Parametric Study of Steel-Lined Pressure Shafts in Anisotropic Rock

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1 SEE Tunnel:Promotng Tunnelng n SEE Regon May 22-28, 2015, Lacroma Valamar Congre Center, Dubrovnk, Croata Parametrc Study of Steel-Lned Preure Shaft n Anotropc Rock Alexandre J. PACHOUD, Laboratory of Hydraulc Contructon (LCH), Ecole Polytechnque Fédérale de Lauanne (EPFL), Swtzerland, Emal : alexandre.pachoud@epfl.ch Anton J. SCHLEISS, Laboratory of Hydraulc Contructon (LCH), Ecole Polytechnque Fédérale de Lauanne (EPFL), Swtzerland, Emal : anton.chle@epfl.ch Topc: Plannng and Degnng Tunnel and Underground Structure Summary: Steel-lned preure tunnel and haft are ued to convey water from the reervor to the turbne n hgh-head hydroelectrc power plant. An axymmetrcal multlayer model often condered by engneer for the degn of the teel lner. Stree and deformaton can then be computed wth a cloed-form analytcal oluton n otropc rock. When the rock anotropc, the lowet elatc modulu of the rock meaured n tu often condered n the analytcal oluton, whch regarded a conervatve. In th work, the behavour of teel lner n anotropc rock wa tuded ytematcally by mean of the Fnte Element Method (FEM). The materal, namely teel, backfll concrete, near-feld rock and far-feld rock were modelled a lnear and elatc, and a ted contact wa aumed between the layer. The nfluence of geometrcal and materal parameter under a qua-tatc nternal water preure wa tuded through an extenve parametrc tudy. It wa oberved that, compared to the correpondng reult n otropc rock, maxmum tree n the teel-lner can be reduced up to 25% when anotropy condered. On the contrary, the maxmum tree n the far-feld rock can be largely underetmated, namely up to 65%. Keyword: Preure tunnel and haft, Anotropc rock, Steel lner, Stree, Fnte Element Method, Degn 1. Introducton The development of hgh-capacty Pelton turbne and hgh-trength teel (HSS) allow the degn of hgh-head power plant up to 2000m gro head to meet the ncreang demand for peak energy [1]. Wth the re of volatle new renewable energe, torage hydropower plant are ubjected to more dratc operatonal requrement to balance the electrcty grd. HSS ha been developed to addre the mechancal requrement, but they are more dffcult to weld than ductle grade. Furthermore they are more entve to fatgue and brttle falure [2-3]. The common method for the degn of teel-lned preure tunnel and haft conder an axymmetrcal model n otropc rock to compute the tree and deformaton, takng the lowet elatc modulu meaured n tu. Such an approach commonly condered a conervatve for the maxmum tree n the teel lner [4]. In Europe, the CECT (1980) recommendaton [5] are followed for the degn and the contructon of teel-lned preure tunnel and haft. Thee gudelne are applcable for ductle teel grade, but are no longer adequate for the degn wth HSS. Intead, degner ue a range of horzontal falure aement method and/or recommendaton, but there tll a need for pecfc recommendaton for teellned preure haft embedded n rock. Several author have tuded the behavour of lnng n anotropc rock (e.g. [6-11]). However, the cae of teel-lned preure tunnel or haft ha not yet been tuded ytematcally n anotropc rock. There nether analytcal oluton nor extenve parametrcal tudy of thee tructure avalable n anotropc rock. In th work, the behavour of teel lner conderng the anotropc behavour of the rock tuded by mean of the FEM. The oluton n otropc rock ntroduced n Secton 2. The conttutve model are dcued n Secton 3. Fnally, the numercal model, the parametrc tudy and the reult are preented n Secton 4.

