Modified minimum principal stress estimation formula based on Hoek–Brown criterion and equivalent Mohr–Coulomb strength parameters

www.nature.com/scientificreports OPEN Modified minimum principal stress estimation formula based on Hoek–Brown criterion and equivalent Mohr–Coulomb strength parameters Yanhui Song 1, Man Feng 1* & Peng Chen 1,2 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 (σ3,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 σ3,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 σ3,max while the influence of intact rock strength and the material constant mi are relatively small. According to the variation ′ ′ of σ3,max with different factors, two new formulas for estimating σ3,max are proposed. Finally, the proposed two equations were applied to 31 real case studies to illustrate the applicability and validity. Abbreviations σ1 Major effective principal stress σ3 Minor effective principal stress σci Unconfined compressive strength of intact rock σcm Unconfined compressive strength of rock mass σt Tensile strength of rock mass σv Gravitational stress ′ Normalized upper limit of confining stress σ3n ′ σ3,max Upper limit of confining stress over the equivalent Mohr–Coulomb and Hoek–Brown criteria are considered a σ3,max Appropriate value of σ3,max obtained from elastic stress analysis p σ3,max Predicted value of σ3,max mb Hoek–Brown constant for rock mass mi Hoek–Brown constant for intact rock s Hoek–Brown constant for rock mass a Hoek–Brown constant for rock mass c′ Equivalent cohesion φ′ Equivalent friction angle GSI Geological strength index D Disturbance factor γ Unit weight H Slope height 1 School of Geology Engineering and Geometrics, Chang’an University, Xi’an 710054, China. 2JiNan Design Institute of China Railway Engineering Design and Consulting Co. LTD, Jinan 250022, Shandong, China. *email: Scientific Reports | (2023) 13:6409 | https://doi.org/10.1038/s41598-023-33053-x 1 Vol.:(0123456789) www.nature.com/scientificreports/ β Slope angle Erm Deformation modulus of rock mass At present, limit equilibrium method based on Mohr–Coulomb (MC) failure criterion is still the main method for slope stability analysis. However, some studies show that the nonlinear Hoek–Brown (HB) failure criterion more correctly represents rock failure for almost all rock types1–5. Multiple methods for evaluating the equivalent MC friction angle and cohesion have been p roposed5–15. Iamael and K onietxky16 modified the HB criterion to consider the anisotropy of rock by applying an explicit function of the rock parameter mi with orientation β. The Hoek–Brown criterion was firstly proposed for intact rock by Hoek and Brown in 1 98017, and the latest version for rock mass is as follows5: a  σ3 +s σ1 = σ3 + σci mb (1) σci where σ1 and σ3 are the major and minor principal stresses, σci is the unconfined compressive strength, and mb, s, and a are rock mass material constants given by Eqs. (2), (3), and (4), respectively.   GSI − 100 mb = mi exp (2) 28 − 14D s = exp a=  GSI − 100 9 − 3D   20 1 1  − GSI e 15 − e− 3 + 2 6 (3) (4) where mi is a material constant for intact rock, GSI is the geological strength index which depends on rock mass characterization and commonly varies from 0 to 100; D is a factor which depends upon the degree of disturbance due to blast damage and stress relaxation and varies from 0 to 1. The GSI classification system is based upon the assumption that the rock mass contains sufficient number of ‘randomly’ oriented discontinuities such that it behaves as an isotropic mass. Therefore, the control failure of a single discontinuous structure is beyond its range, which will lead to highly anisotropic mechanical behavior. In line with the above discussion, it is important to realise the research in this paper will be subject to the same limitations that underpin the Hoek–Brown yield criterion itself. Hoek9 proposed a method to calculate the equivalent Mohr–Coulomb parameters based on instantaneous rock mass properties for: (1) a specified effective normal stress, (2) a specified minor principal effective stress, and (3) a condition in which the rock mass uniaxial compressive strength is the same for both the Hoek–Brown and Mohr–Coulomb criteria. In 1997, Hoek and B rown8 revised the method of calculating the equivalent Mohr–Coulomb parameters according to the generalized Hoek–Brown criterion. It is recommended that the maximum value of the minimum effective principal stress generally be 0.25σci , and the estimated c value using this method be decreased by 25% to avoid overestimating the rock mass strength. Also, for rock slopes, the effective normal stress on the potential failure surface of the slope may be small, so the maximum value of the minimum effective principal stress 0.25σci should be applied cautiously, otherwise the rock mass shear strength mass may be overestimated. For rock slopes, a minor principal stress range of 0 < σ3 < σv can be used, where σv = depth × unit weight of the rock mass18. In this case, depth is defined as the average depth of a failure surface in which a circular type can be assumed. The equations for determining the equivalent cohesion and friction angle proposed by Hoek et al. in 2002 and 2018 a re5,19:    ′ ′ a−1 σci (1 + 2a)s + (1 − a)mb σ3n s + mb σ3n  c′ =     (5) ′ a−1 /(1 + a)(2 + a) (1 + a)(2 + a) 1 + 6amb s + mb σ3n φ′ = sin with −1     ′ a−1 6amb s + mb σ3n   ′ a−1 2(1 + a)(2 + a) + 6amb s + mb σ3n ′ σ3n = σ3′ max σci   σ3′ max σcm −0.91 = 0.72 σcm γH (6) (7) (8) where γ is the rock mass unit weight, H is the slope height, and σcm is the rock mass global strength, which is expressed as follows: Scientific Reports | Vol:.(1234567890) (2023) 13:6409 | https://doi.org/10.1038/s41598-023-33053-x 2 www.nature.com/scientificreports/ σcm = σci a−1  [mb + 4s − a(mb − 8s)] m4b + s (1 + a)(2 + a) (9) Li et al.11 found that, for steep slopes (i.e. greater than 45°), the safety factors calculated using the equivalent friction angle and cohesive strength obtained from Eqs. (5–9) are significantly higher due to the deviation of the ′ ′ estimated σ3,max ; therefore, they suggested the following modified power functions to estimate σ3,max :   σ3′ max σcm −1.23 = 0.41 for β < 45◦ (10) σcm γH   σ3′ max σcm −1.07 = 0.2 for β ≥ 45◦ σcm γH (11) where βis t (...truncated)