Estimating moment capacity of ferrocement members using self-evolving network

Abstract

In this paper, an empirical model based on self-evolving neural network is proposed for predicting the flexural behavior of ferrocement elements. The model is meant to serve as a simple but reliable tool for estimating the moment capacity of ferrocement members. The proposed model is trained and validated using experimental data obtained from the literature. The data consists of information regarding flexural tests on ferrocement specimens which include moment capacity and cross-sectional dimensions of specimens, concrete cube compressive strength, tensile strength and volume fraction of wire mesh. Comparisons of predictions of the proposed models with experimental data indicated that the models are capable of accurately estimating the moment capacity of ferrocement members. The proposed models also make better predictions compared to methods such as the plastic analysis method and the mechanism approach. Further comparisons with other data mining techniques including the back-propagation network, the adaptive spline, and the Kriging regression models indicated that the proposed models are superior in terms prediction accuracy despite being much simpler models. The performance of the proposed models was also found to be comparable to the GEP-based surrogate model.

Keywords

ferrocement / moment capacity / self-evolving neural network

Introduction

Ferrocement is a composite material made up of a cement mortar with closely spaced layers of wire mesh embedded as reinforcement (see Fig. 1). As in the case of reinforced concrete, the resistance to compression is provided by the cement mortar, while the mesh reinforcement takes care of tensile stresses. Compared to the reinforced concrete, ferrocement elements are easier to fabricate to a variety of shapes and require less formwork. They are also less prone to cracking and offer a better resistance to impact and vibrations. One of the key factors to consider in the design of ferrocement structural elements is the flexural behavior, which is difficult to predict given the complex interaction between the mesh network and the mortar matrix. The currently available methods of estimating flexural capacity include the plastic analysis and the mechanism approach. The simplifying assumptions made in the methods make them quite convenient to use, but compromise their reliability because of the inadequate representation of the behavior of the composite material.

Recently, soft-computing techniques have been used to develop experiment-based alternative models to predict flexural capacity of ferrocement members. These include the use of back-propagation neural (BPN) and neuro-fuzzy networks [¹], as well as Gene expression programming (GEP) [²]. The key advantages of these approaches to modeling is their ability to generate empirical models for predicting behavior of systems based on the observed performance rather than a prior knowledge on how the systems work. In this study, self-evolution algorithm will be used to develop a neural network model of the moment capacity of ferrocement members. The idea is obtaining a robust empirical formula capable of making accurate predictions with a minimum number of constants. The relevant information necessary for the prediction are: the cross-sectional geometry of the ferrocement specimen, mortar compressive strength, as well as the tensile strength and volume fraction of wire mesh. To test the viability of the proposed model as alternative capacity predictor, a comparative study will be carried out between the prediction results of the proposed method and those of the analytical and empirical models reported in the in the literature.

Existing methods of moment capacity estimation

The currently available methods of predicting moment capacity of ferrocement elements are outlined in the following sub-sections.

Plastic analysis method

Plastic analysis method [³] involves evaluation of moment capacity on the basis of the condition of equilibrium of forces. The method is based on the assumption that the ferrocement is a homogeneous elasto-plastic material. The resulting expression for moment capacity M_{\mathrm{u}} is given as:

(1)M_{\mathrm{u}}\mathrm{}={\sigma }_{\mathrm{t}\mathrm{u}}b\left(h-x_{\mathrm{1}}\right){\displaystyle \frac{h}{\mathrm{2}}},

where M_u, b, h, and x₁ are, respectively, the moment capacity, the section width, the section height, and the depth of neutral axis. σ_tu is defined as A_{\mathrm{s}}{f}_{\mathrm{u}}/\left(bh\right) in which f_u is the ultimate tensile strength of mesh reinforcement. The drawback of the method is that the simplifying assumptions made are likely to lead to a very poor prediction of the flexural response of ferrocement as confirmed by the lack of a good agreement with experimental data in a previous study [¹].

Mechanism approach

This approach is proposed in Ref. [⁴] based on the simplified plastic analysis where the distance from neutral axis to the top surface is assumed to be small enough to regard the mesh reinforcement to be purely in tension. The moment capacity based on the mechanism approach is expressed as follows:

(2)M_{\mathrm{u}}=\mathrm{}{\sigma }_{\mathrm{t}\mathrm{u}}b{\displaystyle \frac{h^{\mathrm{2}}}{\mathrm{2}}}.

