An experimental investigation and machine learning-based prediction for seismic performance of steel tubular column filled with recycled aggregate concrete

2.1 Specimen preparation

The research object of this test is RCSTC. Taking the axial compression ratio, steel pipe wall thickness, and slenderness ratio as the variable parameters, a total of 23 RCSTC and 3 concrete-filled steel tubular comparison columns were designed and fabricated. Table 1 shows the detailed design parameters of the various test pieces. The naming convention for the test pieces is as follows: L stands for steel pipe with total length of 2,700 mm and effective length of 1,980 mm, M stands for steel pipe with length of 2,200 mm and effective length of 1,480 mm, and S stands for steel pipe with length of 1,700 mm and effective length of 980 mm; E stands for steel pipe wall thickness of 8 mm, T stands for steel pipe wall thickness of 10 mm, and W stands for steel pipe wall thickness of 12 mm; C stands for concrete-filled steel tubular comparison column. The structural drawing of the test piece is shown in Figure 1.

Figure 1

Schematic diagram.

This test abandons the traditional concrete base and adopts a 1,300 mm × 600 mm × 500 mm steel base to reduce the amount of concrete. A hollow pipe with a diameter of 205 mm is reserved in the center of the steel base to facilitate the installation and disassembly of the steel pipe column and to save a lot of test preparation time. This steel base is the invention being patented by our research group. After numerous tests, the deformation of the hollow pipe in the center of the base is very small and can be ignored. A photograph of the test site and setup is shown in Figure 2.

Figure 2

Loading diagram of test site of specimen assembly (unit: mm).

2.2 Raw materials and mix proportions

2.3 Basic parameters of test pieces

According to the calculation method in ref. [35], the design value N _u of the axial compression bearing capacity of concrete-filled steel tubular column can be obtained. The calculation results are shown in Table 8; f _a is the yield strength of steel pipe, f _cu is the cube compressive strength of concrete, f _c is the design value of concrete compressive strength, θ is the hoop index of the concrete-filled steel tube, N _u is the design value of axial compression bearing capacity, n is the axial compression ratio, and N is the actual applied axial force during the test.

2.4 Test device and loading system

2.4.1 Test device

This test is to simulate the seismic conditions, and the loading mode of low-cycle reciprocating load is adopted, i.e., constant axial pressure applied vertically downward and the horizontal force of round-trip cycle are applied to the specimen. The horizontal force of the round-trip cycle is applied by the loading actuator installed on the reaction wall (the rated loading capacity is 500 kN). The constant axial pressure is provided by an LTA hydraulic control system (purchased from Liuzhou Litian prestressed equipment factory), which is composed of a hydraulic jack and console with a range of 4,000 kN, as shown in Figures 3 and 4.

Figure 3

Hydraulic jack.

Figure 4

Hydraulic control system.

2.4.2 Loading system

The loading method used in this test is full displacement loading, which was loaded according to the relevant provisions in ⟪Seismic test code for buildings⟫ (JGJ/t101-2015) [36].

Preloading was carried out before the formal test, so that the structure could quickly enter the normal working state, and the reliability of loading equipment and devices could be checked. Then, it was loaded to full load, and LTA hydraulic control system was used to keep the shaft pressure at the set value during the experiment. In addition, preloading was carried out before the horizontal load was applied to test whether the loading equipment and devices work normally.

The horizontal repeated load was loaded in the way of displacement control. To determine the yield load of the steel pipe, it was loaded level by level with 1 mm (1, 2, 3 mm, etc.). When there is an obvious inflection point in the resultant load displacement curve, it is regarded as the yield of the steel pipe, and the corresponding displacement value when the steel pipe yield is the yield displacement of the steel pipe. After the yield point is reached, the displacement control method varies, the value of the displacement increment is an integral multiple of the yield displacement. Each level of loading is repeated three times. When the specimen has obvious damage or the horizontal load drops to 85% of the maximum, the loading process should be stopped, as shown in Figure 5.

Figure 5

Loading system diagram.

Figure 6

Typical failure mode of test piece.

