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1. Introduction
With the rapid advancement of e-commerce, automated storage and retrieval systems (AS/RS) have been so widely applied in China that high-rise steel storage pallet racks (SPR) have exhibited an explosive growth in production and logistics system (Figure 1). Acting as one of the most important infrastructures for AS/RS, structural design for SPR needs the elaborate decision-making between structural systems and a variety of cold-formed steel members in such a way that the stability and safety behave as intended by the designer and satisfies the constraints imposed by capital investment, environment, and so on. Opposite to traditional civil engineering structures, the material of steel members in storage pallet racking is thin and lightweight while the racking system itself can usually carry live load many times larger than the dead load with an extraordinary height.
[figure omitted; refer to PDF]
Of all the members in the SPR, the beam-to-column connections (BCCs) constitute the most critical part of the assembly which largely determines the overall stability of SPR in the down-aisle direction [1]. The details of the boltless BCCs with the three rivets mostly used in industrial racking system are shown as an example in Figure 2. Due to the great variety of connector types and connected members, a generalized analytical evaluation of the connection mechanical properties still appears to be very difficult [2]. One of main reasons for this is that the typical boltless steel connections are essentially called “semirigid” or “partial-strength” structure representing a strong nonlinear behavior [3]. Therefore, the most recent design codes, such as those of the EN 15512 [4], RMI [5], and AS4084 [6], recommend physical experiment method of the testing results to assess the moment-rotation (M-θ) behavior of any SPR BCC. Numerous studies in the last few years are available on the experimental testing of SPR BCCs [7–11]. Apparently, these investigations, dependent on experimental results, are relatively accurate and reliable but their arrangements are too expensive, and operations are too complicated to be utilized in industrial production on a large scale. On the other hand, the possibility of random or systematic errors in the experimental investigations and the diversity of beam-end-connectors also directed researchers towards the finite element (FE) modeling of connections [2, 12]. FE modeling, which was made by using different analysis software, has proven itself to be a powerful tool to gain a more predictable performance of the connections and the effects of various parameters on the overall performance of SPR. Furthermore, a suitable solution was proposed to derive a particular uniform M-θ relationship for each connection type on the basis of BCCs geometric parameters through experimental investigations and finite element (FE) modeling [13]. From analytic point of view, Zhao et al. and Gusella et al. proposed a mechanical model based on the component method to predict the initial rotational stiffness of beam-to-upright connections [14, 15]. Those analytical models are based on physical knowledge of stability mechanics, which not only are very appealing but also give a thorough insight into the deformation mechanism of multiple components. However, to cope with the inherent complexity of structural mechanics, some assumptions have to be introduced into these models, among which, the assembly relationship of the beam-to-column joints has not specially been addressed in previous studies. This may ultimately affect the prediction accuracy of analytical models and then result in unexpected deviation from physical tests. The increasing demands for cold-formed thin-walled steel in modern industry need to explore more reliable methods of accurate prediction of the behavior of storage racks, which have a wide range of adaptability and operational convenience in engineering design. The proliferation of industrial “big data” has created many exciting opportunities for those working in various fields such as science, engineering, and business. The machine learning (ML) and data mining (DM) from industrial big data have been rapidly developed as new disciplines of computer science and engineering application [16, 17]. It has been gradually realized that those data from engineering experiments and analysis not only can be used for the engineering practice, but also have the potential to provide insight and knowledge for the designer to improve the construction quality itself. The data-driven approaches focus on analysis and discovery of the potential pattern of design process and can realize precise prediction of complex engineering problems, usually including some metaheuristic optimization algorithms such as the genetic algorithm and particle swarm optimization, artificial neural network (ANN), support vector machines, and Bayesian models [18]. Within the constructional steel fields, the advantage of an ANN was used to propose an intelligent finite element for viscoelastic material behavior in [19]; Shah et al. [2] also proposed a hybrid intelligence model based on linear genetic programming (LGP), artificial neural networks (ANNs), and adaptive neuro-fuzzy inference system (ANFIS) to predict the moment-rotation (M-θ) behavior of boltless steel connections [20].
