# Effects of Flexural Stiffness on Deformation Behaviour of Steel and FRP Stress-Ribbon Bridges
## 4 Numerical Modelling
### Parametric Analysis
The increase of the ribbons' flexural stiffness of the ideal _Model-3_ further increases the compressive stresses in the ribbons. The dead load reduction from 20% to 60% of _Model-3_ is achieved (Table 3) and the thrust force decreases by 2.5 times regarding _Model-2_. The maximum stresses in the ribbon of all models are similar. The simulation results (Table 3) demonstrate that the thrust forces decrease with increasing the ribbon flexural stiffness. That also increases the compressive stresses in the cross-section because of the bending effect. As mentioned in Section 2, the limitations of compressive stresses can be necessary in some cases (e.g., applying FRP profiles). Figure 13 shows functional relationships between the flexural stiffness of the ribbon-bands and the extremum stresses in the cross-section having the maximum displacement \(u_{tot}\). Second-order polynomials approximate the calculated points. This figure demonstrates the application example of condition (9) in the engineering design of the flexural-stiff steel ribbons. The figure shows that _Model-2_ fulfils the stress-limitation requirement--the compressive stresses in the ribbon do not exceed 20 MPa. A bridge _Model-4_ is introduced to investigate the applicability of the proposed methodology for designing stress-ribbons made of glass fibre-reinforced polymer (GFRP). The loading conditions of _Model-2_ and _Model-4_ are the same. The choice of cross-section geometry is based on Equation (3). The GFRP profile elasticity modulus \(E=28\) GPa and strength \(f_{d}=190\) MPa are assumed by following the producer recommendations [30]. Equations (3), (8) and (7), define the moment of inertia, the profile height, and the cross-section area. The inertia moment \(l_{y}\) is increased, compensating for the effect of a relatively low elasticity modulus of GFRP, for maintaining the same flexural stiffness of the ribbon, as _Model-2_ (Figure 14a). Figure 14b shows the resultant dimensions of the cross-section. Table 4 gives the physical characteristics of alternative cross-sections. Figure 13: Extremum stresses in the cross-section having the maximum vertical displacement. Table 3 presents the FE simulation results of _Model-4_ in the grey-filled row. As expected, the minimal compressive stresses among the analysed bridge systems were induced in the GFRP ribbons. Although the flexural stiffness (\(El_{y}\)) of both bridges (i.e., _Model-2_ and _Model-4_) is the same, more significant vertical displacements are characteristic of the bridge model with GFRP ribbons. It is a consequence of the high deformability of _Model-4_, calling for an excessive increase to the axial stiffness of GFRP ribbons because of the absence of standard profiles [30], satisfying both the axial stiffness and flexural stiffness limitations. A non-standard rectangular \(127\times 27\) mm (depth \(\times\) width) cross-section can be an example satisfying both the axial stiffness and the flexural stiffness conditions. The unit weight, \(m\), of this profile is equal to 6.18 kg/m. Further reduction of the weight is impossible since it reduces the stiffness \(EA\). That makes the GFRP profiles inefficient for this structural application. The application of carbon fibre-reinforced polymers (CFRP) presents an acceptable solution to the above problem [4,15]. The elasticity modulus of such profiles can vary from 88 GPa to 245 GPa depending on the carbon filament content; the tensile strength can reach 2.5 GPa [31]. Thus, an alternative _Model-5_ has ribbons made of CFRP profiles. In this example, the CFRP elasticity modulus and strength are equal to 148 GPa and 1.1 GPa. Figure 14c shows that the cross-section composed following the design procedure described in Section 2. The cross-section shape was changed because of the CFRP composite manufac  
Figure 14: Alternative cross-sections: (**a**) steel; (**b**) GFRP; (**c**) CFRP. GFRP: glass fibre-reinforced polymer; CFRP: carbon fibre-reinforced polymer.