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8/18/2019 Cyclic Loading Using Intelligent Finite Element Method
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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/240032782
Analysis of Behaviour of Soils Under CyclicLoading Using EPR-based Finite Element
Method
ARTICLE in FINITE ELEMENTS IN ANALYSIS AND DESIGN · OCTOBER 2012
Impact Factor: 2.02 · DOI: 10.1016/j.finel.2012.04.005
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3 AUTHORS:
Akbar Javadi
University of Exeter
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Asaad Faramarzi
University of Birmingham
30 PUBLICATIONS 133 CITATIONS
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Alireza Ahangar-Asr
University of Salford
25 PUBLICATIONS 81 CITATIONS
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Available from: Asaad Faramarzi
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8/18/2019 Cyclic Loading Using Intelligent Finite Element Method
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Analysis of behaviour of soils under cyclic loading using EPR-based finite
element method
Akbar A. Javadi a,n, Asaad Faramarzi b,1, Alireza Ahangar-Asr a,2
a Computational Geomechanics Group, College of Engineering Mathematics and Physical Sciences, University of Exeter, Harrison Building, North Park Road, Exeter EX4 4QF, UK b Department of Civil Engineering, School of Engineering, University of Greenwich, Central Avenue, Chatham Maritime, Kent ME4 4TB, UK
a r t i c l e i n f o
Article history:
Received 9 April 2011Received in revised form
17 April 2012
Accepted 18 April 2012Available online 16 May 2012
Keywords:
Finite element
Evolutionary computation
Material modelling
Cyclic loading
EPR
a b s t r a c t
In this paper, a new approach is presented for modelling of behaviour of soils in finite element analysis
under cyclic loading. This involves development of a unified approach to modelling of complexmaterials using evolutionary polynomial regression (EPR) and its implementation in the finite element
method. EPR is a data mining technique that generates a clear and structured representation of
the system being studied. The main advantage of an EPR-based constitutive model (EPRCM) over
conventional models is that it provides the optimum structure and parameters of the material model
directly from raw experimental (or field) data. The development and validation of the method will be
presented followed by the application to study of behaviour of soils under cyclic loading. The results of
the analyses will be compared with those obtained from standard finite element analysis using
conventional constitutive models. It will be shown that the EPR-based models offer an effective
and unified approach to modelling of materials with complex behaviour in finite element analysis of
boundary value problems.
& 2012 Elsevier B.V. All rights reserved.
1. Introduction
Finite element method has, in recent decades, been widely
used as a powerful tool in the analysis of engineering problems. In
this numerical analysis, the behaviour of the actual material is
approximated with that of an idealised material that deforms
in accordance with some constitutive relationships. Therefore, the
choice of an appropriate constitutive model that adequately
describes the behaviour of the material plays an important role
in the accuracy and reliability of the numerical predictions. During
the past few decades several constitutive models have been
developed for various materials including soils. Among these
models there are simple elastic models [1], plastic models (e.g.,
[2]), models based on critical state theory [3], and single or double
hardening models [4,5], etc. Most of these models involve deter-mination of material parameters, many of which have little or no
physical meaning [6]. In conventional constitutive material mod-
elling, an appropriate mathematical model is initially selected and
the parameters of the model (material parameters) are identified
from appropriate physical tests on representative samples to
capture the material behaviour. When these constitutive models
are used in finite element analysis, the accuracy with which theselected material model represents the various aspects of the
actual material behaviour and also the accuracy of the identified
material parameters affect the accuracy of the finite element
predictions.
In the past few years, the use of artificial neural networks
(ANN) has been introduced as an alternative approach to con-
stitutive modelling of materials. The application of ANN for
modelling of the behaviour of concrete was first proposed by
Ghaboussi et al. 1991 [7]. Ghaboussi and Sidarta [8] presented an
improved technique of ANN approximation for learning the
mechanical behaviour of drained and undrained sand. Ghaboussi
et al. 1998 and Sidarta and Ghaboussi [9,10] presented a new
autoprogressive approach to train ANN constitutive model
(autoprogressive ANN). In this approach initially, a finite elementmodel of an available experimental test (with the measured
boundary forces and displacements) is created using a pre-trained
ANN model as the constitutive material model. Then the mea-
sured forces and displacements data are applied incrementally to
the FE model and through the increments the ANN model is
updated with more data. The data for training ANN come from the
stresses and strains at gauss points of all elements. In this method
the main idea is that the FE model of the experimental tests
usually contain a large number of stresses and strains with a wide
range of different values that can be used for training of the ANN
model. Hashash et al. [11] continued and extended the autopro-
gressive training methodology in a new framework, self learning
Contents lists available at SciVerse ScienceDirect
journal homepage: www .elsevier.com/locate/finel
Finite Elements in Analysis and Design
0168-874X/$ - see front matter & 2012 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.finel.2012.04.005
n Corresponding author. Tel.: þ44 1392 723640; fax: þ44 1392 217965.
