Ultimate Bending Strength Evaluation of MVFT Composite Girder by using Finite Element Method and Machine Learning Regressors

This paper has evaluated the bending performance of a novel prefabricated MVFT steel-concrete composite girder. 9 meters pilot MVFT girder was analyzed by validated finite element model. In the pilot test, the height of web, the length of grouted concrete in the girder and net spacing between webs were parametrically modeled to discuss their effect to the bending strength. An ultimate bending strength formula has been obtained, which was based on the regression of parametric results. In the meantime, the two Machine Learning (ML) models, BP neural network and Least Squares Support Vector Machine, have been also implemented to train and then predict the ultimate strength of MVFT girder. Three factors were selected as input in ML models: the distance between steel girder’s Tensile Centroid(TC) and slab’s Compressive Centroid(CC), the distance between steel girder’s TC and its CC, the compressive area of steel girder. After the completion of the ML training, the ultimate strength predictions of 30 meters MVFT girder by BP model and the formula have been compared, which agrees well with each other and validates their accuracy.


INTRODUCTION
Steel-concrete composite bridge has the merits of light weight and excellent fatigue performance in long-span bridges, therefore, they are widely employed in Bridge Engineering (Svensson, 2013;Liu et al., 2015).With respect to the small-span bridges, German scholars have reformed the shear connections of conventional steel-concrete composite girder and proposed VFT (Verbund-Fertigteil-Trager) composite girder, which has been used in Germany and other European countries since 1998 (Petzek and Bancila, 2010).Hechler et al. (2011) and Kołakowski and Lorenc (2015) introduced the construction technology and its engineering practices.Zanon et al. (2021) proposed the VFT-RS(Rolled Section) composite girder on the basis of VFT technology, which can further improve the structural efficiency and take full advantages of high strength of steel.The composite dowel as shear connector is the innovation of VFT girder, which is different from typical steel-concrete composite girder.Harnatkiewicz et al. (2011) and Berthellemy et al. (2018) studied the fatigue performance of composite dowel, and they suggested an optimized dowel's shape to improve its fatigue performance.
For the cold and high-altitude region, the authors proposed a small-span prefabricated MVFT steel-concrete composite girder, which is evolved from VFT girder.The steel girder and the concrete slab of MVFT girder are both prefabricated in the mill, without secondary casting (Xiong et al., 2018;Xiong, 2021;Chen et al., 2021).MVFT composite girder has the merits of convenient fabrication, light weight, fast construction and time-saving.
The ultimate bending capacity of steel-concrete composite girders is the focus of theoretical analysis and engineering design.Non-plastic and plastic analysis are the two typical methods to calculate the flexural capacity of the composite girder.Yang et al. (2018) proposed a formula for calculating the flexural capacity of composite girders in the sagging moment region by adopting the elastoplastic section analysis method and introducing the reduction coefficient of flexural capacity.Liang et al. (2005) studied the flexural and shear bearing capacity of simply supported composite girders under combined moment and shear; Liu et al. (2019) studied the flexural strength of steel-concrete simply supported composite girders under hogging bending moment.Ryu et al. (2006) studied the stiffness and strength of composite girders with Class 3 section under bending moment through 4-point flexural test.Zhang et al. (2020) studied the degradation process of flexural capacity of composite box girders under fire through numerical simulation.In this paper, the plastic method is adopted to calculate the ultimate bending capacity of MVFT girder due to its clear concept, concise form and extensive use.
In recent years, machine learning (ML) has developed rapidly and been applied to damage detection and fire resistance evaluation of composite girders.Abdeljaber et al. (2018) estimated the actual amount of vibration-based structural damage by using an enhanced CNN-based approach.Tan et al. (2020) used the normalized value of modal strain energy-based damage index Z as the input layer to locate and quantify the damage of composite girders, and the feasibility of this method through several numerical examples were verified.Hakim and Razak (2013) used the first five natural frequencies as the input layer to train neural networks, and then used them to predict the severity of damage.Tadesse et al. (2012) proposed three neural networks with the number of input layer parameters of 3, 7 and 8 respectively, to predict mid-span deflection of simply supported, two-span and three-span composite girder bridges.Li et al. (2021) used the neural network with 7 inputs, 3 outputs and 2 hidden layers to predict the fire resistance of concrete encased steel (CES) composite columns with concrete grade up to C120.In addition, machine learning has also been applied in other fields (Bağcı Daş and Birant, 2021;Calderón et al., 2020).In light of these previous research, the ML approaches are implemented in this paper to predict the bending strength of the MVFT girder.
The high-performance construction material also gives rise to the development of the steel-concrete composite girders.Especially on the issue of ultra high-performance concrete (UHPC)-steel composite member, these researches mainly focus on the subjects: negative bending moment of steel-UHPFRC composite girders (Qi et al., 2020;Hamoda et al., 2017), flexural strength of UHPC-concrete composite members (Shirai et al., 2020).In addition, there are some findings on new type of composite girders, such as the post-installed shear connector aiming to strengthen composite bridge (Hällmark et al., 2019), bending capacity of U-shaped steel-concrete composite girders (Zhou et al., 2019) and straight-side U-shaped steel-encased concrete composite girders (Yan et al., 2021).Besides, some researchers have performed the dynamic analysis of the plate structure by using FE method (Das and Gonenli, 2022;Gonenli and Das, 2021;Das et al., 2020;Sahoo and Barik, 2020;Jafarpour and Khedmati, 2020).
In this paper, a series of pilot MVFT composite girders are established numerically and are analyzed theoretically to obtain the formula of the ultimate bending capacity, the capacity of MVFT girder is then predicted by the machine learning regressors (MLR).The accuracy of the two methods is verified by comparing the results of the fitting formula and the prediction results by machine learning approach.Therefore, this study combining FE method and MLR has reliable results and can avoid a large number of numerical calculations, which provides a new approach for MVFT girder's engineering design.

