## 4 Results
### Analysis of the Stiffness of RC Beams Strengthened with CFRP Strips
Figure 12: Idealization of M–\(\kappa\) relationship for reinforced concrete element according to [38]. Figure 13 presents the curvature and stiffness diagrams both depending on the bending moment for the tested beams. These plots show the curvature changes for the sections in the middle of the spans of these beams (Figure 13a). For these cross-sections, the stiffnesses were then calculated and shown in Figure 13b. Boow-type beams (relieved before the moment of their strengthening using CFRP strips) always represent two curves in the drawings. The first one presents the characteristics of the beams taking into account their initial deflections arising after stage 1 of the test (after the loading and relieving process before the strip application), whereas the second one neglects the initial state, assuming the end of stage 1 of the tests as the beginning of the analyses, i.e., the moment when the beam is already strengthened with the strip. For strengthened elements that are part of the structures in operation, the stiffness changes obtained in the load ranges corresponding to the actual operation of these structures are of the greatest importance. A structure without strengthening is assumed to safely operate up to a load level that corresponds to its design capacity. Therefore, this load range has been analyzed in more detail. Table 7 lists the curvature and stiffness of the beams for the load P = 60 kN, which corresponds to the design load capacity of a reinforced concrete beam without any strengthening (Br type). Table 8 summarizes the relative curvature reductions and relative stiffness of beams strengthened with the strip compared with the curvature and stiffness of the beams without strengthening (the value of 1 for the beam with no strengthening is some reference level for the analyzed parameters of beams strengthened with the strip). The summary shows that for the load level that corresponds to the design load-carrying capacity of the reference beam with no CFRP strips, the most efficient strengthening is obtained for a fully relieved beam before strengthening with the CFRP strips. For Boow-type beams analyzed from the moment the strip is glued, the greatest increases in stiffness and the greatest reduction in curvature values are obtained. Figure 13: Stiffness analysis based on experimental results. (**a**) Curvature-bending moment diagram at the center of the span of beams; and (**b**) bending moment–stiffness diagram for the cross-section at the midspan of beams. For all strengthened beams, the presence of the CFRP strip begins to play a decisive role in increasing the stiffness of the beams only from the load level at which the main reinforcement in the beam without the strip becomes plasticized (\(\sim\)M = 40 kNm). Figure 14 shows the dependence of the beam curvatures on the bending moments, calculated on the basis of the beam deflection lines obtained in the FEM analyses. It is worth noting that the beam stiffness, calculated on the basis of numerical analyses, qualitatively confirms the results obtained in laboratory tests, especially when the analysis covers the entire load range of the beams. Nevertheless, the results of the quantitative analysis (summarized in Table 9) show a large discrepancy between the stiffness values obtained in the experiment and numerical analyses, especially when it relates to a specific value of the applied load. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Beam Type** & \
\begin{tabular}{c} **Curvature \(\kappa\)** \\ **[1/m]** \\ \end{tabular} &
\begin{tabular}{c} **Stiffness B** \\ **[kNm2]** \\ \end{tabular} \\ \hline
1 & Br & 0.008125 & 3692.4 \\
2 & Bz & 0.007812 & 3840.7 \\  & Boow—Stage 1 + 2 & 0.008184 1  & 3666.2 \\
3 & Boow—Stage 2 & 0.006692 2  & 4483.6 \\
4 & Bow & 0.008659 & 3464.7 \\
5 & Bopw & 0.009002 & 3333.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Values of curvature and stiffness of CFRP strengthened beams at load of 60 kN (which corresponds to design load-carrying capacity of reference beam) based on experimental results.