Dr C.M. Martin Department of Engineering Science University of Oxford

1. 11th International Conference of IACMAG, Torino21 Giugno 2005Exact bearing capacity calculations using the method of characteristics Dr C.M. Martin Department of Engineering Science University of Oxford

2. Outline • Introduction • Bearing capacity calculations using the method of characteristics • Exact solution for example problem • Can we solve the ‘Ng problem’ this way? • The fast (but apparently forgotten) way to find Ng • Verification of exactness • Conclusions

3. Bearing capacity • Idealised problem (basis of design methods): Central, purely vertical loading qu = Qu/B Rigid strip footing D B Semi-infinite soil c, f, g, y = f

4. Bearing capacity • Idealised problem (basis of design methods): Central, purely vertical loading qu = Qu/B q = gD q = gD Rigid strip footing B Semi-infinite soil c, f, g, y = f

5. Classical plasticity theorems • A unique collapse load exists, and it can be bracketed by lower and upper bounds (LB, UB) • LB solution from a stress field that satisfies • equilibrium • stress boundary conditions • yield criterion • UB solution from a velocity field that satisfies • flow rule for strain rates • velocity boundary conditions • Theorems only valid for idealised material • perfect plasticity, associated flow (y = f) } Statically admissible Plastically admissible } Kinematically admissible

6. Method of characteristics • Technique for solving systems of quasi-linear PDEs of hyperbolic type • Applications in both fluid and solid mechanics • In soil mechanics, used for plasticity problems: • bearing capacity of shallow foundations • earth pressure on retaining walls • trapdoors, penetrometers, slope stability, … • Method can be used to calculate both stress and velocity fields (hence lower and upper bounds) • In practice, often gives LB = UB  exact result • 2D problems only: plane strain, axial symmetry

7. Method of characteristics • Technique for solving systems of quasi-linear PDEs of hyperbolic type • Applications in both fluid and solid mechanics • In soil mechanics, used for plasticity problems: • bearing capacity of shallow foundations • earth pressure on retaining walls • trapdoors, penetrometers, slope stability, … • Method can be used to calculate both stress and velocity fields (hence lower and upper bounds) • In practice, often gives LB = UB  exact result • 2D problems only: plane strain, axial symmetry

8. Outline • Introduction • Bearing capacity calculations using the method of characteristics • Exact solution for example problem • Can we solve the ‘Ng problem’ this way? • The fast (but apparently forgotten) way to find Ng • Verification of exactness • Conclusions

9. Lower bound stress field • To define a 2D stress field, e.g. in x-z plane • normally need 3 variables (sxx, szz, txz) • if assume soil is at yield, only need 2 variables (s, q) x t X q c s s3 s1 sn f 2q s3 = s–R Z M-C s1 = s+R [ ] z general

10. Lower bound stress field • To define a 2D stress field, e.g. in x-z plane • normally need 3 variables (sxx, szz, txz) • if assume soil is at yield, only need 2 variables (s, q) x t b X b q c s s3 s1 sn f a 2q s3 = s–R Z a a M-C s1 = s+R b [ ] z general

11. Lower bound stress field • To define a 2D stress field, e.g. in x-z plane • normally need 3 variables (sxx, szz, txz) • if assume soil is at yield, only need 2 variables (s, q) x t b X b q e = p/4–f/2 c 2e s s3 s1 sn f a 2q s3 = s–R 2e Z a a e e M-C s1 = s+R b [ ] z general

12. Lower bound stress field • Substitute stresses-at-yield (in terms of s, q) into equilibrium equations • Result is a pair of hyperbolic PDEs in s, q • Characteristic directions turn out to coincide with a and b ‘slip lines’ aligned at qe • Use a and b directions as curvilinear coords  obtain a pair of ODEs in s, q(easier to integrate) • Solution can be marched out from known BCs

13. Lower bound stress field • Substitute stresses-at-yield (in terms of s, q) into equilibrium equations • Result is a pair of hyperbolic PDEs in s, q • Characteristic directions turn out to coincide with a and b ‘slip lines’ aligned at qe • Use a and b directions as curvilinear coords  obtain a pair of ODEs in s, q(easier to integrate) • Solution can be marched out from known BCs > 0

14. x z Lower bound stress field • Marching from two known points to a new point: (xB, zB, sB, qB) B (xA, zA, sA, qA) A

15. x z Lower bound stress field • Marching from two known points to a new point: (xB, zB, sB, qB) B (xA, zA, sA, qA) A   C (xC, zC, sC, qC)

16. x z Lower bound stress field • Marching from two known points to a new point: (xB, zB, sB, qB) B (xA, zA, sA, qA) A   C (xC, zC, sC, qC)

17. x z Lower bound stress field • Marching from two known points to a new point: • ‘One-legged’ variant for marching from a known point onto an interface of known roughness (xB, zB, sB, qB) B (xA, zA, sA, qA) A   C (xC, zC, sC, qC)

18. x z Lower bound stress field • Marching from two known points to a new point: • ‘One-legged’ variant for marching from a known point onto an interface of known roughness (xB, zB, sB, qB) B (xA, zA, sA, qA) A   C (xC, zC, sC, qC) FD form FD form

