Machine learning can correct mechanics-based estimates of the global buckling capacity of built-up cold-formed steel columns, but a study of 148 laboratory tests shows that its apparent accuracy falls sharply when asked to predict columns from experimental programmes it has never seen. A random division of the data produced test mean absolute percentage errors of about 13–16% and coefficients of determination above 0.96, yet nested validation that held out entire source groups yielded much weaker results: the best knowledge-informed model, Lasso regression, achieved an R² of 0.787, a root mean squared error of 68.5 kN and a mean absolute percentage error of 73.0%. The results indicate that machine learning can interpolate effectively among related tests without reliably transferring to new testing campaigns or unfamiliar combinations of geometry, materials and connections.
The research focuses on a curated database of built-up cold-formed steel columns whose failure mode was classified as global flexural buckling. The subset contains 148 tests drawn from 15 experimental source identifiers and 13 built-up section families, with member lengths ranging from 502.7 to 3,500 mm, thicknesses from 1.00 to 3.00 mm, yield strengths from 287.5 to 615.0 MPa and measured peak loads from 7.5 to 753 kN. Most specimens had pinned boundary conditions: 134 of the 148 tests were pinned, eight semi-fixed and six fixed.
The records came from one open repository but not from a single laboratory programme, a distinction that became central to the validation strategy because specimens from the same source may share geometry, material batches, boundary conditions and connection detailing. The mechanics-based starting point was the Direct Strength Method, which links nominal axial resistance to elastic buckling calculations and is included in the ANSI/SDI AISI S100-24 and AS/NZS 4600:2018 specifications. Its global slenderness parameter separates inelastic and elastic buckling regimes at a value of 1.5.
For non-composite members, whose connected components are assumed to buckle largely independently, nominal resistance is calculated from the yield load and Euler elastic buckling load. A partial-composite formulation modifies member slenderness to reflect limited shear transfer through fasteners or welds. This added term depends on fastener spacing, the radius of gyration of the individual component and the radius of gyration of the complete built-up section, while an interaction criterion limits the spacing for which effective partial action is assumed.
Both formulations displayed substantial discrepancies from the experiments. The non-composite model generally underestimated resistance when substantial interaction was present, particularly around the intermediate-slenderness region near λc = 1.5, whereas the partial-composite model reduced systematic bias but retained wide dispersion. For the non-composite formulation, the mean ratio of measured to predicted resistance was approximately 1.40 and the coefficient of variation was about 52%.
The partial-composite mean was approximately 0.90, with a coefficient of variation still above 50%. These observations motivated a method that does not replace the mechanics calculation with an unconstrained black box, but instead asks machine learning to estimate a positive multiplicative correction for the residual model error. The resulting knowledge-informed machine-learning framework expresses measured resistance as the product of a Direct Strength Method prediction and a model-uncertainty factor, ΘR, which is equal to the ratio of experimental capacity to analytical capacity.
The algorithms learn the logarithm of this ratio, preserving positivity and dimensional consistency when the corrected capacity is reconstructed. Inputs include length, thickness, web height, flange width, cross-sectional area, fastener spacing, yield load, global slenderness, elastic critical load, derived geometric ratios, section family and boundary condition. Six algorithms were tested: Ridge, Lasso and Elastic Net regularised linear models, a decision tree, k-nearest neighbours and AdaBoost.
The framework is mechanics guided rather than physically constrained: it includes mechanics-derived variables and imposes dimensional and positivity structure, but it does not enforce monotonicity or require the regressors themselves to reproduce the governing equations. A conventional random 80/20 split divided the specimens into 118 training cases and 30 test cases and produced mean absolute percentage errors between 13.44% and 15.72%, with R² values from 0.966 to 0.975. That result describes interpolation within the assembled database, however, because related specimens from the same experimental programme could appear on both sides of the split.
