PSI - Issue 83

Mohamed Rjilatte et al. / Procedia Structural Integrity 83 (2026) 208–217

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1. Introduction Cable-stayed bridges are among the most widely used structures worldwide, particularly in modern cities, to span natural or artificial obstacles. They are characterized by large spans and are designed to carry heavy and moving loads. The design of such structures is highly complex, as it must ensure not only user safety but also the stability, strength, and durability of the bridge. Engineers must therefore take all these factors into account to design bridges that meet both performance and safety standards. However, analyzing and calculating the behavior of such structures is far from simple. Because of their large spans and the considerable loads they carry, cable-stayed bridges often display nonlinear vibration behavior, where displacements become significant enough to invalidate the assumptions of classical linear theory. Given the importance of this subject, numerous researchers have explored different aspects of modeling and dynamic analysis of these structures. Nevertheless, only a few studies have specifically addressed the nonlinear dynamic behavior of such bridges, particularly in relation to geometrically nonlinear vibrations. Before presenting the context of the present work, several previous studies related to this topic are reviewed. In [1], Cong et al developed a simplified nonlinear single-degree-of-freedom model of a cable–beam system using the stationary Hamiltonian functional method. In [2], Guo et al proposed an asymptotic reduced model of a double-inclined cable–beam system to analyze the dynamic interactions in cable-supported structures. Assuming that the beam is much heavier and exhibits smaller displacements than the cables, the study revealed two distinct types of coupled dynamics depending on the excitation source (cable or beam), results that were confirmed through a Galerkin numerical model. A linear dynamic theory of a cable-stayed beam model was established and solved using the Transfer Matrix Method (TMM) by Cao et al [3], to investigate the dynamic behavior of cable-stayed bridges. Considering both axial and lateral vibrations in the plane of the cables and the beam, the method of separation of variables was employed to solve the governing differential equation. In [4], Ma et al have investigated the influence of cable modeling on the static and dynamic responses of a cable stayed bridge subjected to strong winds. The study demonstrated that wind loads acting on the cables contribute as much as those on the deck to lateral deformation. The MECS model, which accounts for the nonlinear sag effect of the cables, was found to reproduce experimental results more accurately, revealing increased vertical and torsional deformations as well as a reduction in the vertical frequency, with little influence on torsional frequency. A study on the nonlinear behavior of a cable-stayed beam equipped with a Tuned Mass Damper (TMD) was conducted by Su et al. [5]. The results indicated that the TMD, even without direct contact with the beam, significantly affects the system response by inducing energy transfer among the beam, the cable, and the TMD. As the excitation level increases, the softening behavior becomes more pronounced, leading to bifurcations and amplified response peaks, which reveal the complexity of the dynamic interactions. The main goal of this study is to conduct a parametric analysis to examine how material properties, boundary conditions, and cable positions affect the nonlinear vibrational behavior of a cable–beam system. The adopted methodology first involves a linear modal analysis, which allows the determination of natural frequencies and linear mode shapes that serve as a basis for modeling the nonlinear behavior of the system. The resulting nonlinear system is solved using the implicit method proposed by Benamar and further refined by El Kadiri and Benamar in [6], [7]. 2. MATHEMATICAL FORMULATION 2.1 Linear formulation This study focuses on the geometrically nonlinear free vibrations of a beam suspended by two elastic cables (cable stayed beam), fixed at both ends as illustrated in Fig. 1.