Optimized Computational Intelligence Model for Estimating the Flexural Behavior of Composite Shear Walls

Abstract

This article presents a novel approach to estimate the flexural capacity of reinforced concrete-filled composite plate shear walls using an optimized computational intelligence model. The proposed model was developed and validated based on 47 laboratory data points and the Transit Search (TS) optimization algorithm. Using 80% of the experimental dataset, the optimized model was selected by determining the unknown coefficients of the network-based computational structure. The remaining 20% of the data was used to evaluate the accuracy of the model, and the best-performing structure was selected. Furthermore, the final neural network details were subjected to statistical analysis to extract a user-friendly formula, making it easier to apply in practice. The proposed ANN model and the proposed user-friendly formula were then compared with the AISC 341-16 and experimental results and demonstrated their efficacy in predicting the flexural behavior of composite shear walls. Overall, the proposed approach provides a more reliable and efficient framework for estimating the flexural behavior of composite shear walls.

1. Introduction

A composite shear wall (CSW) comprises a matrix material surrounding walls with different elastic constants bonded to it [1]. There are three main classes of CSW, i.e., reinforced concrete (RC) panel infilled in composite frame, RC panel with embedded steel band bracing, and steel panel with or without covering the RC panel [2]. The resisting mechanism of these structural systems has been studied by Yamada and Osaka [2]. The resisting mechanism of a CSW combines two fundamentally opposite mechanisms [2]: the diagonal compression field of the infilled concrete and the diagonal tension field of the buckled steel panel. The diagonal compression field mechanism contributes to the CSW’s initial stiffness and ultimate resistance by forming interconnected voids when the infilled concrete cracks diagonally under shear loading, allowing it to compress and resist shear forces. Achieving an optimal balance between these mechanisms involves designing the CSW with the right thickness of concrete and steel panel. The research by Yamada and Osaka [2] explores various factors affecting the CSW’s resisting mechanism, such as shear span ratio, boundary conditions, and reinforcement details, concluding that CSWs offer versatility in meeting diverse seismic performance requirements.

The potential of CSW, compared to conventional shear wall systems, makes it a priority for choosing lateral bearing systems by structural designers, especially in tall buildings [3,4,5]. The CSW with concrete-filled steel tube (CFST) columns is one of the common lateral resistance systems. The steel plates in the CSW act as reinforcement once the concrete has hardened and transmitted the lateral loads via the floors and frame to the walls in tall buildings [6]. The column stiffness has a significant role in the overall behavior of CSW [7], and the presence of a steel plate deep beam in CSW can bring the deformation closer to that of the entire shear wall [8]. The hysteretic models for the CSW were proposed and evaluated for different loading configurations and cross-section shapes [9,10,11]. The effective stiffness of a CSW system consisting of a composite shear wall with double plates and filled with concrete has been numerically calculated, and the results were verified by a series of experimental tests [12]. Figure 1 represents a schematic view of three different composite shear walls (CSWs). Different section examples of a CSW can be found in the literature [9,10,11,12,13].

The effects of several parameters on the behavior of CSW have been evaluated in the literature. Previous studies indicated that the material properties of the bonded plates have high and low effects on the deflection and frequency of the RC shear walls strengthened with thin plates of carbon-fiber-reinforced polymer (CFRP), respectively [14]. The use of CFRP plates as externally bonded material can improve damaged RC shear walls [15], mainly due to the damping properties of the polymer. The CSW embedded with steel sections at the boundaries and middles of the RC walls enhances the ductility and energy dissipation than those with steel sections only at the boundaries [16]. The seismic behavior of CSW improves with an increase in steel areas at the end columns [16] and when considering the bending stiffness of the columns [17]. Furthermore, the ductility is influenced by the “bolt space” to “plate thickness” ratio [17] since it can prevent local buckling before yielding [18].