The study therefore added nested leave-one-source-out validation: each of the 15 source groups was withheld in turn, and all its specimens were excluded from model development. Preprocessing, imputation, scaling, categorical encoding and hyperparameter tuning were confined to the remaining groups, with 25 tuning trials per model in each outer fold. A further robustness analysis repeated grouped holdout validation 30 times, each time withholding about 20% of source groups.
The source-independent results were markedly worse. In the nested analysis, decision trees achieved R² = 0.434, while the other knowledge-informed algorithms ranged from 0.025 to 0.345, apart from k-nearest neighbours at 0.152. Between 49.3% and 57.4% of the knowledge-informed predictions were unconservative, meaning that the predicted capacity exceeded the measured capacity, depending on the algorithm.
The maximum unconservative capacity ratio ranged from 6.37 to 10.65. Lasso had the strongest R², but its source-cluster bootstrap interval extended from approximately −1.79 to 0.89, while its mean absolute percentage error had an interval of about 18.7–147.6%. Across the 30 repeated grouped splits, mean R² values were negative for all six knowledge-informed models, and mean percentage errors ranged from 67.7% to 111.1%, with large variation between splits.
A matched comparison tested whether retaining the mechanics baseline was responsible for the performance. Direct machine-learning models predicted measured capacity from the same geometric, material, categorical and mechanics-derived predictors, while the knowledge-informed models learned a correction to the Direct Strength Method result. Both approaches used identical source-group partitions, and the direct models also received the non-composite Direct Strength Method resistance because the knowledge-informed method uses it explicitly.
Discrepancy learning substantially reduced source-independent percentage error for Ridge, Lasso and Elastic Net and modestly for the decision tree, but it did not improve mean absolute percentage error for k-nearest neighbours or AdaBoost. The advantage was therefore model dependent rather than a universal consequence of introducing a mechanics prior. Reliability calibration confirmed the difficulty of transferring the models.
Using only the concatenated out-of-source predictions, with each of the 148 specimens receiving one prediction from a model trained without its experimental source, the mean model bias ranged from 1.007 to 1.172 and the coefficient of variation from 0.552 to 0.898. The resulting Load and Resistance Factor Design resistance factors ranged from φc = 0.191 to 0.399, while corresponding Allowable Strength Design safety factors ranged from Ωc = 4.01 to 8.38. Explainability analysis also required caution: several mechanics-derived predictors were strongly correlated, including global slenderness with the ratio of yield load to elastic critical load, and grouped resampling showed that some feature rankings changed with the available sources.
SHAP values were therefore interpreted as associations within fitted models, not causal effects. Overall, the study supports combining mechanics-based resistance equations with structured machine-learning corrections, source-grouped validation, explainability diagnostics and reliability analysis. It does not validate unrestricted use of the resulting models across new experimental programmes.
Their demonstrated domain is limited to the represented section families, material strengths, lengths, thicknesses, fastener-spacing records, boundary conditions and global flexural-buckling cases; local buckling, distortional buckling and interacting instability modes were excluded and would require separate mode-specific data and validation. The central lesson is that high scores from random train–test splits can conceal substantial programme dependence, and that reliability calibration based on genuinely unseen sources reveals a much more demanding uncertainty profile than within-database interpolation alone.
Subject of Research: Knowledge-informed machine learning and reliability calibration for global buckling capacity of built-up cold-formed steel columns
Article Title: Reliability calibration of knowledge informed machine learning for global buckling capacity of built up cold formed steel columns
Article References: Mwanangombe, J., Sumbwanyambe, M., Poologanathan, K., Ikotun, B. D., & Simwanda, L. (2026). Reliability calibration of knowledge informed machine learning for global buckling capacity of built up cold formed steel columns. Discover Artificial Intelligence, 6(1), Article 1569. https://doi.org/10.1007/s44163-026-02504-5
Image Credits: AI Generated
DOI: 10.1007/s44163-026-02504-5
Keywords: cold-formed steel, global buckling, built-up columns, machine learning, reliability calibration, model discrepancy, source-grouped validation, structural engineering, explainable AI, uncertainty, Reliability, calibration
