# On the Use of the Effective Width for Simply Supported Generally Loaded Box-Girders
## 4 Displacements  
To evaluate the deflections of the beam, three different models are used, of increasing refinement, and always in the framework of the Fourier analysis. The models are  
1. The _total Euler-Bernoulli_ model (\(EB_{t}\)), in which the shear lag is ignored, so that the cross-section is the integer; this model is used only for comparison purposes, since it is expected it overestimates the stiffness of the beam;
2. The _total Timoshenko_ model (\(TM_{t}\)), in which the shear lag is still ignored, but the shear deformability of the web is accounted for by taking a shear area approximately equal to the web area, namely \(A^{*}=A_{web}\); also this model is expected to give better results of the former, but they are still unsatisfactory;
3. The _effective Euler-Bernoulli_ model (\(EB_{e}\)), in which the shear-lag is included by exploiting the concept of wavelength-dependent effective width; this model is believed to be sufficiently accurate. The shear lag could also be introduced in the Timoshenko model as, e.g., done in [35] in a dynamic context. There, it was proven that the Timoshenko model is more accurate than the Euler-Bernoulli model, when considering modes of short wavelength. In statics, however, since regular distribution of equiverses forces, which are of interest in bridge analysis, admit Fourier series in which higher harmonics bring a small contribution, the difference between the two models can be checked to be negligible, as proven ahead in the numerical analysis. The \(EB_{t}\) model is considered first. The equation of the elastic curve and boundary conditions of the simply supported beam read  
\[ \begin{array}{l}EIv^{\prime\prime\prime\prime}(z)=\sum_{n}\dot{\rho}_{n}\sin \big{(}\frac{n\pi}{\ell}z\big{)}\\ v(0)=v(\ell)=0\\ v^{\prime\prime}(0)=v^{\prime\prime}(\ell)=0\end{array}  \tag{16}\]  
where \(v(z)\) is the transverse displacement at the abscissa \(z\), \(I\) is the second-order inertia moment of the _integer_ cross-section and use has been made of Equation (5) to express the load. After integration, the following displacement field is found:  
\[v(z)=\frac{\ell^{4}}{\pi^{4}EI}\sum_{n}\frac{1}{n^{4}}\dot{\rho}_{n}\sin\Big{(} \frac{n\pi}{\ell}z\Big{)} \tag{17}\]  
Concerning, the \(EB_{e}\) model, a different inertia moment must be taken for any \(n\), so that the \(n\)th problem reads  
\[ \begin{array}{l}EI_{n}v_{n}^{\prime\prime\prime\prime}(z)=\dot{\rho}_{n}\sin \big{(}\frac{n\pi}{\ell}z\big{)}\\ v_{n}(0)=v_{n}(\ell)=0\\ v_{n}^{\prime\prime}(0)=v_{n}^{\prime\prime}(\ell)=0\end{array}  \tag{18}\]with \(v_{n}(z)\) the response to the \(n\)th harmonic load component. From these equations, \(v_{n}(z)=\hat{v}_{n}\sin\bigl{(}\frac{n\pi}{\ell}z\bigr{)}\) is found, with  
\[\hat{v}_{n}=\frac{\hat{p}_{n}}{EI_{n}}\biggl{(}\frac{\ell}{n\,\pi}\biggr{)}^{4} \tag{19}\]  
so that the response to \(p(z)\), i.e., \(\sum_{n}\hat{v}_{n}\sin\bigl{(}\frac{n\pi}{\ell}z\bigr{)}\), is  
\[v(z)=\frac{\ell^{4}}{\pi^{4}E}\sum_{n}\frac{1}{n^{4}I_{n}}\hat{p}_{n}\sin \Bigl{(}\frac{n\pi}{\ell}z\Bigr{)} \tag{20}\]  
The solutions in Equations (17) and (20) are similar, but while the inertia \(I\) is the same in each term of the former series, the inertia \(I_{n}\) decreases with the number \(n\) in each term of the latter series. Therefore, the \(EB_{e}\) is softer than the \(EB_{t}\) model. Finally, the \(TM_{t}\) model is addressed. The equilibrium equations in terms of the (independent) transverse displacement \(v(z)\) and rotation \(\varphi(z)\) are (see, e.g., [35])  
\[ \begin{array}{l}-GA^{*}(\varphi^{\prime\prime}-\varphi^{\prime})=\sum_{n} \hat{p}_{n}\sin\bigl{(}\frac{n\pi}{\ell}z\bigr{)}\\ -EI\varphi^{\prime\prime}-GA^{*}(\varphi^{\prime}-\varphi)=0\end{array}  \tag{21}\]  
with the boundary conditions  
\[ \begin{array}{l}v(0)=v(\ell)=0\\ \varphi^{\prime}(0)=\varphi^{\prime}(\ell)=0\end{array}  \tag{22}\]  
and symbols already introduced. These equations are exactly solved by the series as follows:  
\[