Seismic response of circular tunnels: Numerical validation of closed form solutions

Transcription

1 1 st Civil and Environmental Engineering Student Conference 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 seismic response of circular tunnel. The dynamic response in non-linear conditions due to seismic shaking is analysed numerically making use of 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 flexibility ratios. The results both thrust forces and bending moments acting in lining are compared with existing analytical solutions and quasi-static numerical linear analyses. The use of reduced or initial stiffness of soil in 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 nonlinear has been assessed, which lead to discrepancies arose among different models used. 1. INTRODUCTION Dynamic effects on underground structures have often been neglected on assumption that ir 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 original design have been studied. According to Hashash (2001), several tunnels suffered damages during strong earthquakes. One example is Kobe earthquake of 1995 in Japan, where collapse of center columns of Daikay subway station lead to collapse of ceiling slab as well as soil settlement; a second example is 1999 Chi-Chi earthquake in Taiwan, where portals of several highway tunnels were damaged due to slope instability. According to Yashiro (2007), or four or historical earthquakes hit deep tunnels in Japan, all of which experienced serious damage. Previous analysis has been made of damages suffered by underground structures during se events (Corigliano et al., 2011). It has been shown that a careful definition of seismic input is required for seismic assessment of se 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 surrounding soil., The deformations, and inertial response of underground structures are controlled by that of 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 se structures have to be designed to withstand significant seismic forces in addition to static loads. 1 This report discusses various approaches used in seismic design and analysis of underground structures. The discussion focuses on use of analytical and numerical analysis tools for pseudo static and dynamic analysis. To evaluate performance of se structures during earthquake shaking and to compute seismic induced loads on lining. Static design of tunnels has achieved high levels of refinement, whereas same level in seismic design it is yet to be achieved. There are few codes specifically address issue of seismic design (ISO and French AFPS/AFTES 2001). This study is, refore, 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 dynamic response of a tunnel to deformations imposed by surrounding soil. According to previous studies (Owen & Scholl 1981) underground structures suffer two types of s seismically induced deformations: along longitudinal axis of 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 tunnel axis Figure 1.

2 The construction of an underground structure modifies free field deformations by resisting deformation. By considering soil-structure interaction, a more realistic determination of tunnel response is achieved. The parameters which influence this are compressibility and flexibility ratio (C and F) of tunnel, overburden pressure (γh) and lateral earth pressure coefficient (K 0 ). The seismic loads are Figure 1 Ovaling of tunnel section (Owen & Scholl 1981) simulated by replacing overburden pressure with free-field shear stress (γ 2.2. Free-field analyses max ) and lateral! Fig. 6. Deformation modes of tunnels due to seismic waves Ž after Owen and Scholl, Fig 2.1 Deformation modes of tunnels due to seismic earth pressure waves (Owen coefficient & Scholl, is 1981) assumed to be equal Shear strain in ground caused by vertical shear to -1 to approximate simple shear condition. waves severely affects structure. To compute The tunnel s stiffness relative to surrounding nearly normal this free-field to tunnel shear axis, strain, resulting without intaking a distortion of cross-sectional shape of tunnel lining. shaking. into ground response is quantified of ground by and compressibility structure and to such account soil-structure interaction, it was used flexibility ratios C and F. Respectively, se are equivalent linear 1D analysis making use of extensional Table 1stiffness summarizes and a systematic flexural stiffness