Configuration of the pilot MVFT girder
The steel girder and concrete plate of MVFT girder are both prefabricated in the mill, without secondary casting.The section near the support of MVFT girder is grouted.To reduce the dead weight, there is no grouted concrete in the mid-span section.The general section of MVFT girder is shown in Figure 1.
In this paper, 3:10 scaled pilot MVFT simply supported composite girder was investigated numerically.The reduced scale span of MVFT girder is 9m.The cross-section of the model is shown in Figure 2, and Figure 3 presents the detailed geometry of steel dowel.The overall and detailed finite element model are demonstrated in Figure 4 and Figure 5, respectively.To explore the influence of web height(h w ), clear spacing between webs(w), and length of concrete filled steel tube(l c ) on the bending capacity of MVFT girders, the parametric studies are conducted based on the validated FE model as tabulated in Table 1.(Note: All dimensions in Table 1 are full-scale values.)In this paper, the ultimate load and bending strength of MVFT girder were obtained through the 3-point flexural test.And the failure mode of MVFT girder were identified by the concrete slab's load-strain curve and the steel girder's load-strain curve.

Validation of FE model
In the start of numerical test, the FE model was validated by the pull-out experimental data of composite dowels in UHPC slabs, which was conducted by Gallwoszus and Claßen (2015).The test steel dowels had a web thickness of 20 mm and were made of S460 structural steel.The dimensions of the puzzle-shaped steel dowel and the UHPC slab are shown in Figure 6, where h is the embedment depth of steel dowel.The FE analysis was performed in ABAQUS to simulate the pull-out test.The steel dowel and UHPC slab were discretized with a uniform mesh of solid elements C3D8R.Surface-to-surface contact was employed to describe the interaction between the steel dowel and UHPC slab.The surface of steel dowel and UHPC slab were chosen as the master surface and the slave surface, respectively.Contact properties were defined along with both the normal and tangential directions.The penalty friction algorithm with the friction coefficient of 0.3 was used to characterize the tangential behavior between the steel dowel and UHPC slab.Hard contact algorithm was employed in the normal direction.The numerical simulation results are demonstrated in Figure 7.The comparison between test and numerical simulation results is listed in Table 2.It can be found from Table 2 that the simulation results by FE agree with the test results.

