1. Predicting non-linear ground movements Malcolm Bolton Cambridge University, UK

2. What is the aim? Single calculation to verify safety and serviceability. Direct non-linear ground displacement calculation based on a bare minimum of soil element data, without using constitutive equations or FEA. Mobilisable Strength Design (MSD) offered as an improvement to Limit State Design (LSD) in that it deals properly with serviceability. Focus: construction-induced displacements in clay. We will show 2 examples: rigid pads / rafts under vertical loading multi-propped excavations

3. Mobilisable Strength Design (MSD) MSD defines a local zone of finite plastic deformation. The ideal location of a representative element is selected at the centroid of the plastic zone. Stresses are derived from plastic equilibrium. Stress-strain data is treated as a curve of plastic soil strength mobilised as strains develop. Strains are deduced from raw stress-strain data. Ground displacements are obtained by entering strains back into the plastic deformation mechanism.

4. Example 1: circular (square) footing on clay Focus on undrained settlement under load. Use Prandtl’s plane strain geometry to select the plastic zone of deformation. Select a kinematically admissible displacement field. Use plastic work equation to find equilibrium stress factor (familiar as bearing capacity factor). Use plastic displacement field to find compatible strain factor (unfamiliar, to be explained). Convert triaxial stress-strain curve, using the two factors, into a foundation load-settlement curve.

5. Plastic deformation mechanism D  u,ru,r v,zv,z

6.  c mob  N c =5.81 (5.69)  Stresses and strains for circular footing

7. Design procedure c mob    = M c  /D 0.3D  c mob

8. Relation to a triaxial test Foundation stress  mob  N c c mob  5.7 c mob Triaxial deviator stress  q mob  2 c mob   mob /2.85 Foundation distortion  D  Triaxial axial strain  a  D q OR  mob /2.85  a OR 0.9  /D

9. Validation by non-linear FEA

10. G max =Ap’ n1 OCR m1 G=Bp’ n2 OCR m2  q b2 MCC flow rule ln  q G Very small strains Small Strains Large Strains  q ~10 -5  q ~10 -2 Soil model: SDMCC Bolton M.D., Dasari G.R. and Britto A.M. (1994)

12. Soil profile around the representative element

13. Soil displacements by FEA

14. MSD versus FEA

15. More FE validation: BRICK model  /D or  q (%)  or q (kPa) Many soil profiles and realistic stress-strain curves have been checked, all with the same high quality of fit.

16. Why does it work so well? Soil stress-strain curves resemble power curves over the significant range (see Bolton & Whittle, 1999) with shear strain roughly proportional to the square of shear stress. So the significant deformation zone is close to the perturbing boundary stress. And the equation  /  ref = (  /  ref )  is self-similar at all stress levels, ensuring that the deformation mechanism at “small” strains is identical to that at “large” strains.

17. Field validation: Kinnegar test Kinnegar site Lehane (2003) Stiff square pad footing treated here as a circle of diameter 2.26m

18. Kinnegar soil profile

19. Normalised stress-strain behaviour

20. (Triaxial compression data) (Triaxial extension data) MSD predictions for Kinnegar Also predicts Jardine’s Bothkennar test rather well, and matches Arup’s observations of large rafts on London Clay. But most field tests are not accompanied by the necessary stress-strain data from a shallow sample. This is a lesson well taught by MSD methodology.

21. Example 2: ground movements around braced excavations

22. Stability calculations

23.  max Soil excavated to cause  max y Incremental displacements 1 1 0  /  max y/L Supports L (Incremental displacement profile after O’Rourke 1993)

24. Comparison of incremental displacement profile between field data and cosine function (after O’Rourke 1993)

25. s L=  S Plastic deformation mechanism

26. s s L =  S  = 2 Wavelength L: free-end condition

27. L =  S  = 1 s s Wavelength L: fixed-end condition

28. s 1 <  <2 L = S ~ 2 S s Wavelength L: intermediate end condition

29. Estimation of the mobilised shear strength  = c mob /c u

30. Shear strength cucu Depth c mob =  c u Assumption of a mobilisation ratio

31. Calculation procedure for bulging movements ss

32. Surface settlement MSD

33. Effect of cantilever movement

34. Plastic deformation mechanism for cantilever retaining walls H D 45    s =2  ss

35. Permissible stress field aa  a pp pp  a =  v -2c mob  p =  v +2c mob D H 2c u pp aa vv Limiting pressures in undrained conditions

36. C mob /  D Mobilised strength versus excavation depth for cantilever retaining walls

37. Calculation procedure for cantilever retaining walls aa  a pp pp  a =  v -2c mob  p =  v +2c mob D H ss H D   s =2  ss

38.  ' log scale  ' log scale  log scale Whittle’s data of Boston Blue Clay

39. FE validation comparing with Hashash and Whittle (1996) Boston blue clay

40. Stability calculations for braced excavations – props placed at 2.5m intervals to failure at excavation depth H f Boston blue clay

41. Case history: Boston Post Office Square Garage (Whittle et al. 1993) The 1400 car parking underground garage was constructed with seven levels of below-grade structure in the heart of the downtown financial district of Boston in late 1980s. The garage occupies a plan area of 6880 m 2.

43. Measured and predicted displacements Boston Post Office Square Garage

44. Measured and predicted settlements

45. Braced excavation in Singapore soft clay The sub-structure consists of a two-level basement in soft marine clay surrounded by Gairnill Garden (a 12 storey residential block of flats), Scotts Road and Cairnhill Road. The excavation was 110m by 70 m. The depth of excavation varies from 6.4m to 7.5m. The sheetpile wall was supported by three levels of bolted struts. The vertical spacing varies from 1.4m to 1.8m. The sheetpile lengths range from 12m to 24m.

46. Soil profile at Moe Building

47.  a (%) Stress-strain response of Singapore Soft Marine Clay (after Wong and Broms 1989)

48. Measured and predicted displacements Singapore soft marine clay

49. Measured and predicted displacements Singapore soft marine clay

50. Conclusions Raw stress-strain data from a triaxial test on a representative sample taken from a selected location in the plastic zone of influence can be used directly to predict displacements. No need for constitutive laws or parameters. Plastic deformation mechanisms with distributed plastic strains can provide a unified solution for design problems. This application can satisfy approximately both safety and serviceability requirements and can predict stresses and displacements under working conditions; without the need for FE analysis.

51. The future Extend MSD to predict consolidation settlements from drained / creep stages carried out during the representative element or pressuremeter test. Verify using centrifuge model tests on foundations with long-term PIV monitoring providing ground strain contours at 0.01% intervals. Attempt to extend to sand, referenced to pressuremeter test rebound loops.

52. Thank you for inviting me!