approach for Design considerations software EERA. for Analyses this type were of deformation carried out are with evaluating of medium seismic relative to response lining. ofit underground is necessary structures. to highlight This approach that consists flexibility of ratio, three indicates major steps: in transverse various earthquake direction. The scenarios, general to behavior account for of both lining may be influence simulatedof as a buried kind of structure soil as well subject as to resistance to ovaling. (Hashash, 2001) (Merritt, ground deformations spectral content under on a two-dimensional results. These analyses planestrain condition. ment of seismic parameters! for analysis ). Definition of seismic environment and develop- allowed maximum free-field shear strain γ max at tunnel depth to be estimated. Different soil E! R 1 ν! Diagonally C = (1) layouts propagating were modeled waves subject by varying different parts soil s 2. Evaluation E! t (1 of + ground ν! ) 1 response 2ν to shaking, which! of structure stiffness. to A linear out-of-phase elastic model displacements was used for ŽFig. includes ground failure and ground deformations. 6d., resulting soil, whereas in a longitudinal stiffness compression rarefaction wavewas traveling modeled along by using structure. Daraendeli In general, model degradation (G/G max ) 3. Assessment of structure behavior due to seismic shaking including (2001). Furrmore, were considered two F = E Ž! R! a. development! of seismic de- 1 ν larger displacement amplitudes are associated with! (2) separated layers were considered in order to take sign loading criteria, Ž b. underground structure re- 6E! I 1 + ν longer wavelengths,! into account while variation maximum in confining curvatures pressure are (σ ) sponse to ground deformations, and Ž c. special produced at by different shorter wavelengths depths. The with soil s relatively or assumed small Where seismic E m = design Young s issues. modulus of medium; displacement parameters amplitudes are Ž listed Kuesel, in 1969 Table.. 1. The relevant E l = Young s modulus of lining; ν l = The assessment results obtained of underground are shown in Table structure 2. seismic Poisson s Steps 1 ratio and of 2 aremedium: described t = thickness in Sections of 4 and 5, lining; I = Moment of Inertia R = Tunnel radius. response, refore, requires an understanding of respectively. Table 1 Soil's Parameters In order to Sections compute 6 8 provide stresses acting details on of Steps anticipated ground shaking as well as an evaluation of Plasticity Index 0 φ 30 ν m 0.2 OCR 1 σ (z=15m, z=40m) [KPa] Vs [] ρ [ Mg/m 3 ] 2.0 Table 2 Relevant results obtained for all scenarios V s=100 V s=150 V s=200 V s=300 V s=350 V s=500 V s=1000 E [Kpa] Northridge e/q γ max G/G max Sierra Madre e/q γ max G/Gmax Kocaeli e/q γ max G/Gmax Approaches taking into account soilstructure interaction 3a, lining, 3b and several 3c. different analytical solutions were used. Two conditions were assumed: full-slip (with 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 work by Hoeg (1968), and Corigliano (2007). The analyses were carried out keeping same parameters as those used in free-field case (i.e. earthquakes scenario, soil s properties). In addition geometry of tunnel was defined using 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 E lining [Kpa] ν lining 0.2 Cross sectional area [m 2 /m] 0.3 Moment of Inertia) [m 4 /m]

3 using relation:!!! #!! (4.4) The values of γ max computed with 1D analysis were employed in closed form solutions. Different values of F were computed for each soil layout in order to capture influence of soil s stiffness. Attention was focused on no-slip assumption, which was considered more conservative than full-slip condition. Furrmore, both reduced and initial stiffness were employed, Comparison obtaining between extended considerably Hoeg s Method and different Corigliano s values. Method The Comparison results, between obtained extended Hoeg s with Method and Corigliano s Northridge Method earthquake scenario (Figure 2; Figure 3) were This section presents a comparison between two selected different analytical compared in order to show discrepancies between solutions: This section extended presents Hoeg s a comparison solution and between Corigliano s two selected solution. different All previous analytical two methods. Note that re is a perfect match solutions: analyses