New data-driven model predicts shear behavior of grouped studs in composite bridges

Engineers designing steel–precast concrete composite bridges have long struggled with one deceptively small component: the humble headed stud. In modern bridges that combine steel girders with precast concrete slabs, these connectors cannot be placed continuously along the beam. Instead, they must be clustered in groups inside pockets that are later filled with high-strength concrete, and their behavior under load becomes far more complicated than that of a single stud. A new study published in Advances in Bridge Engineering tackles this problem with a data-driven modeling approach that produces explicit, engineer-friendly equations for the entire nonlinear shear force–slip relationship of grouped studs.

The research team, led by Viet Hai Do and Hoang Nam Phan, compiled an experimental database of 249 push-out tests drawn from 24 previous studies, covering 19 input parameters that describe the geometry and materials of the studs, the steel beam, the cast-in-place concrete in the pockets, and the surrounding precast concrete. The dataset spans normal-strength, high-strength, and ultra-high-strength concretes, with compressive strengths ranging from 25 to 150 megapascals, stud diameters from 8 to 30 millimeters, and groups of 2 to 12 studs arranged in 2 to 4 rows. Only conventional push-out configurations with an H-shaped steel section and two concrete slabs were included; modified setups, pull-out tests, and tests under cyclic, dynamic, or elevated-temperature loading were excluded to keep the load-transfer mechanisms consistent.

Before building their own model, the researchers evaluated the existing tools of the trade. Design codes such as Eurocode 4, AASHTO LRFD, JSCE, and GB50017, along with empirical formulas from earlier studies, were compared against the compiled experimental data. The results were sobering: correlation coefficients for shear capacity predictions ranged only from 0.655 to 0.711, and most formulas underestimated the measured capacity of grouped studs, with regression slopes between 0.50 and 0.89. The authors attribute this to the fact that code-based equations were developed primarily from tests on individual studs in normal-strength concrete, and their validation datasets were typically small, often fewer than 50 specimens with a narrow parameter range.

The picture was even worse for slip behavior. Ultimate slip predictions from existing models showed essentially no meaningful correlation with the experimental data, because some models treat ultimate slip as a constant while others tie it only to stud diameter. When the researchers assessed initial stiffness, defined as the slope from the origin to the point at one-third of the shear capacity, they found that some models underestimated it by half or more while others overestimated it by up to 16 times, with standard deviations between 0.4 and 2.3 indicating wide dispersion. Since the serviceability limit state in Japanese design practice is defined by whether connectors return to their original position after unloading, inaccurate slip predictions directly undermine rational fatigue-resistant design.

To close these gaps, the team adopted symbolic regression rather than black-box machine learning. While artificial neural networks and support vector machines have achieved high accuracy in predicting single-stud capacity, their lack of transparency makes them difficult for practicing engineers to use in iterative design calculations. The new approach assumes a first-order multivariate polynomial structure, determines coefficients by ordinary least squares, and automatically discards coefficients below 0.00001 as negligible. The result is a set of four explicit equations predicting shear capacity, ultimate slip, and the slips at one-third and two-thirds of capacity, each expressed in terms of the physical design variables that engineers actually control.

The predictive performance marks a clear improvement. The shear capacity model achieved coefficients of determination exceeding 0.86 on both training and testing subsets, with regression slopes close to unity and calculated-to-experimental ratios concentrated around 1.0, with a mean of about 1.03, a median of 0.99, and a standard deviation of only 0.22. The slip models performed more moderately, with R-squared values of 0.69 for ultimate slip and 0.70 to 0.76 for the intermediate slip points. The authors note that slip is inherently more variable: even specimens with identical input parameters showed different slip responses, because interface friction, surface roughness, casting quality, and local aggregate distribution around the studs are difficult to quantify and rarely reported in experimental datasets.

From these four predicted characteristic points, the researchers constructed a continuous piecewise curve for the full shear force–slip relationship. The first segment is linear, connecting the origin to the point at one-third of capacity, representing the elastic stage governed by stud deformation and the bond between steel flange and concrete. The second segment is a rational fractional function constrained to pass through the one-third, two-thirds, and peak points with slope continuity at the transition and monotonic single curvature, avoiding the artificial oscillations that polynomial fits produced. Notably, the model deliberately stops at peak load: post-peak softening is not captured, because push-out tests mainly report pre-peak behavior and post-peak records are often incomplete or highly variable due to sudden stud fracture.

A sensitivity analysis confirmed that the equations behave physically. Shear capacity is dominated by stud diameter and the tensile and yield strengths of the stud material, with moderate contributions from flange geometry and group arrangement. Slip predictions respond most strongly to stud tensile strength, yield strength, diameter, and the elastic modulus of the pocket concrete. Parametric studies showed that increasing the number of studs in a group or tightening their spacing reduces the per-stud capacity, because concentrated load transfer accelerates concrete bearing failure within the pocket; in a typical configuration, a nine-stud group showed roughly half the per-stud capacity of a two-stud group, and halving the spacing from 100 to 50 millimeters reduced total capacity by about 10 percent.

The practical payoff came in a case study under assumed design constraints referencing median database values. An automated framework generated 504 feasible stud configurations and evaluated each with the new equations. Total shear capacity per pocket ranged from about 400 to 750 kilonewtons, meaning a poorly chosen design could sacrifice nearly half the available strength. The optimal solution used only four large-diameter 22-millimeter studs with ultra-high-strength pocket concrete, achieving up to 50 percent higher total shear capacity than non-optimized alternatives, demonstrating that fewer large studs can outperform many small ones within the same pocket.

The authors caution that the model applies only within the parameter ranges of the underlying database and to monotonic static loading; fatigue, cyclic loading, and dynamic effects are not covered and remain topics for future work. Scale effects from extrapolating push-out results to full-scale girders also deserve attention, although previous studies indicate that such relationships can still consistently predict overall composite response. Within those limits, the study delivers what the field has lacked: a transparent, equation-based tool that captures grouped-stud behavior across a broad design space and plugs directly into automated optimization workflows for steel–precast concrete bridge construction.

Subject of Research: Data-driven modeling of the nonlinear shear force–slip behavior of grouped stud connectors in steel–precast concrete composite bridges

Article Title: Data-driven nonlinear shear force–slip modeling for automated optimization of grouped studs in steel–precast concrete composite bridges

Article References: Do, V. H., Dang, D. N. N., Nguyen, M. H., Ngoc, T. P. H., Pham, X. Y., Nguyen, H. T., & Phan, H. N. (2026). Data-driven nonlinear shear force–slip modeling for automated optimization of grouped studs in steel–precast concrete composite bridges. Advances in Bridge Engineering, 7(1), Article 31. https://doi.org/10.1186/s43251-026-00221-8

Image Credits: AI Generated

DOI: 10.1186/s43251-026-00221-8

Keywords: grouped studs, shear connectors, steel–precast concrete composite bridges, push-out tests, symbolic regression, shear force–slip relationship, shear capacity, ultimate slip, design optimization, high-strength concrete, structural engineering, machine learning