Abstract
The paper is focused on the application of artificial neural networks (ANNs) in predicting the natural frequency of basalt fiber reinforced polymer (FRP) laminated, variable thickness plates. The author has found that the finite strip transition matrix (FSTM) approach is very effective to study the changes of plate natural frequencies due to intermediate elastic support (IES), but the method difficulty in terms of, a lot of calculations with large number of iterations is the main drawback of the method. For training and testing of the ANN model, a number of FSTM results for different classical boundary conditions (CBCs) with different values of elastic restraint coefficients (${K}_{T}$) for IES have been carried out to training and testing an ANN model. The ANN model has been developed using multilayer perceptron (MLP) Feedforward neural networks (FFNN). The adequacy of the developed model is verified by the regression coefficient (${R}^{2}$) and Mean Square error (MSE) It was found that the R2 and MSE values are 0.986 and 0.0134 for train and 0.9966 and 0.0122 for test data respectively. The results showed that, the training algorithm of FFNN was sufficient enough in predicting the natural frequency in basalt FRP laminated, variable thickness plates with IES. To judge the ability and efficiency of the developed ANN model, MSE has been used. The results predicted by ANN are in very good agreement with the FSTM results. Consequently, the ANN is show to be effective in predicting the natural frequency of laminated composite plates.
1. Introduction
Continuous laminated plates and laminated composite plates with intermediate stiffeners are one of very common in composite structures that used in many engineering fields such as aerospace, civil and marine industries.
Natural frequencies of laminated composite structures are the classical tools for the intelligent diagnosis of composite defects (e.g. fiber breakage, matrix cracks, debonding and delamination). Vibration response has become an important tool in Structural health monitoring (SHM) in the last decades, the natural frequencies of composite structure are extracted and analyzed under each damage scenario as damage detection and localization technique based on the changes in vibration parameters. In the recent years, several approaches to find the natural frequencies and the mode shapes of laminated composite plates became a field that has attracted a lot of interest in the scientific community.
In order to find the natural frequencies and the mode shapes for different boundary conditions with IES, a numerical approach or an approximate method must be used. Several researchers are attracted to vibration of plates with IES. FSTM is one of a semianalytical methods are welcomed in the many literatures as an alternative to the exact solution. In our previous paper by Altabey [1], a semianalytical method, the FSTM approach has been used to investigate the free vibration of basalt FRP laminated variable thickness rectangular plates with IES. Since all of the coefficients in the FSTM method can be obtained, a lot of calculations with large number of iterations must be performed to obtain a frequency parameters, i.e. the time of solutions will be increased, and the large amount of data are required to study the changes of plate natural frequencies due to IES. This is the main drawback of the method identified so far.
In the present study, the laminated composite plate shown in Fig. 1 has been modeled and analyzed using FSTM method. The laminated composite plate was manufactured using five symmetrically, angleply, laminates with the fiber orientations [45º/45º/45º/45º/45º] of basalt fiber and a polymer resin matrix. A predictive model for natural frequency in terms of fiber orientations is then developed using artificial neural networks. The developed model is tested with the FSTM data which were never used for developing the model. The FSTM results show that R2 and MSE are 0.986 and 0.0134 for train and 0.9966 and 0.0122 for test data respectively. Hence ANN model predicted results are in very good agreement with the FSTM results. Consequently, ANN are shown to be very effective in predicting the natural frequency of laminated composite plates under Four different CBCs are used in the analysis with different KT for IES.