2.5 Experimental phenomenon

1. The test phenomena and failure forms of the 23 specimens are similar. Before the steel pipe yields, the lateral displacement of the specimen is small, and there is no obvious phenomenon on the steel pipe surface. After the steel pipe yields, the lateral displacement of the specimen gradually increases, and a small drum begins to appear at the bottom of the steel pipe column. The reason for the bulging deformation is that when the load is about 3–5 times that of the yield displacement, the micro buckling starts to appear. With the forward and reverse loading and unloading of the horizontal load, the micro buckling on one side of the steel pipe gradually recedes, and the buckling on the other side starts to occur. During this repeated loading and unloading process, the degree and range of buckling of the steel pipe gradually increase [37,38]. After the horizontal thrust reaches the peak value, the bulging at the bottom of the steel pipe column gradually becomes more obvious. When the load is about five times the yield displacement, the buckling failure develops rapidly, and the buckling degree and range in the steel pipe increase rapidly. In the final stage, the buckling surface of each specimen forms an “elephant leg” shape of drum failure. Under the axial compression ratio of 0.6, the outer diameter of the drum is at its largest. The failure mode of the test piece is shown in Figure 6.

2. The drum height of each specimen is within 10–80 mm from the column bottom, and there is a small difference in the drum degree on the front and rear sides of the specimen. This is mainly caused by the loading system error. During the loading process, because the test piece has only front and rear constraints and no lateral constraints, when the horizontal displacement reaches a certain degree, the test piece tends to offset to the side, and the vertical axial force applied by the jack fluctuates slightly, resulting in inconsistent stress on the cross section of the steel pipe in all directions. In addition, when pouring the core concrete into the steel pipe, it is impossible to achieve complete uniformity of the concrete aggregate, which is also the reason for the asymmetry of the degree and range of steel pipe bulging.

3.1 RF

RF is a decision tree (DT) model based on bagging algorithm. RF adopts a sampling with replacement method to randomly collect samples from the original dataset to form multiple sample sets of the same size, and each sample set is used in the training process of different DTs. The random nature includes random sample extraction and random feature extraction. The final result is obtained by voting in the classification problem, and the mean value of the above model is used as the final result in the regression problem. The bagging method proposed by Breiman [39] not only effectively reduces the prediction variance but also improves the prediction performance and stability. The RF algorithm is described in Figure 7. The training set can be expressed as R _n in equation (1), where X is the input vector with m features ( X = { x 1 , x 2 , … , x m } ), and Y is an output scalar.

Figure 7

Construction of an RF model.

In the training process, samples are randomly selected from the training set by repeated sampling, and the probability of each sample being selected in each selection is 1/n. The abovementioned sample collection process is called “bootstrap,” and the resulting dataset can be expressed as R n θ . Afterward, the DT is split from the root node to the leaf node based on the dataset R n θ . If an RF consists of k DTs, then k prediction functions a ˆ (X, R n θ ) are constructed after the training process, where k = 1 , 2 , … , k . Finally, the RF outputs the final prediction result Y through the output ( { Y ˆ 1 , Y ˆ 2 , … , Y ˆ k } ) of an average of k DTs, and the output equation is shown in equation (3).

3.2 Baseline models

Constructing a baseline model as a comparison object helps to evaluate the target model from multiple perspectives. When dealing with regression problems in the construction field, multiple linear regression (MLR) and logical regression (LR) are relatively popular models [40]. As mentioned above, this study proposes these two baseline models (LR and MLR) as the comparison objects of the RF and back propagation neural network models, and evaluates their prediction differences from the perspective of performance evaluation. The LR and MLR models suitable for multivariate prediction are shown in equations (4) and (5).

where x _k is an independent variable and p is a dependent variable; b ₀ and b _k are the constant coefficients, Y is the output, x _n is the multiple predictive variables, β 1 , … , β n denote the regression coefficients.

