Mohr-Coulomb Failure Envelope

Original published in: Hack, H.R.G.K., 2018. Mohr-Coulomb Failure Envelope. In: Bobrowsky, P.T., Marker, B. (Eds), Encyclopedia of Engineering Geology. Springer, Cham, Switserland. ISBN: 9783319735665. DOI: https://doi.org/10.1007/978-3-319-73568-9_207. pp. 667-668. Mohr-Coulomb Failure Envelope 1 Mohr-Coulomb Failure Envelope Robert (H.R.G.K.) Hack Engineering Geology, ESA, Faculty of Geo-Information Science and Earth Observation (ITC), University of Twente, Enschede, The Netherlands Synonym Mohr-Coulomb Failure Criterion Definition A constitutive model that describes the shear strength of ground. Exceeding the shear strength results in failure of the ground which can be described by the MohrCoulomb Failure Envelope. This is also used, sometimes, as a constitutive model for the shear strength along surfaces of, for example, discontinuities. Ground materials are diverse and may be gasses, fluids, solids (i.e., minerals, grains, and aggregates of grains or minerals), and any mixture of these and also include man-made ground, such as fills and waste dump materials. Ground is commonly differentiated in soil and rock; soil being an aggregate of loose or weakly-bonded particles, and rock consisting of particles cemented or locked together, giving rock a tensile strength. Soil and rock are, by some, differentiated based on a compressive strength difference with soil being weaker than 1 MPa and rock being stronger. A differentiation is made between ‘intact’ and ‘discontinuous’ ground, i.e., ground without respectively with distinct planes of mechanical weakness such as faults, joints, bedding planes, fractures, schistosity, etc. A groundmass consists of (blocks of) intact ground with discontinuities, if present. The stresses in the ground should be in terms of total and effective stresses. Mohr-Coulomb Failure Envelope Fig. 1a shows the foundation of a surface object, the reactions in the ground (Fig. 1b,c), and the stress configuration between two particles (Fig. 1d). The stress configuration on the contact between the two ground particles is shown in Fig. 1e in the Mohr-circle diagram. The maximum shear stress sustainable between the ground particles can be formulated with a Mohr-Coulomb failure envelope (Coulomb 1776). The envelope gives the boundary condition of the shear and normal effective stress configuration at which the shear stress equals the shear strength and is formulated in eq. [1]. The stresses are effective stresses and for uniformity also the parameters are accented to indicate that the envelope is in terms of effective stresses. Fig. 1f shows the stress configuration when the stress circle touches the MohrCoulomb failure envelope (in point σ'f, τf). 𝜏 = 𝑐′ + 𝜎 ′ tan 𝜑 ′ 𝑐 = (effective) cohesion (in rock mechanics also denoted Si) 𝜑 ′ = (effective) angle of internal friction (for material strength) or (effective) angle of friction (for a surface) 𝜏 = shear strength 𝜎 ′ = effective normal stress on shear plane ′ [1] The parameter ‘cohesion’ (eq. [1]) is not the same as the ‘tensile’ strength of ground. Ground with tensile strength will also have cohesion, but not necessarily the same value, and ground without tensile strength may or may not have cohesion. The Mohr-Coulomb failure envelope is a constitutive model suitable for describing the strength of many soils, intact rock, and rock masses. Values of cohesion' and φ' for different grounds including the range of confining pressure for which these apply are given in the chapter on Mechanical Properties (Hack, 2018 - Table 3). The Mohr-Coulomb failure envelop can also Original published in: Hack, H.R.G.K., 2018. Mohr-Coulomb Failure Envelope. In: Bobrowsky, P.T., Marker, B. (Eds), Encyclopedia of Engineering Geology. Springer, Cham, Switserland. ISBN: 9783319735665. DOI: https://doi.org/10.1007/978-3-319-73568-9_207. pp. 667-668. Mohr-Coulomb Failure Envelope 2 be formulated in