# A Minimal GBT Model for Distortional-Twist Elastic Analysis of Box-Girder Bridges
## 6 Conclusions  
A minimal linear and elastic model of rectangular box-girder, undergoing twist and distortion, was formulated in the framework of GBT. The model is useful for bridge analysis in skew-symmetric loading conditions caused by the eccentricity of live loads. The displacement field was approximated as the superposition of two cross-sectional modes, one describing twist and the other describing distortion, all given in closed form. Modes are modulated by two amplitude functions, defined on the beam axis, governed by a set of two coupled fourth-order ordinary differential equations, which generalize the well-known uncoupled equations by Vlasov for torsion and of beam on elastic soil for distortion. The method, being analytic, is able to supply closed-form expressions for stresses in the whole domain. Algorithmic aspects related to these equations were tackled, leading to the following conclusions. 1. The coefficients of the derivatives of fourth-order are mainly generated by nonuniform warping both in twist and distortion. Therefore, among them, the contribution of the flexural nature of the plates are negligible. In contrast, warping terms, although small, cannot be neglected when the displacement wavelength is short. It is argued that they could be relevant in describing boundary layers, e.g., produced by constraints preventing free warping. 2. The distortion-to-twist ratio was proven to be of order 1 for thin and short girders. 3. The Fourier analysis is a convenient and efficient tool to analyze simply supported girders warping free at the ends; for other boundary conditions, exact integration of the differential equations should be carried out. By following the literature, a simplified procedure was illustrated, which calls for neglecting all coupling terms, solving two independent problems and superimposing the effects. The two problems are (i) the Vlasov beam under torsion and (ii) the Winkler soil equation-like beam for distortion. A sample system was considered, for which both Fourier analysis and uncoupled equation integration were carried out. The following conclusions were drawn. 1. Fourier analysis works well even for non-smooth loading conditions, provided that a sufficient number of terms is accounted for in the series. The exact integration of the uncoupled equations gives reasonably good results, with errors of about 10% with respect to the coupled Fourier representation. 2. Stresses due to torsion mainly consist of (i) normal longitudinal components equilibrating the bimoment; (ii) active tangential stresses, as given by the Bredt theory; and (iii) reactive tangential stresses equilibrating the complementary torsional moment due to warping. All these effects are significant, except for the reactive tangential  
Figure 20: Torsion coupled to distortion: (**a,b**) Transverse normal stresses at bottom edges (continuous lines) and top edges (dashed lines): (**a,b**) stresses at \(z=7.5\,\)m vs. \(s_{i}\), (**c,d**) stresses at the corners vs. \(z\) ; (**a,c**) element 1, (**b,d**) element 2. Fourier analysis with \(N=50\) terms, FEM analysis (bullets). stresses, of which the influence is appreciable only close to the discontinuity points of the load. 3. Stresses due to distortion consist of (i) membrane normal stress in the longitudinal direction, triggered by nonuniform warping, kinematically compatible with the loss of the shape of the cross section; (ii) membrane normal stresses in the transverse direction, generated by the frame-like behavior of the cross-section; (iii) flexural normal stresses in the longitudinal direction, generated by the flexure of the plates associated with the longitudinal modulation of the frame-deflection; and (iv) tangential stresses, generated by the torsion of the plates, made of an active and a reactive component. Among these stresses, (i) and (ii) are the most important. However, among the tangential stresses, the reactive component cannot be neglected. Finally, all the results have been validated by a Finite Element Analysis.