Rock Engineering Practice & Design - ISRM

1 Rock EngineeringRock EngineeringPractice & DesignPractice & DesignLecture 8: Lecture 8: Stress Analysis around Stress Analysis around Stress Analysis around Stress Analysis around Underground OpeningsUnderground Openings1 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionAuthor s Note:Author s Note:The lecture slides provided here are taken from the course Geotechnical Engineering Practice , which is part of the 4th year Geological Engineering program at the University of British Columbia (V C d ) Th k ii d (Vancouver, Canada). The course covers rock Engineering and geotechnical Design methodologies, building on those already taken by the students covering Introductory Rock Mechanics and Advanced Rock Mechanics Rock Mechanics. Although the slides have been modified in part to add context, they of course are missing the detailed narrative that accompanies any l l d h h l lecture.

2 It is also recognized that these lectures summarize, reproduce and build on the work of others for which gratitude is extended. Where possible, efforts have been made to acknowledge th v ri us s urc s ith list f r f r nc s b in pr vid d t th the various sources, with a list of references being provided at the end of each lecture. Errors, omissions, comments, etc., can be forwarded to the 2 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionErrors, omissions, comments, etc., can be forwarded to the author at: Instability MechanismsControlled Instability Mechanismsllll d ibili i ll d i b Structurally-controlled instabilities are generally driven by a unidirectionalbody force, gravity. Stress-controlled instabilities, however, are not activated by a single force, but by a tensorwith six independent components Hence the a tensorwith six independent components. Hence, the manifestations of stress-controlled instability are more variable and complexthan those of structurally-controlled of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionStressStress--Controlled Instability MechanismsControlled Instability MechanismsKaiser et al(2000)4 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionKaiser et al.

3 (2000)StressStress--Controlled Instability MechanismsControlled Instability MechanismsAlh h h f dl l i f h f h b Although the fundamental complexity of the nature of stresshas to be fully considered in the Design of an underground excavation, the problem can be initially simplifiedthrough the assumptions of continuous, homogeneous, isotropic, linear elastic behaviour (CHILE).homogeneous, isotropic, linear elastic behaviour (CHILE).CHILE: Continuous, Homogeneous, Isotropic, Linear ElasticDIANE: Discontinuous, Inhomogeneous, Anisotropic, Non-ElasticTh ii ti i h th l ti b d th CHILE The Engineering question is whether a solution based on the CHILE assumption are of any assistance in Design . In fact though, many CHILE-based solutions have been used successfully, especially in those excavations at depthwhere high stresseshave closed the fractures and pgthe rock mass is relatively homogeneous and isotropic.

4 However, in near-surface excavations, where the rock stresses are lower, the fractures more frequent, and the rock mass more disturbedand weathered, there is more concern about the validityof the CHILE model 5 of 37 Erik Eberhardt UBC Geological Engineering ISRM Editionmore concern about the validityof the CHILE model. StressStress--Controlled Instability MechanismsControlled Instability MechanismsA stress analysis begins with a knowledge of the magnitudesand directionsof the in situstressesin the region of the excavation. This allows for the calculation of the excavation disturbed or induced (2006)There exists several close form solutionsfor the induced stresses Brady & BrowThere exists several close form solutionsfor the induced stresses around circular and elliptical openings (and complex variable techniques extend these to many smooth, symmetrical geometries), and with numerical analysis techniquesthe values of the induced stresses can be determined accurately for any three-dimensional B6 of 37 Erik Eberhardt UBC Geological Engineering ISRM Editionstresses can be determined accurately for any three-dimensional excavation geometry.

5 Stresses & Displacements Stresses & Displacements -- Circular ExcavationsCircular ExcavationsThe Kirsch equationsare a set of closed-form solutions, derived from the theory of elasticity, used to calculate the stresses and displacements around a circular excavation. pPkPk PStress ratio:Brady & Brown (2006)7 of 37 Erik Eberhardt UBC Geological Engineering ISRM Editionk = h/ vStresses & Displacements Stresses & Displacements -- Circular ExcavationsCircular ExcavationsFrom these equations we can see that the stresses on the boundary ( when r = a) are given by: = p[(1+k) + 2(1-k)cos2 ] = p[(1+k) + 2(1k)cos2 ] rr= 0 r = 0 h h dl r Note that the radial stresses are zero because there is no internal pressure, and the shear stresses must be zero at ttinf bunda traction-free of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionExample 1: Stresses around a Circular OpeningExample 1.

