UPPER BOUND LIMIT ANALYSIS OF SOIL SLOPE STABILITY BASED ON RPIM MESHLESS METHOD

Transcription

1 he 5th Cross-strat Coferece o Structural ad Geotechcal Egeerg (SGE-5) Hog Kog, Cha, July 011 UPPER BOUND LM ANALYSS OF SOL SLOPE SABLY BASED ON RPM MESHLESS MEHOD F.. Lu 1, J. D. Zhao 1, Y. F. Fa, ad J. H. Y 3 1 Departmet of Cvl ad Evrometal Egeerg, he Hog Kog Uversty of Scece ad echology, Hog Kog, Cha. School of Natural Sceces ad Humates, Harb sttute of echology She Zhe Graduate School, She Zhe, Cha. Emal: yhfa@ht.edu.c 3 Departmet of Cvl ad Structural Egeerg, he Hog Kog Polytechc Uversty, Hog Kog, Cha. ABSRAC Lmt aalyss s wdely used to evaluate the stablty of structures cvl egeerg. comparso wth elasto-plastc aalyss, lmt aalyss ca avod the complcated computato of cremetal aalyss. A soluto procedure based o radal pot terpolato method for upper boud lmt aalyss of structures s preseted. For evaluatg the tegratos of the exteral work rate ad teral power dsspato rate, a ew meshless tegrato techque based o Cartesa rasformato Method (CM) was used to trasform the doma tegral to a boudary tegral ad a 1D tegral. Fally, the olear optmzato problem derved from the upper boud lmt aalyss ca be solved based o dstgushg rgd/plastc zoes. Ad some examples of stablty aalyss show that ths approach s a vald ad smple techque. KEYWORDS Slope stablty; Upper boud lmt aalyss; Radal pot terpolato method; olear programmg; Cartesa rasformato Method (CM) NRODUCON Lmt aalyss s a powerful method for stablty aalyss ad lmt bearg capacty of egeerg structures. geotechcal egeerg, upper boud lmt aalyss s wdely used to aalyze the slope stablty. Drucker (195) frstly preseted lmt aalyss based o plastc lmt theorem, ad the Che (1975) troduced lmt aalyss to the geotechcal egeerg for aalyzg the bearg capacty, earth pressure o retag wall ad slope stablty. t takes advatage of the lower ad upper theorems of classcal plastcty to bracket the true soluto from a lower boud to a upper boud. However, t s dffcult to obta aalytcal soluto for practcal egeerg, ad umercal approaches are ofte requred for lmt aalyss. the past three decades, may studes have bee devoted to developg umercal methods of lmt aalyss. May researchers (Lysmer 1970; Aderhegge ad Kopfel 197; Bottero et al. 1980; Sloa 1988, 1989; Sloa ad Kleema 1995) costructed umercal lmt aalyss based o fte elemet method ad lear programmg theory, where the geeral yeld crtero ofte was learzed to a covex polyhedro, ad the olear equaltes were approxmated by a set of lear equaltes. Especally for the slope stablty aalyss, followg the related work by Sloa ad Kleema (1995), some researchers (Yu et al. 1998; Km et al. 1999; Km et al. 00, etc) have appled the lower ad upper boud approach to evaluate the slope stablty. O the other had, followg the work of Zoua et al. (1993), Lyam ad Sloa (00) proposed a olear umercal method to perform upper ad lower boud lmt aalyss based o lear fte elemets ad olear programmg. he results showed that ther approach s vastly superor to a wdely used lear programmg formulato, especally for large scale applcatos. However, ths approach has a potetal dffculty applyg these formulatos s that specal stress or dsplacemet fte elemets eed to be used. herefore, a alteratve olear techque whch amed the drect teratve algorthm s used to perform lmt aalyss of o-frctoal materals (Zhag et al., 1991; Lu et al., 1995; Capso ad Corrad, 1997). Followg these deas, L ad Yu (006) exteded the drect teratve algorthm to calculate plastc collapse loads of D ad 3D structures obeyg the ellpsod yeld crtero. these approaches, upper boud lmt aalyss of structures s formulated as a olear optmzato problem wth a sgle equalty costrat, ad a techque based o dstgushg rgd/plastc zoes was adopted to solve ths specal olear costraed optmzato problem