Composite plates greatly impact the behavioral characteristics of the CSW, such as displacement or frequency [19]. The seismic behavior of these structural elements depends on the geometrical factors of the shear wall and the frequencies of the earthquake [20]. Concrete thickness is one of the parameters that when increased reduces shear steel plate buckling [21] and enhances the ultimate strength of the CSW [22]. Moreover, reducing the distance between the connectors may cause more energy dissipation [21]. Dey and Bhowmick proposed two equations to estimate the connector spacing and thickness of reinforced concrete panels in CSW [23]. They also introduced a formulation to predict the period of the CSW system. Damage states and repairing techniques of CSW were also studied by researchers [24]. The seismic performance of embedded CSW containing high-strength concrete can improve the deformation, load, and energy dissipation capacities [25]. Seo et al. [26] defined a shear force–shear strain model considering the in-plane shear behavior of SCW. In another study, displacement–force equations were obtained [27] for CSW systems. Previous studies indicated that at the beginning of loading in an infill composite wall (CSW), the interaction between concrete and steel increases until the shear yield of the steel plate [28]. The encased steel plates, e.g., horizontal and vertical corrugated, in CSW systems represent different shear-sharing ratios [29]. Furthermore, the critical buckling stress and flexibility in CSW can be limited considering tie spacing, plate slenderness, and flexural rigidity of the steel plates [30]. Finite elements models are one of the most important techniques that have been discussed in the analytical studies of CSW. Some researchers developed these models to investigate the structural behavior of the CSW [31,32,33,34,35].

In recent years, numerous studies have been conducted to investigate the performance of various structures derived from the different classes of CSW; some of which have led to the introduction of specific models of such systems. For example, shear walls infilled with concrete having bundled channels [36], CSW with concrete-filled steel tube (CFST) columns connected to beams only [37], precast slender CSW with CFST boundary elements [38], multi-partition CSW [39], RC wall web having square CFST columns as boundary elements incorporating CFRP-confined concrete core [40], encased steel plate-reinforced gangue CSW [41], double-skin profiled wall [42], CSW infilled with concrete having dual steel plates with vertical stiffening steel plates [43], CSW with high-strength concrete and steel rebars with consideration of concrete-encased CFST columns as boundary elements [44], and CSW with FRP-confined concrete cores [45] or enhanced C-channels [46] can be mentioned.

The shear strength is one of the key parameters that has been evaluated in the literature for CSW. To avoid shear failure in these systems, ensure reliable stiffness, and control the inter-story drifts, researchers recommended designing the RC walls for the shear force of the capable bending moment [47]. In some cases, mathematical formulations and new methods were introduced to predict the shear capacity of such lateral-resistant systems [48,49,50,51]. The value of this parameter can be decreased with increasing initial axial tension or in the presence of out-of-plane eccentricity in CSW [52]. Mo et al. reviewed the design techniques and seismic studies for CSW systems [53].

2. Research Significance

In recent years, there have been numerous studies examining the performance of composite shear wall (CSW) systems. Many researchers have utilized experimental results to develop hysteretic and finite element behavioral models. Among the critical aspects of structural elements, determining their ultimate strengths is of utmost importance. Accurate estimation of these values can enhance our understanding of structural behavior and aid in the selection of retrofit methods and strengthening of elements. Despite the significant research on estimating the shear strength of CSW systems, there is limited information available on their flexural capacity. This gap can be attributed to the scarcity of adequate laboratory data required for statistical modeling. Therefore, this article proposes a novel method that employs computational intelligence techniques, such as neural networks and meta-heuristic optimization algorithms, to perform and validate flexural capacity prediction models of CSW systems using laboratory results. A continuous statistical sequence is created using the resulting artificial neural network (ANN) system to formulate a user-friendly prediction formula. The proposed ANN model and the proposed user-friendly formula are compared with the AISC 341-16 [54] results and existing experimental tests, and the results are thoroughly discussed, providing valuable insights into the behavior of CSW systems.