[figures omitted; refer to PDF]
In recent years, a variety of machine learning methods have been applied on a large scale in the modern industrial and civil engineering field. Among them, convolutional neural network (CNN) is one of the representatives of deep learning algorithm, which is suitable for multipixel and audio processing. Cha et al. [21] and Wang and Cha [22] used these novel deep learning methods for damage detection in structural-health monitoring for civil structures. In addition, Santos et al. [23] compared four kernel-based algorithms for damage detection under varying operational and environmental conditions, namely, based on one-class support vector machine, support vector data description, kernel principal component analysis, and greedy kernel principal component analysis. Langone et al. [24] came up with a technique called adaptive kernel spectral clustering (AKSC) which unifies the data normalization and damage detection steps. Inspired by the idea of unsupervised feature learning that uses artificial intelligence techniques to learn features from raw data, a two-stage learning method is proposed, with Moving Kernel Principal Component Analysis (MKPCA) and Nyström methods, by Ghiasi and Ghasemi [25] for intelligent health monitoring of civil engineering structures. Besides, support vector machines (SVMs) are also receiving increasing attention in different application domains for which artificial neural networks (ANNs) have had a prominent role, due to their many attractive features and promising empirical performance. This systematic approach, motivated by statistical learning theory, led to a class of algorithms characterized by the use of kernels, the absence of local examples, the sparseness of the solution, and the capacity control obtained by acting on the margin. Unlike traditional ANN models, SVM models are based on the principle of structure risk minimization (SRM), which equips the latter with greater potential to generalize. Since the foundation of the SVMs paradigm was laid down by Vapnik in mid-1998 [26], applications in many engineering fields have emerged, such as architecture [27], communication system [28], geology [29], and even financial management [30]. However, reports about which SVMs are used for predicting the M-θ behavior of SPR BCCs have not been seen so far.
Taking the riveted BCCs as our research object, we present a novel data-driven model, using an integrated experimental-FEM-SVM methodology to overcome many difficulties associated with the mechanical performance of semirigid beam-to-upright joint modeling, which is the main contribution of this paper. The objective of data-driven based predictive models is the development of enabling tools for designers to make rapid and effective decision when big datasets are available on prediction and reasonable number of predictors. Compared with existing references, the obvious distinctions of our work lie in the fact that the finite element simulation data based on physical test are utilized to train SVM model and predict the bending strength of the complex boltless steel connections with data mining method. The results have undergone comparative analysis with those of the traditional FEM and ANN. The preliminary investigation demonstrates that the data-driven models have a reasonably good accuracy in most of the cases and are more suitable for the nonlinear mechanical behaviors. The outline of the remaining content of this paper is as follows. Section2 briefly describes data-driven model framework and integrated methodology. Based on the data from physical performance tests of BCCs, Section 3 develops a finite element model to exactly simulate the flexural behavior under monotonic loads. Section 4 introduces the SVM regression algorithm and data mining process. The results and discussion of the case study are presented in Section 5. Finally, the conclusions and future work are summarized in Section 6.
2. Data-Driven Based Methodology Framework
Generally speaking, the current models available to solve the mechanical performance problem of the SPR BCCs can be mainly categorized into two types: physical experiment based analytical models and FE based numerical model. However, the use of the two techniques for analyzing this wide range of beam-to-upright assembly in massive engineering practices could be inappropriate due to the great amount of time and economic resources required. The methodology proposed in our paper is based on a hybrid approach of experimental, numerical (finite element method), and machine learning (support vector machines) techniques, which allows the obtainment of computational efficient results for various design solutions to make rapid and accurate evaluation. As shown in Figure 3, the data-driven modeling framework includes three stages, and the general task in each stage is described as follows.