E-mail addresses: [email protected] (A.A. Javadi),
[email protected] (A. Faramarzi), [email protected] (A. Ahangar-Asr).1 Tel.: þ44 1634 883126.2 Tel.: þ44 1392 723909.
Finite Elements in Analysis and Design 58 (2012) 53–65
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protected]:[email protected]:[email 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8/18/2019 Cyclic Loading Using Intelligent Finite Element Method
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simulation (SelfSim), to extract soil constitutive behaviour from a
sequence of construction stages of a braced excavation using
measurements of lateral wall deflection and surface settlements.
The role of autoprogressive and self-learning simulation was also
studied by other researchers [12–19]. These works indicated that
neural network based constitutive models can capture nonlinear
material behaviour with a high accuracy. The developed ANN
models are versatile and have the capacity to continuously learn
as additional material response data become available.On the subject of studying material behaviour under cyclic and
hysteric loading using ANNs, a few works have been reported so
far. Furukawa and Hoffman [20] proposed an approach to mate-
rial modelling using neural networks, which can describe mono-
tonic and cyclic plastic deformation and its implementation in a
FEA system. They developed two neural networks, each of which
was used separately to represent the back stress and the drag
stress. After training and validation stages of neural networks
(NNs), the neural network constitutive models (NNCMs) were
implemented in FEA to update the stiffness matrix ( D). In this
approach D is made of an elastic matrix De, and a plastic matrix
Dp. In the proposed approach the elastic matrix was derived from
Young’s modulus and Poisson’s ratio and only the plastic matrix
was updated using the developed ANNs.
Tsai and Hashash [21] showed the application of SelfSim
method in dynamic soil behaviour. They described the imple-
mentation of SelfSim to integrate data from field measurements
and numerical simulations of seismic site response to obtain the
underlying cyclic soil response. They applied the SelfSim to study
1D seismic site response.
Yun et al. [22] introduced an approach for ANN-based model-
ling of the cyclic behaviour of materials. They focused on the issue
in the hysteric behaviour of material where one strain value may
correspond to multiple stresses and this can be a major reason
that stops NNs from learning hysteretic and cyclic behaviour. To
overcome this issue, they introduced two new internal variables
in addition to the other ordinary inputs of ANN-based constitutive
material models to help the learning of the hysteretic and cyclic
behaviour of materials. The ANN models trained in this way were
implemented in a FE model to analyse a boundary value problem
(steel beam–column connection) under cyclic loading.
Yun et al. [23] extended the ANN-based cyclic material model
developed by [22] to beam–column connections by adding the
mechanical and design parameters as inputs of the ANN model.
Moreover Yun et al. [24] used self-learning simulation to char-
acterize cyclic behaviour of beam–column connections in steel
frames.
Ghaboussi et al. [25] developed a hybrid modelling framework
to analyse engineering systems. The hybrid method combines the
mathematical models of engineered systems (derived based on
physics and mechanical laws) with artificial neural network
models using autoprogressive and self-learning simulation. They
applied this hybrid method to model and analyse a steel jointunder cyclic load.
The implementation of the ANN based constitutive models in FE
codes has been the interest of many researchers. Shin and Pande
[26] used a self learning code to identify elastic constants for
orthotropic materials from a structural test. Hashash et al. [27]
described some of the issues related to the numerical implementa-
tion of NNCM in finite element analysis and derived a closed-form
solution for material stiffness matrix for the neural network-based
constitutive model. Jung and Ghaboussi 2006 [28] presented a rate
dependant ANN material model for creep behaviour of concrete
and its implementation in finite element software (ABAQUS),
through its user material subroutine (UMAT). Kessler et al. [29]
demonstrated the implementation of an ANN material model in
ABAQUS, through user subroutine VUMAT. Haj-Ali and Kim [30]
presented a neural network based constitutive model for fibre
reinforced polymeric (FRP) composites. The developed ANN model
was implemented in ABAQUS user material subroutine to analyse
a notched composite plate with an open hole. Yun et al. [22,23]
also implemented the developed ANN models for materials under
cyclic load in ABAQUS.