terms of total stresses, i.e. effective stresses plus pore gas and fluid pressure (Fig. 1f). This is normally only applicable to situations where the pore gas and fluid pressure cannot or not fast enough dissipate, for example, for fast loading of a low-permeable clay (so-called ‘undrained’ situation). Although the Mohr-Coulomb failure envelope is often suitable it is not always valid over the entire range of confining pressures. For many types of ground the envelope is not a perfectly straight line, but is curved (Fig. 2). The parameters of the Mohr-Coulomb envelope are then only applicable for the range of confining pressure where the curved envelope can be approximated by a straight line. The Mohr-Coulomb failure envelope may also be applied to the shear strength along a plane (i.e. a discontinuity). The mathematical formulation is similar to eq. [1], however, mostly more sophisticated constitutive models are used for discontinuities, that include, for example, roughness and strength of asperities (Hencher 2015). Fig. 1. Mohr-Coulomb failure envelope; a) Baptisterium, Pisa, Italy; b) schematized foundation of the wall, the arrows indicate the load of the foundation; c) foundation load (yellow); confining pressure (green) in part due to sideway expansion of the ground under the foundation load, and in red the reaction stress of the subsoil due to the foundation load; d) enlargement showing the configuration of normal and shear stresses between two ground particles; e) the geometry, stresses and position in the Mohr-circle diagram; f) Mohr-circle diagram with MohrCoulomb failure envelope (photo courtesy Arnoldus, 2017) Curved failure envelope Many types of ground and, in particular, groundmasses do not fit the Mohr-Coulomb failure envelope very well so other empirical relations between failure and stress configuration have been proposed (e.g. Hoek and Brown 1980; Bieniawski 1974; Hoek et al. 2002; Hack et al. 2003). Fig. 2 shows the empirical relation of Bieniawski (1974) which is formulated as: 𝑀 𝜎 ′3 𝜎 ′1 = 1+𝑁[ ] 𝑈𝐶𝑆 𝑈𝐶𝑆 𝜎 ′1 , 𝜎 ′ 3 = major, minor effective principal stress 𝑁, 𝑀 = material constants 𝑈𝐶𝑆 = Unconfined Compressive Strength) [2] Original published in: Hack, H.R.G.K., 2018. Mohr-Coulomb Failure Envelope. In: Bobrowsky, P.T., Marker, B. (Eds), Encyclopedia of Engineering Geology. Springer, Cham, Switserland. ISBN: 9783319735665. DOI: https://doi.org/10.1007/978-3-319-73568-9_207. pp. 667-668. Mohr-Coulomb Failure Envelope 3 Fig. 2. Empirical failure envelope following eq. [2] Cross-references Mechanical Properties Shear Strength References Arnoldus JGAM (2017) Photograph: JGAM Arnoldus, Leiden, The Netherlands Bieniawski ZT (1974) Estimating the strength of rock materials. Journal of the South African Institute of Mining and Metallurgy (March):312-320 Coulomb CA (1776) Essai sur une application des regles des maximis et minimis a quelquels problemesde statique relatifs, a la architecture. Mem Acad Roy Div Sav 7:343–387 Hack HRGK, Price DG, Rengers N (2003) A new approach to rock slope stability - A probability classification (SSPC). B Eng Geol Environ 62 (2):167-184 Hack, HRGK, 2018. Mechanical Properties. In: Bobrowsky, P.T., Marker, B. (Eds), Encyclopedia of Engineering Geology. Springer, Cham, Switserland. ISBN: 9783319735665. DOI: https://doi.org/10.1007/978-3-31973568-9_197. pp. 604-618. Hencher SR (2015) Practical Rock Mechanics. CRC, Taylor & Francis Group, Boca Raton, FL, USA Hoek E, Brown ET (1980) Underground excavations in rock. Rev. edn. Institution of Mining and Metallurgy, London Hoek E, Carranza-Torres C, Corkum B (2002) Hoek-Brown criterion – 2002 edition. In: Hammah R, Bawden WF, Curran J, Telesnicki M (eds) Mining and Tunnelling Innovation and Opportunity; 5th North American Rock Mechanics Symposium; 17th Tunnelling Association of Canada Conference; NARMSTAC 2002, Toronto, Canada, 7-10 July 2002. University of Toronto, Canada, pp 267-273