AN INVESTIGATION ON SEISMIC ANALYSIS OF SHALLOW TUNEELS IN SOIL MEDIUM

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1 The 4 th October -7, 8, Beijing, China AN INVESTIGATION ON SEISMIC ANALYSIS OF SHALLOW TUNEELS IN SOIL MEDIUM J. Boouri Bazaz and V. Besharat Assistant Professor, Dept. of Civi Engineering, Ferdowsi University, Mashhad. Iran Ph.D. Student, Dept. of Engineering, Isaic Azad University, Science and research branch, Tehran, Iran Eai: boouri@u.ac.ir, v.besharat@gai.co ABSTRACT : Buried tunnes with circuar cross section, such as urban undergrounds are very popuar in transportation engineering. The seisic behavior of such structures during operation is of iportance. There exist different cosed for soutions, such as Penzien and Wang ethods for seisic evauation of shaow tunnes. In the present research, these anaytica ethods are epoyed to anaysis tunne forces, constructed in soft to hard soi with different echanica properties and constant shear strain. A nuerica ode, based on finite eeent ethod, was then deveoped to copare the anaytica and nuerica ethod outputs when the tunne is under seisic oads. For this purpose seven types soi, soft soi to very dense sand, was seected and anaysis was carried out. The anaysis shows reative error in anaytica ethods. Aso it is shown that with increasing soi stiffness, the induced circuar stress in is reduced KEYWORDS:. INTRODUCTION Buried tunne, Cosed for soution, Shear strain, Seisic anaysis The buried structures depend on geoetry, depth of buried and other geotechnica and seisic paraeters behave in different anners during an earthquake. The researches indicate buried structures experience a ower rate of daages than surface structures. Nevertheess, the buried structures aso faced significant daages in recent arge earthquakes. When designing such structure under earthquake conditions, seisic paraeters and physica and echanica properties if soi pay iportant roe. In this study a specified circuar tunne was anayzed, using various anaytica soutions based on work of Wang (993 and Hashash ( & 5 for various soi groups. Wang in 993, based on cosed for soutions recoended by Peck et a (7, proposed odified anaytica equations in ters of axia force, bending oents and dispaceents under externa oading conditions. Penzien and Wu (998 deveoped siiar anaytica soutions for thrust, shear, and oent in the tunne due to racking deforation. Later, Penzien ( provided a copeentary anaytica procedure for racking deforation evauation of rectanguar and circuar tunnes. Recenty, Hashash in and 5 with coparison these two ethods, found out that the cacuated forces and dispaceents are identica for the fu-sip assuption, however, Penzien s soution resuts in uch ower estiation of axiu thrusts copared to Wang s soution for the no-sip assuption. This difference is aso reported by Park (6. In the present investigation a series of nuerica anayses were perfored, using finite eeent code PLAXIS (PLAXIS-B.V., and outputs were copared with resut of anaytica soutions. On the other hand sensitivity anaysis on the echanica paraeters of soi was perfored and the effect of various soi properties and axiu shear strain was evauated on the tunne response.. ANALYTICAL SOLUTION The response of a of buried circuar tunne against the earthquake is a function of the copressibiity and fexibiity ratios of the structure, the in-situ overburden pressure and at-rest coefficient earth pressure, K, of soi. The trust, bending oent and deforations can be cacuated using cosed for soution [Peck et a., 97]. The cosed for soutions, reported by Wang and Penzien are based on sip between soi and tunne.

2 The 4 th October -7, 8, Beijing, China. Wang foruation Wang (993 proposed the soutions for the axiu thrust and oent in the due to the equivaent static ovaing deforations. The soutions for the fu-sip condition at the soi interface, to evauate thrust and oent due to seisic oading can be expressed as: T M Dd d ax = K 6 KF g 3 ax E r g ( -n ax (. = (. E ax = K r g (.3 ax 6 ( -n Where υ, r and F are Poisson ratio of soi, radius and fexibiity ratio of tunne respectivey. K ( -n = (.4 F + 5-6n 3 E ( -n r F = (.5 6E I( + n Where E, E and υ are oduus of easticity of soi, oduus of easticity and Poisson ratio of respectivey. The axiu shear strain in the ediu for both constant shear strain and variabe shear strain ay be obtained as: G C E = (.6 ( +n G = (.7 r ax Vs = g (.8 Where r = gs / g and G is shear oduus. Fu-sip assuption under sipe shear deforation, however, ay cause significant underestiation of the axiu thrust. It has been therefore, recoended to evauate T ax by the foowing equation for the for fu-sip assuption case [Wang, 993]: C Where τ ax is axiu shear strain and K is: ( F [( -n - ( - n C ]- - n K = + 5 F [(3 - n + ( - n C] + C[ - 8n + 6n + ] + 6-8n (.9 (. (.. Penzien foruation Penzien & Wu (998 and Penzien ( deveoped siiar anaytica soutions for the thrust oent and shear in tunne. For fu-sip condition at the soi- interface we have: D (. n n d = R Dd free- fied

