# Influence of Girder Flaring on Load Effect in Girders of Composite Steel Bridges
## 2 Background
The equations of the GDF include parameters that are related to the stiffness and geometry of the bridge, such as the girder spacing, bridge length, deck slab thickness, and the girder's moment of inertia. The girder distribution factor for flexure in the interior girder, \({(GDF_{M})}^{int}\), of slab-on-girder bridges can be computed as follows:  
For one live loaded lane  
\[{(GDF_{M})}^{int}_{1-lane}=0.06+\left(\frac{S}{4300}\right)^{0.4}\left(\frac{S }{L}\right)^{0.3}\left(\frac{K_{g}}{L\ t_{s}^{3}}\right)^{0.1} \tag{1}\]  
For two or more live loaded lanes  
\[{(GDF_{M})}^{int}_{2-lane}=0.075+\left(\frac{S}{2900}\right)^{0.6}\left(\frac{ S}{L}\right)^{0.2}\left(\frac{K_{g}}{L\ t_{s}^{3}}\right)^{0.1} \tag{2}\]  
where \(S\) = girder spacing (mm), \(L\) = bridge span length (mm), \(t_{s}\) = deck slab thickness (mm), and \(K_{g}\) = girder longitudinal stiffness factor (mm\({}^{4}\)), obtained as follows:  
\[K_{g}=n\left(I+Ae^{2}\right) \tag{3}\]  
in which \(I\) = moment of inertia of the non-composite girder (mm\({}^{4}\)), \(A\) = bare girder cross-sectional area (mm\({}^{2}\)), \(e\) = eccentricity between centroid of the girder and the deck slab (mm), and \(n\) = modular ratio between the girder and the deck slab materials. The girder distribution factor for shear in the interior girder, \({(GDF_{V})}^{int}\), of slab-on-girder bridges is represented as follows:  
For one live loaded lane  
\[{(GDF_{V})}^{int}_{1-lane}=0.2+\frac{S}{7600} \tag{4}\]  
For two or more live loaded lanes  
\[{(GDF_{V})}^{int}_{2-lane}=0.2+\frac{S}{3600}-\left(\frac{S}{10700}\right)^{2} \tag{5}\]  
With regard to the load effect in the exterior girder, the GDF for such girder is usually computed in terms of the GDF of the interior girder as follows:  
\[{(GDF)}^{ext}=e*{(GDF)}^{int}\]  
where for the case of flexure  
\[e=0.77+(d_{e}/2800) \tag{6}\]  
and for the case of shear  
\[e=0.60+(d_{e}/3000) \tag{7}\]  
in which \(d_{e}\) = distance from the face of the edge parquet to the center of the exterior girder (mm). Other requirements on the subject are included in the AASHTO LRF Bridge Design Specifications [1]. Many researchers working on live load distribution in girder bridges have found the GDF formulas presented by the AASHTO LRFD specifications to be reasonably accurate for bridges without cross-bracing or diaphragms. However, it appears that the GDF expressions can be more conservative for bridges with large girder spacings and long spans, while they can be somewhat unconservative for bridges with small girder spacings and short spans [14]. Limitations and restrictions on using these equations are applied, and in some cases, other methods such as the lever rule or rigid body rotation should be used to determine the girder distribution factor. The lever rule is a simple method that is based on assuming an internal hinge develops in the deck slab over each interior girder and then using statics to solve for the reaction of the considered girder as a fraction of the truck load. The rigid body rotation approach is a way of determining a lower bound on the girder distribution factor of exterior girders in bridges containing diaphragms or cross-bracing by assuming an infinitely rigid deck slab on elastic supports represented by the girders. Complex analytical methods of analysis for slab-on-girder bridges, such as the finite element method, finite difference procedure, or grillage analogy, are the most accurate methods for establishing the distribution of live load on bridges. Three-dimensional finite element models can provide very accurate results if the model is reasonable, detailed and representative of the actual structural behavior. However, it can be time-consuming for the designer and prone to error in modeling and interpretation of the output because of the sensitivity of the results to the type of elements in use, nature of connections, and meshing intensity.