# Optimization of Glulam Regular Double-Tapered Beams for Agroforestry Constructions
## 2 Materials and Methods
### Structural Calculation
#### 2.1.1 Verification of the Ultimate Limit States  
The verification of the ULS assesses the strength of the cross-sections of the structural elements. For this purpose, Eurocode 5 [26] establishes the following requirements:  
* Beam verification for shear strength (I\({}_{\mathrm{v}}\)): the most unfavorable cross-sections are those corresponding to the supports, as the maximum shear forces act on minimum surfaces. Equation (1) shall be satisfied. \[\mathrm{I}_{\mathrm{v}}=\frac{\mathrm{\tau_{d}}}{\mathrm{f}_{\mathrm{v,d}}} \leq 1,\] (1) where \(\mathrm{\tau_{d}}\) is the design shear stress (kN/mm\({}^{2}\)) and \(\mathrm{f}_{\mathrm{v,d}}\) is the design shear strength parallel to the grain (kN/mm\({}^{2}\)). For a rectangular cross-section, the value of design shear stress is calculated by Equation (2). \[\mathrm{\tau_{d,max}}=1.5\times\frac{\mathrm{V_{d}}}{h\times\mathrm{b_{ef}}},\] (2) where \(\mathrm{V_{d}}\) is the maximum shear force (kN), \(h\) is the height of the cross-section (mm) and \(\mathrm{b_{ef}}\) is the effective width of the beam (i.e., the product of the cross-section width and a coefficient that reduces that value, taking into account the possibility of drying cracks, which in the case of glued laminated timber has a value of 0.67). * Beam verification for bending strength (I\({}_{\mathrm{m}}\)): as an element with variable cross-section, one of the edges will be parallel to the direction of the fibers (i.e., parallel edge), and the other forms a certain angle relative to the grain (i.e., tapered edge). Therefore, this verification should be independently assessed for the parallel edge (I\({}_{\mathrm{m},0}\)) and tapered edge (I\({}_{\mathrm{m},\alpha}\)), according to Equations (3) and (4), respectively. \[\mathrm{I}_{\mathrm{m},0}=\frac{\mathrm{\sigma_{m,0,d}}}{\mathrm{f}_{\mathrm{m },d}}\leq 1,\] (3) where \(\mathrm{\sigma_{m,0,d}}\) is the maximum stress caused by the design bending moment at the parallel edge (N/mm\({}^{2}\)) and \(\mathrm{f}_{\mathrm{m},d}\) is the design bending strength (N/mm\({}^{2}\)). \[\mathrm{I}_{\mathrm{m},\alpha}=\frac{\mathrm{\sigma_{m,d}}}{\mathrm{f}_{m,d} \times\mathrm{f}_{m,\alpha}}\leq 1,\] (4) where \(\mathrm{\sigma_{m,d}}\) is the maximum stress caused by the design bending moment at the tapered edge (N/mm\({}^{2}\)), \(\mathrm{f}_{\mathrm{m},d}\) is the design bending strength (N/mm\({}^{2}\)) and \(k_{m,\alpha}\) is the reduction factor considering the redistribution of bending stresses in the cross-section (N/mm\({}^{2}\)), which could be determined as indicated in Equation (5) when the tapered edge is subjected to tensile stresses, or Equation (6) when it is subjected to compressive stresses. \[\mathrm{k}_{\mathrm{m},\alpha}=\frac{1}{\sqrt{1+\left(\frac{\mathrm{f}_{ \mathrm{m},d}}{\mathrm{0.75\times f}_{\mathrm{v,d}}}\times\mathrm{t}\mathrm{g} \alpha\right)^{2}+\left(\frac{\mathrm{f}_{\mathrm{m},d}}{\mathrm{f}_{\mathrm{v },0,d}}\times\mathrm{t}\mathrm{g}^{2}\alpha\right)^{2}}},\] (5) \[\mathrm{k}_{\mathrm{m},\alpha}=\frac{1}{\sqrt{1+\left(\frac{\mathrm{f}_{ \mathrm{m},d}}{\mathrm{1.5\times f}_{\mathrm{v,d}}}\times\mathrm{t}\mathrm{g} \alpha\right)^{2}+\left(\frac{\mathrm{f}_{\mathrm{m},d}}{\mathrm{f}_{\mathrm{v },0,d}}\times\mathrm{t}\mathrm{g}^{2}\alpha\right)^{2}}},\] (6)where \(\rm f_{m,d}\) is the design bending strength (N/mm\({}^{2}\)), \(\rm f_{vd}\) is the design shear strength parallel to the grain (N/mm\({}^{2}\)), \(\rm\alpha\) is the angle relative to the grain (\(\rm\circ\)), \(\rm f_{t,90,d}\) is the design tensile strength perpendicular to the grain (N/mm\({}^{2}\)) and \(\rm f_{c,90,d}\) is the design compression strength perpendicular to the grain (N/mm\({}^{2}\)). It is worth noting that Eurocode 5 [26] concedes that the stress caused by the design bending moment at each edge (\(\rm\sigma_{m,0,d}\) and \(\rm\sigma_{m,0,\alpha}\)) is equal, as per Equation (7).