Being a simplified version of plastic analysis approach, the method suffers from the similar drawback highlighted previously.

Second order polynomial regression formula

This empirical formula was developed by Naaman and Homrich [⁵], as a simplified method of computing the flexural capacity of ferrocement sections. The equation is as follows:

(3a)y=0.005+\mathrm{0.422}x-\mathrm{0.0772}{x}^{\mathrm{2}},

(3b)x=\frac{v_{\mathrm{f}}{\sigma }_{\mathrm{y}}}{f_{\mathrm{c}}},\hspace{0.17em}\hspace{0.17em}y={\displaystyle \frac{M_{\mathrm{u}}}{\mathit{\xi }{f}_{\mathrm{c}}b{h}^{\mathrm{2}}}},

in which v_f is the ratio of cross-sectional area of mesh reinforcement to the area of section; σ_y is the ultimate tensile strength of mesh reinforcement; f_c compressive strength of the mortar; and ξ is the global efficiency factor for mesh reinforcement.

The downside of the formula is inability to adequately capture the complexity of ferrocement behavior as confirmed by the high scatter observed when the estimates of the formula are compared with the experimental data [¹].

Models based on data mining techniques

To tackle the modeling difficulties associated with analytical methods, a number of meta-heuristic data mining tools have recently been used to develop empirical models for predicting the flexural capacity of ferrocement. For instance, Mashrei et al. [¹] successfully utilized BPN network and neuro-fuzzy system (ANFIS) to predict the moment capacity of ferrocement members. Their results showed that BPN and ANFIS are far superior to the methods based on plastic analysis. The major downside of conventional BPN and ANFIS network models is the difficulty in expressing the complex network of computational units in form of easy-to-use equations, which tends to make them less appealing to end-users. A more transparent model based on Gene Expression programming (GEP) technique was proposed by Ghandomi et al. [²]. This model, which is in form of a relatively simple formula, is much more convenient to use than ANN-based models proposed in Ref. [¹], while at the same time gives a comparatively accurate estimate of flexural capacity. Other data mining techniques that could be used for the same purpose include the methods of polynomial regression, moving least squares, penalised spline regression, and Kriging regression. Their successful usage in previous studies on structural and material modeling [⁶–⁹] makes them worthy of consideration for empirical modeling of the behavior of ferrocements.

The major limitation of data mining techniques is that their reliability depends on the quality and coverage of the training and testing data. Therefore, to ensure that accurate and robust models are developed, the database should comprise of good quality data and data division should be made in such a way that sufficiently large data are allocated for testing the prediction quality of the models.

Self-evolving Neural network (SEANN)

Self-evolving networks are a class of artificial neural networks whose training involve not only optimising the synaptic weights but also the network topology. Self-evolution algorithm minimises user intervention during training by allowing the network topology to evolve as it interacts with the mined data. The main aim of the algorithm is to attempt to address some of the shortcomings of conventional neural networks which include lack of transparency and poor generalisation. In this study, ferrocement moment capacity data are used to develop a self-evolving network model in order to benefit from the ability of the later in producing a relatively simple but accurate empirical model to estimate the moment capacity of ferrocement sections. The key distinction between the ANN and ANFIS models developed previously and the proposed model is that the algorithm used in this work allows a much simpler but accurate model to be developed.

Self-evolving network algorithm

The network training algorithm adopted in this work involves topology optimisation process based on jumping particle swarm optimisation (JPSO) technique and network parameter optimisation based on a combination of particle swarm optimisation and linear regression methods. To minimise computation cost, the algorithm adopts a bottom to top strategy, where the network topologies are initially generated as very simple architectures with a single hidden node, then gradually evolve in complexity as they interact with the knowledge base. The topology of the networks is updated using JPSO, a discrete optimisation technique proposed by Ref. [¹⁰]. The self-evolution procedure starts by generating a population of neural nets, each having a randomly generated synaptic connections and synaptic parameters. The connection parameters C_i,j are binary, assuming a value of 1 if the connection between two nodes i and j exist and 0 otherwise (refer to Fig. 2). The binary connection parameters are updated using the JPSO algorithm, while the synaptic weights p_i,j of connections between the input nodes and the nonlinear hidden nodes are updated using the continuous PSO algorithm. With the output node being represented by a linear model, the weights of connections to the output node, represented by w_i,j, are optimised using the method of least squares.

