# Effects of Flexural Stiffness on Deformation Behaviour of Steel and FRP Stress-Ribbon Bridges
## 4 Numerical Modelling
### Parametric Analysis  
The effects of flexural stiffness are investigated using three FE models with different flexural stiffness of the stress-ribbons. The horizontal movements of the supports are neglected to eliminate the influence of the boundary conditions. Figure 11 shows the considered cross-sections. The cross-section area, the material parameters, and FE type remain the same as in the previous section. Figure 10: The bridge models’ vertical displacements. The bridge sag extent, altering the magnitude of the thrust force (\(H\)), affects the choice of the cross-section parameters by the iterative procedure described in Section 2. In this section, the initial sag of _Model-2_ was increased to 135 mm for illustrating the above statement by changing the compressive stresses in the ribbons. The same sag\(f_{0}\) is used in all FE models considered in this section for comparison purposes. An additional _Model-3_ is investigated as a hypothetical case, representing a maximal flexural stiffness of the ribbon, though the cross-section area \(A\) does not change (Figure 11). In reality, this cross-section layout is impossible due to the low lateral-torsional buckling resistance of such a slender element. The modelling of the band half with symmetry condition along the longitudinal surface (Figure 12) ensures the buckling stability of _Model-3_. Since the stress-ribbon bridges are susceptible to asymmetric loading conditions, the numerical analysis focuses on the deformation behaviour of the bridge models under the asymmetrical loading conditions. In the same manner as Section 4.1, the FE analyses employ the linear-elastic material model of steel, assuming the elasticity modulus \(E\) = 208 GPa and Poisson's ratio \(\
u\) = 0.30. The design displacement (\(u_{q,\text{max}}\)) induced by the live load is used for the analysis, as shown in Figure 10. The load situation of two alternative models (_Model-1_ and _Model-3_) are optimised regarding the deformation response of _Model-2_. The similarity of the displacements \(u_{q,\text{max}}\) of the models is the optimisation condition. This process assumes the constant design load \(q\) and alters the dead load component to satisfy the optimisation criterion (i.e., \(u_{q,\text{max}}\) = const). The reference _Model-2_ is subjected to the dead load \(g\) = 2 kN/m combined with various live load components (\(q\) = 2 kN/m, 3 kN/m, and 4 kN/m) applied to the bridge half span, as shown in Figures 3 and 10. The vertical displacement due to the live load (\(u_{q}\)) is estimated by subtracting the dead load-induced displacement component (\(u_{g}\)) from the total displacement predicted by the model (\(u_{tot}\)). The magnitude of the dead load applied to _Model-1_ and _Model-3_ is tailored iteratively to ensure the same displacement \(u_{q,\text{max}}\) as the reference _Model-2_. Table 3 presents the calculation results in terms of the maximum vertical  
Figure 11: The ribbons’ cross-sections assumed for the parametric analysis. Figure 12: Symmetry conditions applied in the bridge _Model-3_. displacement (\(u_{tot}\)) and the corresponding design component \(u_{q}\). This table also includes the thrust force (\(H\)) in the ribbon and the extremum stresses in the cross-section (\(\sigma_{min}/\sigma_{max}\)) estimated in the section where the maximum vertical displacement appeared. In _Model-1_, that section was close to the quarter of the span. On the contrary, the maximum vertical displacement was localised near the mid-span of _Model-2_ and _Model-3_. Bold numbers in Table 3 indicate the design values of the vertical displacement. _Model-1_ with flexible bands requires a considerable amount of the dead load to stabilise the displacement induced by the asymmetrically distributed load \(q\). The triple dead load doubles the thrust force regarding _Model-2_. In both models (_Model-1_ and _Model-2_), the maximum stresses in the ribbons are similar; however, the minimum stresses are much lower in the reference case because of the bending effect in the flexural-stiff ribbon.