Application of Neural Network models on analysis of …...Vlasov’s thin-walled beam theory to analyze box girders. Razaqpur and Li (1994) developed a straight multi-cell box girder

Scientific Research and Essays Vol. 5(9), pp. 978-989, 4 May, 2010 Available online at http://www.academicjournals.org/SRE ISSN 1992-2248 © 2010 Academic Journals Full Length Research Paper

Application of Neural Network models on analysis of prismatic structures

N. Tayi

Department of Civil Engineering, Faculty of Engineering, University of Gaziantep, 27310 Gaziantep, Turkey.

E-mail: [email protected]. Fax: +90 3423172422.

Accepted 6 April, 2010

A Neural Network based design system is presented in this paper for preliminary analysis and design of prismatic structures. Because of the broad diversity of prismatic structures encountered in practice, it becomes clear that this study is concentrated on fundamental frequencies of single cell box girder bridge with straight planform. To provide a wide range of dataset for neural network training, fundamental frequencies are determined using variable thickness Mindlin-Reissner finite strips which offers an accurate and inexpensive tool for the production of database. The results of proposed neural network model to compute the fundamental frequency are furthermore compared with the results of finite strip analysis and found to be quite accurate. The trained neural network model proposed in this study and finite strip methods are used to conduct an extensive parametric study to investigate its generalization capability and the effect of various parameters on the fundamental frequency. Key words: Soft computing, Finite strip method, free vibration, Neural Networks.

INTRODUCTION Because of severe economic constraints and stringent deadlines coupled with the enormous growth in computer speed and power, engineers are resorting to numerical methods for the analysis of prismatic structures. Among the various numerical methods, the Finite Element (FE) method and its variance, such as the Finite Strip (FS) method, have become firmly established as engineering tools for the linear elastic analysis of prismatic structures. The predominant advantage of the FE method lies in its ability to analyze complex shell structures with varying thickness, difficult boundary conditions and arbitrary loading. However, from the engineering point of view, the use of the FE method for structural analysis drastically increases the computational time which cannot be affordable. The introduction of the FS method, which in its most usual form makes use of a combination of the FE method and Fourier series, has provided a very useful and economical analysis tool for this class of prismatic structures.

The FS method was initially developed by Cheung (1976). Since its initial introduction, many authors have investigated the applicability of the FS method and have developed many useful extensions. One area of research has been concerned with the development of FS models for plates and shells based on Mindlin-Reissner

assumptions. Hinton and Rao (1993); Özakça et al. (1994) have presented a comprehensive study covering static and free vibration analyses of variable thickness prismatic folded plates and curved shells using linear, quadratic and cubic strips. In practice of several methods with various degrees of rigor are available for analysis. These range from the elementary or engineer’s beam theory to complex shell FE analyses; other methods of analysis utilize folded-plate and FS methods Scordelis (1974); Cheung and Cheung (1972). Some researchers and designers have been using an advanced form of Vlasov’s thin-walled beam theory to analyze box girders. Razaqpur and Li (1994) developed a straight multi-cell box girder FE with exact shape functions based on this extended version of Vlasov’s thin walled beam.

In the absence of FE and FS programs or preliminary design stage use of empirical formulas may be more useful. These empirical formulas can be obtained by alternative soft computing techniques such as Neural Networks (NN) which have been widely applied to a wide range of engineering problems. NNs can be treated as an information processing technique that imitates function of human’s brain built on processing elements, called neurons which are connected to each other. NNs are well known universal approximators. However, for high

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Figure 1. Definition of curved Mindlin-Reissner FSs.

accurate open form formula which are obtained by NNs huge amount of training data is required.

This data set can be produced by FS method with an inexpensive and accurate way. As far as authors knowledge, NN have not been used for computation of fundamental frequencies of prismatic structures. The other studies on NN applications for box-girder bridges can be grouped under 2 main headings: First NN applications for preliminary design of tubular girder bridge decks under static loads Pathak and Gupta (2006); Zhao et al. (2001), second NN applications is bridge performance monitoring Chen (2005); Maria et al. (2004). The main aim of the present paper is to present NNs as an alternative tool for the analysis of fundamental frequency of prismatic structures in straight planforms such as single cell box girder bridges. In this study a parametric study is also carried out in order to investigate the effect of various parameters on the fundamental frequency. The proposed results for the investigated samples can used preliminary analysis and design of box girder section. The strategy described here consists of restricting the equation solver to a particular problem domain, e.g, a particular type of bridge construction, and then using NNs to approximately encode the basic structural behaviour of that class of structures.