3.3 FA

Naturally inspired algorithm is one of the powerful optimization algorithms. FA comes from the unique social behavior of fireflies [41,42]. Fireflies can not only emit short and regular flashes but also be attracted by other flashes. This is because fireflies rely on flashes to find mating partners. It is worth noting that the flash of fireflies decreases with the increase in the distance between two fireflies, and the attraction of the flash also decreases. FA idealized the flash characteristics of the firefly: low brightness firefly will spontaneously move toward the brighter firefly; the fireflies with the highest brightness will move randomly; although attractiveness is proportional to brightness, both are inversely proportional to distance; the brightness of fireflies depends on the objective function of the problem. Equation (6) represents the position change of a low-brightness firefly i as it moves toward a high-brightness firefly j. The pseudocode of FA is shown in Figure 8.

where the positions of fireflies i and j at the t-th iteration are x i t and x j t , respectively; r _ij is the linear distance between two fireflies, as shown in equation (7); β ₀ represents the maximum attractiveness of fireflies when r _ij is equal to zero; γ is the absorption coefficient, controlling the influence of distance and medium on brightness, ranging from zero to one. α is a random parameter derived from Gaussian distribution, and rand is a random vector; the values of both are from zero to one.

Figure 8

The pseudocode of FA.

The limitation is that the FA is a metaheuristic optimizer, thus it possesses inherent stochastic properties. Thereby, this procedure fails to provide a highest prediction accuracy if the program is only carried out one time. Several run times are conducted and the statistical outcomes are compared to verify the feasibility of the proposed procedure. In this article, the results of only one run time are presented.

3.4 Cross validation (CV)

For the RF model, two hyperparameters (numTree and minNumLeaf) have a significant impact on the final model performance. The numTree parameter represents the number of regression trees in the model, which affects the computational efficiency and generalization ability of the model. The minNumLeaf parameter represents the minimum sample number of leaf nodes, which is an important parameter for constructing regression trees. The empirical ranges of numTree and minNumLeaf are from 1 to 50 and from 1 to 5, respectively. The initial values of both are 40.

The dataset is first randomly divided into two parts, one of which accounts for 30% for the test model and the other 70% for the training and verification model. 70% of the datasets were divided into 5 blocks, 4 of which were used to train the model, while the other one was used to verify the model. According to the above division method, repeated division for 5 times, you can get 5 groups of validation sets that do not overlap between the training set of partially overlapping datasets. Then, the model completes 50 iterations in the 5 groups of divided datasets, and the FA will dynamically select the optimal hyperparameters in the iteration process. Finally, the one with the optimal root mean square error (RMSE) is selected from the five models completed by iteration as the final model, and the test set is used to evaluate its final effect. Figure 9 shows the iterative process of the RF model.

Figure 9

RF model training by 5-fold CV and FA.

3.5 Performance evaluation

Model evaluation is an important part of building solutions using ML. In this study, we selected the common and effective evaluation indexes in ML, namely, RMSE and correlation coefficient (R). Their definitions are as follows:

where n represents the number of samples involved in the evaluation; y i ⁎ means the prediction result of the model; y _i means the actual result of the sample set; y ⁎ ¯ and y ¯ are the average of the model prediction results and the average of the actual value of the sample set, respectively.

3.6 Variable significance determination

Sensitivity analysis is an analytical method to study the relationship between input variables and output results. This method can evaluate the impact of inputs on outputs by changing the value of input variables in a feasible range [43]. Sensitivity analysis includes global sensitivity analysis (GSA) and local sensitivity analysis (LSA). Because LSA lacks the ability to search for uncertainty, this study uses the GSA method as it can evaluate all input variables. Finally, the impact of input variables on the output results is quantified as a percentage and visualized in the form of bar charts. Equation (10) shows a gradient metric used to estimate the resulting change and relative importance of the output [44].

where ε is the analyzed input variable; y ε , j ^ represents the susceptibility reaction indicator for x _ε,j ; and R _ε is the relative importance of the variable.

4.1 Hysteresis behavior

Due to the large number of specimens and limited space, only representative specimens are listed in the hysteresis curve.