The complexity of the network is increased gradually by adding more nodes, one node at a time. At the point of nodal addition, only the information on the current particles best positions are retained by zeroing the values of additional synaptic weights and switching off the additional connections. For the rest of particles (networks), the synaptic weights and topologies are randomly regenerated. The rationale here is to preserve the so-far acquired information as the change takes place. The self-evolution algorithm is summarized in the following steps:

1)‚ Initialize a particle swarm population of size N, with each particle representing neural networks with a single hidden node and randomly generated set of synaptic weights (p) and connection parameters (C).

2)‚ Determine the synaptic weights (w) using least square method and evaluate the fitness of each particle and update the best particle and global positions.

3)‚ Use PSO/JPSO to update particle coordinates for certain number of iterations in the following sub-steps:

(a) use PSO to update the weight vector (p) of each particle,

(b) use JPSO to update the connection parameters (C) of each particle,

(c) update the particle best position and the best swarm position.

4)‚ If convergence is sufficient then go to step 8. Else continue.

5)‚ Reset randomly the binary and continuous parameters of duplicate particles. Also, reset in the same manner, the binary parameters of certain fraction of the swarm with poor fitness then go to step 3.

6)‚ If number of iterations is less than the maximum number then go back to step 3. Else continue.

7)‚ Generate N particles with a number of nodes which is one greater than the current number of nodes. Replace all current particles with the newly generated particles while retaining the current particle best positions (topology and synaptic weight). Go back to step 3.

8)‚ Terminate algorithm and return result.

The self-evolution algorithm is also represented by the flow chart shown in Fig. 3. In order to complement the effort by the swarm of partially connected networks in the search for global best network, a parallel swarm of fully connected networks but with the same number of nodes is simultaneously optimized, with the former learning from the later whenever the best swarm position lies in the latter.

Activation function

The performance and complexity of networks is largely dependent on the type of activation function selected. Depending on the characteristics of the training data, some transfer functions yield better accuracy with simpler network topologies than others. In this work, the product unit function is used as a model for the hidden nodes in the network. The choice of this function is informed by its greater information capacity, which could significantly reduce the required number of hidden nodes [¹¹]. The function is represented by the following equation:

(4)f\left(x\right)=\mathrm{}{\displaystyle \prod _{i=\mathrm{1}}^{i=n}{x}_i^{p_{i,j}}},

where n is the number of input variables, x and p are the input signal and the synaptic weight, respectively. In this study, the values of p are limited to a range between -3 to +3 in order to prevent over-fitting the data. Also for the sake of obtaining a relatively simple empirical equation, the p values are rounded up to the nearest 0.5.

SEANN model for flexural capacity prediction

Database description

The database gathered for this study is obtained from the results of a series of experimental studies compiled by Mashrei et al. [¹] from a number of sources [³,⁴,¹²–¹⁵]. The database consists of 75 data sets, with each set including a measurement of ultimate moment of a ferrocement section for given section dimensions (depth and width) and area ratio of mesh reinforcement. Also included in the database are values of compressive strength of cement mortar and tensile strength of reinforcement. The summary of the database characteristics is provided in Table 1.

Input parameters

Due to dependence of neural networks on experimental measurements, the input parameters to a network must sufficiently represent the factors controlling the behavior of the system under consideration for a reasonable prediction to be done by the network. The moment capacity, the parameter under investigation in this paper, depends on parameters such as the section geometry, area of reinforcement, and the strength properties of the cement mortar and reinforcement. In a mathematical form, the moment capacity can be related to the controlling variables as:

(5)M_{\mathrm{u}}\mathrm{}=f\left(b,h,f_{\mathrm{c}\mathrm{u}},f_{\mathrm{u}},v_{\mathrm{f}}\right).