Fig. 1The geometrical model of Basalt FRP laminated variable thickness rectangular plate with IES
2. Governing Equations
The normalized partial differential equation governing the vibration of symmetrically, angleply laminated, variable thickness, rectangular plates under the assumption of the classical deformation theory in terms of the plate deflection ${w}_{o}(x,y,t)$ using the nonDimensional variables $\xi $ and $\eta $ is given by [1, 2]:
where $\beta =a/b$ is the aspect ratio, also:
${W}_{\xi \xi \xi \xi}=\frac{{\partial}^{4}{w}_{o}}{\partial {\xi}^{4}},{W}_{\eta \eta \eta \eta}=\frac{{\partial}^{4}{w}_{o}}{\partial {\eta}^{4}},{W}_{\xi \xi \eta \eta}=\frac{{\partial}^{4}{w}_{o}}{\partial {\xi}^{2}\partial {\eta}^{2}},$
${W}_{\xi \xi \xi \eta}=\frac{{\partial}^{4}{w}_{o}}{\partial {\xi}^{3}\partial \eta},{W}_{\eta \eta \eta \xi}=\frac{{\partial}^{4}{w}_{o}}{\partial {\eta}^{3}\partial \xi},{W}_{\eta \eta}=\frac{{\partial}^{2}{w}_{o}}{\partial {t}^{2}},{m}_{o}=\rho {h}_{o}$
${D}_{ij}$ is the flexural rigidities matrix of the plate.
Since the treatment of IES conditions are the main objective of this paper we presented it in more details. At IES, $y=b/2$, the displacement must vanish and the moment must be continuous, i.e. [1]:
3. Finite strip transition matrix (FSTM) with artificial neural network (ANN) method
3.1. Finite strip transition matrix (FSTM) method
The method is made when such a shape function is not conveniently obtained in case of discussing the plate problems by series. The plate may be divided into N discrete longitudinal strips spanning between supports as shown in Fig. 2. By basic displacement interpolation functions may then be used to represent displacement field within and between individual strips.
For a plate striped in the $\xi $direction as shown in Fig. 2, the shape function $W(\xi ,\eta ,t)$ may be assumed in the form:
where: ${Y}_{i}\left(\eta \right)$ is unknown function to be determined and ${X}_{i}\left(\xi \right)$ is chosen a priori, the basic function in $\xi $direction.
Fig. 2Finite strip simulation on plate
3.2. Artificial neural network (ANN) modeling
Artificial neural network (ANN) is an attractive inductive approach for modeling nonlinear and complex systems without explicit physical representation and thus provides an alternative approach for modeling hydrologic systems. Artificial neural network was first developed in the 1940s. Generally speaking, ANNs are information processing systems. In recent decades, considerable interest has been raised over their practical applications. Training of artificial neural network enables the system to capture the complex and nonlinear relationships that are not easily analyzed by using conventional methods such as linear and multiple regression methods and the network is built directly from experimental or numerical data by its selforganizing capabilities. Based on the different applications, various types of neural network with various algorithms have been employed to solve the different problems. In this work will be using the preferred ANN structures to predict the frequency parameter of presented numerical conditions.
3.2.1. ANN configuration
Since the ANN configuration has a great influence on the predictive quality, various arrangements have been considered in previous work. It is necessary to define a simple code to describe the ANN configuration, as follows:
where ${N}_{in}$ and ${N}_{out}$ are the element numbers of input and output parameters, respectively, and $e$ is the number of hidden layers. ${N}_{h1}$ and ${N}_{h2}$ are numbers of neurons in each hidden layer, respectively. For example, $\left\{7{\left[21\right]}_{1}1\right\}$ means a one hidden layer ANN with seven input and one output parameters, with the hidden layer containing 21 elements (neurons); $\left\{9{\left[\begin{array}{ll}15& 10\end{array}\right]}_{2}1\right\}$ denotes a nine input and one output ANN, with 15 and 10 neurons, respectively, in two hidden layers.
The powerful function of an ANN is due to the neurons within the hidden layers, as well as to the related interconnections. Networks are also sensitive to the number of neurons in their hidden layers. It is believed that an ANN can represent any reasonable relationship between input and output if the hidden layers have enough neurons. However, for the practical case, more hidden neurons bring more interconnections, which require, in turn, larger training datasets for learning the relationships. It is therefore always necessary to optimize the number of neurons of the ANN hidden layers, as demonstrated by Demuth and Beale [3].
