Predicting the Crushing Behavior of Axially Loaded

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Predicting the Crushing Behavior of Axially Loaded

Elliptical Composite Tubes Using Artificial NeuralNetworks

Hany El Kadi

Received: 11 September 2008 / Accepted: 30 October 2008 /

Published online: 13 November 2008

# Springer Science + Business Media B.V. 2008

Abstract In this research work, the artificial neural networks (ANN) technique is used in

predicting the crushing behavior and energy absorption characteristics of axially-loaded

glass fiber/epoxy composite elliptical tubes. Predictions are compared to actual experimental

results obtained from the literature and are shown to be in good agreement. Effects of

parameters such as network architecture, number of hidden layers and number of neurons per

hidden layer are also considered. The study shows that ANN techniques can effectively be

used to predict the crushing response and the energy absorption characteristics of elliptical

composite tubes with various ellipticity ratios subjected to axial loading.

Keywords Composite tubes . Artificial neural networks . Crushing behavior. Ellipticity ratio

1 Introduction

Impact energy absorbers are used to protect automobile passengers and pedestrians from the

effects of sudden impact caused by collisions. This is accomplished by converting the

impact energy into many different types of deformation energy keeping the peak forceexerted on the individual below the level causing damage [1, 2]. The absorbers must also

provide a long deformation path to slow the deceleration of the protected person. These

impact energy absorbers will mostly rely on a crushable energy absorber to cushion the

passenger compartment during impact.

The use of advanced materials in the design of energy absorber devices has been

hindered by a lack of experimental and numerical simulation work that would guide

designers to the optimum energy absorber device. Extensive utilization of advanced

composites in energy absorber design will mainly depend on finding accurate techniques to

predict their response to different loading conditions. To that goal, many studies have lately

investigated the quasi-static crushing behavior of composite tubes both experimentally and

Appl Compos Mater (2008) 15:273285

DOI 10.1007/s10443-008-9074-2

H. El Kadi (*)

Mechanical Engineering Department, American University of Sharjah, Sharjah,

United Arab Emirates

e-mail: [email protected]


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numerically using finite elements analysis [112]. In these cases, the load was either

applied to the tube in a transverse or in an axial direction. In these studies, both the load-

carrying capacity and the energy-absorption capability of composite tubes were

investigated.

The behavior of axially-loaded elliptic composite tubes under compression loading hasrecently been investigated both experimentally and numerically [11]. In this study, the

effect of the ellipticity ratio (a/b; a=inner major radius, b=inner minor radius) on the load-

carrying capacity of the tubes as well as the energy absorption until failure were also

investigated. Finite elements analysis was used for the numerical investigation of elliptic

tubes with ellipticity ratios varying from 1 to 2 (1 signifying a circular tube). Although

good agreement was obtained from the finite element analysis compared to the

experimental results, the authors emphasized that typical imperfections existing in the

manufacturing process of the tubes could not possibly be accounted for by the analysis.

They suggested using a non-linear finite element analysis to physically include these

imperfections in the model.

One way of including specimen irregularities and material inhomogeneities in modeling

is to use the results of the available experiments to predict the behavior of composite tubes

subjected to different loading. Artificial neural-networks (ANN) is a technique that uses

existing experimental data to predict the behavior under a variety of testing conditions.

Using this method, details regarding bonding properties between fiber & matrix, strength

variation of fibers and any manufacturing-induced imperfections are implicitly incorporated

within the input parameters fed to the neural network.

Caliskan [13] has one of the few published studies dealing with the use of these

networks in predicting the crushing energy absorption of carbon fiber-reinforced circulartubes under axial loading. A simple neural network with back-error-propagation algorithm

was trained using 84 data sets of crush energy absorption of circular tubes obtained from

the literature. Ten input parameters were fed into the network; these included material and

geometric properties of the tubes. The network was then used to predict the specific

energy absorption of a single tube. Comparing the average experimental results obtained

from six tests to the ANN predictions resulted in an error of 14%. The author suggested

that neural networks could more accurately predict the crushing behavior of these tubes if

a more complex network was used or if the properties of all input parameters were

experimentally measured rather than calculated using micromechanics and laminate plate

theory.In a recent work, Mahdi and El Kadi [14] evaluated the prediction of both load-carrying

capacities and energy absorption of elliptical composite tubes using artificial neural

networks (ANN). In this study, the experimental behavior and corresponding ANN

predictions of circular and elliptical tubes subjected to lateral compressive loads were

presented and discussed. The ANN was shown to successfully predict the crushing

behavior of tubes for a wide range of ellipticity ratios. The predicted results obtained from

the neural network were compared with actual experimental data in terms of load-carrying

capacity and energy absorption capability, showing an excellent agreement. It was

concluded that ANN techniques could effectively be used to predict the response ofcollapsible composite energy absorber device subjected to different loading conditions.