have been extended carried Hoeg s out also solution with and Corigliano s Corigliano s method; solution. in order to All be compared previous between analyses with Hoeg s have solution, been carried showing solutions out a also perfect with for match Corigliano s thrust concerning with method; thrust each in order both to for value be initial compared and of F. with reduced On Hoeg s stiffness solution, (Fig or 3.12). showing Regarding hand, a perfect re moment match concerning stress, is a some relevant differences thrust both were influence for found; initial and of reduced reasons F with for stiffness se respect (Fig lie in 3.12). initial Regarding to assumption, magnitude moment according stress, to, some of moment differences thrust is same were and found; for both reasons full-slip for and se no-slip lie in assumption. initial assumption, Using Corigliano s according formula to, se two is stresses same for are both bending moments. As for bending moments some full-slip considered and separately. no-slip assumption. Therefore, Using values Corigliano s are different as formula shown se in next two (Fig stresses 3.13). are It discrepancies can be seen how Corigliano s arose, method tends showing to obtain lower slightly values for higher moment stress values considered separately. Therefore, values are different as shown in next (Fig 3.13). in noslip assumption. This by difference Hoeg s tends solution. to decrease for higher F It obtained values Quasi-static approach The response of tunnel was analysed numerically with finite element code ICFEP (Potts & Zdravkovic, 1999). The model layout was implemented in ICFEP assuming vertical boundaries 50m away from centre of tunnel. In order to be consistent with analytical solutions earthquake motion was modelled as a uniform displacement u applied along top boundary (Figure 4). Vertical and horizontal displacements were restricted along bottom boundary, while a restricted vertical movement was implemented for side and top boundaries (Avgerinos & Kontoe, 2011). The magnitude of displacements applied (u) is related to γ max values (Table 2) and to height (H) of model according to following relationship: 3 u = γ # H (3) Figure 4 Schematic representation of model used. Fig 3.16 Schematic representation of mesh configuration in quasi-static analysis The same analysis was carried out in free-field conditions and with analytical solutions. Both were performed with linear and non-linear Comparisons approaches. of numerical results with analytical solutions The use of a non-linear model is necessary to capture soil s real behaviour that is far from can be seen how Corigliano s method tends to obtain lower values for moment stress in noslip assumption. This difference tends to decrease for higher F values. being linear. In order to do this, hyperbolic This paragraph degradation presents comparisons model (Taborda, between 2011), two selected opportunely analytical solutions '$! ()*+",-."/*012*0" ()*+",-."345" and results of calibrated, numerical was analyses, implemented in order to in describe analyses. discrepancies The and to '$! '#! 6)/7+874-)",-."345" ()*+",-."/*012*0" results obtained from both linear and non-linear '!! 6)/7+874-)",-."/*0" ()*+",-."345" '#! better understand differences between se solutions and ir causes. 6)/7+874-)",-."345" analyses are compared in Figure 5. A general '!! &! 6)/7+874-)",-."/*0" A numerical simulation of circular tunnel ovaling is performed with closed-form trend is noticed with values obtained in &! %! solution assumption, under no-slip conditions. After establishing initial stress conditions, linear numerical analyses, as y are in very good %! $! racking deformations are applied corresponding to a wide range of soil shear strain and agreement with analytical ones. As for nonlinear $! #! flexibility ratios. For each values, earthquake se scenario show seven a different good analyses match with have been carried #! #! $! %! '" &! '!! '#! '$! '%! out using different ones shear obtained wave velocity, using hence different soil s soil s reduced stiffness stiffness. for which soilstructure interaction The is graph important. below The shows input parameters matching such as relationship shear strains values, Figure 3.12 Comparison 2 Comparison Hoeg s-corigliano s of Thrust no-slip - Northridge No-slip scenario with Hoeg's and #! $! %! '" &! '!! '#! '$! '%! Corigliano's solutions Fig '# 3.12 Comparison Hoeg s-corigliano s Thrust no-slip - Northridge scenario flexibility ratios and between soil s stiffness are linear same numerical as those results used with (x axis) analytical and solutions, '# '! computed non-linear numerical results (y axis). A 45 +! with EERA. Soil s stiffness, flexibility ratios, shear strains values are listed below & '! ()*+"9-."