1

1st Civil and Environmental Engineering Student Conference 25-26 June 2012 Imperial College London

Seismic response of circular tunnels: Numerical validation of

closed form solutions

M.A. Zurlo ABSTRACT This paper presents a parametric investigation of the seismic response of circular tunnel. The dynamic response in non-linear conditions due to seismic shaking is analysed numerically making use of the finite element program ICFEP in non-linear conditions. The hyperbolic non-linear model was used to give a better representation of real soil’s behaviour. Plane strain analyses were conducted on various soils stiffness by, parametrically varying the flexibility ratios. The results both thrust forces and bending moments acting in the lining are compared with existing analytical solutions and quasi-static numerical linear analyses. The use of the reduced or the initial stiffness of the soil in the linear solutions, is also object of investigation. As a case study, a real earthquakes scenario has been analysed in different kinds of soils, The influence of the non-linear has been assessed, which lead to discrepancies arose among the different models used. 1. INTRODUCTION

Dynamic effects on underground structures have often been neglected on the assumption that their response to earthquake loading is relatively safe compared to that of surface structures. Despite this, several examples of recorded damage to underground structures for which seismic forces were not considered in the original design have been studied. According to Hashash (2001), several tunnels suffered damages during strong earthquakes. One example is the Kobe earthquake of 1995 in Japan, where the collapse of the center columns of the Daikay subway station lead to the collapse of the ceiling slab as well as soil settlement; a second example is the 1999 Chi-Chi earthquake in Taiwan, where the portals of several highway tunnels were damaged due to slope instability. According to Yashiro (2007), other four other historical earthquakes hit deep tunnels in Japan, all of which experienced serious damage. Previous analysis has been made of the damages suffered by underground structures during these events (Corigliano et al., 2011). It has been shown that a careful definition of the seismic input is required for the seismic assessment of these structures. This is important as a possible tunnel collapse could result in life and economical losses. The seismic response of underground structures is dominated by that of the surrounding soil., The deformations, and inertial response of the underground structures are controlled by that of the surrounding soil mass (Wang, 1993; Hashash et al., 2001). Unlike above ground structures, underground structures do not experience free vibration as a result of seismic shaking. However, in seismically active areas these structures have to be designed to withstand significant seismic forces in addition to static loads.

This report discusses the various approaches used in seismic design and the analysis of underground structures. The discussion focuses on the use of analytical and numerical analysis tools for pseudo static and dynamic analysis. To evaluate the performance of these structures during earthquake shaking and to compute seismic induced loads on the lining. Static design of tunnels has achieved high levels of refinement, whereas the same level in seismic design it is yet to be achieved. There are few codes specifically address the issue of seismic design (ISO 23469 2005 and the French AFPS/AFTES 2001). This study is, therefore, not only academically important, but will also have a practical use in understanding circular tunnel’s behaviour under seismic conditions.

2. ANALYTICAL SOLUTIONS

2.1. Dynamic response of tunnels

An elastic circular beam is usually used to model the dynamic response of a tunnel to the deformations imposed by the surrounding soil. According to previous studies (Owen & Scholl 1981) underground structures suffer two types of s seismically induced deformations: along the longitudinal axis of the tunnel and ovaling of circular cross-sections, which will be investigated in this paper. In particular ovaling deformation is caused by vertical seismic waves propagating perpendicular to the tunnel axis Figure 1.

2

Figure 1 Ovaling of tunnel section (Owen & Scholl 1981)

2.2. Free-field analyses

Shear strain in the ground caused by vertical shear waves severely affects the structure. To compute this free-field shear strain, without taking into account soil-structure interaction, it was used the equivalent linear 1D analysis making use of the software EERA. Analyses were carried out with various earthquake scenarios, to account for both the influence of the kind of soil as well as the spectral content on the results. These analyses allowed the maximum free-field shear strain γmax at the tunnel depth to be estimated. Different soil layouts were modeled by varying the soil’s stiffness. A linear elastic model was used for the soil, whereas the stiffness degradation (G/Gmax) was modeled by using the Daraendeli model (2001). Furthermore, were considered two separated layers were considered in order to take into account the variation in confining pressure (σ’) at different depths. The soil’s other assumed parameters are listed in Table 1. The relevant results obtained are shown in Table 2. Table 1 Soil's Parameters