2 SEE Tunnel:Promotng Tunnelng n SEE Regon May 22-28, 2015, Lacroma Valamar Congre Center, Dubrovnk, Croata 2. Soluton n otropc rock 2.1 Axymmetrcal model The tandard axymmetrcal model for the degn of teel-lned preure tunnel and haft ntroduced n Fg. 1. Fg. 1 Axymmetrcal model for teel-lned preure tunnel and haft n otropc rock In the multlayer model fve zone are dtnguhed (e.g. [12-13]): (1) the teel lner; (2) an ntal gap between the lner and the backfll concrete; (3) the backfll concrete; (4) the near-feld rock; and (5) the far-feld rock. All materal are condered lnear elatc. The ntal gap r 0 an annular pace between the teel lner and the backfll concrete due to the thermal hrnkng of the teel durng the frt fllng of the haft. The backfll concrete regarded a a radally cracked materal due to t qua-brttle nature. The near-feld zone a looened zone of the rock reultng from the excavaton method and the dturbance of the n tu tre feld. Fnally, the far-feld rock the undturbed zone of the rock, aumed a homogeneou and otropc n the axymmetrcal model. 2.2 Analytcal oluton The model decrbed n Fg. 1 ha an analytcal oluton derved from the compatblty condton on the dplacement at the nterface between the layer [14]. They can be wrtten a u (r ) r u (r ), c r c 0 r c u (r ) u (r ) u, c r r rm (r ) u (r ) r rm r rm (1) where r potve, 0 u r denote a radal dplacement, and the upercrpt and rad are related the layer accordngly to Fg. 1. Aumng that cracked materal cannot tranmt tenle tree and that the teel lner modelled accordng to the thck-walled ppe theory [15], tree and dplacement can be derved analytcally [13,14].

3 SEE Tunnel:Promotng Tunnelng n SEE Regon May 22-28, 2015, Lacroma Valamar Congre Center, Dubrovnk, Croata 3. Conttutve model 3.1 Anotropc rock In the followng a tranverely otropc rock condered. Tranvere otropy a partcular cae of orthotropy,.e. wth a plane of otropy. Such a materal can be decrbed by fve ndependent contant, namely the elatc modul n the plane of otropy and perpendcular to t repectvely (E and E repectvely), the Poon coeffcent whch characterze the reducton n the plane of otropy for the tenon n the ame plane and the tenon n a drecton normal to t repectvely ( and ' repectvely) and the hear modulu normal to the plane of otropy ( G ' ) [16]. The hear modulu parallel to the plane of otropy computed a n an otropc materal wth E and. G ' can be computed by an emprcal relaton wdely ued n the lterature: G' emp E '. 1 E'/ E 2 ' (2) 3.2 Cracked layer To model the cracked layer (namely the backfll concrete and the near-feld rock) a calar damage parameter D ntroduced a 1 D R (3) where denote a materal parameter and R a calar factor to be multpled to a materal property. Thu a radally damaged materal ha to be modelled. Snce cylndrcal coordnate are ued, the elatc modulu hould be dratcally decreaed n the tangental drecton by a large factor a: R E / E (4) E where the tlde ymbol denote a property of the damaged materal and the tangental drecton. Accordngly, the approprate hear modul and Poon rato are decreaed to account the effect of damage. 4. Sytemc parametrc tudy 4.1 Fnte element model The FEM code Mechancal APDL (ANSYS) 14.0 wa ued [17]. The FE model wa bult parametrcally n order to run a parametrc tudy ung the Probabltc Degn Sytem wth a Uer-Defned Samplng. It aume the ame hypothe than decrbed n the analytcal oluton. An example of a meh hown n Fg. 2. The model dcretzed n approxmately element, dependng on the generated parameter. The far-feld rock condered a nfnte. It wa taken nto account by modellng a large far-feld rock medum of 30 r tartng from r rm. Conttutve law for the materal where mplemented a decrbed n Secton 3. Fve materal parameter have been aumed a contant, namely the teel properte ( E 210 GPa) and 0.29 ), the backfll concrete properte ( Ec 20 GPa and c 0.20 ) and the Poon rato of the near-feld rock ( 0.20 ). In addton one geometrcal parameter wa condered a contant, namely the thckne of the backfll concrete ( tc 0.5 m). 4.2 Varaton of the parameter There are 10 varable n the parametrc model a defned n Table 1. A o-called reference et of cae defned wth gven parameter a hown n Table 2. Startng from the reference et of cae, 7 parameter are changed ndvdually a preented n Table 2. Wthn thee et of cae, 3

4 SEE Tunnel:Promotng Tunnelng n SEE Regon May 22-28, 2015, Lacroma Valamar Congre Center, Dubrovnk, Croata Fg. 2 Example of a meh of the FE model around the openng Table 1 Defnton of the varable parameter n the FE model Parameter r t Defnton Internal radu of the teel lner Thckne of the teel lner t Thckne of the near-feld rock (looened) p Qua-tatc nternal water preure Elatc properte of the tranverely otropc far-feld rock E, E ', G ',, ' (ee Secton 3.1) E Elatc modulu of the near-feld rock (looened) Table 2 Varaton of the parameter wth repect to the reference et of cae Reference Parameter Unt Value 1 Value 2 Value 3 value t m E GPa G'/ G (-) ' emp p bar (-) / ' (-) E / E ' (-)

