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Lower Bound Limit Analysis Using Power Cone Programming for Solving Stability Problems in Rock Mechanics for Generalized Hoek–Brown Criterion

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Abstract

This study introduces a methodology to solve plane strain stability problems in rock mechanics, following the generalized Hoek and Brown yield criterion, by employing the lower bound finite elements limit analysis in conjunction with the power cone programming. The efficacy of the proposed approach has been demonstrated by solving three different types of stability problems: (1) finding the bearing capacity of strip footings on rock media, (2) assessing the stability of finite rock slopes, and (3) the stability analysis of unlined rectangular tunnels in rock mass. In all the cases, the results obtained from the analysis have been compared thoroughly with that computed using (1) nonlinear programming, and (2) semi-definite programming technique. The present approach has been found to be computationally very robust and it generates very accurate solutions.

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Abbreviations

\(\sigma_{1}\) :

Major principal stress

\(\sigma_{3}\) :

Minor principal stress

\(\sigma_{\text{ci}}\) :

Uniaxial compressive strength

\(m_{\text{b}}\) :

Hoek–Brown material constant

\(m_{i}\) :

Hoek–Brown material constant

s :

Hoek–Brown material constant

α :

Hoek–Brown material constant

GSI:

Geological strength index

D :

Disturbance factor

\({\mathcal{K}}\) :

Cone

\({\mathcal{K}}^{*}\) :

Dual cone

\({\Re }^{n}\) :

n-Dimensional real space

\({\Re }^{m \times n}\) :

Real matrices of size m\(\times\)n

\(Q^{n}\) :

n-Dimensional quadratic cone

\(Q_{r}^{n}\) :

n-Dimensional rotated quadratic cone

\(P_{n}^{\beta}\) :

n-Dimensional power cone

\({\mathbf{S}}^{n}\) :

Set of \(n \times n\) symmetric matrices

\({\mathbf{S}}_{ + }^{n}\) :

Set of \(n \times n\) positive definite matrices

\(\varvec{\sigma}\) :

Stress tensor

\(\lambda_{\hbox{max} }\) :

Auxiliary variable

\(\lambda _{{\min }}\) :

Auxiliary variable

t :

Auxiliary variable

\(\sigma_{x}\) :

Normal stress on x-plane

\(\sigma_{y}\) :

Normal stress on y-plane

\(\tau_{xy}\) :

Shear stress on x-plane in y-direction

\(\varvec{c}\) :

Vector which contains the objective function

\(\bar{\varvec{\sigma }}\) :

Global unknown vector

NN :

Number of nodes

\({\mathbf{A}}_{\text{equi}}\) :

Matrix containing the left-hand side of all the equilibrium equations

\({\mathbf{A}}_{\text{dis}}\) :

Matrix containing the left-hand side of all the discontinuity equations

\({\mathbf{A}}_{\text{bc}}\) :

Matrix containing the left-hand side of all the boundary conditions

\({\mathbf{b}}_{\text{equi}}\) :

Vector containing the right-hand side of all the equilibrium equations

\({\mathbf{b}}_{\text{dis}}\) :

Vector containing the right-hand side of all the discontinuity equations

\({\mathbf{b}}_{\text{bc}}\) :

Vector containing the right-hand side of all the boundary conditions

B :

Width of foundation

\(Q_{\text{u}}\) :

Maximum vertical load

\(N_{\sigma }\) :

Bearing capacity factor of strip footing in weightless media

\(N_{\sigma \gamma }\) :

Bearing capacity factor of strip footing

\(q_{\text{u}}\) :

Ultimate bearing pressure

γ :

Unit weight of rock mass

L :

Parameter which defines the boundary

H :

Parameter which defines the boundary

p :

Parameter which defines the state of stress

q :

Parameter which defines the state of stress

β :

Slope angle

\(N_{\text{s}}\) :

Stability number for slope

h :

Height of tunnel

w :

Width of tunnel

\(N_{\text{t}}\) :

Stability number for tunnel

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Authors and Affiliations

  1. Department of Civil Engineering, Indian Institute of Science, Bengaluru, 560012, India

    Jyant Kumar & Obaidur Rahaman

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  1. Jyant Kumar
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  2. Obaidur Rahaman
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Correspondence to Jyant Kumar.

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Appendix: Implementation of the PCP in MOSEK

Appendix: Implementation of the PCP in MOSEK

The present investigation utilizes the optimization toolbox: MOSEK in MATLAB since it can handle the PCP which is found to be robust and computationally very efficient (Makrodimopoulos and Martin 2006; Ukritchon and Keawsawasvong 2018; Mohapatra and Kumar 2019a). To solve the PCP in MOSEK, the input needs to be specified in a particular way. The objective function, as given in Eq. (22), is defined by the command ‘prob.c’. The matrices containing inequality or equality constraints are defined with ‘prob.a’ command. The lower and upper bounds of the constraints are specified by ‘prob.blc’ and ‘prob.buc’, respectively. The corresponding bounds on the variables are defined by ‘prob.blx’ and ‘prob.bux’. The cones are prescribed using four different cell arrays: ‘prob.cones.type’, ‘prob.cones.conepar’, ‘prob.cones.sub’ and ‘prob.cones.subptr’. The array ‘prob.cones.type’ specifies the type of cone and it takes the command (1) ‘res.symbcon.MSK_CT_QUAD’ for the second-order cone, (2) ‘res.symbcon.MSK_CT_RQUAD’ for the rotated quadratic cone, and (3) ‘res.symbcon.MSK_CT_PPOW’ for the power cone. The value of the power term in the power cone is specified by ‘prob.cones.conepar’. The variables related to a particular cone are specified by ‘prob.cones.sub’ and the change in the cone is indicated in ‘prob.cones.subptr’. The optimal solution of the objective function is obtained through ‘res.sol.itr.pobjval’ command. The primal as well as dual optimal solutions of the variables are finally reported in ‘res.sol.itr.xx’ and ‘res.sol.itr.y’, respectively.

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Kumar, J., Rahaman, O. Lower Bound Limit Analysis Using Power Cone Programming for Solving Stability Problems in Rock Mechanics for Generalized Hoek–Brown Criterion. Rock Mech Rock Eng 53, 3237–3252 (2020). https://doi.org/10.1007/s00603-020-02099-y

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