Geology 3120 Powerpoint notes available online at http

Geology 3120 Powerpoint notes available online at: http: //www. colorado/edu/geolsci/courses/GEOL 3120 1

Geology 3120 The Mohr Stress Diagram ss 0 2 Q Stress Space sn 2

Outline • Setting up the Problem • The Mohr Stress Diagram • Mohr-Coulomb Law of Failure • Exercises 3

Setting up the Problem Q is defined as the angle between the plane and the force vector. Clockwise is positive (+). 4

Decomposing Stresses After several trigonometric and algebraic simplifications, the two equations left are …… 5

Fundamental Stress Equations n. Normal Stress sn = (s 1 + s 3) - (s 1 - s 3)cos 2 Q 2 2 n. Shear Stress ss = (s 1 - s 3)sin 2 Q 2 6

Physical and Stress Space ss s 1 s 3 Q s 3 2 Q sn 0 s 1 Physical Space Stress Space 7

Conventions - Physical Space s 1 Q is defined as the angle between the plane and the s 1 stress direction. A clockwise direction is s 3 +Q positive (+). s 1 8

Conventions - Stress Space ss (+) 0 (-) sn = tension s 3 2 Q s 1 sn (+) sn = compression 9

Conventions - Stress Space ss sn(p), ss(p) 0 s 3 2 Q sn(p) s 1 sn(p) = (s 1 + s 3) - (s 1 - s 3)cos 2 Q 2 2 ss(p) = (s 1 - s 3)sin 2 Q 2 sn 10

Conventions - Stress Space ss sn(p), ss(p) (s 1 - s 3)sin 2 Q 2 0 s 3 2 Q s 1 sn 11

Conventions - Stress Space ss 0 sn(p), ss(p) s 3 2 Q (s 1 + s 3) 2 s 1 sn Mean Stress center of circle 12

Conventions - Stress Space ss 0 sn(p), ss(p) s 3 2 Q (s 1 - s 3) 2 s 1 sn Deviatoric Stress radius of circle 13

Conventions - Stress Space ss 0 sn(p), ss(p) s 3 2 Q (s 1 - s 3) s 1 sn Differential Stress diameter of circle 14

Conventions - Stress Space ss 0 sn(p), ss(p) s 3 2 Q s 1 sn (s 1 - s 3)cos 2 Q 2 Difference between mean stress 15 and normal stress on plane

Laboratory Experiments in Rock Deformation Deformed marble rock cylinders 16

Repeated Failure Experiments 17

Stress Requirements for Rock Failure 18

Mohr-Coulomb Law of Failure sc Y X m = ( ( +b sn sn f = angle of internal friction tan f = coefficient of internal friction [slope; m] sn = normal stress [X] sc = critical shear stress required for faulting [Y] s = cohesive strength [y-intercept; b] 19

Influence of Pore Fluid Pressure Effective Stress Applied Stress pf Pore fluid pressure decreases normal stresses by the fluid pressure amount. Rock can then fail under the Mohr-Coulomb Law. 20

Problems 1 & 2 1. For the maximum and minimum principal stresses of 600 and 200 MPa oriented as a vertical vector and a horizontal, E-W striking vector, respectively, determine the normal and shear stress on a plane oriented N 0°E, 45°E. 2. For the stress state in the problem above determine the deviatoric stress and mean stress. 21

Problems 1 & 2 s 1 W E s 3 +Q 1. For the maximum and minimum principal stresses of 600 and 200 MPa oriented as a vertical vector and a horizontal, E-W striking vector, respectively, determine the normal and shear stress on a plane oriented N 0°E, 45°E. sn = 400 Mpa, ss=200 MPa 2. For the stress state in the problem above determine the deviatoric stress and mean stress. Deviatoric Stress = 200 MPa, Mean Stress = 400 MPa 22

Problem 3 3. Given two planes P 1 and P 2 oriented where Q equals 90° (P 1) and 45° (P 2), P 1 has a normal stress of 500 MPa and P 2 has a normal stress of 300 MPa and a shear stress of 200 MPa, determine the magnitudes of the principal stresses, the deviatoric stress and the 231? mean stress. Is this stress state more or less likely to produce failure as that in Problem

Problem 3 3. 4. 5. 6. Given two planes P 1 and P 2 oriented where Q equals 90 (P 1) and 45 (P 2) degrees, P 1 has a normal stress of 500 MPa and P 2 has a normal stress of 300 MPa and a shear stress of 200 MPa, determine the magnitudes of the principal stresses, the deviatoric stress and the mean stress. s 1 = 500 MPa and s 3 = 100 MPa Deviatoric Stress = 200 MPa Mean Stress = 300 MPa 24

Problem 3 3. Is this stress state more or less likely to produce failure as that in Problem 1? 4. The stress state of Problem 3 is more likely to produce failure than in Problem 1 since the Mohr circle is closer to the failure envelope. 25

References Slide 15 Twiss, R. J. and E. M. Moores, Structural Geology, W. H. Freeman & Co. , New York, 532 p. , 1992. Slides 16 -19 Davis. G. H. and S. J. Reynolds, Structural Geology of Rocks and Regions, 2 nd ed. , John Wiley & Sons, New York, 776 p. , 1996. 26