5 SEE Tunnel:Promotng Tunnelng n SEE Regon May 22-28, 2015, Lacroma Valamar Congre Center, Dubrovnk, Croata parameter are changed accordngly to Table 3. Conequently, a total of 22 et of cae were condered (7 x 3 + 1), each one contanng 325 cae for a total of 7150 mulated cae. The parameter of each cae repect the thermodynamc contrant of tranverely otropc materal and realtc practcal range of varaton (e.g. [16], [18]). Table 3 Varaton of the varable parameter n each et of cae Parameter Unt Mnmum Maxmum Increment r m t m E/ E ' (-) In addton, otropc cae were randomly ampled wth a unform dtrbuton of the parameter n order to valdate the FE model compared to the analytcal oluton (ee Secton 2.2). The reult howed very good agreement. 4.3 Maxmum tree n the teel lner and n the far-feld rock For the followng dcuon, all the reult are normalzed. They are preented n a dmenonle form by dvdng the numercal reult of the anotropc cae by the numercal reult of the mot conervatve o-called correpondng otropc cae. In the anotropc cae, the maxmum frt prncpal tree n the teel lner alway occur n the plane of otropy. Denotng the angle wth repect to the otropc plane, the maxmum normalzed frt prncpal tree n the teel lner are thu computed a: ˆ 1,max ˆ (r r, 0). (5) ˆ (r r ) 1, ano 1, o In the far-feld rock, the maxmum tree do not occur at a contant angle of locaton, unlke n the teel lner, dependng on the rock ma properte. The locaton of the maxmum tree not dcued heren. The maxmum normalzed frt prncpal tree n the far-feld rock are computed a: ˆ rm 1,max ˆ (r r rm, ) ˆ (r r ) rm 1, ano rm 1, o rm (6) where denote the drecton of the maxmum frt prncpal tre for each cae. In Fg. 3(a), the nfluence of the near-feld rock thckne on the maxmum normalzed frt prncpal tree n the teel lner hown a a functon of the degree of anotropy. Reult are lghtened for readng purpoe. The maller the near-feld rock thckne, the greater the nfluence of the degree of anotropy. Fg. 3(a) alo how that the greater the nternal radu and the maller the lenderne (.e. t /r), the greater the nfluence of the degree of anotropy. For all the teted cae, the maxmum tree were lower than n the correpondng otropc cae (normalzed reult lower than unty). The maxmum tree can be reduced up to 22% compared to the correpondng otropc cae for the teted cae. In Fg. 3(b), the correpondng maxmum normalzed tree n the far-feld rock are preented. Small thckne of the near-feld rock, mall nternal radu and great lenderne nduce great nfluence of the degree of anotropy. Unlke n the teel lner, the maxmum frt prncpal tree n the far-feld rock are up to 34% larger than n the correpondng otropc cae for the teted cae. Fg. 4(a) llutrate the nfluence of the rock tffne parallel to the plane of otropy on the maxmum tree n the teel lner. The lower the tffne, the lower the nfluence of the degree of anotropy.

6 SEE Tunnel:Promotng Tunnelng n SEE Regon May 22-28, 2015, Lacroma Valamar Congre Center, Dubrovnk, Croata (a) In the teel lner (b) In the far-feld rock Fg. 3 Maxmum normalzed frt prncpal tree n (a) the teel lner and n (b) the far-feld rock a a functon of the degree of anotropy E/ E ' for dfferent near-feld rock thckne t and by varyng the lenderne t / r. t t t t = Black refer to = 0.50 m, blue refer to = 1.00 m, red refer to = 2.00 m and green refer to 3.00 m Indeed, n the cae of a weak rock the partcpaton of the concrete-rock ytem to wthtand the nternal preure low, and conequently anotropy ha a lower nfluence on the lner whch take a great part of the nternal preure. Smlarly, the maller the nternal radu and the greater the teel thckne, the maller the nfluence of the anotropy. Th alo due to the fact that a tffer lner le nfluenced by change n the rock ma modulu. The correpondng maxmum normalzed tree n the far-feld rock are hown n Fg. 4(b). In a general manner, the greater the tffne of the teel lner compared to the rock ma (.e. low rock tffne, low nternal radu and great lner tffne) the greater the nfluence on the maxmum tree n the rock ma. Unlke the teel lner, the maxmum tree n the rock ma can be gnfcantly ncreaed compared to the correpondng otropc cae, up to 47% for the teted cae.