The authors have also carried out extensive research on applica-
tion of neural networks in constitutive modelling of complex
materials in general and soils in particular. They have developedan intelligent finite element method (NeuroFE code) based on the
incorporation of a back propagation neural network (BPNN) in finite
element analysis (e.g., [31,32]). The intelligent finite element model
has been applied to a wide range of boundary value problems
including cyclic loading and has shown that ANNs can be very
efficient in learning and generalising the constitutive behaviour of
complex materials [33].
In this paper a fundamentally different approach is presented
for constitutive modelling using Evolutionary Polynomial Regres-
sion (EPR). In the proposed EPR approach the optimum structure
for the material constitutive model representation and its para-
meters are determined directly from raw data. Furthermore, it
provides a transparent and structured representation of the
constitutive relationships that can be readily incorporated in a
finite element code. Javadi and Rezania [34] and Javadi et al. [35]
presented the application of the EPR-based constitutive models in
material modelling under monotonic loading conditions. How-
ever, this paper focuses on the application of the EPR-based
constitutive models, to the simulation of behaviour of soils under
cyclic loading and its integration in a FE model. The development
and validation of the EPRCM and its integration in FEA are
presented and the efficiency of the methodology is examined by
application to the complex problem of cyclic loading of soils. It is
shown that the proposed methodology can simulate the real
behaviour of complex materials under cyclic loading with very
high accuracy. The main advantages of using an EPR approach are
highlighted.
In what follows, the main principles of EPR will be outlined. The
application of EPR in modelling of nonlinear constitutive relation-
ships and the implementation of developed EPRCMs in FE analysis
will be illustrated with two examples. An EPR will be trained with
data from results of a synthetic triaxial cyclic loading tests. The
trained EPR will then be incorporated into a finite element model
which will in turn be used to analyse the behaviour of the soil
under cyclic loading. The training and generalisation capabilities of
the EPR in extending the learning to cases of multiple, variable and
irregular cycles will be investigated.
2. Evolutionary polynomial regression
Evolutionary polynomial regression (EPR) is a data-driven
method based on evolutionary computing, aimed to search forpolynomial structures representing a system. A general EPR
expression can be presented as [36]
y ¼Xn j ¼ 1
F ð X , f ð X Þ,a jÞþa0 ð1Þ
where y is the estimated vector of output of the process; a j is a
constant; F is a function constructed by the process; X is the
matrix of input variables; f is a function defined by the user;
n is the number of terms of the target expression. The general
functional structure represented by F is constructed from ele-
mentary functions by EPR using a genetic algorithm (GA) strategy.
The GA is employed to select the useful input vectors from X to be
combined. The building blocks (elements) of the structure of F can
be defined by the user based on understanding of the physical
A.A. Javadi et al. / Finite Elements in Analysis and Design 58 (2012) 53 –6554
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processes. While the selection of feasible structures to be com-
bined is done through an evolutionary process the parameters a jare estimated by the least square method.
EPR is a technique for data-driven modelling. In this technique,
the combination of the genetic algorithm to find feasible struc-
tures and the least square method to find the appropriate
constants for those structures implies some advantages. In parti-
cular, the GA allows a global exploration of the error surface
relevant to specifically defined objective functions. By using suchobjective functions some criteria can be set in order to (i) avoid
overfitting of models, (ii) push the models towards simpler
structures, and (iii) avoid unnecessary terms representative of
the noise in data [36]. Selecting an appropriate objective function,
assuming pre-selected elements in Eq. (1) based on engineering
judgment, and working with dimensional information enable
refinement of final models. Application and capability of EPR in
modelling and analysing different civil and geotechnical engineer-
ing problems have been investigated by the authors, [37–39].
Detailed explanation of the method is out of the scope of this
paper and can be found in [36,40].
3. Application of EPR for modelling of material behaviour
In modelling using EPR, the raw experimental or in-situ test
data are directly used for training the EPR model. In this approach,
there are no mathematical models to select and as the EPR learns
the constitutive relationships directly from the raw data it is the
shortest route from experimental research to numerical model-
ling. In this approach there are no material parameters to be
identified and as more data become available, the material model
can be improved by re-training of the EPR using the additional
data. Furthermore, the incorporation of an EPR in a finite element
procedure avoids the need for complex yield/failure functions,
flow rules, etc. An EPR equation can be incorporated in a finite
element code/procedure in the same way as a conventional
constitutive model. It can be incorporated either as incremental
or total stress–strain strategy [35]. In this study the incrementalstrategy has been successfully implemented in the EPR-based
finite element model.
