A robust method for safety evaluation of steel trusses using Gradient Tree Boosting algorithm
Introduction
In conventional design methods of steel structures, e.g., Allowable Stress Design (ASD), Plastic Design (PD), and Load and Resistance Factor Design (LRFD), member forces obtained from an elastic analysis are separately checked using design equations provided in design codes. This is a two-stage, member-based design process which ignores the interaction of members in the whole structure, and thus cannot capture the overall buckling and failure mode of the whole system. To overcome these limitations, advanced analysis or nonlinear inelastic analysis methods have been developed [1], [2], [3], [4], [5], [6], [7], [8], [9] enabling the accurate capture of all sources of nonlinearities of the whole structural systems. Therefore, separate member capacity checks encompassed by the specification equations are not required. However, advanced analysis is time-consuming compared to the elastic analysis which ignores all sources of nonlinearities. For simple design tasks where the number of required structural analysis performance is small, this matter is acceptable. For complicated design problems such as optimization design using metaheuristic algorithms, structural reliability analysis, and structural damage assessment, a large number of structural analyses are required, and thus the use of advanced analysis is the extremely computational cost [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. To solve this problem, metamodels using machine learning (ML) methods such as artificial neural network (ANN) [20], Kriging [21], polynomial response surface method (RSM) [22], and support vector machines (SVM) [23] have been widely used as a promising alternative.
A metamodel is an approximate mathematical representation used to describe high-level abstractions in data. In metamodel, the complex nonlinear responses of structures can be approximated without performing time-consuming advanced analyses. A typical application of metamodels in structural design optimization and structural reliability analysis can be listed here as the following. Yang and Hsieh [23] used SVM to evaluate the reliability constraint for the reliability-based design optimization of a ten-bar truss. The results indicated that using SVM can significantly reduce the number of reliability constraint evaluation in the optimization process. Xue et al. [24] improved the metamodel method for efficient reliability analysis by accounting for the approximation error due to metamodel approximation. Bucher [25] discussed the metamodels of optimal quality for stochastic structural optimization. Suprayitno and Yu [26] developed a Kriging-based metamodel for highly nonlinear optimization problems, whilst Chen et al. [27] and Liu and Elishakoff [28] proposed a hybrid Kriging-based metamodel for small failure probabilities. Cheng et al. [29] developed a new adaptive method for structural reliability analysis based on ensemble learning of multiple competitive metamodels. Li and Wang [30] proposed an optimization framework to enable the quantification of metamodel model uncertainty in reliability assessment. Gholizadeh [31] used a back-propagation (BP) neural network to estimate the structural safety in the optimization design problem of steel frames under seismic loading. In addition, metamodels have been used to identify structural risk factors. For example, Oh et al. [32] used Bayesian learning to classify the damage states of structures subjected to seismic loading. Hasni et al. [33] applied SVM to perform the health monitoring of steel plates. Lee and Lee [34] proposed an ANN-based method for the shear strength prediction of fiber-reinforced polymer concrete. A review of the application of artificial intelligence (AI) methods in structural engineering can be found in Ref. [35]. Deep learning (DL), a new branch of the ML methods, has advanced in addressing various design and analysis problems of structures such as damage detection [36, 37], health monitoring [38, 39], etc. However, it requires big data to get a highly accurate result.
Among various ML methods, the GTB method proposed by Friedman [40, 41] has been considered as a powerful DL model with many successful applications for both classification and regression problems in different fields ([42], [43], [44], [45], [46], among others) except for structural design. GTB is an ensemble-based algorithm that consists of multiple base models (learners). Each of base learner is a single tree model built by training the model using the bootstrap sample from the training data, partitioning the feature space into a set of regions, and fitting a simple model for each region [47]. In the analysis process of a GTB model, the base models are generated sequentially where samples of the additional base models that are incorrectly estimated or misclassified have more chances to be selected due to correcting the predictive mistakes made by the previous base models. The final output of the model is predicted by combining different solutions of the base models. Consequently, GTB models are effective to handle missing data, outlier effects, inaccurate training, unbalanced datasets, and predictors’ interactions based on their stochastic characteristic in modelling, and capability to quantify variable importance. However, to the best knowledge of authors, the application of GTB for predicting structural responses of nonlinear inelastic trusses has not been studied yet in the literature.
