Application of artificial neural networks to evaluation of ultimate strength of steel panels

Engineering Structures 28 (2006) 1190–1196 www.elsevier.com/locate/engstruct

Application of artificial neural networks to evaluation of ultimate strength of steel panels Yongchang Pu ∗ , Ehsan Mesbahi School of Marine Science and Technology, University of Newcastle Upon Tyne, Newcastle Upon Tyne, NE1 7RU, UK Received 3 December 2004; received in revised form 9 December 2005; accepted 13 December 2005 Available online 28 February 2006

Abstract Structural design of ships and offshore structures has been moving towards limit state design or reliability-based design. Improving the accuracy and efficiency of predicting the ultimate strength of structural components, such as unstiffened panels and stiffened panels, has a significant impact on our daily structural design. Empirical formulations have been widely used because of their simplicity and reasonable accuracy. In the past, empirical formulations were generally developed by using regression analysis. The model uncertainties of good empirical formulations are around 10%–15% in terms of coefficients of variation. In this paper, artificial neural networks (ANN) methodology is applied to predict the ultimate strength of unstiffened plates under uni-axial compression. The proposed ANN models are trained and cross-validated using the existing experimental data. Different ways to construct ANN models are also explored. It is found that ANN models can produce a more accurate prediction of the ultimate strength of panels than the existing empirical formulae. The ANN model with five (original) input variables has slightly better accuracy than the model with three input variables. This demonstrates the capacity of the ANN method to establish successfully a functional relationship between input and output parameters. c 2006 Elsevier Ltd. All rights reserved.  Keywords: Artificial neural networks; Ultimate strength of panels; Empirical formulae for ultimate strength; Regression analysis

1. Introduction Plates are important components in ships and offshore structures. Accurate prediction of the ultimate strength of plates has been a very important task for ship designers. Three types of methods are normally used to estimate the ultimate strength of panels, namely experimental method, empirical formulations, and numerical methods such as finite element method. Running experiments is very expensive, so it is usually used in the final stage to validate the results of other methods. Numerical methods are quite time-consuming. If the finite element method is adopted in the context of the prediction of the ultimate strength of panels, non-linear analysis considering both geometrical and material non-linearity has to be carried out to provide accurate results, hence numerical methods are not the kinds of methods for routine design. Empirical formulations have been the preferred method, because they ∗ Corresponding author. Tel.: +44 191 222 6243; fax: +44 191 222 5491.

E-mail address: [email protected] (Y. Pu). c 2006 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter  doi:10.1016/j.engstruct.2005.12.009

are generally simple to use and provide reasonably accurate results. Most of the existing empirical formulae were derived by regression analysis, in which important parameters are included. Much work has been carried out to develop analytical formulae for predicting the ultimate strength of panels since the 1970s, such as [1–3], among many others. A comprehensive review was carried out by Faulkner [4]. Pu [5] has carried out a comparative study of several existing empirical formulae. The model uncertainties of these formulae were calculated by comparing them with experimental results. ISSC [6] has reviewed the results of other researchers relating to model uncertainties of some existing formulae. These results show that the coefficient of variation (C.O.V.) of the good existing formulae is around 10%–15%. The artificial neural networks (ANN) technique was initially developed for adapting or learning, for generalising or clustering, and organising data, in which an operation is based on parallel processing. It has been applied to the fields of robotics, pattern identification, psychology, physics, computer

