## 2 Working Conditions and Mathematical Modelling of Masts
### Limitations
#### 2.4.2 Static Strength Checks and Constraints  
The static strength analysis of the mast is conducted based on the principle of the most critical working conditions. In the practical application of rotary drilling rigs, the most dangerous working condition is the in-hole lifting condition B3 or the drilling condition B1.  
Figure 3: Location of cross-section dangerous points.  
Figure 2: Box section parameters for rotary drilling rigs.  
Under drilling conditions, the mast experiences a combination of tensile, bending and torsion deformations [21], as shown in Figure 4.  
Specifically, under the B1 condition, points 1 and 3 are only subject to maximum free bending positive stress, which is calculated as shown in Equation (2).  
\[\sigma_{0}=\frac{N}{A_{\text{j}}}+\frac{M_{\text{x}}}{(1-(N/N_{Ex}))W_{\text{jx }}}+\frac{M_{\text{y}}}{(1-(N/N_{Ey}))W_{\text{jy}}} \tag{2}\]  
The points 2 and 4 for B1 are subjected to a combination of the following stresses: free bending positive stress, maximum restrained bending positive stress, restrained torsional positive stress (fan positive stress), bending shear stress as shown in Equation (3), and torsional shear stress as shown in Equation (4). To calculate the constrained bending positive stress and constrained torsion positive stress, the equivalence method is employed, as shown in Equation (5) below.  
\[\tau_{\text{s}}=\frac{F_{\text{shearing}}}{(2\times\text{h}\times\delta)} \tag{3}\]  
\[\tau_{\text{n}}=\frac{T_{\text{n}}}{(2\times A_{0}\times\delta_{\text{min}})} \tag{4}\]  
\[\sigma=1.15\sigma_{0} \tag{5}\]  
Under in-hole lifting conditions, the mast is subjected to a combination of both tensile (compression) and bending deformations [21]. Points 1 and 3 under the B3 condition are subjected to maximum free bending positive stress only, which is calculated as Equation (2).  
Points 2 and 4 under the B3 condition are subjected to free bending positive stresses as in Equation (2), maximum restrained bending positive stresses as in Equation (6), and bending shear stresses as in Equation (3). The constrained bending positive stress is calculated using the equivalent method. The stresses experienced by hazardous points in dangerous working conditions are shown in Table 2.  
\[\sigma=1.1\sigma_{0} \tag{6}\]  
Figure 4: Diagram of the internal forces in section of the mast.  
For unidirectional positive stresses, the static strength calibration equation is as follows:  
\[\sigma\leq[\sigma] \tag{7}\]  
For a spatial stress state, its principal stress can be determined according to the stress tensor. Subsequently, the complex stress state is evaluated and calibrated based on the fourth strength theory.  
The stress tensor is as follows:  
\[\sigma=\begin{bmatrix}\sigma_{\text{x}}&\tau_{xy}&\tau_{xz}\\ \tau_{yx}&\sigma_{y}&\tau_{yz}\\ \tau_{zx}&\tau_{zy}&\sigma_{z}\end{bmatrix} \tag{8}\]  
According to the tensor, its principal stress is as follows:  
\[\sigma_{1,2,3}=\text{eig}(\sigma) \tag{9}\]  
This can be converted to a unidirectional state of stress by the fourth theory of strength, as shown below:  
\[\sigma_{r4}=\sqrt{\frac{1}{2}\Big{[}\big{(}\sigma_{1}-\sigma_{2}\big{)}^{2}+ \big{(}\sigma_{2}-\sigma_{3}\big{)}^{2}+\big{(}\sigma_{1}-\sigma_{3}\big{)}^{2 }\Big{]}} \tag{10}\]  
\[\sigma_{r4}\leq[\sigma] \tag{11}\]  
Through the above analysis, the stress at the largest point among the four hazardous points is calculated and the static strength constraint is applied to it, and the static strength constraint condition of the mast under working condition B1 is as follows:  
\[\text{g}(1)=\sigma_{B1\text{max}}-[\sigma]\leq 0 \tag{12}\]  
The static strength constraints of the mast at operating condition B3 are as follows:  
\[g(2)=\sigma_{B3\text{max}}-[\sigma]\leq 0 \tag{13}\]