To fill in the above research gap, a GTB-based method is developed for the safety evaluation of steel trusses. Four GTB models are proposed. The first model is used to estimate the load-carrying capacity, whilst the second one is developed for strength safety evaluation of the structure based on the ultimate load factor developed in the first model. The third model is used to predict the displacement which is used in the fourth model developed for serviceability safety evaluation of the structure. Advanced analysis is used to accurately generate the outputs of datasets (i.e., ultimate load-carrying capacity and maximum displacement) for the developed GTB models. To demonstrate the efficiency of the proposed procedure, three numerical examples of planar truss, spatial truss, and planar truss bridge are carried out. Although it is common to consider discrete design variables, it is reasonable to consider continuous design variables in the viewpoint of practice and theory. Therefore, both continuous and discrete design variables are considered in this study. Four other commonly used ML methods including SVM, DT, RF, and DL are also used for comparison purposes.
Section snippets
Advanced analysis of steel trusses
In the advanced analysis of truss structures, the accuracy of the structural behavior is strongly dependent on the constitutive model used in the modeling. A typical response of a truss under cyclic is shown in Fig. 1 exhibiting several failure modes such as buckling, yielding, inelastic post-buckling, unloading, and reloading [5]. When a member under tension, it will not be buckled, and thus its stress-strain relationship can be represented by a bilinear model including parts (e) and (f) of
Gradient tree boosting
Considering the vector input X ={x1,..., xn} that has the output y. The goal of a training process of a metamodel is to find a relationship, that can represent a function Fopt(X), to map X to y based on a given training data set, , so that the predefined loss function, Loss(y, F(X)), of expected value F(Xi) and the exact value yi is minimized. Fopt(X) can be written as
To solve the problem presented in Eq. (2), a tree ensemble model is developed by
Estimation of the ultimate load-carrying capacity of steel trusses
The GTB-based procedure for estimating the ultimate load-carrying capacity of steel trusses is developed with the following main steps.
Step 1: Initialization of problem
The input is cross-sectional areas of the structural elements that are divided into several design groups. Other information on structure such as geometry, material, and applied loads is constant in this work. The output is the ultimate load factor which is obtained from the advanced analysis. Dataset for the GTB-based procedure
Numerical examples
In this section, three steel trusses (planar 10-bar, spatial 72-bar, and 113-member plane bridge) are discussed. The design variables are the cross-sectional areas of the truss members with both continuous and discrete design spaces. Four well-known ML methods including SVM, DT, RF, and DL are used for comparison and are developed using the programming language Python and the open-source software libraries Tensorflow, sklearn, and Keras. The information on these methods, which are chosen by
Conclusions
This study presents an efficient GTB-based method for safety evaluation of steel trusses. In the proposed method, advanced analysis is adopted to predict the outputs of the dataset including the ultimate load-carrying capacity and displacement of the structure. The accuracy and efficiency of the proposed GTB model are examined in three numerical examples including a planar truss, a spatial truss, and a planar truss bridge. Compared with other well-known ML methods such as SVM, DT, and RF, the
CRediT authorship contribution statement
Viet-Hung Truong: Conceptualization, Methodology, Software, Formal analysis, Investigation, Resources, Data curation, Writing - original draft, Writing - review & editing. Quang-Viet Vu: Writing - original draft, Formal analysis. Huu-Tai Thai: Writing - review & editing, Supervision, Formal analysis. Manh-Hung Ha: Writing - review & editing, Formal analysis, Supervision, Project administration.
Declaration of Competing Interest
None.
Acknowledgment
This research is funded by National University of Civil Engineering (NUCE) under grant number 30-2020/KHXD-TĐ.
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