The 4 th October -7, 8, Beijing, China n E IDd T( q = Cos( q + 45 3 d ( -n (.3 n 6E IDd M ( q = Cos ( q + 45 3 d ( -n (.

3 The 4 th October -7, 8, Beijing, China n E IDd T( q = Cos( q d ( -n (.3 n 6E IDd M ( q = Cos ( q d ( -n (.4 n 4E IDd V ( q = Sin ( q d ( -n Where R n and θ are inertia oent and wave incident ange respectivey, i.e. (.5 n 4( -n R = n a + (.6 n E I(5-6n a = 3 d G ( - g (.7 The preceding equations take the foowing for for the no-sip condition at the soi- interface,: D (.8 d = RD d free - fied 4E IDd T( q = Cos( q d ( -n (.9 6E IDd M ( q = Cos ( q d ( -n (. 4EIDd V ( q = Sin ( q d ( -n (. 3. COMPARISON BETWEEN ANALYTICAL SOLUTIONS AND NUMERICAL ANALYSIS FOR THE CASE OF CONSTANT SHEAR STRAIN As stated above, there exists a significant difference between anaytica and nuerica soution in case of no-sip assuption. Here, a nuerica ode, caed origin one, deveoped to copare anaytica and nuerica resuts. 3. Nuerica ode deveopent In order to evauate anaytica soutions for seisic-induced ovaing deforation of the in a circuar tunne, the nuerica anaysis has been perfored, using PLAXIS Code. The nuerica anaysis is based on the foowing assuptions: Pane-strain condition exists. Soi and tunne behave ineary and easticay. In this ode the axiu shear strain is converted to ovaing deforation and these deforations are appied to the ode. It shoud be noted that in PLAXIS, ony no-sip condition between the tunnes and ground can be siuated. A 5 node eeent and aso a fine esh in geoetry of the ode were used to obtain higher accuracy. Figure Typica odeing esh and 5 node eeent For ore accuracy the eshes around the tunne are ore refined. Figure shows a typica odeing esh used

The 4 th October -7, 8, Beijing, China in this study. Now, the ovaing deforations were appied at the two vertica boundaries of the ode to siuate pure shear condition.

4 The 4 th October -7, 8, Beijing, China in this study. Now, the ovaing deforations were appied at the two vertica boundaries of the ode to siuate pure shear condition. This shear oading is aso appied at horizonta boundaries so that the haf of the ovaing deforation in positive sign is appied at the top and other haf this deforation with negative sign is appied at the botto boundaries. The deforation contours due to the appied shear force is shown in figure. These contours show the unifory distributed deforations, indicating pure shear. Figure Deforation contours in fina anaysis The induced ovaing deforations in the ground can be cacuated in two cases, naey perforated ground and non perforated ground. In the case of non-perforated ground, the axiu diaetric deforation is a function of axiu free-fied shear strain ony. In the case of perforated ground, however, it is further reated to the Poisson s ratio of the ediu. The diaetric strains can be stated by Eqn. 3. and 3.. Ddnon - perforated g = ± ax (3. d Dd = ± g ax -n (3. perforated d ( Now, based on the cosed for soution and aso considering the geoetry and physica properties of the origina tunne incuding d=6.6, g ax =. 6 and n =. 3 fro origina tunne, the diaetric deforation can be cacuated, i.e. Ddnon- perforated None perforated ground: g = ± ax.6 Ddnon - perforated = ± 6.6 Þ Dd non- perforated =. 5 d Dd perforated Perforated ground: = ±g ( -n d ax Dd perforated = ±.6( Þ Dd perforated =. 558 It is obvious that the diensions of the ode, i.e. the height and width of the edia around the tunne can affect the resut. To verify the ode, odes with various heights and widths were anayzed and resuts copared with anaytica soution. The ode is shown in figure 3 and the resuts are gathered in tabe. Tabe The variations of the diaeter in non-perforated and perforated ground Mode Diension Ddnon- perforated Dd perforated Fro tabe, it can be observed that the ode with 4 eter height and eter width produces the east error which can be cacuated as foow: None perforated ground Error = ( /.43» 3% Perforated ground Error = ( /.558» -.9% It can be seen the error between anaytica and nuerica ethods (ode 4 is ess than 3%, indicating identica deforation for anaytica and nuerica ethods.