Y. Pu, E. Mesbahi / Engineering Structures 28 (2006) 1190–1196

science and biology etc. One of the applications of ANN is its capacity for mapping the relation between input data and output data. Nevertheless, the application of ANN to marine structures is quite limited. An attempt was made to apply ANN to the ultimate strength of stiffened plates [7]. The findings of this work clearly indicate the potential of ANN, which generally produces better results than those empirical formulae derived from conventional regression analysis. Shao and Morutsu [8] first applied ANN to predict the reliability of structures. In their paper, the ANN model was first used to approximate the limit state function. Then the reliability of the structure was evaluated by a First Order Second Moment method (FORM). This methodology was applied to a couple of simple examples and a portal frame. The results obtained from ANN models are reasonably accurate. However, there is room for further improvement in the efficiency, which is a very important issue when this method is applied to a complex structure. So far, to the knowledge of the authors, the ANN has not been applied to the prediction of the ultimate strength of unstiffened plates. In addition, ANN was applied as a sort of ‘black box’ without much discussion about its mathematical expression in the past. The objective of this paper is to apply ANN to predict the ultimate strength of unstiffened plates. Its accuracy, efficiency and advantages over conventional regression analysis will be fully discussed. The mathematical expressions of the developed ANN models will be provided so that they can be used in the same way as other empirical formulae. 2. Artificial neural networks Artificial neural networks can be most adequately characterised as computational models with particular properties such as the ability to adapt or to learn, to generalise or to cluster or organise data, in which an operation is based on parallel processing. It can be applied to many fields. In this study, ANN is used to approximate the ultimate strength of unstiffened plate in uniaxial compression. Hence the general description of ANN in the following sections will be concentrated on its capacity to establish a functional relationship between input and output data. Firstly, the general structure of ANN will be introduced. The methods used for training, cross validating and testing ANN models are then presented. Thirdly, a mathematical expression of ANN models will be described, and its inherent advantages over the conventional regression analysis will be discussed. 2.1. Architecture of neural network A typical architecture (structure) of a feed-forward ANN model can be demonstrated by Fig. 1, in which the left column is the input layer, the right most column is the output layer, and in between the input and output layers is a hidden layer. Generally, there could be more than one hidden layer. In the present study, only one hidden layer is used, because the efficiency and accuracy of this topology is sufficient for the present application. In each layer there are several Processing Elements (PE) (also called ‘neurons’). The number of PEs in the input layer

1191

Fig. 1. General structure of an artificial neural network.

is equal to the number of input variables, while the number in the output layer is equal to the number of output variables. The important task in a one-hidden-layer ANN model is to determine the number of PEs in the hidden layer, which in turn affects the accuracy of the model. Each processing element has several inputs and one output. The relation between input, x j , and output, y p , of a single PE can be expressed as:
 T  j yp = f wpx j + ap ( j = 1, 2, . . . , T ) (1) j =1 j

where w p are weights, a p is a constant (normally referred to as threshold), f (·) is called the activation function, which could be a sigmoid function or hyperbolic tangent function, etc. The PEs in the hidden layer will receive data from only the input layer. Similarly, the PEs in the output layer will only receive data from the hidden layer. For this reason, this ANN is called feed-forward network. 2.2. Development of ANN models Once the topology of an ANN model is decided, the next task is to determine the values of all the weights in Eqs. (5) and (6), which will be derived in the following section. This process consists of three steps, namely training, cross validation and test. The details of this can be seen in the reference [9]. 2.3. Mathematical background From the mathematical point of view, developing an empirical formula from experimental or numerical data is to find an approximate function, which can best represent the relation between input and output variables. Suppose that the transpose of the vector of input variables is XT = (x 1 , x 2 , . . . , x n ), and the transpose of the vector of output variables is YT = (y1 , y2 , . . . , y L ). For an ANN model with one hidden layer, as shown in Fig. 1, with m number of PEs in the hidden layer, the mathematical expression of the j th

1192

Y. Pu, E. Mesbahi / Engineering Structures 28 (2006) 1190–1196

output variable is: 
m  i who j ψi (X) + c j yj = f

( j = 1, . . . , L)

