Composite Structures 246 (2020) 112407

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Composite Structures
journal homepage: www.elsevier.com/locate/compstruct

Theory-guided machine learning for damage characterization of composites T

a,⁎ b c
Navid Zobeiry , Johannes Reiner , Reza Vaziri
a
Materials Science & Engineering Department, University of Washington, 302 Roberts Hall, Seattle, WA, USA
b
School of Engineering, Faculty of Science Engineering & Built Environment, Deakin University, Australia
c
Composites Research Network, Departments of Civil Engineering and Materials Engineering, The University of British Columbia, Vancouver, Canada

A R T I C LE I N FO A B S T R A C T

Keywords: A novel approach for damage characterization through machine learning is presented where theoretical
Damage characterization knowledge of failure and strain-softening is linked to the macroscopic response of quasi-isotropic composite
Machine learning laminates in over-height compact tension tests. A highly efficient continuum damage finite element model en-
Neural network ables the training of a system of interconnected Neural Networks (NNs) in series solely based on the macroscopic
Progressive damage modeling
load-displacement data. Using experimental results, the trained NNs predict suitable damage parameters for
Failure
progressive damage modeling of IM7/8552 composite laminates. The predicted damage properties are validated
successfully using experimental measurements obtained through cumbersome non-destructive data analysis. The
proposed strategy demonstrates the effectiveness of machine learning to reduce experimental efforts for damage
characterization in composites.

1. Introduction represent physically realistic damage mechanisms. However, these

models suffer from significantly high computation time and im-
Over the last five decades, numerous computational models have plementation complexity, as well as characterization cost. Some notable
been developed to predict damage evolution and failure in fibre re- applications include simulation of delamination migration [9], fatigue
inforced composite laminates [1–4]. However, despite the tremendous degradation [10–12], matrix cracking-induced delamination [13,14],
efforts and partial success in establishing suitable theories, the search and mesh-independent matrix cracking [15]. In contrast to discrete
for physics-based damage models, validated for industrial applications, approaches, Continuum Damage Models (CDM) smear the effects of
is still ongoing. Christensen [5,6] recently outlined the reasons and damage over the volume of a finite element (usually via reduction in
possible solution strategies to overcome the lack of progress in this mechanical properties) to gain computational efficiency, while redu-
field. As one of the reasons, he identified the need for high quality cing characterization cost. Some successful examples of CDM are the
physical tests of composites in order to develop and validate proposed simulation of axial crushing [16,17], the prediction of size-effects in
theories [5]. In the absence of robust and validated damage models, notched CFRP laminates [18], and the structural analysis of laminates
industry must rely on the building-block approach to substantiate where kinking is the pre-dominant failure mechanism [19]. By com-
composite structures [7] for which parallel to numerical activities, paring two CDMs with different underlying assumptions, Reiner et al.
comprehensive testing programs must be undertaken. For statistically- [20] concluded that CDMs are able to predict the structural response of
based damage properties (i.e. A-basis or B-basis [7]), a testing program composites in loading scenarios that lead to fibre-dominated damage
may include thousands of tests at the coupon-level [8]. Depending on mechanisms.
the simulation approach, this includes a combination of standard ASTM Unlike discrete models, the application of continuum models re-
tests and non-standard tests with complex data reduction schemes. quires physical testing at the laminate level. For example, the Over-
Despite efforts by both industry and academia to reduce testing, the height Compact Tension (OCT) test developed by Kongshavn and
building-block approach has yet to undergo any major changes. The Poursartip [21] produces stable and self-similar crack growth in a pre-
testing program to characterize damage properties is influenced by the notched laminate. This enables characterization of fracture energy at
choice of modeling approach. In the context of Finite Element Analysis the laminate or sub-laminate level [22,23]. Using Digital Image Cor-
(FEA), distinct failure modelling approaches can be classified as con- relation (DIC) technique, Zobeiry et al. [24] were able to extract
tinuum and discrete. Fully discrete models are promising tools to complete stress-strain curves within the fracture process zone to

⁎
Corresponding author.
E-mail address: navidz@uw.edu (N. Zobeiry).