It is often useful for the sake of dimensionality reduction and for the sake of doing away with scale effects to build models based on non-dimensional parameters. For instance, the moment capacity can be transformed to a non-dimensional parameter by normalizing it with f_{\mathrm{c}\mathrm{u}}b{h}^{\mathrm{2}}. If it is, in turn, expressed as a function of non dimensional parameters, namely α and v_f, the resulting relationship takes the following form:

(6)\eta \mathrm{}=f\left(v_{\mathrm{f}},\alpha \right),

where \eta =M_{\mathrm{u}}/f_{\mathrm{c}\mathrm{u}}b{h}^{\mathrm{2}}, \alpha =f_{\mathrm{u}}/f_{\mathrm{c}\mathrm{u}}, and v_f is the ratio of reinforcement area to the sectional area of ferrocement in percentage. In this work, two self-evolving networks were trained to approximate the functions described in Eqs. (5) and (6). The SEANN model based on Eq. (5) is referred to as SEANN-I, while SEANN-II model refers to the model predicting the normalized moment capacity based on the parameters enclosed in the right bracket of Eq. (6).

Networks training and testing

To develop the two versions of SEANN, the database was divided into two sets: i) training set to optimise the network parameters; ii) testing data to assess the prediction quality of the network models. A total of 36 data sets were used for the purpose of training, while the remaining 39 sets were reserved for testing. Note that roughly half of the data was allocated for testing in order to ensure a more robust assessment of the models accuracy given the limited number of data points. The data division also takes into consideration the presence of some repeated tests in the data. All Repeated tests of are lumped into groups and then allocated to either training or testing set. This is to avoid having a false impression of a good prediction when a lump of repeated tests is divided between training and testing data. For the purpose of comparison, conventional BPN network with a combination of linear and sigmoid activation functions was also trained alongside the two versions of self-evolving network. The network training was carried out in accordance with the algorithm described in sub-section 3.1. To avoid the tendency of over-fitting the data and unnecessary complexity, the training is based on the error function proposed by Ref. [¹⁶], which penalises both the prediction inaccuracy and topology complexity. The particular advantage of this method over other regularization methods such as weight decay method is that the former pays attention to reducing the number of network connections rather than reducing the magnitude of the synaptic weights, which often doesn’t correlate positively with network complexity reduction [¹⁷]. Other parameters used to assess the prediction quality of the proposed models include the mean (µ_β) and standard deviation (σ_β) of the ratio of predicted capacity to actual value β = M_u (predicted) /M_u (observed). The mean value of β ratio indicates whether a model, on average, underestimates or overestimates the value in question, while the standard deviation gives an idea about the extent of scatter in the prediction. A perfect model with 100% accuracy will have a mean value of 1.0 and a standard deviation of zero. The coefficient of determination and the non-dimensional root mean square error (N-RMSE) were also used as performance indicators alongside those described previously. The N-RMSE is defined using the Eq. (7):

(7)N-RMSE=\mathrm{}{\displaystyle \frac{\sqrt{{\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \sum _{i=\mathrm{1}}^{i=n}{\left(O_i-y_i\right)}^{\mathrm{2}}}}}{O_{\mathrm{rms}}}},

where O_i, y_i, and O_rms are, respectively, the experimental value, the predicted value, and the root mean square value of the observed data.

Results and discussions

The training and testing results of SEANN-I model are displayed in Figs. 4(a) and 4(b) scattergrams, respectively. Similarly, Figs. 5(a) and 5(b) show the scattergrams comparing SEANN-II predictions with training and testing data, respectively. As can be seen, the predictions of the two network models are in a good agreement with the experimental data, given the low scatter level in both cases of training data (σ_β = 0.2482 for SEANN-I and σ_β = 0.2457 for SEANN-II) and the testing data (σ_β = 0.2919 for SEANN-I and σ_β = 0.18708 for SEANN-II). This is indicative of success of the learning process undergone by the networks not only because the network parameters are fine-tuned enough to yield a good approximation of the experimental observations but more crucially because of the ability of the networks to reasonably estimate the moment capacity in comparison with the data that was excluded from the training set. The result also showed that the SEANN-II model generalizes far better than SEANN-I as the former returns much smaller value of σ_β when compared with the testing data.

