Using BESO method to optimize the shape and reinforcement of the underground openings

Transcription

1 Usng BS method to optmze the shape and renforcement of the ndergrond openngs. Ghabrae, Y.M. Xe & X. Hang School of Cvl, nvronmental and Chemcal ngneerng, MI Unversty, Melborne, Astrala ABSAC: In excavaton desgn, optmzng the renforcement and fndng the optmal shape of the openng are two sgnfcant challenges. Both of these problems can be vewed as searchng for the optmm dstrbton of materal n the desgn doman. In strctral desgn, the state-of-the-art topology optmzaton technqes have been sccessflly sed to deal wth sch problems. ne of these technqes, known as bdrectonal evoltonary strctral optmzaton (BS), s employed here to mprove the shape and renforcement desgns of ndergrond openngs. he BS algorthm s extended to smltaneosly optmze the shape of the openng and the topology of the renforcement. he valdty of the proposed approach s tested throgh a smple example. INDUCIN Drng the last two decades topology optmzaton has attracted consderable attenton and the technqes n ths feld have been mproved sgnfcantly. Several physcal problems have been tackled n ths context. However the applcaton of topology optmzaton tools n geotechncal problems has not been stded thoroghly. In spte of the great potental n ths class of problems there are only a few pblshed works n ths area. Among these works some attempted to optmze the shape of ndergrond openngs (en et al. 2005, Ghabrae et al. 2007) whle others tred to optmze the topology of renforcement arond a tnnel wth a predefned shape (L et al. 2008, Yn et al. 2000, Yn & Yang 2000a,b). Both of these optmzatons can lead to consderable savngs n excavaton desgns. In ths paper, attempts have been made to optmze the shape of the openng and the topology of the srrondng renforcement smltaneosly. hs coplng can mprove the soltons leadng to greater savngs. It wll be shown that the senstvtes of these two optmzaton problems only dffer n constant vales. Hence the two optmzaton problems can be solved together wth almost no extra comptatonal effort. he BS method was proposed n late 90s (Qern et al. 998, Yang et al. 999) as an mproved verson of the S method whch was orgnally ntrodced n early 90s by Xe and Steven (Xe & Steven 993, 997). he S method mproves the desgn by gradally removng the neffcent elements. In the BS method, on the other hand, a b-drectonal evoltonary strategy s appled whch also allows the strengthenng of the effcent parts by addng materal. he effcency of elements can be calclated by senstvty analyss of the consdered objectve fncton or can be assgned nttvely (L et al. 999). In ths paper the BS method s sed for solvng both problems of shape optmzaton of the openng and topology optmzaton of renforcements. hese two problems can both be modeled as two-phase materal dstrbton problems. For shape optmzaton the materal s changng between solds and vods. In renforcement optmzaton, on the other hand, the materal can be swtched between orgnal rock and renforced rock. In the orgnal BS neffcent elements are completely elmnated from the mesh. Sch topology optmzaton technqes are sometmes referred to as hard kll methods as oppose to soft kll methods. In hard kll methods only the non-vod elements wll reman n the mesh and so the fnte element analyss can be performed faster. However n these methods the senstvty of vod elements cannot be calclated drectly from the analyss reslts and shold be extrapolated from the srrondng sold elements. In ths paper a soft kll BS has been adopted where the vod elements are represented by a very soft materal. In ths manner the senstvtes of vods are drectly calclable.

