29 4 2010 12 GLOBAL GEOLOGY Vol. 29 No. 4 Dec. 2010 1004-5589 2010 04-0633 - 07 Mohr-Coulomb 1 2 3 1. 130026 2. 130031 3. 100024 Tresca Mises Mohr-Coulomb Mohr-Coulomb Mohr-Coulomb Mohr-Coulomb Mohr-Coulomb TV698. 11 TU452 A doi 10. 3969 /j. issn. 1004-5589. 2010. 04. 015 Application of Mohr-Coulomb yield criterion in geo-technical engineering LIU Ying 1 2 YU Li-hong 3 1. College of Construction Engineering Jilin University Changchun 130026 China 2. Changchun Vanke Co. Ltd Changchun 130031 China 3. HydroChina Beijing Engineering Corporation Beijing 100024 China Abstract The yield curves and yield stress of Tresca Mises Double shear and Mohr-Coulomb yield criterions were discussed and compared to find the differences among them. The relation among Mohr-Coulomb yield criterion and the others were discussed and comfirmed the safety of Mohr-Coulomb yield criterion in geo-technical engineering and the relationships between fracture plane location and slip curves when the rock and soil fractured. The different yield criterions were analyzed and compared taking Yabiluo hydropower station geo-stresses for example and demonstrated the safety of Mohr-Coulomb yield criterion and the difference between failure surface and slip curves. Key words Mohr-Coulomb yield criterion safety Yabiluo hydropower station geo-stresses slip curves 0 1-4 Tresca Mises Drukle-Plager Mohr-Coulomb Mohr- 2010-04-28 2010-10-25 40872170
634 29 Coulomb Mohr-Coulomb S -D Strength Difference Effect τ c τ max Mohr-Coulomb Tresca φ 10 σ Tresca σ Mises σ DJ Mohr-Coulomb 11 π Mohr-Coulomb Mohr-Coulomb Mohr-Coulomb Mohr-Coulomb Tresca 1 Mohr-Coulomb Mises 1. 1 Mohr-Coulomb σ 1 σ 2 σ 3 3 Tresca 4 Mohr-Coulomb τ max = σ 1 σ 3 /2 = σ s /2 1 3 Mises 4 σ 1 - σ 2 2 + σ 2 - σ 3 2 + σ 3 - σ 1 2 = 6C = 2σ 2 s J 2 = C 2 5 6 τ 13 + max τ 12 τ 23 = c c c = σ s τ 13 + max τ 12 τ 23 = σ s 3 Mohr-Coulomb 6-9 τ n = σ n tgφ + c 4 σ 1 - σ 3 = σ 1 + σ 3 sinφ + 2ccosφ 5 f = 1 2 I 1sinφ + cosθ σ 1 槡 3 sinθ sinφ J σ 槡 2 - ccosφ = 0 6 - π /6 θ σ π /6-1. 2 Mohr-Coulomb σ m = σ 1 + σ 2 + σ 3 /3 σ 2 = σ m 1 σ m M M τ max = ± σ 1 σ 3 /2 Tresca π /4 α β Mohr-Coulomb τ max = ± σ 1 σ 3 /2 τ = τ max sinφ τ max τ max = c σ s = σ 1 + σ 2 /2 - σ 1 + σ 2 /2 sinφ 7 1 Fig. 1 Mohr-Coulomb Relations between failure surface and slip curves of Mohr-Coulomb yield criterion
4 Mohr-Coulomb 635 π /4 + φ /2 φ = 0 φ /2 1 2 Mohr-Coulomb 3. 2 Mohr-Coulomb 37 km π 8 3. 2. 1 Mohr-Coulomb km 10 km 6 km V 25 ~ 60 20 ~ 60 30 ~ 80 m NE62 80 ~ 100 m Mohr-Coulomb 0. 24% Drukle- 3. 2. 2 12 Plager 13 π Mohr- 45% ~ 70% Coulomb 20% ~ 40% 5% ~ 25% Zienkiewicz and Pande 250 30 ~ 45 14 15 π Mohr-Coulomb Mohr- Coulomb 10 ~ 30 m π Mohr-Coulomb 1 ~ 3 15 16 Mohr- m Coulomb Mohr-Coulomb 3. 2. 3 Mohr-Coulomb 3 30 km 3. 1 7 km 1. 6 km 70 ~ 80 300 m 63 km 616 km 152. 5 m 1 075 m 1 950 MW 32 km 4. 713 m 3 1 ZK10