3. Database

In order to determine the appropriate model for estimating the flexural capacity of composite shear walls based on the approaches used in this article, it is first necessary to collect a set of laboratory data. For this purpose, experimental results related to CSW tests (47 tests) have been gathered from the literature [55]. For each of the data, 11 variables including the wall height, the section length, the thickness of the wall web, the length of the boundary element (BE), the thickness of the boundary element, the steel plate thickness in the boundary element, the steel faceplate thickness, the yield strength of steel faceplate, the yield strength of steel in boundary element, the concrete strength, and also the axial stress (axial load divided to the wall section area) were considered as input while the flexural capacity was selected as the target. The statistical details of the parameters and their distribution are given in

Table 1. Statistical details of the database.

Variable	Unit	Name in This Research	Minimum	Maximum	Mean	St. Dev.	Median
Wall height	mm	X1	750.00	3850.00	2379.34	658.06	1200
Section length	mm	X2	750.00	1284.00	1055.41	175.93	2016
Thickness of the wall web	mm	X3	90.00	214.00	143.24	42.82	2350
Length of the boundary element	mm	X4	0.00	280.00	165.16	71.13	1020
Thickness of the boundary element	mm	X5	100.00	219.10	152.50	42.05	140
Steel plate thickness in boundary element	mm	X6	2.94	10.00	5.01	2.12	150
Steel faceplate thickness	mm	X7	1.83	10.00	4.78	2.27	140
Yield strength of steel faceplate	MPa	X8	245.00	443.00	335.49	56.12	4.71
Yield strength of steel in boundary element	MPa	X9	245.00	443.00	331.01	55.37	4.08
Concrete strength	MPa	X10	25.30	92.60	49.18	21.42	322
Axial stress	MPa	X11	0.00	30.04	12.59	9.67	306
Flexural capacity	kN.m	Y	858.00	6929.00	2868.19	1942.09	41.3

and Figure 2, respectively. Distribution plots of the input variables against the targets are also presented in Figure 3.

In order to use the above data, their values were normalized into the range of 0 and 1 by Equation (1), and then, 37 data points were randomly chosen to be used in the process of determining the optimal architecture of the models. In this equation, X_i,n, X_min, and X_max are the normal, minimum, and maximum values of the X_i, respectively. The remaining 10 data points were used to evaluate the proposed model’s ability to predict the output.

$$X_{i,n}=\frac{X_i-X_{min}}{X_{max}-X_{min}}$$

(1)

4. The Proposed Approach

Nowadays, the use of soft computing (SC) methods in solving various engineering problems is widespread [56,57,58,59]. The reason for this can be attributed to flexibility, high accuracy, application in nonlinear and complex models, and the ability to combine different methods in order to take advantage of them together. In this article, the authors have used artificial neural network (ANN) models [60] to provide a computational framework to predict flexural capacity. The Transit Search (TS) optimization algorithm [61] was considered to determine the unknown parameters of the neural networks. The TS algorithm is an innovative optimization technique inspired by the astrophysical exoplanet exploration method known as “transit”. Transit has proven to be a highly effective approach for detecting exoplanets, with more than 3800 planets discovered using this method from space telescope databases. Detecting these planets is challenging due to their minute size within the vast cosmos. Capitalizing on the efficiency of the transit method in astrophysics, the TS optimization algorithm was introduced in the literature for optimization purposes. In the transit algorithm, changes in luminosity are meticulously examined by monitoring the light received from stars. By using the neural network with the best performance, a step-by-step approach has been performed to extract mathematical and user-friendly relationships from the selected neural network. This increases the efficiency of the proposed model and provides the ability to use it in a simple form for researchers and engineers. Details of the model and its determination process are provided in the following sections.

4.1. Optimization of the Neural Networks

The operation of artificial neural networks (ANNs) is based on the fundamental principle of human neural networks. Information is directed to the network’s cells, where it is processed, remembered, and utilized in the future. Just like in the human brain, artificial neural systems operate in a similar manner. They receive, process, and store information and can even correct themselves. Similar to the interconnected cells in a human neural network, ANNs can communicate with each other, forming a multilayer network that enhances the computational capabilities of the algorithm.

