⚡ electrical engineering

TimoDS: A comprehensive and synthetically generated dataset of Timoshenko steel beams with general elastic supports and responses to elementary unit load cases

This paper introduces TimoDS, a comprehensive synthetic dataset of 60,000 Timoshenko steel beam instances with general elastic supports and elementary load cases, which serves as a validated benchmark for training machine learning surrogates in structural mechanics.

Original authors: Youssef Derrazi, Diego Hernán Peluffo-Ordóñez, Juan Carlos Torres Cantero

Published 2026-08-13

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CC BY 4.0

Original authors: Youssef Derrazi, Diego Hernán Peluffo-Ordóñez, Juan Carlos Torres Cantero

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect or an engineer trying to build a bridge, a skyscraper, or even a simple bookshelf. Before you pour a single drop of concrete or weld a single beam, you have to guess how it will behave. Will it bend too much? Will it snap? Will it wobble in the wind? Traditionally, engineers use powerful computer programs to simulate these scenarios. These programs are like incredibly detailed video game engines that calculate the physics of every single piece of metal. They are accurate, but they are also slow. If you want to test a million different designs to find the perfect one, or if you need to know how a building will react to an earthquake in real-time, waiting for the computer to crunch the numbers for every single try is like trying to count every grain of sand on a beach by picking them up one by one.

To speed things up, scientists have started using "surrogate models." Think of these as super-smart shortcuts. Instead of calculating the physics from scratch every time, you train a computer brain (a machine learning model) on a massive library of examples. Once trained, this brain can guess the answer almost instantly. But here's the catch: to teach this brain, you need a huge, perfect library of examples. In the real world, getting real-world data is expensive and dangerous (you can't just drop a building to see what happens). So, researchers have to create these libraries using simulations. The big challenge has been making sure these simulated libraries are diverse enough to cover every possible situation—from a beam held loosely at the ends to one clamped down tight—and that they are physically correct, so the shortcut doesn't lead to a disaster.

This is where a new dataset called TimoDS comes in. It's like a massive, synthetic recipe book for steel beams, created entirely by computers but designed to be as realistic as possible. The paper introduces TimoDS, a collection of 60,000 different "what-if" scenarios for steel beams. The researchers didn't just pick random numbers; they built a sophisticated, automated pipeline to generate these scenarios. They simulated beams made of standard European steel (specifically IPE, HEA, and HEB profiles) and tested them under six different types of basic "pushes" and "twists" (unit loads and moments). The magic of TimoDS is that it covers the entire spectrum of how a beam can be held up. Instead of just testing beams that are either completely free or completely stuck, TimoDS tests beams held by "elastic supports"—imagine the ends of the beam resting on springs that can be as soft as a mattress or as hard as a rock. This allows the dataset to cover every possible middle ground.

For each of the 60,000 beams, the dataset records exactly how the beam reacts. It doesn't just give a single number like "it bent 5 centimeters." Instead, it maps out the entire story of the beam's movement, force, and energy at 21 different points along its length. This creates a massive output of 253 data points for every single beam, capturing everything from how much it twists to how much energy is stored inside it. The researchers verified that their computer simulations obeyed the laws of physics perfectly, checking that forces balanced out and energy was conserved. They even fixed a small glitch in the software they used to ensure the energy numbers were 100% accurate.

To prove this dataset is actually useful, the authors trained a simple machine learning model on just one of the six load cases. The results were impressive: the model learned to predict how the beam would bend and twist with high accuracy. Even better, they showed that the model could use a principle called "superposition" to figure out how a beam would react to a complex mix of loads it had never seen before, simply by adding up the answers from the basic cases it had learned. This suggests that TimoDS isn't just a static collection of numbers; it's a powerful tool that can help engineers train AI to design safer, more efficient structures in the future, turning a slow, heavy calculation into a lightning-fast guess that still respects the laws of physics.

Technical Summary: TimoDS Dataset

Problem Statement

The development of data-driven surrogate models for structural mechanics is currently hindered by a scarcity of high-quality, publicly accessible training data. Existing experimental datasets are often costly, narrow in scope, and fail to cover the full parameter space required for generalization. Furthermore, many existing synthetic datasets suffer from several limitations: they frequently focus on isolated members with standard boundary conditions (e.g., simply supported or fully clamped) rather than the continuum of intermediate elastic restraints found in real structures; they are often generated via ad hoc procedures lacking full automation and reproducibility; and they typically provide only scalar outputs rather than full spatial distributions of structural responses. These gaps limit the utility of datasets for advanced architectures like Physics-Informed Neural Networks (PINNs) and Graph Neural Networks (GNNs) that operate on spatially distributed fields.

Methodology

The paper presents TimoDS, a synthetically generated dataset comprising 60,000 instances of Timoshenko steel beams with general elastic supports. The generation process is a fully automated, reproducible pipeline executed within the Grasshopper parametric modeling environment using the Karamba3D finite element plug-in.