To further assess the prediction quality of the proposed models, the values of β are plotted against the test data in Figs. 6(a)–6(b). It can be noticed from both figures that the scatter around the horizontal line representing β = 1 has no clear trend, thus indicating that the prediction errors are more likely to be as a result of noise in measurements rather than model limitations. The results presented so far are based on 73 data points. The remaining two data points are considered as outliers and therefore removed from the database. This is due to lack of agreement (100%–200% difference) between the removed points and all models considered in this work (both proposed and existing).

Comparisons with spline and Kriging regression models

To provide a basis for further assessment of the performance of the proposed models, penalized adaptive spline and Kriging regression models are built using the same training data used in developing the SEANN models. Brief description of the two methods is given as follows:

Penalised Spline regression

Spline regression is a surrogate modeling tool for nonlinear function approximation that has been used in a range of engineering applications [⁸,⁹,¹⁸,¹⁹]. The approximated function based on adaptive spline regression is expressed as follows:

(8)f\left(x\right)=\mathrm{}{B}_{\mathrm{0}}+B_{\mathrm{1}}x+\cdots +B_n{x}^n+{\displaystyle \sum _{i=\mathrm{1}}^{i=k}{B}_{ni}\left(x-{\tau }_i\right)},

where n is the degree of spline polynomial, k is the number of knots, and B are coefficients. The regression is performed by minimising the cost (CCV) function:

(9)CCV\left(ϵ\right)=\mathrm{}{\displaystyle \frac{{\displaystyle \sum _{i=\mathrm{1}}^{i=N}{\left[y_i-f\left(x_i\right)\right]}^{\mathrm{2}}}}{N{\left[\mathrm{1}-C\left(ϵ\right)/N\right]}^{\mathrm{2}}}},

where, N is the number of samples; C stands for the complexity penalty term meant to facilitate trimming the size of the approximating function. ϵis the smoothening parameter. In this study, piecewise linear and cubic splines will be compared with the proposed self evolving network model.

Kriging regression method

Kriging is another type of nonlinear function approximating scheme based on Gaussian process interpolation principle. The method is implemented based on the following equation:

(10)y_{\mathrm{p}}\left(\mathit{x}\right)=\mathrm{}\mathit{\mu }+{\mathit{\psi }}^{\mathrm{T}}{\left({\mathit{\Psi }}^{-\mathrm{1}}+F\mathit{I}\right)}^{-\mathrm{1}}\left(\mathit{y}-\mathit{\mu }\right),

where y_p is the predicted variable for a given input vector x. µ is the mean of the observed values of y; y is the vector of the observed values of y. ψ and \mathit{\Psi } are arrays defined by the following expressions:

(11)\mathit{\psi }=\left[\begin{array}{l}cor\left(y\left(x_{\mathrm{1}}\right),y\left(x\right)\right)\hfill \\ ⋮\hfill \\ cor\left(y\left(x_N\right),y\left(x\right)\right)\hfill \end{array}\right],

(12)\mathit{\Psi }=\left[\begin{array}{lll}cor\left(y\left(x_{\mathrm{1}}\right),y\left(x_{\mathrm{1}}\right)\right)\hfill & \cdots \hfill & cor\left(y\left(x_{\mathrm{1}}\right),y\left(x_N\right)\right)\hfill \\ ⋮\hfill & \hfill & ⋮\hfill \\ cor\left(y\left(x_N\right),y\left(x_{\mathrm{1}}\right)\right)\hfill & \cdots \hfill & cor\left(y\left(x_N\right),y\left(x_N\right)\right)\hfill \end{array}\right].

The correlation function, cor(), is defined as follows:

(13)cor\left(y\left({\mathit{x}}_q\right),y\left({\mathit{x}}_r\right)\right)\mathrm{}={\sigma }^{\mathrm{2}}\exp \left(-{\displaystyle \sum _{i=\mathrm{1}}^{i=k}{\theta }_i{\left(x_{qi}-x_{ri}\right)}^{\mathrm{2}}}\right),

where σ is the variance; k is the number of inputs, and q_i is the weighing parameter indicating the correlation strength. The parameters F and θ are fine-tuned to minimise the error of estimation.