(2)

i=1

where whoij is the weight of the i th PE in the hidden layer to the j th PE in the output layer, c j is a constant, ψi (X) is the output of the i th PE in the hidden layer, which is expressed as: 
n  k (i = 1, . . . , m) (3) ψi (X) = f wihi x k + bi k=1

where wihki is the weight of the kth input variable in the input layer to the i th PE in the hidden layer, and bi is a constant. Eqs. (2) and (3) can be expressed in matrix form as:    I     Y = f WT2 ×  I (4) f W1 ×   X in which I is a one by one unit matrix, and the expressions of matrices W1 and W2 are:   b1 wih11 wih21 · · · wihn1 · · · ·  W1 =  · (5) 1 2 n bm wihm wihm · · · wihm   c1 who11 who21 · · · whom 1 · · · · . WT2 =  · (6) 1 2 c L who L who L · · · whom L So, the development of an ANN model is to determine the values of matrices W1 and W2 . When ANN is used to approximate a function, it has inherent advantages over the polynomial based response surface method. To explain these ideas, a simple but general case is used. Assume that there is only one output variable. Let g(x) be a real function that is square integrable over the real numbers. Developing an empirical formulation is to describe the behaviour of g(x) by a combination of simpler functions ϕi (x), which will be called base functions, in the form: g(x, ˆ a) =

n 

ai ϕi (x)

(7)

i=1

where g(x, ˆ a) is the approximation of g(x), a = (a1 , a2 , . . . , an ) are coefficients to be determined so that |g(x) − g(x, ˆ a)| < ε

(8)

in which ε is a small number. There are many ways to choose base functions, for example trigonometric polynomials (Fourier series) and polynomials. Selecting appropriate base functions is very important in function approximation. If the choice is not appropriate, there will be a non-vanishing error, no matter how big the number of base functions is. In the context of developing empirical formulae in structural design, the polynomial is a popular form. The Response Surface Method [10] is based on this. It is proved that polynomials can approximate any continuous real function in an interval

arbitrarily well. One of the problems of the polynomialbased response surface method is that the number of datasets required to develop the mathematical expression is increasing exponentially with an increase in the number of input variables. This seriously limits its application to complex structures. For a one-hidden-layer ANN model, the base functions are expressed by Eq. (3). This expression clearly shows that the base functions are dependent on the first layer’s weights of the ANN, which are determined by the input and output data. So this means that the base functions are not predefined as in response surface methods. This is an adaptive system. The ANN model has advantages over the response surface methods in the following ways: • ANN models are universal approximators, and their asymptotic accuracy is approximately independent of the dimension of the input space. However, the accuracy of the polynomial-based response surface method is exponentially related to the number of dimensions of the input. • In response surface methods, base functions are predetermined, so that it is possible that non-vanishing error would exist in the approximate function, while the ANN model has adaptive base functions that are determined on the basis of the input and output data. So, it is possible to adjust the coefficients in ANN models to eliminate the non-vanishing error, or at least reduce it to a minimum. This is why the ANN model tends to achieve better accuracy than polynomial-based response surface methods. 3. ANN models for the prediction of ultimate strength of steel panels The ANN models used for this study are multilayer feedforward neural networks, which have only one hidden layer. The number of PEs in the input layer is equal to the number of input variables, and the number of PEs in the output layer is 1, which returns the ultimate strength of panels. The number of PEs in the hidden layer is a key parameter to determine in the development of ANN models. The number of PEs in the hidden layer has a direct effect on the model quality in terms of accuracy. Unfortunately, there is no general rule (or algorithm) to determine this number. In practice, a trial-and-error process is used. All the ANN models are trained by a back-propagation algorithm with momentum learning, which can speed up and stabilise convergence during the training process. The stop criterion for the training is called ‘stop with cross-validation’. This means that training is stopped at the point of the smallest error in the validation set, which is the point of maximum generalisation for the given data and topology. To implement this method, the whole dataset should be divided into two sets: the training set and the cross-validation set. The crossvalidation set is normally taken as being about 10%–15% of the total training samples. Every so often (i.e., 5 to 10 iterations), the learning machine performance with the present weights is tested against the cross-validation set. Training is stopped when the error in the cross-validation set starts to increase.