Plasticity Index 0

φ ’ 30°

νm 0.2

OCR 1

σ ’ (z=15m, z=40m) [KPa] 200 - 533

Vs [m/s] 100-1000

ρ [ Mg/m3] 2.0

Table 2 Relevant results obtained for all the scenarios

Vs=100

m/s Vs=150

m/s Vs=200

m/s Vs=300

m/s Vs=350

m/s Vs=500

m/s Vs=1000

m/s

E [Kpa] 48000 110400 194400 440400 600000 1224000 4891200

Northridge e/q γ max 0.0011 0.0019 0.0018 0.00061 0.00055 0.00046 0.0001

G/Gmax 0.41 0.29 0.31 0.54 0.57 0.61 0.86

Sierra Madre e/q γ max 0.00065 0.00081 0.0006 0.00044 0.00041 0.00023 0.00014

G/Gmax 0.53 0.49 0.55 0.62 0.63 0.68 0.82 Kocaeli e/q γ max 0.002624 0.001394 0.00092 0.0007 0.0005 0.00038 0.00013

G/Gmax 0.24 0.36 0.45 0.52 0.58 0.65 0.83

2.3. Approaches taking into account soil-structure interaction

The construction of an underground structure modifies the free field deformations by resisting deformation. By considering soil-structure interaction, a more realistic determination of the tunnel response is achieved. The parameters which influence this are the compressibility and flexibility ratio (C and F) of the tunnel, the overburden pressure (γh) and the lateral earth pressure coefficient (K0). The seismic loads are simulated by replacing the overburden pressure with the free-field shear stress (γmax) and the lateral earth pressure coefficient is assumed to be equal to -1 to approximate the simple shear condition. The tunnel’s stiffness relative to the surrounding ground is quantified by the compressibility and flexibility ratios C and F. Respectively, these are the extensional stiffness and the flexural stiffness of the medium relative to the lining. It is necessary to highlight that the flexibility ratio, indicates the resistance to the ovaling. (Hashash, 2001) (Merritt, 1985).

𝐶 =  𝐸! 𝑅 1 − 𝜈!!

𝐸!𝑡 (1 + 𝜈!) 1 −  2𝜈! (1)

𝐹 =  𝐸! 𝑅! 1 − 𝜈!!

6𝐸!𝐼 1 +  𝜈! (2)

Where Em = the Young’s modulus of the medium; El = the Young’s modulus of the lining; νl = the Poisson’s ratio of the medium: t = thickness of the lining; I = Moment of Inertia R = Tunnel radius.

In order to compute the stresses acting on the lining, several different analytical solutions were used. Two conditions were assumed: full-slip (with the absence of friction between structure and medium) and no-slip (with perfect adherence between medium and structure). Various, different analytical solutions were used and compared based on the work by Hoeg (1968), and Corigliano (2007). The analyses were carried out keeping the same parameters as those used in the free-field case (i.e. earthquakes scenario, soil’s properties). In addition the geometry of the tunnel was defined using the same parameters as those presented by Hashash (2001) and listed in Table 3. The tunnel radius is assumed to be equal to 5 m. Table 3 Tunnel's parameters

Depth [m] 15

Elining [Kpa] 24800000

νlining 0.2

Cross sectional area [m2/m] 0.3

Moment of Inertia) [m4/m] 0.00225

usually related to these structure allows neglecting the surface waves, as suggested by

Wang (1993).

!

Fig 2.1 Deformation modes of tunnels due to seismic waves (Owen & Scholl, 1981)

( )Y.M.A. Hashash et al. ! Tunnelling and Underground Space Technology 16 2001 247!293 253

! .Fig. 6. Deformation modes of tunnels due to seismic waves after Owen and Scholl, 1981 .

nearly normal to the tunnel axis, resulting in a distor-tion of the cross-sectional shape of the tunnel lining.Design considerations for this type of deformation arein the transverse direction. The general behavior of thelining may be simulated as a buried structure subject toground deformations under a two-dimensional plane-strain condition.

