# Safety Design of Rotary Drilling Rig Mast Based on Multi-Condition Analysis
## 2 Classification of Working Conditions and Structural Mechanics Model
### Load-Bearing Capacity Analysis
#### 2.3.1 Strength Analysis  
* Drilling Model  
In the drilling condition, the cross-section is subjected to axial forces, shear forces, bending moments, and torques, resulting in a complex stress state, as illustrated in Figure 5.  
Figure 4: Critical points. Figure 5: Drilling conditions: (**a**) internal forces in the cross-section; (**b**) cross-sectional normal stress distribution; (**c**) cross-sectional torsional shear stress distribution; (**d**) cross-sectional bending normal stress distribution; and (**e**) cross-sectional bending shear stress distribution. During the pressurized drilling phase with the hydraulic cylinder, the drilling mast acts as a tension member. The formula for calculating the normal stress [21] is given in Equation (1). \[\sigma_{\text{BI}}=\frac{N}{A_{\text{j}}}+\frac{M_{x}}{W_{\text{j}x}}+\frac{M_{y }}{W_{\text{j}y}} \tag{1}\]  
In the equation, \(N\) represents the axial force on the calculated cross-section; \(A_{\text{j}}\) denotes the net cross-sectional area of the member; and \(M_{x}\) and \(M_{y}\) are the bending moments about the \(x\)-axis and \(y\)-axis at the calculated cross-section, respectively. The moment about the \(x\)-axis primarily originates from the moment generated by the applied or lifting force acting on the centroid, while the \(y\)-axis primarily arises from the eccentric load of the main pulley frame; and \(W_{\text{j}x}\) and \(W_{\text{j}y}\) are the section modulus for bending about the \(x\)-axis and \(y\)-axis for the net cross-sectional area of the member. For box-section structures, the constrained bending stress should be calculated. To simplify, a coefficient method is used, taking it as 0.1 times the free bending stress. In addition, it is also subjected to bending shear stress, which is calculated using the following formula, as shown in Equation (2):  
\[\tau_{\text{s}}=\frac{F}{(2\times h\times\delta)} \tag{2}\]  
In the equation, \(F\) represents the shear force; \(h\) is the height of the web; and \(\delta\) denotes the thickness of the web. In addition to the aforementioned normal stress and shear stress due to bending, the section under pressurized drilling also experiences torque. The simplified calculation formula for the torsional shear stress is given by Equation (3) as follows:  
\[\tau_{\text{n}}=\frac{T_{\text{n}}}{(2\times A_{0}\times\delta_{\text{min}})} \tag{3}\]  
In the equation, \(A_{0}\) represents the area enclosed by the midline of the closed section plate thickness, while \(\delta_{\text{min}}\) denotes the minimum plate thickness of the section. Due to constrained torsion, both sector shear stress and sector normal stress are present. The sector shear stress is generally negligible compared to the torsional shear stress. For the sector normal stress, a coefficient method is applied, assuming it to be 0.05 times the free bending normal stress. The types of stress at the critical points are shown in Table 4.  
For points 2 and 4, which are under a three-dimensional stress state, it is necessary to calculate the von Mises stress to convert the three-dimensional stress state into a uniaxial stress state, as shown in Equations (4) and (5). \[\sigma= \begin{bmatrix}\sigma_{1}&0&0\\ 0&\sigma_{2}&0\\ 0&0&\sigma_{3}\end{bmatrix}  \tag{4}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Section** & **Stress Categories** & **Formula** & **Point 1** & **Point 2** & **Point 3** & **Point 4** \\ \hline \multirow{4}{*}{Critical Section} & Normal Stress & (1) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) \\  & Constrained Bending Normal Stress & & & \(\surd\) & & \(\surd\) \\  & Constrained Torsional Normal Stress & & & \(\surd\) & & \(\surd\) \\  & Bending Shear Stress & (2) & & \(\surd\) & & \(\surd\) \\  & Torsional Shear Stress & (3) & & \(\surd\) & & \(\surd\) \\ \hline \end{tabular}
\end{table} 
Table 4: Types of stress at critical points under drilling conditions.