The layout of the paper is now discussed. The following sections summarize the theory and implementation of the FS method based on Mindlin-Reissner shell theory. Fundamentals of NNs are presented following the

numerical application of NNs. Furthermore, the explicit formulation of the proposed NN model is presented. In the penultimate section box girder bridge parameters are presented which is used in NN methods. Parametric studies are later performed to demonstrate the generalization capability and accuracy of the proposed NN model compared to FS results. Finally, some conclusions are presented. Finite strip formulation Total potential energy Structures, which are simply supported on diaphragms at two opposite edges with the remaining edges arbitrarily restrained (where the cross section does not change between the simply supported ends) can be analyzed accurately and inexpensively using the FS method in cases where a full FE analysis could be considered extravagant. In a way that, FS method combines the use of Fourier expansions and one-dimensional FEs to model the longitudinal and transverse structural behaviour, respectively.

Consider the Mindlin-Reissner curved shell strip in straight planform shown in Figure 1. Displacement components

vu , and

w are translation in the local ,

y and n directions, respectively. Note that, y varies from 0

L

980 Sci. Res. Essays to b in straight planform. The displacement components

u and

w may be written in terms of global

displacements u and w in the x and z directions as

αα sincos wuu +=

α+α−= cossin wuw

(1) where α is the angle between the x and axes; Figure 1. The radius of curvature R may be obtained from the expression

Rdd 1−=

α . (2)

Note that, the displacement components

v and v

coincide. In the absence of external loads and damping effect, the virtual work for a typical curved Mindlin-Reissner strip e is given in terms of the global displacements u, v, w and the rotations φ and ψ of the midsurface normal in the ln and yn planes, respectively, by the expression

eTss

Tsbb

Tb

V

mmTme dVI

e

)(21 uPuDDD δδδδ +++= εεεεεεεεεεεεεεεεεεεεεεεε

(3) Where, mεεεε , bεεεε and sεεεε are the membrane, bending or curvatures and transverse shear strains, respectively, and given in Özakça (1993) for box girder bridge in straight and curved planforms. Finite strip discretization If we list the nodal displacements and accelerations in a vector u and u respectively,

0dMK =− ][ 2ω (4)

Where, 2ω is the frequency and d the associated mode shape. K and M are the global stiffness and mass matrices, respectively, and contain submatrices contributed from each strip e linking nodes i and j and harmonics p and q. These submatrices have the form

[ ] eqsjs

Tpsi

qbjb

Tpbi

qmjm

Tpmi

pqeij d

e

Ω++= Ω

BD][BBD][BBD][BK

(5)

eqj

Tpi

pqeij d

e

Ω= Ω

][][ NPNM (6)

and p

bipmi BB , and p

siB are the membrane, bending and shear strain displacement matrices associated with harmonic p, node i and Jacobian J and given in Özakça et al. (1994). 5IN ii N= in which I5 is the 5 × 5 identity matrix.

pqeij ][K and 0M =pqe

ij ][ if qp ≠ because of the

orthogonality conditions. Thus, for each harmonic p we are left with a compact eigen value problem for

0dMK =− ppp2

ppp ][ ω

(7)

The matrix ppeij ][M is independent of the harmonic

number p and therefore, the same matrix can be used for all the different harmonic equations. Neural networks Haykin (2000) defines NNs aptly as “NN is a massively parallel distributed processor that has a natural propensity for storing experiential knowledge and making it available for use”. It resembles the brain in two respects: Knowledge is acquired by the network through a learning process and interneuron connection strengths known as synaptic weights are used to store the knowledge.

The basic element of a NN is the artificial neuron as shown in Figure 2 in which it consists of three main components namely; as weights, bias, and an activation function. Each neuron receives inputs nxxx ,...,, 21 , attached with a weight wi which shows the connection strength for that input for each connection. Each input is then multiplied by the corresponding weight of the neuron connection. A bias ib can be defined as a type of connection weight with a constant nonzero value added to the summation of inputs and corresponding weights given in Equation (8).