Ultimate load of MVFT girder
Similarly with the previous validation model, the concrete slab, steel girders, stiffening ribs and concrete filled steel tube were simulated by three-dimensional eight-node solid elements (C3D8R) with one integration point.And three-dimensional two-node truss elements (T3D2) were used for the rebars in concrete slab.The loading device and supports were set as rigid bodies.The bilinear constitutive model was adopted for the steel with the value of yield strength(f y ) of 345MPa, Young's modulus (E s ) of 2.06x10 5 MPa, and tangent modulus of strengthening stage of 0.01E s .The material characteristics of concrete can be represented by the concrete damage plasticity (CDP) model in ABAQUS.The design value of concrete compressive strength(f c ) is 23.1MPa, the value of Young's modulus (E c ) is 3.42x10 4 MPa.
According to the numerical simulation results, the P-δ curve is plotted in Figure 8, where P is the loading force, δ is the mid-span vertical deflection.To further exhibit the effect of w and l c on the ultimate load(P u ) more directly, Table 3 summaries the results by controlling variables.It can be observed from Figure 8, when the h w increases from 700mm to 800mm and 900mm, the ultimate load (P u ) shows an obvious increase.As shown in Table 3, with the increase of w, the P u increases first and then decreases overall; and the P u increases gradually with increasing l c .

Failure mode of MVFT girder
The bending failure modes of steel-concrete composite girders with full shear connection are mainly classified into concrete slab compressive failure and steel girder tensile failure.In this paper, the failure mode is determined under the assumption of plastic theory.The concrete will be crushed when the maximum compressive strain of concrete slab(ε cc,max ) exceeds the ultimate compressive strain of concrete(ε cu ), and ε cu is set to be 0.0033, according to the Code for design of concrete structures (GB 50010-2010); the steel will be yielded when the maximum tensile strain of steel girder(ε st,max ) exceeds the ultimate tensile strain of steel(ε su ), and ε su =15ε y , which is defined by the Eurocode 3.For the steel involved in the analysis, ε su is 0.02803.In order to discuss the failure mode of MVFT girder, load-concrete slab compressive strain curves (P-ε cc curves) and load-steel girder tensile strain curves (P-ε st curves) are plotted in Figure 9 and Figure 10.As shown in Figure 9 and Figure 10, ε cc,max exceeds 0.0033, and ε su is less than 0.02803, when the web height of MVFT girders is 700mm, 800mm and 900mm, respectively.Therefore, the failure mode of MVFT composite girder is due to the concrete slab's crush.The compression damage of concrete slab is shown in Figure 11, and the tensile strain of steel girder corresponding is shown in Figure 12, while steel girder has turned into plastic stage.4 Formula of ultimate bending strength For MVFT girder, there is no design formula for its ultimate bending strength at present, while the design formulas for ultimate flexural capacity of conventional steel-concrete composite girder under the assumption of plastic theory are provided by the Code for design of composite structures (JGJ 138-2016) and the Standard for design of steel structures (GB 50017-2017).There is no essential or formal difference between the two formulas.Considering that the steel girder is inserted into the concrete and cooperates with the concrete, the concrete will be strengthened based on the formula from the code (JGJ 138-2016, GB 50017-2017).Therefore, a concrete strengthening coefficient(α) is proposed to modify the existing code formula.With the assumption that the plastic neutral axis is located in the steel girder, and the calculation model is shown in Figure 13.The calculation formula for ultimate bending capacity of MVFT girder is proposed by Equation (1): where α is the concrete strengthening coefficient, f c is the design value of concrete compressive strength, f a is the design value of steel compressive and tensile strength, b e is the effective width of MVFT composite girder, h c1 is the thickness of concrete slab, y 1 is the distance between steel girder's tensile centroid and slab's compressive centroid, y 2 is the distance between steel girder's tensile centroid and its compressive centroid, A ac is the compressive area of steel girder, A a is the area of steel girder.
The ultimate bending moments of MVFT girders were calculated by FEM(M NSU ) , design formula (M codeU ) are presented in Table 4.
Latin American Journal of Solids and Structures, 2022, 19(3), e438 12/20 As shown in Table 4, the length of grouted concrete in the girder ( l c ) has little influence on the ultimate flexural strength of MVFT girder.Therefore, all 60 groups of M NSU can be used to fit the proposed correction formula of MVFT girder, and the concrete strengthening coefficient α=1.221 was obtained.In terms of the coefficient of determination R 2 =0.9483, the fitting results were precise.In addition, it can be seen from Table 4 that the calculation results of the ultimate moment resistance using the present code are slightly conservative.