2 Recetly, as the developmet of fte elemet method, a so-called meshless method has attracted more atteto the feld of umercal method. Recetly, Che et al. (008) ad Le et al. (009, 010) costructed lower ad upper formulato of lmt aalyss based elemet-free Galerk (EFG) method whch frst proposed by Belytschko et al. (1994). Although the EFG method has bee successfully appled to the lower ad upper boud approaches, two ssues are stll ot well studed: 1) dffcultes the eforcemet of essetal boudary codtos. hs s because ts shape fucto whch calculated based o the movg least square method (MLS) s lack of Kroecher delta fucto property,.e., where s the Kroecker delta fucto; ) complexty umercal algorthms for calculatg shape fucto ad ts dervatves. For two ssues, the oe of approach s so-called radal pot terpolato method (RPM) proposed by Wag ad Lu (00). he RPM shape fuctos have the Kroecker delta fucto property ad parttos of uty. herefore, the essetal boudary codtos could be easly eforced. Furthermore, the accuracy of RPM s hgher tha that of the MLS (Lu ad Gu 005). O the other had, regardg the tegral strategy, some truly meshless methods commoly rely o the odal tegrato techque. However, drect odal tegrato s ustable because of uder-tegrato ad vashg dervatves of shape fuctos at the odes. o overcome ths dffculty, Bessel ad Belytschko (1996) added a resdual of the equlbrum equato terms to the potetal eergy fuctoal for stablzg odal tegrato. However, Bessel ad Belytschko (1996) stated that the accuracy of ths method s less tha that of the orgal EFG method. Up-to-date stablzed coformg odal tegrato techque s proposed by Che et al. (001, 00), they modfed the shape fuctos pror to odal tegrato, eve though ths method s cosdered as a robust tegrato techque, t s based o the costructo of a oroo dagram. So t caot be cosdered a true meshless method. Recetly, Khosravfard ad Hematya (010) proposed a ew meshless tegrato techque based o Cartesa rasformato Method (CM), ther method a doma tegral s trasformed to a boudary tegral ad a 1D tegral. ths paper, we reformulated the upper boud lmt aalyss of structures usg the olear programmg theory ad the RPM method, ad the ew tegrato techque proposed by Khosravfard ad Hematya (010) to calculate the teral dsspato power ad exteral work rate. Ad the preset method was used to calculate the lmt loadg parameter of a vertcal slope. he layout of ths paper s as follows: Secto brefly descrbes the upper boud lmt aalyss formulato for a ellpsod yeld fucto usg RPM method ad CM tegrato. A drect teratve algorthm based o Lagrage method s used to solve the olear programmg problem Secto 3. Numercal example for vertcal slope s provded Secto 4 to llustrate the valdty of the preset method. NUMERCAL FORMULAON FOR UPPER BOUND APPROACH BASED ON RPM MESHLESS MEHOD he upper boud theorem he upper boud theorem of lmt aalyss states: amog all kematcally admssble veloctes (that s the plastc admssble stras), the real oe yelds the lowest rate of plastc dsspato power (Drucker ad Prager 195) * * * D d ud f ud (1) where s the lmt load multpler, s the basc load vector of surface tractos, f s the body force vector, u * s the kematc admssble velocty vector, D( * ) deotes the fucto for the rate of the plastc dsspato power * terms