To develop an effective ANN model, a substantial database is required. These data are used to determine the unknown coefficients of the computational model and evaluate its performance. A larger amount of available data typically leads to suitable accuracy in the model. However, in this research, the database contains only 47 data points, which may not be sufficient to achieve an efficient ANN model. Therefore, an optimization algorithm has been applied instead of a common learning process to determine the coefficients of the ANNs, as shown in Figure 4.

In each iteration of the algorithm, the initial values of the weights and biases of the network are replaced with the best values obtained so far. Subsequently, the input vector of each of the 37 data points is applied to the network, and the corresponding output is calculated (as shown in Figure 5). By comparing the network outputs with the laboratory values, the network error is determined. The objective function of the optimization algorithm in this method is to minimize the root mean squared error (RMSE).

The proposed method in this article employs a neural network with one middle layer. Initially, the number of neurons in this layer is set to 1, and the optimization algorithm is applied with 400 iterations to obtain the optimal values for the weights and biases of the network. Then, the number of neurons is incrementally increased by one, and the algorithm is applied again to determine the network coefficients. This process is repeated for a range of 1 to 30 neurons, allowing for a thorough exploration of different network configurations.

4.2. Model Selection

As mentioned earlier, a total of 30 optimized neural networks were obtained to determine the target parameter, which is the flexural capacity of CSW members in this article. The architectural difference among these networks lies in the number of neurons in the middle layer, ranging from 1 to 30. For the identification of each of these networks, a 400-iteration optimization algorithm was employed, utilizing 37 datasets to determine the associated costs. The progression of the objective function (RMSE) values for each iteration is visually presented in Figure 6. After thorough analysis, it was observed that the network with the most favorable cost performance boasted 28 nodes within its middle layer.

To further investigate the performance of each of the 30 neural network models, both training (37 data) and test (10 data) datasets were applied to the networks, and RMSE values were obtained for each. The results are illustrated in Figure 7. It is evident from Figure 7 that the network with 28 neurons in the middle layer has the lowest error values for both datasets. As a result, this network is selected as the final neural network model in this article, taking into consideration its superior performance in predicting the flexural capacity of CSW systems.

In the proposed model for estimating the flexural capacity of RC composite shear walls based on the proposed ANN directly, it is first necessary to calculate the normalized values of each input variable according to Equation (1) or Figure 5. Then, the final output of the middle layer (Y₁) will be determined using Equation (2). By applying the layer weight and corresponding bias values to the corresponding node, the normalized output of the ANN is obtained by Equation (3). Given that this value is normalized (between 0 and 1), it should be converted to its real value with consideration of Equation (4). The values of input weight (IW), layer weight (LW), and bias (b₁ and b₂) for the proposed ANN are given in

Table 2. Input weights for the best ANN.