In the current work, the prediction of both load-carrying capacities and energy

absorption for axially loaded elliptic composite tubes is evaluated using artificial neural

networks (ANN). To test the validity of using ANN in determining the crushing behavior of

these tubes, the study will compare the predictions obtained using ANN to the experimental

results obtained from the literature [11] for various ellipticity ratios.

274 Appl Compos Mater (2008) 15:273285


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2 Experimental Investigation

The current work will make use of the comprehensive experimental program developed by

Alkoles et al [11] which deals with the crushing behavior and energy absorption

characteristics of axially loaded glass fiber/epoxy composite elliptical tubes. Axial quasi-static crushing tests were carried out on thin-walled glass/epoxy composite elliptical tubes

with various ellipticity ratios by compressing them between parallel rigid plates. The load

and displacements were recorded by an automatic data acquisition system and, as the quasi-

static crushing tests were carried-out, instant photographs were taken throughout the test.

Figures 1, 2, 3, 4, 5 show the typical deformation history and corresponding load-end

shortening path in elliptical composite tubes for the various ellipticity ratios considered. For

more specific details about the load deformation relation of each of the tubes, one could

refer to the work of Alkolose et al [11]. The experimental results are shown here for the

sake of completeness and for proper comparison with the predictions obtained by the

artificial neural networks introduced later in this work. Since the purpose of the current

work is to gauge the effectiveness of ANN in predicting the crushing behavior of axially

loaded composite tubes rather than compare the experimental results obtained in [11] to

other published experimental result, only the experiments published in [11] will be used

Fig. 1 Typical load-end shortening path histories of axially loaded elliptical composite tube with a/b ratio of

1.00 [11] (With kind permission of Springer Science and Business Media)

Appl Compos Mater (2008) 15:273285 275


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here. Using ANN to predict other experimental results published in the literature will

necessitate training the networks using the corresponding experimental data.

3 Artificial Neural Networks (ANN)

Artificial Neural Networks have emerged as one of the useful artificial intelligence concepts

successfully used over the past decade in modeling the mechanical behavior of fiber-

reinforced composite materials (see for example [15, 16].)

In general, ANN consist of a layer of input neurons, a layer of output neurons and one or

more layers of hidden neurons [1719]. Neurons in each layer are interconnected to

preceding and subsequent layer neurons with each interconnection having an associated

connection strength (or weight).

A training algorithm is commonly used to iteratively minimize the following cost

function with respect to the interconnection weights and neuron thresholds:

E 1

2

XM

1

XN

i1

di zi 2

where M is the number of training patterns and N is the number of output nodes. di and ziare the desired and actual responses for output node i, respectively.





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The training process is terminated either when the mean-square-error (MSE) between

the observed data and the ANN outcomes for all elements in the training set has reached

a pre-specified threshold or after the completion of a pre-specified number of learning

epochs.

Although all neural network models share common operational features, their underlyingstructures, input requirements and modeling and generalization abilities are different.

Consequently, each paradigm would have advantages and disadvantages depending on the

particular application. Hence, selecting the appropriate network class with suitable parameters

is vital to ensure a successful application. The following neural network architectures will be

considered here in predicting the crushing behavior in composite tubes:

Feedforward neural networks (FNN) This is the most known and commonly used class of

neural networks. Although the main success of neural networks has been in the application

of the multilayer FNN with back-propagation training, they suffer from some drawbacks

such as local convergence and the need for large training cases in order to make adequate

modeling generalization [19].

Recurrent Neural Networks (RNN) RNN distinguish themselves from FNN in that the

outputs from some neurons are fed back to the same neurons or to the neurons in preceding

layers. Thus signal can flow in both forward and backward directions [ 18]. The Elman





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neural networks (ENN) are multi-layer back-propagation partially recurrent networks, with

the addition of a feedback connection from the output of the hidden layer to its input. In

partially recurrent networks, the main network structure is feedforward. These feedforwardconnections are trainable. The feedback connections are formed through a set ofcontext

units and are not trainable. The context units memorize some past states of the hidden units,

and so outputs of the network depend on an aggregate of the previous states and the current

input [20].