=

+=H

jijiji bxwU

1 (8) The summation iU is transformed using a scalar-to-scalar function called an "activation or transfer function",

)( iUF yielding a value called the unit's "activation", given as:

)( ii UfY = (9)

Activation functions serve to introduce nonlinearity into NN which makes them so powerful. NN are commonly

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1iw

2iw

3iw

ijw

1x

2x

3x

ix

=

+=H

jijiji bxwU

1

Bias Activation Function

Output • •

)( ii UfY =

Figure 2. Basic elements of an artificial neuron, Çevik (2006).

classified by their network topology, (that is, feedback, feedforward) and learning or training algorithms (that is, supervised, unsupervised). For example, a multilayer feedforward NN with back propagation indicates the architecture and learning algorithm of the NN. Back propagation Neural Networks Back propagation algorithm is one of the most widely used supervised training methods for training multilayer NN due to its simplicity and applicability. It is based on the generalized delta rule and was popularized by Rumelhart et al. (1986). As it is a supervised learning algorithm, there is a pair of inputs and corresponding output. The algorithm is simply based on a weight. It consists of two passes: a forward pass and a backward pass. In the forward pass, first, the weights of the network are randomly initialized and an output set is obtained for a given input set where weights are kept as fixed. The error between the output of the network and the target value is propagated backward during the backward pass and used to update the weights of the previous layers Zupan and Gasteiger (1993). Optimal Neural Networks model selection The performance of a NN model mainly depends on the network architecture and parameter settings. One of the most difficult tasks in NN studies is to find this optimal network architecture which is based on determination of numbers of optimal layers and neurons in the hidden layers by trial and error approach or genetic algorithm optimization techniques. The assignment of initial weights and other related parameters may also influence the performance of the NN in a great extent. However, there

is no well defined rule or procedure to have optimal network architecture and parameter settings where trial and error method still remains valid. This process is very time consuming.

In this study, Matlab NN toolbox is used for NN applications. Various Back propagation Training Algo-rithms are used. Matlab NN toolbox randomly assigns the initial weights for each, run each time which considerably changes the performance of the trained NN, even all parameters and NN architecture are kept constant. This leads to extra difficulties in the selection of optimal network architecture and parameter settings. To over-come this difficulty a program has been developed in Matlab which handles the trial and error process automatically. The program tries various number of layers and neurons in the hidden layers both for first and second hidden layers for a constant epoch for several times and selects the best NN architecture with the minimum MAPE (Mean Absolute % Error) or RMSE (Root Mean Squared Error) of the testing set, as the training of the testing set is more critical. For instance, a NN architecture with 1 hidden layer with 7 nodes is tested 10 times and the best NN is stored, while in the second cycle the number of hidden nodes is increased up to 8 and the process is repeated. The best NN for cycle 8 is compared with cycle 7 and the best one is stored as best NN.

This process is repeated nn times where nn denotes the number of hidden nodes for the first hidden layer. This whole process is repeated for changing number of nodes in the second hidden layer. Moreover, this selection process is performed for different back propagation training algorithms such as Levenberg-Marquardt back propagation (trainlm), Scaled conjugate gradient back propagation (trainscg) and BFGS quasi-Newton back propagation (trainbfg). The optimum learning algorithm for modeling of box girder bridges was found by using Levenberg-Marquardt back propagation algorithm. The

982 Sci. Res. Essays

Figure 3. Cross section and position of design variables of single cell box girder bridge.

Table 1. Ranges of variables used in the database.

t/b s1/b s2/b s3/b s4/b t1/b t2/b t3/b Initial Final

1.0 9.0

0.05 0.25

0.5 0.9

0.25 0.75

0.05 0.2

0.015 0.025

0.005 0.025

0.005 0.02

program begins with simplest NN architecture, that is, optimal NN architecture. Further information can be found in Çevik (2006). This algorithm has been successfully applied to a wide range of engineering problems Cevik and Guzelbey (2007). Numerical example of Neural Networks applications Because of the diversity of structures in practice, this study is concentrated on fundamental frequencies of single cell box girder bridge with straight planform. Box girder bridges are increasingly constructed for modern highways. Because of its well-known structural advantage of torsional stiffness, the box girder has become a popular solution for medium and long span bridges. Maisel (1982) conducted a detailed survey of the box girder bridges built worldwide. The usual types of bridges were not economical for long spans because of the rapid increase in the ratio of dead load to total design load as the span lengths increased. The hollow girder concrete bridge was developed as a solution to this problem. Box girder bridges are suitable for railroads as well as highways. Dynamic loads caused by live loads are a critical issue for bridges and, box girder bridges are more resistance to vibration effects with respect to classical bridges.

The main purpose in this study is to model the elastic free vibration analysis of box girder bridge with straight platform using NN. The database used for NN training has been generated by the program developed by Özakça and coworkers explained in section 2. A non-dimensional parametric database has been formed where variables have been given as a ratio of “b” which is the width of the geometry shown in Figure 3. The ranges of variables of the database are shown in Table 1. Note that L is the length of box girder bridge and the structures are simply supported at end conditions at 0=y and Ly = , other ends are free. The following material properties are used: 6100.200 ×=E , Poisson’s ratio =0.2 and mass density =7.8. The FS analyses are carried out using quadratic strips. Note that, all units are consistent. The database is divided into train and test sets where the patterns in test set are randomly selected with a ratio 80 and 20%, respectively.

