–15
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