Diagonally propagating waves subject different parts!of the structure to out-of-phase displacements Fig.

.6d , resulting in a longitudinal compression!rarefac-tion wave traveling along the structure. In general,larger displacement amplitudes are associated withlonger wavelengths, while maximum curvatures areproduced by shorter wavelengths with relatively small

! .displacement amplitudes Kuesel, 1969 .The assessment of underground structure seismic

response, therefore, requires an understanding of theanticipated ground shaking as well as an evaluation of

the response of the ground and the structure to suchshaking. Table 1 summarizes a systematic approach forevaluating the seismic response of underground struc-tures. This approach consists of three major steps:

1. Definition of the seismic environment and develop-ment of the seismic parameters for analysis.

2. Evaluation of ground response to shaking, whichincludes ground failure and ground deformations.

3. Assessment of structure behavior due to seismic! .shaking including a development of seismic de-

! .sign loading criteria, b underground structure re-! .sponse to ground deformations, and c special

seismic design issues.

Steps 1 and 2 are described in Sections 4 and 5,respectively. Sections 6!8 provide the details of Steps3a, 3b and 3c.

3

The values of γmax computed with the 1D

analysis were employed in the closed form solutions. Different values of F were computed for each soil layout in order to capture the influence of the soil’s stiffness. Attention was focused on the no-slip assumption, which was considered more conservative than full-slip condition. Furthermore, both the reduced and the initial stiffness were employed, obtaining considerably different values. The results, obtained with the Northridge earthquake scenario (Figure 2; Figure 3) were compared in order to show discrepancies between the two methods. Note that there is a perfect match between the solutions for thrust with each value of F. On the other hand, there is a relevant influence of F with respect to the magnitude of thrust and the bending moments. As for bending moments some discrepancies arose, showing slightly higher values obtained by Hoeg’s solution.

Figure 2 Comparison of Thrust No-slip with Hoeg's and Corigliano's solutions

Figure 3 Comparison of Bending moment No-slip with Hoeg's and Corigliano's solutions

3. NUMERICAL SIMULATION

3.1. Quasi-static approach

The response of the tunnel was analysed numerically with the finite element code ICFEP (Potts & Zdravkovic, 1999). The model layout was implemented in ICFEP assuming vertical boundaries 50m away from the centre of the tunnel. In order to be consistent with the analytical solutions the earthquake motion was modelled as a uniform displacement u applied along the top boundary (Figure 4). Vertical and horizontal displacements were restricted along the bottom boundary, while a restricted vertical movement was implemented for the side and the top boundaries (Avgerinos & Kontoe, 2011). The magnitude of the displacements applied (u) is related to the γmax values (Table 2) and to the height (H) of the model according to the following relationship:

𝑢 = 𝛾!"#𝐻 (3)

Figure 4 Schematic representation of the model used. The same analysis was carried out in free-field conditions and with the analytical solutions. Both were performed with linear and non-linear approaches.

The use of a non-linear model is necessary to capture the soil’s real behaviour that is far from being linear. In order to do this, the hyperbolic degradation model (Taborda, 2011), opportunely calibrated, was implemented in the analyses. The results obtained from both the linear and non-linear analyses are compared in Figure 5. A general trend is noticed with the values obtained in the linear numerical analyses, as they are in very good agreement with the analytical ones. As for the non-linear values, these show a good match with the ones obtained using the soil’s reduced stiffness. The graph below shows the matching relationship between the linear numerical results (x axis) and the non-linear numerical results (y axis). A 45° sloped line (1:1) marks the perfect match between the different results. The discrepancies for thrust, computed by the non-linear approach, are considerably lower than the ones computed with linear (square) and analytical approaches (triangle). The reason why these differences arose depends on the different models used to highlight the non-linear features of the soil affects. This considerably varies the way in which load is induced on the tunnel lining. Therefore it is possible to argue that the use of reduced stiffness is more reasonable to compute the stresses acting on the lining.

