# Experimental Study of 3D Concrete Printing Configurations Based on the Buildability Evaluation
## 4 Results and Discussion
### Height Reduction Test
#### 4.4.3 Hrr Printing Simulation  
To verify Equation (6), a VBA code in Excel simulated the printing process and the slump steps when \(BI=n\) ranging from 1 to 100, by finding the minimum \(\prod_{i=1}^{n}x_{i}\), via traversing \(x_{i}\), where \(i\in\left(1,n\right)\), \(n\in\left(1,100\right)\), and \(x_{i}\in\left(0,1\right)\).  
As shown in Figure 16, the slump steps and the \(x_{i}\) are in columns A and B, while the printed filaments are simulated and shown in columns J to ZZ. Each layer takes two steps to finish its slump and stop losing height, with \(HRR\)\(x_{1}\) and \(x_{2}\), where \(0<x_{1},x_{2}<1\), as shown in cells B8 and B9. Then, the total height reduction ratio can be calculated as in cell D7 by \(r=1-x_{1}x_{2}\). Twenty layers are listed in the Excel file. Column F lists the layer counting number, column G lists the accumulated height for the current layer, and the cells of column H are the gap heights reserved for the new layer.  
In the first printing cycle, layer 1 was printed and shown in column J, having the \(x_{1}\) slumped in J8. In the second printing cycle, layer 2 was printed and shown in column K with \(x_{1}\) slumping in K9, while layer 1 continues with \(x_{2}\) slumping in J9. In the third printing cycle, layer 3 was printed and shown in column L, and has \(x_{1}\) slumping in L10, while layer 2 continues its \(x_{2}\) slumping in cell K10 and layer 1 remains at its height. The subsequent simulation continues the same pattern. After each layer has been \"printed\", the accumulated height and the gap height for the next layer are calculated.  
The VBA \"printing\" code runs for every combination of \(x_{1}\) and \(x_{2}\), each ranging from 0 to 1.0, with an increment of 0.001. One \"printing\" succeeds only if each gap height \(H<1.2\), and the minimum \(x_{1}\times x_{2}\) with succeeding \"printing\" can be found for the maximum \(HRR=1-x_{1}\times x_{2}\). Eventually, the simulated printing calculation found that \(HRR\) is 0.167 for this two-step slump material when GHT \(\delta=0.2\). The same \(HRR=0.167\) was then calculated for \(BI>2\).  
Figure 15: Printing process with a two-step slump material.  
In addition, several simulations were conducted for \(BI>0\) with \(\delta\in(0,0.3)\) and obtained the same result with Equation (8), proving that the multi-layer slump can be considered as a one-layer slump to characterize the \(HRR\) criteria.  
Conclusively, based on the \"printing\" simulation and theoretical calculation, the \(HRR\) of PM can be correlated with the printing GHT \(\delta\) as shown in Equation (9), regardless of the multi-step slump and the compression from self-weight and weight from subsequent layers in a printing process.  
\[HRR=r=\frac{\delta}{\delta+1} \tag{9}\]  
For the GHT \(\delta\) = 0.2, the criteria value of \(HRR\) is 0.167. For printing complex geometry, the GHT should be smaller, and the material with low \(HRR\) is then required.  
In actual printing practice, the height reduction ratio test will be conducted on the maximum \(BI\) layers, and the \(HRR\) can be calculated as:  
\[HRR_{BI}=r_{BI}=1-\frac{\textit{real height measured on BI layers}}{h\times BI} \tag{10}\]  
The slump of the cementitious material cannot be avoided since the force of gravity on the particles is stronger than the cohesion in the material matrix at the beginning of the hydration, even if most of the PMs have admixtures of thickener to increase their viscosity. Like \(BI\), \(HRR\) is not for the material only, but related to the GHT \(\delta\), the requirement for the printing accuracy of PO, a reasonable and direct basis for the specific slump data of the PM. Thus, the \(HRR\) is more suitable than the height deformation tests of 20 layers [20], 7 layers [21], or 2 layers [7; 19]. Furthermore, with the precise slumping data, the height of the printed filaments can be predicted, and a more complex toolpath can be realized, such as by varying the layer height [39]; then the height tolerance can be larger than needed.