3.1. Input and output parameters
The choice of input and output quantities is determined by both
the source of the data and the way the trained EPR model is to be
used. A typical scheme to train most of the neural network based
material models includes an input set providing the network with
the information relating to the current state units (e.g., current
stresses and current strains) and then a forward pass through the
neural network yields the prediction of the next expected state of
stress and/or strain relevant to an input strain or stress increment [9].
The same idea has been utilised in this work. Thus depending on theproblem and the available data, typically the mean stress p0i,
deviatoric stress qi, volumetric strain eiv and shear strain eiq are used
as the input parameters representing the current state of stress and
strain in a load increment i, and the devatoric stress qiþ1and/or
volumetric strain eiþ1v corresponding to the input incrementaldeviatoric strain Deiq are used as the output parameters.
The database is divided into two separate sets. One set is used
for training to obtain the EPR model and the other one is used for
validation to appraise the applicability of the trained model.
3.2. EPR procedure and developed EPRCMs
Before starting the training procedure, a number of constraints
can be implemented to control the evolutionary process in terms
of length of equations, type of functions used, number of terms,
range of exponents, number of generations etc. Therefore, there is
a potential to achieve different models for a particular problem
which enables the user to gain additional information for differ-
ent scenarios [40]. Applying the EPR procedure, the evolutionary
constitutive material modelling starts from a constant mean of
output values. By increasing the number of evolutions it gradually
picks up the different participating parameters in order to form
equations representing the constitutive relationship. Each pro-posed model is trained using the training data and tested using
the testing data. The level of accuracy at each stage is evaluated
based on the coefficient of determination (COD), i.e., the fitness
function as
COD¼ 1
PN ðY aY pÞ
2PN ðY að1=N Þ
PN Y aÞ
2 ð2Þ
where Y a is the actual target value; Y p is the EPR predicted value;
N is the number of data points on which the COD is computed. If
the model fitness is not acceptable or the other termination
criteria (in terms of maximum number of generations and max-
imum number of terms) are not satisfied, the current model
should go through another evolution in order to obtain a new
model [40,34].
4. Incorporating of EPRCM in FEA
The developed EPRCMs are implemented in the widely used
general-purpose finite element code ABAQUS through its user
defined material module (UMAT). UMAT updates the stresses and
provide the material Jacobian matrix for every increment in every
integration point [41]. The manner, in which EPRCM is incorpo-
rated in UMAT is shown in Fig. 1. In the developed methodology,
the EPRCM replaces the role of a conventional constitutive model.
The source of knowledge for EPR is a set of raw experimental (or
in situ) data representing the mechanical response of the material
to applied load. When EPR is used for constitutive description, the
physical nature of the input–output data for the EPR is deter-mined by the measured quantities, e.g., stresses, strains, etc.
The constitutive relationships are generally given in the
following form [42]
Dr¼DDe ð3Þ
where D is material stiffness matrix, also known as the Jacobian.
For an isotropic and elastic material, matrix D is given in terms of
Young’s modulus, E , and Poisson’s ratio, n.Therefore the function of an EPR-based constitutive model in a
FE model (at every element’s integration point) is as follows:
(i) For the iþ1th load increment, the input pattern for the
EPRCM contains (1) the values of ( p0i,
qi,
eiv,
eiq) which have
already been calculated in the previous load increment and
(2) an arbitrary value of Deiq. The new value of qiþ1 and e iþ1v
are then calculated for the next step.
(ii) For each load increment the material Young’s modulus, E EPR ,
and the Poisson’s ratio, nEPR can be calculated from therelationship between the relevant stresses and strains. For
example for axisymetric condition:
E EPR ¼ Dqi
Dei1ð4Þ
nEPR ¼ 1DeivDei1
!=2 ð5Þ
where e1 is the axial strain.
A.A. Javadi et al. / Finite Elements in Analysis and Design 58 (2012) 53–65 55
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Therefore with D being recalculated in the FE procedure, the
element stiffness matrix for every single element is updated at
each load increment as
Z O
BT DBdO ð6Þ
where B is the strain matrix and O is the elemental area.
Consequently the global stiffness matrix for a particular problem
is refined for each load increment. The whole procedure ensures
that the constitutive model follows the actual behaviour of
the material, both at the element level and at the global level.
This is primarily because EPRCM avoids the errors associated
with idealisation of the material behaviour and captures and
accurately represents (as will be shown in the examples) the
constitutive behaviour of the material directly from data.