As shown in Figure 10, the hysteresis curves of each specimen are relatively full, without bow, shuttle, and anti-s shape and pinch shrinkage, which shows that the RACSTC has good hysteresis and energy dissipation performance. Before yielding, the specimen is in the elastic stage. In the process of horizontal force loading and unloading, the lateral displacement of the specimen is small. The hysteresis loop surrounded by forward and reverse loading and unloading is shuttle shaped, with a small area, less energy consumption, and small residual deformation. After yielding, the specimen enters the elastic–plastic stage. With the improvement in the displacement loading level, the lateral displacement and residual deformation of the specimen gradually increase, the stiffness degradation accelerates, the area of the hysteresis loop surrounded by forward and reverse loading and unloading increases, and the energy consumption increases.

Figure 10

Hysteresis curve of typical specimens.

Comparing the hysteresis curves of each specimen in Figure 10, the peak horizontal load shows an increasing trend with the increase in axial compression ratio. Under the same axial compression ratio, with the increase in wall thickness, the area proportion of steel increases, and the peak horizontal load of the specimen noticeably increases. Under the same axial compression ratio and wall thickness, the peak horizontal load of the specimen decreases significantly with the increase in slenderness ratio.

4.2 Skeleton curve

Due to the large number of specimens and limited space, and for the convenience of comparison, only representative specimens are listed in the skeleton curve.

1. As shown in Figure 11, the skeleton curve trend of each specimen is similar, with obvious elastic, strengthening, and strength degradation sections. Before the steel pipe yields, the specimen is in the elastic stage, and the load displacement curve is approximately an oblique line. After the steel pipe yields, plastic deformation occurs, and the gradient of the load displacement curve decreases, but it still shows an upward trend until it reaches the peak load. Then, the curve, and the specimen, enters the downward strength degradation section, resulting in residual deformation.

2. For the specimens SE1, SE2, and SE3 with a single parameter change in axial compression ratio, the elastic stiffness of the specimen increases slightly with the increase in axial compression ratio. The decline speed of the falling section of the skeleton curve is faster, indicating that the specimen with a large axial compression ratio has small failure displacement and poor deformation capacity, but the peak water level bearing capacity of the specimen tends to increase.

3. For the specimens SE1, ST1, and SW1 with a single parameter change in wall thickness, the elastic stiffness increases slightly with the increase in wall thickness. The horizontal bearing capacity of the specimen increases and the displacement at failure decreases, which shows that increasing wall thickness is conducive to improving the bearing capacity of RACSTC.

4. For the specimens with a single parameter change in slenderness ratio (ST1, LT1, and MT1), which has a great impact on the mechanical properties of the specimens, the elastic stiffness and horizontal bearing capacity decrease significantly with the increase in the slenderness ratio, but the displacement at failure increases slightly.

Figure 11

Skeleton curve of typical specimens.

4.3 Ductility analysis

The RACSTC do not display an ideal elastic–plastic relationship. The yield displacement value observed according to the load displacement curve will be affected by systematic error. At present, the commonly used methods to determine the yield displacement of components are the energy equivalence method and geometric drawing method. In this work, the geometric drawing method is used to determine the yield displacement of RACSTC, Δ _y . The corresponding displacement value when the peak load decreases to 85% is taken as the limit displacement, Δ _u . Then, the ductility coefficient, μ = Δ _u ∕Δ _y , is calculated. Yield displacement angle, Θ _y = Δ _y /L, and ultimate displacement angle, Θ _u = Δ _u ∕L, are also calculated. The calculation results are shown in Table 9.

Analysis of Table 9:

1. The ductility coefficient of each specimen is greater than 3, indicating that the RACSTC has good ductility.

2. For multi-storey and high-rise steel structures, the limit value of the elastic interlayer displacement angle subject to frequent earthquakes is 1/250, and the limit value of the elastic–plastic interlayer displacement angle of the weak layer subject to rare earthquakes is 1/50 [45]. In this work, the elastic interlaminar displacement angle of each specimen is 0.007–0.012, and the elastic–plastic interlaminar displacement angle is 0.042–0.072, which are greater than the limit value specified in the code. This indicates that the deformation capacity of the specimen in the elastic stage and the anti-collapse capacity in the elastic–plastic stage meet the design requirements.