3.2.2. Performance evaluation measures
It is very useful from the designer point of view to have a neural system aids to decide whether his suggested design is suitable or not by Compute the MSE from equation:
where: ${\left(\mathrm{\Omega}\right)}_{nn}$ is the predicted frequency parameter from ANN, $\mathrm{\Omega}$ is the target or computed values from FSTM method of frequency parameter, and $n$ is the number of FSTM computed data values.
Thus, the performance index will either have one global minimum, depending on the characteristics of the input vectors. Local minimum is the minimum of a function over a limited range of input values. Local minimum is an unavoidable when the ANN is fitted. So, a local minimum may be good or bad depending on how close the local minimum is to the global minimum and how low an MSE is required. In any case, the method applied to solve this problem and descent the local minimum with momentum. Momentum allows a network to respond not only to the local gradient, but also to recent trends in the error surface. Without momentum a network may get stuck in a shallow local minimum.
The estimation performances of frequency parameter $\left(\mathrm{\Omega}\right)$ is evaluated by the lack of fit with the regression coefficient of the multiple determination ${R}^{2}$; ${R}^{2}$ is defined as:
The value of ${R}^{2}$ is equal to or lower than 1.0. A higher value of ${R}^{2}$ implies a better fit. When the ANN shows a very good fit, ${R}^{2}$ approaches 1.0. A good fit of the ANN means that the ANN gives good estimations for the dielectric properties change used for the regression. Lower ${R}^{2}$ values means poorer estimations and the error band of the estimated result is wider.
3.3. Feedforward neural networks (FFNN)
In this work we will use the suggested feed forward neural network, FFNN, to predict natural frequency of basalt FRP composite plate, because it has minimum mean square error (MSE), in composite structure applications [4, 5]. FFNN in general consist of a layer of input neurons, a layer of output neurons and one or more layers of hidden neurons [6]. Neurons in each layer are interconnected fully to previous and next layer neurons with each interconnection have associated connection strength or weight. The activation function used in the hidden and output layers neurons is nonlinear, where as for the input layer no activation function is used since no computation is involved in that layer. Information flows from one layer to the other layer in a feedforward manner. Various functions are used to model the neuron activity such as liner transfer function ($purelin\left(n\right)$), TanSigmoid transfer function ($\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{g}\left(n\right)$) or Radial Basis (Gaussian) transfer functions ($radbas\left(n\right)$).
The training process is terminated either when the (MSE) between the observed data and the ANN outcomes for all elements in the training set has reached a prespecified threshold or after the completion of a prespecified number of learning epochs.
4. Results and discussion
In this work, The FSTM with ANNs techniques are used to predict the free vibration of the laminated composite plate shown in Fig. 1. The plate was manufactured using five symmetrically, angleply, laminates with the fiber orientations [$\theta $/$\theta $/$\theta $/$\theta $/$\theta $] is [45°/–45°/45°/–45°/45°] of basalt fiber and a polymer resin matrix. Physical and mechanical properties of the basalt FRP laminate composite plate are shown in Table 1.
Table 1Physical and mechanical properties of the basalt FRP
${E}_{1}$  ${E}_{2}={E}_{3}$  ${G}_{1}={G}_{3}$  ${G}_{2}$  ${\upsilon}_{1}={\upsilon}_{3}$  ${\upsilon}_{2}$  $\rho $ 
96.74 GPa  22.55 GPa  10.64 GPa  8.73GPa  0.3  0.6  2700 kg/m^{3} 
The nondimensional frequency parameter $\mathrm{\Omega}$ are addressed in form [1]:
The plate has linear deformation in thickness $h\left(y\right)$ woth nondimensional form [1]:
where: $\mathrm{\Delta}$ is the tapered ratio of plate given by $\mathrm{\Delta}={(h}_{b}{h}_{o})/{h}_{o}$, (${h}_{o}$) is the thickness of the plate at $\eta =$0 and (${h}_{b}$) is the thickness of the plate at $\eta =$1.