5 The 4 th October -7, 8, Beijing, China Figure 3 The created ode in PLAXIS for non-perforated condition 3. The resut of anaysis for origina tunne The specifications of the reference tunne used in this study are based on Tabriz Underground tunne with 5 depth fro ground eve and 3.3 radius. Other specifications are g =.5 kn/ 3, E =767 kn/, n =.3, t=.3, E =.48 7 kn/, n =. a ax =.35g. Aso the interna forces in the caused by changes in shear deforation ust be studied. Epoying equations.6 to.8, the cacuated shear strain is.6 which can be appied to the nuerica ode after inverting to the shear deforation. The anaysis resuts, based on Penzien, Wang and nuerica ethod, which are shown in figure 4 indicate that the shear forces and bending oents cacuated by anaytica and nuerica ethod are we coincide but axia forces cacuated by Wang forua, especiay in axiu q shows high differences. 3 3 Moet (kn- - - Wang Penzien Nuerica Axia Force (kn - - Shear Force (kn q (Degree 3.3. The anaysis resuts for cassified sois To investigate the soi strength paraeters on axia forces, shear forces and bending oent of, cassified sois with different eastic oduus and Poisson ratio were considered. Tabe suarizes the upper and ower iits of eastic oduus and Poisson ratio for different sois incuding cay, sit, sand and grave. The eastic oduus varies fro to 5 Pa and Poisson ratio fro. to.4. To investigate the soi strength paraeters changes, the axiu shear strain was considered constant and equa to g ax =.6 which is an average vaue for soft to hard sois. Tabe Specifications of cassified sois Upper iit of Soi Lower iit of Soi Mode S S S3 S4 S5 S6 S7 Soi type Soft cay Mediu cay Hard cay Dense sand Sand & grave Dense sand & grave Highy dense sand q (Degree ν paraeters E γ (kn/ 3 (kn/ q (Degree Figure 4 Coparison of nuerica anayses and anaytica soutions for tunne ground interaction ν paraeters E γ (kn/ 3 (kn/ The anaysis resuts, using Penzien, Wang and nuerica ethod are shown in tabe 3. In this tabe T, V and M 6 6 8

6 The 4 th October -7, 8, Beijing, China are axia force, shear force and bending oent of respectivey. Subscripts P, W and N refer to Penzien, Wang and nuerica ethod. Mode S S S3 S4 S5 S6 S7 Tabe 3 The cosed for and nuerica ethod anaysis resuts Penzien ( Wang (993 Nuerica ethod T P (kn V P (kn M P (kn. T W (kn M W (kn. T N (kn V N (kn M N (kn Considering anaysis resuts, the cacuation error between anaytica and nuerica ethods are given in tabe 4. In this tabe T, V and M are the agnitude of error in axia force, shear force and bending oent of between cosed for soution and nuerica ethod. Subscripts P-N and W-N refer to Penzien and Wang and nuerica ethod. To copare the errors for different ethods, the resuts are iustrated in figure 5. It can be seen that with increasing fexibiity ratio, i.e. increasing soi stiffness, the error agnitude increases, ipying that for soft cay to dense sand the error between anaytica and nuerica ethods wi be ess than % which ay be considered satisfactory. For had sois or soi with fexibiity ratio higher than 4 cosed for soution ead to conservative resuts. Tabe 4 The agnitude of error between cosed for soutions and nuerica ethod Mode S S S3 S4 S5 S6 S7 Penzien ( T P-N (% V P-N (% M P-N (% Wang (993 T W-N (% M W-N (%

