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Transcription

1 Dscussons and Closures Closure to Lmt-Equlbrum Metod for enforced Sloe Stablty and Otmum Desgn of ntslde Mcrole Parameters by Deng Dong-ng, L Lang, and Zao Lan-eng DOI:.6/(SCE)GM Deng Dong-ng ; L Lang 2 ; and Zao Lan-eng ssstant Professor, Scool of Cvl Engneerng, Central Sout Unv., Cangsa 47, Cna (corresondng autor). E-mal: dengd82@26.com 2 Professor, Scool of Cvl Engneerng, Central Sout Unv., Cangsa 47, Cna. E-mal: llang_csu@26.com Professor, Scool of Cvl Engneerng, Central Sout Unv., Cangsa 47, Cna. E-mal: zl876@6.com In ter dscusson, te dscussers ave rased two man roblems. Problem : Mecancal Mode of a Mcrole n te Lmt Equlbrum State of a Sloe In te stablty analyss of a sloe renforced wt te antslde mcrole, erformed usng te lmt equlbrum metod (LEM) of te orgnal aer, two assumtons are made for te mcrole: () te mcrole s not easly broken because of ts suffcent nerent strengt, and (2) te art of te mcrole tat s embedded below te sl surface lays a full role n renforcng te sloe. ccordng to tese assumtons, te lmt workng state of te antslde mcrole also wll be reaced wen te sloe s n te lmt equlbrum (LE) state. s te art of te mcrole embedded below te sl surface works, te maxmum renforcement force wll be roduced by te art of te mcrole tat s above te sl surface to act on te sloe; terefore, te otmum renforcement effect of te mcrole on te sloe s aceved. However, wen te FLCD software (Itasca Consultng Grou 22) s used to analyze te stablty of te sloe renforced wt te antslde mcrole, te numercal results gve te mnmum factor of safety (FOS) of te sloe wen te mcrole s n te actual workng state. In te actual workng state of te mcrole, te art of te mcrole tat s embedded below te sl surface may not lay a full role n renforcng te sloe. Terefore, comared wt te renforcement force of te mcrole n te lmt workng state, te renforcement force roduced by te mcrole n te actual workng state s smaller. ccordngly, te LEM of te orgnal aer as larger results tan te numercal smulaton metod (refer to te dscusson). In fact, te lmt workng state of te mcrole s not necessarly reaced wen te sloe s n te stable state. Neverteless, wen te sloe s n te unstable state, te lmt workng state of te mcrole s reaced to revent te sloe sldng so tat te maxmum renforcement force s roduced by te mcrole. Hence, t s a smle and effectve ractce to assume tat te art of te mcrole tat s embedded below te sl surface would work wt a full role durng te stablty analyss of te sloe renforced wt te antslde mcrole. To verfy te feasblty of te earler assumton, te followng study nvestgates te stablty of te sloe renforced wt te antslde mcrole under dfferent workng states of te mcrole. ccordng to te numercal smulaton result of te dscusson, one art of te mcrole, wc s embedded below te sl surface, may not work n renforcng te sloe wen te mcrole s n te actual workng state. Ts s sown n Fg., n wc te corresondng art as te lengt z H, were z s te roortonalty coeffcent. Here, te mecancal mode of te mcrole s consstent wt tat used n te LEM of te orgnal aer. Furter, te mcrole s dvded nto tree oter arts,.e., art ef wt te effectve mcrole sde ressure P, artfg wt eart ressure on te back of te mcrole, and art gt wt eart ressure on te front of te mcrole. Tus, te force equlbrum of all forces on te mcrole n te orzontal drecton can be exressed as 2 λk g 2 þ 2 k g 2 ð z ÞH z z ¼ 2 k g þ ð z ÞH z ð z ÞH z () were l = oter roortonalty coeffcent; k = eart ressure coeffcent; = egt of te mcrole located n te sldng body; and z = actng det of te eart ressure on te front of te mcrole. Solvng Eq. () for te roortonalty coeffcent l yelds λ ¼ 2 ½ ð z ÞH zš 2 ½ð z ÞH Š (2) Te moment equlbrum of all forces on te mcrole around ont f can be exressed as 6 λk g þ 6 k g þ 2½ð z ÞH zš ½ð z ÞH z Š 2 ¼ 6 k g ½ð z ÞH Š ½2 ð z ÞH zšz ½ ð z ÞH 2zŠz 2 g () Substtutng Eq. (2) ntoeq.