of the admssble stra rate, deotes the tracto boudary, ad deotes the space doma of the structure. Here, the kematc admssble velocty vector u * must satsfy the followg two codtos (Che 00): 1) Compatblty ad velocty boud codtos * 1 * * u u (a) * u u o u (b) ) he yeld crtera fucto f 0 (3) he above two codtos ca be related by the assocated flow rule,.e., * f (4) where u deotes the dsplacemet boudary; s the o-egatve plastc multpler. herefore, the soluto of lmt load multpler based o upper boud theorem ca be formulated as the followg mathematcal -338-

3 programmg problems (the surface tracto s omtted) * m D d * st.. f u d 1 (5) * 1 * * u u * u u o u From the optmum of lmt loadg multpler opt, the lmt loadg ca be computed accordg to the followg equato: f f (6) lm opt Nolear optmzato problems for the plae stra vo-mses yeld crtero geeral, the slope stablty problems are treated as the pla stra problems geotechcal egeerg. For the pla stra codto, the vo-mses (or resca) yeld crtero ca be wrtte as (Pastor, 000) f 1 x y xy c 0 4 (7) where c s the coheso. Accordg to the assocated flow rule, the power of dsspato ca be formulated as a fucto of stra rates as (Capso ad Corrad, 1997) D (8) where c c 0 c c 0 (9) 0 0 c herefore, the mathematcal programmg problem (5) of fdg the upper boud soluto of lmt loadg multpler ca be formulated as the followg olear optmzato problem: m d st *.. f u d 1 * 1 * * u u * u u Radal pot terpolato method o u he approxmato of the feld varables of terest x usg radal pot terpolato method (RPM) ca be expressed the followg form (Lu ad Gu, 005): 1 Us u( x) Rq x Pm x G xus (7) 0 where u(x) s the fucto of feld varables, U s ={u 1, u,, u } s the vector of fucto values, (x) s the RPM shape fuctos correspodg to the odal value ad gve by x 1 R x P q m x G 1 x x x (8) whch, R q s the momet matrx of the radal bass fucto (RBF) gve by R1x1 Rx1 R x1 R1x Rx R x R q (9) R1x Rx Rx ad P m the polyomal momet matrx s defed as follows (10) -339-

4 x x m x x x x p p p p p p Pm p p p ad the matrx G s defed as Rq Pm G Pm m x x x 1 m m (10) (11) herefore, the k-th elemet of shape fucto ca be expressed as follows, R G, k p G jk, k j 1 j1 m x x x (1) where G (,k) s the elemet of matrx G -1. A classcal RBF s multquadrc bass (MQ), whch has the followg form (Gu ad Lu, 005): q R x xx y y cdc (13) where c ad q are two shape parameters, d c s the character legth that relates to the odal spacg the local support doma. addto, the complete polyomal bass of order p for two-dmesoal domas ca be wrtte the followg form (Gu ad Lu, 005): P x 1 x y x xy y x p y p (14) f the fucto u(x) stads for the dsplacemet feld for two-dmesoal domas, t ca be terpolated from the vectors of odal fucto value ad RPM shape fucto correspodg to the odal value,.e., 0 ux u ( ) ( ) 1 0 v xu (15) 1 where (x) s the matrx of shape fucto of ode, ad u s the odal dsplacemets. Ad, the dervatves of the RPM shape fuctos ca be formulated as follows (Wag ad Lu 00): m k R p j G k, G jk, (17a) x 1 x j1 x m k R p j G k, G jk, (17b) y y y 1 j1 Accordg to the approxmato of dsplacemet feld fucto, the plastc admssble stras ca be expressed as 1 0 u u1 x v x x 1 v 1 1 ( x) 0 ( x) 0 1 Lu( x) y 0 1 ( ) 0 ( ) Bu (18) x x y y u u 1 1 v v y x y x y x where B s the stra matrx. herefore, substtutg Eq. (18) to olear programmg problem, the dscretzed formulato of upper boud approach based o RPM meshless method ca be expressed as follows: m ubbu d st.. f u d 1 u u o A ew tegrato method based o CM u the umercal formulato of olear programmg problem (19) based o RPM, the ma task s to calculate the tegrato the objectve fucto ad costraed equatos. Recetly, Khosravfard ad Hematya (010) proposed a ew meshless tegrato techque based o Cartesa rasformato Method (19) -340-