Stage I: Data acquisition
The task of this stage is to collect and transform the data from the beam-to-column physical experiment and finite element simulation into engineering database. Because the physical experiments are so costly that volume of real dataset is relatively limited, the finite element simulation is employed to expand engineering data as machine learning required. On the basis of the test data, the finite element model in the commercial software ANSYS is repeatedly calibrated and validated in order to exactly simulate the blending process of the cantilever beam experiment; then, using the so-called virtual testing method, the different rotational stiffness from finite element simulation for the existing joint solutions is obtained instead of the real physical test. Finally, the substantial data such as the geometric features, assembly relationship, and corresponding mechanical behavior on the diverse BCC joints are stored in engineering analysis database.
Stage II: Machine learning
This stage is the core module of data-driven modeling which can cover the full machine learning pipeline from data processing to result evaluation. In most cases, those modeling data from the engineering database fall within different ranges. It is highly essential to preprocess the input data before applying them to the machine learning models, so as not to affect the obtained results. On the other hand, these raw data and engineered features probably have a large number of independent or redundant variables, which often make models more complex and incomprehensible. There are two main dimensionality reduction methods for data: one is to extract the main features of the data by destroying the original structure of the data. The other is to conduct correlation analysis on the data and select the attributes of the data according to certain rules to achieve the purpose of dimensionality reduction. Kernel methods belonging to the first type, such as the kernel principal component analysis (KPCA), have the ability to find nonlinear patterns from the data while keeping the computational elegance of matrix algebra, but they often take up a lot of memory and the calculation is more complicated [31]. Here, the correlation coefficient after Pearson R falls into the second type as an easier feature extraction method is used to reduce the data dimensionality and improve the generalization performance of a predictive model. In the model training, the normalized dataset is randomly divided into separate train and test sets; on the basis of those data, the control parameters of the SVM model are continuously adjusted and optimized through iterative loop mode until the predictive accuracy satisfies the need of engineering practice as a whole.
Stage III: Design decision
Once the predictive model is trained well, new design solutions for the BCC joint are input into it one by one, and their mechanical performance can be quickly obtained so that structural engineers can make more reasonable decisions.
Unlike the existing programs and methods [2, 14, 32, 33], the merits of data-driven model framework lie in the following:
(i) SVM/ANN is a self-adaptive and data-driven method in nature, so there is no need to make some rigorous assumptions about the statistical distribution on real engineering data.
(ii) SVMs are good at handling data with much more features than samples, which makes it more accurate for modeling complex data patterns, as opposed to traditional modeling approaches based on a large amount of test data.
(iii) Along with the growth of engineering data, the proposed framework is very expandable and has the capability of improving prediction accuracy by system self-learning.
(iv) Robust reasoning machine in the intelligent prediction model is utilized to optimize design parameters on the SPR BCCs as predictive model calculating without consideration of the potential rule collision from explicit design knowledge.
[figure omitted; refer to PDF]
At the beginning of the test, an initial load F of 10% of the expected failure load was preloaded at 400 mm from the beam flange surface to the column. The purpose is to make the rivets on the beam-end-connector fully contact the column grooves, then fix the components, and then unload. The measuring instrument was reset, and then the force F was gradually increased to the maximum load value until the BCCs failed. During the test, load F was measured by a load cell, and the vertical components of the displacements d1 and d2 at the loaded section were directly monitored by the linear variable displacement transducer (LVDT) of the testing machine. LVDTs and wire-actuated encoders were connected to a computer-aided data recording system and load cells.
