Geotechnology Fundamental Theories of Rock and Soil Mechanics

1. GeotechnologyFundamental Theories of Rock and Soil Mechanics

2. Geotechnology • Theory of Rock and Soil Mechanics • Stress 1. Concept Stress = Pressure = ???

3. Geotechnology • Theory of Rock and Soil Mechanics • Stress 1. Concept Stress = Pressure = Force Area

4. Geotechnology • Theory of Rock and Soil Mechanics • Stress 1. Concept Stress = Pressure = Force Area versus

5. A. Stress 2. Primary Forces (natural)

6. A. Stress 2. Primary Forces (natural) a. Gravitational Forces (overlying materials and upslope activity)

7. A. Stress 2. Primary Forces (natural) b. Tectonic Forces “Important for Virginia and the Eastern Seaboard?”

8. A. Stress 2. Primary Forces (natural) c. Fluid Pressures (‘quick conditions’)

9. Geotechnology • Theory of Rock and Soil Mechanics • Stress 3. Secondary Forces (Human Induced)

10. Geotechnology 3. Secondary Forces (Human Induced) a. Excavation and Mining

11. Geotechnology 3. Secondary Forces (Human Induced) b. Loading

12. Geotechnology 3. Secondary Forces (Human Induced) c. Other * Blasting * Tunneling * Pumping of Fluids

13. 4. Stress (σn ) on a plane normal to Force σn = Force / Area Where n = ‘normal’, or stress perpendicular To the cross sectional area

14. σ = Force / Area 5. Stress on an inclined plane to Force Where inclined area = A = An/cos Θ Θ = angle to normal

15. σ = Force / Area 5. Stress on an inclined plane to Force Where is 1)Normal Force and 2)Shear Force = ??

16. σ = Force / Area 5. Stress on an inclined plane to Force Where Normal Force and Shear Force = ?? cos Θ = a h sin Θ = o h

17. σ = Force / Area 5. Stress on an inclined plane to Force Where Normal Force and Shear Force = ?? cos Θ= a =Fn h = F sin Θ = o = Fs h=F

18. σ = Force / Area 5. Stress on an inclined plane to Force Where Normal Force and Shear Force = ?? cos Θ = a = Fn h = F sin Θ = o = Fs h = F Fn = F cos Θ Fs = F sin Θ

19. A reminder… Fn = F cos Θ Fs = F sin Θ A = An/cos Θ 5. Stress on an inclined plane to Force

20. A reminder… Fn = F cos Θ Fs = F sin Θ A = An/cos Θ 5. Stress on an inclined plane to Force Stress Normal = Force Normal / Area σn = {F cos Θ} / {An/cos Θ} Stress Shear = Force Shear / Area τ = {F sin Θ} / {An/cos Θ}

21. 5. Limits: (max) σn when Θ = 0 (min) σn when Θ = 90 (max) τ when Θ = 45 (min) τ when Θ = 0 or 90

22. Example Problem Given: Force = 10 lbs, Area (normal) = 5 in2 Determine σn and τ when Θ = 0 °, Θ = 30°, Θ = 45°, and Θ = 60°

23. Example Problem Given: Force = 10 lbs, Area (normal) = 5 in2 Determine σn and τ when Θ = 0 °, and Θ = 30° σn = {F cos Θ} / {An/cos Θ} = (10 lbs * cos 0)/(5 in2/cos 0) = τ = {F sin Θ} / {An/cos Θ} = (10 lbs * sin 0)/(5 in2/cos 0) =

24. Example Problem Given: Force = 10 lbs, Area (normal) = 5 in2 Determine σn and τ when Θ = 0 °, and Θ = 30° σn = {F cos Θ} / {An/cos Θ} = (10 lbs * cos 0)/(5 in2/cos 0) = 2 lbs/in2 τ = {F sin Θ} / {An/cos Θ} = (10 lbs * sin 0)/(5 in2/cos 0) = 0 lbs/in2

