# Numerical Design and Optimisation of Self-Compacting High Early-Strength Cement-Based Mortars
## 2 Central Composite Design
### Experiment Definition and Planning  
The present work sought to analyse 50 mortar mixes published by Maia [25] using DoE, the statistical methodology to analyse data and predict properties. It was necessary to integrate simple and conventional statistical methods into the experimental design methodology to draw statistically sound conclusions from the experiment. Subsequently, the commercial software Design-Expert was employed in the current study to develop a polynomial regression model. Maia's data set [25] corresponds to a full central \(2^{5}\) composite design augmented by 10 axial runs plus 8 central runs. It allows for the development of a polynomial regression model for each response variable as a function of the five design variables considered in that study (key factors): (i) Vw/Vc--water to cement volume ratio; (ii) Sp/p--superplasticiser to powder mass ratio; (iii) Vw/Vp--water to powder volume ratio; (iv) Vs/Vm--sand to mortar volume ratio; and (v) Vfs/Vs--fine sand to total sand volume ratio. In addition to the 50 mixes previously described, 14 extra mixes were produced to allow for comparing and validating the results of the response models. The effect of each key factor was evaluated at five levels \(-\alpha\), \(-1\), \(0\), \(+1\), and \(+\alpha\), as shown in Table 1. To make the design rotatable, the value of \(\alpha\) was equal to nF\({}^{1/4}\), where nF is the number of points in the factorial part of the design. In the current study, this corresponds to \(\alpha\) equal to 2.378. The self-compacting mortars were produced with the following commercial materials available (in Portugal): (i) cement CEM I 42.5 R with a specific gravity of 3100 g/m\({}^{3}\), (ii) limestone filler with a specific gravity of 2680 kg/m\({}^{3}\), (iii) siliceous natural sands with a maximum particle size of 4 mm, specific gravity of 2620 kg/m\({}^{3}\), and water absorption of 0.40%, (iv) fine siliceous sand with a maximum particle size of 2 mm, specific gravity of 2630 kg/m\({}^{3}\), and water absorption of 0.20%, and (v) a polycarbonate based superplasticiser with a specific gravity of 1080 kg/m\({}^{3}\) and solid content of 40% [25]. Four dependent variables (i.e., response variables) were considered: (i) slump-flow diameter (D-flow); (ii) the time in the V-funnel (T-funnel); (iii) flexural strength at 24 h, determined by the three-point loading method (F,24 h); and (iv) uniaxial compressive strength at 24 h (Rc,24 h). Table 2 summarises the response variables and is followed by a brief explanation of the measurement procedures to access the dependent variables. Immediately after production, the (mini) slump flow was assessed following the procedures of EFNARC [26]. The procedure consists, in brief, in filling the mini cone with the fresh composite and then taking the cone off and observing the flow capacity on a plane surface. When the fresh composite stops moving, two perpendicular diameters are measured, and the average value is considered the flow diameter (D-flow) in mm. Additionally, the V-funnel test was performed to assess the viscosity and passing ability using the mini funnel according to EFNARC [26]. In brief, the fresh composite was poured into the funnel after production. Then, the trap door was opened, allowing the fresh composite to flow out under gravity, and the time for the discharge to complete (the flow time), i.e., when light is seen from above through the funnel, was recorded. These results showed a typical trend of an increase in slump-flow diameter with a reduction in flow time. After fresh tests, three specimens 40 \(\times\) 40 \(\times\) 160 [mm\({}^{3}\)] were moulded. The specimens were demoulded at 24 h, and the mechanical strength was assessed following EN 196-1 [27].