# Effects of Flexural Stiffness on Deformation Behaviour of Steel and FRP Stress-Ribbon Bridges
## 4 Numerical Modelling
### The Numerical Models' Verification  
Two FE models of the bridge prototypes described in Section 3 are built up; the FE models and the prototypes have the same notation. Figure 8 shows the FE models and boundary conditions. Non-linear isoparametric shell-brick elements with nine integration points in the shell plane are used. The 3D brick elements have 20 nodes and a layered structure [29]; six internal layers are distributed in the FE shell plane. Figure 9 shows the layered FE structure. The linear-elastic isotropic material model of the steel is used for the FE analysis, assuming the elasticity modulus \(E\) = 208 GPa (experimental value) and [PERSON]'s ratio \(\
u\) = 0.30. Figure 8: The bridge prototypes’ FE models: (**a**) model with flexible bands (_Model-1_); (**b**) model with flexural-stiff ribbon-profiles (_Model-2_). A uniaxial deformation state is characteristic of _Model-1_. Therefore, the 20 \(\times\) 20 mm FE mesh size is applied. The total number of finite elements in the model is 2000. However, the reference [29] recommends at least four finite elements in the bending plane to simulate the bending effects. Thus, the 18 \(\times\) 18 mm FE mesh size is used for _Model-2_. Six elements compose the vertical wall of the section (Figure 9). The same element size is used in the flanges. Steel braces connect the flexural-stiff ribbons, preventing out-of-plane displacements of the structure. The same system was used in the corresponding bridge prototype (Section 3). The total number of finite elements in _Model-2_ is 14,702. Table 2 compares the numerical predictions and test results. The readings of the strain gauges \(T_{i}\) (Figure 5) determine the thrust forces under the assumption of elastic behaviour of the steel ribbons (\(E_{s}\) = 208 GPa). A high confidence level of the deformation predictions is evident: the prediction error does not exceed 7%. The disagreement can result from a relatively coarse FE model discretisation and the geometry inaccuracies of the physical prototypes. The actual difference between the initial sag (\(f_{0}\)) of two ribbons of _Model-2_ was equal to 8.7 mm, whereas the FE model assumes the average sag value. The acceptable adequacy of the numerical predictions enables applying the FE methodology to analyse the ribbon geometry effect on resisting the kinematic displacements. Figure 10 shows the modelling example of the idealised bridge prototypes. The idealisation assumes the identical initial sag (\(f_{0}\) = 135 mm) and loading conditions (\(g\) = 2 kN/m + \(q\) = 2 kN/m) for both _Model-1_ and _Model-2_. The horizontal movements of the supports are neglected to eliminate the boundary condition influence. The experimental values (Section 3) determine the remaining parameters of the numerical models. The modelling results (Figure 10) reveal two aspects. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & \multicolumn{4}{c}{_Model-1_} & \multicolumn{4}{c}{_Model-2_} \\ \cline{2-9}
**Source** & \multicolumn{2}{c}{**Symmetrical Load**} & \multicolumn{2}{c}{**Asymmetrical Load**} & \multicolumn{2}{c}{**Symmetrical Load**} & \multicolumn{2}{c}{**Asymmetrical Load**} \\  & & \multicolumn{2}{c}{**(\(q\)/\(g\) = 0.49)**} & & & & \multicolumn{2}{c}{**(\(q\)/\(g\) = 0.49)**} \\ \cline{2-9}  & \(u_{max}\)**mm** & \(H\)**, **kN** & \(u_{max}\)**mm** & \(H\)**, **kN** & \(u_{max}\)**mm** & \(H\)**, **kN** & \(u_{max}\)**, **mm** & \(H\)**, **kN** \\ \hline Test & 29.2 & 36.0 & 29.3 & 29.5 & 49.9 & 41.3 & 44.8 & 30.2 \\ Finite element simulations & 30.5 & 37.9 & 29.5 & 31.7 & 48.8 & 43.2 & 42.0 & 30.6 \\ Difference & –4.5\% & –5.0\% & –0.7\% & –6.9\% & 2.2\% & –4.4\% & 6.2\% & –1.3\% \\ \hline \end{tabular}
\end{table} 
Table 2: Test and numerical results: maximum vertical displacements (\(u_{max}\)) and thrust forces (\(H\)). Figure 9: Shell-brick finite element and corresponding layered model. On the one hand, the total areas (\(S\)) under the live load-induced displacement curve, determining the system's deformation energy, are almost identical for both prototype models.