Investigation of the critical buckling load of a column with linearly varying moment of inertia using analytical, numerical, and hybrid machine learning approaches

This study investigates the buckling behavior of columns with variable cross-sections using analytical, numerical, and hybrid machine learning (ML) approaches. Initially, the power series method is employed to calculate the buckling loads of columns with both constant and varying cross-sections under diverse boundary conditions. Then a finite element (FE) analyses of the columns are performed to obtain the buckling loads and the results are validate by comparing them with results from power series method. Once validated, the FE model is used to generate a large dataset encompassing a wide range of cross-sections, lengths, and material properties, as per the samples obtained through the Sobol sampling method. A hybrid ML model is then developed by integrating the XGBoost algorithm with the particle swarm optimization (PSO) technique for hyperparameter tuning. This hybrid PSO-XGBoost model is trained to predict the buckling loads of columns with varying cross-sections. Its performance for input parameters outside the training dataset is evaluated using statistical metrics and scatter plots. The results demonstrate excellent agreement between the FE analysis and the power series method, confirming the reliability of both approaches. The PSO-XGBoost model achieved remarkable predictive accuracy, with R 2 values of 0.999 and 0.996 for the training and testing datasets, respectively. Furthermore, SHapley Additive exPlanations (SHAP) analysis is conducted to explore the influence and interactions of input parameters on buckling loads, providing valuable insights into the model’s interpretability and the underlying mechanics of column buckling.