5. Numerical examples
To illustrate the developed computational methodology,
three numerical examples of application of the developed
EPR-based finite element method to engineering problems are
presented. In the first example, the application of the methodol-
ogy to a simple case of linear elastic material behaviour is
examined. In the second example, the method is applied to a
boundary value problem involving the analysis of stresses and
strains around a tunnel considering nonlinear and elasto-plastic
Start
Input Data (Geometry,
Applied Load, Initial and
Boundary Condition)
Increase the Applied Load
Incrementally
Convergence
I t er a t i onL o o p
Output Result
L o a d I n c r em en t
L o o p
STOP
Start
Input Data (Experimental
Data, Physical Insight)
Genetic
Algorithm
Mathematical
Structure
Least Square
Symbolic
Function
Fitness
Check based on fitness criteria
and or generation number
YESNO
NO
YES
Whole load
applied?
YES
NO
EPR
Constitutive
equation
EPR FEA
Current state of
stresses and strains
1- Next state of
Stresses
2- Jacobian Matrix
EPRCM(s)
Solve the Main
Equation
UMAT
Fig. 1. The incorporation of EPR-based material model in ABAQUS finite element software.
A.A. Javadi et al. / Finite Elements in Analysis and Design 58 (2012) 53 –6556
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material behaviour. In the third example, the application of
the method to the analysis of the behaviour of soil under cyclic
loading is presented.
5.1. Example 1
This example involves a thick circular cylinder conforming to
plane strain conditions. Fig. 2 shows the geometric dimensions
and the element discretization employed in the solution where 12
eight-node isoparametric elements have been used. The cylinder
is made of linear elastic material with a Young’s modulus of
E ¼2.1105 N/mm2 and a Poisson’s ratio of 0.3 [42]. This example
was deliberately kept simple in order to verify the computationalmethodology by comparing the results with those of a linear
elastic finite element model. The compressibility of the material
is assumed to be negligible and hence the EPR model for
volumetric strain is not considered in this example. The loading
case considered involves an internal pressure of 8.0104 kPa as
shown in Fig. 2.
Fig. 3a shows a linear elastic stress–strain relationship with a
gradient of 2.1105 MPa. The slope of this line represents the
elastic modulus, E , for the material. The data from this figure were
used to train the EPR model in order to capture the linear stress–
strain relationship for the material.
After training, the selected EPR model to represent the stress–
strain:
qiþ1 ¼1:5 108
Deq3:4662 106
ffiffiffiffiqi
q þ2:4231 1011Deq
þ2:4231 1011eq0:00504eq
ffiffiffiffiqi
q þ0:01197 ð7Þ
Fig. 3(b) shows the stress–strain relationship predicted by the
EPR model, together with the original one. It is seen that after
training, the EPR model has successfully captured the stress–
strain relationship with a precise accuracy.
100 mm
200 mm
P
Fig. 2. FE Mesh in symmetric quadrant of a thick cylinder.
0
50
100
150
200
250
0
Strain
S t r e s s ( M P a )
0
50
100
150
200
250
0 0.001
Strain
S t r e s s ( M P a )
0.0002 0.0004 0.0006 0.0008 0.001
0.0002 0.0004 0.0006 0.0008
Fig. 3. (a) Linear stress–strain relationship used for training and (b) the results of
EPR predictions for stress–strain values.
0
40
80
120
160
200
0
radius (mm)
R a d i a l S t r e s s ( M P
a )
Standard FEM
IFEM (EPRCM)
0
0.02
0.04
0.06
0.08
radius (mm)
R a d i a l D i s p l a c e m e n t ( m m )
Standard FEM
IFE (EPRCM)
40 80 120 160 200
0 40 80 120 160 200
Fig. 4. Comparison of the results of the EPR-FEM and standard FEM solution in
terms of (a) radial stress and (b) radial displacement.
A.A. Javadi et al. / Finite Elements in Analysis and Design 58 (2012) 53–65 57
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The EPR-based finite element model incorporating the trained
EPR model was used to analyse the behaviour of the cylinder
under applied internal pressure. The results are compared with
those obtained using a standard linear elastic finite element
method. Fig. 4 shows the radial displacements and radial stresses
along a radius of the cylinder, predicted by the two methods.
Comparison of the results shows that the results obtained using
the EPR-based FEM are in excellent agreement with those
attained from the standard finite element analysis. This showsthe potential of the developed EPR-based finite element method
in deriving constitutive relationships from raw data using EPR
and using these relationships to solve boundary value problems.