4.1. Convergence study and accuracy
In this subsection, the author has carried out for convergening the proposed method, first six frequencies are calculated and compared with available results in literatures [1]. He compared the computational results of FSTM method from previous work [1] with values available from literatures [7, 8] for Eglass/ epoxy, square ($\beta =$1.0), uniform thickness ($\mathrm{\Delta}=$0) plates with elastic foundation support, the mechanical properties of the plate material are ${\upsilon}_{1}=$${\upsilon}_{2}=$ 0.23, $D=E{h}^{3}/\left[12\left(1{\upsilon}^{2}\right)\right]$, and ${D}_{66}=\left(1\upsilon \right)D/2$. Two different CBCs (SSSS and CCCC) are used in calculations, KT is taken equal to 500, 1390.2 for SSSS and CCCC respectively. In this study are addressed in form $\mathrm{\Omega}={\left(\rho h{\omega}^{2}{a}^{4}/D\right)}^{1/2}$. He found the results by FSTM method are in a very close agreement with results in References [7, 8] and the convergence speed can be achieved with terms of series solution ($N=$ 3 to 7).
4.2. Finite strip transition matrix (FSTM) results
In the present study, the numerical computations using FSTM approach is applied with an ANNs, which are combined to decrease detection effort to discern for different KT by minimizing the number of FSTM computations in order to keep save the time of calculations to a minimum. The method has successfully computations of the frequency parameter using only two different ${K}_{T}$, the first one is located at ${K}_{T}=$ 50 the second is ${K}_{T}=$ 750, respectively, as shown in Table 2.
Table 2 presents the first six frequencies of the laminated composite plate shown in Fig. 1. The plate has the parameters of aspect ratio ($\beta $) and thickness tapered ratio ($\Delta $) are 0.5. The Four different type of CBCs (SSSS, CCCC, SSFF and CCFF) and two different values of KT of IES are used in the calculations to study the changes of natural frequencies due to IES. The locations of the IES is at midline of the presented plate.
Table 2The first six frequencies of laminated, plate shown in Fig. 1 for two different values of KT, (Δ=0.5), (β=0.5)
${K}_{T}$  ${\mathrm{\Omega}}_{1}$  ${\mathrm{\Omega}}_{2}$  ${\mathrm{\Omega}}_{3}$  ${\mathrm{\Omega}}_{4}$  ${\mathrm{\Omega}}_{5}$  ${\mathrm{\Omega}}_{6}$  
SSSS  50  22.1450  36.2210  53.5870  78.2360  105.5870  138.6970 
750  55.0692  69.1452  86.5112  111.1602  138.5112  171.6212  
CCCC  50  28.4310  46.5025  68.7979  100.4437  135.5584  178.0668 
750  61.3551  79.4267  101.7221  133.3678  168.4826  210.9910  
SSFF  50  12.2750  20.0773  29.7033  43.3663  58.5270  76.8799 
750  45.1992  53.0015  62.6275  76.2905  91.4511  109.8041  
CCFF  50  19.5928  32.0466  47.4112  69.2194  93.4182  122.7123 
750  52.5170  64.9707  80.3353  102.1436  126.3424  155.6365 
4.3. A feedforward neural network (FFNN) design for predicting frequency parameters
A FFNN configuration in this case was $\left\{2{\left[\begin{array}{ll}5& 5\end{array}\right]}_{2}1\right\}$, with tansigmoid neurons for the first layer while the second layer has pure linear ones. FFNN is trained by measuring values of ${K}_{T}$, $\theta $ to predict $\mathrm{\Omega}$ for four different CBCs are SSSS, CCCC, SSFF and CCFF. Determination of the hidden layer, in addition to the number of nodes in the input and output layers, for providing the best training results, was the initial phase of the training procedure. The target for MSE to be reached at the end of the simulations was 0.001. Since the second step was largely a trialanderror process, and involved FFNNs with the number of hidden layer neurons more than five, it did not show any sizeable improvement in prediction accuracy. Thus, the number of neurons for the single hidden layer was selected as five neurons. Selection of the number of hidden layer neurons, with respect to the MSE term is shown in Fig. 3. In the first FFNN structure is applied for training the results data of FSTM. Fig. 4 shows the training performance of suggested FFNN.
