Multi-objective design framework under uncertainties for strengthening tubular truss by partially filling with grout

Abstract

Due to the growing needs of strengthening steel tubular truss, a new method for enhancing tubular joints by partially filling the chord with grout is proposed. However, the strengthening design of a whole truss is a challenging task, mainly because of multiple design objectives and various fabrication uncertainties. Current practice based on empirical or simple rule-based strategies is not able to handle the task. To address this challenge, a design framework for tubular truss strengthening is developed. The proposed framework can reduce the maximum deflection, improve the load capacities of the truss, and minimize the usage of grout. Furthermore, it considers geometric and modeling uncertainties through Monte Carlo simulation and predict intervals, thereby preventing over-idealization during practical optimization. To demonstrate the proposed design framework, a comparative structural analysis was conducted on a typical Warren truss between pre- and post- optimal strengthen. The results show that, by building upon the Machine Learning models, the proposed framework can produce an effective strengthening scheme. After considering uncertainties in optimization, some idealized samples are filtered out, resulting in a more practical strengthening scheme. The proposed framework is versatile and can be applied to other similar optimal strengthening designs with minimal additional effort.

Graphical abstract

Keywords

tubular truss strengthening / design framework / machine learning model / fabrication uncertainties / structural analysis

1 Introduction

Steel tubular structures built in early periods are increasingly overburdened by aging, natural disasters, and/or increasing demands (e.g., load increase due to growing population), making it necessary to strengthen the key parts, particularly joints. Traditionally, the joints are strengthened by welding stiffeners [1–5] or gluing Carbon Fiber Reinforced Polymer (CFRP) [6–9] to improve load capacity. However, the above-mentioned strengthening methods each have their unique disadvantages. For example, it is difficult to ensure the quality of construction of prefabricated stiffener plates; while the bonding performance of CFRP is often unstable due to variability in adhesive effectiveness. Inspired by Concrete-Filled Steel Tube (CFST) structures [10–25], the authors proposed a new method for strengthening steel tubular joints by partially filling the chord with grout [26,27].

How to design the most efficient strengthening scheme is a challenging task to engineers, particularly achieving multiple objectives simultaneously, such as minimizing deflection [28], maximizing load capacity, and reducing material usage. The current state-of-the-art solution relies on calculating the internal forces of joints using empirical formulas. However, this method is time-consuming and cannot capture the overall response of the truss. To address this limitation, multi-objective optimization algorithms are considered in this study, including Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) [29,30], Multi-Objective Particle Swarm Optimization [31–33], and Evolutionary Multiobjective Optimization Algorithms (SMS-EMOA) [34]. Among these, NSGA-II maintains diversity through fast non-dominated ranking and crowding distance, ensuring efficient convergence and global search performance. Wang et al. [35], Leyva et al. [36], Bakhshinezhad and Mohebbi [37] applied NSGA-II to shape and size optimization of large free-form spatial structures and seismic optimization design of various structures respectively. Grubits et al. [38] applied bi-directional evolutionary structural optimization to get the optimal structural topology of steel I-beams, showing a more comprehensive understanding to the plastic-limit behavior I-beams. Grubits et al. [39] also applied genetic algorithm to accurately evaluate the structural performance of bolted T-stub configurations. To avoid the time-consuming process of nonlinear finite element (FE) analysis in the design problem, Fang et al. [40] proposed a design framework that combined machine learning (ML) and NSGA-II, leveraging pre-trained ML models to replace FE models. However, to the author’s best knowledge, though ML has been used to predict load capacity calculation of tubular structures [41–43] or design of structure [44–47], no existing studies are found in the literature to predict and design strengthening steel tubular structures, not to mention about using ML and NSGA-II algorithms.

One of the most critical gaps herein to apply the above strategies for strengthening structures is the presence of various uncertainties, e.g., the dimensions of existing structures are not exact the same as initial design values. To fill in the gap, this study considers uncertainties of the features to avoid overidealization. These uncertainties typically arise from the construction and modeling processes. On one hand, construction uncertainties arise from variations in the parameters of existing structures (e.g., the thickness of steel tubes) and the strengthening materials (e.g., grout strengthening length (SL)), which are often neglected or avoided by conservative design for the sake of simplification. On the other hand, errors in the modeling process (whether in FE models or ML models) are inevitable, making it necessary to account for these uncertainties in practical analysis [48]. To simulate such uncertainties, statistical techniques such as the Gaussian Process [49], adaptive sampling [50], and Monte Carlo simulation [51] can be employed. However, to date, no existing studies have been identified to consider different types of uncertainties for multi-objective optimization of steel truss strengthening.