7 SEE Tunnel:Promotng Tunnelng n SEE Regon May 22-28, 2015, Lacroma Valamar Congre Center, Dubrovnk, Croata (a) In the teel lner (b) In the far-feld rock Fg. 4 Maxmum normalzed frt prncpal tree n (a) the teel lner and n (b) the far-feld rock a a functon of the degree of anotropy E/ E ' for dfferent rock ma elatc modulu E and by varyng the lenderne t / r. Black refer to E = 10 GPa, blue refer to E = 1 GPa, red refer to E = 15 GPa and green refer to E = 25 GPa Fg. 5(a) how the nfluence of the devaton of the cro-hear modulu from the emprcal relaton (Eq. 2). The greater the cro-hear modulu compared to the emprcal relaton, the greater the nfluence of the degree of anotropy. For weak cro-hear modulu (e.g. 70% of the emprcal relaton n Fg. 5(a)) and for low degree of anotropy, the maxmum tree n the teel lner are lghtly hgher than the correpondng otropc cae. Otherwe the maxmum tree n the teel lner are generally lower than n the correpondng otropc cae, down to 28% for the teted cae. Fg. 5(b) alo llutrate that, the greater the lenderne of the teel lner, the lower the nfluence of the anotropy on the maxmum tree n the lner. Fg. 5(b) llutrate the correpondng maxmum normalzed frt prncpal tree n the far-feld rock. It can be oberved that the nfluence of the degree of anotropy hgh for ether low or

by varyng the lenderne t / r. Black refer to G'/ G ' emp = 1.0 (-), blue refer to G'/ G ' emp = 0.7 (-), red refer to G'/ G ' emp = 0.9 (-) and green refer to G'/ G ' emp = 1.

8 SEE Tunnel:Promotng Tunnelng n SEE Regon May 22-28, 2015, Lacroma Valamar Congre Center, Dubrovnk, Croata (a) In the teel lner (b) In the far-feld rock Fg. 5 Maxmum normalzed frt prncpal tree n (a) the teel lner and n (b) the far-feld rock a a functon of the degree of anotropy E/ E ' for dfferent cro-hear modulu to emprcal relaton rato G'/ G ' emp and by varyng the lenderne t / r. Black refer to G'/ G ' emp = 1.0 (-), blue refer to G'/ G ' emp = 0.7 (-), red refer to G'/ G ' emp = 0.9 (-) and green refer to G'/ G ' emp = 1.3 (-) hgh value of the cro hear-modulu (e.g. 70 or 130% of the emprcal relaton n Fg. 5(b)). For medum value (e.g. 90 or 100%), the nfluence lower, although gnfcant. Maxmum tree n the far-feld rock can be up to 64% hgher than n the correpondng otropc cae. 5. Concluon For the degn of teel-lned preure tunnel and haft, an axymmetrcal model wth the mallet rock ma modulu oberved n tu often condered a conervatve n term of maxmum tree n the teel lner. The anotropy not commonly taken nto account nce the mechancal behavour of the teel-concrete-rock ytem tll not well undertood. In th work, the behavour of teel-lned preure tunnel and haft n anotropc rock wa tuded by mean of the FEM. The rock ma wa aumed a tranverely otropc. An extenve parametrc tudy wa performed over 10 varable materal and geometrcal parameter. Maxmum

