Geotechnical and Geological Engineering

, Volume 25, Issue 1, pp 65–77 | Cite as

Reliability analysis of drilled shaft behavior using finite difference method and Monte Carlo simulation

  • Anil Misra
  • Lance A. Roberts
  • Steven M. Levorson
Original Paper


Load displacement analysis of drilled shafts can be accomplished by utilizing the “t-z” method, which models soil resistance along the length and tip of the drilled shaft as a series of springs. For non-linear soil springs, the governing differential equation that describes the soil-structure interaction may be discretized into a set of algebraic equations based upon finite difference methods. This system of algebraic equations may be solved to determine the load–displacement behavior of the drilled shaft when subjected to compression or pullout. By combining the finite difference method with Monte Carlo simulation techniques, a probabilistic load–displacement analysis can be conducted. The probabilistic analysis is advantageous compared to standard factor of safety design because uncertainties with the shaft–soil interface and tip properties can be independently quantified. This paper presents a reliability analysis of drilled shaft behavior by combining the finite difference technique for analyzing non-linear load–displacement behavior with Monte Carlo simulation method. As a result we develop probabilistic relationships for drilled shaft design for both total stress (undrained) and effective stress (drained) parameters. The results are presented in the form of factor of safety or resistance factors suitable for serviceability design of drilled shafts.


Drilled shaft Probabilistic analysis Finite difference “t-z” methods Monte Carlo simulation 