https://doi.org/10.1016/j.compstruct.2020.112407
Received 15 January 2020; Received in revised form 14 April 2020; Accepted 20 April 2020
Available online 23 April 2020
0263-8223/ © 2020 Elsevier Ltd. All rights reserved.
N. Zobeiry, et al. Composite Structures 246 (2020) 112407

calibrate a sub-laminate based continuum damage model to simulate with a loading rate of 0.25 mm/min, the Pin Opening Displacement
the progression of damage in quasi-isotropic laminates. However, full (POD) and the corresponding loads are recorded. Fig. 2 shows the load-
characterization of strain-softening response at the laminate level re- POD curves from three conducted OCT tests (CTA1, CTA2 and CTA3).
quires conducting many tests with complex data reduction techniques. For the purpose of this study, these load-POD curves provide sufficient
This makes the implementation of such modeling techniques for in- amount of experimental data to calibrate a continuum damage model
dustrial applications quite challenging and costly. by ML as outlined below. Zobeiry [24,36] collected additional valuable
One potential solution is to take advantage of Machine Learning data during and after the OCT testing. Utilization and analysis of DIC
(ML) to reduce the number of characterization tests required at the data on the surface of the outer 90° ply identified the shape and gov-
coupon-level, as well as to address the difficulties associated with data erning parameters of the stress-strain response within the fracture
reduction. With the recent advances in ML, many studies have explored process zone of the laminate. Furthermore, Zobeiry et al. [24] de-
ML applications in engineering problems [25–29]. Specifically for termined the damage height experimentally by cumbersome and de-
composites, ML techniques have been proposed for damage detection structive methods involving de-plying and detailed Scanning Electron
and failure analysis (among others) in recent years (e.g. [30–33]). Microscopy (SEM). This dataset will be used here for the validation of
However, shortcomings of applying theory-agnostic black-box ML ap- the proposed method as a means of calibrating the continuum damage
proaches for scientific and engineering problems are now well ad- model through machine learning techniques.
dressed [34]. One major issue revolves around difficulties in acquiring
physical data for training models such as Deep Neural Networks (DNN) 2.2. Simulation
within the acceptable range of accuracy. Physical data is often small,
fragmented, and suffers from under-represented classes which makes it The commercial explicit finite element software, LS-DYNA, contains
unsuitable for ML applications [34]. As opposed to physical data, vir- several built-in continuum damage models for composite materials.
tual data generated by relatively fast FE models does not suffer from Most of these models are formulated to incorporate ply-based input
similar problems. However, this limits the application of ML to either data to build up a composite laminate such as MAT58 [37], MAT219
speeding-up existing FE simulation tools or increasing their fidelity. [38] or MAT261 [19]. Here, we are aiming to utilize an efficient
Moreover, this usually requires conducting many simulations for well- macroscopic damage model in order to generate sufficient data points
defined and well-understood engineering problems. To address these as input to machine learning algorithms. Since the presented OCT tests
issues, more recently, a new paradigm known as Scientific Machine of the quasi-isotropic [90/45/0/-45]4S IM7/8552 laminates pre-dom-
Learning (SciML) or Theory-Guided Machine Learning (TGML) inantly leads to intra-laminar damage modes, we can use the overall
[27,29,34,35] has emerged. TGML not only relies on a combination of laminate properties to simulate damage progression. Note that the si-
virtual and physical data, but also on the underlying physical laws mulation of inter-laminar delamination typically requires the compu-
governing the problem (i.e. domain knowledge or theory). For example, tationally costly use of cohesive interfaces between the different plies of
during training of Neural Networks (NNs), theory may be used to define the laminate. Therefore, for the purposes of this study we adopt the
physics-based features (i.e. inputs), model architecture, loss function isotropic coupled damage-plasticity material model in LS-DYNA,