25. Example Problem Given: Force = 10 lbs, Area (normal) = 5 in2 Determine σn and τ when Θ = 0 °, and Θ = 30° σn = {F cos Θ} / {An/cos Θ} = (10 lbs * cos 30)/(5 in2/cos 30) = lbs/in2 τ = {F sin Θ} / {An/cos Θ} = (10 lbs * sin 30)/(5 in2/cos 30) = lbs/in2

26. Example Problem Given: Force = 10 lbs, Area (normal) = 5 in2 Determine σn and τ when Θ = 0 °, and Θ = 30° σn = {F cos Θ} / {An/cos Θ} = (10 lbs * cos 30)/(5 in2/cos 30) = 1.50 lbs/in2 τ = {F sin Θ} / {An/cos Θ} = (10 lbs * sin 30)/(5 in2/cos 30) = 0.87 lbs/in2

27. Example Problem Given: Force = 10 lbs, Area (normal) = 5 in2 Determine σn and τ when Θ = 0 °, and Θ = 45° σn = {F cos Θ} / {An/cos Θ} = (10 lbs * cos 45)/(5 in2/cos 45) = τ = {F sin Θ} / {An/cos Θ} = (10 lbs * sin 45)/(5 in2/cos 45) =

28. Example Problem Given: Force = 10 lbs, Area (normal) = 5 in2 Determine σn and τ when Θ = 0 °, and Θ = 45° σn = {F cos Θ} / {An/cos Θ} = (10 lbs * cos 45)/(5 in2/cos 45) = 1.00 lbs/in2 τ = {F sin Θ} / {An/cos Θ} = (10 lbs * sin 45)/(5 in2/cos 45) = 1.00 lbs/in2

29. Example Problem Given: Force = 10 lbs, Area (normal) = 5 in2 Determine σn and τ when Θ = 0 °, and Θ = 60° σn = {F cos Θ} / {An/cos Θ} = (10 lbs * cos 60)/(5 in2/cos 60) = 0.5 lbs/in2 τ = {F sin Θ} / {An/cos Θ} = (10 lbs * sin 60)/(5 in2/cos 60) = 0.87 lbs/in2

30. Do your answers conform to the trends shown here? 5. Limits: (max) σn when Θ = 0 (min) σn when Θ = 90 (max) τ when Θ = 45 (min) τ when Θ = 0 or 90

31. Stress at any point can be ‘resolved’ via 3 mutually perpendicular stresses: σ1 , σ2 , σ3 Where σ1 > σ2 > σ3 6. Stress (σ) in 3 dimensions

32. B. Strain “your ideas??”

33. B. Strain 1. Strain Effects

34. B. Strain 1. Strain Effects a. Stress produces deformation Strain = dL L

35. B. Strain 1. Strain Effects a. Stress produces deformation “phi”

36. B. Strain 1. Strain Effects a. Strain Ellipse Maximum Shear Stress: Where σ1 - σ3 2

37. 2. Stress – Strain Diagrams σ “which material is stronger?” ε

38. II. Elastic Response A. Young’s Modulus (E) “best shown in rocks” E = stressσ strain ε “elastic limit” “The greater E is, ……?

39. II. Elastic Response A. Young’s Modulus (E) “best shown in rocks” E = stressσ strain ε “The greater E is, the less deformation per unit stress OR “the stronger the material”

40. An Example:

41. II. Elastic Response B. Poisson’s Ratio (ν) ν = lateral strain length strain In compression In tension

42. II. Elastic Response C. Ideal Elastic Behavior

43. II. Elastic Response D. Non-Ideal Elastic Behavior Strain hardening

44. ‘delayed feedback’ II. Elastic Response E. Hysteresis Hard Rock “under repeated loads” Soft Rock

45. Assumes OM, MD II. Elastic Response F. Stress-Strain in Soils Limits of Proportionality (how much of the strain is Elastic?)

46. “under repeated loads” II. Elastic Response G. Repeated Loading of Soils (when rolled) Increment of permanent strain decreases (densification)

47. III. Time-Dependent Behavior – Strain A. Creep – under static loads Elastic response occurs instantaneously

49. Collapsed Culvert, Cincinnati, OH

50. III. Time-Dependent Behavior – Strain A. Creep – under static loads