/*0" sloped line (1:1) marks perfect match between for all *! earthquakes analysed (Tab 3.5). Soil s stiffness and F ratios are referred to % ()*+"9-."345" & different results. The discrepancies for thrust, 6)/7+874-)"9-."345" ()*+"9-."/*0" initial maximum )! stiffness. $ % 6)/7+874-)"9-."/*0" ()*+"9-."345" computed by non-linear approach, are (! # 6)/7+874-)"9-."345" $ considerably lower than ones computed with 16 6)/7+874-)"9-."/*0" '! # linear (square) and analytical approaches #! $! %! '" &! '!! '#! '$! '%! &!,-./" (triangle). The reason why se differences arose Fig 3.13 Comparison Hoeg s-corigliano s bending moment no-slip - Northridge scenario %! 0-12/3245-" #! $! %! '" &! '!! '#! '$! '%! $! depends on different models used to highlight " Figure 3.13 Comparison 3 Comparison Hoeg s-corigliano s of bending Bending moment no-slip moment - Northridge No-slip scenario with non-linear features of soil affects. This Hoeg's and Corigliano's solutions 12 #! " " considerably varies way in which load is 12 induced $! on &! tunnel (! '" lining. *! Therefore #!! #$! it is #&! 3. NUMERICAL SIMULATION Fig 4.3 Thrust No-Slip assumption - Comparison among Analytical, Linear Numerical, Non-Linear possible to argue that use of reduced stiffness #$%&" #$%&" ("#)%*&" ("#)%*&" where:! max is maximum free field shear strain at tunnel axis depth, computed using EERA; H is height of mesh, H=50m. #$%&" Numerical results varying F is more reasonable to compute stresses acting on lining. %()*+,)-./"!0/123"#$%&" *! )! (! '! &! %! $! #! "9:;" " 0-12/3245-"9:;"7-5"625.41" $! &! (! *! +,)-./"!0/123"#$%&" (a) Fig 4.4 Thrust for No-Slip assumption - Linear vs. Non-Linear results compared by 1:1 line

4 # $! &! (! *! #!! #$! #&! )" Fig 4.5 Thrust No-Slip assumption - Comparison among Analytical, Linear Numerical, Non-Linear Numerical results varying F * ovaling deformation can be seen at 45 (Figure 7). ) *+&,-.&/01"!+'/&23"#$*'(" ( ' & % $ "789"5, " +,-./0.12,"789"5,2"3.241-" # Figure 7 $ & ( * -.&/01"0&4"5& "!+'/&23"#$*'(" (b) Figure 5 Comparison of calculated thrust (a) and bending moments (b) by analytical solutions, linear and non-linear numerical analysis for no-slip assumption. Fig 4.6 Bending moments for No-Slip assumption - Linear vs. Non-Linear results compared by 1:1 line 3.2. Dynamic approach As before ICFEP finite element code has been used to analyse dynamic response of tunnel. The non-linear model previously mentioned has also been used in this set of analyses. Four different kinds of soil were investigated, maintaining same parameters listed in Table 1 and Table 3. Shear wave velocity was varied, in order to vary F, covering a wide range of soils from soft to very stiff. The dynamic excitation was based on data obtained from Northridge earthquake (Figure 6), opportunely filtered with a cut-off frequency f=25hz and scaled to a PGA=0.2g. The model used was refined to be employed in soft soil by resizing mesh. The analyses were conducted in consecutive steps: a) tunnel excavation, b) lining construction in static conditions and c) dynamic excitation. Restricted vertical displacement boundary conditions were assigned at lateral edges of model. #$%&'()*+,%+ ("$!# ("!!# '"$!# '"!!# &"$!# &"!!# %"$!# %"!!# $!# Figure 6 Response spectra of Northridge e/q 6!!#!!# &!# (!# )!# *!# %"!!# %"&!# %"(!# %")!# %"*!# &"!!# -$)+.()/0+ First of all, results obtained were compared with 1D equivalent linear analyses in free-field conditions in order to validate numerical model employed. As a result of this comparison a good match was obtained, as shown in Table 4. An example of different cases studied, one with a shear wave velocity V s =1000 was selected to show deformed shape at point where largest increment in stress occurs. As expected, For a better representation of complete set of data obtained, a hoop stress distribution graph is plotted. Recall relationship used to combine thrust and moment: σ = T A + My I (4) Where A=0.3m 2 is cross sectional area, I= m 4 is Inertia moment, y=0.15m is distance between neutral axis and boundary of lining. Figure 8 represents combination of all hoop stresses computed for each case study. It is evident that softer soil is, bigger stresses are which affect lining. Furrmore, it is possible to appreciate a general trend, where peak of stresses for each case study happen at values of θ=45 +kπ/2, as predicted by analytical solutions. "####$ %###$ &###$ '###$ (###$ #$!