9 SEE Tunnel:Promotng Tunnelng n SEE Regon May 22-28, 2015, Lacroma Valamar Congre Center, Dubrovnk, Croata tree n the teel lner and n the far-feld rock were llutrated and trend of behavour were outlned. When conderng anotropc rock behavour, the maxmum frt prncpal tree n the teel lner are lower than n the correpondng otropc cae that would be condered for the degn. They can be reduced down to 25% for the teted cae. Conderng otropc rock wth the lowet rock ma modulu thu a conervatve aumpton n term of maxmum tree n the teel lner. Unlke the teel lner, conderng an otropc cae not conervatve n term of maxmum frt prncpal tree n the far-feld rock, whch can be up to 65% greater than the predcton conderng otropc rock. Maxmum tree n the rock are thu underetmated. The preented reult allow a better undertandng of the mechancal behavour of teel-lned preure tunnel and haft n anotropc rock. Th an mportant ue n the vew of optmzed degn gudelne for teel-lned preure tunnel ung hgh-trength teel, whch are entve to fatgue and brttle falure ue. Acknowledgement Th tudy part of the reearch project conortum HydroNet 2: Modern methodologe for degn, manufacturng and operaton of hydropower plant, funded by the Sw Competence Center Energy and Moblty (CCEM-CH) and Swelectrc Reearch. Reference [1] Schle, A. J. and Mano, P. A. (2012). Degn of preure relef valve for protecton of teel-lned preure haft and tunnel agant bucklng durng emptyng. Rock Mechanc and Rock Engneerng 45(1): [2] Cerjak H. and Enznger, N. and Pudar, M. (2013). Development, experence and qualfcaton of teel grade for hydro power condut. In: Proceedng of the Conference on Hgh Strength Steel for Hydropower Plant, Graz Unverty of Technology, Graz, Autra. [3] Grener, R. and Innerhofer en, G. and Sterng, W. (2013). New degn apect for teel lnng of preure haft made of hgh trength teel. In: Proceedng of the Conference on Hgh Strength Steel for Hydropower Plant, Graz Unverty of Technology, Graz, Autra. [4] Brekke, T. L. and Rpley, B. D. (1987). Degn gudelne for preure tunnel and haft. Tech. rep., Unverty of Calforna at Berkeley, Department of Cvl Engneerng, Berkeley, Calforna 94707, EPRI AP-5273, Project [5] CECT. (1980). Recommendaton for the degn, manufacture and erecton of teel pentock of welded contructon for hydropower ntallaton. European Commttee of Boler, Veel and Ppe Work Manufacturer. [6] Értov, V. S. (1967a). Computaton of preure tunnel lnng n anotropc rock. Hydrotechncal Contructon 1(5): [7] Értov, V. S. (1967b). Expermental tude of preure-tunnel lnng n anotropc formaton. Hydrotechncal Contructon 1(12): [8] USACE (1997). Tunnel and haft n rock (EM ). U.S. Army Corp of Engneer. [9] Balavk, I. A. (1973). Stree n the lnng of a preure tunnel drven n an nhomogeneou rock ma. Sovet Mnng 9(6): [10] Potol kaya, O. K. (1986). Effect of the tructure of a rock ma and properte of rock on the tre tate of hydraulc preure tunnel. Hydrotechncal Contructon 20(1): [11] Kumar, P. and Sngh, B. (1990). Degn of renforced concrete lnng n preure tunnel, conderng thermal effect and jonted rock ma. Tunnellng and Underground Space Technology 5(1/2): [12] ASCE (2012). Steel Pentock, Manual and Report on Engneerng Practce, vol 79. Amercan Socety of Cvl Engneer, Reton, Vrgna. [13] Schle, A. J. (1988). Degn crtera appled for the lower preure tunnel of the North Fork Stanlau Rver hydroelectrc project n Calforna. Rock Mechanc and Rock Engneerng 21(3): [14] Hachem, F. E. and Schle, A. J. (2011). A revew of wave celerty n frcton-le axymmetrcal teel-lned preure tunnel. Journal of Flud and Structure 27(2): [15] Tmohenko, S. P. and Gooder, J. N. (1970). Theory of elatc tablty, 3rd edn. McGraw-Hll

10 SEE Tunnel:Promotng Tunnelng n SEE Regon May 22-28, 2015, Lacroma Valamar Congre Center, Dubrovnk, Croata Internatonal. [16] Amade, B. and Savage W. Z. and Swolf, H. S. (1987). Gravtatonal tree n anotropc rock mae. Internatonal Journal of Rock Mechanc and Mnng Scence 24(1):5-14. [17] ANSYS Inc (2011). ANSYS Academc Reearch, Help Sytem, Documentaton Releae ANSYS Inc. [18] Amade, B. and Swolf, H. S. and Savage, W. Z. (1988). Gravty-nduced tree n tratfed rock mae. Rock Mechanc and Rock Engneerng 21(1):5-14.

Lateral stresses caused by uniform rectangular area loads on a cross-anisotropic backfill

Lateral stresses caused by uniform rectangular area loads on a cross-anisotropic backfill Wang, C. D. (). Géotechnque 5, No. 9, 5 [do:./geot..5.9.5] TECHNICAL NOTE Lateral tree caued by unform rectangular area load on a cro-anotropc backfll C. D. WANG* KEYWORDS: anotropy; degn; earth preure;