Neuron	Input 1	Input 2	Input 3	Input 4	Input 5	Input 6	Input 7	Input 8	Input 9	Input 10	Input 11
N1	−0.011503	−0.004022	−0.002823	−0.000002	0.000000	0.001313	−0.007224	−0.018706	0.001310	−0.001246	−0.001350
N2	−0.003010	0.000005	0.002786	−0.000007	−0.001349	0.013977	0.000000	−0.002840	0.000001	−0.006054	−0.001337
N3	0.000966	0.003575	−0.003528	0.000143	0.000007	0.003397	−0.000949	−0.002801	0.000005	−0.002819	−0.004304
N4	0.000000	−0.000019	−0.000019	−0.002874	0.000002	0.000215	0.000621	0.001310	−0.001844	0.000003	−0.004217
N5	0.006638	−0.004013	0.000002	0.000000	−0.000002	−0.001310	0.000000	0.000002	−0.001262	0.001310	−0.001307
N6	0.000923	0.001884	−0.000049	0.002092	0.000000	−0.000927	−0.010491	0.003001	0.001871	0.004747	0.000000
N7	0.001925	−0.004841	0.000002	0.001310	0.000000	0.002856	0.000042	0.000000	0.006810	0.002394	0.004729
N8	0.000002	0.001655	−0.000318	−0.001310	0.002144	−0.000018	0.000041	−0.001530	0.001662	−0.001723	0.000000
N9	−0.000983	−0.000197	−0.001304	−0.001331	0.004399	0.001291	−0.002109	0.002820	0.003538	0.000162	−0.001579
N10	−0.000019	0.000217	−0.000090	0.000007	−0.001309	1.955827	−0.001317	0.000000	0.010353	−0.000019	−0.008409
N11	−0.008876	−0.006932	−0.012903	0.000000	−0.000006	0.001730	0.000000	0.001313	0.000019	−0.001313	−0.005812
N12	−0.001287	−0.000219	−0.000018	−0.000001	−0.000002	−0.000057	−0.001692	−0.001350	−0.000004	0.000002	−0.000004
N13	0.000000	−0.009142	−0.000192	0.000010	−0.001310	0.001310	0.000000	0.000000	0.000001	−0.008685	−0.004695
N14	0.000000	−1.305027	−0.000010	−0.003954	0.001310	0.002722	−0.000010	0.005397	−0.000004	−0.000199	−0.000303
N15	0.000000	0.000219	0.009769	−0.000267	0.013826	0.000002	−0.001490	−0.000042	0.001307	−0.000002	−0.001310
N16	0.000947	−0.002846	−0.001979	−0.001330	0.000000	−0.006541	−0.006685	−0.002095	0.000218	0.000218	0.004712
N17	−0.003684	−0.000005	0.000000	−0.000104	0.000000	0.008035	0.001310	0.008684	0.000018	−0.001307	−0.001655
N18	−0.000218	0.004706	−0.000002	0.000000	0.001828	0.000000	0.000000	0.001929	−0.000960	−0.000226	1.931129
N19	0.002160	0.001310	0.000219	−0.006635	0.002873	−0.001222	0.009734	−0.000691	−0.002075	0.000004	−0.001307
N20	−0.000966	−0.000018	−0.001310	0.001327	−0.001313	−0.000049	0.000234	0.000003	−0.000019	−0.002786	0.001000
N21	0.000000	0.001307	−0.000961	0.000000	−0.000003	0.000002	−0.000019	−0.000019	0.000003	0.006654	0.000962
N22	−0.000752	0.006077	−0.001316	0.000000	−0.001313	0.001307	0.000000	0.001389	0.001310	0.002226	0.001248
N23	−1.970320	−0.006568	−0.001291	−0.005663	−1.814332	0.000002	−0.864744	0.000002	−0.000002	0.001310	−0.001318
N24	0.002908	0.006066	0.000000	−0.002092	−0.001304	−0.004567	−0.000166	−0.004562	0.002824	0.000000	0.003524
N25	0.000097	−0.000923	−0.001290	−0.000064	0.005243	−0.000004	0.000000	−0.000984	−0.001854	−0.000009	0.000219
N26	0.001313	0.008380	−0.001642	0.002823	0.002791	−0.002075	0.000004	0.001310	0.001200	0.000004	−0.000007
N27	0.000218	0.000019	0.000004	−0.000255	0.001335	−0.001310	−0.001403	−0.000234	−0.000040	0.000006	0.000000
N28	−0.000028	−0.000019	0.000019	0.000019	−0.000028	0.006925	−0.000745	0.001313	−0.002839	0.000000	0.001295

and

Table 3. Biases and the layer weights for the best ANN.