Results comparisons

Results of training and testing for the proposed network models are compared with those of spline and Kriging regression techniques in Fig. 7. As the results showed, the SEANN-II model produced the best result, not only because it returns least value of N-RMSE with respect to the testing data, but also because it achieved a reasonably high level of accuracy with smallest number of constants (see Table 2). It can also be seen from the results that the model based on cubic spline returns lower training N-RMSE values and higher testing N-RMSE values compared with linear spline predictions, which highlights the tendency of the former to over-fit the training data due to its higher polynomial order. It is also worthy to note that the predictions of Kriging regression based on Eq. (6) (Kriging-II) is significantly better than those of Kriging-I and the splines. It is, however, essential to mention that Kriging’s ever-dependence on the training raw data goes along with a relatively huge computational burden on the end user. As shown in Table, the Kriging models require excessively large number of parameters (112 for Kriging-I and 220 for Kriging-II) to make their predictions. In a nutshell, SEANN-II model is the best predictor of moment capacity based on the comparisons made. Also, the models based on Eq. (5) generally produced better predictions compared with those based on Eq. (6).

Comparisons with existing models

Based on testing data, the values of root mean square error and coefficient of determination for the proposed models, the BP network and other methods are computed and compared in Table 3. The models are also compared on the basis of mean and standard deviations of β with the aid of the chart in Fig. 8. The results showed that the SEANN-II model is the best predictor of moment capacity, having returned the lowest value of σ_β and N-RMSE with a relatively simple network topology (number of parameters= 4) in addition to having an average value of β close to 1.00. It can also be observed from the results that the GEP model performs remarkably well as it surpasses that of the SEANN-I model despite having only three parameters. It is, however, important to mention that the former was developed based on 64 data points in contrast with the 36 data points used to develop the models in the present work. The BP-ANN predictions trailed behind the SEANN models and GEP in accuracy despite having a much larger number of constants. The low performance of the BP-ANN developed in this work compared to the ANN and ANFIS models developed in Ref. [¹] is due to more complex network topologies and excessively large number of parameters used in the previous work (88 constants in the case of ANN and over 70 constants in the case of ANFIS). With regards to the performance in the case of plastic analysis method, mechanism approach and the parabolic formula, the results indicate a relatively poor prediction quality, thus confirming the findings of previous studies [¹,²].

In summary, the SEANN-II model is selected as the best among the models examined based on relative performance and the topology complexity. The following expression represents the selected network:

(14a)\eta ={10}^{-\mathrm{5}}\left(\mathrm{319.84}+\mathrm{27.66}{v}_{\mathrm{f}}{\alpha }^{\mathrm{2}}-\mathrm{0.4492}{v}_{\mathrm{f}}^{\frac{\mathrm{3}}{\mathrm{2}}}{\alpha }^{\mathrm{3}}\right),

(14b)M_{\mathrm{u}}\mathrm{}=\eta f_{\mathrm{c}\mathrm{u}}b{h}^{\mathrm{2}},

where M_{\mathrm{u}} is in \mathrm{N·mm}, f_{\mathrm{c}\mathrm{u}} is in N/mm². Both b and h are in mm.

Parametric study

The parametric study is meant to assess the extent to which model predictions agree with the underlying behavior of the system considered. The study is conducted by adopting a commonly used approach [²⁰,²¹], which involves keeping all input variables, except one (the subject of study), to values around the mean, while the subject parameter is allowed to vary between the minimum and the maximum values in the database. In this manner, the controlling parameters are investigated, one after the other. The results of the parametric study carried out on the SEANN-II model by varying the percentage of mesh reinforcement (v_f) and the ratio of the yield strength of reinforcement to mortar compressive strength of the mortar (α) are displayed in Figs. 9(a)–9(b). It can be seen from the figures that the predicted values of non-dimensional flexural capacity vary positively with both v_f and α, which implies that the model predictions are in accordance with the current knowledge about the flexural behavior of rectangular RC sections. The reasonable conformity between the experimental data and the contours of normalized flexural capacity versus α (see Fig. 10) further supports this assertion. It is, however, important to mention that the validity of the proposed model predictions, like any empirical model, is limited to the range of data that is used its development. The values of moment capacity cannot therefore be extrapolated using the model beyond the range covered by the database.