3. For the specimen with a single parameter change in axial compression ratio, the influence law of axial compression ratio is different between the specimen groups with different slenderness ratios. Among the low slenderness ratio specimens, the middle axial compression ratio specimens have the best ductility; the low axial compression ratio is the second and the high axial compression ratio is the worst. In the group of medium and high slenderness ratio specimens, the specimens with low axial compression ratio have the best ductility, the medium axial compression ratio is the second, and the high axial compression ratio is the worst. The ductility coefficients of the specimens with medium and high axial compression ratios under the condition of high slenderness ratio are not different. The above rules can show that, in general, the specimens with low axial compression ratio have the best ductility, and the specimens with high axial compression ratio have the worst ductility. However, when the slenderness ratio is large, the influence of axial compression ratio on ductility is significantly reduced.

4. For the specimen with a single parameter change in wall thickness, the ductility coefficient of the specimen with steel pipe wall thickness of 8 mm is larger and better. With the increase in the wall thickness, the ductility of the specimen decreases. However, under the condition of high slenderness ratio, the ductility coefficient of the specimens with wall thicknesses of 10 and 12 mm is almost the same. It also shows that when the slenderness ratio is large, the influence of steel pipe wall thickness on ductility decreases significantly.

5. For the specimen with a single parameter change in slenderness ratio, the high slenderness ratio specimen has the worst ductility under the condition of medium and low axial compression ratio, while it has the best ductility under the condition of high axial compression ratio.

4.4 Energy dissipation performance

The area of the hysteresis loop obtained in this test directly reflects the energy dissipation capacity of the specimen, and the equivalent viscous damping coefficient, ζ _eq. The ζ _eq calculation results of each typical specimen are shown in Figure 12.

Figure 12

Curves of ζ _eq vs Δ of typical specimens.

According to the analysis in Figure 12:

1. The equivalent viscous damping coefficients of the specimens are above 0.4, while the equivalent viscous damping coefficients of ordinary reinforced concrete columns are only 0.1–0.2, indicating that the RACSTC have good energy dissipation performance.

2. For the specimen with a single parameter change in axial compression ratio, the ζ _eq−Δ curve of the specimens with a high axial compression ratio is always located above the ζ _eq−Δ curve of the specimens with a low axial compression ratio. This shows that the energy consumption of RACSTC increases with the increase in axial compression ratio.

3. For the specimen with a single parameter change in wall thickness, the ζ _eq−Δ curve of the specimen with a wall thickness of 8 mm is always located above the ζ _eq−Δ curve of the specimens with wall thicknesses of 10 and 12 mm. This shows that the equivalent viscous damping coefficient of the specimen decreases significantly with the increase in wall thickness.

4. For the specimen with a single parameter change in slenderness ratio, in the initial stage of loading, the ζ _eq−Δ curve of the specimen with a low slenderness ratio is located above the ζ _eq−Δ curve of the specimen with a high slenderness ratio. Also, when the displacement cyclic loading grade is about 4–5 times the yield load, the ζ _eq−Δ curve of the specimens with medium and low slenderness ratios is located below the ζ _eq−Δ curve of the specimen with a high slenderness ratio. This shows that the specimen with a high slenderness ratio has higher energy dissipation performance in the later stage of loading.

4.5 Stiffness degradation

In this work, the secant stiffness is used to represent the stiffness of the specimen according to the provisions in the paper [44], and the secant stiffness K i = ( ∣ + F i ∣ + ∣ − F i ∣ ) / ( ∣ + X i ∣ + ∣ − X i ∣ ) , where +F _i and −F _i represent the load value of the i-th forward and reverse peak points, respectively; +X _i and −X _i represent the displacement value of the ith forward and reverse peak points, respectively. The secant stiffness degradation curve of the typical specimens is shown in Figure 13.

Figure 13

Stiffness degradation curve of typical specimens.

