## 2 Analytical Model and the Development of the Mathematica Package
\begin{array}{c}U=\frac{1}{2}\int\limits_{0}^{L}E_{I}(z)\left[\frac{d^{2}u( z)}{dz^{2}}\right]^{2}dz+2\int\limits_{0}^{L}GJ(z)\left[\frac{d\phi(z)}{dz} \right]^{2}dz\\ +\frac{1}{2}\int\limits_{0}^{L}E\left[C_{w}(z)\left[\frac{d^{2}\phi(z)}{dz^{2 }}\right]^{2}+2\left[\frac{d^{2}C_{w}(z)}{dz^{2}}\right]\left[\frac{d\phi(z)} {dz}\right]^{2}+2\left[\frac{d\mathcal{C}_{w}(z)}{dz}\right]\left[\frac{d\phi (z)}{dz}\right]\left[\frac{d^{2}\phi(z)}{dz^{2}}\right]\right]dz\end{array}  \tag{11}\]  
where \(E\) and \(G\) denote the Young's modulus and the shear modulus, respectively. The external work associated with transverse loading is formulated in Equation (12) [32; 44]:  
\[V=\frac{1}{2}\int\limits_{0}^{L}M_{x}(z)\left[2\phi(z)\left[\frac{d^{2}u(z)}{ dz^{2}}\right]+\beta_{x}(z)\left[\frac{d\phi(z)}{dz}\right]^{2}\right]dz+\frac{1}{2} \sum PH_{p}\phi_{p}^{2}+\frac{1}{2}\int\limits_{0}^{L}qH_{q}\phi(z)^{2}dz \tag{12}\]  
The function of the bending moment distribution along the beam's length is defined as \(M_{x}(z)\). Out-of-shear center loading of concentrated load \(P\) causes extra torque of \(PH_{p}\Phi_{p}\) because the concentrated load moves due to the cross-section rotation during LTB. When the application point of \(P\) is below the shear center, the extra torque that occurs opposes the twist rotation of \(\Phi_{p}\) and increases the LTB load. However, the LTB load decreases when the application point of \(P\) is above the shear center since, in that case, extra torque tends to enhance twist rotations. Therefore, extra work due to the out-of-shear center loading of \(P\) can be written as the second term of Equation (12). Similarly, the extra work done by the out-of-shear center loading of the distributed load \(q\) can be defined as the last term of Equation (12). The distances between the shear center and the application points of the concentrated force \(P\) and distributed load \(q\), denoted by \(H_{p}\) and \(H_{q}\), are taken as positive when the transverse loads act below the shear center. The variable \(\Phi_{p}\) represents the twist angle at the location where the point load is applied. The LTB behavior of monosymmetric I-sections is more complex than that of symmetric I-sections. Different from the symmetric I-sections, an additional torque develops as a result of the section's monosymmetric geometry, induced by the interaction between longitudinal bending stresses and twisting during the buckling process. This extra torque substantially influences the torsional stiffness of the beam. Torsional stiffness should be modified considering the major axis bending moment and the beam's monosymmetry parameter for accurate LTB assessment [29]. The coefficient \(\beta_{x}\), introduced by [PERSON], quantifies the cross-sectional monosymmetry and is formulated in Equation (13) [32]. In Equation (13), \(x\) and \(y\) represent the Cartesian coordinates corresponding to the differential area element (\(dA\)). \(lx\) represents the major axis moment of inertia and is expressed for web-tapered mono-symmetric I-section beams as a function of the z-axis in Equation (14). \(y_{0}\) is taken as positive when the shear center lies below the centroid of the cross-section. \[\beta_{x}=\frac{1}{I_{x}}\int_{A}y\Big{(}x^{2}+y^{2}\Big{)}dA-2y_{0}(z) \tag{13}\] \[I_{x}(z)=\frac{b_{f1}t_{f1}^{3}+h^{3}(z)t_{w}+b_{f2}t_{f2}^{3}}{12} +A_{f1}(z)\Big{[}y(z)-\frac{t_{f1}}{2}\Big{]}^{2}+A_{w}(z)\Big{[}t_{f1}+\frac{h( z)}{2}-y(z)\Big{]}^{2}\] (14) \[+A_{f2}(z)\Big{[}d(z)-\frac{t_{f2}}{2}-y(z)\Big{]}^{2}\]  
\(\beta_{x}(z)\) can be written for the tapered mono-symmetric I-section beams in Figure 1a as follows:  
\[\beta_{x}(z)=\frac{1}{I_{x}(z)}\left[[h_{0}(z)-\overline{y}(z)]\left[\frac{b_ {f1}^{3}t_{f2}}{12}+b_{f2}t_{f2}\left[h_{0}(z)-\overline{y}(z)\right]^{2}+ \frac{t_{w}[h_{0}(z)-\overline{y}(z)]^{3}}{4}\right]\right]-2y_{0}(z) \tag{15}\]  
Based on [PERSON]'s assumptions, the total potential energy (\(\Pi=U+V\)) for mono-symmetric tapered I-beams is expressed in Equation (16).