Sensitivity analysis

Sensitivity analysis was carried out with the aim of studying how strongly various input parameters affect the output of the proposed model. The method of analysis adopted in this work is the local based sensitivity analysis requiring the computation of partial derivatives of outputs with respect to the input parameter under investigation. As described in Ref. [⁷,²²], the method is implemented in two ways, the first is to keep all input variables around the average values except the studied variable. The sensitivity index is computed in this case as follows:

(15)S_{x_i}=\mathrm{}{\displaystyle \frac{\mathrm{1}}{N}}{{\displaystyle \sum _{j=\mathrm{1}}^{j=N}{\displaystyle \frac{\partial y\left({\overline{x}}_{\mathrm{1}},{\overline{x}}_{\mathrm{2}},\mathrm{...},x_i,\mathrm{...},{\overline{x}}_k\right)}{\partial x_i}}}}_{x_i=x_i^j},

where S_{x_i} is the sensitivity index, y\left(x\right) is the function of a number of variables x. \overline{x} are the average values of all input variable except the subject variable x_i. The second approach involves varying all variables in the process. The corresponding sensitivity index is computed as follows:

S_{x_i}^v\mathrm{}={\displaystyle \frac{\mathrm{1}}{N^k}}{\displaystyle \sum _{j_{\mathrm{1}}=\mathrm{1}}^{j_{\mathrm{1}}=N}\mathrm{....}}{{\displaystyle \sum _{j_k=\mathrm{1}}^{j_k=N}{\displaystyle \frac{\partial y\left(x_{\mathrm{1}}^{j_k},\mathrm{...},x_m^{j_k},\mathrm{...},x_k^{j_k}\right)}{\partial x_i}}}}_{x_i=x_i^{j_k}},

(16)m=\mathrm{1},\hspace{0.17em}\mathrm{2},\mathrm{...},k.

Since the selected model for moment capacity is expressed as a formula represented by Eq. (8), the partial derivatives with respect to the input parameters α and v_{\mathrm{f}} are as follows:

(17a){\displaystyle \frac{\partial \eta }{\partial \alpha }=}{10}^{-\mathrm{5}}\left(\mathrm{55.32}{v}_{\mathrm{f}}\alpha -\mathrm{1.3476}{v}_{\mathrm{f}}^{\frac{\mathrm{3}}{\mathrm{2}}}{\alpha }^{\mathrm{2}}\right),

(17b){\displaystyle \frac{\partial \eta }{\partial v_{\mathrm{f}}}=}{10}^{-\mathrm{5}}\left(\mathrm{27.66}\alpha -\mathrm{0.6738}{v}_{\mathrm{f}}^{\frac{\mathrm{1}}{\mathrm{2}}}{\alpha }^{\mathrm{3}}\right).

For the sake of eliminating the effect of units, the indices are non-dimensionalised by having the partial derivatives normalized with y/x_i. The computed values of non-dimensional sensitivity indices of the input parameters are plotted in Fig. 11, which indicated that the ratio of steel-to-mortar strength ratio α is the dominant parameter affecting the magnitudes of normalized flexural strength regardless of the adopted method of computing the sensitivity index. The chart also showed that although the sensitivity of reinforcement ratio trails behind the α parameter, the impact of the former on the estimated flexural capacity is significant (S_{x_i} = 0.623; S_{x_i}^v = 0.819).

Conclusions

Self-evolving network training algorithm was used to develop empirical models for estimating the moment capacity of ferrocement sections. Two variants of SEANN models developed, both of which were trained using experimental data consisting of wide range of section sizes, mesh reinforcement proportions and varying compressive strengths of cement mortar and tensile strengths of mesh reinforcement.

From the results obtained, the models optimised using the proposed algorithm correlate better with experimental data than analysis, mechanism approach and parabolic formula. Compared with the predictions of conventional ANN model, the two SEANN models returned more accurate predictions. However, in contrast with GEP-based model, SEANN-II model produced more accurate results, while SEANN-I model produced less accurate estimates. With regard to model complexity, the SEANN-II model, apart from being the best in term of accuracy, yields a remarkably simple and user friendly formula that was found to respond rationally to input parameters based on the parametric and sensitivity studies carried out.

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