Limit analysis solutions for the bearing capacity of rock masses using the generalised Hoek–Brown criterion

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Abstract

This paper applies numerical limit analyses to evaluate the ultimate bearing capacity of a surface footing resting on a rock mass whose strength can be described by the generalised Hoek–Brown failure criterion [Hoek E, Carranza-Torres C, Corkum B. Hoek–Brown failure criterion—2002 edition. In: Proceedings of the North American rock mechanics society meeting in Toronto, 2002]. This criterion is applicable to intact rock or heavily jointed rock masses that can be considered homogeneous and isotropic. Rigorous bounds on the ultimate bearing capacity are obtained by employing finite elements in conjunction with the upper and lower bound limit theorems of classical plasticity. Results from the limit theorems are found to bracket the true collapse load to within approximately 2%, and have been presented in the form of bearing capacity factors for a range of material properties. Where possible, a comparison is made between existing numerical analyses, empirical and semi-empirical solutions.

Introduction

The ultimate bearing capacity is an important design consideration for dams, roads, bridges and other engineering structures, particularly when large rock masses are the foundation materials. With the exception of some very soft rocks and heavily jointed media, the majority of rock masses provide an excellent foundation material. However, there is a need to accurately estimate the ultimate bearing capacity for structures with high foundation loads such as tall buildings and dams.

Rigorous theoretical solutions to the problem of foundations resting on rock masses do not appear to exist in the literature. This may be attributed to the fact that rock masses are inhomogeneous, discontinuous media composed of rock material and naturally occurring discontinuities such as joints, fractures and bedding planes. This makes the derivation of simple theoretical solutions based on limit equilibrium methods very difficult. In addition, fractures and discontinuities occurring naturally in rock masses are difficult to model using the displacement finite element method without the addition of special interface or joint elements. The upper and lower bound formulations of Lyamin and Sloan [1], [2] are ideally suited to analysing jointed or fissured materials due to the existence of discontinuities throughout the mesh. These discontinuities allow an abrupt change in stresses in the lower bound formulation and in velocities in the upper bound formulation. Moreover, employing discontinuities when modelling geotechnical problems enables great flexibility as they can be assigned different material properties and/or yield criteria. This unique feature was recently exploited by Sutcliffe et al. [3] and Zheng et al. [4] who used the formulations of Sloan [5] and Sloan and Kleeman [6] to analyse the bearing capacity of jointed rock and fissured materials respectively.

The purpose of this paper is to take advantage of the ability of the limit theorems to bracket the actual collapse load by computing both lower and upper bounds for the bearing capacity of strip footings on a broken rock mass. These solutions are obtained using the numerical techniques developed by Lyamin and Sloan [1], [2] which have been modified to incorporate the well-known Hoek–Brown yield criterion [7]. The applicability and background of the Hoek–Brown criterion will be discussed in the following section in more detail.

Section snippets

Applicability

It is well known that the strength of jointed rock masses is notoriously difficult to assess. The behaviour of a rock mass is complicated greatly because deformations and sliding along naturally occurring discontinuities can occur in addition to deformations and failure in the intact parts (blocks) of the rock mass. Unfortunately, laboratory tests on specific core samples is often not representative of a rock mass at field scale, while in situ strength testing of the rock mass is seldom

Problem definition

The plane strain bearing capacity problem to be considered is illustrated in Fig. 3. A strip footing of width B rests upon a jointed rock mass with an intact uniaxial compressive strength σci, geological strength index GSI, rock mass unit weight γ, and intact rock yield parameter mi. The ultimate capacity can be written asqu=σciNσ,where Nσ is defined as the bearing capacity factor. For a weightless rock mass (γ=0), the above expression is valid but the bearing capacity factor Nσ is replaced

Previous studies

A review of the literature reveals that very few thorough numerical analyses have been performed to determine the ultimate bearing capacity of shallow foundations on rock. Of the numerical studies that have been presented, few can be considered as rigorous. The ultimate tip bearing capacity of pile foundations, on the other hand, has received much more attention and is discussed by Serrano and Olalla [11], [18].

Carter and Kulhawy [19] and Kulhawy and Carter [20] proposed a simple lower bound

Results and discussion

The computed upper and lower bound estimates of the bearing capacity factor Nσ for both the weightless and ponderable rock analyses were found to be within 5% of each other. This indicates that, for practical design purposes, the true collapse load has been bracketed to within ±2.5% or better. As a consequence, average values of the upper and lower bound bearing capacity factor have been calculated and will be used in the following discussions.

Typical upper and lower bound meshes for the

Comparison with previous numerical studies

As a preliminary comparison, several limit analyses were performed for weightless rock masses using equivalent Mohr–Coulomb parameters as determined by Eqs. (16) and (15). Table 2 presents the results obtained for three different quality rock masses. The equivalent Mohr–Coulomb parameters were obtained over two separate ranges of the minor principal stress σ3; namely, 0<σ3<0.25σci and 0<σ3<0.75σci. This table, along with Fig. 16, indicates just how sensitive the interpreted values of c and φ

Conclusions

The bearing capacity of a surface strip footing resting on a rock mass whose strength can be described by the generalised Hoek–Brown failure criterion has been investigated. Using powerful new formulations of the upper and lower bound limit theorems, rigorous bounds on the bearing capacity for a wide range of material properties have been obtained. The results have been presented in terms of a bearing capacity factor Nσ in graphical form to facilitate their use in solving practical design

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