and activation function [34]. Compared to theory-agnostic ML models, MAT81, which allows the versatility of supplying local stress–strain
this results in a physically consistent model which requires far less data curves of general shapes that include strain-softening features. In ad-
for training. Notable recent examples are applications of TGML in de- dition to the elastic properties such as Young’s modulus E and Poisson’s
veloping high-fidelity turbulence models [25], high-fidelity molecular ratio ν , the model requires as input a yield stress σpeak and the evolution
dynamics models [26], and near real-time process simulation tools for of damage as a function of the effective plastic strain. In order to sim-
manufacturing of composites [28,35]. plify the generation of input data, we assume that damage initiation
In this study, we explore the application of TGML to reduce ex- coincides with the onset of yielding with the strain at damage initiation
perimental efforts in characterizing damage properties and strain-soft- being denoted by εi . This is similar to the manner in which the MAT81
ening response of laminated composites. By combining FE simulation material model in LS-DYNA was used in previous studies involving
results, limited experimental results, and domain knowledge, a TGML damage simulation of quasi-isotropic composite laminates [24,36,39].
model consisting of physics-based features and several NNs in series Fig. 3a shows a typical curve for damage evolution as a function of the
were constructed and trained. A highly efficient smeared damage model effective plastic strain used as input to MAT81. Here, we consider a bi-
in the explicit finite element software, LS-DYNA, was used to create linear relation where the first linear part is characterized by the initial
sufficient data points for training of the TGML model. Using the global slope through the angle θ and an intermediate effective plastic strain of
load-displacement results from limited number of OCT tests, the trained ε1 − εi . This is followed by a second linear increase until complete da-
model was able to predict the failure properties of quasi-isotropic IM7/ mage saturation at effective plastic strain of εs − εi , where εs denotes the
8552 CFRP laminates leading to construction of the entire strain-soft- damage saturation strain. Fig. 3b shows a typical resulting tri-linear
ening curve. Predictions were validated using both experimental and stress–strain response of the laminate based on the input damage evo-
numerical tests. This study shows how ML can be effectively used to lution in Fig. 3a. Stress increases linearly until reaching σpeak (which
calibrate a continuum damage model by only considering the load- coincides with both the peak stress and yield stress). The bi-linear post-
displacement curves from OCT tests without reliance on cumbersome peak response is governed by ε1 and the damage saturation strain εs . The
analysis of data generated by DIC or destructive sectioning. area under the resulting stress–strain curve (fracture energy density, gf)
is related to the overall fracture energy, Gf . Bazant’s crack-band scaling
2. Damage progression in composite laminates [6] is applied to maintain the fracture energy of the material invariant
with respect to the finite element size. Overall, we consider the fol-
2.1. Testing lowing five input parameters to be calibrated by the proposed machine
learning algorithm: E , εi, ε1, θ and εs .
The geometry and dimensions of the OCT test are illustrated in The finite element model of the OCT test sample is shown in Fig. 1b.
Fig. 1a. A quasi-isotropic [90/45/0/-45]4S laminate made from Hexcel A prescribed displacement is applied to the rigid loading pins (green) in
HexPly IM7/8552 CFRP with nominal ply thickness of 0.125 mm is opposite vertical directions. The mesh only consists of one shell element
used to evaluate damage progression. In this laminate configuration, through the thickness. This coarse discretization is adequate to model
the OCT loading geometry produces stable and self-similar crack the current specimen geometry and quasi-isotropic laminate for which
growth with delamination confined to a narrow fracture process zone delamination is not a major failure mode. If delamination becomes a
ahead of the crack. In the displacement-controlled quasi-static tests major failure mode, multiple shell elements stacked through-the-