Ultimate bending strength by Machine Learning
In the calculation formula for ultimate bending capacity of MVFT girder, A ac , y 1 , y 2 are variables that have influence on the M NSU .Therefore, the three factors that A ac , y 1 , y 2 are determined as the independent variables of the prediction model while the corresponding M NSU is set as the dependent variable of the model.A total of 60 samples were listed from the former study.48 sets of samples are randomly selected as training set, and the remaining 12 sets of sample data served as the test part for the reliability of prediction models.Considering that the units of different uncertain parameters and the numerical magnitude may have different degrees of influence on the prediction results, all the extracted samples are normalized and collated.The normalization formula is expressed in Equation (2).(2) where P i is the normalized data of a variable x i in the training sample; x min is the minimum value of that group of data in the sample; x max is the maximum value of that group of data in the sample.

BP neural network-based ultimate bending strength prediction model
BP neural network is a multi-layer feed-forward network trained by back propagation of error.The single hidden layer network structure is chosen for the prediction model in this paper, due to the fact that the neural network can approximate any complex continuous mapping if the single hidden layer feed-forward neural network is continuous and the transfer function is sigmoid (Hornik et al., 1989).It will lead to underfitting or overfitting respectively because of deficiency or surplus hidden layer neurons.And the error of the prediction model can be minimized when the number of hidden layer neurons is 3 according to a large number of artificial neural network modeling experience and computational data comparison results.Hence, the number of hidden layer neurons is determined to be 3 here, and the flow chart of the BP neural network is shown in Figure 14.This model applies the L-M algorithm to optimize the search direction of the network weight vector so that the network quickly approaches the objective function.The iterative equation of the L-M algorithm is expressed in Equation ( 3), ( 4).
where x (k) , x (k+1) are the vectors composed of weights and thresholds among the layers in the k th and k+1 th iterations of the neural network, respectively; e is the error vector of each layer of the network; u is the coefficient, Equation ( 4) is the Newton method when u is 0; Equation ( 4) is the gradient descent method; H is the Hessian matrix when u is large.

LSSVM-based ultimate bending strength prediction model
The primary principle of least squares support vector machine (LSSVM) regression is to map the input data to a highdimensional feature space through certain nonlinear mapping, and then construct the optimal linear regression equation in the high-dimensional space.The LSSVM approach's advantages include high accuracy (Wang and Hu, 2005), a fastsolving speed, and consumes less computational resources.According to the principle of Structure Risk Minimization (SRM), the training objective of LSSVM can be expressed as Equation ( 5), (6).
. .( ) 1, 2, , where γ is the regularization parameter that controls the degree of penalty on the error; ω is the weight vector; φ(x i ) is the kernel function; b is the offset; and e i is the error variable.
The normalized data was used to build a system numerical model in MATLAB for simulation analysis, where LSSVM offline training was implemented with algorithmic programming.To ensure the general vadility of the prediction results, the LSSVM is randomly performed five times as well.The predicted values are listed in Table 6.The mean absolute percentage error (MAPE) and root mean square error (RMSE) are assigned to measure the accuracy of the prediction model for training and prediction of existing data.In the light of the error judgment rule, the prediction effect of the model is more accurate as MAPE and RMSE get closer to zero.Defining the predicted output value to be Pre i , the true value to be P, then the MAPE and RMSE are calculated as follows in Equation ( 7), (8).The error analysis of the prediction results by the above equations is listed in Table 7.As shown in Table 7, it is clear that the mean value of MAPE and RMSE of the two prediction models are both close to 0, but the prediction model based on the BP neural network obtains smaller mean MAPE and RMSE, while the prediction model based on LSSVM obtains smaller mean maximum relative error.And both their results show high precision and strong stability.

