## 2 Analytical Model and the Development of the Mathematica Package
Therefore, it can be used to calculate the elastic LTB loads of the web-tapered beams with doubly symmetric and mono-symmetric I-sections subjected to various transverse loads, considering their positions along the cross-section. The equations that formed the present analytical model were defined in the Mathematica software v11.0. In the proposed analytical procedure, integration was performed using a custom implementation of [PERSON]'s numerical method rather than relying on the software's built-in symbolic integration capabilities. In the present analytical approach, all derivative operations were carried out using the software's built-in symbolic differentiation algorithm. The Mathematica package includes twelve input parameters; the first eight are related to geometric dimensions, and the next two are for the definition of elastic material, so they are elastic modulus \(E\) and Poisson's ratio \(\
u\). The last two inputs correspond to the transverse load type and its position. It should be noted here that shear modulus is determined from the formula of \(G=E/2(1+\
u)\). Equation (22) shows the input parameters of the Mathematica package:  
\[f_{INPUT}(b_{f1}\text{, }b_{f2}\text{, }t_{f1}\text{, }t_{f2}\text{, }t_{w} \text{, }d_{i}\text{, }d_{m}\text{, }L\text{, }E\text{, }\upsilon\text{, }loadType\text{, }loadPosition) \tag{22}\]  
Six common transverse load types, as shown in Figure 3, have been defined in the Mathematica package. The \"loadType\" input accepts an array of transverse loads, ranging from 1 to 6. For example, when 1 is entered as the input for \"loadType\", the LTB load is calculated for a concentrated load at mid-span. Users can practically add new transverse load types to the Mathematica package, considering the moment distribution function along the beam existing in the \(M_{x}(z)\) term. The \"loadPosition\" input allows for the definition of the transverse loads' position along the cross-section. Three loading positions have been considered in the Mathematica package. The variable \"loadPosition\" is assigned values of \(-1\), \(0\), or \(1\) to indicate loading applied at the top flange, shear center, and bottom flange, respectively. The only output of the present Mathematica package is the elastic LTB load of a web-tapered beam with an I-section. However, users can add more outputs using basic Mathematica commands. For instance, the variation of the mono-symmetry parameter \(\beta_{x}\) along the beam can be included in the outputs. Eventually, the present Mathematica package, called LTBWTB, can calculate the elastic LTB load of a web-tapered beam with an I-section, requiring only geometric dimensions, the elastic modulus, and Poisson's ratio, under the effect of six load types and three load height levels along the cross-section. The LTBWTB tool provides structural designers with a rapid means of evaluating the lateral-torsional buckling (LTB) load of web-tapered I-section beams, offering a practical alternative to shell-based finite element analysis (FEA). Unlike FEA, which involves multiple stages such as 3D geometry creation, section assignment, meshing, load and boundary condition definition, and result interpretation, LTBWTB simplifies the process and eliminates the need for specialized expertise. For some examples, the inputs and outputs of LTBWTB are presented in Figure 4. Additionally, Figure 5 illustrates the software's flowchart. Figure 3: The considered load cases in the numerical analysis.