Y. Pu, E. Mesbahi / Engineering Structures 28 (2006) 1190–1196

In many physical or experimental sets of data, input and output parameters could vary within a very wide range. Input/output data normalisation is a common practice to linearly transform these sets into a similar range of variability. The non-linear normalisation of input/output data sets has also been recommended and exercised in some engineering applications [11]. This could lead to an improvement in ANN training stability and, consequently, a higher degree of accuracy for generalisation purposes. Higher and lower bounds of normalisations are dictated by saturation limits of the activation functions: [0, 1] for sigmoidal and [−1, 1] for tangential hyperbolic functions. In certain applications, where extrapolation outside the boundaries of the input data could be an advantage or a requirement, input/output data normalisation within a closer range of [0.2, 0.8] or [0.3, 0.7] is recommended [12]. So, both input and output variables are normalised in this study. The formula used to normalise variables to a range of [U, L] is expressed as: x normalised = ax + b

(9)

U −L x max −x min ,

and b = U − ax max , in which x max and where a = x min are the maximum and minimum values of the variable. In this study, U = 0.95 and L = 0.05. Five thousand epochs, defined as one complete presentation of all the data, are used to train both ANN models. 3.1. Selection of variables for ANN In order to develop the ANN model for predicting the ultimate strength of unstiffened panels, parameters that influence the ultimate strength should be determined. Based on existing knowledge, the major parameters are the plate width (B) (mm), plate thickness (t) (mm), yield stress of the material (σ y ) (N/mm2 ), Young’s modulus of the material (E) (N/mm2 ), and the residual stresses (σr ) (non-dimensional) and initial deflection (δ) (non-dimensional) of the plate due to welding. Two kinds of ANN model will be developed. In the first model, all the six major parameters, except for Young’s modulus, will be used as input variables. The reason for not including Young’s modulus in the first model is that the experimental results are for steel, which has a constant Young’s modulus. Therefore, how the change in Young’s modulus would affect the ultimate strength of panels could not be reflected in the experimental results. In the second model, a new variable, plate slenderness, will be used to replace B, t, and σ y . Plate slenderness is defined as:  B σy . (10) β= t E So, only three input variables, namely plate slenderness, residual stresses and initial deflection, will be used in the second ANN model. The reason for choosing two sets of input variables is to investigate the difference between the two philosophies of selecting input variables. On the one hand, ANN is a universal estimator from a mathematical point of view, as discussed in

1193

the previous sections. In principle, using the original variables as input variables would not cause a loss of accuracy, even if the relation between input variables and output variables has strong non-linearity. On the other hand, from past experience, the most important parameter is plate slenderness. In most empirical formulae, it is the dominant (or, in some cases, the only) parameter. Faulkner [13] has suggested that a strength of materials approach should be used, in which all important terms and parameters reflect the mechanics of failure. Curve fitting should be restricted to secondary terms. This guidance proved to be very useful when an empirical formula was developed for perforated plates [14]. Even from a purely mathematical point of view, introducing new variables that are a combination of the original variables is a very useful technique for developing an accurate formulation. By introducing new variables, finding the relation between the original variables and the output variables is transformed to finding the relation between the new variables and the output variables. A good transformation could reduce the non-linearity in the relation between input and output, so that a more accurate mathematical expression could be obtained. 3.2. Experimental data for developing ANN models The experimental data that are used to develop ANN models are from the published papers [15–17]. The numbers of data in each source are 33 from Dwight, 56 from Moxham, and 54 from Ueda, giving a total of 143 sets of data. Since the ANN models used in this study are strictly data-driven models, their overall quality in terms of accuracy in interpolation and validity in prediction are highly dependent on the number and the quality of data sets used for training and testing purposes. When a large set of data is available, depending on the range, coverage and quality of the data, up to 50% of the data may be chosen for training and the remaining for cross-validation and testing purposes. Generally speaking, to achieve a valid model, the data selected for training must be “representative” of the overall behaviour of the input or output data space. Since the number of data available for this study is limited to those provided in [15–17], the selection of the number of data for training, cross validation and testing needs to follow a more careful approach. In this case, it is also necessary to observe the behaviour (degree of non-linearity) of each input and output data within the available range. After careful observation of the nature of available data, it was decided to randomly select 16% of the data (24) for testing and the rest (119) for training purposes. The training data set was also randomly shuffled before the start of the training procedure to avoid overtraining of the model for a particular region of the data set. The ranges of all the variables of the experimental data are shown in Table 1. A ‘Neural Solution’ [9] is used to develop ANN models in this study. A sigmoid function is used as an activation function, which is expressed as: f (x) = sig(x) =

