A machine learning approach for buckling analysis of a bi-directional FG microbeam

This study investigates the buckling analysis of a bi-directional functionally graded nanobeam (BD-FGNB) on a Winkler foundation through machine learning (ML) methodologies and semi-analytical solution based on Fourier series and Stokes’ transform. Buckling is investigated via nonlocal strain gradient theory that incorporates the effects of both nonlocal theory and strain gradient theory into the problem. The nonlocal strain gradient theory is employed to model the nanobeam and generate the dataset for training ten distinct ML models. The predictive capabilities of models are evaluated and the ML model with best predictive accuracy is identified by comparing their outcomes against analytical results. Results indicate the exceptional performance of the XGBoost (XGB) model in precisely predicting buckling loads while maintaining high computational efficiency. The R², MAE, and RMSE evaluation metrics demonstrate remarkable values of 0.999, 2.05, and 3.58, respectively, affirming the model's accuracy. Utilizing the SHAP approach, it is found that the foundation parameter has the highest impact on the initial buckling mode, while its impact reduces in subsequent modes. The results from SHAP are validated using the analytical solution where both approaches show that higher values of foundation and material length scale parameters increases the buckling load, however higher values of nonlocal parameter and material grading coefficient in y and z directions decreases the buckling load.