Fig. 5 represents the comparison between the FSTM data and the feedforward neural network FFNN predicted data ($\mathrm{\Omega}$) for ${K}_{T}=$50 of four different CBCs are SSSS, CCCC, SSFF and CCFF. The results of this NN show much satisfactory predication quality for this case study. Fig. 6 shows the comparison between the FSTM data and the feed forward neural network FFNN expected (tested) data for ${K}_{T}=$750 of four different CBCs are SSSS, CCCC, SSFF and CCFF. From Fig. 6, it noted that the expected data from the suggested FFNN are applicable with the FSTM data.
Fig. 3Plot of MSE terms corresponding to the number of hidden layer neurons, used for selecting the optimum number of hidden layer neurons
Fig. 4Training performance of suggested FFNN
Fig. 5Comparison between the FSTM data and FFNN predicted data for Ω
Fig. 6Comparison between the FSTM data and FFNN expected data for Ω
Fig. 7The performances of the present FFNN to predict data of Ω
Fig. 8The performances of the present FFNN to expect data of Ω
Table 3Mean square error (MSE) and regression coefficient (R2) values
Data  MSE  ${R}^{2}$ 
Predicted  0.0134  0.986 
Expected  0.0122  0.9966 
Table 3 and Figs. 7 and 8 represent the performances of the present FFNN by calculating the values of MSE and ${R}^{2}$ (see Eqs. (5, 6)) between FSTM data for both FFNN predicted and expected data for $\mathrm{\Omega}$. As shown in Fig. 7 the value of MSE and ${R}^{2}$ between the FSTM and FFNN predicted data are 0.0134 and 0.986 respectively. From Fig. 8 the value of MSE and ${R}^{2}$ between the FSTM and FFNN expected data are 0.0122 and 0.9966 respectively.
4.4. The use of present FFNN for predicting nonFSTM data of nondimensional frequency parameter $\mathbf{\Omega}$
The main goal of the artificial neural network design is predicting non FSTM data. In this section we will use the suggested FFNN to predict some non FSTM data not included in FSTM evaluation. It is selected to use seven different ${K}_{T}$, for four types of CBCs (SSSS, CCCC, SSFF and CCFF). The previous parameters ${K}_{T}$, $\theta $ are the input vectors for artificial neural network, while the output is the signal vector is $\mathrm{\Omega}$.
Fig. 9The FFNN predicted nonFSTM results of nondimensional frequency parameter Ω
a) SSSS
b) CCCC
c) SSFF
d) CCFF
Table 4The predicted nonFSTM results of the first six frequencies by FFNN
${K}_{T}$  ${\mathrm{\Omega}}_{1}$  ${\mathrm{\Omega}}_{2}$  ${\mathrm{\Omega}}_{3}$  ${\mathrm{\Omega}}_{4}$  ${\mathrm{\Omega}}_{5}$  ${\mathrm{\Omega}}_{6}$  
SSSS  150  Ref [1]  34.6580  48.7340  66.1000  90.7490  118.1000  151.2100 
FFNN  34.8971  48.1343  66.6159  90.5891  118.8782  148.7286  
400  Ref [1]  45.8453  59.9213  77.2873  101.9363  129.2873  162.3973  
FFNN  45.8453  60.1268  77.2873  100.2624  129.2873  162.3973  
1500  Ref [1]  62.4709  76.5469  93.9129  118.5619  145.9129  179.0229  
FFNN  65.5428  76.7674  93.799  117.6289  147.1225  178.4941  
2500  Ref [1]  67.5270  81.6030  98.9690  123.6180  150.9690  184.0790  
FFNN  67.6715  80.0752  98.248  122.7383  152.2729  183.4709  
5000  Ref [1]  70.7832  84.8592  102.2252  126.8742  154.2252  187.3352  
FFNN  71.5277  83.8535  101.5647  125.5413  154.9449  186.8396  
10000  Ref [1]  72.7065  86.7825  104.1485  128.7975  156.1485  189.2585  
FFNN  72.8107  86.2942  105.2392  128.2564  156.1388  189.2703  