Neuron	Bias 1	LW	Bias 2
N1	0.000049	0.463061	0.006269
N2	−0.002855	−0.000004	-
N3	0.001655	0.018138	-
N4	0.004712	0.000000	-
N5	−0.000166	−0.000002	-
N6	−0.001330	0.145684	-
N7	0.000019	0.008551	-
N8	0.000959	−0.000219	-
N9	0.000018	0.002786	-
N10	0.001301	−0.000065	-
N11	−0.001646	−0.000307	-
N12	0.000002	−0.000215	-
N13	0.002143	0.000000	-
N14	−0.001266	−0.815577	-
N15	−0.004587	−0.000090	-
N16	−0.004343	−0.002167	-
N17	−0.003749	−0.000019	-
N18	0.000019	0.974563	-
N19	0.000000	0.001357	-
N20	−0.002839	−0.000019	-
N21	−0.003281	−0.000019	-
N22	−0.001925	0.000000	-
N23	0.001316	−1.290456	-
N24	−0.000009	0.022915	-
N25	0.001086	0.001310	-
N26	−0.000060	0.000253	-
N27	0.000000	−0.001530	-
N28	0.002856	0.001310	-

.

[Y₁]_28×1 = [IW]_28×11 [X_i,n]_11×1 + [b₁]_28×1

(2)

0 ≤ (Y₂ = [LW]_1×28 [1/(1 + exp(−[Y₁]_28×1))]_28×1 + b₂) ≤ 1

(3)

Y_ANN (kN.m) = 6071 Y₂ + 858

(4)

4.3. Extraction of a User-Friendly Equation from the Proposed ANN

In this section, the final neural network model, as defined in the previous section, is utilized. The primary objective here is to extract mathematical relationships that can determine the flexural capacity of CSW members from the proposed ANN structure. To achieve this, the first step is to identify the input variable that has the most significant impact on the output. This is performed through sensitivity analysis, which involves investigating the effect of changes in each input on the output. The results of the sensitivity analysis are presented in Figure 8, illustrating the findings obtained. To perform this analysis for variable X₁, first, the values of parameters X₂ to X₁₁ were considered equal to their average value (see Table 1). Then, the network output was obtained for different values of X₁. The above process has been carried out for other input variables as well. According to Figure 8, it is clear that changes in the input parameters directly impact the output for all input variables. Therefore, the authors used the relationship introduced by Milne [62] to determine the most effective input parameter. This relationship [62], which is shown by Equation (5), calculated the relative importance of each variable on the output based on the weight values of the neural network.

$$Q_{ik}=\frac{{{\displaystyle \sum }}_{j=1}^{n_{hidden}}\frac{W_{ji}}{{{\displaystyle \sum }}_{l=1}^{n_{input}}\left|W_{jl}\right|}{W}_{oj}}{{{\displaystyle \sum }}_{k=1}^{n_{input}}\left({{\displaystyle \sum }}_{j=1}^{n_{hidden}}\left|\frac{W_{jk}}{{{\displaystyle \sum }}_{l=1}^{n_{input}}\left|W_{jl}\right|}{W}_{oj}\right|\right)}$$

(5)

In this equation, Q_ik, W, j, i, k, n_hidden, W_oj, and n_input are the percentage of the impact of each parameter, the weight, layer node number in the middle layer, input node number, output node number, number of nodes in the middle layer, and the number of inputs, the connection weight between the middle layer nodes and the output node, respectively.

Using Equation (5), the relative importance percentages of each input variable are calculated, the values of which can be seen in Figure 8. Accordingly, the most effective parameter is the X₂, with 13.78% impact on the output. This variable, which represents the section length of the RC composite shear wall, is known as the chart parameter in this article.

The relative importance parameter is an input variable that has the most significant impact on the output, which in this case is the section length of the RC composite shear wall. This parameter is derived from a sensitivity analysis, which involves systematically investigating how changes in each input variable affect the output of our neural network model. By performing regression in the results of sensitivity analysis for this variable, Equation (6) was determined to calculate the Y_chart.

$$Y_{chart}=2.827 X_2-247.19$$

(6)

After determining the chart equation, a new database was created to determine the outputs using the proposed ANN. For this purpose, 50 datasets were generated each time the network was implemented. In the first step, all input values were set to their averages (as displayed in Table 1), except for the first input, which was varied from its minimum to maximum values in 50 steps. This process was repeated for 10 different sets of 50 databases, with the second variable being changed 10 times and one value from each set being applied to each of the 10 collections. After completing the 10 sets of 50 databases for the first variable, the network output for each database was determined. The results are presented in Figure 9, depicting the changes in the entire input variables. The parameter C(X) in Figure 9 represents the correction for input variable X, which indicates the ratio between the ANN output and the corresponding value obtained from the network for the dataset with average input values.