(###$ #$ ')$ *#$ "+)$ "%#$ (()$ (,#$ +")$ +&#$!'###$!&###$!%###$ ####$ "#!#$%&! Figure 5.31 Envelope 8 Envelope of hoop stress of for different hoop case stress study during earthuake different cases studied +! *! Finally )! a comparison was made with results (! obtained from previous approaches. The same '! graph &! used in section 2.3 was adopted to,-./012" show %! that results obtained for thrust with dynamic 3./4-512/4" $! analysis, #! presents a very good agreement with non-linear values. The results are considerably $! &! (! *! #!! #$! #&! lower than analytical '" ones found using Fig 5.32 Thrust against flexibility ratios using reduced stiffness soil s reduced stiffness. The values obtained when #& using maximum soil stiffness are not #$ comparable with those from dynamic analysis, as #! y are dramatically greater. The results obtained *,-./012" for bending moments with reduced or initial soil ( 3./4-512/4" stiffness are not comparable with results from & dynamic analysis. In fact bending moments $ computed with last set of analyses are $! &! (! *! #!! #$! considerably greater than previous ones. Table #$%&" ("#$%)&" Fig 5.33 Bending moment against flexibility ratios using reduced stiffness '" -./(##$ -./+##$ -./)##$ -./"###$ 4 22

5 4 summarizes 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.!#$%&!!#$%&! F Hoeg vs. Dynamic Reduced stiffness T no slip M no slip [%] [%] Hoeg vs Dynamic Initial stiffness T no slip M no slip F [%] [%] "####$ "####$ %###$ %###$ &###$ &###$ '###$ (###$ (###$!(###$ #$ #$ ')$ *#$ "+)$ "%#$ (()$ (,#$ +")$ +&#$!(###$!'###$!&###$ '###$ /(##$ /+##$ #$ !'###$ #$ ')$ 33.3 *#$ "+)$ "%#$ (()$ (,#$ +")$ &#$ /"###$ !&###$!%###$ !%###$ ####$ ####$ Fig 5.31 Envelope of hoop stress for different case study during earthuake Fig 5.31 Envelope of hoop stress for different case study during earthuake +! #$%&" #$%&" Figure 5.32 Thrust 9 Thrust against flexibility against ratios F using reduced '" reduced stiffness stiffness $ $! &! (! *! #!! #$! '" $! &! (! *! #!! #$! Fig 5.33 Bending moment against flexibility ratios using '" reduced stiffness Figure 5.33 Bending 10 moment Bending against flexibility moments ratios using against reduced stiffness F using reduced stiffness 4. CONCLUSIONS "# /(##$ -./+##$ -./)##$ -./)##$ -./"###$ +! *! *! )! )! (! (! '! '! &!,-./012" &! %!,-./012" 3./4-512/4" %! $! 3./4-512/4" $! #! #! $! &! (! *! #!! #$! #&! '" $! &! (! *! #!! #$! #&! Fig 5.32 Thrust against flexibility ratios using reduced stiffness #& ("#$%)&" ("#$%)&" #& #$ #$ #! #! * * ( ( & & $ "#,-./012",-./012" 3./4-512/4" 3./4-512/4" 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. Furrmore, a relevant influence in F was noticed, with respect to magnitude of thrust and bending moment. This makes F a key parameter in proper seismic design. Finally dynamic analyses, carried out using a nonlinear model, were found to compare well with analytical solutions, with regards to thrust using a soil s reduced stiffness. As for bending moments, comparison does not match well regardless of stiffness used. A final recommendation is to make use of 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 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 Sourn 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 Sourn California. Einstein, H., & Schwartz, C. (1979). Simplifield analysis for tunnel support. Journal Geotechnical engineering division, 105, Hashash. (2001). Seismic design and analysis of underground structures. Tunneling and Underground space technology, 16, 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 Hoeg, K. (1968). Stresses against underground structural cylinders. Journal of Soil Mechanics and Foundations Division, 94 (4), p ISO (2005). Bases for design of structuresseismic actions for designing geotechncal works. Merritt, J. (1985). Seismic design of underground structures. Proceedings of 1985 Rapid excavation tunneling conference, 1, p 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

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