Mechanical Model

The dataset is based on a linear static Timoshenko beam model of length $L$ with a constant cross-section $\Sigma$ . The model accounts for:

Material: Isotropic linear elastic steel ( $E = 210$ GPa, $\nu = 0.33$ ).
Cross-sections: European steel profiles from the IPE, HEA, and HEB families (66 distinct sections).
Boundary Conditions: General elastic supports at both ends (A and B), defined by 12 stiffness parameters (3 translational $k$ and 3 rotational $c$ per end). These parameters span a continuum from free to quasi-clamped conditions, sampled in logarithmic space relative to the beam's characteristic stiffness.
Load Cases: Six elementary unit load cases are considered independently: three unit point loads ( $\vec{F} = \vec{e}_x, \vec{e}_y, \vec{e}_z$ ) and three unit point moments ( $\vec{C} = \vec{e}_x, \vec{e}_y, \vec{e}_z$ ). Each load is applied at a relative position $\alpha$ along the beam axis.

Data Generation Pipeline

Sampling: Latin Hypercube Sampling (LHS) with a fixed seed (2) is used to ensure uniform coverage of the 15-dimensional input space (beam length, load position, section selection, and 12 dimensionless stiffness multipliers).
Simulation: For each of the 60,000 instances, a parametric graph in Grasshopper/Karamba3D assembles the beam, applies the specific boundary conditions and unit load, and solves the linear static problem.
Output Retrieval: The pipeline retrieves kinematic quantities (displacements and rotations) and internal forces (axial force, shear forces, torsion, bending moments) at 21 equally spaced positions along the beam axis. It also calculates the total elastic strain energy.
Correction: A systematic discrepancy in Karamba3D's internal energy output (which neglects energy stored in support springs for rigid supports) is addressed. The dataset includes a corrected strain energy value ( $\hat{U}_D$ ) recomputed via numerical integration of the internal force fields to satisfy the work-energy theorem.

Key Contributions

Comprehensive Dataset: TimoDS provides 60,000 fully defined beam problems with general elastic supports, covering a wide range of geometric and boundary condition parameters.
Spatial Resolution: Unlike many datasets providing scalar outputs, TimoDS delivers full spatial distributions of 12 physical quantities sampled at 21 points, resulting in an output vector of 253 quantities per instance.
Reproducibility: The entire dataset is generated via a deterministic, self-driving pipeline with a fixed random seed, allowing for exact regeneration. The code and scripts are publicly available.
Superposition Capability: By providing responses to elementary unit load cases, the dataset functions as a discrete influence function library. Due to the linearity of the Timoshenko model, responses to arbitrary loading distributions can be reconstructed via superposition without additional solves.
Physical Consistency: The dataset undergoes rigorous validation, confirming equilibrium residuals at machine precision and verifying energy balance through the corrected strain energy metric.

Results and Validation

Statistical Characterization: The input space shows effective decorrelation between geometric, loading, and boundary condition parameters. The output space exhibits expected physical behaviors, such as the dominance of bending over axial deformation and the influence of section families (IPE vs. HEA/HEB) on torsional and bending flexibility.
Physical Consistency: Equilibrium checks confirm that force and moment residuals are within machine precision ( $\sim 10^{-16}$ to $10^{-15}$ ). The corrected strain energy $\hat{U}_D$ satisfies the energy balance equation $2\hat{U}_D = W_{ext} - W_k$ across 22 orders of magnitude.
Baseline Machine Learning Experiment: A feedforward neural network was trained on Load Case 2 (unit transverse load) to predict the response fields.
- The model achieved high $R^2$ scores for internal forces ( $V_y, M_z$ ) and bending rotation ( $\theta_z$ ).
- Transverse displacement ( $u^G_y$ ) showed a lower relative accuracy in quasi-clamped regimes due to the wide dynamic range of the data, though the overall $R^2$ remained high (0.9945).
- Crucially, the trained model successfully reconstructed responses to combined loading scenarios (two simultaneous point loads) via superposition, demonstrating that a model trained on elementary cases can generalize to complex loading without retraining.

Significance and Claims

The paper positions TimoDS as a viable benchmark for surrogate modeling, physics-informed learning, and operator learning in structural engineering. Its primary significance lies in addressing the lack of diverse, high-quality, and reproducible synthetic data in the field. By providing full spatial response fields and covering the continuum of elastic boundary conditions, the dataset enables the training of models that can generalize across varying structural configurations. The authors explicitly state that the dataset is designed to facilitate the development of data-driven models that can serve as influence function libraries, allowing for the reconstruction of responses to arbitrary loading distributions. The work is modest in its scope, acknowledging limitations such as the restriction to linear material behavior, single-span prismatic beams, and the neglect of warping effects in torsion, framing these as areas for future extension rather than current capabilities.