1E+06  Ref [1]  73.1195  87.1955  104.5615  129.2105  156.5615  189.6715  
FFNN  72.6947  87.0396  105.4209  128.5854  156.8348  189.6596  
CCCC  150  Ref [1]  40.9440  59.0155  81.3109  112.9567  148.0714  190.5798 
FFNN  41.0549  57.0992  80.3422  111.2997  148.2179  186.4952  
400  Ref [1]  52.1313  70.2028  92.4983  124.1440  159.2587  201.7672  
FFNN  55.4754  70.442  92.6468  123.0928  160.6131  201.1848  
1500  Ref [1]  68.7569  86.8284  109.1239  140.7696  175.8843  218.3928  
FFNN  69.7659  85.3865  109.1148  141.4497  179.6854  218.0006  
2500  Ref [1]  73.8130  91.8845  114.1800  145.8257  180.9404  223.4489  
FFNN  73.4808  92.2119  113.7449  143.3314  180.9021  222.5692  
5000  Ref [1]  77.0692  95.1407  117.4361  149.0819  184.1966  226.7050  
FFNN  77.8273  94.092  117.0825  147.6263  184.9599  226.1968  
10000  Ref [1]  78.9925  97.0640  119.3594  151.0052  186.1199  228.6283  
FFNN  79.3118  96.3864  119.7441  150.1071  187.0576  228.5796  
1E+06  Ref [1]  79.4055  97.4770  119.7724  151.4182  186.5329  229.0413  
FFNN  79.1422  97.2499  120.7791  150.5119  186.7872  229.1948  
SSFF  150  Ref [1]  24.7880  32.5903  42.2163  55.8793  71.0400  89.3929 
FFNN  24.4094  32.9047  41.7941  54.6268  71.2814  89.0171  
400  Ref [1]  35.9753  43.7777  53.4036  67.0666  82.2273  100.5802  
FFNN  35.9713  42.5469  52.8278  67.4484  83.9108  98.563  
1500  Ref [1]  52.6009  60.4033  70.0293  83.6922  98.8529  117.2058  
FFNN  52.904  60.2689  70.3874  83.7121  100.3213  119.6639  
2500  Ref [1]  57.6570  65.4594  75.0853  88.7483  103.9090  122.2619  
FFNN  57.9028  64.964  75.361  89.2589  105.6161  122.2164  
5000  Ref [1]  60.9132  68.7155  78.3415  92.0045  107.1652  125.5181  
FFNN  61.3157  69.5938  79.1805  91.7539  107.5848  125.4066  
10000  Ref [1]  62.8365  70.6388  80.2648  93.9278  109.0885  127.4414  
FFNN  63.0778  70.2994  80.2805  93.3565  109.3353  127.2809  
1E+06  Ref [1]  63.2495  71.0518  80.6778  94.3408  109.5015  127.8544  
FFNN  63.3054  70.5596  80.69  93.9744  110.0274  127.6158  
CCFF  150  Ref [1]  32.1058  44.5596  59.9242  81.7324  105.9312  135.2253 
FFNN  32.1827  43.2269  59.686  81.9174  108.2676  135.0946  
400  Ref [1]  43.2931  55.7469  71.1115  92.9197  117.1185  146.4127  
FFNN  45.562  57.8813  73.8431  93.8403  117.8631  145.3086  
1500  Ref [1]  59.9187  72.3725  87.7371  109.5453  133.7442  163.0383  
FFNN  59.6275  72.4773  88.4423  108.7844  134.0482  163.8665  
2500  Ref [1]  64.9748  77.4286  92.7932  114.6014  138.8002  168.0944  
FFNN  65.0645  76.7108  92.8809  114.0148  139.5554  167.6554  
5000  Ref [1]  68.2310  80.6848  96.0494  117.8576  142.0564  171.3505  
FFNN  68.3337  79.7057  96.0889  117.8545  143.8554  171.2413  
10000  Ref [1]  70.1543  82.6081  97.9727  119.7809  143.9797  173.2738  
FFNN  70.9017  82.7021  98.6556  119.3321  144.5419  173.0477  
1E+06  Ref [1]  70.5673  83.0211  98.3857  120.1939  144.3927  173.6868  
FFNN  70.6424  81.6968  98.1541  120.3744  146.7246  173.5624 
Fig. 10The performances of the present FFNN to predicted nonFSTM results of nondimensional frequency parameter Ω
a) SSSS
b) CCCC
c) SSFF
d) CCFF
Fig. 9 shows the FFNN predicted results of the first six frequencies of the laminated composite plate are presented in this study. The plate has $\beta =$0.5 and $\mathrm{\Delta}=$0.5 with four different types of CBCs.