(a)

12

3.3.4. Comparison between extended Hoeg’s Method and Corigliano’s Method

This section presents a comparison between the two selected different analytical

solutions: the extended Hoeg’s solution and the Corigliano’s solution. All the previous

analyses have been carried out also with the Corigliano’s method; in order to be compared

with Hoeg’s solution, showing a perfect match concerning the thrust both for initial and

reduced stiffness (Fig 3.12). Regarding the moment stress, some differences were found; the

reasons for these lie in the initial assumption, according to, the moment is the same for both

full-slip and no-slip assumption. Using Corigliano’s formula these two stresses are

considered separately. Therefore, the values are different as shown in the next (Fig 3.13). It

can be seen how Corigliano’s method tends to obtain lower values for moment stress in no-

slip assumption. This difference tends to decrease for higher F values.

Fig 3.12 Comparison Hoeg’s-Corigliano’s Thrust no-slip - Northridge scenario

Fig 3.13 Comparison Hoeg’s-Corigliano’s bending moment no-slip - Northridge scenario

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3.3.4. Comparison between extended Hoeg’s Method and Corigliano’s Method

This section presents a comparison between the two selected different analytical

solutions: the extended Hoeg’s solution and the Corigliano’s solution. All the previous

analyses have been carried out also with the Corigliano’s method; in order to be compared

with Hoeg’s solution, showing a perfect match concerning the thrust both for initial and

reduced stiffness (Fig 3.12). Regarding the moment stress, some differences were found; the

reasons for these lie in the initial assumption, according to, the moment is the same for both

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slip assumption. This difference tends to decrease for higher F values.

Fig 3.12 Comparison Hoeg’s-Corigliano’s Thrust no-slip - Northridge scenario

Fig 3.13 Comparison Hoeg’s-Corigliano’s bending moment no-slip - Northridge scenario

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leading to a triangular shape distribution of displacements along the vertical boundaries over

the 100 increments of the quasi-static part. The applied displacement, u, was computed

using the relation:

! ! !!"#!! (4.4)

where:

!max is the maximum free field shear strain at tunnel axis depth, computed using EERA;

H is the height of the mesh, H=50m.

Fig 3.16 Schematic representation of the mesh configuration in quasi-static analysis

3.4.3. Comparisons of the numerical results with the analytical solutions

This paragraph presents the comparisons between the two selected analytical solutions

and the results of the numerical analyses, in order to describe the discrepancies and to

better understand the differences between these solutions and their causes.

A numerical simulation of the circular tunnel ovaling is performed with the closed-form

solution assumption, under no-slip conditions. After establishing initial stress conditions,

racking deformations are applied corresponding to a wide range of soil shear strain and

flexibility ratios. For each earthquake scenario seven different analyses have been carried

out using different shear wave velocity, hence different soil’s stiffness for which the soil-

structure interaction is important. The input parameters such as shear strains values,

flexibility ratios and soil’s stiffness are the same as those used with the analytical solutions,

computed with EERA. Soil’s stiffness, flexibility ratios, shear strains values are listed below

for all the earthquakes analysed (Tab 3.5). Soil’s stiffness and F ratios are referred to the

initial maximum stiffness.

5

Fig 4.3 Thrust No-Slip assumption - Comparison among Analytical, Linear Numerical, Non-Linear Numerical results varying F

Fig 4.4 Thrust for No-Slip assumption - Linear vs. Non-Linear results compared by 1:1 line

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3.2. Dynamic approach

As before the ICFEP finite element code has been used to analyse the dynamic response of the tunnel. The non-linear model previously mentioned has also been used in this set of analyses. Four different kinds of soil were investigated, maintaining the same parameters listed in Table 1 and Table 3. Shear wave velocity was varied, in order to vary F, covering a wide range of soils from soft to very stiff. The dynamic excitation was based on the data obtained from the Northridge earthquake (Figure 6), opportunely filtered with a cut-off frequency f=25Hz and scaled to a PGA=0.2g. The model used was refined to be employed in soft soil by resizing the mesh. The analyses were conducted in consecutive steps: a) tunnel excavation, b) lining construction in static conditions and c) dynamic excitation. Restricted vertical displacement boundary conditions were assigned at the lateral edges of the model.

