# On the Use of the Effective Width for Simply Supported Generally Loaded Box-Girders
## 5 Average Effective Width
### Normal Stresses Equivalence  
It is also possible to obtain an average value of the effective width by using the equivalence of the normal stresses instead of the displacements (Section 5.1). For this purpose, the following strategy, similar to that pursued in Section 5.1, is followed: (a) for a given load \(p(z)\), the maximum normal stress at the midspan (or at another significant point) is computed via the proposed algorithm, through the series expansion Equation (10a), i.e.,  
\[\sigma_{z}\left(\frac{\ell}{2},-\frac{b_{2}}{2}\right)=\frac{\ell^{2}b_{2}}{2 \pi^{2}}\sum_{n}\frac{1}{n^{2}I_{n}}\hat{p}_{n}\sin\left(\frac{n\pi}{2}\right) \tag{30}\]  
(b) such a normal stress is equated to \(\tilde{\sigma}_{z}\left(\frac{\ell}{2},-\frac{b_{2}}{2}\right)\),experienced by a beam whose cross-section has flanges of average effective width \(\tilde{b}_{e}\), i.e., average inertia moment \(\bar{I}\); (c) from the equality \(\tilde{\sigma}_{z}\left(\frac{\ell}{2},-\frac{b_{2}}{2}\right)=\sigma_{z} \left(\frac{\ell}{2},-\frac{b_{2}}{2}\right)\), the unknown \(\tilde{b}_{e}\) is derived. In the three cases, uniformly distributed load, segment of load centered at the midspan and concentrated load applied at the midspan, the normal stresses \(\tilde{\sigma}_{z}\left(\frac{\ell}{2},-\frac{b_{2}}{2}\right)\) read, respectively,  
\[ \begin{array}{l}\tilde{\sigma}_{z}\left(\frac{\ell}{2},-\frac{b_{2}}{2} \right)=\frac{p_{04}\ell^{2}b_{2}}{16\ell}\\ \tilde{\sigma}_{z}\left(\frac{\ell}{2},-\frac{b_{2}}{2}\right)=\frac{p_{05} \alpha(2\ell-\Delta)b_{2}}{16\ell}\\ \tilde{\sigma}_{z}\left(\frac{\ell}{2},-\frac{b_{2}}{2}\right)=\frac{p\ell b_{ 2}}{8\ell}\end{array}  \tag{31}\]  
In the three cases, it is found, respectively, that  
\[ \begin{array}{l}\frac{\tilde{b}_{e}}{\tilde{b}_{1}}=\frac{1}{6}\frac{\pi^{3 }}{\sum_{n}\frac{(1-(-1)^{n})}{n^{2}\sin\left(\frac{n\pi}{2}\right)}}\\ \frac{\tilde{b}_{e}}{\tilde{b}_{1}}=\frac{1}{32}\frac{\pi^{3}\Delta(2\ell- \Delta)}{\ell^{2}\sum_{n}\frac{1}{n^{2}\sin\left(\frac{n\pi}{2}\right)}\sin^{ 2}\left(\frac{n\pi}{2}\right)}\\ \frac{\tilde{b}_{e}}{\tilde{b}_{1}}=\frac{1}{8}\frac{\pi^{2}}{\sum_{n}\frac{1} {n^{2}\tilde{a}_{n}}\sin^{2}\left(\frac{n\pi}{2}\right)}\end{array}  \tag{32}\]  
Figure 5 shows the average shear-lag coefficient \(\bar{\alpha}=\frac{\tilde{b}_{e}}{\tilde{b}_{1}}\) vs. \(\frac{b_{1}}{\ell}\) for the uniformly distributed load, concentrated load at the midspan and segment of distributed load, centered at midspan, for different extensions \(\frac{\Delta}{\ell}=\frac{1}{10},\frac{1}{4},\frac{1}{2}\), by both the displacement equivalence (dashed line) and the stresses equivalence (continuous line). Results obtained by the displacement equivalence does not appear able to take into account the loading conditions, therefore it is considered to be not sufficiently accurate. The shear-lag coefficient for the two load cases, concentrated force at midspan and uniformly distributed load, have been studied by the BS5400 [34] and by a variational analysis in [27], where the graph of the shear-lag coefficient \(\frac{b_{e}}{\tilde{b}_{1}}\) vs. \(\frac{b_{1}}{2\ell}\) is obtained numerically. Figure 6 shows results by BS5400, by [27] and by the present study all together, for the two load cases mentioned above. They are very close to each other; however, it is important to underline that our study, unlike the other studies mentioned, provides an analytical formula for the relationship, and therefore, it is applicable to different load conditions, and this character of generality constitutes an added value. In support of the above, the calculation of the average shear-lag coefficient using the equivalence of the normal stresses for two load conditions not present in the literature is proposed here: a concentrated force applied at \(\xi=\frac{\ell}{4}\) and a segment of distributed load, centered at the abscissa \(\xi=\frac{\ell}{4}\).