2 2 MAIAL MDL Althogh the geomechancal materals are natrally nhomogeneos, non-lnear, ansotropc, and nelastc (Jng 2003), n excavaton desgn n rocks, modelng them as an sotropc, homogeneos materal and assmng lnear elastc behavor can be nstrctve and sometmes can predct the real behavor wth acceptable accracy (Brady & Brown 2004). In fact the smplfed lnear elastc materal model s stll the most common materal model sed n geomechancs (Jng 2003). Moreover the reslts of a lnear analyss can be sed as a frst-order approxmaton of non-lnear cases. Most of prevos works (en et al. 2005, L et al. 2008, Yn et al. 2000, Yn & Yang 2000a,b) whch appled the topology optmzaton technqes n excavaton desgn have adopted the lnear elastc materal model. For smplcty and n order to prodce comparable reslts wth prevos works and verfyng the crrent approach, the lnear elastc materal model has been sed here. he optmzed reslts can be sed n cases where the lnear elastc materal behavor can be assmed. ven n elasto-plastc meda the proposng optmzaton technqe can provde the startng gess desgn. he homogenety assmpton s vald n cases of ntact rocks and hghly weathered rocks. In case of massve rocks wth few major dscontntes, the overall behavor of the rock mass s hghly nflenced by the dscontntes. In ndergrond excavaton t rarely happens that the grond materal can adeqately resst the conseqences of stress relef. he se of spports s ths sally navodable. A common technqe to spport the ndergrond excavatons s throgh rock bolts and grotng. Usng rock bolt systems the rock mass can effectvely be renforced only where t s not strong enogh. o smplfy the nmercal model, homogenzed propertes can be sed to model the behavor of the renforced parts of rock mass (Bernad 995). In ths paper, n lne wth prevos pblcatons n ths context (L et al. 2008, Yn et al. 2000, Yn & Yang 2000a,b), a lnear elastc behavor has been assmed for renforced rock. Frther dscsson on valdty of ths type of analyss can be fond n (Yn et al. 2000). he modls of elastcty of host rock and renforced rock are represented by and respectvely. As mentoned before the vod areas are consdered to be made of a very weak materal. he modls of elastcty of ths weak materal s represented by and t s assmed that = It s also assmed that all these materals have same Posson's rato eqal to BJCI FUNCIN AND PBLM SAMN Consder a smple desgn case depcted n Fgre. In ths fgre, Γ represents the bondary of the openng. he mnmm dmensons, shown n the fgre, can be de to some desgn restrctons. he placement, orentaton and the length of rock bolts has been depcted by sold lne segments n ths fgre. he dark shaded area Ω wth the oter bondary of Ω and nner bondary of Γ s the renforced area of the desgn. Havng fond ths renforced area one can choose the proper locaton and length of the renforcng bars and vce versa. he smltaneos shape and renforcement optmzaton can be vewed as fndng the optmal Ω when both ts nner and oter bondares Γ and Ω are changng. Fgre. A smple desgn case. In ths paper the mean complance has been selected as objectve fncton whch s the most common objectve fncton sed n topology optmzaton problems. Consderng a volme constrant on renforcement materal and restrctng the sze of the openng, the problem of concern can be expressed as x, x2,, xn s.t. mn c = f, = () where x, x, 2, xn are desgn varables, c s the mean complance, f s the nodal force vector, stands for nodal dsplacement vector, and,, and are renforcement and vod volme and ther correspondng lmts respectvely. Mnmzng complance wll be eqvalent to maxmzng the stffness of strctre. he problem ths wll be fndng the stffest desgn wth prescrbed openng sze and predefned pper lmt for the volme of renforcement materal. For problems wth constant load (where load s not a fncton of desgn varables) senstvtes of mean complance can be easly calclated va adjont

3 method (Bendsøe & Sgmnd 2004) or drect dfferentaton (anskanen 2002) as c = x x (2) where stands for stffness matrx and x s the - th desgn varable. he stffness matrx n element level can be related to desgn varables by ( x ) ( x ) = (3) where s the elastcty modls of the element whch s a fncton of the element's desgn varable. s the stffness matrx of element as f t was made of orgnal rock. In order to mantan the topology of the hole for shape optmzaton, the bondary of the hole shold be determned and only the bondary elements shold be allowed to change. In ths paper t s assmed that there s a shotcrete lnng arond the openng wth materal propertes smlar to that of renforced rock. In ths manner, n the shape optmzaton of the openng, the materal can only swtch between vod and renforced rock. In the renforcement optmzaton, on the other hand the two materal phases are orgnal rock and renforced rock. 4 MAIAL INPLAIN SCHM For a general two-phase materal case, the nterpolated modls of elastcty can be defned as ( ) ( x) = + x (4) 2 where and 2 are Yong's modl of the two materals. he vale of x = 0 reslts n = and ths represents the frst materal. Smlarly the second materal can be represented by x =. Usng qaton 4 and 3 n qaton 2, the latter can be rewrtten as c 2 = x (5) where ndcates local nodal dsplacements at the element level for the -th element. he change n objectve fncton de to a change n an element can then be approxmated as c c = x (6) x If the materal of an element changes, one can calclate the approxmate change n objectve fncton by sbstttng correspondng x vale n qaton Shape optmzaton of the tnnel In shape optmzaton of the openng the two phases of the materal are vod and renforced rock so qatons 4 and 6 can be rewrtten as = + ( ) ( x) x (7) and c = x (8) respectvely. Note that n qaton 7 the vod and the renforced rock phases are represented by vales of 0 and for x respectvely. Now for an element changng from vod to renforced rock ( x = x x = + ) one can wrte c =, (9) wth standng for the set of the nmbers of crrently vod elements. Note that n qaton 9 the -th element s vod so =. Smlarly for an element changng from renforced rock to vod c =, (0) where stands for the set of the nmbers of crrently renforced elements. Based on qatons 9 and 0 we defne the followng senstvty nmbers for shape optmzaton of the openng ( ), α S = () ( ), Here the senstvty nmber s defned as the change n complance mltpled by the sqare of the Yong's modls. hs defnton prevents nfnte senstvty nmbers for the case of = 0 (hard kll). Consderng ths defnton, the renforced elements wth the lowest senstvty nmbers are the least effcent elements and shold be change to vods whle the vod elements wth the hghest senstvty nmbers are the most effcent ones and shold be swtched to renforced rock. 4.2 enforcement optmzaton In topology optmzaton of renforcements, the two materal phases are orgnal and renforced rock. he senstvty nmbers can ths be easly obtaned by replacng by nto qaton : ( ), α = (2) ( ),

