# Optimization of Glulam Regular Double-Tapered Beams for Agroforestry Constructions
## 3 Results and Discussion
### Relationship between the Purlin Height, Beam Width, Snow Load and Span  
Table 5 shows the ANCOVA results, which indicate that the variable Snow and the Hp factor, independently, have a significant effect on the variable Wb (\(p\) < 0.05). Additionally, the combined effects of the Hp factor and Snow variable (F (1.202) = 12.498, \(p\) < 0.05), Snow and Span (F (1.202) = 1013.022, \(p\) < 0.05), as well as the Hp factor with the Span and Snow variables (F (1.202) = 115.553, \(p\) < 0.05), significantly affect variable Wb. This suggests that the relationship between the variables is that of a model with interaction. There was no heteroscedasticity or significant deviation from the normal distribution ([PERSON]'s test, \(p\) > 0.05 and Shapiro-Wilk test, \(p\) > 0.05).  
Table 6 displays the estimates of Snow, Span and Hp effects on Wb. Snow and Span are independent variables, and their relationship with Wb determines the slope of the plane. The Snow coefficient (\(-\)0.121 mm\(\cdot\)m\({}^{2}\)/kN for Hp = 160 mm and \(-\)0.182 mm\(\cdot\)m\({}^{2}\)/kN for Hp = 200 mm) and the coefficient of the product of Span and Snow (0.008 mm\(\cdot\)m/kN for Hp = 160 mm and 0.013 mm\(\cdot\)m/kN for Hp = 200 mm) determine the plane's slope in this case.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multirow{2}{*}{**Effect**} & \multirow{2}{*}{**Estimate**} & \multirow{2}{*}{**SE**} & \multicolumn{2}{c}{**95\% Confidence Interval**} \\ \cline{5-6}  & & & & **Lower** & **Upper** \\ \hline \multirow{2}{*}{(Intercept)} & \multirow{2}{*}{(Intercept)} & 2.428 & 0.010 & 2.409 & 2.447 \\  Snow & Snow & \(-\)0.121 & 0.009 & \(-\)0.138 & \(-\)0.104 \\  Hp & 200–160 & \(-\)0.046 & 0.020 & \(-\)0.084 & \(-\)0.007 \\  Snow \(\times\) Hp & Snow \(\times\) 200–160 & \(-\)0.061 & 0.017 & \(-\)0.095 & \(-\)0.022 \\  Snow \(\times\) Span & Snow \(\times\) Span & 0.008 & 0.000 & 0.008 & 0.009 \\  Snow \(\times\) Hp \(\times\) Span & Snow \(\times\) 200–160 \(\times\) Span & 0.005 & 0.001 & 0.004 & 0.006 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Estimation of the effects of the considered variables on the response variable Wb.  
Figure 5: Multiple linear regression plot of the relationship between Snow, Span and Hp segmented by Hp.  
Figure 6 displays the multiple regression plot of Snow, Span and Wb as a function of Hp. The figure provides evidence that the model of the relationship between the variables is a model with interaction.  
To determine the optimal value of Wb based on the Snow, Span and Hp values, multiple regression planes were constructed using Equation (23).  
\[\begin{split}\text{For}\ \text{Hp}=160\ \text{mm}\to\text{Wb}=(2.428-0.121 \times\text{Snow}+0.008\times\text{Snow}\times\text{Span})^{5},\\ \text{For}\ \text{Hp}=200\ \text{mm}\to\text{Wb}=(2.382-0.182\times \text{Snow}+0.013\times\text{Snow}\times\text{Span})^{5},\end{split} \tag{23}\]  
where Hp is the height of the purlin (mm), Wb is the width of the beam (mm) with a minimum value of 90 mm, Snow is the applicable snow load (kN/m\({}^{2}\)) and Span is the full dimension of the regular double-tapered beam (m).  
An analysis of the data suggests that when the purlin height is 160 mm, the optimal beam width is roughly 90 mm, except for spans between 26 and 30 m, where it increases to 100-110 mm. On the other hand, when the purlin height is 200 mm, the optimal beam width is 90-100 mm for 15-20 m spans, 110-120 mm for 20-25 m spans and 130-160 mm for 25-30 m spans.