# Optimization of Glulam Regular Double-Tapered Beams for Agroforestry Constructions
## 3 Results and Discussion
### Relationship between the Purlin Height, Beam Height, Snow Load and Span  
The ANCOVA model presented in Table 3 indicates that the variables Span and Snow, as well as the factor Hp, independently have a significant effect on the variable Hp (\(p\) < 0.05). Furthermore, the factor Hp in combination with the variable Span (F (1.202) =10.7, \(p\) < 0.05), and the variable Snow with the variable Span (F (1.202) =14.4, \(p\) < 0.05), were found to significantly affect the variable Hp. 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: 6.58 \(\times\) 10\({}^{-4}\), \(p\) > 0.05; Shapiro-Wilk test: 0.994, \(p\) > 0.05).  
Table 4 presents the estimates of the effects of Snow, Span and Hp on Hb. Snow and Span are the independent variables, and their relationship with Hb determines the slope of the plane. In this case, the coefficients of Snow (1.0755 mm\(\cdot\)m\({}^{2}\)/kN), Span (0.0946 mm/m for Hp = 160 mm and 0.0703 mm/m for Hp = 200 mm) and their product (\(-\)0.0304 mm\(\cdot\)m/kN) determine the slope of the plane. The positive values of the higher coefficients indicate that as the snow load and span increase, the height of the beam also increases.  
Figure 5 displays the multiple regression plot of Snow, Span and Hb 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 Hb based on Snow, Span and Hp values, multiple regression planes were constructed using Equation (22).  
\[\begin{split}\text{For Hp}=160\text{ mm}\rightarrow\text{Hb}=\ (4.2752\ +1.0755\times\text{ Snow}+0.0946\times\text{Span}-0.0304\times\text{Snow}\times\text{Span}+\varepsilon)^{\frac{1}{ 0.28}},\\ \text{For Hp}=200\text{ mm}\rightarrow\text{Hb}=\ (4.9298+1.0755\times\text{ Snow}+0.0703\times\text{Span}-0.0304\times\text{Snow}\times\text{Span}+\varepsilon)^{\frac{1}{ 0.28}},\end{split} \tag{22}\]  
where Hp is the height of the purlin (mm), Hb is the height of the beam (mm), Snow is the applicable snow load (kN/m\({}^{2}\)), Span is the full dimension of the regular double-tapered beam (m) and \(\varepsilon=\pm 0.0161\).  
The equations indicate that as the span of the beam increases by 1.2 m (for Hp = 160 mm) or 1.5 m (for Hp = 200 mm), and the snow load increases by 0.2 kN/m\({}^{2}\), the beam height increases by 40 mm, which is equivalent to the addition of one timber laminate to its composition.  
\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 (Intercept) & (Intercept) & 4.2752 & 0.24322 & 3.7956 & 4.75481 \\ Snow & Snow & 1.0755 & 0.18408 & 0.7125 & 1.43843 \\ Span & Span & 0.0946 & 0.01058 & 0.0737 & 0.11544 \\ Hp & 200–160 & 0.6546 & 0.17105 & 0.3173 & 0.99189 \\ Hp \(\times\) Span & Span \(\times\) 200–160 & \(-\)0.0243 & 0.00744 & \(-\)0.0390 & \(-\)0.00965 \\ Snow \(\times\) Span & Snow \(\times\) Span & \(-\)0.0304 & 0.00801 & \(-\)0.0462 & \(-\)0.01456 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Estimation of the effects of the considered variables on the response variable Hb.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Sum of Squares** & **df** & **Mean Square** & **F** & \(p\) \\ \hline Snow & 0.548 & 1 & 0.5481 & 34.1 & \textless{}0.001 \\ Span & 1.282 & 1 & 1.2820 & 79.8 & \textless{}0.001 \\ Hp & 0.235 & 1 & 0.2352 & 14.6 & \textless{}0.001 \\ Hp \(\times\) Span & 0.171 & 1 & 0.1715 & 10.7 & 0.001 \\ Snow \(\times\) Span & 0.231 & 1 & 0.2306 & 14.4 & \textless{}0.001 \\ Residuals & 3.243 & 202 & 0.0161 & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: ANCOVA results considering the effect of Snow, Span and Hp on Hb.