# Optimization of Glulam Regular Double-Tapered Beams for Agroforestry Constructions
## 3 Results and Discussion
### Relationship between the Cost of the Structure, Snow Load and Span  
The correlation coefficient R has a value of one, which indicates a perfect linear dependence between the variables (Table 1). This finding supports reliable predictions through the proposed multiple regression model of the optimal cost of the roof based on the values of Span and Snow. The coefficient of determination R\({}^{2}\) = 0.999 further suggests that 99.9% of the variability in optimal cost can be explained by the variability in Span and Snow. In addition, when its value is almost 1, it indicates that there is an almost perfect linear fit between the three variables. Table 2 shows the values of the linear correlation and determination coefficients, the test value, and the \(p\)-value that corroborate the good linear fit between the three variables. The data did not differ significantly from the normal distribution (Shapiro-Wilk 0.994, \(p\) > 0.05). The slope of the plane is significantly different from 0, since the values of the \(t\)-test for Snow, Span, and the combination of Snow and Span are \(-\)12.4 (\(p\) < 0.001), 98.2 (\(p\) < 0.001), 48.8 (\(p\) < 0.001), respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**R**} & \multirow{2}{*}{**R\({}^{2}\)**} & \multicolumn{4}{c}{**General Model Test**} \\ \cline{3-6}  & & **F** & **df1** & **df2** & \(p\) \\ \hline
1 & 0.999 & 68,615 & 3 & 204 & \(<\)0.001 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Correlation, determination and test coefficients. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Estimate** & **Standard Error** & **t** & \(p\) \\ \hline (Constant) & 13.837 & 0.15350 & 90.1 & \(<\)0.001 \\ Snow & \(-\)1.431 & 0.11521 & \(-\)12.4 & \(<\)0.001 \\ Span & 0.656 & 0.00668 & 98.2 & \(<\)0.001 \\ Snow \(\times\) Span & 0.245 & 0.00501 & 48.8 & \(<\)0.001 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter estimation (slopes and constant) of the multiple regression model. From the coefficients of the multiple regression shown in Table 2, the Equation (20) is proposed to predict the cost of the timber roof structure:  
\[\text{Cost}= (13.837-1.431\times\text{Snow}+0.656\times\text{Span}+0.245\times \text{Snow}\times\text{Span})^{3.125}, \tag{20}\]  
where: Cost is the structure investment (EUR), Snow is the applicable snow load (kN/m\({}^{2}\)) and Span is the full dimension of the regular double-tapered beam (m). Figure 4 displays the resulting plane of the multiple regression for the Snow, Span, and Cost variables. The increasing slope of the plane suggests that an increase in the snow load and span results in a corresponding increase in the cost of the structure. This observation is expected since as the snow load and span increase, the size of the structural elements also increases, thereby increasing the total mass of timber used. In this regard, to determine the volume of the glulam employed in the roof structure instead of calculating the cost at a price of 900 EUR/m\({}^{3}\), Equation (21) should be considered:  
\[\text{Volume}= (1.56922-0.16229\times\text{Snow}+0.0744\times\text{Span}+0.02778 \times\text{Snow}\times\text{Span})^{3.125}, \tag{21}\]  
where Volume is the amount of glued laminated timber (m\({}^{3}\)), Snow is the applicable snow load (kN/m\({}^{2}\)) and Span is the full dimension of the regular double-tapered beam (m). The equations reveal that when the span of the roof is held constant, an increase of 0.1 kN/m\({}^{2}\) in snow load results in a 2.6% to 4.4% increase in the optimal cost or volume for the 15 m and 30 m spans, respectively. Additionally, when the snow load is held constant, a 1 m increase in roof span leads to a 9.5% to 11.6% increase in optimal cost or volume for snow loads of 0.5 kN/m\({}^{2}\) and 2 kN/m\({}^{2}\), respectively. Upon comparing the cost curves, it is evident that, similar to [PERSON][31], raising the cost to the exponent of 0.32 results in a linearized outcome.