## 2 Materials and Methods
### Structural Beam Prototype
#### 2.3.2 Prediction of Flexural Performance  
Taking into account the mechanical characterisation of the recycled concrete carried out in this work, it is proposed to predict the flexural performance of the manufactured beams in order to subsequently validate the experimental tests. In particular, the flexural strength and deflection of the beams are studied. For this purpose, an approach based on Eurocode 2 for the design of concrete structures [58] is proposed. Figure 5: Dimensions and reinforcement of beam prototypes. According to the design configuration proposed for the beams and their reinforcement, the equilibrium equation that determines the ultimate moment of their section is the following Equation (1):  
\[M_{u}=f_{ck}\cdot\lambda\cdot\kappa\cdot\mathrm{b}\cdot\left(\frac{h}{2}-\lambda \cdot\frac{x}{2}\right)-A_{s}\cdot\sigma_{s}\cdot\left(\frac{h}{2}-\mathrm{d}\right) \tag{1}\]  
where \(f_{ck}\) is the characteristic compressive strength of concrete; \(\lambda\) is the coefficient defining the effective depth of the compression zone (\(\lambda=0.8\) for \(f_{ck}\leq 50\) MPa); \(x\) is the depth of the neutral axis; \(b\) and \(h\) are the base and height of the beam section, respectively; \(A_{s}\) is the area corresponding to the reinforcement; and \(\sigma_{s}\) is the characteristic strength of concrete steel. Based on the test and load configuration, the maximum total load and maximum deflection of the beam under ultimate moment conditions can be calculated using the following Equations (2) and (3):  
\[F_{max}=2\cdot\frac{M_{u}}{a} \tag{2}\]  
\[y_{max}=\frac{M_{u}}{24\cdot E\cdot I_{ef}}\cdot\left(3\cdot L^{2}-4\cdot a^{2}\right) \tag{3}\]  
where \(M_{u}\) is the ultimate moment; \(E\) is the elastic modulus of concrete; \(L\) is the beam length; \(a\) is the shear span and \(I_{ef}\) is the effective moment of inertia, defined by Equation (4):  
\[I_{ef}=\frac{I_{cr}}{1-\left(1-\frac{I_{cr}}{I_{g}}\right)\cdot\left(\frac{M_ {cr}}{M_{u}}\right)^{2}} \tag{4}\]  
Figure 6: Platform and set up for four-point bending test of beam prototypes. where \(I_{cr}\) is the cracked moment of inertia; \(I_{b}\) is the gross moment of inertia; and \(M_{cr}\) is the moment to cause cracking, which depends on the tensile strength of the concrete. In order to provide a more accurate calculation, the usual design safety coefficients for concrete and steel strength are not used. Instead, robust design techniques are implemented to incorporate material uncertainty by adopting a probabilistic approach based on reliability. This approach takes advantage of the larger population of data obtained by the DIC technique to calculate a significant mean and covariance associated with each of the parameters. In this way, a probabilistic density function (PDF) of the main mechanical properties can be extracted. The simulation strategy is based on using a Monte Carlo Sampling (MCS) method to generate a large set of random numbers that match the previously calculated PDFs. The proposed ultimate moment and maximum load equations are then evaluated for each of these parameters to obtain a probability density function for the corresponding results. Considering the reliability analysis, the probability of failure can be evaluated according to the following Equation (5):  
\[P_{f}=Prob[G(Y)\leq 0]=\int\limits_{G(Y)\leq 0}p_{y}(Y)dY \tag{5}\]  
where \(P_{f}\) is the probability of failure; \(G\) is the performance function; \(Y\) is the variable's vector and \(p_{y}\) is the joint probability density function. Taking into account that the analytical theoretical models include several variables and that a probabilistic analysis is chosen, it is important to know the influence of each input on the final result. To this end, the MCS strategy allows an adequate sampling of the parameters used in the model, thus generating equiprobable situations to carry out a sensitivity analysis.