7 The 4 th October -7, 8, Beijing, China 4. COMPARISON BETWEEN ANALYTICAL SOLUTIONS AND NUMERICAL ANALYSIS FOR THE CASE OF VARIABLE SHEAR STRAIN In this section shear strain is not hed constant, i.e. based on the soi properties shear strain is cacuated and its rea vaue is then considered in the anaysis process. Error (% Wang Penzien Nuerica F (Fexibity Ratio Figure 5 Existing errors between anaytica and nuerica ethods ax (% g G ( kn/ Figure 6 Variations of axiu shear strain with shear oduus 4.. Cosed for soution ethods The anaysis reviewed in preceding sections was based on that the axiu shear strain ( g ax, siiar to other structura and earthquake paraeters is constant. It was noted in section 3. that the axiu shear strain is dependent to paraeters such as shear wave veocity and axiu ass veocity in the edia. The shear wave veocity, Vs, is dependent to the axiu veocity and acceeration, which is in turn dependent to the easticity oduus and Poisson ratio. If, therefore, E and n vary, shear oduus, G and C vary as we. With varying C, axiu shear strain ( g ax, wi vary. In this ethod the earthquake entry is the axiu acceeration of earth oveent, which ay be cacuated using ground eve acceeration and soi ayers specifications. In this section based on axiu acceeration at the ground eve assuption to be a ax =.35g and aso soi ayer specifications, g ax can be cacuated. Figure 6 iustrates the variations of g ax with easticity oduus. It can be seen that with increasing easticity oduus, g ax increases initiay sharpy, but reaining roughy constant with higher vaues of easticity oduus. The cosed for soution with constant axiu shear strain, presented in the previous section, can now be appied for an earthquake with specified acceeration gravity for which the axiu shear strain is cacuated with respect to different soi specifications. Here, the anaysis was carried out for Wang's axia force and Penzien's bending oent and shear force which the resuts are shown in figure 7. M & V (kn & kn- ax ax V M F (Fexibity Ratio ax ax T (kn ax F (Fexibity Ratio Figure 7 Variations of anaytica resuts with fexibiity ratio

8 The 4 th October -7, 8, Beijing, China CONCLUSIONS Based on this research considering the origina tunne the foowing points can be concuded: With coparisons cosed for soution with nuerica ethod, there is no significant difference between nuerica and Wang ethod in axia force cacuation, however Penzien ethod eads to underestiation axia force in no-sip condition. Since the anaytica ethods are not abe to evauate shear strain distribution it is recoended to evauate axiu shear strain based on seisic tie history axiu shear, and then epoy this vaue in presented in anaytica ethods. It was observed that the bending oent and axia force variations depend on the fexibiity ratio, it is, therefore, recoended stress distribution to be considered for anaysis and design of of the tunnes. REFERENCES Wang J. N., (993. Seisic design of tunnes: A state of the art approach, Parsons Brinkerhoff Quad & Dougas Inc., New York, NY, Monograph 7. Hashash Y. M. A., Hook J. J., Schidt B. & Yao J. I., (. Seisic design and anaysis of underground structure. Journa of Tunneing and Underground Space Technoogy 6, Hashash Y. M. A., Park D. & Yao J. I., (5. Ovaing deforations of circuar tunnes under seisic oading: an update on seisic design and anaysis of underground structures. Journa of Tunneing and Underground Space Technoogy, Peck R. B., Hendron A. J. & Mohraz B., (97. State of the art in soft ground tunneing. Proceedings of the Rapid Excavation and Tunneing Conference. Penzien J., (. Seisicay induced racking of tunne s. Journa of Earthquake Eng. Structura Dynaics, 9, Park K., Tantayopin K. & Tontavanich B., (6. Anaytica soutions for seisic design of tunne in Bangkok MRT subway. Proceedings of the Internationa Syposiu on Underground Excavation and Tunneing, Bangkok, Thaiand PLAXIS-B.V., (. PLAXIS: Finite eeent package for anaysis of geotechnica structures, Deft, Netherands. Das B. M., (. Principes of geotechnica engineering, nd Edition, The University of Texas (Pubisher. Penzien J., Wu, C., (998..Stresses in s of bored tunnes. Journa of Earthquake Eng. Structura Dynaics 7, 83 3.

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