# Real Geometrical Imperfection of Bow-String Arches--Measurement and Global Analysis
## Pilot Study
### Comparison of Results  
Some results from the presented study are compared in Table 1. Maximum stresses in four chosen cross-sections of the right arch are compared. The values in Table 1 were found as the maximum stresses picked from the FEM analyses at the same corner point of each arch cross-section marked in Figure 9 from A1 to A4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Arch Cross- Section**} & \multirow{2}{*}{**Dominant Elements/Software**} & \multicolumn{5}{c}{**Type of FEM Analysis**} \\ \cline{3-6}  & & **LA** & **LI\({}_{\text{RG}}\)A** & **GN\({}_{\text{RG}}\)A** & **GN\({}_{\text{TG}}\)A** \\ \hline A1 & shells/ADINA & \(-190.8\) & \(-196.6\) & \(-198.9\) & \(-196.4\) \\  & beams/SCIA Eng. & \(-202.7\) & \(-207.2\) & \(-208.8\) & \(-209.8\) \\ A2 & shells/ADINA & \(-181.5\) & \(-193.5\) & \(-194.8\) & \(-180.3\) \\  & beams/SCIA Eng. & \(-214.2\) & \(-228.8\) & \(-230.6\) & \(-219.5\) \\ A3 & shells/ADINA & \(-183.6\) & \(-180.5\) & \(-182.0\) & \(-185.3\) \\  & beams/SCIA Eng. & \(-208.8\) & \(-213.4\) & \(-214.6\) & \(-217.5\) \\ A4 & shells/ADINA & \(-186.1\) & \(-185.9\) & \(-186.6\) & \(-187.2\) \\  & beams/SCIA Eng. & \(-179.0\) & \(-179.8\) & \(-180.0\) & \(-181.0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Maximum corner stresses in the selected arch cross-sections. Figure 8: Horizontal displacement of the arch axis in the assumed global geometric imperfection used in FEM analyses: (**a**) the real geometric imperfection based on the data obtained from the 3D laser scanning; (**b**) the theoretical geometric imperfection in the form of the first buckling mode with the amplitude scaled on the basis of allowed tolerances. Figure 9: Side view of the bridge with the designation of the four selected arch cross-sections, where the stresses were analysed (dimensions in millimetres). From the comparison it can be stated that geometric imperfections had a smaller influence on the stresses than expected. The consideration of the measured imperfection produced stresses not far from analyses, where the theoretical geometrical imperfections derived from the allowed tolerances were taken into account. The graph in Figure 10 shows a percentage increase in stresses in the analyses with imperfections included in comparison with linear analyses without any imperfections (LA). Except for the differences between both software analyses, only a few percent of effect is evident. Surprisingly, the linear analyses with the real geometrical imperfections included (LI\({}_{\text{RG}}\)A) were very close to their more complex geometrically nonlinear alternative (GNI\({}_{\text{RG}}\)A). The absolute values of differences are small, but relative variation between the four chosen arch cross-sections is evident. The real deformation can be sometimes very different from the theoretical shape of buckling, thus it could probably better reflect the increase in stresses in the most affected cross-sections. However, if the values are compared, it can be clearly seen that the geometrical imperfection influenced stresses only slightly in the case of this bridge. At the same time, it has to be pointed that almost three times higher differences were produced by the application of the different FEM model than by the different type of analysis. Such a comparison is given in Figure 11, where the percentage differences in the stress obtained by the SCIA Engineering model are presented in comparison to the stresses produced by the analyses in the ADINA software. In the cross-sections near the arch foot and in the middle of the span the difference are smaller. The influence of using 2D versus 1D finite elements, respectively, is evident even though both the models acted statically very similarly in global comparison. For instance, the differences in the calculated deformations (deflections) between the models were only about 1%.