Modular neural networks (MNN) The central idea behind such networks is task

decomposition, where in this case the concept of using a combined (or averaged) estimator

may be able to exceed the limitation of a single estimator [18, 19]. Using a modular

network, the task of predicting the crushing behavior of composite tubes is split up amongseveral local neural networks (sub-networks) not communicating with each other.

Combination weights that determine the degree by which each sub-network should

contribute to the final composite material properties are estimated with an integrating unit.

Furthermore, this integrating unit decides which module should learn which training

pattern.





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4 Prediction of Axial Load-Carrying Capacity

The three neural network architectures introduced were used to predict the axial load

carrying capacity of the elliptic composite tubes. The Neurosolution-5 software [21] wasused to construct, train and test the networks. In all cases, the input experimental data to the

network was the ellipticity ratio and the specimen deformation. The output was the tube

load-carrying capacity. In each case, the network was trained using all but one of the

ellipticity ratios obtained experimentally. The network was then required to predict the

behavior of the composite tube for the ellipticity ratio it was not trained for. The predictions

obtained were then compared to the experimental results for this ellipticity ratio. Once

assured that the predictions obtained are reliable, the network could be used in the future to

predict the behavior of a tube with any ellipticity ratio for which experimental results do not

exist. Ellipticity ratios of 1.0, 1.25, 1.5, 1.75 and 2.0 were used in this study. Since the

ANN cannot be accurately used to predict behavior outside the area of training, predicting

the behavior of the elliptic tubes with 1.0 and 2.0 ellipticity ratios was not attempted.

In the work done for transverse loading [14], the effect of the number of hidden layers

and the number of neurons per layer was not considered. This is because the main goal of

that study was to establish the feasibility of using ANN to predict the crushing behavior of





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composite tubes. In the current study, the effect of varying the number of hidden layers as

well as the number of neurons per layer was also examined. Prediction was attempted using

the three above-mentioned ANN configurations with one and two hidden layers. The

number of training epochs was held constant at 5,000 epochs.

Tables 1, 2, 3 show the mean square error obtained from the predicted values of the load-displacement relation of the tubes compared to the experimental data for each of the three

eccentricity ratios (1.25, 1.5, and 1.75). The tables show the effect of varying the neural

network architecture, the number of hidden layers and the number of neurons per hidden

layer. Although no single neural network configuration consistently resulted in the best

load-displacement predictions, predictions using ANN with two hidden layers consistently

lead to lower MSE values when compared to experimental data. Figures 6, 7, 8 graphically

compare the load-displacement relationship obtained experimentally to typical predictions

from the three neural networks for each of the three eccentricity ratios investigated. In these

figures, FFN1 and FFN2 relate to predictions obtained using feedforward neural networks

with one and two hidden layers respectively, ENN1 and ENN2 relate to predictions

obtained using Elman neural networks with one and two hidden layers respectively, and

MNN1 and MNN2 relate to predictions obtained using modular neural networks with one

and two hidden layers respectively. The predictions obtained using the Elman neural

network with one and two hidden layers were the only type of ANN able to match the

experimental behavior at low deformations; all other ANN structures predicted an initial

load carrying capacity at the start of the test. From these typical results, it can be concluded

that artificial neural networks can, in general, be used to predict the load-displacement

relationship for the composite elliptical tubes subjected to axial loading.

5 Crushing Behavior

In order to study the effects ofa/b ratio on the crashworthiness of elliptical composite tubes,

the instantaneous load is normalized with respect to the cross section area of the tube. Crush

stress were chosen to eliminate the influence of different cross-section area so that the effect

of ellipticity ratio remains. Accordingly, Fig. 9 describes the variation of the instantaneous

crush stress with ellipticity ratio. Experimental results show that the load carrying capacity

at pre-crush failure stage is independent of the ellipticity ratio. On the other hand, the load

carrying capacity at post crush failure stage is strongly sensitive to the ellipticity ratio. Thesame figure also shows the predicted crush stress using the ANN. The recurrent neural

network (ENN) resulted in the best predictions and was therefore used here. The figure

shows that these networks accurately predict the crushing stress behavior. They also

accurately show the independency of the crush behavior of the elliptic tubes at the pre-crush