The optimal NN architecture was found to be 8 - 17 - 1 NN architecture with logistic sigmoid transfer function (logsig) shown in Figure 4. The training algorithm was Levenberg-Marquardt back propagation (trainlm). Statistical parameters of test and training sets are presented in Table 2. The performance of the proposed NN model versus FS results are shown in Figure 5. As shown in Tables 3 and Figure 6, the performances of the

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Figure 4. Proposed NN architecture.

Table 2. Statistical parameters of test and training sets.

Test set Training set Mean (FS/NN) 1.01 1.00 Standard deviation (FS/NN) 0.07 0.06 Correlation coefficient (R) 0.98 0.99

proposed NN model is quite high which indicates the accuracy of the NN model to map the relationship between input and output variables. It should be noted that the proposed NN model is valid for ranges of variables show in Table 1. Derivation of explicit formulation of proposed Neural Network model In general, NN are often treated as "black box" modelling tools, but they can be explicitly formulated using the parameters (weights and biases) of the trained NN as carried out in this article. The fundamental frequency

2ω is computed as:

+= −We1

10.17672ω

(10)

Figure 5. Performance of the proposed NN model vs. FS results for training set.

W =Σ (w2i* )( iUf ) + b2 (11) Where w2i is the weight vector to the output layer and b2 is the bias added to the output which is


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1 (13) where wij

is the weight matrix of the first hidden layer and xj is the corresponding input parameter vector and ib is the bias matrix to the first hidden layer given as. “

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bt

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bL

(14) Parametric study A well-trained NN model can also be used as a robust tool to conduct parametric studies due to generalization capability of NNs. It is obvious from statistical results (R = 0.99) shown in Table 2 that the proposed NN model accurately learned to map the relationship between inputs and output. Four different group parametric study is done by which three of the input parameters shown in Figure 7 (b = 1.0, s1 = 0.15, s2 = 0.7, t1 = 0.0015) are kept constant whereas, the other inputs are taken as variable. The trend of this input parameters with the output is obtained graphically. The same graphs were also obtained by (Finite Strip Method) FSM to verify the NNs generalization capability. In Figures 8 - 11, interesting outcomes are observed on the graphs of trends. To demonstrate the accuracy of the present methods, fundamental frequencies obtained by the present NN procedure for some benchmark examples are compared with FS solutions in TableS 3 - 6. CONCLUSIONS In the present work NNs are presented as an alternative tool for the computation of fundamental frequency of single box girder bridges in straight platforms for preliminary design. The proposed NN model performs quite well as compared to FS results. Although, NNs are generally treated as “black-box” tools, the proposed NN is presented as an explicit function to be used by other researchers as a so-called “white-box” for further applications.

Parametric studies are also performed to prove the generalization capability of the explicit formulation obtained by NNs. The parametric studies show that the the bias matrix to the first hidden layer given as linear relationship among fundamental frequency and other parameters are existing.

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Table 3. Values of parameters and corresponding frequencies of group 1.

L/b s1/b s2/b s3/b s4/b t1/b t2/b t3/b Freq (NN) Freq (FS)

1.0 0.15 0.7 0.25 0.2 0.015 0.005 0.005 712.25 725.51

1.0 0.15 0.7 0.25 0.2 0.015 0.015 0.015 903.04 896.5

1.0 0.15 0.7 0.25 0.2 0.015 0.005 0.02 719.29 746.05

1.0 0.15 0.7 0.25 0.2 0.015 0.015 0.01 894.6 902.83

1.0 0.15 0.7 0.25 0.2 0.015 0.025 0.02 1048.6 1036.9

1.0 0.15 0.7 0.25 0.2 0.015 0.025 0.005 1039.1 1045.6

1.0 0.15 0.7 0.25 0.2 0.015 0.025 0.015 1045.2 1043.6

1.0 0.15 0.7 0.25 0.2 0.015 0.005 0.015 718.68 721.41

1.0 0.15 0.7 0.25 0.2 0.015 0.015 0.02 910.14 905.4

1.0 0.15 0.7 0.25 0.2 0.015 0.015 0.005 888.24 913.32

1.0 0.15 0.7 0.25 0.2 0.015 0.005 0.01 717.25 721.45

1.0 0.15 0.7 0.25 0.2 0.015 0.025 0.01 1042.1 1049.5

Figure 6. Performance of the proposed NN model vs. FS results for test set.

s3

s4

t3

0.015 t2

0 .7

1

0 .15

Figure 7. Dimension of parametric study.