cross-sectional area of drilled shaft, m2


curve fitting parameter


curve fitting parameter




drilled shaft diameter, mm


drilled shaft elastic modulus, MN/m2


tip soil elastic modulus, kN/m2


expected value of dead load


expected value of live load


shear modulus of shaft–soil interface sub-grade reaction, kN/m2


coefficient of lateral earth pressure


initial tangent shear modulus of shaft–soil interface sub-grade reaction, kN/m2


drilled shaft axial stiffness, MN


modulus number


in-situ horizontal stress coefficient


drilled shaft tip soil stiffness, kN/m


initial tangent tip soil stiffness, kN/m


shaft interaction zone length, m


shaft non-interaction zone length, m


probability of drilled shaft failure


drilled shaft tip force, kN


ultimate capacity of the tip soil, kN


deterministic drilled shaft load, kN


shear force per unit length, kN/m


failure strength of the shaft–soil interface, kN/m


ultimate (asymptotic) strength of shaft–soil interface, kN/m


unit toe bearing resistance, kN/m2


drilled shaft load capacity, kN


failure ratio


displacement, mm


tip displacement, mm


depth, m


reliability index


side resistance parameter


target reliability index


dead load factor


live load factor


incremental length along the drilled shaft, mm


drained friction angle for the shaft–soil interface, deg


bias of the dead load


bias of the live load


bias of the resistance


tip soil Poisson’s ratio


atmospheric pressure, MPa


vertical effective stress, kPa


ultimate shear strength of shaft–soil interface, kN/m2


resistance factor


drained friction angle of soil, deg


coefficient of variation of dead load


coefficient of variation of the live load; and


coefficient of variation of the resistance


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  1. AASHTO (2004) LRFD bridge design specifications. 3rd edn. American Association of State Highway and Transportation Officials, Washington, DCGoogle Scholar
  2. Baecher GB, Christian JT (2003) Reliability and statistics in geotechnical engineering. Wiley, West Sussex, UKGoogle Scholar
  3. Barker RM, Duncan JM, Rojiani KB, Ooi PSK, Tan CK, Kim SG (1991) NCHRP report 343: manual for the design of bridge foundations, Transportation Research Board, National Research Council, Washington, DCGoogle Scholar
  4. Burland J (1973) Shaft friction in piles in clay: a simple fundamental approach. Ground Eng 6(3):30–42Google Scholar
  5. Chandler RJ (1968) The shaft friction of piles in cohesive soils in terms of effective stress, Civil Eng Public Works Rev, 63:48–51Google Scholar
  6. Coduto DP (2001) Foundation design: principles and practices. Prentice-Hall, New JerseyGoogle Scholar
  7. Das BM (2004) Principles of foundation engineering. Thomson, New YorkGoogle Scholar
  8. Duncan JM, Chang CY (1970) Nonlinear analysis of stress and strain in soils. J Soil Mech Found Div, ASCE, 96(SM 5):1629–1653Google Scholar
  9. Duncan JM, Byrne P, Wong KS, Marby P (1980) Strength, stress–strain, and bulk modulus parameters for finite element analyses of stresses and movements in soil masses. Report No. UCB/GT/80-01, College of Engineering Office of Research Services, University of California – Berkeley, Berkeley, CAGoogle Scholar
  10. FHWA (1982) Tolerable movement criteria for highway bridges, vol. 1 – Interim Report. Report No. FHWA-RD-81-162, Federal Highway Administration, U.S Department of Transportation, McLean, VAGoogle Scholar
  11. FHWA (1999) Drilled shafts: construction procedures and design methods. Report No. FHWA-IF-99-025, Federal Highway Administration, U.S. Department of Transportation, Mclean, VAGoogle Scholar
  12. Harr ME (1996) Reliability-based design in civil engineering. Dover Publications Inc., Mineola, New YorkGoogle Scholar
  13. Hildebrand FB (1974) Introduction to numerical analysis. Dover Publications Inc., Mineola New YorkGoogle Scholar
  14. Janbu N (1963) Soil compressibility as determined by oedometer and triaxial tests, vol. I, European Conference on Soil Mechanics and Foundations Engineering, Wiesbaden, Germany, pp 19–25Google Scholar
  15. Johnson KL (1985) Contact mechanics. Cambridge University Press, London UKGoogle Scholar
  16. Kondner RL (1963) Hyperbolic stress–strain response: cohesive soils, J Soil Mech Found Div, ASCE, 89(SM␣1): 115–143Google Scholar
  17. Kraft LM, Ray RM and Kagawa T (1981) Theoretical t-z curves. J Geotech Eng, ASCE 107(11):1543–1561Google Scholar
  18. Kulhawy FH (1991) Drilled shaft foundations. In: Fang HY (ed) Foundation Engineering Handbook, 2nd edn. Van Nostrand Reinhold, pp 537–552Google Scholar
  19. Kulhawy FH, Phoon KK (1996) Engineering judgment in the evolution from deterministic to reliability-based foundation design, Uncertainty’ 96, Geotechnical Special Publication No. 58, ASCE, New York, 1, pp 29–48Google Scholar
  20. Lacasse S, Nadim F (1996) Uncertainties in characterizing soil properties, Uncertainty’ 96, Geotechnical Special Publication No. 58, ASCE, New York, 1, pp 49–75Google Scholar
  21. Mathcad version 11 (2002). Mathsoft Engineering & Education, Inc., Cambridge, MAGoogle Scholar
  22. Meyerhof GG (1976) Bearing capacity and settlement of pile foundations. J Geotech Eng, ASCE, 102(GT3): 197–228Google Scholar
  23. Misra A, Chen C-H (2004) Analytical solutions for micropile design under tension and compression. J␣Geotech Geol Eng 22(2):199–225CrossRefGoogle Scholar
  24. Misra A, Roberts LA (2005) Probabilistic axial load–displacement relationships for drilled shafts, Proceedings Geofrontiers 2005, Austin, TX, ASCE, Arlington, VAGoogle Scholar
  25. Misra A, Roberts LA (2006) Axial service limit state design of drilled shafts using probabilistic approach. J␣Geotech Geol Eng 24(5):1--20Google Scholar
  26. Olson RE (1990) Axial load capacity of steel pipe piles in sand. Proceedings Offshore Technology Conference, Houston, TX, pp 17–24Google Scholar
  27. Paikowsky SG, Birgisson B, McVay M, Nguyen T, Kuo C, Baecher G, Ayyub B, Stenersen K, O’Malley K, Chernauskas L, O’Neill M (2004) NCHRP report 507: load and resistance factor design (LRFD) for deep foundations, Transportation Research Board, National Research Council, Washington, DCGoogle Scholar
  28. Phoon K-K, Kulhawy FH, Grigoriu MD (1995) Reliability-based design of foundations for transmission line structures, Report TR-105000, Electric Power Research Institute, Palo Alto, CA, July 1995, p 380Google Scholar
  29. Phoon KK, Kulhawy FH (1999) Characterization of geotechnical variability. Can Geotech J 36(4):612–624CrossRefGoogle Scholar
  30. Reese LC, O’Neill MW (1987) Drilled shafts: construction procedures and design methods, in Report No. FHWA-HI-88-042, Federal Highway Administration, U.S. Department of Transportation, McLean, VirginiaGoogle Scholar
  31. Rollins KM, Clayton RJ, Mikesell RC, Blaise BC (2005) Drilled shaft side friction in gravelly soils. J Geotech Geoenviron Eng, ASCE 131(8):987–1003CrossRefGoogle Scholar
  32. Scott RF (1981) Foundation analysis. Prentice-Hall, New JerseyGoogle Scholar
  33. Terzaghi K, Peck RB, Mesri G (1996) Soil mechanics in engineering practice. John Wiley and Sons Inc., New YorkGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  • Anil Misra
    • 1
  • Lance A. Roberts
    • 2
  • Steven M. Levorson
    • 3
  1. 1.Professor of Civil EngineeringUniversity of Missouri-Kansas CityKansas CityUSA
  2. 2.Geotechnical Engineer, Terracon Consultants, Inc.LenexaUSA
  3. 3.Senior Geotechnical Engineer, Terracon Consultants, Inc.LenexaUSA

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