## 2 Theoretical Analysis  
### The Average Shear Stress Yield Criterion  
In many limit analysis cases, the results of the Tresca and Mises yield criteria are not accurate enough for the neglect of the intermediate principal stress and the difference in material tensile and compression properties. However, some materials may have different tensile and compression properties, and most of them work under complex stress conditions. Many scholars have put forward different models as solutions and have achieved excellent results, the most representative of which is the unified strength theory proposed by [PERSON] et al. [35]. In this theory, the difference in material strength and stress in three directions is cleverly considered with two parameters [35]. The principal stress in three directions (\(\sigma_{1}\), \(\sigma_{2}\), \(\sigma_{3}\)) is transformed into the principal shear stress state (\(\tau_{13}\), \(\tau_{23}\), \(\tau_{23}\)) (Figure 3). The principal shear stress is converted into the double shear stress state (\(\tau_{12}\), \(\tau_{13}\), \(\sigma_{12}\), \(\sigma_{13}\)) or (\(\tau_{23}\), \(\tau_{23}\), \(\sigma_{13}\), \(\sigma_{13}\)) for only two independent quantities appearing in the latter. Its expression is obtained by the following:  
\[ \begin{split} F&=\tau_{13}-b\tau_{12}+\beta(\sigma _{13}+b\sigma_{12})=C,\tau_{12}+\beta\sigma_{12}\geq\tau_{23}+\beta\sigma_{23} \\ F^{\prime}&=\tau_{13}+b\tau_{23}+\beta(\sigma_{13} +b\sigma_{23})=C,\tau_{12}+\beta\sigma_{12}\leq\tau_{23}+\beta\sigma_{23} \\ \beta&=\frac{\sigma_{c}-\sigma_{t}}{\sigma_{c}+ \sigma_{t}}=\frac{1-\alpha}{1+\alpha},C=\frac{(1+b)\sigma_{c}\sigma_{t}}{\sigma _{c}+\sigma_{t}}=\frac{1-\alpha}{1+\alpha}\sigma_{t},b=\frac{(1+\alpha)\tau_ {0}-\sigma_{t}}{\sigma_{t}-\tau_{0}}\end{split}  \tag{1}\]  
where \(b\) is a material parameter reflecting the effect of intermediate principal stress; \(\beta\) and \(C\) are the coefficients which are determined by the material tensile strength limit \(\sigma_{\text{t}}\) and compressive strength limit \(\sigma_{\text{c}}\); \(\tau_{0}\) is the shear strength of the material; \(\sigma_{\text{t}}\) and \(\sigma_{\text{c}}\) are the uniaxial tensile strength and uniaxial compressive strength, respectively; and \(\alpha=\sigma_{\text{t}}/\sigma_{\text{c}}\) represents the tensile and compression strength ratio. The shear stress \(\tau_{13}\), \(\tau_{23}\), or \(\tau_{12}\) and the normal stress \(\sigma_{13}\), \(\sigma_{12}\), or \(\sigma_{23}\) on the acting surface are  
\[\begin{split}\tau_{13}&=\frac{1}{2}(\sigma_{1}- \sigma_{3})\sigma_{13}=\frac{1}{2}(\sigma_{1}+\sigma_{3})\\ \tau_{12}&=\frac{1}{2}(\sigma_{1}-\sigma_{2})\sigma _{12}=\frac{1}{2}(\sigma_{1}+\sigma_{2})\\ \tau_{23}&=\frac{1}{2}(\sigma_{2}-\sigma_{3})\sigma _yield (TSSY) criterion established by [PERSON] is a relatively common yield criterion applicable to most materials [35]. When \(\alpha=1\) and \(b=1\), TSSY is expressed as Equation (5). \[ \begin{array}{l}\sigma_{\rm T}=\max(|\sigma_{1}-\sigma_{2}|,|\sigma_{2}-\sigma _{3}|,|\sigma_{3}-\sigma_{1}|),\ \ \text{Tresca yield criterion}\\ \sigma_{M}=\frac{1}{2}\Big{[}(\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma _{3})^{2}+(\sigma_{3}-\sigma_{1})^{2}\Big{]}^{\frac{1}{2}},\ \ \text{Mises yield criterion}\\ \sigma_{\rm TS}=\left\{\begin{array}{l}\sigma_{1}-\frac{1}{2}(\sigma_{2}+ \sigma_{3}),\sigma_{2}\leq\frac{1}{2}(\sigma_{2}+\sigma_{3})\\ \frac{1}{2}(\sigma_{1}+\sigma_{2})-\sigma_{3},\sigma_{2}\geq\frac{1}{2}(\sigma _{1}+\sigma_{3})\end{array}\right.,\ \ \text{Twin shear stress yield criterion}\end{array}  \tag{5}\]  
where \(\sigma_{\rm T}\), \(\sigma_{\rm M}\), and \(\sigma_{\rm TS}\) are the equivalent stress based on Tresca, Mises, and Twin shear stress yield criterion, respectively. When the Tresca and von Mises yield criterion are used in material plasticity analysis, many studies have confirmed that the initial yield and post-yielding occur between those two theories' predictions [28; 31].