For the first time, a design methodology is proposed for strengthening steel tubular trusses (by partially filling the chords with grout) by considering uncertainties and combining ML models with NSGA-II, and this is demonstrated by means of a case study. Specifically, this study introduces a new joint-strengthening method first and then present the proposed design framework. Extensive FE nonlinear analyses are further conducted to build a database for a typical Warren steel truss structure. Based on the database, several ML models are trained to predict the deflection and load capacity of the strengthened truss, while also assessing the uncertainties in the analyses. Finally, NSGA-II is employed to optimize three objectives from the joint-strengthened truss solution considering a probability constraint to obtain the best scheme, and the influence of uncertainties on the three objectives are investigated.

The contributions of this study include: 1) to the best of author’s knowledge, this is the first comprehensive design strategy developed for strengthening tubular structures; 2) the proposed design strategy addresses various uncertainties in the strengthening scheme of existing structures by employing Monte Carlo simulation; 3) the proposed design framework has been validated through a case study of a truss structure.

2 Strengthening steel tubular truss joints by partially filling with grout

This section provides a brief review of the new strengthening method proposed by our team. The method has been published in Refs. [26,27], but there is still limited research on the overall strengthening optimization strategy for trusses in practical engineering. Figure 1 takes a T-joint as an example to illustrate the comprehensive flowchart. A detailed description of the strengthening process is as follows.

Step 1: To strengthen a circular hollow section (CHS) T-joint, it is necessary to cut two plate holes, four locating holes, one filling hole and one air hole in the chord. The first two types of holes are used to insert and fix the circular steel plates (CSPs), while the remaining holes are used for the subsequent filling process with materials.

Step 2: The placement of CSPs can be divided into three sub-steps as shown in Fig. 1, including: 1) inserting the CSP into the chord; 2) pinching one end of the screw and rotating the CSP by 90 degrees; and 3) inserting two screws to tighten onto the CSP.

Step 3: To avoid the leakage of filling materials, the gap between the CSP and the inner wall of the chord tube is filled with glass glue.

Step 4: First, the filling material is mixed in appropriate proportions, then placed in a funnel and pumped into the joint using a screw rod connected to an electric drill that slowly dispense the material. The inside of the chord is considered to be filled once the filling material spill out of the air hole.

Step 5: After the grout solidifies, all the holes are welded shut.

3 Design framework for strengthening steel tubular joints

This section provides a brief summary of the proposed design framework for strengthening steel truss joints by partially filling with grout. Figure 2 illustrates the entire process, which can be divided into four different parts: data acquisition from parametric FE Models (FEM) simulations, ML model training and explanation, uncertainties quantification, and design with NSGA-II considering the uncertainties. The details of each part are described in the following subsections.

3.1 Data acquisition and machine learning models

Data acquisition is fundamental to the subsequent steps in ML model training. FE simulations are conducted to obtain the truss response under various strengthening schemes and load combinations. To manage computational complexity and avoid an excessively large sample size, this framework adopts the Latin Hypercube Sampling (LHS) technique [52]. LHS ensures that each feature is thoroughly covered across its defined range by obtaining sample values from the tails, even with a limited number of samples. Furthermore, the features conform to a normal distribution, which enhances the efficacy of the ML model and contributes to more accurate results.

The success and performance of ML models depend heavily on the quality of the data and the selection of features. Therefore, it is essential to focus on feature engineering before training the ML models. In this study, interval scaling is employed, which transforms the data into a standardized range, typically between 0 and 1, based on the maximum and minimum values of each feature. For truss strengthening problems, several key features are considered, including section properties, span length, shape characteristics, materials, grout SL, Load Magnitude (LM), and internal forces acting on the truss. As the framework is demonstrated using a specific truss configuration, only a subset of SL and LM is adopted in the model training.

ML is a powerful technique that involves the use of algorithms and mathematical models. In this study, the features are defined as SL and LM, and the objective is to predict new data based on these inputs. Therefore, supervised learning algorithms are employed. There are many existing supervised ML algorithms, such as Linear Regression [53], Ridge Regression [54], Decision Tree algorithms [55], Naive Bayes, K-Nearest Neighbor (KNN), Support Vector Machines (SVM) [56], Artificial Neural Network [57], and Ensemble Algorithms (Gradient Boosting Decision Tree (GBDT), eXtreme Gradient Boosting (XGBoost), Light Gradient Boosting Machine (LightGBM), etc.) [58]. In this study, Ridge, KNN, Decision Tree, GBDT, XGBoost, and LightGBM are selected for training. The first three algorithms are single learning models, while the latter three are ensemble learning models.