4 Here represents the set of the nmbers of orgnal rock elements. Implementng qaton 2, the renforced elements wth the lowest senstvty nmbers are the least effcent elements and shold be changed to orgnal rock. n the other hand the rock elements wth the hghest senstvty nmbers are the most effcent ones and shold be renforced. Senstvty nmbers defned n qatons and 2 only dffer n constant coeffcents and both can be obtaned by mltplyng the stran energy of the elements by the calclated coeffcents. hs the comptatonal tme to solve these two problems s nearly same as that of a sngle optmzaton problem. 5 FILING SNSIIIIS It s known that some topology optmzaton methods, ncldng the BS method, are prone to nmercal nstabltes sch as the formaton of checkerboard patterns and mesh dependency (Sgmnd & Petersson 998). ne of the smplest approaches known to be capable of overcomng these two nstabltes s flterng the senstvtes (Sgmnd & Petersson 998, L et al. 200, Hang & Xe 2007). In flterng technqe a new senstvty nmber s calclated based on the senstvty nmbers of the element tself and ts srrondng elements. he followng flterng scheme has been sed n ths paper to calclate the fltered senstvty nmbers n j= α j H j ˆ α = (3) n H j= j where αˆ s the fltered senstvty nmber of the -th element, n s the nmber of elements and H j { 0 r d } = max, (4) f j Here r f s the flterng rads and d j s the dstance between the centers of the -th and the j-th elements. he qaton 3 s actally a weghted average whch reslts n greater vales n elements near the areas of hgh senstvty and vce versa. Usng ths flterng scheme wll reslt n stable reslts and smoother topologes. 6 BS PCDU he BS procedre teratvely swtches elements between dfferent materals (and vods) based on ther senstvty nmbers. If n the ntal desgn the materals' volmes are not wthn the constrants n qaton, then these volmes shold be adjsted gradally to meet the constrants. hs can be acheved by controllng the nmber of swtches between dfferent materals. Hang & Xe (2007) proposed an algorthm for gradally adjstng the materals' volmes. However, f one starts from a feasble desgn there s no need to change the volme. In ths case the nmber of addng elements shold be eqal to the nmber of removng elements n order to keep the volme constant. In the examples solved here a feasble ntal desgn s sed. hs redces the complexty of the algorthm and eases the verfcaton of the reslts. At every teraton a nmber of elements wll swtch between renforcements and vods to optmze the shape of the openng based on qaton. hen some other swtches wll be appled between normal and renforced rocks to optmze the topology of renforcement's dstrbton based on qaton 2. By restrctng the program to swtch only a few elements each tme, one can prevent sdden changes to the desgn. he maxmm nmber of swtches between dfferent elements at each teraton s referred to as move lmt. Usng larger move lmts one can obtan faster convergence bt may lose some optmm ponts. Wth a small move lmt, the evolton of the objectve fncton shold show a relatvely monotonc trend wth a steep descent at the ntal teratons reachng a flat lne at the end ndcatng convergency. Gettng sch evolton trend one can ensre that the optmzaton procedre s workng well. o keep p wth the shotcrete lnng the elements on the bondary of the hole shold be changed to shotcrete elements after each pdate n the hole's shape. herefore the nmber of shotcrete elements mght change drng optmzaton whle the total volme of the renforced rock and the shotcrete lnng s constraned. In order to satsfy ths volme constrant, n renforcement optmzaton the nmber of renforcng and weakenng elements shold be adjsted. 7 XAMPLS A smple example has been consdered to verfy the proposed BS algorthm. he relatve vales of modl of elastcty of renforced rock, orgnal rock, and vod elements have been consdered as 0000:3000:3 respectvely. It s assmed that the tnnel s long and straght enogh to valdate plane stran assmpton. he oter bondares of the desgn doman have been consdered as non-desgnable rock elements n order to prevent renforcng of far felds. Becase the dscretzed doman s very large n compare to the sze of the openng, changes n the openng's shape wll not have a consderable effect on the overall complance. he objectve fncton s ths lmted to the complance of desgnable doman only. he flterng rads s consdered eqal to twce of the elements' sze. he move lmt has been lmted to fve elements. It s also assmed that the