5 (CM), ther method a doma tegral s trasformed to a boudary tegral ad a 1D tegral. Accordg to the CM tegral techque, the tegral ca be calculated terms of the followg formulato, for D problems: G D D W F W x f x (0) where 1 W x J J w w D x y x y ad x s the Gaussa pots, =1,, G, where G s the umber of the Gaussa pots. Furthermore, by troducg the trasformato matrx C e, the odal velocty vector u for each ode ca be expressed by the global odal velocty vector U for the slope,.e u C U (1) e herefore, the objectve fucto of olear programmg problem (19) ca be reformulated as follows: where G D W x U KU () 1 K C D DC e e O the other had, the tegral of exteral work rate ca be calculated by usg 1-D Gaussa quadrature method, ad the odal tracto force vector F ca be expressed as follows ct Gt F Jlwx fx C e (3) l 1 herefore, the RPM formulato of olear programmg (19) ca be fally expressed as follows: G D m W x U KU 1 st.. F U=1 KU v 0 u 0 o u (4) where K v U=0 s the plastc compressblty should be satsfed for the materals wth vo Mses or Hll s yeld crtero, the matrx K v ca be expressed as follows: Kv DvC e; Dv Dv1 Dv Dv ; Dv, x, y ; 1,, addto, t should be poted out the velocty boudary codtos ca be mposed by meas of the covetoal fte elemet techque due to the use of radal pot terpolato shape fucto ths study. HE DREC ERAE MEHOD For the olear programmg problem (4), there s a calculato of square root whch could make the objectve fucto usmooth ad odfferetable. hs causes some dffcultes solvg the olear programmg problem. Followg the work of L ad Yu (006), t ca be overcome usg a teratve algorthm for dstgushg rgd/plastc zoes. At frst, the NLP (4) are trasformed to a ucostraed optmzato problem usg Lagrage method. he Lagrage fucto s the followg form: G G D D, v v L U W x UKU W x KU KU F U (5) where s the Lagrage multpler. Followg the work of L ad Yu (005), a teratve cotrol parameter CP was defed as follows: CP UKU (6) Ad the, the Lagrage fucto ca be reformulated as: UKU G G D D, v v 1 CP 1 1 L U W x W x K U K U F U (7) For fdg all rgd regos, the followg teratve process s eeded

6 Step1: talzg the olear objectve fucto Let teratve cotrol parameter CP =1, the, the tal odal dsplacemet velocty ca be estmated by solvg the followg equato system: G D W xk KvKvU0 0F 1 (8) f U0 1 ad the tal load multpler ca be calculated by usg: G D 0 W x U0KU 0 (9) 1 Step k+1 (k=0, 1, ): dstgushg the odfferetable areas to revse the objectve fucto Based o the results at step k, the value of CP eed to be calculated at very Gaussa tegral pot of CM, k 1 the the Gaussa tegral pot set S wll be subdvded to two subsets: the subset S r where the object k 1 fucto s ot dfferetable ad the subset S P where the object fucto s dfferetable,.e., k 1 CP k 1 CP S S, 0 ; S S, 0 (30) r p For CP =0, the orgal optmzato problem ca be solved terms of the followg problem: G D Uk 1KUk1 m W x U 1 S UKU k p st.. f U 1 k 1 KU 0 S U v k1 K U 0 k1 k1 k1 r k k S he revsed NLP problem ca be solved terms of the followg equato system: D KU k1 D D W x W xkuk1 W xkvkvuk1 k 1F k1 k1 k1 Sp UKU k k Sr Sr FUk11 (31) (3) By solvg the Eq. (3), we ca obta the odal velocty ad lmt load multpler at ths step G k 1 k 1 k 1 1 D W x U K U (33) he above teratve process s repeated utl the followg covergece crtera are satsfed k 1 k Uk 1Uk 1; k 1 Uk 1 where 1 ad are the computatoal error toleraces. (34) UPPER BOUND FOR HE HEGH LM OF A ERCAL SLOPE he heght lmt of a vertcal slope s a classcal problem of lmt aalyss or yeld desg theory. he vertcal slope (See Fgure 1) s subjected oly to ow weght. he sol s homogeeous ad sotropc, ad ts coheso s c, ut weght s. Accordg to the research by Pastor et al. (000), the ew bouds of lmt loadg parameter s as follows: H Q (35) c -34-