The average of the graph of the change for each input is shown in Figure 10. The relationship for each curve is shown in Equations (7)–(16):

$$C\left(X_1\right)=0.856 X_1+0.173$$

(7)

$$C\left(X_3\right)=8.75\times {10}^{-5} X_3+1$$

(8)

$$C\left(X_4\right)=0.0033 X_4+0.997$$

(9)

$$C\left(X_5\right)=1.39 X_5-0.371$$

(10)

$$C\left(X_6\right)=-7.26\times {10}^{-4} X_6+1$$

(11)

$$C\left(X_7\right)=0.331 X_7+0.674$$

(12)

$$C\left(X_8\right)=-0.01 X_8+1.01$$

(13)

$$C\left(X_9\right)=1.1\times {10}^{-4} X_9+1$$

(14)

$$C\left(X_{10}\right)=-6.1\times {10}^{-4} X_{10}+1$$

(15)

$$C\left(X_{11}\right)=0.373 X_{11}+0.639$$

(16)

Once the relationships of the chart equation (Equation (6)) and the average curves (Equations (7)–(16)) are determined, the flexural capacity of CSW members, Y, can be simply calculated (in kN.m) using the user-friendly Equation (17):

$$858 \le Y \left(\mathrm{kN}.\mathrm{m}\right)=Y_{chart} C\left(X_1\right) C\left(X_3\right) C\left(X_4\right) C\left(X_5\right) C\left(X_6\right) C\left(X_7\right) C\left(X_8\right) C\left(X_9\right) C\left(X_{10}\right) C\left(X_{11}\right) \le 6929$$

(17)

5. Comparison Study

In this section, the results obtained for the proposed neural network model and the formula derived from it (Equation (17)) are compared with the AISC 341-16 [54]. In order to determine the flexural capacity of the CSW members, 47 laboratory data points used in this article were considered. The input vector of each dataset, which includes 11 variables, was applied to the three computational models mentioned above to calculate the flexural capacity. Subsequently, the results were compared with the values reported by the laboratory. The findings indicate that both models proposed in this article exhibit an acceptable accuracy compared to the composite flexural strength provided by the AISC 341-16 seismic provisions, which underestimated the experimental test results (as depicted in Figure 11).

In order to comprehensively assess the performance of each method, the root mean squared error (RMSE) and mean absolute error (MAE) were evaluated separately for the train and test datasets. The results are presented in Figure 12 (as radar graphs) and

Table 4. A summary of the comparison results.

Parameter	The Proposed ANN			The Proposed Formula			AISC
	Train	Test	All	Train	Test	All	Train	Test	All
RMSE	395.50	350.83	386.43	557.75	590.42	564.86	613.12	943.98	696.80
MAE	299.45	260.48	291.16	454.22	380.69	438.58	417.55	653.71	467.80

. For the proposed artificial neural network (ANN), the RMSE and MAE for the test data were found to be 350.83 and 260.48, respectively. In contrast, the RMSE and MAE values for the proposed formula (Equation (17)), which was extracted from the neural network structure using a step-by-step statistical approach, were 590.42 and 380.69, respectively. Furthermore, the error values for the AISC 341-16 seismic provisions, which yielded higher values compared to both proposed models, were also calculated. Based on the obtained error parameters, it can be concluded that the two models proposed in this article exhibit good accuracy in estimating the flexural capacity of CSW members.

6. Discussion

The utilization of an optimization algorithm to determine the unknown parameters of the artificial neural network model addresses the limited availability of laboratory data and the complexity of multiple input variables, ensuring accurate predictions. The sensitivity analysis conducted by the authors provides valuable insights into the influence of individual input variables on the output parameter. Identifying the sixth, eighth, and tenth variables as significant contributors allows for targeted adjustments in design and analysis. The relative importance of section length emphasizes its significance in affecting the target flexural capacity.

