731
Undrained bearing capacity of spatially random clays by finite elements and limit
analysis
Capacité portante des argiles non drainées des champs aléatoires par éléments finis et analyse
limite
Huang J., Lyamin A.V.
The University of Newcastle, Australia
Griffiths D.V.
Colorado School of Mines, USA
Sloan S.W., Krabbenhoft K.
The University of Newcastle, Australia
Fenton G.A.
Dalhousie University
ABSTRACT: This paper combines the random field methodology with the upper and lower bound finite element limit analysis
algorithms (Sloan 1988, 1989) to study the bearing capacity of undrained clays with spatially varying shear strength. The results of
the Random Field Limit Analysis (RFLA) analyses are compared with existing results obtained by elastic-plastic Random Finite
Element (RFEM) analyses (Griffiths and Fenton 2001). It is shown that RFEM results are bounded by RFLA ones. The difference
(Nd) between the upper (Nu) and lower (Nl) bound bearing capacities in random soils is shown to be a lognormally distributed
random variable. The effects of spatial correlation length and coefficient of variation of undrained strength on Nu and Nl are also
studied.
RÉSUMÉ : Ce document combine la méthode des champs aléatoires avec les limites inférieure et supérieure des algorithmes
d
’
analyse par éléments finis limites (Sloan 1988, 1989) pour étudier la capacité portante des argiles non drainées variant dans l
’
espace
avec la résistance au cisaillement. Les résultats de l
’
analyse de limiter le champ aléatoire (RFLA) des analyses sont comparés avec les
résultats actuels obtenus par élasto-plasticité des éléments finis (Random RFEM) analyses (Griffiths et Fenton, 2001). Il est montré
que les résultats RFEM sont délimités par les RFLA. La différence (N
d
) entre la tige (N
u
) et inférieure (N
l
) lié capacités portantes dans
les sols aléatoires se révèle être une variable aléatoire une distribution lognormale. Les effets de la longueur de corrélation spatiale et
coefficient de variation de la résistance non drainée sur N
u
et N
l
sont également étudiés.
KEYWORDS: bearing capacity, limit analysis, finite element method, random field.
1 INTRODUCTION
Limit analysis has been used in geotechnical practice for
decades as a means of estimating the ultimate strength of
structures. Starting from early 80s (e.g., Sloan 1988, 1989),
Sloan and his colleagues combined the bound theorems with
finite element method and mathematical programming
techniques. The resulting methods inherit all the benefits of the
finite element approach and are applicable to a wide range of
problems involving arbitrary domain geometries, complex
loadings and heterogenous material properties. The Random
Finite Element Method (RFEM) (Fenton and Griffiths 2008)
combines elastoplastic finite elements and random field theory
in a Monte-Carlo framework. It has been proved to be able to
assess the reliability of a wide range of geotechnical problems
including settlement, seepage, consolidation, bearing capacity,
earth pressure and slope stability.
In this paper, we combines the finite element limit analysis
method developed by Sloan and his colleagues with random
field theory. The framework is very similar to RFEM, but three
components are combined together, namely, bound theorems,
finite element method and random field theory. The finite
element limit analysis utilizes recent developments of convex
optimization algorithms. The random field is generated by the
Local Averaging Subdivision method developed by
Fenton and
Vanmarcke (1990).
The method is then used to investigate the
statistical bounds of the bearing capacity of a smooth rigid strip
footing (plane strain) at the surface of an undrained clay soil
with a shear strength
u
c
(
0
u
) defined by a spatially varying
random field.
The study starts with a deterministic analysis which shows
the bearing capacity obtained by finite element method is
bounded by the ones obtained by limit analysis. By introducing
spatial variability, the robustness of finite element limit analysis
involving heterogenous soil properties is tested. It is shown that
the limit analyses always bounds the finite element analysis no
matter how heterogenous the soils are. Although the RFEM
always gives estimations lie between the lower and upper
bounds, RFLA gives quantitative error estimation which RFEM
cannot offer. The probabilistic analysis is then carried out. It is
shown that even the mean upper bound bearing capacity factors
are lower than the Prandtl solution in all cases. This confirms
that using mean soil strength with deterministic analysis or first
order probabilistic estimate will be on the unconservative side.
In addition, a worst case spatial correlation length is observed
where mean bearing capacity is minimized. This suggests that
the spatial variability of soil strength has to be taken into
account properly.
2 REVIEW ON FINITE ELEMENT LIMIT ANALYSIS
The lower and upper bound theorems of classical plasticity
theory is a powerful tool for analysing the stability of problems
in soil mechanics. The theory assumes a perfectly plastic soil
model with an associated flow rule. The lower bound theorem
states that any statically admissible stress field will furnish a
lower bound (or ‘safe’)
estimate of the true limit load.