According to the analysis in Figure 13:

1. For the specimen with a single parameter change in axial compression ratio, the stiffness degradation curve of the specimen with a high axial compression ratio is always above the stiffness degradation curve of the specimens with medium and low axial compression ratios. This indicates that the initial stiffness of the specimen with a high axial compression ratio is larger and the stiffness degradation rate is faster. When the displacement loading level is four times the yield displacement, the stiffness decline amplitude and rate slow down, and reach the later stage of loading. There is little difference in the residual stiffness of each specimen.

2. For the specimen with a single parameter change in wall thickness, the stiffness degradation curve of thick-walled specimens is always above the stiffness degradation curve of thin-walled specimens. This indicates that the specimens with a thicker steel wall have greater initial stiffness and faster stiffness degradation rate, which gradually slows down in the later stage of loading, and there is little difference in residual stiffness during failure.

3. For the specimen with a single parameter change in slenderness ratio, the stiffness degradation curve of the specimen with a low slenderness ratio is always above the stiffness degradation curve of the specimens with medium and high slenderness ratios. This shows that specimens with smaller slenderness ratios have larger initial stiffness, faster stiffness degradation rate, and the overall stiffness of the specimen in the whole loading process is larger, while the stiffness of the specimens with medium and high slenderness ratios show little difference.

5.1 Hyperparameter tuning

The model FA–RF in this study ran 50 iterations in the 5-fold CV method and generated 5 corresponding RMSE values, which were determined according to the balance between the training time and RMSE reducing efficiency [46,47].

Figure 14a shows the results of the CV iteration of the model for predicting the displacement ductility coefficient, which shows that the RMSE of the 1st fold is the smallest. Figure 14b represents the iterative process of the 1st fold. Similarly, Figure 15a shows the results of the 5-fold CV iteration for the model used to predict equivalent damping coefficient. It can be observed that the RMSE obtained at the 5th fold is optimal. Figure 15b represents the detailed iterative process of the 5th fold. The RMSE value’s reduction can be clearly observed within the first 10–20 iterations and basically maintains the minimum value after 20 iterations (or occasionally changes), illustrating that the RMSE reaches the local minimum.

Figure 14

The CV results of displacement ductility coefficient prediction and the optimal fold iterative process. (a) 5-fold CV results. (b) The iterative process of optimal fold.

Figure 15

The CV results of equivalent damping coefficient prediction and the optimal fold iterative process. (a) 5-fold CV results. (b) The iterative process of optimal fold.

According to the above experiments, the optimal hyperparameters (numTree and minNumleaf) of the two models are finally obtained. For the displacement ductility coefficient prediction model, the optimal hyperparameters are numTree = 15 and minNumleaf = 1. For the equivalent damping coefficient prediction model, the optimal hyperparameters are numTree = 16 and minNumleaf = 1.

5.2 Performance of FA–RF modeling

Figures 16–18 are the prediction errors of the FA–RF model, including the errors on the training set and the errors on the test set. The model uses the optimal hyperparameters obtained by FA adjustment.

Figure 16

Scatter plot of simulated and actual displacement ductility coefficient for training and test sets. (a) Training dataset. (b) Testing dataset.

Figure 17

Scatter plot of simulated and actual equivalent damping coefficient for training and test sets. (a) Training dataset. (b) Testing dataset.

Figure 18

Scatter plot of simulated and actual values for training and test sets: (a) displacement ductility coefficient prediction and (b) equivalent damping coefficient prediction.

Figures 16 and 17 are the error diagrams of the prediction of displacement ductility coefficient and equivalent damping coefficient, respectively, and their differences are expressed by the difference between the actual value and the predicted value. The prediction is relatively accurate since no outlier or large error is observed. Thus, the mapping relationships from input variables (i.e., concrete strength, pipe length, etc.) to output results (i.e., equivalent damping and displacement ductility coefficients) are properly acquired by the ML model.