(a) 3D Response of NN (b) 3D Response of FSM

Figure 8. Comparison of NN and FS models (group 1:L/b=1.0, S3=0.25).


Figure 9. Comparison of NN and FS models (group 2:L=5.0, S3=0.25, S4=0.2).

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Figure 10. Comparison of NN and FS models (group 3:S3=0.25, S4=0.05, t2=0.005).


Figure 11. Comparison of NN and FS models (group 4:L=9.0, t2=0.005, t3=0.0015).



L/b t1/b s2/b s3/b s4/b t1/b t2/b t3/b Freq (NN) Freq (FS)

5.0 0.15 0.7 0.25 0.2 0.015 0.015 0.015 152.69 152.85

5.0 0.15 0.7 0.25 0.2 0.015 0.005 0.02 157.93 159.54

5.0 0.15 0.7 0.25 0.2 0.015 0.005 0.005 119.67 136.05

5.0 0.15 0.7 0.25 0.2 0.015 0.025 0.02 155.37 162.09

5.0 0.15 0.7 0.25 0.2 0.015 0.025 0.005 137.09 136.39

5.0 0.15 0.7 0.25 0.2 0.015 0.015 0.01 144.53 144.74

5.0 0.15 0.7 0.25 0.2 0.015 0.005 0.015 150.01 149.32

5.0 0.15 0.7 0.25 0.2 0.015 0.015 0.005 133.47 138.22

5.0 0.15 0.7 0.25 0.2 0.015 0.025 0.015 150.63 154.37

5.0 0.15 0.7 0.25 0.2 0.015 0.015 0.02 158.83 161.79

5.0 0.15 0.7 0.25 0.2 0.015 0.025 0.01 144.67 144.68

5.0 0.15 0.7 0.25 0.2 0.015 0.005 0.01 138.21 142.58



1.0 0.15 0.7 0.25 0.05 0.015 0.005 0.005 509.85 538.31

1.0 0.15 0.7 0.25 0.05 0.015 0.005 0.01 534.08 525.95

1.0 0.15 0.7 0.25 0.05 0.015 0.005 0.015 545.04 532.2

1.0 0.15 0.7 0.25 0.05 0.015 0.005 0.02 552.19 559.29

5.0 0.15 0.7 0.25 0.05 0.015 0.005 0.005 30.702 31.643

5.0 0.15 0.7 0.25 0.05 0.015 0.005 0.01 35.926 35.606

5.0 0.15 0.7 0.25 0.05 0.015 0.005 0.015 39.321 40.03

5.0 0.15 0.7 0.25 0.05 0.015 0.005 0.02 41.698 45.609

9.0 0.15 0.7 0.25 0.05 0.015 0.005 0.005 9.609 9.0942

9.0 0.15 0.7 0.25 0.05 0.015 0.005 0.01 11.321 10.284

9.0 0.15 0.7 0.25 0.05 0.015 0.005 0.015 12.461 11.589

9.0 0.15 0.7 0.25 0.05 0.015 0.005 0.02 13.278 13.762



9.0 0.15 0.7 0.25 0.1 0.015 0.005 0.015 24.499 21.23

9.0 0.15 0.7 0.5 0.05 0.015 0.005 0.015 14.449 14.922

9.0 0.15 0.7 0.5 0.2 0.015 0.005 0.015 55.662 51.848

9.0 0.15 0.7 0.5 0.15 0.015 0.005 0.015 42.145 37.376

9.0 0.15 0.7 0.25 0.2 0.015 0.005 0.015 48.557 46.336

9.0 0.15 0.7 0.25 0.05 0.015 0.005 0.015 12.461 11.589

9.0 0.15 0.7 0.25 0.15 0.015 0.005 0.015 36.561 32.702

9.0 0.15 0.7 0.5 0.1 0.015 0.005 0.015 28.378 25.181

The main decrease in the fundamental frequency is obtained when the span length of the box girder bridge increases. On the other hand, fundamental frequency increases, when the height of the web increases. ACKNOWLEDGMENTS I acknowledge the mechanic group members Dr. A. Çevik and Dr. M. T. Göü for the contributions to the development of this paper. REFERENCES Çevik A (2006). New Approach for Elastoplastic Analysis of Structures:

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Application of Neural Network models on analysis of …...Vlasov’s thin-walled beam theory to analyze box girders. Razaqpur and Li (1994) developed a straight multi-cell box girder