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A Physics-informed Neural Network Approach for Robust Buckling Load Prediction and Reliability-Based Design of Thin Truncated Conical Shells
Devasmit Dutta, Budhaditya De, Rohan Majumder, Sudip Kumar Mishra
TL;DR
Imperfection-sensitive buckling makes conventional deterministic knockdown factors limited for uncertainty-aware conical-shell design. This study develops a physics-guided neural surrogate with reliability-based design, achieving 97.1% test R2 and calibrating higher safety-consistent KDFs than conventional recommendations.
Problem
Axially compressed conical shells are highly imperfection-sensitive, while conventional deterministic KDFs do not explicitly reflect geometry, fabrication quality, uncertainty, or target reliability.
Method
The framework combines shell-geometry and material descriptors with LRSM-derived stability features and a loss constraint preventing predictions above the theoretical elastic buckling load.
Results
97.1% test R2, RMSE = 25.06 N, and MAE = 12.46 N were achieved, versus 90.6% R2, RMSE = 44.76 N, and MAE = 17.47 N for the DNN.
Takeaways & Limitations
PiNN-based KDFs were consistently higher than conventional recommendations while targeting prescribed reliability indices, indicating additional usable buckling capacity within the evaluated framework.
Takeaways & Limitations
The experimental database is limited in size and scope to Mylar conical shells.
Abstract
from arXiv · showhide
Thin-walled truncated conical shells are widely used in aerospace, marine, offshore, and lightweight infrastructure systems due to their high strength-to-weight ratio and geometric efficiency. Their buckling resistance under axial compression, however, is highly sensitive to geometric imperfections, manufacturing tolerances, material variability, and nonlinear instability effects. Conventional design procedures rely on conservative knockdown factors (KDFs), such as those recommended in NASA SP-8019, which do not explicitly account for shell geometry, fabrication quality, data uncertainty, or target reliability. This study develops a physics-informed neural network (PiNN) framework for predicting critical buckling loads of thin truncated conical shells and integrates the trained surrogate within a reliability-based design (RBD) formulation. The model combines geometric and material descriptors with mechanics-informed features derived from shell stability theory and the localized reduced stiffness method (LRSM). A physics-informed loss function penalizes mechanically inadmissible predictions exceeding the theoretical elastic buckling load. The framework is trained and evaluated using 133 experimental Mylar conical shell tests under axial compression. Compared with a conventional deep neural network (DNN), the PiNN improves predictive accuracy, reduces mean absolute error, and enhances physical consistency. The trained PiNN is then used to evaluate reliability indices and calibrate safety-consistent KDFs for prescribed target reliability levels. Results demonstrate that the PiNN-RBD framework provides an efficient approach for uncertainty-aware design of imperfection-sensitive shell structures.
1. Introduction
The introduction frames thin truncated conical shells as efficient but imperfection-sensitive structures whose axial-compression buckling challenges conventional design, motivating a physics-informed, reliability-based approach. The study combines mechanics-informed features and a constraint-based loss within a PiNN surrogate for robust buckling-load prediction.
- Motivation: Thin-walled shells provide high strength and stiffness at low structural mass, while truncated conical shells serve aerospace, marine, offshore, pressure-vessel, and lightweight infrastructure applications.Their design is often governed by stability rather than material strength.
- Motivation: Axial-compression buckling is highly sensitive to geometric imperfections, boundary-condition deviations, residual stresses, material variability, and nonlinear instability effects.Buckling can occur substantially below the theoretical elastic critical load.
- Existing design practice: Conventional shell design relies on empirical knockdown factors, including NASA SP-8019 recommendations for truncated conical shells, to address imperfection sensitivity.These factors provide a practical design basis for stability-related uncertainty.
- Methodological context: Physics-informed machine learning embeds physical and mechanical laws through model architecture, input features, governing-equation residuals, boundary constraints, or loss functions.Prior work has applied PiNNs to structural and shell-response prediction.
- Study contribution: The proposed PiNN predicts axial-compression buckling loads using geometric, material, and fabrication-quality descriptors plus shell-stability and LRSM features, while penalizing predictions exceeding the theoretical elastic buckling load.The trained surrogate is integrated into a reliability-based design formulation.
2. Physics-informed Machine Learning Model Architecture
The model is a physics-guided regression surrogate that combines shell-stability knowledge with experimental descriptors to predict axial buckling loads of thin truncated conical shells. Its architecture augments original features with LRSM-based indicators and uses a one-sided physics-informed loss enforcing the elastic buckling bound.