Figure 6 Response spectra of the Northridge e/q

First of all, the results obtained were compared with the 1D equivalent linear analyses in free-field conditions in order to validate the numerical model employed. As a result of this comparison a good match was obtained, as shown in Table 4. An example of the different cases studied, one with a shear wave velocity Vs=1000 m/s was selected to show the deformed shape at the point where the largest increment in stress occurs. As expected,

the ovaling deformation can be seen at 45° (Figure 7).

Figure 7

For a better representation of the complete set of data obtained, a hoop stress distribution graph is plotted. Recall the relationship used to combine thrust and moment: 𝜎 =

𝑇𝐴+𝑀𝑦𝐼

(4)

Where A=0.3m2 is the cross sectional area, I=0.00225m4 is the Inertia moment, y=0.15m is distance between the neutral axis and the boundary of the lining. Figure 8 represents the combination of all the hoop stresses computed for each case study. It is evident that the softer the soil is, the bigger the stresses are which affect the lining. Furthermore, it is possible to appreciate a general trend, where the peak of the stresses for each case study happen at values of θ=45°+kπ/2, as predicted by the analytical solutions.

Figure 8 Envelope of the hoop stress the different cases studied Finally a comparison was made with the results obtained from the previous approaches. The same graph used in section 2.3 was adopted to show that the results obtained for thrust with the dynamic analysis, presents a very good agreement with the non-linear values. The results are considerably lower than the analytical ones found using the soil’s reduced stiffness. The values obtained when using the maximum soil stiffness are not comparable with those from dynamic analysis, as they are dramatically greater. The results obtained for bending moments with reduced or initial soil stiffness are not comparable with results from dynamic analysis. In fact bending moments computed with the last set of analyses are considerably greater than the previous ones. Table

6

With regards to the bending moments the matching with the linear solutions is very good

for all the flexibility ratios as shown in Fig 4.5. This is likely to be explained by low magnitude

of the moments. However, some discrepancies arose for a specific range of flexibility ratios,

where the computed values are greater than the linear results as shown in Fig 4.6 where, for

a middle range of flexibility ratios, the match of the non-linear results is not as good as for the

other values. In Tab 4.1 shown the computed results and is clear how the matching of the

bending moments is considerably better than thrust forces.

Fig 4.5 Thrust No-Slip assumption - Comparison among Analytical, Linear Numerical, Non-Linear Numerical results varying F

Fig 4.6 Bending moments for No-Slip assumption - Linear vs. Non-Linear results compared by 1:1 line

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Fig 5.31 Envelope of the hoop stress for different case study during the earthuake

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4 summarizes the results obtained and compares values for both initial and reduced soil stiffness. Table 4 Percentage differences of obtained results with initial and reduced soil stiffness.

Hoeg vs. Dynamic Reduced stiffness

Hoeg vs Dynamic Initial stiffness

F Tnoslip[%]

Mnoslip[%] F Tnoslip

[%]Mnoslip

[%]

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71.95 23.2 ‐111.1 131.54 56.2 -108.6

223.01 33.3 ‐155.4 365.59 55.8 -154.5

1256.40 ‐0.9 28.2 1460.93 6.6 28.2

Figure 9 Thrust against F using reduced stiffness

Figure 10 Bending moments against F using reduced stiffness

4. CONCLUSIONS

This study shows that soil layer properties in combination with seismic excitation characteristics, may have important affects on a tunnel lining. The main stresses considered where thrust and bending moments. Furthermore, a relevant influence in F was noticed, with respect to the magnitude of thrust and bending moment. This makes F a key parameter in proper seismic design. Finally dynamic analyses, carried out using a non-linear model, were found to compare well with analytical solutions, with regards to thrust using a soil’s reduced stiffness. As for bending moments, the comparison does not match well regardless of the stiffness used. A final recommendation is to make use of the reduced soil stiffness in analytical solutions when calculating thrust, whereas an amplification coefficient is recommended when calculating bending moments.