5 tnnel shold have a flat floor. o flfll ths reqrement a layer of non-desgnable renforced rock has been consdered at the bottom of the openng. he ntal gess desgn together wth nondesgnable elements has been depcted n Fgre 2. he mnmm sze of the openng s 2.4m.6m. hs area s restrcted to vod elements by settng a rectanglar area of non-desgnable vods. he sze of the openng s 7.92m 2. he pper lmt for the volme of the renforcement materal s chosen eqal to 4.8m 2. he nfnte doman has been replaced by a large fnte doman of sze 20m 20m srrondng the openng. Becase of symmetry only half of the desgn doman has been consdered n fnte element analyss wth proper symmetry constrants. A typcal 2D mesh consstng of eqally szed qadrlateral 4-node elements has been sed to dscretze the half model. Fgre 2. An ntal gess desgn llstratng the desgn doman, non-desgnable elements, loadng, and restrants. he tnnel s consdered nder baxal stresses. o model the stress condtons nform dstrbted loads wth consstent magntdes have been appled on top, rght and left sdes and the bottom s restraned aganst vertcal dsplacement (Fg. 2). hree cases wth dfferent vales of horzontal to vertcal stress rato ( λ ) has been consdered. Fgre 3 shows the fnal topologes for λ = 0. 4, λ = 0. 7, and λ =. 2. It can be seen that the fnal shape of the openng and the fnal topology of renforcements change dramatcally wth the appled load rato. he aspect rato of the optmm openng shapes show a correlaton wth the appled load ratos whch s also reported n (en et al. 2005) and (Ghabrae et al. 2007). he evoltons of the objectve fnctons have been depcted n Fgre 4. In all cases the objectve fncton changes almost monotoncally and smoothly. he ntal and the fnal vales of the objectve fncton are reported n able. able. he ntal and fnal objectve fncton's vales for the three load cases. Case Intal vale Fnal vale Dfference λ= % λ= % λ= % 8 CNCLUSIN he topology optmzaton of renforcement arond an ndergrond openng n rock mass and shape optmzaton of the openng tself have been solved smltaneosly. Among dfferent topology optmzaton methods the BS method has been chosen de to ts clear topology reslts and ts fast convergence. he bnary natre of the BS method makes t stable for solvng shape optmzaton problems. However nlke the reglar BS, n ths paper a soft kll approach has been followed and a weak materal has been sed to model vod elements. Mean complance has been consdered as objectve fncton for the optmzaton procedres together wth constrants on maxmm volme of renforcements and on the sze of the openng. he problem then redced to two two-phase materal dstrbton problems. he frst problem represents the shape optmzaton of the openng where the materal s changng between renforced rock and vod. he second one relates to the renforcement optmzaton where the two materal phases are orgnal and renforced rock. he senstvtes of the objectve fncton wth respect to the desgn varables have been calclated for these problems. wo dfferent senstvty nmbers have then been defned based on the calclated senstvtes. It has been shown that the two senstvty nmbers only dffer n some constant coeffcents. Hence the two optmzaton problems can be solved sng nearly same comptatonal effort as reqred by a sngle problem. A shotcrete lnng has been assmed arond the openng wth mechancal propertes smlar to that of renforced rock. A flterng scheme has been sed to prevent nmercal nstabltes sch as checkerboard patterns. he flterng approach also smoothes ntermateral bondares, resltng n a topology free of jagged edges. he proposed approach has been verfed by solvng a smple example. he evolton of the objectve fncton shows a smooth, relatvely monotonc and convergng crve. he proposed method can be sed to mprove the desgn of ndergrond excavatons n lnear elastc and homogeneos rocks. It can also be sed to provde ntal desgns for excavatons n elasto-plastc meda.

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