Figure 18 is a linear regression graph, which shows the difference between the coefficients by the correlation between the actual value and the predicted value. Figure 18a shows the experimental results of the model for predicting the displacement ductility coefficient, and Figure 18b shows the experimental results of the model for predicting the equivalent damping coefficient. The coordinates in the figures are composed of predicted values and actual values. Blue points and red points are data on the training set and test set, respectively. The black diagonal line represents the highest correlation, meaning that the predicted value equals the actual value. The distance between the points and the black diagonal is inversely proportional to the model prediction ability. The RMSE and R of the displacement ductility coefficient prediction model on the training set are 0.5476 and 0.924, respectively, and the RMSE and R on the test set are 0.7826 and 0.9276, respectively. The RMSE and R of the equivalent damping coefficient prediction model on the training set are 0.0254 and 0.0457, respectively, and the RMSE and R of the test set are 0.0457 and 0.9218, respectively. It can be observed that the points in the image are mostly distributed near the black diagonal, indicating that the predicted results are close to the actual results, and there is no underfitting. The distribution of blue dots and red dots is basically the same, indicating that the model does not have overfitting.

5.3 Comparison of RF, LR, and MLR

To further evaluate the necessity of the selected model in prediction, this study selects the baseline model MLR and LR as the comparison objects.

Figure 19 presents a performance evaluation of the RF, MLR, and LR models as a boxplot diagram. The red cross represents the abnormal point, and the upper and lower black horizontal lines represent the upper and lower limits, respectively. The upper and lower boundaries of the rectangular box represent the upper quartile and the lower quartile, respectively, and the red line inside indicates the median. In the prediction of the displacement ductility coefficient, the FA–RF model has fewer outliers, a narrower quartile range, and lower median. In the prediction of equivalent damping coefficient, the FA–RF model performs slightly worse, but it still has fewer outliers and the difference is not large.

Figure 19

Performance evaluation of three models by (a) displacement ductility coefficient prediction boxplot and (b) equivalent damping coefficient prediction boxplot.

Figure 20 is the Taylor diagram, which evaluates the model from three aspects: standard deviation, correlation coefficient, and RMSE. The predicted results are closer to the real data (i.e. higher accuracy) if the distance between the ML points and the black point “Actual” is closer. For both displacement ductility coefficient prediction and equivalent damping coefficient prediction, the model FA–RF showed better results in standard deviation, correlation coefficient, and RMSE than the MLR and LR models.

Figure 20

Performance evaluation of three models by (a) displacement ductility coefficient prediction Taylor and (b) equivalent damping coefficient prediction Taylor.

Table 10 shows the evaluation results of the model on the test set for the prediction of displacement ductility coefficient and equivalent damping coefficient. It can be observed that the FA–RF model has better in RMSE and R than other models.

Combined with the box plot and Taylor plot analysis, the FA–RF model is more effective than LR and MLR in displacement ductility coefficient prediction and equivalent damping coefficient prediction. Table 10 also supports this conclusion. The results show that the FA–RF model performs the best in displacement ductility coefficient prediction and equivalent damping coefficient prediction.

5.4 Variable significance analysis

In sensitivity studies, input variables are analyzed to obtain the impact of each input variable on prediction results (usually shown in percentage). Figures 21 and 22 are the analysis results of the variable importance of the FA–RF model for the prediction of displacement ductility coefficient and equivalent damping coefficient, respectively. It can be observed in Figure 21 that the axial pressure ratio, solid waste content, and concrete design strength are of high importance for displacement ductility coefficient prediction; 41.72, 36.53, and 17.35%, respectively. Steel pipe thickness and effective steel pipe length are less important; 3.36 and 1.04%, respectively. Figure 22 shows that steel pipe thickness is the most important variable for equivalent damping coefficient prediction, accounting for 41.26%. Less important variables were effective steel pipe length, concrete design strength, solid waste content, and axial pressure ratio; 18.56, 15.38, 14.95, and 9.85%, respectively.

Figure 21

Variable importance analysis of FA–RF model for displacement ductility coefficient prediction.

Figure 22

Variable importance analysis of FA–RF model for equivalent damping coefficient prediction.