# On the Use of the Effective Width for Simply Supported Generally Loaded Box-Girders
## 2 Effective Flange Width  
The formulation developed in [35] to obtain the effective flange width of rectangular box-girders undergoing sinusoidal deflections is summarized here.  
A double-symmetric box-girder of dimensions \(b_{1}\times b_{2}\) and span \(\ell\), illustrated in Figure 1, is considered. The box-girder comprises four thin-walled elements: the bottom and upper flanges, represented by \(i=1,3\) and having a thickness \(h_{1}=h_{3}\), and the right and left webs, represented by \(i=2,4\), and having a thickness \(h_{2}=h_{4}\). The box-girder is referenced to the cross-section inertia principal axes \(x,y\) at the centroidal axis \(z\in(0,\ell)\). The curvilinear abscissa are represented by \(s_{i}\in\left(-\frac{b_{i}-h_{2}}{2},\frac{b_{i}-h_{2}}{2}\right)\) along the flanges and by \(s_{i}\in\left(-\frac{b_{i}-h_{1}}{2},\frac{b_{i}-h_{1}}{2}\right)\) along the webs. Furthermore, the box-girder is simply supported and torsionally clamped at the ends. Additionally, the cross-section of the box-girder is assumed to remain undeformable within its own plane. In-plane displacements of the cross-section, along the \(x,y\) axes, are denoted by \(u,v\) respectively and out-of-plane displacements by \(w\).  
A simplified analytical model based on the following hypotheses is used: (a) the vertical deflection of the box-girder is sinusoidal; (b) the flange and the web are inextensible in the cross-section plane; and (c) the web is shear undeformable. It is assumed that the box-girder deflects in the \(yz\) plane with a sinusoidal law  
\[v(z)=\hat{v}\sin\left(\frac{n\pi}{\ell}z\right) \tag{1}\]  
of amplitude \(\hat{v}\). By virtue of the hypotesis (b), it is \(u(z,s_{i})=0\). The out-of-plane displacement of the bottom flange is taken as \(w(z,s_{1})=\hat{v}(s_{1})\cos\left(\frac{n\pi}{\ell}z\right)\), with aplitude \(\hat{v}(s_{1})\) to be determined by equilibrium conditions together with symmetry considerations (the amplitude should be symmetric with respect to the \(y\) axis).  
The membrane strains on the flange are written, and then, the active stresses in the flange are derived by the elastic law (taking zero, for the moment, the Poisson ratio). By enforcing the indefinite equilibrium in the \(z\) direction and integrating ordinary differential equations, the normal stresses are evaluated. Finally, the result of these abscissa-dependent stresses acting on the flange is equated to the product of the maximum stress occurring at the flange-web junction by the unknown effective width \(b_{en}\). The following result is drawn:  
\[b_{en}=\frac{2}{\beta}\tanh\left(\frac{b_{1}}{2}\beta\right) \tag{2}\]  
where  
\[\beta:=\frac{n\,\pi}{\ell}\sqrt{\frac{E}{G}} \tag{3}\]  
is the wave number. The shear-lag coefficient \(\alpha_{n}=\frac{b_{en}}{b_{1}}\) is defined as follows:  
Figure 1: Box-girder bridge: cross-section and curvilinear abscissa.  
\[\alpha_{n}=\frac{2}{n\,\pi\frac{b_{1}}{\ell}\sqrt{\frac{E}{G}}}\tanh\left(n\,\pi \frac{b_{1}}{2\ell}\sqrt{\frac{E}{G}}\right) \tag{4}\]  
in which \(E\) and \(G\) are the Young modulus and the tangential elastic modulus, respectively. This is a ready-for-use formula, giving the effective width as a function of (i) the width-to-length ratio of the box-girder, (ii) the material property ratio \(E/G=2(1+\
u)\), with \(\
u\) the Poisson coefficient, and (ii) the number \(n\) of halfwaves which the beam deflects. For a beam of given geometric and mechanical characteristics, this latter is the only variable. The shear-lag coefficient in Equation (4) is plotted in Figure 2a vs. \(n\), for a fixed \(\frac{b_{1}}{\ell}\) and different values of \(\
u\). It is noticed that the material parameter weakly affects the results. Figure 2b shows the shear-lag coefficient _vs._\(n\), for different values of \(\frac{b_{1}}{\ell}\). For fixed \(\frac{b_{1}}{\ell}\), the graph makes it possible to associate with each \(n\) the corresponding value of \(\frac{b_{2n}}{b_{1}}\) and therefore the known \(b_{1}\), the relevant \(b_{en}\).