BP neural network-based ultimate bending strength extrapolation prediction model
It can be seen from Table 5, 6 and 7 that the two ML models can fit the historical data with high accuracy, in general, the BP neural network model is more precise than LSSVM.On this basis, the BP model is developed to predict the ultimate bending capacity of MVFT girders with new section, and to compare the calculation results of the fitting formula.The section properties of 30 meters MVFT girder are shown in Table 8.The ultimate bending strength prediction of 30 meters MVFT girder by BP neural network model (M EU ) and the formula (M FU ) are listed in Table 9 and have been compared, which correlates well with each other and validates their precision.Assuming the accuracy of the formula, it is employed to calculate the ultimate flexural capacity of MVFT girder with other span length, without using the BP neural network to make predictions.Then, the fitting formula is used to predict the ultimate flexural strength of 40 meters MVFT girder, and the results are shown in Table 10.(Note: Where b is the width of concrete slab, h c1 is the height of concrete slab, h w is the height of web, t w is the thickness of web, w f is the width of flange, t f is the thickness of flange, w is the clear spacing between webs.)

CONCLUSION
In this paper, a series of pilot MVFT composite girders are established numerically and the finite element models are validated against background experiment.The ultimate bending capacity and failure mode of MVFT girder are studied through the verified numerical models, the capacity of MVFT girder is then predicted by MLR.The following conclusions are drawn: 1. Owing to the steel web of MVFT girder embedding in concrete, the concrete resistance part will be reinforced compared with typical steel-concrete composite girder.Therefore, a concrete strengthening coefficient is proposed based on the existing code formula, and the formula of ultimate bending strength of MVFT girder is proposed accordingly.According to the fitting results, the concrete strengthening coefficient α=1.221 has been obtained.In terms of the coefficient of determination R 2 =0.9483, the fitting formula is precise.
2. With the ascending clear spacing between webs, the ultimate load shows a trend of initial increasing and then decreasing.The rule can be used as a reference for the preliminary section design of MVFT girder.Under the assumption of plastic theory, both the load-strain curves of concrete and steel girder disclose that: the failure of MVFT girder under bending is owing to the concrete crushing.
3. The two ML models, BP neural network and LSSVM, can fit the historical data of the ultimate bending moment of the MVFT girder with high precision: The mean maximum relative errors of the two ML models are less than 3%, and the mean values of MAPE and RMSE of the two ML models are close to 0. On this basis, the BP neural network is developed to predict the ultimate bending capacity of MVFT girders with new section, and the prediction results are in good agreement with the calculation results of the fitting formula, which validates their accuracy.The ML models are capable of an accurate prediction of the strength of MVFT section with a sufficient database, which is essential for the steel-concrete composite bridge design.Furthermore, this approach combining FE method and MLR provides a reliable result and can avoid a large number of numerical simulations, which is highly efficient in engineering design.

Figure 5
Figure 5 Details of FE model.

Figure 7
Figure 7 Results of numerical modeling.

Figure 11 20 Figure 12
Figure 11 Contour plot of concrete slab compression damage.

Figure 13
Figure 13 Calculation model of ultimate bending strength.

Figure 14
Figure 14 Flow chart of the BP neural network

Table 1
Schemes for the parametric study.

Table 2
Comparison between the results obtained by numerical simulation (NS) and static pull-out tests.

Table 3
Effect of w and l c on the ultimate load.

Table 4
Comparison of FEM and design formula predicted ultimate moment capacity of MVFT girders.

Table 7
Error comparison of two ML models.

Table 8
Section Properties of 30 meters MVFT girder.

Table 9
Ultimate bending strength prediction of 30 meters MVFT girder.

Table 10
Ultimate bending strength prediction of 40 meters MVFT girder.