3.1.4. The Experimental Results and Moment-Rotation Response
The stiffness of beam-to-column is obtained by moment-rotation (M-θ) curve. The rotation may be measured by displacement transducers bearing onto a plate tack-welded to the beam close to the connector, but with enough clearance to allow for connector distortion. The moment M and rotation θ were calculated by the following equations:
According to the code EN 15512 [4], the yield stress and thickness of the materials of the beam, upright and connector in Table 2 are used to calibrate the observed value of M and θ of the test. The acquisition of stiffness requires an over coordinate origin line at the M-θ polynomial fitting curve (Figure 7), which, with the line of design moments
[figure omitted; refer to PDF]
A comparison of the M–θ graphs plotted for the experimental and finite element studies is provided in Figures 13(a)–13(d). Four specimens with varying column thickness values and column cross-sectional areas were compared to illustrate the agreement between the experimental and FE analysis results. It was found that the stiffness of the specimens was on the verge of that in the experimental result even though the ultimate moment capacity of the connection obtained from the FE model for specimen was slightly higher than that from the experiments. This is because the imperfections from material and fabrication are not considered by the FE model [1]. Moreover, due to the assembly defect between rivets and columns and the small applied load at the initial stage of the test, there is a deviation between the two M-θ curves near the origin. According to formulas (2) and (3), the stiffness values of M-θ are calculated, and it is found that the stiffness values are not much different, and the average error of the four groups is about 4.6%.
[figures omitted; refer to PDF]
During the mechanical performance test and finite element simulation, three failure modes of beam-column joints were observed. The failure modes are shown in Table 4: (i) yielding of the beam-end-connectors, (ii) tearing of the column material, especially the holes, and (iii) fracture or yielding of the rivets. When the beam-column joint is under compression, the rivet will bear the shear force opposite to the direction of the connector and the column, resulting in plastic deformation. The three simulated failure modes basically agree with the mechanical performance test, which verifies the validity of the simulation results. It is shown from the above diagrams that the FE model can predict the experimental behavior well as a whole, and the physical test on the BCCs can be replaced by the FE simulation from the perspective of engineering structure design. Consequently, with similar FE simulation method, 432 bending tests of beam-to-column joints were carried out by selecting representative columns including M90 B, M100 A, and M100 B, which are the most widely used in industrial application. The detailed parameters and specific simulation data are shown in the Supplementary Material (available here).
Table 4
Comparison of failure modes between the experimental test and FEM.
Failure mode | Experiment | FEM |
Yielding of the connectors | ||
Tearing of the column | ||
Fracture of the rivets |
4. Empirical Studies
To validate prediction performances on the SPR BCCs used by the proposed data-driven methods, some empirical cases are conducted in this study. This section first argues the mapping relationship of predictive model between the input data and output data. Then, it describes how the empirical cases were carried out. Finally, it provides a description of how the search for the parameter that achieves the best possible performance was made.
5. SVM Regression Algorithms
Support vector machines (SVMs) are based on principles of convex optimization and statistical learning theory proposed by Vapnik and Izmailov [36]. The main idea of the SVM regression algorithm is to estimate the output variable y from original input data vector x mapped into a higher-dimensional feature space through nonlinear transformation, and extract the information and regularity contained among the data. The SVM regression function is defined as
[figure omitted; refer to PDF]
After training a regression model, the predicted vs. simulated response plot (as shown in Figure 17) is used to check model performance, which is used to understand how well the regression model makes predictions for different response values. A perfect regression model has a predicted response equal to the true response, so all the points lie on a diagonal line. The vertical distance from the line to any point is the error of the prediction for that point. A good model has small errors, and so the predictions are scattered near the line.
[figure omitted; refer to PDF]
The residual plot (as shown in Figure 18) is used to check model performance. The residual plot displays the difference between the predicted and simulated responses. Usually a good model has residuals scattered roughly symmetrically around zero.
[figure omitted; refer to PDF]5.2.2. ANN Model Train
Artificial neural network (ANN) is a powerful data-modeling tool that is able to capture and represent any kind of input-output relationships. BP is widely used in engineering because of its simple model and high prediction accuracy [40].
The design of BP model program adopts the artificial neural network app in Matlab2017a. The model training uses the same 400 sets of training data as SVM model. Similarly, the datasets from BCCs virtual test are divided into two groups of training data and testing data. The training process of the model includes the determination of hidden layer number, the selection of transfer function, and the preset number of neurons in hidden layer. The error of observation results can be modified by adjusting the above parameters, until the expected results are obtained.