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N. Zobeiry, et al. Composite Structures 246 (2020) 112407

Fig. 1. (a) Schematic of the geometry of the OCT test specimen, and (b) Finite element mesh for the numerical simulation of OCT test.

thickness, and connected using cohesive elements may be required. This

however increases the computation time significantly and hence such
higher fidelity models are not suited to generate large amounts of data
for machine learning purposes. The in-plane element size in the ex-
pected fracture process zone is 1 mm × 1 mm. This results in a total of
about 4000 elements. Performing these numerical simulations, which
takes under four minutes on 12 CPUs, is considerably faster than con-
ducting physical OCT tests and such efficiency proves to be essential for
generating sufficiently large dataset required for the machine learning
algorithm.

3. Machine learning for damage characterization

Fig. 2. Load-POD curves obtained from three OCT experiments on quasi-iso- 3.1. Data generation
tropic [90/45/0/-45]4S IM7/8552 laminates. CTA1 to CTA3 refer to three dif-
ferent OCT samples. Using the mesh described previously, and the MAT81 material
model, 10,000 simulations were conducted with LS-DYNA. These si-
mulations were used to create a dataset for training of several neural
networks. In each simulation, LS-DYNA input parameters were varied
randomly from the pre-defined ranges below:

Fig. 3. (a) Damage as a function of plastic strain defined for MAT81 material model in LS-DYNA, and (b) Strain-softening response constructed based on the
definition of the damage for MAT81 material model in LS-DYNA.

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N. Zobeiry, et al. Composite Structures 246 (2020) 112407

• 50GPa < E < 65GPa

• 0.8% < ε < 2% i

• 1.1% < ε < 57%

1

• 65 < θ < 85
° °

• 10% < ε < 65%

s

These ranges were selected based on the typical elastic and damage
properties reported in the literature for quasi-isotropic IM7/8552 la-
minates. It should be noted that by using random input parameters,
there will be no bias in FE results. Any bias in input parameters, may
create under-presented classes in FE results which in turn promotes bias
in behavior of machine learning models. Using the above 5 parameters,
other damage parameters including σpeak , σ1 and Gf can be calculated.
For the 1 mm mesh used to represent the localized zone for damage
progression ahead of the notch, the computed overall fracture energy of
the laminate falls between 50 and 150 kJ/m2. Only 5 of the above 8
parameters are needed to create the complete stress–strain curve. From
initial ML exercises, it was determined that the best combination of Fig. 5. Process of extracting damage properties from the Load-POD curve of an
parameters with the highest accuracy are E , Gf , σpeak , θ and σ1 (i.e. OCT test using four trained neural networks in series.
lowest prediction errors by NN). These will be the outputs of the ma-
chine learning model. because in the current OCT geometry, effects of fracture energy and
For each simulation, several parameters were recorded from nu- damage initiation strain are much more pronounced compared to the
merical results to be used as features (i.e. inputs) in ML: effects of other parameters of the strain softening response (i.e. ε1, θ and
εs ) [36].
• Slopes of the Load-POD curve at POD = 0.1 and 0.3 mm were re- For training of NNs, optimization was performed based on mini-
corded as Slope1 and Slope2 mizing the Mean Square Error (MSE). Root Mean Square Error (RMSE)
• Maximum load, and POD at the maximum load were recorded as was recorded at the end of each training as the representative error
F and d respectively as shown schematically in Fig. 4
max 100 value. Out of 10,000 LS-DYNA simulations, 70% were used for training
• Based on the value of d , several points were recorded on the load-
100 and 30% for validation. Reported RMSE values are based on predicting
POD curve such that: the validation set only. A combination of 4 NNs in series were trained to
n characterize damage properties based on the parameters of load-POD
dn = d100, n ∈ [80, 90, ⋯, 170] curve only. Each NN has 4 hidden layers and 10 nodes per hidden layer.
100 (1)
NNs in series are shown in Figs. 5 and 6 and described below:
In this equation, n/100 is the ratio of deformation at a given load, to
deformation at the maximum load (i.e. d100 ). For each of these points, - Modulus, E : For all simulations, it was determined that for the
load and area below the load-POD curve were recorded as Fn and Un , current quasi-isotropic lay-up, modulus has a linear correlation with
respectively as defined in Fig. 4 slope of the load-POD curve before damage initiation (Slope1 and
Slope2). Therefore, it was possible to determine modulus with a
3.2. Training using theory-guided machine learning simple linear fitting without the need to use ML.
- Fracture energy, Gf : To guide ML, a highly correlated parameter (i.e.
For ML, a high-level API in Python (version 3.6.8), Tensorflow an engineered feature) to fracture energy is defined for training.
(version 1.8.0) [40], was used to train feed-forward dense neural net- Fracture energy can be defined as:
works. For theory-guided machine learning (TGML), we use our un-
derstanding of how parameters of the stress–strain curve affect the load-
POD response to select appropriate features. Also, we use a progressive
approach to determine damage parameters step by step. This is done

Fig. 4. Definition of features extracted from the Load-POD curve used in ma-
chine learning and training of neural networks. The shaded area below the load- Fig. 6. Architecture of the Neural Network #1 in Fig. 5 trained to extract
displacement curve marked as Un , represents the total energy loss due to da- fracture energy based on engineered features from the Load-POD response in
mage and failure at POD = dn . OCT test.

