Geotechnical and Geological Engineering

, Volume 25, Issue 1, pp 65–77

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

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

## 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.

## Keywords

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

## Notation

Am

cross-sectional area of drilled shaft, m2

a

curve fitting parameter

b

curve fitting parameter

c

constant

D

drilled shaft diameter, mm

Em

drilled shaft elastic modulus, MN/m2

Es

tip soil elastic modulus, kN/m2

E(QD)

E(QL)

K

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

Kep

coefficient of lateral earth pressure

Kinit

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

Km

drilled shaft axial stiffness, MN

Kmod

modulus number

Ko

in-situ horizontal stress coefficient

Kt

drilled shaft tip soil stiffness, kN/m

Kti

initial tangent tip soil stiffness, kN/m

Lb

shaft interaction zone length, m

Ld

shaft non-interaction zone length, m

pf

probability of drilled shaft failure

Pt

drilled shaft tip force, kN

Putip

ultimate capacity of the tip soil, kN

Q

q

shear force per unit length, kN/m

qf

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

qo

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

qt

unit toe bearing resistance, kN/m2

R

Rf

failure ratio

u

displacement, mm

ut

tip displacement, mm

z

depth, m

β

reliability index

βep

side resistance parameter

βT

target reliability index

γD

γL

Δz

incremental length along the drilled shaft, mm

δ

drained friction angle for the shaft–soil interface, deg

λQD

λQL

λR

bias of the resistance

μs

tip soil Poisson’s ratio

σatm

atmospheric pressure, MPa

σz

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

ΩQL

coefficient of variation of the live load; and

ΩR

coefficient of variation of the resistance

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## 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