# A Minimal GBT Model for Distortional-Twist Elastic Analysis of Box-Girder Bridges
## 1 Introduction
In this class of papers falls, e.g., References [11; 12], in which warping and ovalization of a multi-layered beam is analyzed, and References [13; 14; 15], where nonlinear effects are accounted for. The Finite Element Method (FEM) is the most popular tools today to solve problems. In the last decades, however, alternative numerical instruments have also been used, tailored to Thin-Walled Beam (TWB) structures, namely (i) the Finite Strip Method (FSM) [16; 17; 18] and (ii) the Generalized Beam Theory (GBT) [19]. The latter is a numerical version of the well-known Kantarovitch semi-variational method, in which the displacement field is taken as a linear combination of known functions defined on the cross-sectional domain (usually said _modes_) and unknown modulating amplitudes defined on the beam axis. The original two-dimensional plate problem is thus brought to a simpler one-dimensional problem in amplitude functions, for which Fourier series expansions, or 1D-FEM, can be applied. The idea had [PERSON] himself as a forerunner, who used it to analyze closed TWB; some numerical applications of the Vlasov procedure have been recently illustrated in [20] by never mentioning GBT. The modern version of the method was implemented by many authors, following an original idea by [PERSON], successively disseminated in English by References [19; 21; 22]. It had in [PERSON] and its school [23; 24; 25] a huge impulse in the last two decades, through a series of papers in which the choice of modes (cross-section analysis) and various algorithmic aspects were progressively refined. [PERSON], [PERSON], and coworkers [26; 27; 28; 29; 30] launched the idea to combine GBT with a dynamic boundary value problem to automatize the cross-sectional analysis. [PERSON] et al. [31; 32] analyzed, in particular, the shear effects. More recently, in References [33; 34], GBT was combined with FEM to solve problems in which localized strains and stresses occur. Moreover, the use of the Boundary Element Method (BEM) has also been proposed [35] to analyze beams with general (not necessarily thin-walled) beams. All such philosophies, based on purely numerical approaches, are more oriented towards describing _how_ a phenomenon manifests itself rather than towards explaining _why_ it occurs, which are the essential parameters that govern them. Therefore, it is believed by the authors of this paper that ancient analytical methods, oriented towards hand-solving a problem, should be rethought in the light of the new knowledge and instruments available. In this view, a simplified model of the box-girder in torsion-distortion is proposed here. It is developed by reconsidering the approach of Reference [8] by framing it in the modern GBT. Accordingly, just two modes (and therefore two configuration variables) describing torsion and distortion, respectively, are shown to be accurate enough to capture the mechanical behavior based on an orthogonal decomposition of the planar cross-sectional displacement field. Strong differences from Reference [8], however, exist, namely (a) generalized configuration variables are not a priori assumed but, in the spirit of GBT, they naturally appear as the amplitude of the deformation modes; (b) distortion involves warping on a kinematic ground, entailing active normal stresses and passive tangential stresses (the reverse occurs in Reference [8]); (c) the plate behavior of the single elements is accounted for, allowing evaluation of the stresses within the thickness; (d) shear strains are not simply piece-wise constant on the directrix, allowing for satisfaction of the equilibrium at joints; and (e) an eighth-order (instead of sixth-order) differential system is derived, as a consequence of the accounted plate behavior and the different treatment of warping.