19 5 19 (5): 545 550 2000 9 Ch inese J ou rna l of R ock M echan ics and E ng ineering S ep t. 2000 3 20 M ohr-coulom b 100 1 1 2 3 4 ( 1 710049) ( 2 710054) ( 3 639798 ) ( 4 221008) 1900 (O. M oh r) 2 (M oh r2coulom b) 100 ( ) ( ) 100 ( 2 1900) ( 1985) ( 1991) 100 ( 2 ) TU 451 O 346 A 1000 26915 (2000) 05 20545 206 1 [ 3 1905 ] 1913 1928 [ 3 20 2 (M oh r2cou lom b) ] ( ) 1882 1900 100 1900 100 2 20 [ 4 5 (J. Bau sch inger ] 1833 1893) (C. Bach) (W. V o igt 1850 1919) (A. Fopp l 1854 1924) (L. P randtl 1875 1953 ) (V on ; ( Karm an 1881 1963) ) ; (1835 1918) 1882 [ 1 ] [ 6 8 ] 65 1900 [ 2 2 ] 2000 1 11 2000 3 20 2000 4 10 3 (59779028) : 1934 1955
546 2000 2 (Hoek2B row n) [ 12 14 ] 30 Ρ1 = Ρc + aρ b 3 (3) 60 Σ n 13 = Ρc + aρ b 3 (4) ; 80 Σ13 = Ρc + aρ13 (5) 90 (1) 2 2 2 [ 16 ] (R. Bo rker) 1901 20 2 : 100 2 Ρ2 100 [ 9 ] ( D. Sandel) 1919 Ρ2 [ 3 ] (Ρ1 - Ρ3) + n (Ρ1 + Ρ2 + Ρ3) = 2Σ0 (6) 1910 (H uber 1904) (M ises 1913) (H encky 1925 ) (Ro s [ 10 ] E ich inger 1927) [ 11 ] 1911 Ρ2 40 50 1915 [ 10 11 ] (D rucker) (P rager) 1952 2 Ρ2 [ 12 14 ] (Griffith) 2 ; ;. 3 Ρ2 1971 4 2 (H um pheson2n yalo r) (Zienk iew icz O. C. W. F. Chen) 2 [ 26 27 ] Σ= f (Ρ) Ρ1 - Ρ3 = f (Ρ1 + Ρ3) (1) 2 [ 28 29 ] [ 29 ] 2 : 2 Ρ1 - ΑΡ3 = Ρt Σ= Σ0 + ΒΡ (2) : Ρt ; Α Α= ΡtgΡc 1911 (1) Ρ2 60 [ 15 18 ]
19 5. 20 M oh r2coulom b 100 547 2 Ρ2 15% Ρ2 (Sangha) 2 7 20% [ 3 ] 2 50% [ 17 ] Ρ2 [ 19 20 ] Ρ2 (M ichelis) Ρ2 [ 21 22 ] Ρ2 1985 [ 18 23 ] [ 31 32 Ρ2 ] Ρ3 Ρ1 Α F = [ 23 24 ] Ρ1-2 (Ρ2 + Ρ1 + ΑΡ3 Ρ3 ) = Ρt Ρ2 1 + Α (7a) [ 24 25 ] F = 1 2 (Ρ1 + Ρ1 + ΑΡ3 Ρ2 ) - ΑΡ3 = Ρt Ρ2 1 + Α 1 1 F ig. 1 L im iting m eridian curves at different stress angles 5 Ρ2 (7b) Π 2 2 2 1900 1985 85 a 2 2 70 80 2 2 1995 [ 30 ] 2 F ig. 2 A series of lim it loci of the united strength theo ry
548 2000 [ 39 ] 2 [ 39 [ 24 25 33 ] ] [ 34 35 ] 2 2 21 [ 40 43 ] [ 44 48 ] 6 20 100 1991 ( 2 1900) 2 [ 36 37 ] ( 1985) ( 1991) 100 Α F = Ρ1-1 + b (bρ2 + Ρ1 + 100 ΑΡ3 Ρ3 ) = Ρt Ρ2 1 + Α (8a) 7 (1) 1900 2 1 F = 1 + b (Ρ1 + Ρ1 + ΑΡ3 bρ2 ) - ΑΡ3 = Ρt Ρ2 1 + Α (2) 1985 (8b) (3) Π [ 36 ] (4) 1991 Π 2 (b = 0) (b = 1) (5) (b < 0 2 ) (b > 1 ) (0 < b < 1 ) (6) 100 2 2 2 80
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550 2000 45 Guow ei M Iw asaki S M iyano to Y. P lastic lim it analyses of circular p lates w ith respect to unified yield criterion [J ]. Inṫ J. M ech. Sci. 1990 40 (10) : 963 976 46. [J ]. arbitrary load[j ]. J. A pp lied M echanics A SM E 1999 66 (2) : 1996 17 (1) : 1 8 47. [J ]. 