In the light of Bishop’s report, more than one hidden layer is usually not necessary. Therefore, the ANN architecture for thin-walled steel design has only one hidden layer. As proposed in the literature [41], the node number of hidden layers was obtained as 8 by
The BP network toolbox in Matlab2017a has a variety of transfer functions for modeling, including linear function, nonlinear function, and other error surface functions. In this paper, “logsig” is selected as the model transfer function, which is a differentiable logarithmic s-type transfer function, which maps the input range of neurons (−∞, +∞) to the interval of (0, +1), and its equation is
The detailed parameter settings of the ANN prediction model are summarized in Table 7.
Table 7
The selection of ANN model parameters.
Name | Parameters |
Hidden layer node number | 8 |
Hidden layer number | 1 |
Transfer function | logsig |
5.3. Results and Discussion
After the establishment of the model, it still needs to be verified, so it is compared with the results of the four groups of mechanical properties tests, as shown in Table 8. It can be seen that both SVM model and BP model are close to the test value, with mean absolute error (AE) of over 3% and correlation coefficient R close to 1. The accuracy of the model is preliminarily verified. Because the training set of the prediction model comes from the calculation results of the numerical model, the overall value is higher than the experimental value. The final results are listed in Table 9, where “FEM,” “SVM,” and “ANN” refer to the finite element values, the SVM predicted values, and ANN predicted values, respectively. Statistical parameters, such as the mean absolute error (%) and correlation coefficient R between the expected and real value, are used to judge the predictive power of the data-driven models. It is evident that the accuracy of all the predictive models is relatively high (R > 95%), while the SVM model, in terms of the mean absolute error and the ratio of the cases with more than 5% error, is lower than the ANN model. It is evident from Table 9 that the predictive power of the SVM model the predictive power of the SVM model is the better of the two models considered here.
Table 8
Comparison between the test values, SVM predicted values, and ANN predicted values.
Number | Test | SVM | AE (%) | ANN | AE (%) |
1 | 34870.00 | 35033.05 | 0.47 | 36607.54 | 4.98 |
2 | 49890.00 | 51735.48 | 3.70 | 50863.15 | 1.95 |
3 | 60640.00 | 60463.71 | 0.29 | 62736.40 | 3.46 |
4 | 65170.00 | 65553.39 | 0.59 | 64236.76 | 1.43 |
MAE | 40.7625 | 434.385 | |||
RMSE | 950.06 | 1519.20 | |||
MAPE | 0.12% | 1.19% | |||
Correlation coefficient R | 0.9978 | 0.9959 | |||
R-squared | 0.9956 | 0.9919 | |||
Cases with an error of more than 3% | 1 | 2 |
Note. The unit of bending strength is kN mm/rad.
Table 9
Comparison between the FEM values, SVM predicted values, and ANN predicted values.
Number | FEM | SVM | AE (%) | ANN | AE (%) |
1 | 43945.22 | 45042.03 | 2.50 | 48907.54 | 11.29 |
2 | 52853.68 | 53755.41 | 1.71 | 52873.51 | 0.04 |
3 | 56572.25 | 56453.72 | 0.21 | 56736.45 | 0.29 |
4 | 53116.89 | 56553.71 | 6.47 | 54236.05 | 2.11 |
5 | 56792.16 | 56853.3 | 0.11 | 57766.89 | 1.72 |
6 | 62064.84 | 61920.47 | 0.23 | 62083.12 | 0.03 |
7 | 47064.88 | 47399.29 | 0.71 | 45548.75 | 3.22 |
8 | 51237.39 | 50781.86 | 0.89 | 49606.38 | 3.18 |
9 | 55195.33 | 54033.37 | 2.11 | 54038.4 | 2.10 |
… | |||||
31 | 51897.03 | 51306.61 | 3.59 | 50754.98 | 4.63 |
32 | 55933.86 | 54525.76 | 1.78 | 55121.01 | 0.71 |
MAE | 25.51 | 115.40 | |||
RMSE | 2329.36 | 1806.92 | |||
MAPE | 0.06% | 0.24% | |||
Correlation coefficient R | 0.9560 | 0.9651 | |||
R-squared | 0.9140 | 0.9315 | |||
Cases with an error of more than 5% | 3 | 7 |
Note: the unit of bending strength is kN mm/rad.