5.2. Example 2
This example involves the analysis of deformations around a
tunnel subjected to excavation and gravity loadings. The geometry of
the tunnel and the finite element mesh are shown in Fig. 5. The finite
element mesh includes 142 eight-node isoparametric elements and
451 nodes. The depth of the tunnel crown from the ground surface is
12 m. The analysis is done in two steps. The first step includes a
geostatic analysis where all the elements are subjected to gravity
loading. In the second step 46 elements representing the tunnelelements, are removed to simulate the excavation process.
The results from a series of drained triaxial tests from
literature [43], containing information on both shear and volu-
metric behaviour, were used for the training of the EPR based
constitutive model with an incremental stress–strain (tangential
stiffness) strategy. It was assumed that the soil tested was
representative of the soil material around the tunnel. The test
data were arranged as shown in Table 1 and used to train an
EPRCM to model the stress–strain relationship for the soil.
The results from five tests conducted at confining pressures
of 50, 100, 300, 400, 600 kPa were used for training of the
EPR models while those for the sixth and seventh tests at the
confining pressure of 200, 500 kPa were used for validation of the trained EPR models. At the end of the training and testing
procedure, the selected best EPR models representing the beha-
viour of the soil are:
qiþ1 ¼7:3384 107 p03ev
qi
5:077eivqiDeq
þ0:416 p0eq
qi
0:47537eqþ0:90403qi0:09154qiDeq
þ0:00011719ðqiÞ2þ0:06081 p0þ0:12364 p0Deq
0:0025518 p0eqDeq0:00006795 p02þ4:4456 ð8aÞ
eiþ1v ¼ 1:0005eivþ0:0010163q
iDeqþ0:000016185q
ieqDeq
0:001178 p0
Deqþ0:0017108 ð8bÞFig. 5. Geometry of the tunnel and the FE mesh.
Table 1
Input and output parameters used for training the
EPR constitutive model for the tunnel example.
Input parameters Output parameter
p0i , qi , eiv , eiq , De
iq q
iþ1
p0i , qi , eiv , eiq , De
iq e
iþ1v
0
100
200
300
400
500
600
700
800
D e v i a t o r i c S t r e s s ( k P a )
Deviatoric Strain %
0
100
200
300
400
500
600
700
0
D e v i a t o r i c S t r e s s ( k P a )
Deviatoric Strain %
5 10 15 20 25 30
0 5 10 15 20 25 30
Fig. 6. (a) Results of training of the ANN and (b) stress–strain relationship
predicted by the trained EPR.
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Figs. 6(a) and 7(a) shows the stress–strain curves predicted by
Eqs. (8a) and (8b) against those expected and used as training
data. It is clearly seen that, the EPR was able to capture the
constitutive (nonlinear) stress–strain relationship for the soil with
very good accuracy. The generalisation capability of the EPRCMs
is shown in Fig. 6(b) and Fig. 7(b). The data from the test
conducted at the confining pressures of 200 and 500 kPa (which
did not form a part of the training data) were used to test the
trained EPRCMs. The predicted output values of the EPR modelsare compared with the experimentally measured values in
Fig. 6(b) and Fig. 7(b). It is seen that the generalisation capability
of the trained EPRCM is excellent.
In addition, the EPR models (Eqs. (8a) and (8b)) are used to
predict the entire stress paths, incrementally, point by point, in
the q : eq and ev : eq spaces. This is used to evaluate the capabilityof the incremental EPR models to predict the behaviour of the soil
during the entire stress paths. Fig. 8 illustrates the procedure
followed for updating of the input parameters and building the
entire stress path for a shearing stage of a triaxial test. At the start
of the shearing stage in a conventional triaxial experiment, the
values of all parameters are known. For example in a test on a
sample of a saturated soil, the values of effective mean stress, p0i,
deviator stress qi
, deviatoric strain ei
q and volumetric strain, ei
vareknown from values of applied cell pressure, pore water pressure
and volume change at the end of the previous stage (e.g., at the start
of shearing stage eiq ¼ 0, eiv ¼ 0 and q
i ¼ 0). Then, for a given
increment of deviatoric strain, Deq, the values of qiþ1 and eiþ1v arecalculated from the EPR models (Eqs. (8a) and (8b), respectively). For
the next increment, the values of p 0i, e iq, eiv q
i and are updated as
qi ¼ qiþ1 ð9Þ
eiv ¼ eiþ1v ð10Þ
p0i ¼ p0iþ qiþ1qi
3
ð11Þ
eiq ¼ eiqþDeq ð12Þ
and the next points on the curves are predicted using the EPR
models. The incremental procedure is continued until all the points
on the curves are predicted. Fig. 9(a) and (b) show the comparison
between two complete curves predicted using the EPR models
following the above incremental procedure and the experimental
results. The predicted results are in good agreement with the
experimental results and the facts that (i) the entire curves have
been predicted point by point; (ii) the errors of prediction of the
individual points are accumulated in this process, and still the EPR
-7
-6
-5
-4
-3
-2
-1
0
1
2
V o l u m e t r i c
S t r a i n %
Deviatoric Strain %
-6
-5
-4
-3
-2
-1
00
V o l u m e t r i c S t r a i n %
Deviatoric Strain %
EPR Prediction (200 kPa)
EPR Prediction (500 kPa)
Actual Data (200 kPa)
Actual Data (500 kPa)
5 10 15 20 25 30
0 5 10 15 20 25 30
Fig. 7. (a) Results of training of the EPR and (b) volumetric strain predicted by the
trained EPR.