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N. Zobeiry, et al. Composite Structures 246 (2020) 112407

possible to predict the initial slope θ by extending features to

E , Gf , d100, F max , σpeak , F110, F120 . NN #3 was trained to predict θ with
RMSE of 1. 7° . The low accuracy might be due to the current OCT
geometry where the load is not markedly affected by changes in θ
[36].
• σ1: After several trials, it was possible to determine this parameter
with an acceptable accuracy. The best results were obtained using a
NN (NN #4) which was trained with
E , Gf , d100, F max , σpeak , θ , F110, F120 as features. RMSE in validation set
was 85 MPa. Similar to θ , the high RMSE value might be because in
the current OCT geometry, the load is not highly affected by changes
in σ1 [36].

Fig. 7. A highly correlated engineered feature to the fracture energy which was The summary of trainings and hyper-parameters are listed in
used to train Neural Network #1 in Fig. 5 and Fig. 6. Correlations shown for Table 1.
10,000 simulations using LS-DYNA.
3.3. Application
ΔU ΔU ΔU
Gf ∝ ∝ ∝ F F
Crack Length f (Loss of Stifness ) f ( d1 , d2 ) The combination of the four trained NNs in series was used to
1 2 (2)
predict damage properties of IM7/8552 laminates. Load-POD curves
where ΔU represents the loss of internal potential energy (strain en- from three OCT tests as shown in Fig. 2 were used to extract the re-
ergy), and Fi / di represents instantaneous stiffness of the specimen. For quired features. As mentioned previously, modulus was determined
the OCT geometry, the function f has quite a complex form [41,42]. For directly using the slope of the load-POD curves without using ML. A
the purpose of ML training, an approximate function suffices. A recently Young’s modulus of 60200 MPa was calculated for the laminate. Using
proposed method based on an iterative correlation analysis to de- this modulus and features from the load-POD curves, four trained NNs
termine highly correlated functions to physical phenomena was used were employed to predict the damage properties. Average values of
here to determine the shape of the above function [43]. It was de- these properties for the three OCT tests are listed in Table 2. Other
termined that fracture energy is highly correlated to the following en- damage properties (ε1, εi, εs ) were calculated from these predicted va-
gineered feature: lues.
U170 U170
Gf ∝ ∝
Fmax / d100 0.75 Fmax 0.75 3.4. Summary
( )
F170 / d170 ( )
F170 (3)
A summary of the developed approach is shown schematically in
This correlation is depicted in Fig. 7 for 10,000 numerical simula-
tions. Based on this, a NN was trained using 4 input parameters to Fig. 8. In the first step, 10,000 FE simulations were conducted based on
U170 randomly selected strain-softening curves (labelled as Data Generation
predict fracture energy: E , Fmax , d100, 0.75 . The architecture of this
⎛ Fmax ⎞ in Fig. 8). Theory-guided machine learning was used to train several
⎝ F170 ⎠
NN is shown in Fig. 6. Hyper-parameters for ML training are listed in NNs as shown in Fig. 5, capable of performing inverse FE analysis (la-
Table 1. Using the trained NN, the fracture energy was determined with belled as Training in Fig. 8). The training was performed by extracting
a negligible RMSE of 1.9kJ / m2 . features from the load-displacement curve as shown in Fig. 4, and
correlating them to features of the strain-softening curve as listed in

• Peak stress, σ peak : After predicting the fracture energy, it was pos-
Table 1. The selection of the features and architecture of NNs were
guided by theoretical knowledge of failure and strain-softening beha-
sible to predict the peak stress with high accuracy based on the
features of the load-POD curve around the maximum force. This viour in composites as described in previous sections. The trained
understanding was developed using previous numerical studies models were then used to extract the material strain-softening response
carried out by Zobeiry [36]. Using E , Gf , d100, F max , U80 and U90 as based on three experimentally obtained load-displacement curves (la-
inputs, a NN was trained (NN #2) to predict the peak stress with belled as Prediction in Fig. 8). The validation of these predictions is
RMSE of 39MPa.Training hyper-parameters were identical to those discussed next.
in the previous training (Table 1). After several iterations, it was
also concluded that it would be more accurate to train NN #2 with 4. Validation
features representing areas below the load-POD curve (Un ) rather
than load values (Fn ). 4.1. Strain-Softening response
• θ : After predicting the fracture energy and maximum stress, it was
As stated before, Zobeiry et al. [24] applied the DIC data obtained

