Physics-Informed Neural Networks for Analysis of Spatial Beam Structures: A Kinematic Decomposition Approach

Abstract

The design of critical steel structures faces a persistent dilemma between computational speed and physical fidelity. While classical 1D beam elements are computationally efficient, they often fail to capture complex spatial effects like non-uniform torsion and cross-sectional distortion, whereas detailed 3D solid finite element models (FEM) offer reference accuracy but come at a prohibitive computational cost, making them unsuitable for real-time generative design or multi-objective optimization. This study proposes a novel Physics-Informed Neural Network (PINN) architecture designed to function as a real-time “AI-Surrogate” capable of predicting the stress-strain state of spatial members with the accuracy of a high-fidelity 3D FEM model but at analytical speeds. The proposed approach utilizes a Kinematic Decomposition strategy, separating the displacement field into a macroscopic “spine” behavior and a field of local cross-sectional deformations. This effectively reduces the dimensionality of the problem and allows for training on a compact dataset of 10,000 samples. To address the “linearization trap” and the vanishing gradient problem associated with predicting higher-order derivatives (curvature and bi-moments), we introduce a Coordinate Scaling technique. This method normalizes the derivative space, ensuring numerical stability and physical consistency of the solution. Validated against nonlinear 3D solid FEM simulations (Ansys), the model demonstrates high precision, achieving a Mean Absolute Error (MAE) of 0.08 mm for deflections and 0.2 mrad for torsion angles. Furthermore, the specialized physics-informed loss function successfully minimizes the curvature error to 1.510^–4 m^–2, ensuring the accurate recovery of internal forces. The results confirm that the proposed PINN architecture effectively bridges the gap between the speed of beam theories and the accuracy of volumetric models. The introduced Coordinate Scaling method proves critical for learning differential relationships in mechanics, paving the way for the next generation of real-time structural analysis tools.