1 . 1 + e−x

(11)

1194

Y. Pu, E. Mesbahi / Engineering Structures 28 (2006) 1190–1196

Table 1 Ranges of the main parameters in the experimental data

Minimum Maximum

B (mm)

t (mm)

σ y (N/mm2 )

σr (Non-dimensional)

δ (Non-dimensional)

β (Non-dimensional)

σm (Non-dimensional)

116.8 609.6

3.2 14.3

206.5 403

0 0.682

0 1.74

0.757 4.103

0.3 1.0

Table 2 Model uncertainties of some existing formulae and ANN models

Mean C.O.V.

Faulkner’s methods

Carlson’s method

Guedes Soares’ method

ANN model 1

ANN model 2

1.159 0.163

1.117 0.138

1.031 0.101

1.022 0.043

1.005 0.061

Fig. 2. The structure of ANN model 1.

3.3. Results of the ANN models ANN models with five input variables Different numbers of PEs in the hidden layer are used to develop the ANN models. The number of PEs that produces the best results is used for the final ANN models. The optimum number of PEs in the hidden layer is eight. The errors for this model in the mean squared error (MSE) term are 0.000719 in training and 0.00108 in cross-validation, both of which are quite small. In addition, the linear correlation is 0.971, which is fairly close to 1. These indicate that the developed ANN model is quite accurate. Fig. 2 shows the structure of this ANN model. The values of weight matrices are: 

−1.704 5.4 6.85  −3.68  6.014e−1  −2.148  −3.636  2.417 W1 =  1.53 −5.854e−1  −16.171  −5.596  −3.976 14.857 −1.875 −3.729  WT = 1.548 −1.807 2.002 2

−8.451e−2 −1.5 −3.284 4.773 −2.14 2.158 −5.531e−2 15.543

−1.071 3.023 3.609e−1 −6.622 −4.622 −2.346 2.008 −1.022

−5.051e−1

2.796

−5.555 −14.355 5.402e−1 −4.316 −1.803 8.009 4.645 2.902 1.544

3.808

 6.879 −2.045   −3.674e−1  −16.123   2.85   −1.763   2.862 −5.415 −3.347

 2.761 .

The normalised input variable X and output variable Y are related by Eq. (4). The normalised input variables are expressed as:  1.826e−3 × B − 1.633e−1   8.108e−2 × t − 2.095e−1    X = 4.580e−3 × σ y − 8.958e−1 .   1.32σr + 5.000e−2 5.172e−1 × δ + 5.000e−2

The non-dimensional ultimate strength (σm ) is defined as the ratio of ultimate strength to yield stress. It is expressed as: σm =

Y + 3.357e−1 . 1.286

Fig. 3. Correlation of experimental and predicted results of ANN model 1.

Fig. 4. The structure of ANN model 2.

The mean and C.O.V. of the model uncertainty, which is defined as the ratio of experimental strength to predicted strength, of this ANN model is presented in Table 2. Fig. 3 shows the correlation of the experimental and predicted results. Clearly, the points are closely scattered around the straight line, which represents a full correlation. ANN models with three input variables Similarly, the ANN model is developed in the same way as the above case. The optimum number of PEs in the hidden layer is four. MSEs of this model are 0.00162 in training and 0.00210 in cross-validation. The linear correlation is 0.973. Again, these indicate that this model is also quite accurate. Fig. 4 shows the structure of the ANN model. The values of the weight matrices are:   2.379 −7.387e−2 −19.986 −8.953  −3.698 1.052 6.897 2.349   W1 =   −7.002 3.018 9.496e−1 11.823  1.292e−1 −10.117 −1.089 −0.987   T W2 = 3.784e−1 4.5 −1.236 −1.657 1.915 . The normalised input variable X and output variable Y are related by Eq. (4). The normalised input variables are expressed