19. Upper bound velocity field • Substitute velocities u, v into equations for • associated flow (strain rates normal to yield surface) • coaxiality (princ. strain dirns = princ. stress dirns) • Result is a pair of hyperbolic PDEs in u, v • Characteristic directions again coincide with the a and b slip lines aligned at qe • Use a and b directions as curvilinear coords  obtain a pair of ODEs in u, v(easier to integrate) • Solution can be marched out from known BCs

20. Upper bound velocity field • Marching from two known points to a new point: x,u (xB, zB, sB, qB, uB, vB) B (xA, zA, sA, qA, uA, vA) A z,v

21. Upper bound velocity field • Marching from two known points to a new point: x,u (xB, zB, sB, qB, uB, vB) B (xA, zA, sA, qA, uA, vA) A z,v   C (xC, zC, sC, qC, uC, vC)

22. Upper bound velocity field • Marching from two known points to a new point: x,u (xB, zB, sB, qB, uB, vB) B (xA, zA, sA, qA, uA, vA) A z,v   C (xC, zC, sC, qC, uC, vC)

23. Upper bound velocity field • Marching from two known points to a new point: • ‘One-legged’ variant for marching from a known point onto an interface of known roughness x,u (xB, zB, sB, qB, uB, vB) B (xA, zA, sA, qA, uA, vA) A z,v   C (xC, zC, sC, qC, uC, vC)

24. Upper bound velocity field • Marching from two known points to a new point: • ‘One-legged’ variant for marching from a known point onto an interface of known roughness x,u (xB, zB, sB, qB, uB, vB) B (xA, zA, sA, qA, uA, vA) A z,v   C FD form FD form (xC, zC, sC, qC, uC, vC)

25. Outline • Introduction • Bearing capacity calculations using the method of characteristics • Exact solution for example problem • Can we solve the ‘Ng problem’ this way? • The fast (but apparently forgotten) way to find Ng • Verification of exactness • Conclusions

26. Example problem Rough base qu q = 18 kPa q = 18 kPa B = 4 m c = 16 kPa, f = 30°, g = 18 kN/m3 after Salençon & Matar (1982)

27. Example problem: stress field (partial) a s known (passive failure); q = p/2 b

28. Example problem: stress field (partial) a s known (passive failure); q = p/2 b Symmetry: q = 0 on z axis (iterative construction req’d)

29. Example problem: stress field (partial) a • Shape of ‘false head’ region emerges naturally • qu from integration of tractions • Solution not strict LB until stress field extended: s known (passive failure); q = p/2 b Symmetry: q = 0 on z axis (iterative construction req’d)

30. Example problem: stress field (complete) Minor principal stress trajectory

31. Example problem: stress field (complete) • Extension strategy by Cox et al. (1961) • Here generalised for g > 0 • Utilisation factor at start of each ‘spoke’ must be  1 Minor principal stress trajectory

32. Extension technique q z0 s1 z s3 s1 gz0 + q s1 + g(z z0) gz + q

33. Extension technique q z0 s1 z s3 s1 gz0 + q Critical utilisation is here: s1 + g(z z0) gz + q

34. Example problem: velocity field Rigid Rigid Rigid Rigid Rigid

35. Example problem: velocity field • Discontinuities are easy to handle – treat as degenerate quadrilateral cells (zero area) Rigid Rigid Rigid Rigid Rigid

36. Some cautionary remarks • Velocity field from method of characteristics does not guarantee kinematic admissibility! • principal strain rates may become ‘mismatched’ with principal stresses s1, s3 • this is OK if f = 0 (though expect UB  LB) • but not OK if f > 0: flow rule violated  no UB at all • If f > 0, as here, must check each cell of mesh • condition is sufficient • Only then are calculations for UB meaningful • internal dissipation, e.g. using • external work against gravity and surcharge

37. Example problem: velocity field • qu from integration of internal and external work rates for each cell (4-node , 3-node ) • Discontinuities do not need special treatment Rigid Rigid Rigid Rigid Rigid

38. Convergence ofqu (kPa) in example

39. Convergence ofqu (kPa) in example LB

40. Convergence ofqu (kPa) in example LB UB

41. Outline • Introduction • Bearing capacity calculations using the method of characteristics • Exact solution for example problem • Can we solve the ‘Ng problem’ this way? • The fast (but apparently forgotten) way to find Ng • Verification of exactness • Conclusions

42. Why not? The solutions obtained from [the method of characteristics] are generally not exact collapse loads, since it is not always possible to integrate the stress-strain rate relations to obtain a kinematically admissible velocity field, or to extend the stress field over the entire half-space of the soil domain. Hjiaj M., Lyamin A.V. & Sloan S.W. (2005). Numerical limit analysis solutions for the bearing capacity factor Ng. Int. J. Sol. Struct.42, 1681-1704.

43. Ng problem as a limiting case qu d q q B c = 0, f > 0, g > 0, y = f

44. Ng problem as a limiting case qu d q q B c = 0, f > 0, g > 0, y = f

45. Ng problem as a limiting case qu d q q B c = 0, f > 0, g > 0, y = f

46. Ng problem as a limiting case qu d q q B c = 0, f > 0, g > 0, y = f

47. Stress field as gB/q  c = 0, f = 30°, Rough (d = f)

48. Stress field as gB/q  c = 0, f = 30°, Rough (d = f)

49. Stress field as gB/q  c = 0, f = 30°, Rough (d = f)

50. Stress field as gB/q  c = 0, f = 30°, Rough (d = f)