() and smlfyng, a cubc equaton of about [( z )H z] s obtaned as ð z ÞH z ð z ÞH z 2 2 ð z ÞH þ 2 ð z ÞH 2 ¼ (4) Solvng Eq. (4) for te arameter z yelds z ¼ ð z ÞH þ b qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff a þ ffffffffffffffffffffffffffffff qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 2 þ q ffffffffffffffffffffffffffffff 2 þ q were a =;b = ; c =;d¼ =2 ð z ÞH þ =2½ð z ÞH Š 2 ; ¼ 27a 2 d 9abc þ 2b =4a ; and q ¼ ac b 2 = 9a 2. Substtutng te arameter z from Eq. () nto Eq. (2), te roortonalty coeffcent l can be solved, and te effectve mcrole sde force s ten calculated as () P ¼ 2 λk g 2 (6) SCE 786- Int. J. Geomec. Int. J. Geomec., 28, 8(): 786

2 Tus, substtutng te mcrole sde force P from Eq. (6) nto te formula of sloe FOS from te orgnal aer, te stablty of te sloe renforced wt te antslde mcrole can be solved for dfferent workng states of te mcrole. Usng te FLCD examles gven by te dscussers, te results of dfferent metods are comared n te followng analyss. Sloe Examle Te sloe egt s H =.7 m; te sloe angle s b = ; and te sol arameters are gven by te natural unt wegt g = 9.6 kn/m, coeson c = 2.94 kpa, and te nternal frcton angle w =. Sngle-row antslde mcroles are arranged on te sloe, and te forces actng on te mcroles are assumed to be unt dstrbuton ressures n te longtudnal secton of te sldng body. Te condtons of unt dstrbuton ressures are also aled on te followng sloe examles. In te mcrole, x s ts orzontal dstance from te sloe toe, H s ts lengt, k s te eart ressure coeffcent, and z s te roortonalty coeffcent. Here, k =.6,x P =.7 m, and z reflects te rato of te lengt of te nonworkng art,.e., te lengt of art to to H n te mcrole (Fg. ). In ts examle, te dscussers comared te results of te LEM roosed n te orgnal aer wt tat obtaned usng FLCD. However, te LE analyss as used te effectve mcrole sde ressure P, gven by te revous study (uslo et al. 2), to ensure tat te mnmum sloe FOS s.. Terefore, te results of te LEM (orgnal aer) are obtaned wtout consderng te sze of te mcrole. Here, accordng to te numercal model n FLCD, t s assumed tat H,.e., te lengt of te mcrole, s.2,., and.8h, resectvely. Wt dfferent values of te coeffcent z, te stablty of te sloe renforced wt te antslde mcrole s analyzed, and te results are sown n Fg. 2. In Fg. 2, te curve of te mnmum sloe FOS versus z and te curve of te effectve mcrole sde ressure P versus z are drawn usng te LEM of te orgnal aer. Fg. 2 sows tat te numercal result gves te mnmum FOS of te sloe for te mcrole n te actual workng state, and ts s consstent wt te revous analyss. In te actual workng state, te art of te mcrole tat s embedded below te sl surface does not work well; n oter words, te coeffcent z s larger tan zero. Tus, te renforcement force of te mcrole does not reac te maxmum value n te numercal smulaton analyss. Furtermore, accordng to te curve of te mnmum sloe FOS versus z,te results of te LEM are closer to tose of te numercal smulaton metod for te mcrole n te same workng state,.e., te coeffcent z as te same value n te two metods. Terefore, te feasblty of te LEM roosed n te orgnal aer s verfed. In Fg. 2, accordng to te numercal smulaton results, te sloes renforced by te antslde mcrole wt dfferent lengts ave essentally te same stablty. Hence, te otental renforcement caablty of te mcrole wt an ncrease n ts lengt wll not be reflected n te numercal smulaton metod. In addton, te urose of te otmzaton desgn of te antslde mcrole s to obtan te otmal arrangement oston or lengt of te mcrole for te sloe wt certan safety requrements. To ensure tat te mnmum sloe FOS s., te antslde mcrole wt a lengt of.2h s referred (Fg. 2). Ten, for te mcrole wt a lengt of.2h, te coeffcent z s close to zero, ndcatng tat te actual workng state of te