Table 1 MSE for eccentricity ratio=1.25

One Hidden Layer Two Hidden Layers

Number of neurons per layer Number of neurons per layer

6 8 10 12 16 6 8 10 12 16

FFNN 0.0298 0.0243 0.0279 0.0242 0.0237 0.0181 0.0242 0.0146 0.0192 0.0173

ENN 0.0218 0.0219 0.0194 0.0233 0.0294 0.0192 0.0174 0.0172 0.0194 0.0163

MNN 0.0266 0.0238 0.0169 0.0266 0.0150 0.0258 0.0259 0.0165 0.0171 0.0142



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failure stage as well as the dependency on the ellipticity ratio during the post-crush failure

stage; the same behavior identified by the experimental data presented in [ 11]. In spite of

the fact that the ANN accurately predict the overall crush stress-strain behavior at various

ellipticity ratios, one can notice a slight deviation in the predicted trend for the ellipticityratio of 1.25; this is especially clear for the crush strain range of 0.1 to 0.45. The deviation

between the experimental behavior and the ANN prediction can be related to the dissimilar

trend of the experimental data for the ellipticity ratio of 1.25 compared to all other

experimental results. Since the ANN depends on the training data to predict the behavior at

a certain ellipticity ratio, having this particular behavior different from all data used in the

training set would understandably result in a deviation in the prediction.

6 Prediction of Energy Absorption Capability

Energy absorption capability during the structural crash is a requirement for the complete

spectrum of passenger transport vehicles. The total work done during the axial crushing of

the tubes is equal to the area under the load-displacement curve. Instantaneous specific

energy absorption capability of the elliptical composite tube defined as the energy absorbed

per unit mass was computed. Figure 10 shows the experimental relation between specific

energy absorption and the deformation for the elliptic tubes with different eccentricity ratios

[11]. The figure also shows typical predicted results obtained using artificial neural

networks. Since, as mentioned before, load-deformation predictions using ENN were more

accurate than those obtained with other types of ANN structures; they also resulted in the

best energy absorption predictions and were therefore used here for comparison purposes.

The average error in predicting the energy absorption capability for the ellipticity ratios of

1.25, 1.5 and 1.75 were calculated to be 14.3%, 9.5% and 31.5% respectively. The

percentage error obtained for the ellipticity ratio of 1.75 is greatly exaggerated by the high

error obtained at the very low values of deformations. As the deformation increases, the




6 8 10 12 16 6 8 10 12 16

FFNN 0.0316 0.0237 0.0274 0.0254 0.0311 0.0150 0.0096 0.0192 0.0073 0.0337

ENN 0.0256 0.0293 0.0299 0.0244 0.0188 0.0230 0.0180 0.0167 0.0160 0.0074

MNN 0.0323 0.0311 0.0311 0.0263 0.0300 0.0260 0.0210 0.0107 0.0146 0.0151




6 8 10 12 16 6 8 10 12 16

FFNN 0.0170 0.0180 0.0110 0.0189 0.0114 0.0107 0.0156 0.0107 0.0065 0.0083

ENN 0.0125 0.0180 0.0133 0.0155 0.0166 0.0163 0.0125 0.0188 0.0129 0.0198

MNN 0.0249 0.0133 0.0166 0.0147 0.0211 0.0155 0.0156 0.0113 0.0103 0.0096



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Fig. 6 Typical experimental vs. predicted load-deformation behavior of axially-loaded composite tubes with

an eccentricity ratio of 1.25





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Fig. 9 Crush stress-strain behavior of elliptical composite tubes: experimental results vs. ANN predictions





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error decreases significantly. Also, as expected, the ellipticity ratio of 1.5 which falls

midway through the range of tested data gave the most accurate predictions. Figure 10

shows that, in general, ANN can suitably predict the energy absorption characteristics of

axially-loaded elliptic composite tubes.

7 Conclusion

The experimental behavior and corresponding ANN predictions of elliptical composite

tubes subjected to axial compressive load were presented and discussed. The ANN has been

shown to successfully predict the crushing behavior of a wide range of elliptic tubes. The

predicted results obtained from the ANN were compared to actual experimental data in

terms of load-carrying capacity, energy absorption capability and crushing load prediction,

showing a very good agreement. In particular, the Elman Neural Network was shown to

consistently lead to the best predictions of the experimental data. Additional work might be

needed to determine whether ANN can also be used to accurately predict the crushing

behavior of non-elliptical composite tubes.

From the current work, it could be concluded that ANN techniques can be used toeffectively predict the response of composite energy absorber devices with elliptical cross-

sections subjected to axial loading conditions.

Acknowledgment The author would like to thank Dr. El-Sadig Mahdi, Associate Professor in the Kulliyyah

of Engineering at the International Islamic University in Malaysia for providing the experimental data used in

this work.

Fig. 10 Specific energy absorption-deformation curves of elliptical composite tubes: experimental results vs.

ANN predictions



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