The predicted nonFSTM results of the first six frequencies by FFNN and a comparison with the previous work results in literature of FSTM method by Altabey [1] are presented in Table 4. As a result, a FFNN gave good prediction for nonFSTM data even for extrapolations $\mathrm{\Omega}$ in presented laminated composite plate.
From Table 4 and Fig. 9, we can see, the first six frequencies increase with the increasing of the value of ${K}_{T}$ and we are observed that the gap between values of frequencies in the small ${K}_{T}$ and next values of ${K}_{T}$ are higher than the gap between values of frequencies in the large ${K}_{T}$, and the frequencies at high values of ${K}_{T}$ are almost constant, for all conditions (SSSS, CCCC, SSFF and CCFF). On the other hand, the fully clamped (CCCC) and semisimply supported (SSFF) condition have the higher and lower values of frequencies respectively and the other two conditions (SSSS) and (CCFF) are lie between them with an intermediate value. This conclusion was found in other work [1].
${R}^{2}$ of nonFSTM results are 0.9995, 0.9994, 0.9992 and 0.9993 for SSSS, CCCC, SSFF and CCFF plate respectively. All of the predicting results are plotted on the diagonal line (Fig. 10) to observe the performances of the present FFNN to predict nonFSTM of $\mathrm{\Omega}$. The MSE is defined of the predicted non FSTM. The MSE is of nonFSTM results are 0.006714, 0.0084, 0.007428 and 0.0086 for SSSS, CCCC, SSFF and CCFF plate respectively.
5. Conclusions
The natural frequency values were found by analyses which were done of basalt FRP laminated variable thickness rectangular plates with IES by FSTM method and ANNs algorithm, which are combined to decrease computational effort to discern natural frequency in basalt FRP laminated composite plates, in order to save the time of the FSTM computational data to a minimum with high accuracy and easy. Based on the FSTM results and the results predicted by artificial neural networks, the following conclusions are drawn for laminated composite material plates.
1) The FFNN model has been developed by considering the ${K}_{T}$ and ply angle ($\theta $) as the input for predicting $\mathrm{\Omega}$. The developed FFNN model could predict $\mathrm{\Omega}$ with the ${R}^{2}$ and MSE are 0.986 and 0.0134 for training data set and 0.9966 and 0.0122 for test data respectively.
2) The FFNN model could predict nonFSTM of $\mathrm{\Omega}$ with ${R}^{2}$ of nonFSTM results are 0.9995, 0.9994, 0.9992 and 0.9993 for SSSS, CCCC, SSFF and CCFF plate respectively. The MSE is defined of the predicted nonFSTM. The MSE is of nonFSTM results are 0.006714, 0.0084, 0.007428 and 0.0086 for SSSS, CCCC, SSFF and CCFF plate respectively.
3) The ANN predicted results are in very good agreement with the FSTM results.
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