To quantitatively evaluate the prediction performance of different ML models, five evaluation indices are employed: Coefficient of Determination (R²), Root Mean Square Error (RMSE), Mean Absolute Percentage Error (MAPE), Mean Absolute Error (MAE), and Coefficient of Variation (COV). R² indicates how well the predicted results fit the true value. The closer the result is to 1, the higher the reliability. RMSE is a measure of the deviation of the predicted value from the true value. Its result is greater than zero, and the closer it is to zero, the smaller the deviation of the training results. MAPE is sensitive to relative errors and does not change owing to the global scaling of the true value. MAE avoids the problem that overestimation and underestimation errors can offset each other, thereby it can fairly reflect the actual forecast error. COV is used to evaluate the discreteness of different models.

(1)\begin{array}{rl}& R^2=1-\frac{{\displaystyle \sum _{i=1}^n{\left(y_i-{\hat{y}}_i\right)}^2}}{{\displaystyle \sum _{i=1}^n{\left(y_i-{\overline{y}}_i\right)}^2}},RMSE=\sqrt{\frac{{\displaystyle \sum _{i=1}^n{\left(y_i-{\hat{y}}_i\right)}^2}}{n}},\\ & MAPE=\frac{1}{n}\sum _{i=1}^n\frac{\left|y_i-{\hat{y}}_i\right|}{y_i}\times 100\mathrm{\%},\\ & MAE=\frac{1}{n}\sum _{i=1}^n\left|y_i-{\hat{y}}_i\right|,COV=\sqrt{\frac{{\displaystyle \sum _{i=1}^n{\left(y_i-{\hat{y}}_i-{\overline{y}}_i\right)}^2}}{n}}/{\overline{y}}_i\end{array},

where y_i is the true value, {\hat{y}}_i is the predicted value, {\overline{y}}_i is the mean value of the true value, and n is the amount of data. To identify the best combination of hyperparameters, a 5-fold cross-validation grid search technique is used. The grid search is an exhaustive search for tuning parameters, as illustrated in Fig. 3.

Although the ML models with integrated algorithms demonstrate high training accuracy and a good fit, their internal computational principles and processes often lack interpretability. To address this problem, the SHapley Additive exPlanations (SHAP) [59] technique can be used. SHAP offers a theoretically sound method to quantify each feature’s contribution and provides the importance of features to a ML model’s prediction, based on cooperative game theory. It enables consistent, model-agnostic interpretation of complex models such as LightGBM employed in this study. This method interprets the prediction results as the contribution of each input feature to that output and provides the importance of features. Specifically, SHAP interprets the prediction results as the additive contribution of each input feature to the output, viewing each feature as a player in a cooperative game and the prediction as the payout for completing a task. It interprets the predicted value of the model as a linear function of binary variables:

(2)g(\mathrm{z})={\varphi }_0+\sum _{j=1}^M{\varphi }_j{\mathrm{z}}_j,

where g is the ML model to be explained, z indicates whether the feature is observed (0 means not observed, 1 means observed), M is the number of input features, {\varphi }_j is the SHAP value of each feature (the contribution of the features to the predictions), and {\varphi }_0 is the mean value of all predicted values. SHAP values are obtained by weighting and summing all possible combinations of feature values:

(3)\begin{array}{rl}{\varphi }_i(val)=& \sum _{S\subseteq \{x_1,\dots ,x_n\}\setminus \{x_i\}}\frac{|S|!(p-|S|-1)!}{p!}\\ & \cdot \left(val(S\cup \{x_i\})-val(S)\right),\end{array}

where S is the subset of features in the model, \mathrm{val}(S)is the predicted value of the subset S, p is the number of features, and {\displaystyle \frac{|S|!(p-|S|-1)!}{p!}} is the weight of the subset S.

3.2 Uncertainties quantification

Construction uncertainties may lead to discrepancies between the actual SL and the expected designed SL. To comprehensively evaluate the risk associated with uncertainties, both the geometric uncertainties of SL and the modeling uncertainties of the ML model are considered. However, material uncertainty has not been explicitly included in this study. This consideration is based on three main reasons: 1) since this study focuses on an existing truss with fixed material properties, the primary uncertainties arise from the strengthening process; 2) geometric imperfections and modeling assumptions have a more significant influence on the structural response; 3) material properties involve various variables, such as the yield strength and elastic modulus of each steel tube, as well as the compressive strength and elastic modulus of the grout, making the consideration of material uncertainty complex and deserving of a comprehensive study in future work. Therefore, only modeling and geometric uncertainties are considered in this study. The combination of geometric and modeling uncertainties ensures a more robust and accurate evaluation of the overall risk associated with the optimization framework for strengthening. By addressing and accounting for these uncertainties, this study aims to provide a more practical and reliable approach for optimization and mitigating potential risks in real-world applications. Geometric uncertainties of SL are assessed through Monte Carlo simulation. Specifically, the geometric uncertainties are regarded as variations in SL, involving decrease or increase within a range of {\xi }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\times d_0, where d₀ is the diameter of chord and the coefficient {\xi }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}} follows a normal distribution. The discrepancies between the designed values and the actual values are attributed to as geometric uncertainties. Modeling uncertainties are evaluated using Confidence Intervals (CI) or Prediction Intervals (PI) [60]. In this study, PI is selected to consider the modeling uncertainties:

(4)\begin{array}{rl}& P{I}_{\mathrm{l}}=\hat{y}-t_{\alpha /2,v=n-2}\cdot \hat{\sigma }\cdot \sqrt{1+\frac{1}{n}+\frac{{\left(x_0-\overline{x}\right)}^2}{\left(n-1\right)s^2}}\\ & P{I}_{\mathrm{u}}=\hat{y}+t_{\alpha /2,v=n-2}\cdot \hat{\sigma }\cdot \sqrt{1+\frac{1}{n}+\frac{{\left(x_0-\overline{x}\right)}^2}{\left(n-1\right)s^2}}\end{array},

where \hat{y} represents the linear regression result of x₀, t_{a/2,v = n–2} is defined as the value of the degree of freedom of n – 2 at the significant level α that conforms to the t-distribution, \hat{\sigma } is the standard residual error of the regression, \overline{x} and s² are the mean and variance of x, respectively. In this study, the original data are denoted as y, and the predicted data by the ML model are denoted as x. Based on these, a linear regression model is trained using ordinary least square [61], as shown below:

(5)\begin{array}{rl}& \hat{y}=a+bx,\\ & b=\frac{{\displaystyle \sum _i^n\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)}}{{\displaystyle \sum _i^n\left(x_i-\overline{x}\right)}}\\ & a=\overline{y}-b\overline{x},\end{array},

where \overline{y} and \overline{x} are the average values of original data and predicted data, respectively, a and b are both the coefficients of linear regression.

The combination of Monte Carlo simulation and PI can consider both geometric uncertainties and modeling uncertainties at the same time. When uncertainties are considered, each Monte Carlo simulation generates a set of objectives with deviations resulted from these uncertainties. At this stage, the PI serves as a threshold to determine whether the objectives fall within the PI range. A probability is then calculated to assess whether the result from Monte Carlo simulation is accepted. Only the acceptable results are included in the Pareto solution. In other words, when calculating each target data, the uncertainty probability is determined using Algorithm 1. This means that each of n samples generated through Monte Carlo simulation will have a probability (m) of falling within a specific limit. The overall uncertainty probability P(x) is then calculated as the sum of these probabilities across all n samples. The upper limit U_i and lower limit L_i are defined based on actual project requirements.

3.3 Multi-objective design considering uncertainties

NSGA-II is a genetic algorithm commonly used in multi-objective optimization and design problems. The algorithm process includes population initialization, non-dominated sorting, crowding distances calculation, individual selection, and crossover and mutation. Figure 4 illustrates the specific NSGA-II process as applied to the design of the truss strengthening scheme.

Initially, the population is generated randomly, and the parent population Q_t is fused with the offspring population F_t, and their corresponding fitness values are computed. Subsequently, non-dominated sorting and crowding distance calculations are performed. The former directs the search direction toward the Pareto front, while the crowding distance helps maintain population diversity by favoring solutions in less crowded regions. The concept of non-domination means that the two solutions x₁, x₂ and the corresponding fitness functions f_i(x₁), f_i(x₂) satisfy the following equations:

(6)

Assuming an individual in the population is not dominated by any others, it is considered as one of the best results of the current generation, referred to as the Pareto front. After removing the first group, the sorting is repeated for the remaining individuals to identify all subsequent Pareto fronts. The crowding is determined by calculating the distance between different individuals and giving preference to those with larger distances. This method ensures that the computation results are evenly spread across the target dimension, thereby preserving population diversity. To consider the uncertainties in optimization, the Pareto front will be filtered. The uncertainty probability of each sample in the Pareto front will be calculated considering the geometric uncertainties and the modeling uncertainties, and the probability constraint will be set. When the calculated uncertainty probability falls below the confidence level λ (90% or 95%), the likelihood of the optimization outcome occurring becomes minimal. Consequently, corresponding samples are removed from the data set. The crossover, mutation, and selection strategies are important to ensure the efficiency and accuracy of the search. The uniform crossover method is used to perform the crossover, with a crossover probability of 0.5, which is suitable for large populations and good for exploration capabilities. The mutation calculation method [40] is expressed as:

(7)f^{\prime }=\{\begin{array}{cc}f+k_{\mathrm{m}}\left(R-s\right)\cdot \gamma ,& r_{\mathrm{d}}=0,\\ f-k_{\mathrm{m}}\left(s-L\right)\cdot \gamma ,& r_{\mathrm{d}}=1,\end{array}

where f is the original feature, f^{\prime } is the mutated feature, L and R are the lower and upper boundaries of the feature range, respectively, r_d is a random number of 1 or 0, k_m is mutation coefficient, which is set to 0.9, and γ is a random number between 0 and 1. Employing a tournament-based approach, individuals are selected and subjected to the crossover process. To balance the efficiency and diversity, the tournament size is set to 3. This method allows the offspring to inherit certain characteristics from their parents, preventing them from being identical. Simultaneously, by introducing variation, the uncertainties of the algorithm are amplified, reducing the risk of becoming trapped in local optima. In summary, this process provides effective solutions for multi-objective decision-making, design, and control. The Pareto set represent the optimal solution identified by NSGA-II that satisfy the design requirements.

4 Demonstration of the design framework and results discussion

4.1 Structure description and database generation

In this study, the strengthening optimization is conducted on a typical Warren truss [62], with a total span of 36m and hinged at both ends (Fig. 5). The truss structure consists of members from four different section types, arranged symmetrically. It includes a total of 13 joints, 11 K-joints and two Y-joints, all of which are considered for strengthening.

The FEMs of the truss are developed by using Python and ABAQUS to establish a sample database for ML. Figure 6 presents the meshes, elements, and boundary conditions of the FEM. Randomly varying loads, following a normal distribution pattern, are applied to five joints of the upper chord. Additionally, the FEMs consider the overall self-weight load, with steel and grout densities of 7850 and 2500 kg/m³, respectively. To represent the truss members, 4-node reduced integral shell element (S4R) and 8-node brick elements with reduced integration (C3D8R) are employed for modeling steel and grout, respectively. To balance the calculation efficiency and precision, the mesh size of steel and grout are both set to 25 mm based on a sensitivity analysis. The effects of large deformation are considered in the iteration. The contact between the steel and grout is characterized by “Hard” normal contact and the friction coefficient μ between the tangential contact is set to 0.6 [63]. In addition, the braces and chords are connected using “Tie” contact to simulate their interaction. The material properties used for the steel was Q355 grade, with a modulus of elasticity E_s of 2.06 × 10⁵ MPa and a Poisson’s ratio ν of 0.3. The nonlinearity is considered in the FEMs: The bilinear model is adopted to simulate the stress-strain behavior of the steel material, as shown in Fig. 7.

The stress−strain relationship of the grout was referenced from Han [63], with a strength grade of C60. The compression stress−strain relationship of the Concrete Damage Plasticity (CDP) model is as follows:

(8)y=\{\begin{array}{ll}2x-x^2,& x\le 1,\\ {\displaystyle \frac{x}{{\beta }_0{(x-1)}^2+x}},& x>1,\end{array}

where x = ε/ε₀, y = σ/f_{\mathrm{c}}^{\prime }, ε and σ are the strains and stresses of the grout, respectively, f_{\mathrm{c}}^{\prime } is the axial compressive strength of the grout, and ε₀ is the strain at the maximum stress of the grout. These variables are calculated as follows:

(9)f_{\mathrm{c}}^{\prime }=(0.76+0.2\mathrm{l}\mathrm{g}(f_{\mathrm{c}\mathrm{u}}/19.6))\cdot f_{\mathrm{c}\mathrm{u}},

(10){\epsilon }_{\mathrm{c}}=(1300+12.5f_{\mathrm{c}}^{\prime })\times {10}^{-6},

(11){\epsilon }_0={\epsilon }_{\mathrm{c}}+800{\xi }^{0.2}\times {10}^{-6},

(12){\beta }_0={\left(2.36\times {10}^{-5}\right)}^{\left[0.25+{\left(\xi -0.5\right)}^7\right]}{\left(f_{\mathrm{c}}^{\prime }\right)}^{0.5}\times 0.5⩾0.12,

where f_cu is the cube compressive strength of the grout and ξ is the confinement factor.

The energy failure criterion is used to consider the tensile relationship of the grout, which exhibits improved convergence in tensile softening performance. The elastic modulus E_c and Poisson’s ratio ν_c of the grout are set as 36640 MPa and 0.2, respectively. The dilation angle ψ, eccentricity e, ratio of biaxial to uniaxial compression strength f_b0/f_{\mathrm{c}}^{\prime }, ratio of the second stress invariant, and viscosity parameter are set to 30°, 0.1, 1.16, 0.666, and 0.0005, respectively.