Y. Pu, E. Mesbahi / Engineering Structures 28 (2006) 1190–1196

1195

Fig. 5. Correlation of experimental and predicted results of ANN model 2.

as:



 1.320σr + 5.000e−2 X =  5.172e−1 × δ + 5.000e−2  . 2.690e−1 × β − 1.537e−1 σm is expressed as: σm =

Y + 3.357e−1 . 1.286

The mean and C.O.V. of the model uncertainty of this ANN model is also presented in Table 2. Fig. 5 shows the correlation of the experimental and predicted results. Similarly, the points are fairly closely scattered around the straight line. 4. Comparison of ANN model with empirical formulae As mentioned in the previous sections, many existing empirical formulae were derived by regression analysis. To demonstrate the capacity of the ANN method, some existing formulae for predicting the ultimate strength of unstiffened panels are compared with ANN models. The results of the existing formulae, which are shown in Table 2, are taken from reference [5]. The details of these formulae can be seen in the following published papers: [18–20]. Table 2 shows that Guedes Soares’ method produces the best results among the existing empirical formulations, with a mean of 1.031 and a coefficient of variation of 10.1%. However, ANN model 1 gives a much better prediction, with a mean of 1.022 and a coefficient of variation of 4.3%. It should be pointed out that the mean and C.O.V. of ANN models include not only the data for training the model but also the data, which is not used in the development of the models, for cross-validation and test. Some empirical formulae may lose accuracy when additional data are used to calibrate the accuracy. So the ANN model is quite stable, in the sense that its accuracy would not noticeably change due to the change of dataset. In Fig. 6, the ultimate strength predicted by different formulae is plotted against plate slenderness together with experimental results. It is observed that both ANN models, especially ANN model 1, show a higher level of non-linearity than other formulae. This means that ANN models can adaptively produce highly non-linear curves to achieve a good fit to the experimental data. It should be pointed out that these curves represent the ultimate strength of

Fig. 6. Comparison of different formulae.

plates when both σr and δ are equal to zero. So, the majority of experimental results are below these curves. From the viewpoint of ANN training, it is desirable to have a large number of training data. Some researchers have recommended that the number of training data should be at least 10 times larger than the number of weights of ANN models [9]. In practice, it is difficult to obtain this many experimental data. The two ANN models developed in this study have 57 and 21 free parameters for model 1 and model 2, respectively. So, ideally there should be more training data, especially for model 1. Two ANN models have more or less the same accuracy. The first model has slightly better accuracy than the second model. This means that the ANN method is a good universal estimator, so using parameters derived from mechanics of failure as input variables is not necessary in the development of ANN models. 5. Conclusions Predicting the ultimate strength of unstiffened panels is a very important task in the structural design of ships. In this study, the artificial neural network technique is applied to estimate the ultimate strength of unstiffened panels in uniaxial compression. ANN models with one hidden layer are developed using the existing experimental results. Two types of ANN model are chosen to investigate how the selection of input variables affects the accuracy of ANN models. It is found that the ANN method can more accurately predict the ultimate strength of plates than those empirical formulae derived by regression analysis. This further improvement is significant in the context of the reliability-based limit state design of ships, because the ANN model can also be used for reliability analysis. The findings in this work clearly demonstrate the capacity of the ANN method. Its accuracy and efficiency are superior to conventional regression analysis. Nevertheless, it must be noted that the main objective of this paper was to investigate and demonstrate the applicability, weaknesses and strengths of using ANNs in developing design formulae. The optimisation of ANN structure, which could be directly affected by the number of available data for training, as well as the choice of activation function, number of neurons in the hidden layer, randomisation procedure, training iterations