6 Stresses around a Circular a depth of 750 m, a 10-m diameter circular tunnel is driven in rock having a unit weight of 26 kN/m3and uniaxial compressive and tensile strengths of 80 0 MPa and 3 0 MPa strengths of MPa and MPa, the strength of the rock on the tunnel boundary be exceeded if:(a) k=0 3 andA (a) k= , and(b) k= the tunnel has neither a support pressure nor an internal Harrison & Hudson (2000)A. Since the tunnel has neither a support pressure nor an internal pressure applied to it, the local stresses at the boundary have 3= r= 0 and 1= . The Kirsch solution for the circumferential stress is:For a location on the tunnel boundary 9 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionFor a location on the tunnel boundary ( a = r), this simplifies to:Example 1: Stresses around a Circular OpeningExample 1: Stresses around a Circular Opening1 Hudson (2000)First, we can assume that the vertical stress is equal to the weight of the overburden, giving:MP5197500260 Harrison & 2 The extreme values of induced stress occur at positions aligned with the principal in situstresses, and so in order to compute the stress induced in the crown and invert ( roof and floor) we use = 90 , and for nwnnn(.)

7 Fnf)wu9,nfthe sidewalls we use = 0 .For k= :Crown/invert ( = 90 ): = MPa ( tensile) Sidewalls ( = 0 ): = 52 7 MPa Sidewalls ( = 0): = MPa For k= :Crown/invert ( = 90 ): = MPaSidewalls ( = 0 ): = 19 5 MPa compressive strength is 10 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionSidewalls ( = 0): = MPa strength is exceededStress and Failure CriterionStress and Failure CriterionThe key is that the failure criterion is comparedto the stresses and is not part of the calculation!! We ll see this is different when similar relationships based on Mohr-Coulomb and Hoek-Brown are incorporated into stress-strain constitutive relationships. 11 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionBrady & Brown (2006)Orientation of Orientation of 11& Induced Stresses & Induced Stresses Potential Ground Control Issues:Destressing = wedge failuresConcentration = spalling 11 11dt it destressingstress ttidestressingstress concentrationconcentrationdestressingStr esses can be visualized as flowing around the excavation periphery in the direction of the major principle stress ( 1) Where they diverge relaxation 12 of 37 Erik Eberhardt UBC Geological Engineering ISRM Editiondirection of the major principle stress ( 1).

8 Where they diverge, relaxation occurs; where they converge, stress increases occur. Stresses Away from Opening Stresses Away from Opening 13 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionBrady & Brown (2006)Zone of InfluenceZone of InfluenceThe concept of influence is important in excavation Design , since the presence of a neighbouring opening may provide a significant disturbance to the near-field stresses to the point of causing (a) axisymmetric stress distribution around a circular i i hd tti t opening in a hydrostatic stress field; (b) circular openings in a hydrostatic stress field, effectively isolated by virtue of ff ct y at y rtu f their exclusion from each other s zone of & Brown (2006)14 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionBrady & Brown (2006)Zone of InfluenceZone of Influence(2006)dy & Brown ( illustration of the effect of contiguous openings of different dimensions The zone of influence of excavation I includes excavation dimensions.)

9 The zone of influence of excavation I includes excavation II, but the converse does not of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionStresses Around Elliptical OpeningsStresses Around Elliptical OpeningsThe stresses around elliptical openingscan be treated in an analogous way to that just presented for circular openings There is much presented for circular openings. There is much greater utilityassociated with the solution for elliptical openings than circular openings, because these can provide a first papproximation to a wide range of Engineering geometries, especially openings with high width/height ratios ( mine stopes, power house caverns etc )house caverns, etc.).From a Design point of view, the effects of changing either the orientation within the stress fieldor the aspect ratioof such stress fieldor the aspect ratioof such elliptical openings can be studied to optimize of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionStresses Around Elliptical OpeningsStresses Around Elliptical OpeningsAssuming isotropic rock conditions, an elliptical opening is completely characterized by two parameters: aspect ratio(major to minor axis) which is the eccentricity of the ellipse; and orientationwith axis) which is the eccentricity of the ellipse; and orientationwith respect to the principle stresses.

10 The position on the boundary, with reference to the x-axis, is given by the angle .Hudson & Harrison (1997)17 of 37 Erik Eberhardt UBC Geological Engineering ISRM Edition()Stresses Around Elliptical OpeningsStresses Around Elliptical OpeningsIt is instructive to consider the maximum and minimum values of the stress concentrationsaround the ellipse for the geometry of an ellipse aligned with the principal stresses. It can be easily pgppyestablished that the extremes of stress concentration occur at the endsof the major and minor & Harrison (1997)18 of 37 Erik Eberhardt UBC Geological Engineering ISRM EditionExample 2: Stresses around an Elliptical OpeningExample 2: Stresses around an Elliptical gold-bearing quartz vein, 2 m thick and dipping 90 , is to be exploited by a small cut-and-fill stoping operation. The mining is to take place at a depth of 800 m, and the average unit weight of the r nite h st r ck b ve this level is 29 kN/m3 The strike f the granite host rock above this level is 29 kN/m3.