To validate the FEMs established in this section, two types of experiments were selected: a plane Warren Truss (W-T model) [64] and a T-joint strengthened by partially filling with grout (T-joint model) conducted by the author [26]. The material properties of both FEMs used for validation are identical to those described previously. For the W-T model, the upper and lower chord ends were fixed, and a concentrated force was applied at the midspan of the upper chord. Additionally, the mesh size was set to 25 mm, which was identical to the previous presented FEMs. For T-joint model, both chord ends were pin-connection, and an axial force was applied to the brace. The mesh size of T-joint was set to 5 mm. The comparisons between FEM and test for both models validated the global response as well as the local behavior of the truss, providing comprehensive support for the reliability of the FEMs. Figure 8 exhibits the validation results of the FEMs, demonstrating good agreement between the FEM predictions and experimental load–displacement curves and failure modes.

To assess the effect of different combinations of load distribution, truss joint J-2 to J-6 were subjected simultaneously to randomly distributed loads ranging from 100 to 900 kN simultaneously. This broad load range enables the training of the ML models to predict behavior of the truss both in elasticity and plasticity stages. The SL varied within the range of 0.25 to 5.0 times the diameter of the chord. The selection of this range was carefully considered to ensure both practicality and effectiveness in improving the load capacity of the truss: SL values below 0.25 were ineffective in enhancing the load capacity, and the previous study [10,65] indicated that excessive SL values beyond 5.0 resulted in an unnecessary increase in the self-weight of the truss. Additionally, conflicts due to grouting of adjacent joints will not occur within the given constraints. In summary, to obtain the deflection for different SL under various load conditions, the SL and LM were varied. However, the structural pattern, the number of strengthened joints, the dimensions of chord and brace, the material properties, the boundary condition remained constant.

A total of 13410 simulations were conducted in the FE analysis, generating a database for training and testing the ML models. The data was split, with 70% of the simulation results designated for training and the remaining 30% for testing. The deflection data collected from five distinct loading points were consolidated into two primary Structural Indices (SIs) for the truss analysis. The first SI, DispMax, represents the maximum deflection observed among the five loading points. The second SI, LoadMax, signifies the ultimate load capacity coefficient of the truss under specific loading conditions. The determination of this ultimate load capacity is based on reaching a deflection threshold, where DispMax equals 1/50 of the span length [66]. The selected features include the SL of the 13 joints (J-1 to J-13) and the LM of the five loading points (L-2 to L-6). This resulted in a final set of 18 features for the ML model. For conciseness, solely the histogram and Q-Q plot of J-4 are displayed in Fig. 9, with all features demonstrating analogous outcomes. The data points on the Q-Q plot align linearly, suggesting adherence to a normal distribution.

Before training the ML model, a statistical analysis of the two SIs was also conducted. Figure 10 represents the distribution of the two SIs, indicating that DispMax values were primarily concentrated between 300 and 1700 mm, while LoadMax values range from 0.4 to 1.8. Both SIs exhibit distributions that are approximately normal, demonstrating a reasonably consistent pattern in the data. Furthermore, no samples with exceptionally large deviations were observed, indicating the relative stability of the data set.

4.2 Machine learning model training, optimization, and evaluation

Figure 11 illustrates the grid search results for DispMax as an example, with the corresponding hyperparameter ranges and selected values for each ML model summarized in Table 1. The results indicated that the Ridge model exhibited minimal change in MSE value, with the influence of different hyperparameters remaining within 0.1. For the KNN model, the hyperparameter n_neighbors had the most significant impact on MSE value, with the minimum MSE achieved at n_neighbors = 5. Both min_samples_leaf and min_samples_split significantly influenced the MSE in the Decision Tree model, with min_samples_leaf having a more pronounced influence. The minimum MSE was obtained when min_samples_leaf = 10. The hyperparameter max_depth, max_depth and learning_rate played a vital role in influencing MSE value in training GBDT, XGBoost and LightGBM model, respectively. Larger values of max_depth and smaller values of the learning_rate resulted in the trained model having a smaller MSE value. Overall, the relationship between MSE and hyperparameters was not monotonic for most models, highlighting the importance of grid search for identifying the optimal hyperparameter combination.

Table 2 presents the errors of training two SIs with different ML models, demonstrating that the three integrated algorithms GBDT, XGBoost, and LightGBM all had smaller errors and better training performance. The coefficient of determination values for these models were all greater than 0.978, indicating a good agreement between predicted and true values. By contrast, the Ridge and KNN model performed poorly in predicting LoadMax, with R² values of only 0.719 and 0.684, respectively. Among all models, LightGBM exhibited the highest prediction accuracy for both DispMax and LoadMax. Therefore, the algorithm was selected, and the prediction models would be used for the following analysis. Figure 12 illustrates the correlation between the predicted and true values for the models trained by the corresponding ML algorithms based on the two SIs. This illustrated that the ML models exhibited excellent fit and accuracy in predicting the SIs.

