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Three-dimensional seismic active earth pressure coefficients using upper bound
numerical limit analysis: a few preliminary results
Coefficients de poussée tridimensionels séismiques déterminés avec une application numérique du
theorème cinématique de l’analyse limite: quelques résultats préliminaires
Santana T., Guerra N.M.C., Antão A.N., Vicente da Silva M.
UNIC, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa
ABSTRACT: The finite element program Sublim3D, which is a three-dimensional numerical implementation of the limit analysis
upper bound theorem, is applied to determine seismic active horizontal earth pressure coefficients, using the equivalent static forces
approach to simulate the seismic effect. The calculations are performed for different width/height ratios of the wall, different seismic
horizontal coefficients, two representative values of the friction angle and two values of the soil-to-wall friction ratio. The three-
dimensional influence on the seismic active earth pressures is emphasized. The mechanisms involved in the mobilization of the active
earth pressures are inferred from the plastic deformation zones determined by the calculations. The ratios between the three-
dimensional and two-dimensional seismic horizontal earth pressure coefficients are obtained and found to be, in practical terms,
independent on the soil-to-wall friction ratio.
RÉSUMÉ : Le programme aux éléments finis Sublim3D, une implémentation numérique tridimensionnelle du théorème cinématique
de l’analyse limite, est appliqué à la détermination des coefficients de poussée horizontal séismiques tridimensionnels, utilisant des
forces statiques équivalentes pour simuler l’effet séismique. Les calculs sont réalisés pour différents rapports largeur/hauteur du mûr,
différents coefficients séismiques horizontaux, deux valeurs représentatifs de l’angle de frottement du sol et deux valeurs du rapport
entre l’angle de frottement sol-structure et l’angle de frottement du sol. L’influence tridimensionnelle dans les coefficients de poussée
séismiques est démontrée. Les mécanismes associés à la mobilisation de la poussée des terres sont inférés à partir des zones de
déformation plastique déterminés par les calculs. Les rapports entre les coefficients de poussée tridimensionnels et les coefficients
bidimensionnels sont déterminés. Leur indépendance de l’angle de frottement entre le sol et le mûr est constatée.
KEYWORDS: three-dimensional seismic active earth pressure coefficients; upper bound limit analysis; finite elements
1 INTRODUCTION.
Static and seismic earth pressures are often determined using
plain strain conditions. These conditions are not always
adequate, particularly when the width to height ratio of the wall
is not very large.
In this paper the finite element program Sublim3D (Vicente
da Silva et al. 2012) is used to determine seismic earth pressure
coefficients for different width to height ratios.
Sublim3D is a finite element limit analysis program which
uses a mixed finite-element formulation that implements the
upper-bound theorem. It scales the mechanisms by setting the
work rate of the external forces affected by a load parameter
equal to one and performs the minimization of the difference
between the plastically dissipated work rate and the work rate of
the fixed external forces. The code resorts to parallel computing
techniques (Vicente da Silva and Antão 2008), thus allowing
very large problems to be solved. The program determines strict
upper bound approximations of limit loads of geotechnical
problems and automatically obtains the mechanisms involved in
such loads. An associative flow rule and perfectly plastic
behaviour are assumed.
2 DEFINITION OF THE PROBLEM
The geometry of the analyzed problem is represented
schematically in Figure 1. The wall is vertical, with width b and
height h and assumed as rigid. The free surface of the supported
soil is considered horizontal. The soil yield criterion is Mohr-
Coulomb, with null cohesion.
For the situation described, a seismic coefficient of active
earth pressure, K
as
, is determined, representing the influence of
soil weight; the seismic coefficient of active earth pressures,
K
asq
, representing the influence of constant surcharge loading, is
not determined. The active seismic force per unit width I
as
/b
can be determined through:
2
2
1
h K
b
I
as
as
(1)
where
is the soil unit weight and the other symbols have the
meaning previously described.
Figure 1. Schematic representation of the geometry of the problem.
The determination of the seismic active earth pressure
coefficient was performed for different b/h ratios – 0.25, 0.5, 1,
2, 5 and infinity (two-dimensional), for two values of the soil
friction angle,
’ – 30 and 40º, and for two values of the soil-to-
wall interface friction ratio,
/
’ – 0 and 2/3. The two-
dimensional calculations were also performed using the same
