## 3 Parametric Model Definition
### Stress Evaluation  
Once the grid curves are obtained (lath axes), the algorithm evaluates the bending moments along the laths by applying beam theory and the concepts of curve and surface geometry.  
As is known, the moment value \(M\) that is produced in a cross-section due to curvature can be obtained from:  
\[M=k\ E\ I, \tag{10}\]  
where \(k\) is the modulus of the curvature vector at that cross-section, \(E\) is the Young's modulus of the material, and \(I\) is the second moment of the cross-section area with respect to the bending axis.  
The maximum bending stress at the cross-section occurs at the outer fibres and can be obtained by using:  
\[\sigma_{\rm m}=k\ E\ \frac{h}{2}, \tag{11}\]  
where \(h\) denotes the cross-section depth in the bending plane.  
Since the laths of a gridshell are usually bent with respect to their two main bending axes, it is necessary to know the curvature values associated with each of these axes.  
If it is accepted that the faces of the laths are tangent and normal to the design gridshell surface, then the curvatures of the laths with respect to the two main bending axes coincide with the normal and geodesic curvatures of the laths directrix, \(k_{\rm N}\) and \(k_{\rm G}\), respectively. As seen in Section 3.1.1 (Equations (3) and (4)) the values of \(k_{\rm N}\) and \(k_{\rm G}\) can be easily calculated at any point on the curve from the curvature vector \(\overset{\rightarrow}{k}\) of the curve and the angle formed between it and the vector \(\overset{\rightarrow}{N}\) normal to the surface.  
Denoting the _y-axis_ and the _z-axis_ as the bending axes of the lath cross-section, tangent and perpendicular to the surface of the gridshell respectively, and \(b\) and \(h\) as the cross-sectional dimensions associated with each axis, the corresponding bending moments and the maximum initial bending stresses in a given cross-section due to the existence of \(k_{\rm N}\) and \(k_{\rm G}\) curvatures can be calculated from Equations (12)-(15), where \(\sigma_{\rm m,y}\) is the maximum initial bending stress due to normal curvature \(k_{\rm N}\) and \(\sigma_{\rm m,z}\) is the maximum initial bending stress due to geodesic curvature \(k_{\rm G}\),  
\[M_{\rm y}=k_{\rm N}\ E\ I_{\rm y}, \tag{12}\]  
\[\sigma_{\rm m,y}=k_{\rm N}\ E\ \frac{h}{2}, \tag{13}\]  
\[M_{\rm z}=k_{\rm G}\ E\ I_{\rm z}, \tag{14}\]  
\[\sigma_{\rm m,z}=k_{\rm G}\ E\ \frac{b}{2} \tag{15}\]  
The maximum initial bending stress \(\sigma_{\rm m,total}\) in a cross-section subjected to two curvatures can be obtained from the sum of the maximum initial bending stresses due to each curvature, according to:  
\[\sigma_{\rm m,total}=\frac{E}{2}(k_{\rm N}\ h+k_{\rm G}\ b) \tag{16}\]  
The developed algorithm calculates the \(\overset{\rightarrow}{k}\) and \(\overset{\rightarrow}{N}\) vectors along the curves on the grid and computes \(M_{\rm y}\), \(M_{\rm z}\), \(\sigma_{\rm m,y}\), \(\sigma_{\rm m,z}\), and \(\sigma_{\rm m,total}\) for the user-defined values of \(E\), \(b\), and \(h\).  
Figure 15 shows the visualisation of the results provided by the algorithm on the test surface for a grid of _Eucalyptus globulus_ laths (\(E\) = 18,108 MPa) [7] of 50 mm width and 25 mm depth.