# Structural Behavior of Large-Scale I-Beams with Combined Textile and CFRP Reinforcement
## 5 Discussion
### Prediction of Deflection  
At serviceability limit state (SLS), a deformation at midspan of \(w=L/250=28\) mm is permitted, corresponding to the limit defined in [64]. To compare the quality of the calculation approaches, the moment applied in the test at a deflection of 28 mm is used as input value for the deflection calculation. A manual calculation with the method by [PERSON] [76], a piecewise integration based on the moment-curvature diagram and a non-linear FE-calculation with _ABAQUS_ were performed. In Equation (3), the adaption of the original [PERSON] equation by [PERSON] and [PERSON] [77; 78; 79] for calculation of the effective moment of inertia \(I_{\mathrm{e}}\) is given:  
\[I_{\mathrm{e}}=\frac{I_{\mathrm{cr}}}{1-\gamma\big{(}\frac{M_{\mathrm{cr}}}{M _{\mathrm{a}}}\big{)}^{2}\Big{[}1-\frac{I_{\mathrm{cr}}}{I_{\mathrm{g}}}\Big{]} }\leq I_{\mathrm{g}}\quad\quad\text{for}\quad\quad M_{\mathrm{a}}\ \geq\ M_{\mathrm{cr}} \tag{3}\]  
with:  
\(\gamma=1.72-0.72\big{(}\frac{M_{\mathrm{cr}}}{M_{\mathrm{a}}}\big{)}\), Factor for structural system with continuous loads according to [78; 79]  
\(I_{\mathrm{cr}}\)_and_\(I_{\mathrm{g}}\), moment of inertia of cracked and uncracked section  
\(M_{\mathrm{cr}}\)_and_\(M_{\mathrm{a}}\), moment at first cracking and moment at SLS  
The moment of inertia of the cracked section, \(I_{\mathrm{cr}}\), was calculated considering all longitudinal reinforcement layers including longitudinal yarns from shear reinforcement with their respective lever arm. Using \(I_{\mathrm{e}}\), deflection in mid span is calculated with equations from linear-elastic structural analysis. The moment at first cracking has a large influence on calculation results. Thus, the accuracy of the results depends to a large extent on concrete tensile strength. Eigen stresses (e.g., resulting from drying shrinkage) need to be considered, as they reduce the first cracking moment. Table 6 shows the influence of \(M_{\text{cr}}\) on predicting deflection. Using tensile strength \(f_{\text{ct}}=1.0\cdot f_{\text{ct,sp}}\) as given in Table 4 and the section modulus \(W_{\text{y}}\) of the uncracked cross-section, the theoretical moment at first cracking is 111.3 kNm. [PERSON] and [PERSON] [79] propose a reduction to 80% to account for eigen stresses. As FT01 was pre-cracked, no experimental first cracking moment could be determined. Employing uncracked specimen FT03, the ratio of experimental to theoretical first cracking moment results in 57.6%. Applying this reduction to calculation for FT01 and FT02 gives the best results. The prediction for higher reinforced FT02 with its smaller ratio of \(M_{\text{af}}\)/\(M_{\text{cr}}\) is less affected by changes of \(M_{\text{cr}}\), which is in accordance to statements in [80]. Results of the second calculation method by piecewise numerical integration with [PERSON]'s rule based on moment-curvature (\(M\)-\(\kappa\)) relation are given in Table 6 (center). The theoretical \(M\)-\(\kappa\) curve (Figure 16) is derived using material characteristics and geometrical properties of cracked and uncracked section. The four points defining the yellow curve in Figure 16 (x = \(\kappa\), y = \(M\)) are P0 [0], P1 [0.576% \(M_{\text{cr}}\) / (\(E_{\text{cm}}\cdot I_{\text{g}}\)) \(\mid\) 0.576% \(\cdot\)\(M_{\text{cr}}\)], P2 [1.3 \(\cdot\) 0.576% \(\cdot\)\(M_{\text{cr}}\) / (\(E_{\text{cm}}\cdot I_{\text{cr}}\)) \(\mid\) (1.3 \(\cdot\) 0.576% \(\cdot\)\(M_{\text{cr}}\))] and P3 [\(M_{\text{u,cal}}\) / (\(E_{\text{cm}}\cdot I_{\text{cr}}\)) \(\mid\)\(M_{\text{u,cal}}\)]. \(M\)-\(\kappa\) experimental is taken from strain gauge measurements at the compression chord and strains in the tension flange determined by LVDT (D9C and Bi2 in Figure 3). Both calculations yield good predictions of the obtained experimental values. Taking into account shear strains would further improve prediction of deflection quality. However, the results also rely on a realistic estimate of first cracking moment.