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

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Abstract

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.

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Abbreviations

A m :

cross-sectional area of drilled shaft, m2

a :

curve fitting parameter

b :

curve fitting parameter

c :

constant

D :

drilled shaft diameter, mm

E m :

drilled shaft elastic modulus, MN/m2

E s :

tip soil elastic modulus, kN/m2

E(Q D):

expected value of dead load

E(Q L):

expected value of live load

K :

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

K ep :

coefficient of lateral earth pressure

K init :

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

K m :

drilled shaft axial stiffness, MN

K mod :

modulus number

K o :

in-situ horizontal stress coefficient

K t :

drilled shaft tip soil stiffness, kN/m

K ti :

initial tangent tip soil stiffness, kN/m

L b :

shaft interaction zone length, m

L d :

shaft non-interaction zone length, m

p f :

probability of drilled shaft failure

P t :

drilled shaft tip force, kN

P utip :

ultimate capacity of the tip soil, kN

Q :

deterministic drilled shaft load, kN

q :

shear force per unit length, kN/m

q f :

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

q o :

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

q t :

unit toe bearing resistance, kN/m2

R :

drilled shaft load capacity, kN

R f :

failure ratio

u :

displacement, mm

u t :

tip displacement, mm

z :

depth, m

β:

reliability index

βep :

side resistance parameter

βT :

target reliability index

γD :

dead load factor

γL :

live load factor

Δz :

incremental length along the drilled shaft, mm

δ:

drained friction angle for the shaft–soil interface, deg

λQD :

bias of the dead load

λQL :

bias of the live load

λR :

bias of the resistance

μs :

tip soil Poisson’s ratio

σatm :

atmospheric pressure, MPa

σ ′ z :

vertical effective stress, kPa

τu :

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

ϕ:

resistance factor

ϕ′:

drained friction angle of soil, deg

ΩQD :

coefficient of variation of dead load

ΩQL :

coefficient of variation of the live load; and

ΩR :

coefficient of variation of the resistance

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Authors and Affiliations

  1. Professor of Civil Engineering, University of Missouri-Kansas City, 5100 Rockhill Road, 350H Flarsheim Hall, Kansas City, MO, 64110, USA

    Anil Misra

  2. Geotechnical Engineer, Terracon Consultants, Inc., 13910 W. 96th Terrace, Lenexa, KS, 66215, USA

    Lance A. Roberts

  3. Senior Geotechnical Engineer, Terracon Consultants, Inc., 13910 W. 96th Terrace, Lenexa, KS, 66215, USA

    Steven M. Levorson

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  1. Anil Misra
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  2. Lance A. Roberts
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  3. Steven M. Levorson
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Correspondence to Anil Misra.

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Misra, A., Roberts, L.A. & Levorson, S.M. Reliability analysis of drilled shaft behavior using finite difference method and Monte Carlo simulation. Geotech Geol Eng 25, 65–77 (2007). https://doi.org/10.1007/s10706-006-0007-2

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