# Optimization of Glulam Regular Double-Tapered Beams for Agroforestry Constructions
## 3 Results and Discussion
### Number of Beams Depending on the Roof Length and Snow Load Intervals  
For an analysis at a fixed roof length of 45 m, Table 10 shows the data count for each value of beam spacing as a function of snow load intervals, which is also illustrated in Figure 9.  
Figure 8: Resulting plane for the multiple linear regression of the relationship between Sp, Span and Snow for Hp = 160 mm.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Estimate** & **Standard Error** & **t** & **\(p\)** \\ \hline (Constant) & 0.736 & 0.005 & 141.102 & \textless{}0.001 \\ Snow & 0.209 & 0.003 & 62.219 & \textless{}0.001 \\ Span & 0.001 & 0.000 & 4.209 & \textless{}0.001 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Parameter estimation (slopes and constant) of the multiple regression model.  
Since the beam spacing is constrained by the length of the roof, the variable was considered in order to assess the optimal number of beams, which was studied for 30 m, 45 m, 60 m and 75 m roof length values. The ANCOVA model presented in Table 11 indicates that, independently, the variable Roof length as well as the factor Snow range have a significant effect on Nb (\(p\) < 0.05). Moreover, the factor Snow range in combination with the variables Roof length and Snow (F (1.59) =37.876, \(p\) < 0.05) was also found to significantly affect the variable Nb. These results demonstrate that the relationship between the variables is a model with interaction. It is worth noting that the data did not exhibit heteroscedasticity, and there were no significant deviations from the normal distribution ([PERSON]'s test: 0.212, \(p\) > 0.05; Shapiro-Wilk test: 0.992, \(p\) > 0.05).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Beam Spacing (m)** & **Snow Load** & **Frequency** & **Percentage (\%)** & **Cumulative** \\  & **Range (kN/m\({}^{2}\))** & & & **Percentage (\%)** \\ \hline  & (1.0, 1.4) & 39 & 18.8 & 18.8 \\
4.1 & (1.6, 2.0] & 52 & 25.0 & 43.8 \\  & [0.8, 1.0] & 39 & 18.8 & 62.5 \\
4.5 & [1.4, 1.6] & 39 & 18.8 & 81.3 \\
5.0 & (0.6, 0.8) & 26 & 12.5 & 93.8 \\
5.6 & [0.5, 0.6] & 13 & 6.3 & 100.0 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Beam separation frequencies according to snow load intervals.  
Figure 9: Beam separation depending on the snow load ranges.  
The optimum beam separation ranges from 4.1 m to 5.6 m for the different snow load ranges. It is observed that, as the snow load increases, beam spacing decreases, with the exception of the interval [1.4, 1.6], in which the separation increases up to 4.5 m. As studied in Section 3.4, the purlin height increases from 160 to 200 mm at snow loads of 1.4 kN/m\({}^{2}\), which allows for a greater beam separation up to snow loads of 1.6 kN/m\({}^{2}\). In 81.3% of cases, beam separation is either 4.1 or 4.5 m, similar values to those usually used in practice as well as those obtained in the truss optimization in [29].  
Table 12 presents the estimates of the effects of Roof length, Snow range and Snow on Nb. The relationship of Roof length and Snow with Nb determines the slope of the plane.  
To determine the optimal value of Nb based on Roof length, Snow range and Snow, multiple regression planes were constructed using Equation (25).  
\[\begin{split}\text{For Snow range }1=[0.5,\ 1.0]\text{ or }[1.4,\ 1,6]\rightarrow\text{Nb}=1.203+0.191\times\text{Roof length}+0.027\times\text{Roof length}\times\text{Snow},\\ \text{For Snow range }2=(1.0,\ 1.4)\text{ or }(1.6,\ 2,0]\rightarrow\text{Nb}= 1.958+0.191\times\text{Roof length}+0.025\times\text{Roof length}\times\text{Snow},\end{split} \tag{25}\]  
where Snow range is the load interval considered (kN/m\({}^{2}\)), Nb is the number of beams, Roof length is the full dimension of the structure (m) and Snow is the applicable snow load (kN/m\({}^{2}\)).