6. Conclusion
Due to computational complexity and accuracy, the analytical expressions for the moment-rotation stiffness of thin-walled steel beam-to-column connections are not widely used for steel member design so far. In this paper, the M-θ behavior predictions from a novel data-driven model with the integrated experimental-FEM-SVM methodology are compared with those obtained from the traditional FEM and ANN model. It is noted that the data-driven model based on SVM technique is very efficient because the prediction performance is much closer to the physical test and FEM than those obtained from the ANN models. Here, we only demonstrate that, trained with the engineering datasets from experiment and simulation, the data-driven model is able to predict the M-θ behavior of different BCCs through self-learning, which can help engineers to make quick and effective decisions for complicated rack design. The results of our paper appear to be preliminary and limited to boltless BCCs situations, but it has been found that data-driven models for solving complex semirigid component design problems are very promising. Future research should focus on the following aspects: (1) expansion of the engineering analysis database to improve the flexibility of the data-driven model and then optimize the design configuration among a large number of beam-to-column joints; (2) development of new methodologies that can effectively explain the results of these apparently incomprehensible models. We believe that this research can be finally fused together with other pioneering analytic or experimental studies. With advancement of data mining and cloud computing techniques, many of the producers’ subjective intuitions in steel pallet rack industry will finally be replaced by smart and friendly expert systems in the near future.
Acknowledgments
This paper was financially supported by Technology Innovation Program of Shanghai Municipal Science and Technology Commission (17DZ2283800) and Songjiang District Industrial Development Special Fund on Demonstration Application Project (2018-01). The authors assure that the above-mentioned funding projects have been funded by the Shanghai Government to the authors’ scientific research team of Donghua University. The funding expenditure complies with relevant regulations.
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Abstract
Due to many differences in the material, geometry, and assembly method of the commercially available beam-end-connectors in steel storage pallet racks (SPR), no common numerical model has been universally accepted to accurately predict the M–θ behavior of complex semirigid connections so far. Despite the fact that the finite element method (FEM) and physical experiment have been used to obtain the mechanical performance of beam-to-column connections (BCCs), those methods have the disadvantages of high computational complexity and test cost. Taking, for example, the boltless steel connections, this paper proposes a data-driven simulation model (DDSM) that combines the experimental test, FEM, and support vector machine (SVM) techniques to determine the bending strength of BCCs by means of data mining from the engineering database. First, a three-dimensional (3D) finite element (FE) model was generated and calibrated against the experimental results. Subsequently, the validated FE model was further extended to perform parametric analysis and enrich the engineering case base of structural characterization of BCCs. Based on the M–θ curve of the FE simulation, support vector machines (SVMs) were trained to predict the flexural rigidity of beam-to-column joints. The predictive power of the SVM algorithms is estimated by comparison with traditional ANN models via the root mean square error (RMSE), the mean absolute percentage error (MAPE), and the correlation coefficient R. The results obtained indicate that the SVM algorithms slightly outperform the ANN algorithms, although both of them are in good agreement with FEM and physical test. From the point of view of engineering application, DDM is able to provide much more effective help for structural engineers to make rapid decisions on steel members design.
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1 College of Mechanical Engineering, Donghua University, Shanghai 201620, China; Shanghai Engineering Research Center of Storage & Logistics Equipment, Shanghai 201611, China
2 SAIC General Motors, Shanghai 201206, China
3 Shanghai Engineering Research Center of Storage & Logistics Equipment, Shanghai 201611, China