5. ACKNOWLEDGEMENTS

I would like to thank my supervisor Dr. Stavroula Kontoe for hers support and guidance during this project. Hers constant availability it was priceless. Also my acknowledgements go to PhD candidate Vasilis Avgerinos, his wisdom and calm lead me through the hardest moments with ICFEP.

6. REFERENCES

AFPS/AFTES. (2001). Earthquake design and protection of underground structures. Avgerinos, V., & Kontoe, S. (2011). Seismic design of circular tunnels: Numerical validation of closed formed solutions. 5th International Conference of Earthquake Geotechnical Engineering. Santiago, Chile. Corigliano, & al, e. (2011). Seismic analysis of deep tunnels in near fault conditions: a case study in Southern Italy. Bull Earthquake Eng . Corigliano, M. PhD dissertation. Seismic respons of deep tunnels in near-fault conditions. Politecnico di Torino. Corigliano, M. (2007). Seismic response of deep tunnels in near-fault conditions. Torino, Italia. Darendeli, M. (2001). Development of a new family of normalized modulus reduction and material dumping curves. Austin, Texas, U.S.A.: The University of Texas at Austin. EERA. (2000). A computer program for equivalent linear earthquake site response analyses of layered soil deposits. Barkeley, California: University of Southern California. Einstein, H., & Schwartz, C. (1979). Simplifield analysis for tunnel support. Journal Geotechnical engineering division , 105, 499 - 518. Hashash. (2001). Seismic design and analysis of underground structures. Tunneling and Underground space technology , 16, 435-441. Hashash, Y. M. (2005). Ovaling deformations of circular tunnels under seismic loading, and update on seismic design and analysis of underground structures. Tunneling and Underground Space Technology , p. 435-441. Hoeg, K. (1968). Stresses against underground structural cylinders. Journal of Soil Mechanics and Foundations Division , 94 (4), p. 833-858. ISO23469. (2005). Bases for design of structures-seismic actions for designing geotechncal works. Merritt, J. (1985). Seismic design of underground structures. Proceedings of the 1985 Rapid excavation tunneling conference, 1, p. 104-131. Owen, G., & Scholl, R. (1981). Earthquake engineering of large underground structures. Federal Highway administration and national science foundation. Park, e. a. (2009). Analytical solution for seismic induced ovaling of circular tunnel lining under no slip interface conditions: A revisit. Tunneling and Underground Space Technology , 24, p. 231-235.

22

Fig 5.31 Envelope of the hoop stress for different case study during the earthuake

Fig 5.32 Thrust against flexibility ratios using reduced stiffness

Fig 5.33 Bending moment against flexibility ratios using reduced stiffness

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Fig 5.32 Thrust against flexibility ratios using reduced stiffness

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Penzien, J. (2000). Seismically induced racking of tunnels linings. International Journal of Earthquake Engineering & Structural Dynamics , 29, p. 683-691. Potts, D., & Zdravkovic, L. (1999). Finite element analysis in geotechnical engineering. London: Thomas Thelford. Power, M. (1996). Seismic vulnerability of tunnels revisited. Proceedings of the North American Tunneling Conference. Long Beach, California: Elsevier. St. John, C., & Zahrah, T. (1987). Aseismic design of underground structures. Tunnelling and Underground Space Technology . Taborda, D. (2011). Development of constitutive models for application in soyl dynamics. PhD Thesis . London: Imperial College of London. Wang, J. (1993). Seismic design of Tunnels: A state of earth approach. New York: Parsons,Brinckerhoff,Quade and Douglas Inc. Yashiro, K. (2007). Historical earthquake damage to tunnels in Japan and case studies of railway tunnels in the 2004 Niigataken-Chuetsu earthquake. Quarterly report, Railway technical research institute.