mcrole s close to ts lmt state. Terefore, t s stll relable to solve te stablty of te sloe renforced wt an antslde mcrole usng te LEM of te orgnal aer. Fs H z y z P x P λk γ P k γ[( ζ)h z] k γ( ζ)h f t H =.2H x e o Curve of Fs vs. ζ Curve of P vs. ζ esult from FLC D H =.H H =.8H ζ Fg. 2. Calculated results n Sloe Examle Problem 2: ssumton of te Sear Stress on aslsurface e f g t ζh o k γ In te LEM of te orgnal aer, tree ndeendent varables,.e., l, l 2, and l, are ncluded n te assumton of te stress on a sl surface. However, accordng to te formula of sear stress on a sl surface, te dscussers concluded tat l s related to l and l 2. Terefore, tere sould be two nstead of tree ndeendent varables n te LEM of te orgnal aer. Here, te reason tat tree / H P H g k γ[( ζ)h z] ζh Fg.. Calculaton mode for stablty analyss of sloe renforced wt antslde mcroles: desgn arameters of te antslde mcrole; falure mode of te antslde mcrole P (kn/m) SCE Int. J. Geomec. Int. J. Geomec., 28, 8(): 786

3 ndeendent varables are adoted n te assumtons of stresses on a sl surface s tat te tree varables can be solved on te bass of tree mecancal equlbrum condtons n te two-dmensonal sldng body. ddtonally, accordng to te formula of sear stress on a sl surface, te calculaton of sear stress conssts of two grous of varables,.e., ðc u tan f Þ and fs tan w þ s tan w ðp j =dxþ= ½ þ ðs Þ 2 Šg. For te same calculaton ont on te sl surface, te contrbuton of tese two grous on te sear stress s roortonal to eac oter. However, for dfferent calculaton onts, te contrbuton of tese two grous on te sear stress s not te same. Te ndeendent varables l 2 and l are used to average te contrbuton of tese two grous of varables on te sear stress for eac calculaton ont along te entre sl surface; terefore, l 2 and l are te yotess varables aled to calculate te sear stress on te entre sl surface. Tus, t s feasble for l to be an ndeendent varable. Moreover, t s recsely because of te tree ndeendent varables used n te assumton of stress on te sl surface tat te derved LE equaton s a lnear soluton formula. Furtermore, te calculaton rocess n te LEM of te orgnal aer s smle n solvng for te varables and te sloe FOS, and te obtaned results are unque. To verfy te feasblty of te assumtons adoted n te LEM of te orgnal aer, te formulas for te sloe FOS and stresses on te sl surface n te tradtonal LEM, ncludng te Sweds metod (Fellenus 96), smlfed so metod (so 9), and Morgenstern-Prce (M-P) metod (Morgenstern and Prce 96), are gven n te endx. For dervng te formulas n te endx, te sldng body of te sloe s dvded nto n vertcal slces. Ten, te forces actng on slce are analyzed to establs te mecancal equlbrum condton n eac slce. For slce, b s ts wdt, a s te orzontal nclnaton angle of te tangent to te sl surface, and l s te lengt of te sl surface. Moreover, te forces actng on slce ave te gravty W ; te normal force N and normal stress s on te sl surface; te sear force S and sear stress t on te sl surface; te normal nterslce forces E and E on te two sdes of te vertcal slce, resectvely; and te effectve mcrole sde force P. In addton, F s s te sloe FOS and s te radus of te crcular sl surface. In te M-P metod, f s te correct functon of nterslce forces, gven by f ¼ sn ðx x Þ= ðx x Þ. Here, x s te x-axs coordnate of te mdont of te sl surface n slce,andx and x are te x-axs coordnates of onts and,.e., te lower and uer sldng onts of te sl surface, resectvely. Te followng two sloe examles are lsted and te dfferences n stresses on te sl surface obtaned by te tradtonal LEM and te LEM roosed n te orgnal aer are comared. Moreover, te relatons among te varables l, l 2, and l s also analyzed. Sloe Examle 2 Te sloe egt s H = m; te sloe angle s b = 4 ; and te sol arameters are gven by natural unt wegt g = 7.8 kn/m, coeson c = kpa, and nternal frcton angle w =. Te sloe stablty s analyzed wt te crcular sl surface on te secfc oston, and te curve of stress on te sl surface s drawn. On establsng te x,y-axs coordnate system wt te sloe toe as te orgn, te crcular sl surface on te secfc oston as te followng arameters: te coordnates of ont,.e., te lower sldng ont of te sl surface, are ( m, m); te coordnates of ont,.e., te uer sldng ont of sl surface, are ( m, m); and te radus of te crcular sl surface s = 2 m. Tereafter, te results σ (kpa) τ (kpa) Sweds metod so metod M-P metod Proosed metod of stress on te sl surface are obtaned by te tradtonal LEM and te LEM n te orgnal aer (Fg. ). In Fg., te sloe FOS calculated by te LEM of te orgnal aer, calculated by te Sweds metod, calculated by te smlfed so metod, and calculated by te M-P metod are.4,.294,.44, and.4, resectvely. In Sloe Examle 2, for te sloe renforced wt te antslde mcrole, te mcrole as te arameters x, H, k,andz, were x =m,h =.6H, k =.6, and z =.. Smlarly, te crcular sl surface on te secfc oston used n Sloe Examle 2 s also adoted to analyze te sloe stablty. Tus, te results of stress on te sl surface are obtaned by te tradtonal LEM and te LEM roosed n te orgnal aer (Fg. 4). In Fg. 4, te sloe FOS calculated by te LEM of te orgnal aer, calculated by te Sweds metod, calculated by te smlfed so metod, and calculated by te M-P metod are.448,.97,.4, and.447, resectvely. Fgs. and 4 sow tat te results of te LEM metod of te orgnal aer are closer to tose of te Sweds metod, smlfed so metod, and M-P metod n terms of te stresses on te sl surface and sloe FOS. Ts ndcates te feasblty of assumtons of stresses on te sl surface n te LEM roosed n te orgnal aer. Moreover, t s seen from te endx tat te solutons of Sweds metod so metod. M-P metod Proosed metod Fg.. Comarson of stresses on te sl surface for unrenforced sloe n Examle 2: normal stress on te sl surface; sear stress on te sl surface SCE 786- Int. J. Geomec. Int. J. Geomec., 28, 8(): 786

4 σ (kpa) τ (kpa) stresses on te sl surface are obtaned only wen te sloe FOS as been solved n te tradtonal LEM. In contrast to te tradtonal LEM, te results of stresses on te sl surface can be obtaned before solvng for te sloe FOS n te LEM roosed n te orgnal aer; terefore, te teratve calculaton on te sloe FOS s avoded n te LEM of te orgnal aer. Sloe Examle Te sloe egt s H = m; te sloe angle s b = 4 ; and te sol arameters are gven by natural unt wegt g = 7.8 kn/m, coeson c = kpa, and nternal frcton angle w =. Wt te crcular sl surface, te sloe stablty s also analyzed usng te LEM of te orgnal aer. Moreover, two cases of te sloe are consdered,.e., te unrenforced sloe and te renforced sloe wt antslde mcrole. Wen te sloe s renforced wt te antslde mcrole, te mcrole as te followng arameters: x = m,h =.6H, k =.6, and z =.. For te crcular sl surface, te lower and uer sldng onts of te sl surface,.e., and, are fxed on te secfc oston. On establsng te x,y-axs coordnate system wt te sloe toe as te orgn, te coordnates of 7. x Sweds metod so metod M-P metod Proosed metod x. M-P metod Proosed metod 7. H 2. H Sweds metod so metod Fg. 4. Comarson of stresses on te sl surface for te renforced sloe wt antslde mcrole n Examle 2: normal stress on te sl surface; sear stress on te sl surface λλ2 (or λ) λλ2 (or λ) onts and are ( m, m) and ( m, m), resectvely. Tus, wt dfferent rad of te crcular sl surface, te curve of varable l l 2 versus and te curve of varable l versus are drawn n Fg.. In Fg., tevarablesl, l 2,andl are solved accordng to te global mecancal equlbrum condtons of te sldng body. Hence, te results of l l 2,andl are not necessarly equal to eac oter. Wen te relatons l l 2 = l s satsfed n te global mecancal equlbrum condtons of te sldng body, te results of l l 2 and l are equal to eac oter, suc as at ont s n Fg.. ddtonally, te tradtonal LEM and te LEM