Table 1
Summary of training parameters and Root Mean Squared Errors (RMSE) for the four Neural Networks in Fig. 5.
Parameter (Output) Features (Inputs) Training Hyper-parameters Error (RMSE)

Gf E,
U170
, d100, Fmax Training dataset = 7000 1.9kJ / m2
0.75
⎛ Fmax ⎞
⎜ ⎟
Validation dataset = 3000
⎝ F170 ⎠ Activation function = ReLU
σpeak E , Gf , d100, F max , U80, U90 Hidden layers = 4 39MPa
θ E , Gf , d100, F max , σpeak , F110, F120 Nodes per layer = 10 1.7°
σ1 E , Gf , d100, F max , σpeak , θ, F110, F120 Optimizer = Proximal Adagrad, 0.05 learning rate 85MPa
Training iterations = 100 K
Batch size = 50

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N. Zobeiry, et al. Composite Structures 246 (2020) 112407

Table 2
Summary of damage parameters extracted from application of ma-
chine learning to simulation of the response of quasi-isotropic [90/
45/0/-45]4S IM7/8552 CFRP laminates under OCT loading.
Parameter Value

Laminate modulus, E 60200 MPa

Fracture energy, Gf 136kJ / m2
Peak stress, σpeak 783 MPa
Slope of the damage function, θ 83.75
Strain-softening parameter, σ1 422 MPa

from the outer 90° ply of the quasi-isotropic [90/45/0/-45]4S IM7/8552

laminate to extract an approximation of the stress–strain response
within the fracture process zone. This is used in this study to validate Fig. 9. Comparison of the approximate strain softening curves across the da-
the prediction of strain-softening response by the proposed TGML mage zone obtained experimentally using Digital Image Correlation technique
model. A detailed SEM analysis of the post-mortem samples further [24], and the strain-softening curve obtained using machine learning.
revealed that the damage height was around 5 mm. Since the values in
Table 2 are obtained for an in-plane finite element mesh size of
1 mm × 1 mm, the resulting stress–strain curve was scaled using Ba-
zant’s crack-band law [6] for a 5 mm damage height. Fig. 9 compares
the experimentally generated stress–strain data (grey) [24] with those
of the resulting stress–strain relation obtained by the TGML calibration
and MAT81. The calibrated curve is in fairly good agreement with the
family of experimentally measured curves with respect to the initial
elastic response as well as the peak and post-peak regimes.

4.2. Load-Unloading in OCT tests

The MAT81 damage model in LS-DYNA combines plasticity and

damage. For simplicity, we assumed that the onset of yielding coincides
with the initiation of damage. A possible scenario is to allow for plastic
yielding before (or after) damage initiates. The load-unload OCT test Fig. 10. Comparison of the load-POD curves from three OCT experiments and
results in Fig. 2 show that the residual displacements (POD) after full numerical simulations conducted using the strain-softening curve obtained
unloading do not return to the origin which is indicative of permanent from machine learning (Fig. 9) as input for MAT81 in LS-DYNA.
plastic deformation. As can be seen in Fig. 10, the calibrated plastic-

Fig. 8. A schematic of the approach developed in this study. NNs are trained based on 10,000 FE simulation results to perform inverse FE analysis. The trained NNs
are then used to predict a material strain-softening response based on Load-POD (Pin Opening Displacement) experimental results from three sets of physical tests.

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Declaration of Competing Interest calibration and validation of a nonlocal continuum damage model for laminated
composites. Compos Struct 2017;173:188–95. https://doi.org/10.1016/j.
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The authors declare that they have no known competing financial [23] Malek S, Zobeiry N, Dai C, Vaziri R. Strain-softening response and failure prediction
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ence the work reported in this paper. 10.1061/(ASCE)MT.1943-5533.0002737.
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