Figure 13 illustrates the distribution of the top ten SHAP values obtained from the LightGBM model for two SIs. A large SHAP value indicates greater feature importance. Notably, features L-2 to L-6 and J-3 to J-5 showed significant influence on the DispMax model. However, the SL of other joints showed negligible influence, with SHAP values below 200. Features L-2 to L-6 exhibited positive correlations, meaning that increasing these loads led to higher maximum deflection. In contrast, J-3 to J-5 showed negative correlations, indicating that increasing their SL reduced DispMax. Among the SL features, J-4 had the largest absolute SHAP value, suggesting that strengthening J-4 was most effective in reducing DispMax. In addition, the SHAP values distribution of SL features were symmetric, demonstrating that increase or decrease in these features contributed similarly to for DispMax. However, the load features had a more significant effect when increased compared to when decreased. The absolute SHAP value rankings for the LoadMax model are reversed compared to those of the DispMax model. The value of features J-3 to J-5 exhibited positive correlations with LoadMax, indicating that increasing joints J-3 to J-5 can effectively enhance the ultimate load capacity of the truss. These findings from SHAP analysis results provide valuable guidance for designing the strengthening scheme, especially from a physics standpoint.

4.3 Probabilistic safety risk analysis and assessment

The previous analysis demonstrated that the truss was primarily affected by the upper chord joints J-3 to J-5. To account for the geometric uncertainties, the SL of these three joints was varied in a Monte Carlo simulation with 5000 times (n = 5000) and a significance level of α = 0.05, based on 200 samples in the database. Figure 14 shows the relative and cumulative frequency distributions of the two SIs after considering geometric uncertainties. A notable observation is that, compared to DispMax, the distribution of LoadMax more closely approximates a normal distribution, with data points being more concentrated. Furthermore, the COV of DispMax is larger than that of LoadMax, indicating that DispMax is more sensitive to geometric uncertainties. This result is consistent with the findings from the preceding SHAP analysis.

Figure 15 presents the results of the Monte Carlo analysis with and without uncertainties. The solid black line represents the mean value of the Monte Carlo simulation, while the shaded area indicates the upper and lower limits of the PI. Triangles and circles denote the true and predicted values, respectively. The two SIs exhibited a greater number of samples falling outside the PI range, indicating that geometric uncertainties had a more pronounced effect than modeling uncertainties. These findings highlight the importance of accounting for geometric uncertainties in the analysis of truss strengthening schemes.

4.4 Multi-objective design with uncertainties

The objective of optimization is to obtain a balance between SL and the performance of the truss. A low value of DispMax and a high value of LoadMax indicated that the strengthening effectively reduced the deflection and improved the load capacity of the truss. In addition, to control economic cost and limit the increase in self-weight, the total grout consumption should be reduced as much as possible. The grouted usage is defined as the sum of all SL. By balancing these objectives, the proposed design framework can provide an optimal and practical solution for strengthening truss joints by partially filling with grout.

The combination of ML and NSGA-II methods offers significant advantages, particularly in reducing the computational time required for the entire design process. By using ML models to predict the structural response instead of relying on FE simulations for each individual in the NSGA-II population, the design process became significantly efficient. To illustrate this advantage, a population size of 300 individuals and 200 iterations for the NSGA-II algorithm was assumed. If the response of each individual needed to be calculated by ABAQUS, it would take around 120 to 150 s per individual (based on a 2.10 GHz Intel i7-12700F processor), corresponding to the time required for each truss FE analysis. Without ML assistance, each population and generation require FE calculation, meaning 60000 FEMs would need to be calculated, taking at least 83 d. However, by employing ML models for predicting the structural response, the time needed for each individual became negligible. Once the FEM database was generated and the ML models were trained (requiring about 19 d), the FE calculations were replaced by rapid ML predictions. The entire optimization process took about 45 mins, achieving a 75% reduction in overall computational time. Additionally, the two-stage analysis, involving the training of the ML model in the first stage and the optimization in the second stage, provided flexibility in adjusting the algorithm process and debugging parameters. This approach allowed for easier fine-tuning and design of the overall framework, thereby improving its efficiency and effectiveness.

Based on the optimization method described earlier, the truss strengthening scheme was designed. The fitness of the target was obtained by calculating the value of each objectives under two loading combinations: fully uniform load and half-fully uniform load, as shown in Fig 16. The LM and load combination coefficients were determined according to Ref. [67]. The dead load and live load were assumed to be 1.5 and 2 kN/m², respectively. The wind load, based on the assumption of the truss being located in Shanghai, China, was determined as 0.6 kN/m².