roosed n te orgnal aer ave dfferent assumtons to derve te formulas for calculatng te stresses on te sl surface. However, te mecancal equlbrum condtons of te sldng body are te bass for tese derved formulas n te LEM. Te results n Fgs. and 4 sow tat tese LE solutons are closer to eac oter. Terefore, n te LE stablty analyss of te sloe, satsfyng te mecancal equlbrum condtons of te sldng body s an mortant rerequste for obtanng reasonable results. Conclusons s λ λ λ λ λ λ (m) In te resonse to te dscusson, two man roblems,.e., te mecancal mode of a mcrole n te LE state of a sloe and 2 22 (m) 24 x H 28 Fg.. Calculated results n Sloe Examle : unrenforced sloe; renforced sloe wt antslde mcrole SCE Int. J. Geomec. Int. J. Geomec., 28, 8(): 786

5 te assumton of te sear stress on a sl surface, are studed. To verfy te feasblty of te LEM of te orgnal aer, te results of bot te FLCD software and tradtonal LEMs are used to comare te dfferences wt tat of te LEM of te orgnal aer. Tus, some conclusons can be obtaned as follows:. It s smle and effectve to analyze te stablty of sloe renforced by te antslde mcrole wt te assumton tat te mcrole s n te lmt workng state, and te otental renforcement caablty of te mcrole wt an ncrease of ts lengt wll be reflected under ts assumton. 2. It s ractcal for l, l 2, and l as te ndeendent varables to assume te normal and sear stress on te sl surface n te LEM of te orgnal aer, and te smle lnear soluton formula can be derved to quckly obtan te unque sloe FOS by usng te tree ndeendent varables. endx. Formulas of Sloe FOS and Stresses on Sl Surface n te Tradtonal LEM For te Sweds metod, te formulas of te sloe FOS (F s ), normal stress (s ) on te sl surface, and sear stress (t ) on te sl surface are, resectvely, gven as ¼ ¼ ½cl þ ðw cos a þ P sn a Þtan f Š ðw sn a P cos a Þþ (7) P s ¼ W cos a þ P sn a l (8) t ¼ c þ W ð cos a þ P sn a Þ=l tan f (9) F s For te Smlfed so metod, te formulas of te sloe FOS (F s ), normal stress (s ) on te sl surface, and sear stress (t ) on te sl surface are, resectvely, gven as were ð t ¼ c=f s ðcl cos a þ W tan f Þ ðw sn a P cos a Þþ () P m u ¼ ¼ s ¼ W =l ðc sn a Þ=F s () m u Þ cos a þ ðw =l Þ ðtan f =F s Þ (2) m u m u ¼ cos a þ sn a tan f F s () For te M-P metod, te formulas of te sloe FOS (F s ), normal stress (s ) on te sl surface, and sear stress (t ) on te sl surface are, resectvely, gven as were ¼ ¼ n Q W j!þ n j¼! (4) Q T n W j þ T n j¼ s ¼fW cos a þ ½ðλf E λf E ÞŠcos a þ ½ðE E Þþ P Šsn a g=l () t ¼fW sn a þ ½ðλf E λf E ÞŠsn a ½ðE E Þþ P Šcos a g=l (6) E ¼ ðw E U F s T þ Þ=U (7) U ¼ ðsn a λf cos a Þtan f þ ðcos a þ λf sn a ÞF s (8) U ¼ ðsn a λf cos a Þtan f þ ðcos a þ λf sn a ÞF s (9) W ¼ ½ðsna λf cosa Þtan f þðcosa þλf sna ÞF s Š=U (2) λ ¼ ¼ b ðe þ E Þtan a 2 4 ð x x ÞP tan a ¼ ½b ðf E þ f E ÞŠ (2) ¼ cl þ ðw cos a þ P sn a Þtan f (22) T ¼ W sn a P cos a (2) P ¼ 2 λk g 2 for x b =2 < x x þ b =2 and P ¼ cknowledgments ðfor oter casesþ (24) Ts roject was funded by te Natonal Natural Scence Foundaton of Cna (No. 684), te Cna Postdoctoral Scence Foundaton (No. 2M872), and te Guzou Provncal Deartment of Transortaton (No ). eferences uslo, E., Conte, E., and Dente, G. (2). Stablty analyss of sloes renforced wt les. Comut. Geotec., 28(8), 9 6. so,. W. (9). Te use of te sl crcle n te stablty analyss of eart sloes. Geotecnque, (), 7 7. Fellenus, W. (96). Calculaton of te stablty of eart dams. Proc., Trans. of te 2nd Congress on Large Dams, Internatonal Commsson on Large Dams of te World Power Conference, Wasngton, DC, FLCD. [Comuter software]. Itasca Consultng Grou, Mnneaols. Itasca Consultng Grou. (22). FLCD. manual, Itasca Consultng Grou, Mnneaols. Morgenstern, N.., and Prce, V. E. (96). Te analyss of te stablty of general sl surfaces. Geotecnque, (), SCE 786- Int. J. Geomec. Int. J. Geomec., 28, 8(): 786