The most critical parameter for determining equivalent values for the Mohr–Coulomb friction angle and cohesion from the nonlinear Hoek–Brown criterion is the upper limit of confining stress. For rock slopes, this value is the maximum value of the minimum principal stress ( $$\sigma_{3,\max }^{\prime }$$σ 3 , max ′ ) on the potential failure surface. The existing problems in the existing research are analyzed and summarized. Using the finite element method (FEM), the location of potential failure surfaces for a wide range of slope geometries and rock mass properties are calculated using the strength reduction method, and a corresponding finite element elastic stress analysis was carried in order to determine $$\sigma_{3,\max }^{\prime }$$σ 3 , max ′ of the failure surface. Through a systematic analysis of 425 different slopes, it is found that slope angle (β) and geological strength index (GSI) have the most significant influence on $$\sigma_{3,\max }^{\prime }$$σ 3 , max ′ while the influence of intact rock strength and the material constant $$m_{i}$$m i are relatively small. According to the variation of $$\sigma_{3,\max }^{\prime }$$σ 3 , max ′ with different factors, two new formulas for estimating $$\sigma_{3,\max }^{\prime }$$σ 3 , max ′ are proposed. Finally, the proposed two equations were applied to 31 real case studies to illustrate the applicability and validity.

Coarse-grained soil (CGS) is a vital geotechnical material extensively used in engineering projects. Scale effects pose challenges to accurately assessing its shear strength, and the influences of scaling methods and moisture content remain underexplored. This study investigates the effects of maximum particle size ($d_{\max }$), scaling techniques, and moisture content on the shear strength of CGS through large-scale direct shear tests and in situ direct shear tests. CGS samples from the Jilangtan platform in Qinghai Province, originally with a $d_{\max }$ of 60 mm, were scaled to 30 mm using the Equivalent Substitution Method (ESM) and Similar Gradation Method (SGM). Tests were conducted at moisture contents of 0.91% (air-dried), 2.45% (natural), and 7.66% (saturated). Results show that ESM consistently yields higher shear strength than SGM across all moisture levels. Scale effects are minimal under air-dried conditions but become pronounced at higher moisture contents. While all large-scale laboratory test results deviate somewhat from in situ findings, shear strength parameters derived from SGM align more closely with field observations.

The most critical parameter for determining equivalent values for the Mohr-Coulomb friction angle and cohesion from the nonlinear Hoek-Brown criterion is the upper limit of confining stress. For rock slopes, this value is the maximum value of the minimum principal stress (σ'_3,max) on the potential failure surface. In this contribution, the existing problems in the existing research are analyzed and summarized. Using the finite element method (FEM), the location of potential failure surfaces for a wide range of slope geometries and rock mass properties are calculated using the strength reduction method, and a corresponding finite element elastic stress analysis was carried in order to determine (σ'_3,max) of the failure surface. Through a systematic analysis of 425 different slopes, it is found that slope angle (β) and geological strength index (GSI) have the most significant influence on (σ'_3,max) while the influence of intact rock strength and the material constant mi are relatively small. According to the variation of (σ'_3,max) with different factors, two new formulas for estimating (σ'_3,max) are proposed. Finally, the proposed two equations were applied to 31 real case studies to illustrate the applicability and validity.