Fig. 8. procedure followed for updating of the input parameters and building the
entire stress path for a shearing stage of a triaxial test.
0
100
200
300
400
500
600
700
800
0
D e v i a t o r i c S t r e s s ( k P a )
Deviatoric Strain (%)
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
0
V o l u m e t r i c S t r a i n ( % )
Deviatoric Strain (%)
5 10 15 20 25 30
5 10 15 20 25 30
Fig. 9. Comparison of EPR incremental simulation with the actual data
(a) simulation of deviatoric stress and (b) simulation of volumetric strain.
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models are able to predict the complete stress paths with a good
degree of accuracy are testaments to the robustness of the
developed EPR framework for modelling of soils.
These figures show that the EPR has been able to capture the
general trend of the nonlinear relationship of stress and strain
with a good accuracy. It also shows that the EPR model was
trained sufficiently to adequately model the stress–strain beha-
viour of the soil. The trained EPRCMs were incorporated in the
EPR-based finite element (EPR-FEM) using UMAT in ABAQUS. TheFE incorporating the EPR models was then used to simulate the
behaviour of the tunnel under gravity and excavation loadings.
For the conventional finite element analyses, the results of the
triaxial tests were used to derive the material parameters for the
Modified Cam Clay (MCC) and Mohr–Coulomb (MC) models for
the soil (see Table 2).
Fig. 10 shows the comparison between the displacements in the
tunnel predicted using standard finite element analyses using the
MCC and MC models as well as the EPR-based finite element
method where the raw data from the triaxial tests were directly
used in deriving the EPR-based constitutive model. Deformations of
tunnel is magnified by a factor of 5. The patterns of deformation are
similar in all three analyses. Despite the relatively small difference
between the results from the different analyses, it can be argued
that the EPR-based FE results are more reliable, as this method used
the original raw experimental data to learn the constitutive
relationships for the material and it did not assume a priori any
particular constitutive relationships, yield conditions, etc.
From the results obtained, it is shown that the developed
intelligent finite element method is also capable of capturing
more complex constitutive relationships of materials and can
offer very realistic prediction of the behaviour of structures.
5.3. Example 3: behaviour of soil under cyclic loading
In this example, the behaviour of a soil is studied in triaxial
tests under cyclic axial loading. The test data for this example
were generated by numerical simulation of triaxial experiments.
In general, generating data by numerical simulation has advan-
tages including: (i) it is more economic (ii) it is far less time
demanding, (iii) it can simulate loading paths and test conditions
that cannot be easily achieved in physical testing due to physical
constraints of the testing equipment. The data for training and
validation of the EPR were created by finite element simulation of
triaxial cyclic loading tests at constant cell pressures using the
Modified Cam Clay model. The material parameters assumed for
the soil are:
l¼0.174 (slope of the virgin consolidation line),
k¼0.026 (slope of the unloading/reloading lines in the v:Lnp0
space),
M ¼1 (slope of the critical state line in the q: p0 space),
P 1¼100.0 kPa (isotropic preconsolidation pressure),
Table 2
Material parameters for Modified Cam Clay and Mohr–Coulomb models.
C 0 (kPa) f0 ðdeg:Þ M k e0 n g (kN/m3) l
11.7 21 0.8 0.00715 0.921 0.3 17 0.091
Fig. 10. Comparison of the results of the intelligent FEA and conventional FE
analyses using Mohr–Coulomb and Duncan–Chang models.
