INFLUENCE OF BACKFILL SLOPE ON DYNAMIC RESPONSE OF FLEXIBLE
Sandra Cristina Miranda Reis Lobo
Instituto Superior Tcnico, Department of Civil Engineering
Abstract: Interaction between earth retaining structures and surrounding soil is a complex phenomenon for
both static and seismic loading. In the present paper a brief review of available methodologies for the design
of flexible retaining structures is done, for the seismic load, with a special focus on dynamic numerical
The seismic performance of a retaining wall depends on the total pressures (i.e., static plus dynamic
pressures) that act on it during the earthquake. Pseudostatic analysis are used in current practice to
estimate seismically induced wall pressures, and the wall is designed to resist those pressures without
failing or causing failure of the surrounding soil.
The present document aims to study the behaviour of a flexible braced retaining structure under the seismic
action. 2D numerical analyses using the finite element method were performed to study the response of
flexible retaining wall using SOFiSTiK FE software.
The parametric studies illustrate the effects of the backfill slope and intensity of seismic acceleration on the
stress field, struts forces, displacements and on the design of reinforced retaining walls.
Keywords: Seismic Action, Dynamics, nonlinear analysis, Retaining wall structure, SOFiSTiK.
Deep excavations are widely used in urban areas for the development of underground space, e.g. subway
stations, basements for high-rise buildings, underground car parks, etc.. However, the excavation process
inevitably changes the grounds stress state and may cause significant wall deformations and ground
movements. Observations of the performance of retaining structures in recent earthquakes show that
earthquakes have caused permanent deformations in some cases these deformations were negligible; in
others they have caused significant damage.
This study was undertaken to develop a better understanding of the dynamic performance for retaining
walls, with different backfill inclination and with different intensities of accelerogram.
2. GROUND RESPONSE ANALYSES
There are various methods to analyse retaining wall under seismic loads. The methods used in defining the
dynamic analysis of retaining walls may be categorized into two groups:
pseudo-static methods: these are simplified methods for seismic design, in which the effects of
earthquakes are represented by vertical and/or horizontal accelerations constant throughout the
height of the supported field;
dynamic methods: methods that take into account, approximately, dynamic response of the
structure and supported soil, translated by seismic coefficients variables.
The pseudo-static method normally to estimate the active thrust used is the Mononobe-Okabe method. This
method computes the dynamic pressure on support structures, being an extension of the Coulomb theory.
Mononobe-Okabe (M-O) method is still employed as the first option to estimate lateral earth pressures
during earthquakes by geotechnical engineers.
The response of the structure to seismic phenomena is always a dynamic response. Two and three-
dimensional ground response analyses are usually performed using dynamic finite-element analysis. These
analyses can be performed using equivalent linear or nonlinear approaches. The numerical model used is
based on time history dynamic 2D analysis using SOFiSTiK finite element analysis software.
3. DYNAMIC SOIL PROPERTIES
The mechanical behaviour of soils can be quite complex under seismic loading conditions.
The behaviour of a soil when subjected to a cyclical action, can be represented by stress-strain curve shown
in Figure 1. The hysteresis loop produced from the cyclic loading of a typical soil can be described by the
path of the loop itself or by two parameters that describe its general shape. These parameters are the
inclination and the area of the hysteresis loop, shear modulus and damping, respectively. Figure 1 is a
simplified schematic showing one loop of symmetric cyclic loading and its corresponding parameters.
Figure 1 Secant shear modulus Gsec and tangent shear modulus
As the strain amplitude is varied, different size loops will be developed and the locus of the points
corresponding to the tips of these loops is called the backbone curve (or skeleton). As the strain increases
the secant shear modulus will decrease. Therefore, the maximum shear modulus is developed at low shear
strain where geophysical tests are used. Another way to represent this shear modulus degradation with
cyclic strain is by means of the modulus reduction curve. The modulus reduction curve normalizes the shear
modulus (G) with respect of the maximum shear modulus (Gmax) and is commonly referred to as the
modulus ratio. Figure 2 shows a schematic of the typical cyclic behaviour of soils.
Figure Stress-strain curve with variation of shear modulus and modulus reduction curve
As the soil element loses stiffness with the amplitude of strain, its ability to dampen dynamic forces
increases. This is due to the energy dissipated in the soil by friction, heat or plastic yielding. The relationship
of shear strain to damping is inversely proportional to the modulus reduction curve. Damping is often
expressed as the damping ratio (D), which is defined as the damping coefficient divided by the critical
damping coefficient. This can be obtained from the hysteresis loop by dividing the area of the loop by the
triangle defined by the secant modulus and the maximum strain (energy dissipated in one cycle by the peak
energy during a cycle. Damping ratio represents the ability of a material to dissipate dynamic load or
dampen the system. It should be noted that many factors contribute to the stiffness of soils during cyclic
loading, such as, plasticity index, relative density, mean principal effective stress, overconsolidation ratio,
number of cycles and void ratio. However, for low-strain dynamic behaviour in geophysical tests, the shear
modulus in that range remains constant as Gmax and is commonly used as an elastic parameter.
Most seismic geophysical methods or tests induce shear strains lower that 10-6
and the shear wave velocity
( ) can be used to compute the Gmax using the expression = 2, where is the mass density of the
soil. In this study, using the Eurocode 8 it was possible to define a shear wave velocity and obtain the Gmax
for a soil deposit.
Since the nonlinearity of the soil behaviour is well known, the linearity approach must be modified to provide
reasonable estimates of ground response. The equivalent linear approach provides reasonable results for
many practical problems.
4. ONE-DIMENSIONAL GROUND RESPONSE ANALYSIS
One-dimensional analysis can be performed when the soil layers and the bedrock surface are horizontal and
they extend to infinity, and the seismic waves coincide with shear waves propagating vertically from the
underlying bedrock. This last assumption can be justified considering that the seismic waves, propagating
from the earthquake source through the soil, are bent by successive refractions into a nearly vertical path
(according to Schnabels law of refraction). The problem can be modelled through a soil column with specific
5. TWO-DIMENSIONAL DYNAMIC RESPONSE ANALYSIS
One-dimensional method is the most widely used in ground response analysis because it is more practical to
be used for quantitative analyses compared with 2-D and 3-D methods. In this study, the ground response
analyses are carried out assuming 2-D shear wave propagation.
The calibration of the Finite Element software SOFiSTiK was made through the simulation of the one-
dimensional vertical propagation of S-waves in elastic layers, whose theoretical solutions are available in
literature. The proposed calibration procedure constitutes a useful preliminary step for performing advanced
dynamic analyses of any geotechnical system.
The results of the dynamic analysis performed with SOFiSTiK were compared to a linear site and an
equivalent linear response analysis performed with Strata (see Figure 2 and 3). The results from both
approaches are compared showing a good agreement.
Figure 2 Displacement in the surface obtained in Strata and SOFiSTiK, assuming linear analysis
Figure 3 - Transfer Functions between the movement in the surface and in the base on the "Free-Field" for equivalent linear response analysis
6. ANALYSIS OF PARAMETRIC STUDIES APPLYING THE METHOD OF
6.1. Description Geotechnical Profile for Reference Model
To design a retaining wall an engineer must know the basic soil parameters, which include unit weight, angle
of internal friction, angle of wall friction, cohesion, wall inclination maximum acceleration and height of
The definition of the geotechnical parameters adopted for different soil layers considered in the model
calculation, took into account the criteria set out in Eurocode 8 - Part 1, evaluate the elasticity or shear (E
G0) modulus depending on the speed of S waves propagation. It is admitted a soil