The authors’ introduction of a step-by-step statistical approach to extract a simplified computational structure from the ANN, resulting in Equation (17), enhances the efficiency of the neural network model and simplifies the estimation of flexural strength. Furthermore, the authors’ models demonstrate acceptable accuracy when compared to laboratory values and the AISC 341-16 seismic provisions, suggesting their potential as reliable tools for estimating the flexural capacity of CSW members in seismic regions.

The primary limitation is the scarcity of available laboratory data for CSWs, which constrained the size of the dataset and potentially limited the ANN predictive capabilities. As more experimental data become available, the accuracy of the models could be further improved. Additionally, the complexity of the ANN structures may pose challenges for implementation and interpretation by practitioners who are not familiar with neural networks. The authors’ proposed computational approaches offer several advantages for estimating the flexural capacity of CSWs. These advantages include acceptable predictions, sensitivity analysis insights, and the introduction of a user-friendly formula.

An inherent challenge faced in this research pertains to the limited availability of laboratory data for CSWs. The dataset’s size is notably constrained due to this limitation, which may affect the predictive capabilities of the artificial neural network models. It is essential to recognize that data quantity significantly influences the accuracy of predictive models. The shortage of data represents a fundamental limitation that must be considered when applying these models in practical engineering contexts. Additionally, sensitivity analysis conducted in this research identified specific input variables, such as section length, as having a significant impact on the target flexural capacity. Nevertheless, the extent to which these findings can be extrapolated to real-world scenarios featuring diverse input data remains an open question. The authors recommend future research endeavors to address these constraints by focusing on the following areas:

• Expanding the dataset: Efforts should be directed towards collecting a more extensive dataset of laboratory data for CSWs to bolster the robustness and generalizability of predictive models.
• Real-world validation: Conducting further experiments and sensitivity analyses within real-world applications to validate the models and account for unexplored factors.

Acknowledging these limitations and pursuing research in these directions is expected to contribute to the practical applicability and reliability of these models in the field of structural engineering.

7. Conclusions

In this article, two computational approaches are introduced for estimating the flexural capacity of reinforced concrete-filled composite plate shear walls. Laboratory test results from the literature, comprising 47 datasets, are utilized with 11 input variables to evaluate the flexural behavior of the models. The findings can be summarized as follows:

• Due to limited laboratory data and the complexity of multiple variables in the problem, the number of unknown parameters in the ANN structures exceeded the available data. To overcome this, an optimization algorithm was employed to determine the unknown parameters of the model. The results indicate that a network with 28 neurons in the middle layer demonstrates good accuracy in estimating the target parameter.
• Sensitivity analysis reveals that altering the values of the steel plate thickness in the boundary element, yield strength of the steel faceplate, and concrete strength results in a decrease in the output, while changes in other variables lead to an increase in the output parameter.
• The weight values of the ANN, along with a numerical relationship, reveal that the section length has the most significant impact on the target parameter, with a relative importance value of 13.78%.
• To enhance the efficiency of the neural network model and reduce matrix calculations in estimating flexural strength, a step-by-step statistical approach was employed to extract a computational structure from the proposed ANN, resulting in the user-friendly formula of Equation (17).
• Based on the results obtained from the two proposed models in this article and their comparison with laboratory values, it is concluded that both proposed models exhibit acceptable accuracy compared to the AISC 341-16 seismic provisions. These models can be utilized for estimating the target parameter in this research, which is the flexural capacity of CSW members.

Future research should consider conducting additional experiments and sensitivity analyses to validate the models further and account for any unexplored factors that could affect the flexural capacity estimation for composite plate shear walls. This acknowledgment of limitations and the potential need for further refinement strengthens the robustness of the findings and encourages a more comprehensive exploration of this important topic.