# Shear Lag Effect on Box Steel Beams with Wide Curved Flanges
## 4 Numerical Parametric Analysis of Shear Lag in Steel Beams
### Numerical Analysis of Continuous Beam with Box Cross-Section  
A numerical model of a steel beam (using the Dlubal RFEM 5 software package [34]) with a box cross-section, continuous over spans of L\({}_{1}\) = 8.0 m and L\({}_{2}\) = 10.0 m, and with an overhang of L\({}_{3}\) = 4.0 m (model analog to one in [2]) was created. As visible in Figure 8, the beam is supported on hinged supports in the middle of the web and loaded with continuous loading in the plane of the web along the entire beam. The cross-section of the beam is a box with a web height of 1000 mm and a thickness of 10 mm, and a variable width of the flange with a constant thickness of 10 mm. The model was automatically discretized (\"meshed\") with linear plate finite elements of a minimum size of 100 \(\times\) 100 mm. As a part of the parametric analysis, the mesh density was evaluated in order to optimize both the model performance and accuracy. A selected mesh size emerged as the optimal value from a mesh sensitivity and a computational efficiency standpoint. Only the values of the longitudinal unit forces obtained by integrating the normal stresses of the cross-sections with maximum/minimum bending moments were considered--cross-sections A-A, B-B, and C-C (Figure 9). Effective widths were not calculated at the edge cross-sections (beginning and end of the beam) since the values of longitudinal stresses are negligible (the bending moment is practically zero) and integrating them yields nonsensical effective width factor values. Effective widths were also not sought in the cross-sections in the area between the points of bending moment extreme values (near the areas where the null points are located), which can be partially in tension and compression, because due to the redistribution of longitudinal stresses due to shear lag (and thus the redistribution of bending moments) there is no ideal null point, i.e., a  
Figure 8: FEM model of continuous box beam—isometric view. Figure 7: Processing and post-processing in B + G Toolbox [33]. cross-section perpendicular to the axis of the beam where all the longitudinal stresses are zero. Integrating the stresses of opposite signs (tension and compression) would also yield nonsensical values of the effective width factor \(\beta\). The input parameters (visually shown in Figure 10) are as follows:  
* Constant flange and web plate thickness \(\mathrm{t_{f}}=\mathrm{t_{w}}=\mathrm{t}=10\) mm;
* Web height \(\mathrm{h}=1000\) mm;
* Flange width \(\mathrm{b--variable}\) (dependent on \(\mathrm{\kappa}=\mathrm{b_{0}/L_{e}}\) ratio);
* Flange curvature radius \(\mathrm{R--variable}\) (dependent on \(\mathrm{b/2R}\) ratio);
* Height difference \(\Delta\) between flange end and midpoint (1). \[\Delta=\mathrm{R}\times\left(1-\cos\frac{\alpha}{2}\right) \tag{1}\]  
Figure 10: Schematic display of longitudinal stress distributions as results of Euler–Bernoulli hypothesis (left side) and shear lag (right side). Figure 9: Multi-span continuous beam—analyzed sections A-A, B-B, and C-C. The data are processed in such a way that the results from certain number of finite elements N were analyzed--the total number of mesh elements depends on the width of the flange. For these elements, the unit axial forces \(\mathrm{n_{x,i}}\) are extracted from the results and the vertical eccentricity \(\mathrm{e_{i,A}}\) is measured in relation to the web-flange junction (2) (point A in Figure 11). Then, the total resulting moment is determined (by summing the partial moments) with the actual state of stress obtained by numerical analysis (3). After that, constant axial stress (with maximum value--point A) with varying lever arm (again related to point A) is assumed along the flange and partial moments are integrated (4) along the flange up to the total value obtained by the original stress distribution, where integration stops and the number of effective finite elements \(\mathrm{N_{eff}}\) is obtained and therefore, effective width (angle) factor \(\beta\) can be calculated (5).