7 H Sol 1 # Sol c (kn/m ) (kn/m 3 ) 1 # Fgure 1 he model of crtcal heght of a vertcal slope (a) (b) Fgure he layout of the feld odes: (a) regular; (b) rregular For ths test problem, two types of layout of feld odes as show Fgure ca be used to dscretze the doma. Ad the, the RPM shape fucto ca be costructed based o the dscretzato of feld odes. For a relable RPM shape fucto costructo, a L-Scheme proposed by Lu (010) s used to select local supportg odes. O the other had, the terpolato accuracy of RPM ca also be affected by the dmesoless shape parameters c, q ad umbers of feld odes. herefore, these parameters should be aalysed oe by oe. Frstly, the shape parameters c =4 ad q=0.5 are fxed for aalysg the effect of odal layout o the lmt loadg parameter Q. addto, the optmal parameters for the drect teratve algorthm ca be chose accordg to the research of L ad Yu (006). Ad the computatoal error toleraces 1 = =0.001 are fxed. Lmt Loadg Multpler odes, 576 tegral pots 95 odes, 104 tegral pots 141 odes, 576 tegral pots 141 odes, 104 tegral pots 186 odes, 576 tegral pots 186 odes, 104 tegral pots 4 odes, 576 tegral pots 4 odes, 104 tegral pots 64 odes, 576 tegral pots 64 odes, 104 tegral pots 303 odes, 576 tegral pots 303 odes, 104 tegral pots 376 odes, 576 tegral pots 376 odes, 104 tegral pots 389 odes, 576 tegral pots 389 odes, 104 tegral pots 518 odes, 104 tegral pots 577 odes, 104 tegral pots 701 odes, 104 tegral pots terato Step Fgure 3 he covergece sequece of lmt loadg multpler wth teratve steps for rregular odal layout -343-

8 able 1 he results of lmt load multpler for rregular odal layout (576 tegral pots) Nodes Q errors 58% 4% 37% 33% 3% 1% 15% 13% Rutme (s) able he results of lmt load multpler for rregular odal layout (104 tegral pots) Nodes Q errors 59% 45% 41% 37% 3% 3% 7% 8% 17% 16% 0.5% Rutme (s) Wth the above parameters, the optmal value of lmt loadg multpler ca be foud usg the drect teratve algorthm based o regular ad rregular odal layout. he covergece of lmt loadg multpler wth teratve steps for rregular odal layout s show Fgure 3. he optmal value of lmt multpler ad the correspodg lmt loadg parameter for 576 ad 104 tegral pots are show able 1 ad able respectvely. As the same umbers of tegral pot, the accuracy of lmt loadg parameter wll crease wth the creasg umbers of feld ode (see Fgure 5a). For regular odal layout, the covergece sequece of lmt multpler wth teratve steps s show Fgure 4. Ad the optmal value of lmt multpler ad the correspodg lmt loadg parameter for dfferet tegral pots are show able 3. ths case, the umber of tegral pots depedets o that of feld odes, hece, the accuracy of lmt loadg parameter s aalysed just for the dfferet umbers of feld ode. From the Fgure 5b ad able3, the accuracy of lmt loadg parameter wll also crease wth the creasg umbers of feld ode. Lmt loadg Multpler feld odes 93 feld odes 446 feld odes 631 feld odes terato Step Fgure 4 he covergece sequece of lmt loadg multpler wth teratve steps for regular odal layout Lmt Loadg Multpler tegral pots 104 tegral pots Lmt Loadg Multpler Numbers of Feld Node Numbers of Feld Node (a) (b) Fgure 5 he lmt loadg multpler wth dfferet umbers of feld odes: (a) rregular odal layout, ad (b) regular odal layout -344-