The design process involved initializing 300 individuals per population (n_r = 300) and performing 200 (m_g = 200) iterations. The design results were visualized through three corresponding projection maps for analysis, as shown in Figs. 17 and 18. Several key observations can be drawn from the analysis: after mutiple iterations, the populations gradually converged, and each objective was significantly optimized, resulting in the formation of a Pareto front under both load combination. It also indicates that this design framework is suitable for various load combinations. The design results indicated that the ranges of DispMax, LoadMax and Usage (total grout consumption) were concentrated between 200 to 400 mm, 2.0 to 2.4, and 10⁶ to 8 × 10⁶ mm³, respectively. A trade-off relationship was observed among the three objectives. Notably, increasing the amount of grout effectively reduced the DispMax and increased the LoadMax. Moreover, DispMax and LoadMax exhibited an approximately linear relationship.

To comprehensively consider uncertainties, the Monte Carlo simulation was conducted 5000 times. The upper limit U and low limit L were determined by the hollow section truss and fully grouted truss, respectively. This means that the minimum LoadMax and maximum DispMax were determined by the hollow section truss, while the maximum LoadMax and minimum DispMax were determined by the full-grouted truss. In addition, a robustness factor (1.5) was considered to avoid over-constrained and enhance optimization robustness. After considering the uncertainties, some solutions on the Pareto front were filtered out. As the significance level increased, the number of retained solutions decreased, with the remaining individuals becoming concentrated within a narrower range. For example, under load combination-1 and at a significance level of 0.1, DispMax, LoadMax, and Usage were concentrated within 225 to 275 mm, 2.1 to 2.3, and 2 × 10⁶ to 6 × 10⁶ mm³, respectively. Compared to deterministic results, which led to excessive reinforcement and inefficiency, accounting for uncertainties alleviated this issue. Furthermore, for the deterministic outcomes, the Pareto-optimal set contained 98 solutions. When the significance level was set to 0.1, the number of Pareto optimal solutions decreased slightly to 95. However, as the significance level decreased to 0.05, the number dropped further to 43, meaning that only 45% of the original solutions remained. This filtration process helped to identify more practical and robust solutions that explicitly considered uncertainties in the design.

For an elucidation of the variation in the three SIs following design, Fig. 19 shows the improvement percentage of average value of the Pareto front under both Load combination-1 and 2. Under Load combination-1, the improvement percentages for the deterministic and 10% PI results showed a reduction in Usage. However, this reduction was accompanied by an increase in DispMax and a decrease in the LoadMax, which were undesirable for design objectives. Conversely, the 5% PI results effectively reduced the Usage while avoided an increase in DispMax and a decrease LoadMax. For Load combination-2, as the significance level decreased, the reduction in DispMax and Usage became more pronounced, along with a corresponding increase in LoadMax. Consequently, the Pareto front, which accounted for uncertainties, offered a more efficient and rational solution. Figure 19 also exhibits the specific improvement metrics achieved after optimization compared to the original truss without optimization. For Load combination-1, DispMax and Usage decreased by 0.2% and 20.3%, while the LoadMax increased by 0.1%. For Load combination-2, DispMax and Usage decreased by 7.1% and 6.9%, and the LoadMax increased by 2.0%. Additionally, the design strengthening schemes under different load combinations are shown in Tables 3 and 4 including the recommended SL of each joint. The final optimal design solution was selected from the Pareto front, featuring relatively lower Usage and DispMax, and higher LoadMax. In practical applications, the specific solution can be chosen based on the design requirements.

5 Conclusions

This study proposed a design framework for strengthening truss joints by partially filling with grout while considering uncertainties. Several ML models were trained based on the database generated from numerous FE analyses to accelerate the subsequent design process. The effects of each feature and the influence of uncertainties on the truss response were investigated. By combining the ML models and NSGA-II, an optimal strengthening solution was obtained with uncertainties taken into consideration. The proposed design framework was further demonstrated using a typical Warren truss and the deformation of the truss before and after strengthening was compared. Based on the results, several key conclusions were drawn.

1) It was verified that the selected ML models can accurately and efficiently predict the maximum deflection (DispMax) and ultimate load capacity (LoadMax) of the strengthened tubular truss structures under different strengthened length and load magnitude.

2) The SHAP analysis result demonstrated that both DispMax and LoadMax were highly sensitive to features L-2 to L-6 and J-3 to J-5. Joints with significant influence on performance of truss should be prioritized for strengthening.

3) The effect of geometric uncertainties for both DispMax and LoadMax was greater than that of modeling uncertainties. Geometric uncertainties deserve more attention in practical applications.

4) NSGA-II effectively optimized the strengthening scheme of the truss, achieving simultaneous improvement in all objectives.

5) After considering the uncertainties, over-idealistic solutions were filtered out, making the strengthening scheme that is more practical and applicable to engineering practice.

6) The proposed design framework is versatile and can be extended to other truss strengthening scenarios, such as strengthening with CFRP, without the need to rebuild the framework.

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