- Physics-informed surrogate: The PiNN learns experimentally observed axial buckling loads from shell geometry, material properties, fabrication quality, and mechanics-informed stability indicators.It is a physics-guided surrogate rather than a classical differential-equation-based PiNN solving governing equations over a spatial domain.
- Input features: The input vector combines β, L, R, t, E, ν, FTQC, ρLRSM, and NLRSM.FTQC represents fabrication-related surface imperfections, while ρLRSM is the LRSM-based knockdown factor relating realistic and theoretical buckling loads.
- Input features: LRSM supplies mechanics-informed features by locally reducing membrane stiffness to reproduce imperfection-sensitive membrane-stress redistribution and lower-bound buckling behavior.The ρLRSM relation was fitted from 290 LRSM-based nonlinear finite element simulations, and the resulting LRSM buckling quantities were appended to the original features.
- Training objective: The total objective combines data loss with a physics-informed penalty for predictions exceeding the corresponding theoretical elastic buckling load.The penalty is one-sided and quadratic above the elastic limit, while λ = 1 controls the physics-loss contribution.
- Training setup: The model was trained and tested on 133 Mylar truncated conical shell tests using an 80 : 20 data split.Optimization used Adam with a fixed learning rate of 1e−3 over 500 epochs and minibatch training.
3. Results and Discussion
The PiNN outperformed the conventional DNN in predictive accuracy, generalization, and mechanical consistency by incorporating LRSM-derived features and a physics-informed loss constraint. Its reliability-based design application produced safety-consistent KDFs that increased with target reliability requirements.
- Predictive performance: The DNN showed widening train–test RMSE gaps after convergence, whereas the PiNN demonstrated improved generalization and reduced overfitting.This behavior was identified from the RMSE loss histories across training epochs.
- Predictive performance: The PiNN achieved slightly better training performance than the DNN, with R2 = 0.9925, RMSE =15.74N, and MAE =9.32N versus R2 = 0.9916, RMSE =16.81N, and MAE =9.64N.The comparison used coefficient of determination, root mean square error, and mean absolute error.
- Prediction consistency: PiNN predictions were more tightly concentrated near Npred/Nexp = 1, while the DNN produced greater scatter, severe unconservative over-predictions, and large testing outliers.The comparison covered different shell slenderness levels through the radius-to-thickness ratio R/t.
- Feature sensitivity: The PiNN relied most strongly on NLRSM, which exhibited a strong, nearly monotonic positive relationship with centered predicted buckling load.NLRSM is described as a mechanics-informed lower-bound estimate of shell buckling capacity, while ρLRSM showed a weaker mild non-monotonic trend.
- Reliability-based design: The PiNN-based KDF curve increased from approximately 0.60 to 0.72 for shells with 15% coefficients of variation in average radius Ra and thickness t.The KDFs were obtained for different target reliability indices within the reliability-based design framework.
4. Conclusions and Future Scope of Research
The study developed a mechanics-informed PiNN for buckling-load prediction and reliability-based design of axially compressed thin truncated conical shells. It improved test performance and physical interpretability over a baseline DNN, enabled reliability-consistent KDF calibration, and motivates broader validation and stronger physics constraints.
- Contributions: The PiNN incorporated shell-stability knowledge through LRSM-derived features and a physics-guided surrogate formulation.The framework used the LRSM KDF, ρLRSM, and LRSM buckling load NLRSM as mechanics-informed descriptors.
- Predictive performance: 97.1% R2, RMSE = 25.06 N, MAE = 12.46 N, and 95% confidence interval coverage of 0.963 were achieved on the test dataset.The PiNN showed slightly improved training performance but substantially better testing performance than the conventional baseline DNN.
- Interpretability: The LRSM-based buckling load was the most influential feature, while shell thickness and semi-vertex angle retained meaningful sensitivity.Partial dependence results indicated that the PiNN learned a physically interpretable response structure rather than relying solely on empirical correlations.
- Reliability-based design: The PiNN-based KDFs were consistently higher than conventional code and literature-based recommendations while targeting prescribed reliability indices.The surrogate was embedded in a reliability-based design framework to recover additional usable buckling capacity while maintaining the desired reliability level.
- Future scope: Future research should expand beyond Mylar shells, incorporate measured imperfections and boundary-condition variability, and strengthen constraints based on governing equations, energy barriers, or nonlinear stability criteria.The current database is limited in size and scope and contains only Mylar conical shells.
Data Availability Statement
The study uses previously reported experimental shell buckling tests and derived physics-informed features. Processed data and trained model parameters may be available from the corresponding author upon reasonable request.
- The study’s data comprise previously reported experimental shell buckling tests and derived physics-informed features.
- Processed data and trained model parameters may be made available by the corresponding author upon reasonable request.