# A Minimal GBT Model for Distortional-Twist Elastic Analysis of Box-Girder Bridges
## 1 Introduction  
It is well-known that, when a mono-cellular box-girder undergoes torsion, induced by eccentric loads with respect to its longitudinal axis, its cross section suffers a distortion in its own plane, which modifies the original shape [1]. Such a phenomenon is often referred to, in the technical literature devoted to bridges, as _differential flexure_, since a quota of the external torsional moment not equilibrated by the internal torsional moment is instead bared by internal forces triggered by equal and opposite flexures of the two webs. Here, however, the word _distortion_ is preferred. The phenomenon is dangerous, since flexure of the webs entails longitudinal normal stress, which adds to the tangential stresses of the [PERSON] theory of uniform torsion and to the normal stresses of the [PERSON] theory of nonuniform torsion. Moreover, flanges and webs all suffer transverse bending, which would not be present if the cross section could maintain its shape. To limit such effects, diaphragms or bracings are occasionally introduced into the box-girder, but more often, these devices are omitted for construction simplicity. In these latter cases, a more refined analysis is required, in which the simple model of a beam must be abandoned and the box-girder must be modeled as a plate assembly. The most popular approach to this problem is based on the so-called Beam on Elastic Foundation (BEF) analogy, attributed to [PERSON] [2], after the pioneristic work by [PERSON]. According to the BEF analogy, the effects of distortion are uncoupled from those of torsion and are governed by the classical differential equation of the beam on Winkler soil. The model was derived on a physical rather than a mathematical ground by substituting the continuous model of a beam with an assembly of elastic thin slices of the box-girder, rigidly connected among them. Among old papers, in Reference [3], an alternative physicalmodel was proposed, leading to the same equation, and an example of reinforced concrete monocellular box-girder bridge was worked out; in Reference [4], parametric analyses were carried out and the results were given in different load conditions. The BEF analogy, in spite of its simplistic idea of uncoupling, is still used nowadays. As a few examples, in Reference [5], the analogy was exploited, aimed at supplying a useful tool for designers and accounting for the presence of diaphragms. In Reference [6], the approach is compared with finite element simulations, concluding that the simplified method provides a safe estimation of the response. In Reference [7], an extended version of the equation of BEF, including the second derivative, is used to analyze the influence of shear-deformable diaphragms. In spite of using BEF, since 1990, it was clearly stated, in Reference [8], torsion and distortion cannot be separated since they are intrinsically coupled. The model presented there concerns a rectangular box-girder undergoing torsion-distorsion. It is based on simplified kinematics, in which three generalized configuration variables, depending on the longitudinal abscissa, are a priori introduced: one for twist, two (not independent) for bending of webs/flanges in their own planes, and one for distortion. The relevant strains are of membrane type, piece-wise linear, or constant along the directrix, while flexural and torsional plate contributions are ignored. By following an energy approach, a differential system of sixth-order in the configuration variable is derived and solved. In a successive paper [9], the author extended his theory to trapezoidal cross sections by introducing twelve configuration variables. To analyze coupling effects, we need to resort to more refined models. Recently, higher-order theories of one-dimensional beams have been used, in which some added kinematic descriptors account for warping and distortion of the beam. An overview of different models in the literature is provided in Reference [10].