0
50
100
150
200
250
0.00
Strain
D e v i a t o r S t r e s s ( k P
a )
100 kPa
150 kPa
200 kPa250 kPa
300 kPa
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Axial Strain
V o l u m e t r i c S t r a i n
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
Fig. 11. Typical cyclic loading test data used for training and validation of EPR.
(a) Shear stress and (b) volumetric strain.
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The generated data were used to train and test EPRCMs. The
EPRCMs were then incorporated in the intelligent EPR-based
finite element model to represent the soil behaviour under cyclic
loading. The results of the EPR-based finite element analyses were
compared with those attained using conventional finite element
method. The performance of the model was evaluated for two
separate cases of loading where the soil was subjected to:
1- Multiple and regular cycles of loading and unloading; and2- Multiple and irregular cycles of loading and unloading.
The data generated by numerical simulation of the cyclic
loading tests at confining pressures of 100, 150, 200, 250 and
300 kPa are shown in Fig. 11(a) and Fig. 11(b). In order to
introduce a level of noise that inevitably exists in real triaxial
test data, numerical simulation for each confining pressure was
repeated by changing the total number of load increments in
the simulation and the obtained data were combined and used
in training of the EPR models. Fig. 12 shows typical results of
the tests conducted at confining pressure of 150 kPa with four
different load increments.
The data from the tests at confining pressures of 100, 150, 200and 300 kPa were used for the training of the two EPR models.
The first model was developed to predict the deviator stress qiþ1
and the second one to predict the volumetric strain eiþ1v . Thetrained EPR models were validated using the data from the test at
the confining pressure of 250 kPa. The input and output para-
meters used for training of the EPR models are presented in
Table 3. In the table p0 is net mean stress, q is deviator stress, ev isvolumetric strain and e1 is axial strain. The indices i and iþ1represent the state of stress or strain at current (incremental)
step, and the next step, respectively.
The selected EPR models for q and ev are:
qiþ1 ¼0:0084ðqiÞ3
p02eiv
2667:247ðqiÞ2ðDe1Þ2
p0e1
0:060714ðqiÞ3De1 p0eiv
þ1:8866evDe1
qi
1:9676qiDe1eiv
þ888:4ðqiÞ2ðDe1Þ
3
eiv
þ104:4964 p0De11:4 105 p02þ0:018826 p02qieivDe1
þ1:0525qi0:71525 ð13aÞ
eiþ1v ¼ 0:02369qiDe1
p0e1
0:4217qiDe1 p0
þ9:3 106e1
qi
þ0:45727De1þ0:99eivþ0:000041535 ð13bÞ
Fig. 13(a) and (b) show the results predicted by Eqs. (13a)
and (13b), respectively, for the training data set together with the
actual training data. Fig. 13(a) shows the results of the devitoric
stress model for different confining pressures. Typical results of
the volumetric strain model (Eq. (13b)) at confining pressure of
100 kPa are presented in Fig. 13(b) together with the actual data.
It is seen from the figures that the EPR models were capable of
learning, with very good accuracy, the constitutive relationship of
the soil under cyclic loading paths. The trained EPRCMs were
validated using a data set corresponding to the confining pressureof 250 kPa. The results of the validation tests are shown in
Fig. 14(a) and (b). It is shown that the trained EPR models were
able to generalise the training to loading cases that were not
introduced to the EPR during training. Moreover the incremental
prediction capability (described in Example 2) of the developed
0
10
20
30
40
50
0.000
Strain
D e v i a t o r S t r e s s ( k P a )
4 increments
8 increments20 increments40 increments
0.002 0.004 0.006 0.008 0.010 0.012 0.014
Fig. 12. Typical cyclic loading test results with different load increments at
confining pressure of 150 kPa.
Table 3
Input and output parameters used for training and
testing of the EPR models for shear stress and
volumetric strain cyclic models.
Input parameters Output parameter
p0i, qi , eiv, ei1 , De
i1 q
iþ1
p0i, qi , eiv, ei1 , De
i1 e
iþ1v
0
50
100
150
200
250
300
D e v i a t o r i c S
t r e s s ( k P a )
Axial Strain
Actual Data (100 kPa) Actual Data (150 kPa)
Actual Data (200 kPa) Actual Data (300 kPa)
EPR prediction (100 kPa) EPR prediction (150 kPa)
EPR prediction (200 kPa) EPR prediction (300 kPa)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.00
V o l u m e t r i c S t r a i n
Axial Strain
Actual Dat