1196

Y. Pu, E. Mesbahi / Engineering Structures 28 (2006) 1190–1196

and some other operational parameters such as learning rate and momentum, could be the subject of further research in this area. Two ANN models with a different number of input variables have more or less the same accuracy. This suggests that it is not necessary to transform the original input variables for the sake of improving accuracy. As long as the influential parameters are included in the input, the ANN has the ability to adapt its base-functions to achieve an accurate model. The simplicity and accuracy of ANN models could provide the designer with an effective tool for the optimisation and reliability analysis of complicated structures. Acknowledgements The authors would like to thank the reviewers for their constructive comments, which improved the quality of the paper. References [1] Guedes Soares C. Design equation for the compressive strength of unstiffened plate elements with initial imperfection. Journal of Constructional Steel Research 1988;9:287–310. [2] Paik JK, Thayamballi AK, Kim BJ. Advanced ultimate strength formulations for ship plating under combined biaxial compression/tension, edge shear, and lateral pressure loads. Marine Technology 2001;38(1):9–25. [3] Cui W, Wang Y, Pedersen PT. Strength of ship plates under combined loading. Marine Structures 2002;15(1):75–97. [4] Faulkner D. A review of effective steel plating for use in the analysis of stiffened plating. Journal of Ship Research 1975;19:1–17. [5] Pu Y. Reliability analysis and reliability-based optimisation design of SWATH ships [PhD]. Glasgow: University of Glasgow; 1995. [6] ISSC. Ultimate strength. In: Sumi Y, editor. Committee III.1, 14th international ship & offshore structures congress. Japan: Elsevier; 2000. p. 253–322.

[7] Wei D, Zhang S. Ultimate compressive strength prediction of stiffened panels by counterpropagation neural networks (CPN). In: The ninth international offshore and polar engineering. 1999, p. 280–85. [8] Shao S, Murotsu Y. Structural reliability analysis using a neural network. JSME International Journal, Series A: Mechanics and Material Engineering 1997;40(3):242–6. [9] Principe JC, Neil RE, Curt Lefebrre W. Neural and adaptive system. John Wiley & Sons, Inc; 1999. [10] Bucher CG, Bourgund U. A fast and efficient response surface approach for structural reliability problems. Structural Safety 1990;7:57–66. [11] Mesbahi E. A quantitative approach to the definition of just noticeable difference (JND) for artificial neural networks. In: IEEE international conference on electro-information technology. 2000. [12] Mesbahi E. Application of artificial neural networks in modelling and control of diesel engines [PhD]. University of Newcastle upon Tyne; 2000. [13] Faulkner D. Criteria and guidance for good strength models. Report: Dept. of Naval Architecture & Ocean Engineering, University of Glasgow; July 1991. [14] Pu Y, Godley MHR, Beale RG. Prediction of the ultimate capacity of perforated lipped channels. Journal of Structural Engineering, ASCE 1999;125(2):510–4. [15] Dwight JB, Moxham KE. Welded steel plates in compression. Structural Engineer 1969;47(2):54. [16] Moxham KE. Buckling tests on individual welded steel plates in compression: Report CUED/C-STRUCT/TR.3, Cambridge University; 1971. Report No.: Report CUED/C-STRUCT/TR.3. [17] Ueda Y, Yasukawa W, Yao T, Ikegami H, Ominami R. Ultimate strength of square plates subjected to compression (1st report) — effects of initial deflection and welding residual stresses. Journal of Society of Naval Architects of Japan 1975;137:210–21. [18] Faulkner D, Adamchak JC, Snyder JG, Vetter MF. Synthesis of welded grillages to withstand compression and normal loads. Computers and Structures 1973;3:221–46. [19] Carlsen CA. Simplified collapse analysis of stiffened plates. Norwegian Maritime Research 1977;7(4):20–36. [20] Guedes Soares C. Design equation for the compressive strength of unstiffened plate elements with initial imperfections. Journal of Constructional Steel Research 1988;9:287–310.