## 4 Test on Beams
### Elastic Tests on the Extracted Beams  
Before retrofitting, elastic tests (three-point loading tests) were performed on the extracted beams DB1 and DB2 to measure their deformation shapes and stiffness values. The maximum load applied during the elastic tests should guarantee the elastic behavior of the beam during the tests. This load was calculated for each beam considering about half of the load value that produces a moment at section SM in Figure 1 equal to the section resisting moment. The section resisting moments were calculated considering the steel reinforcement and the beam geometries given in SS2.1 and the steel yield stress indicated in SS2.2 for each beam. The resulting maximum loads were 156 kN and 172 kN for the beams DB1 and DB2. The applied elastic loads on the beams were 92 kN and 72 kN for the beam DB1 and DB2. The load/deflection curves at different points along each beam (at 0.25L, at two points near 0.5L and at 0.75L, Figure 5b) are shown in Figure 6a. The beam stiffness (load/deflection) values were essentially constant at each measurement point during the test on each beam: this confirms that the beam behavior was elastic. The beam deformation shapes corresponding to the maximum applied load are given in Figure 6b where the maximum beam deflections at midspan are \(-\)0.74 mm and \(-\)1.04 mm for the beam DB1 and DB2. The beam stiffness values calculated by the beam load-deflection curve at midspan (Figure 6a), were 124.3 kN/mm and 69.2 kN/mm for the beam DB1 and DB2: the stiffness of DB1 was 80% greater than the one of DB2. Stiffness of beams depends on the Young's modulus of concrete and steel, the beam length, and the area moment of inertia of section. Cracking is expected in section of RC members subjected to load with reduction of the area moment of inertia with respect to the one of the gross sections before cracking. The values of [PERSON]'s modulus of concrete can be estimated by Equation (3) in [65]:  
\[\mathrm{E_{c}}=22000\cdot(\mathrm{f_{cm}}/10)^{0.3} \tag{3}\]  
where \(\mathrm{f_{cm}}\) is the mean compressive cylindrical strength of concrete. The values of \(\mathrm{f_{cm}}\) can be obtained by means of the mean compressive cubic strength given in SS2.2.1 for each beam. The Young's modulus values are 23.3 GPa and 21.3 GPa for the beam DB1 and DB2. The elastic beam deflections at midspan for a given applied load can be calculated by the elastic curve equation considering the concrete gross section (pre-cracking stage) properties, the beam length, and the calculated Young's moduli of concrete. Based on these results, the beam stiffness values before cracking at midspan are 500.0 kN/mm and 349.7 kN/mm for the beams DB1 and DB2. From the beginning of the elastic tests, the experimental stiffness values are smaller than the ones calculated considering the gross section and therefore the beam sections were already cracked before these tests. This is expected because the beams were extracted from a building subjected to loads. The experimental flexural rigidity (\(\mathrm{E_{c}I^{exp}}\)) can be evaluated by Equation (4) based on the elastic curve equation. \[\mathrm{E_{c}\cdot I^{exp}}=\mathrm{F_{emax}\cdot L^{3}}/(48\cdot\mathrm{g_{ emax}}) \tag{4}\]  
where \(\mathrm{F_{emax}}\) and \(\mathrm{g_{emax}}\) are the maximum load and the corresponding beam deflection at midspan during the elastic test. The experimental flexural rigidity is included in the range with:  
1. The lower limit equal to the fully cracked flexural rigidity (\(\mathrm{E_{c}I^{II}}\)) calculated by Equation (5); \[\mathrm{E_{c}\cdot I^{II}}=\mathrm{E_{c}\cdot\left(\frac{1}{3}\cdot\mathrm{b \cdot y_{c}}^{3}+\mathrm{E_{s}/E_{c}\cdot A_{s}\cdot\left(d-y_{c}\right)^{2}} \right)}\] (5)
2.