1997 16 (6) : 550 557 48 M a G H ao H. U nified p lastic analyses of circular p lates under 568 570 AD VANCES IN STRENGTH THEORY OF ROCK IN 20 CENTURY 100 Y EARS IN M EMORY OF THE MOHR-COULOM B STRENGTH THEORY Yu M aohong 1 Zan Yuew en 1 Fan W en 2 Zhao J ian 3 Dong Zhengzhu 4 ( 1 S chool of C iv il E ng ineering &M echanics X i an J iaotong U niversity X i an 710049 Ch ina) ( 2 Op en L aboratory of Geotechnical E ng ineering Chang an U niversity X i an 710054 Ch ina) ( 3 S chool of C iv il & S tructu ral E ng ineering N any ang T echnolog ical U niversity 639798 S ing ap ore) ( 4 Ch ina U niversity of M ining and T echnology X uz hou 221008 Ch ina) Abstract It has been 100 years since P rofesso r O. M oh r estab lished the w ell2know n M oh r2cou lom b strength theo ry (failu re criterion ) in 1900. A great am oun t of effo rts w ere dedicated to the study and developm en t of st reng th theo ry o r fa ilu re criteria of rock under the com p lex st ress sta te and to their co rrelation w ith test data. T en s of failu re criteria of rock under the com p lex stress state w ere p ropo sed. A con siderab le am oun t of experim en tal w o rk w ere done in connection w ith the failu re of rock under the p lane stress and strain asymm etric tri2ax ial and true tri2ax ial stress state. A su rvey of the m ain researches and developm en t of strength theo ry on rock are p resen ted since 1900. T he advances in strength theo ry of rock from the single2shear strength theo ry (M oh r 1900) to the tw in2 shear strength theo ry (Yu 1985) and to the un ified strength theo ry (Yu 1991) are summ arized. T he single2shear strength theo ry of M oh r2cou lom b is the low er ( inner) bound of all the convex lim it su rface; the tw in2shear strength theo ry is the upper (ou ter) bound of the convex lim it su rface. T he lim iting loci of variou s em p irical criteria in Πp lane are situated betw een the single2shear and tw in2shear strength theo rieṡ T he un ified strength theo ry is a system of strength theo ry. It gives a series of new failu re criteria estab lishes a relation sh ip am ong variou s failu re criteria and encom passes p reviou s yield criteria (T resca M ises) failu re criteria (M oh r2cou lom b Tw in2shear etc. ) and o ther sm oo th criteria o r em p irical criteria as special cases o r linear app rox im ation ṡ It can be m atched w ith experim en tal resu lts over a w ide range of stress states of m any k inds of m aterialṡ T he un ified st reng th theo ry ha s been genera lized and app lied in som e field ṡ It is ea sy to u se fo r analytical so lu tion s engineering app lication and im p lem en ting in to com puter codeṡ T he un ified strength theo ry m ay have good p ro spects in the 21st cen tu ry. Key words rock strength theo ry (failu re criterion) single2shear strength theo ry (M oh r2cou lom b failu re criterion) tw in2shear strength theo ry un ified strength theo ry