# On the Use of the Effective Width for Simply Supported Generally Loaded Box-Girders
## 3 State of Stress
### Stresses  
The normal stresses \(\sigma_{z}(z,s)\) and the tangential stresses \(\tau_{zs}(z,s)\), respectively, induced by the bending moment \(M(z)\) and the shear force \(T(z)\), are evaluated according to the technical theory of the beam, which extends the De Saint Venant results to beams loaded along their axis. Consequently, the Navier and the Jourawsky formulas are assumed to hold. However, to account for the shear-lag effects, reference must be made to the _effective cross-section_, in which the true width \(b_{1}\) is replaced by the effective width \(b_{en}=b_{e}(n)\), as illustrated in Figure 4. Due to the dependence on \(n\), it is no possible to define an effective width of the flange valid for any loads, but reference has to be made to the single harmonic component of solicitation, i.e., \(\hat{M}_{n}\sin\left(\frac{n\pi}{\ell}z\right)\) and \(\hat{T}_{n}\cos\left(\frac{n\pi}{\ell}z\right)\) (which, indeed, trigger a sinusoidal deflection of the beam in \(n\) halfwaves). For each of these harmonics, the Navier and Jourawsky formulas are applied to a _different geometry_ of the cross-section, and the stresses are added together to obtain  
\[\sigma_{z}(z,s) =\sum_{n}\hat{\sigma}_{n}(s)\sin\left(\frac{n\pi}{\ell}z\right) \tag{10a}\] \[\tau_{zs}(z,s) =\sum_{n}\hat{\tau}_{n}(s)\cos\left(\frac{n\pi}{\ell}z\right) \tag{10b}\]  
where  
\[\hat{\sigma}_{n}(s)=\frac{\hat{M}_{n}}{I_{n}}y,\qquad\tau_{zs}(s)=-\frac{\hat {T}_{n}}{I_{n}h(s)}S_{n}^{*}(s) \tag{11}\]  
Here,  
\[I_{n}=2\Bigg{(}\bigg{(}\frac{b_{2}-h_{1}}{2}\bigg{)}^{2}b_{e}h_{1}+\frac{1}{1 2}h_{2}(b_{2}-2h_{1})^{3}+\frac{1}{12}b_{e}h_{1}^{3}\Bigg{)} \tag{12}\]  
is the second-order inertia moment (with respect to the \(x\) axis) of the effective rectangular box-girder associated with the \(n\)-th harmonic of the solicitation, i.e., to flanges of width \(b_{en}=b_{e}(n)\). Similarly, \(S_{n}^{*}(s)\) is the static moment of the portion of effective cross-section cuts by a chord at the abscissa \(s\), reading  
\[S_{n}^{*}(s_{1,2})=\left\{\begin{array}{ll}-\hat{s}_{1}h_{1}\Big{(}\frac{b_ {2}-h_{1}}{2}\Big{)},&\hat{s}_{1}\in\Big{[}0,\frac{b_{2}-h_{2}}{2}\Big{]}\\ -\frac{h_{1}(b_{2}-h_{1})(b_{e}-2h_{2})}{4}-\frac{h_{2}}{8}\Big{(}b_{2}^{2}-4 b_{2}h_{1}+4h_{1}^{2}+\frac{s_{2}^{2}}{2}\Big{)},&s_{2}\in\Big{[}-\frac{b_{2}-h_{1}}{2 },\frac{b_{2}-h_{1}}{2}\Big{]},\end{array}\right. \tag{13}\]  
respectively, on the flange and on the web. Here, \(\hat{s}_{1}=s_{1}-\frac{b_{1}-b_{2}}{2}\) is a new translated abscissa.  
Figure 4: Normal and tangential stresses induced by the \(n\)th harmonic of the load.  
The maximum (minimum) normal stress occurs at the top (bottom) of the cross-section, where, by using Equation (8b) in the Navier formula, it assumes the value  
\[\partial_{n,max}=-\bigg{(}\frac{\ell}{n\pi}\bigg{)}^{2}\frac{b_{2}}{2I_{n}}\, \beta_{n} \tag{14}\]  
The relative maximum (in modulus) tangential stresses occur (i) on the flange, at the junction with the web (\(\tilde{\tau}_{n,max}^{fl}\)), i.e., at \(\tilde{s}_{1}=\frac{b_{n}}{2}-h_{2}\), and (ii) on the web at the midpoint (\(\tilde{\tau}_{n,max}^{w}\)), i.e., at \(s_{2}=0\). By using Equation (8a) in the Jourawsky formula, they assume the values  
\[\begin{array}{l}\tilde{\tau}_{n,max}^{fl}=\frac{\ell}{n\pi}\frac{(b_{2}-h_{1 })(b_{n}-2h_{2})}{4I_{n}}\,\beta_{n}\\ \tilde{\tau}_{n,max}^{w}=\frac{\ell}{n\pi}\frac{1}{4I_{n}}\,\bigg{(}\frac{h_{ 1}(b_{2}-h_{1})(b_{n}-2h_{2})}{h_{2}}+\frac{(b_{2}-2h_{1})^{2}}{2}\bigg{)}\, \beta_{n}\end{array} \tag{15}\]