636 29 Ⅱ Ⅲ 25 2 3. 4. 2 1 NE50 ~ 85 NW SE 60 ~ 85 2 NW275 ~ 300 SW 17 σ H NE 40 ~ 65 σ h σ v 3 1 NE60 ~ 85 NW SE 70 ~ 85 2 NW310 ~ 330 NE SW 70 ~ 85 3 NW275 ~ 295 NE SW 80 ~ 90 1 5 ~ 25 8 9 10 3. 5 1 ZK10 10 32. 52 m 45. 85 m 50. 19 m 67. 30 m 74. 23 m 87. 50 m 98. 58 m 109. 49 m 115. 75 m 122. 00 m T 1 0. 139 g VII 3. 3 2 c = 0. 55 MPa φ = 32. 3 30 ~ 60 km NE -NEE 3. 4 3. 4. 1 3. 4. 3 σ H = 3 p s p r - p 0 8 σ h = p s 9 σ v = γd 10 41 ~ 120 m 1 1 10 4 5 1 2 2 Fig. 2 Theory Mohr s circle and its envelope lines 3
4 Mohr-Coulomb 637 Table 1 1 ZK10 Results of hydraulic fracturing stress of hole ZK10 /m /MPa /MPa P b P r P s P H P 0 T S H S h S V / 1 32. 52 7. 92 3. 12 2. 72 0. 32 0. 10 4. 80 4. 93 2. 72 0. 85 2 45. 85 10. 57 3. 65 3. 25 0. 45 0. 23 6. 92 5. 86 3. 25 1. 19 3 50. 19 11. 29 4. 89 3. 69 0. 49 0. 27 6. 40 5. 90 3. 69 1. 30 4 67. 30 / 5. 46 5. 46 0. 66 0. 44 / 10. 47 5. 46 1. 75 5 74. 23 16. 32 8. 72 6. 72 0. 72 0. 51 7. 60 10. 94 6. 72 1. 93 N65 W 6 87. 50 19. 25 6. 85 5. 25 0. 85 0. 64 12. 40 8. 27 5. 25 2. 27 N20 W 7 98. 58 16. 36 5. 76 5. 36 0. 96 0. 75 10. 60 9. 58 5. 36 2. 56 N10 E 8 109. 49 12. 27 9. 07 7. 07 1. 07 0. 85 3. 20 11. 29 7. 07 2. 84 9 115. 75 17. 73 7. 93 6. 73 1. 13 0. 92 9. 80 11. 35 6. 73 3. 01 N32 E 10 122. 00 12. 19 7. 19 6. 79 1. 19 0. 9771 5. 00 12. 21 6. 79 3. 17 P H P 0 P b P r P s T S h S H S v 2 Table 2 Statistics of yield stress results σ s /MPa 1 2 3 4 5 6 7 8 9 10 Tresca 4. 08 4. 67 4. 60 8. 72 9. 01 6. 00 7. 02 8. 45 8. 34 9. 04 Mises 3. 54 4. 05 3. 99 7. 58 7. 81 5. 20 6. 12 7. 32 7. 24 7. 88 2. 98 3. 37 3. 50 6. 22 6. 90 4. 49 4. 91 6. 34 6. 03 6. 33 Mohr-Coulomb 1. 80 2. 28 2. 37 3. 78 4. 03 3. 67 4. 20 4. 80 4. 95 5. 04 1 2 3 7 ZK10 c = 0. 6 ~ 0. 8 MPa φ = 35 ~ 38 2 2 Tresca Mises c φ ZK10 Mohr-Coulomb Mohr-Coulomb Mohr-Coulomb 3 9 Mohr-Coulomb c φ Fig. 3 Relations between rock mass failure surface and slip curves of No. 9 measures
638 29 τ = tan32. 3 + 0. 55 ZK10 9 115. 75 m 3 16. 15 J. 2008 41 31-34. SHE Cheng-xue LIU Jie. Two-parameter parabolic-type yield criterion based on Mohr strength thoery J. Engineering Journal of Wuhan University 2008 41 31-34. 4 7. J. 2006 25 12 2515-2522. GAO Hong ZHENG Ying-ren FENG Xia-ting. Exploration Mohr-Coulomb π on yield an failure of material J. Chinese Journal of Mechanics and Engineering 2006 25 12 2515- Mohr-Coulomb 2522. π /4 8. J. + φ /2 2008 33 4 337-345. Mohr-Coulomb ZHENG Ying-ren GAO Hong. Discussion of strength theory for materials J. Journal of Guangxi University Nature Sciences Edition 2008 33 4 337-345. 9. 2443. 1. GAO Hong ZHENG Ying-ren FENG Xia-ting. Study J. on energy yield criterion of geomaterials J. Chinese 2006 25 1 86-90. Journal of Rock Mechanics and Engineering 2007 26 WANG Ji-song GUAN Ying-bin BAO Shang-xin et 12 2437-2443. al. Application of similar material simulation in research of coal seam floor failure regularity J. Global Geology 2006 25 1 86-90. 