9 able 3 he results of lmt load multpler for rregular odal layout Nodes Q errors 97% 40% 35% 0.04% Rutme (s) By comparg the results of lmt load parameter lsted ables1, ad 3 (see Fgure 5), t s very apparet that the accuracy of lmt load parameter for regular odal layout s hgher tha that of rregular layout. However, the reaso of dfferece betwee two odal layouts s ot aalysed here, ad t wll be further studed the followg research works. CONCLUSONS AND DSCUSSONS ths paper, a ew formulato of upper boud approach based o RPM ad olear programmg s proposed. the preset method, the CM tegrato method s used to calculate the teral dsspato, ad the drect teratve algorthm s used to solve olear programmg for fdg the optmal value of lmt loadg parameter of vertcal slope. By the classcal vertcal slope stablty problem, the valdty of the preset method s verfed ths paper. he accuracy of lmt loadg parameter maly depeds o the umber of feld odes ad tegral pots. O the other had, the accuracy of lmt loadg parameter for regular odal layout s hgher tha that of rregular layout. he reasos of dfferet accuracy eed to be further studed. t may be carred out from the followg two aspects,.e., the CM tegrato method ad terpolato of RPM shape fucto. ACKNOWLEDGMENS he frst author apprecates the useful commets from Prof H.S. Yu of Uversty of Nottgham. he work was partly supported by Research Grats Coucl of Hog Kog (uder grat No ). REFERENCES Aderhegge, E., Köpfel, H. (197). Fte elemet lmt aalyss usg lear programmg, teratoal Joural of Solds ad Structures, 8, Bessel, S., Belytschko. (1996). Nodal tegrato of the elemet-free Galerk method, Computer Methods Appled Mechacs ad Egeerg, 139, Belytschko,., Lu, Y.Y., Gu, L. (1994). Elemet-free Galerk methods, teratoal Joural for Numercal Methods Egeerg, 37, Bottero, A., Negre, R., Pastor, J., urgema, S. (1980). Fte Elemet Method ad Lmt Aalyss heory for Sol Mechacs Problems, Computer Methods Appled Mechacs ad Egeerg,, Capso, A., Corrad, L. (1997). A fte elemet formulato of the rgd plastc lmt aalyss problem, teratoal Joural for Numercal Methods Egeerg, 40, Che, J.S., Wu, C.., Yoo, S., You, Y. (001). A stablzed coformg odal tegrato for Galerk mesh-free methods, teratoal Joural for Numercal Methods Egeerg, 50, Che, J.S, Yoo, S., Wu, C.. (00). No-lear verso of stablzed coformg odal tegrato for Galerk mesh-free methods, teratoal Joural for Numercal Methods Egeerg, 53, Che, W.F. (1975). Lmt Aalyss ad Sol Plastcty, New York: Elsever Scetfc Publshg Co. Che, S., Lu, Y., Ce, Z. (008). Lower-boud lmt aalyss by usg the EFG method ad o-lear programmg, teratoal Joural for Numercal Methods Egeerg, 74, Che, Z. (00). Lmt aalyss of the classc problems of sol mechacs, Chese Joural of Geotechcal Egeerg, 4(1), ( Chese) Cra, H., Perare, J., Boet, J. (008). Mesh adaptve computato of upper ad lower bouds lmt aalyss, teratoal Joural for Numercal Methods Egeerg, 75, Drucker, D.C., Prager, W. (195). Sol mechacs ad plastc aalyss or lmt desg, Quarterly of Appled Mathematcs, 10(), Km, J., Salgado, R., Yu, H.S. (1999). Lmt aalyss of sol slopes subjected to pore-water pressures, Joural of Geotechcal ad Geoevrometal Egeerg, ASCE, 15(1), Km, J., Salgado, R., Lee, J. (00). Stablty aalyss of complex sol slopes usg lmt aalyss, Joural of Geotechcal ad Geoevrometal Egeerg, ASCE, 18(7),

