PSI - Issue 62

Andrea De Flaviis et al. / Procedia Structural Integrity 62 (2024) 871–878 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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1. Introduction The Generalized Beam Theory (GBT) is an analytical formulation to solve problems related to Thin-Walled Beams (TWB) of different nature like static, buckling, dynamic problems (Davies and Leach (1994), Camotim et al. (2008), Bebiano et al. (2008)), linear or geometrically and/or materially non-linear (Silvestre and Camotim (2003), Henriques et al. (2015)). It possibly considers beams with different types of cross-section: open, closed, partially closed, circular (Gonçalves et al. (2010)), made of different materials with isotropic or orthotropic behavior (Silvestre and Camotim (2002)) with classical or non-standard support conditions (Basaglia et al. (2011)) and more. The usefulness of GBT when dealing with TWB is the possibility to take into consideration the deformation of the cross-section in its own plane, thus enabling to study effects that cannot be taken into consideration with other beam theories as Euler-Bernoulli, Timoshenko (Timoshenko and Goodier (1951)) or even Vlasov (Vlasov (1961)). GBT was first developed by Schardt in 1966 as an extension of Vlasov’s theory, but it took hold in the first 90’s mainly thanks to Davies and his collaborators. Then, at the beginning of the 21 st century, this theory has had a great impact in the scientific world thanks to the huge work of researchers from the Technical University of Lisbon which studied many aspects of the theory and strongly contributed to its dissemination all over the world. In this work, GBT has been applied in a modified version, explained in detail in Ranzi and Luongo (2011) and Piccardo et al. (2013), called GBT-D, which simplifies the way deformation modes are found. The main goal of the work is to prove that GBT-D can be used to predict the static and dynamic behavior of simply supported bridge decks, with the final aim of using models based on GBT to monitor the structural behavior over time and to observe modifications that could signify that the structure is being damaged. To validate the model, static results are compared to load-displacement curves derived from real static loading tests and dynamic properties (natural frequencies, mode shapes, damping ratios) are compared to those obtained with Operational Modal Analysis (OMA). 2. Generalized Beam Theory The basic idea of GBT is to describe the displacement field of the TWB as a linear combination of assumed “deformation modes” of the cross - section, depending on its curvilinear abscissa, and unknown “amplitude functions”, depending on the axial coordinate and time. The theory has two main phases, which are developed in the following order: (a) the cross-section analysis, where the deformation modes are found and (b) the member-analysis, where the equations are solved with respect to the amplitude functions. Before introducing the basic equations, it is important to classify the deformation modes. In particular, in GBT-D, 3 main groups are identified:  “ Conventional modes ” for which the 2 Vlasov’s hypotheses of membrane inextensibility in the tangential direction ε s m = 0 and membrane shear indeformability γ zs m = 0 apply (the latter being constant stepwise for closed loops according to Bredt’s theory) . They have both in-plane and out-of- plane components and include “rigid modes” where the cross -section remains undeformed, i.e. Vlasov ’s modes, which are three translations (two flexures and one extension) and one rotation (torsion), “ distort ional modes” where the “natural nodes” show in plane displacement and “local modes” where the “intermediate nodes” show flexural displacement. For many applications this set of modes is enough.  “Extensional modes” for which the inextensibility condition is removed, obtaining a set of purely in-plane modes, orthogonal to the previous ones. They are considered necessary in non-linear analysis, where membrane and flexural behavior are coupled.  “Shear modes” also known as “warping modes”, which are purely out -of-plane and have an important role for short beams that may suffer from shear lag. A short overview of the formulation is here presented, resuming the main aspects of kinematics, constitutive law and equilibrium, with reference to the generic cross-section illustrated in Fig. 1. Detailed information can be found in reference.