2. Biot J. 2003 33 1 71-75. ZHANG Yan-jun ZHANG Yan-ji. The numberical analysis of elastic visco-plastic Biot's consolidation to marine soft soil J. Journal of Jilin University Earth Science Edition 2003 33 1 71-75. 3. M. TONG Xiao-dong GONG Xiao-nan YAO En-yu. The 2006 239-244. property of the yield curves of the stable material on the GONG Xiao-nan. Advanced soil mechanics M. Hangzhou Zhejiang University Press 2006 239-244. 4. M. 2004 59-66. YANG Gui-tong. Introdution to elasticity and plasticity M. Beijing Tsing University Press 2004 59-66. 5. Analysis of stability for the surrounding rock of underground excavations based on different yield criteria J. J. 1990 23 1 34-40. YU Mao-hong LIU Feng-yu LIU Feng et al. A new Chinese Journal of Underground Space and Engineering general strength theory J. Chinese Journal of Civil En- 2008 4 4 635-639. gineering 1990 23 1 34-40. 6. Mohr J. 2007 26 12 2437-10. J. 2006 13 5 14 15. JI Ming GAO Feng LIAO Meng-ke. Comparison of yield stress among several plastic yield criterions J. Journal of Plasticity Engineering 2006 13 5 14 15. 11. π J. 1998 32 5 643-647. stress π plane J. Journal of Zhejiang University 1998 32 5 643-647. 12. J. 2008 4 4 635-639. XIAO Ming ZHANG Zhi-guo CHEN Jun-tao et al. 13. Drucker-Prager
4 Mohr-Coulomb 639 J. 1990 24 1 108-111. XU Gan-cheng ZHEN Ying-ren. Research on the appli- J. cation of the yield criterions in rock and soil engineering 2009 30 4 865-870. YANG Xue-qiang LING Ping-ping XIANG Shenghua. Comment on slope stability based on a series of Drucker-Prager failure criteria J. Rock and Soil Mechanics 2009 30 4 865-870. 14. DAI Zi-hang SHEN Pu-sheng. Simplified from and applications of Mohr-Coulom equivalent area circle yield J. 1989 8 30-33. XU Gan-cheng ZHEN Ying-ren. Based on a synthetic criterion J. Journal of Fuzhou University 2003 31 ananysis and study the quadratic yield criteria J. Site 4 454-459. Investigation Science and Technology 1989 8 30-33. 15. YOU Ming-qing. Study on the geo-stresses measurement J. Chinese Journal of Geotechnical Engineering 1990 24 1 108-111. 16. - J. 2003 31 4 454-459. 17. J. 2005 27 3 350-353. with hydro-fracture of borehole J. Chinese Journal of Geo-technical Engineering 2005 27 3 350-353. 檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿檿 621 11. 12. J. J. 2002 29 3 285-289. 2002 29 1 52-55. LI Chun-yu XIE Yuan LIU Shao-guang et al. Factors KUANG Jun TANG Yong ZHU Guo-hua et al. Basic controlling the very low-porosity and permeability sandstone reservoir of the Yanchang Formation in Fuxian are- reservoirs in Junggar Basin J. Petroleum Exploration characteristics and main controlling factors of Jurassic a North Shaanxi J. Journal of Chengdu University of and Development 2002 29 1 52-55. Technology 2002 29 3 285-289.