## 2 Analytical Model and the Development of the Mathematica Package  
In the initial stage of LTB, the simply supported web-tapered I-section beam bends about its major axis. When the transverse loads reach a critical value, the web-tapered beam with an I-section buckles out of the plane, deflecting laterally and twisting. Figure 1 adepicts the LTB of a simply supported web-tapered beam with a mono-symmetric I-section under a concentrated load at the mid-span of the beam. The tapering angle and the beam's length are shown as \(\alpha\) and \(L\) in Figure 1a,b, which show an m-m section of the web-tapering beam with the I-section. In Figure 1b, \(S\) and \(C\) represent the shear center and the centroid, respectively. The corresponding displacements of the shear center are expressed as \(u\) for lateral, \(v\) for vertical, and \(\varphi\) for torsional rotation. Besides, the \(y_{0}\) refers to the distance between the shear center and the center of gravity. The cross-section dimensions of the mono-symmetric I-section belonging to the web-tapered beam are shown in Figure 2. While \(b_{f1}\) and \(b_{f2}\) are the top and bottom flange widths, the top and bottom flanges' thicknesses are demonstrated with \(t_{f1}\) and \(t_{f2}\), respectively. The parameters \(d\) and \(h\) indicate the overall height of the cross-section and the web depth, respectively. The \(h\) parameter varies with the distance measured from the left support in the z-axis due to the web tapering and can be written as follows:  
\[ \begin{array}{l}h(z)=h_{i}+2z\,\tan(\alpha)\quad\ \ 0\leq z\leq L/2\\ h(z)=h_{m}-2\Big{[}z-\frac{t}{2}\Big{]}\,\tan(\alpha)\quad\ \ L/2\leq z\leq L\end{array}  \tag{1}\]  
where \(h_{i}\) and \(h_{m}\) are the web height at the left support and mid-span of the beam, respectively. The cross-section height \(d\) and its values at left support \(d_{i}\) and mid-span \(d_{m}\) can be defined as follows:  
\[ \begin{array}{l}d(z)=h(z)+t_{f1}+t_{f2}\\ d_{i}=h_{i}+t_{f1}+t_{f2}\\ d_{m}=h_{m}+t_{f1}+t_{f2}\end{array}  \tag{2}\]  
Figure 1: The LTB of the tapered beam with mono-symmetric I-section (**a**) side view, (**b**) m-m section. In Figure 2, \(y(z)\) represents the vertical distance from the centroid of the cross-section to its top edge and is expressed as follows:  
\[y(z)=\frac{1}{A(z)}\Bigg{[}\frac{A_{f1}(z)t_{f1}}{2}+A_{w}(z)\bigg{[}t_{f1}+\frac {h(z)}{2}\bigg{]}+A_{f2}(z)\bigg{[}d(z)-\frac{t_{f2}}{2}\bigg{]}\Bigg{]} \tag{3}\]  
The terms \(A_{f1}(z)\), \(A_{f2}(z)\), and \(A_{w}(z)\) denote the cross-sectional areas of the top and bottom flanges and the web, respectively. \(A(z)\) represents the total cross-sectional area. In the case of the mono-symmetric I-section shown in Figure 1a, the warping constant \(C_{w}(z)\), torsional constant \(J(z)\), and the minor-axis moment of inertia \(I_{y}(z)\) are key sectional properties influencing the LTB. Their expressions are given as follows:  
\[C_{w}(z)=\frac{h_{0}^{2}(z)b_{f1}^{3}t_{f1}s}{12} \tag{4}\]  
\[J(z)=\frac{b_{f1}t_{f1}^{3}+b_{f2}t_{f2}^{3}+h_{0}(z)t_{w}^{3}}{3} \tag{5}\]  
\[I_{y}(z)=\frac{b_{f1}^{3}t_{f1}+b_{f2}^{3}t_{f2}+h(z)t_{w}^{3}}{12} \tag{6}\]  
where \(\overline{y}(z)\) denotes the vertical distance from the centroid of the cross-section to the centerline of the top flange, while \(h_{0}(z)\) represents the distance between the centroids of the flanges. The \(\overline{y}(z)\), \(h_{0}(z)\), and \(y_{0}(z)\) can be given in Equations (7)-(10). Figure 2: The geometric dimensions of the cross-section of the tapered beam. \[\overline{y}(z)=y(z)-\frac{t_{f1}}{2} \tag{7}\] \[h_{0}(z)=d(z)-\frac{t_{f1}+t_{f2}}{2}\] (8) \[\mathrm{s}=\frac{1}{1+\left[\frac{b_{f1}}{b_{f2}}\right]^{3}\left[ \frac{t_{f1}}{t_{f2}}\right]}\] (9) \[y_{0}(z)=sh_{0}(z)-\overline{y}(z) \tag{10}\]  
The strain energy stored in a web-tapered beam with an I-section due to lateral bending, warping, and torsion, respectively, can be written as follows [43]:  
\[