10 Khosravfard, A., Hematya, M.R. (010). A ew method for meshless tegrato D ad 3D Galerk meshfree methods, Egeerg Aalyss wth Boudary Elemets, 34, Krabbehøft, K., Lyam, A.., Sloa, S.W. (007). Formulato ad soluto of some plastcty problems as coc programs, teratoal Joural of Solds ad Structures, 44, Krabbehøft, K., Lyam, A.., Sloa, S.W. (008). hree-dmesoal Mohr-Coulomb lmt aalyss usg semdefte programmg, Commucatos Numercal Methods Egeerg, 4, Le, C.., Glbert, M., Askes, H. (009). Lmt aalyss of plates usg the EFG method ad secod-order coe programmg, teratoal Joural for Numercal Methods Egeerg, 78, Le, C.., Nguye-Xua, H., Nguye-Dag, H. (010a). Upper ad lower boud lmt aalyss of plates usg FEM ad secod-order coe programmg, Computer ad Structures, 88, Le, C.., Nguye-Xua, H., Askes, H., Bordas, S. P. A., Rabczuk,., Nguye-h, H. (010b). A cell-based smoothed fte elemet method for kematc lmt aalyss, teratoal Joural for Numercal Methods Egeerg, 83, L, H.X., Yu, H.S. (006). Lmt aalyss of -D ad 3-D structures based o a ellpsod yeld crtero, Acta Geotechca, 1, Lu, G.R. (010). Meshfree method: movg beyod the fte elemet method, d ed., CRC press: Boca Rato, USA. Lu, G.R., Gu, Y.. (005). A troducto to meshfree methods ad ther programmg, Sprg. Lu, Y.H., Ce, Z.Z., Xu, B.Y. (1995). A umercal method for plastc lmt aalyss of 3-D structures, teratoal Joural of Solds ad Structures, 3(1), Lyam, A.., Sloa, S.W. (00). Upper boud lmt aalyss usg lear fte elemets ad o-lear programmg, teratoal Joural for Numercal ad Aalytcal Methods Geomechacs, 6, Lysmer, J. (1970). Lmt aalyss of plae problems sol mechacs, Joural of the Sol Mechacs ad Foudatos Dvso, ASCE, 96(4), Makrodmopoulos A., Mart, C.M. (006). Lower boud lmt aalyss of cohesve-frctoal materals usg secod-order coe programmg, teratoal Joural for Numercal Methods Egeerg, 66, Makrodmopoulos A., Mart, C.M. (007). Upper boud lmt aalyss usg smplex stra elemets ad secod-order coe programmg, teratoal Joural for Numercal ad Aalytcal Methods Geomechacs, 31, Makrodmopoulos A., Mart, C.M. (008a). Upper boud lmt aalyss usg dscotuous quadratc dsplacemet felds, Commucatos Numercal Methods Egeerg, 4, Mart, C.M., Makrodmopoulos A. (008b). Fte-Elemet Lmt Aalyss of Mohr-Coulomb Materals 3D Usg Semdefte Programmg, Joural of Egeerg Mechacs, ASCE, 134(4), Makrodmopoulos A. (010). Remarks o some propertes of coc yeld restrctos lmt aalyss, teratoal Joural for Numercal Methods Bomedcal Egeerg, 010, 6, Pastor, J., ha,.h., Fracescato, P. (000). New bouds for the heght lmt of a vertcal slope, teratoal Joural for Numercal ad Aalytcal Methods Geomechacs, 4, Sloa, S.W. (1988). Lower boud lmt aalyss usg fte elemets ad lear programmg, teratoal Joural for Numercal ad Aalytcal Methods Geomechacs, 1, Sloa, S.W. (1989). Upper boud lmt aalyss usg fte elemets ad lear programmg, teratoal Joural for Numercal ad Aalytcal Methods Geomechacs, 13, Sloa, S.W., Kleema, P.W. (1995). Upper boud lmt aalyss usg dscotuous velocty felds, Computer Methods Appled Mechacs ad Egeerg, 17, Wag, J.G., Lu, G.R. (00). A pot terpolato meshless method based o radal bass fuctos, teratoal Joural for Numercal Methods Egeerg, 54 (11), Yu, H.S., Salgado, R., Sloa, S.W., Km, J.M. (1998). Lmt aalyss versus lmt equlbrum for slope stablty, Joural of Geotechcal ad Geoevrometal Egeerg, ASCE, 14(1), Zhag, P., Lu, M., Hwag, K. (1991). A mathematcal programmg algorthm for lmt aalyss, ACA MECHANCA SNCA, 3(4), ( Chese) Zoua, N., Herskovts, J., Borges, L.A., Fejóo, R.A. (1993). A teratve algorthm for lmt aalyss wth olear yeld fuctos, teratoal Joural of Solds ad Structures, 30,