Human-Induced Vibration of Long-Span Multi-Layer Composite Beams—Analytical Calculation Method

Abstract

The finite element method (FEM) provides a general framework for analyzing virtually all types of structures under various loading conditions. Nevertheless, in certain situations, designers may identify that the structural system exhibits predominantly beam-like behavior, allowing the use of analytical theories. This is often the case when evaluating the performance of rectangular floor systems where bending stiffness is significantly lower in one direction compared to the other. Additionally, beam theory offers highly accurate structural response predictions for footbridges. The theory of layered beams is extensively utilized in engineering, particularly in the design of composite beams, cross-laminated timber beams, and sandwich structures. When it comes to long-span lightweight floors, addressing human-induced vibrations is a crucial aspect of the design process. This paper derives and solves the exact equation of motion for a simply supported layered beam. The formulation is based on the exact differential equation governing layered beam behavior, ensuring applicability to a wide range of cross-sectional configurations. The solution is given, including both steady-state and transient responses. These combined features distinguish the present study from previous works, where such completeness is rarely achieved within a single paper. This study is structured into two main sections: the first section derives the equation of motion and offers a comprehensive solution for simply supported layered beams using a validated walking function. The second section illustrates the practical application of this solution in the vibration design of lightweight floors. The proposed method clarifies the theoretical foundations of the guidelines and provides a straightforward approach to designing layered beams. A case study is presented, focusing on a timber–steel–timber composite floor beam, where timber plates are bolted to a steel beam. The results show that the proposed method effectively explains the underlying theories and offers a practical design approach for layered beams subjected to human-induced vibrations.