Sloan, S. W. (2013). Ge´otechnique 63, No. 7, 531–572 [http://dx.doi.org/10.1680/geot.12.RL.001] Geotechnical stability analysis S. W. SLOANà This paper describes recent advances in stability analysis that combine the limit theorems of classical plasticity with finite elements to give rigorous upper and lower bounds on the failure load. These methods, known as finite-element limit analysis, do not require assumptions to be made about the mode of failure, and use only simple strength parameters that are familiar to geotechnical engineers. The bounding properties of the solutions are invaluable in practice, and enable accurate limit loads to be obtained through the use of an exact error estimate and automatic adaptive meshing procedures. The methods are very general, and can deal with heterogeneous soil profiles, anisotropic strength characteristics, fissured soils, discontinuities, complicated boundary conditions, and complex loading in both two and three dimensions. A new development, which incorporates pore water pressures in finite-element limit analysis, is also described. Following a brief outline of the new techniques, stability solutions are given for several practical problems, including foundations, anchors, slopes, excavations and tunnels. KEYWORDS: anchors; bearing capacity; excavation; numerical modelling; plasticity; slopes; tunnels STABILITY ANALYSIS Limit equilibrium In geotechnical engineering, stability analysis is used to Limit equilibrium is the oldest method for performing predict the maximum load that can be supported by a stability analysis, and was first applied in a geotechnical geostructure without inducing failure. This ultimate load, setting by Coulomb (1773). In its most basic form, this which is also known as the limit or collapse load, can be approach presupposes a failure mechanism, and implicitly used to determine the allowable working load by dividing it assumes that the stresses on the failure planes are limited by by a predetermined factor of safety. The precise value of this the traditional strength parameters c and ö. The chief factor depends on the type of problem, with, for example, advantages of the limit-equilibrium method are its simplicity lower values being appropriate for slopes and higher values and its long history of use, which have resulted in widely being adopted for foundations. Rather than impose a factor available software and extensive collective experience con- of safety on the ultimate load to obtain the allowable work- cerning its reliability. Its main disadvantage, on the other ing load, it is also possible to apply a factor of safety to the hand, is the need to guess the general form of the failure strength parameters prior to performing the stability analysis. surface in advance, with poor choices giving poor estimates In some procedures – finite-element strength reduction of the failure load. In practice, the correct form of the analysis, for example – the actual safety factor on the failure surface is often not intuitively obvious, especially for strength can be found for a given set of applied loads and problems with an irregular geometry, complex loading, or material parameters (which typically comprise the cohesion complicated stratigraphy. There are other shortcomings of and friction angle). the technique, as follows. Once the allowable load is known, the working deform- (a) The resulting stresses do not satisfy equilibrium at every ations are usually determined using some form of settlement point in the domain. analysis. Historically, these deformations have been predicted (b) There is no simple means of checking the accuracy of the using elasticity theory, but they are now often found from a solution. variety of numerical methods, including non-linear finite- (c) It is hard to incorporate anisotropy and inhomogeneity. element analysis. In some cases, particularly those involving (d) It is difficult to generalise the procedure from two to three dense sands, serviceability constraints on the deformations dimensions. may actually control the allowable load rather than the ultimate load-carrying capacity. Despite these limitations, a multitude of limit-equilibrium Broadly speaking, there are four main methods for per- methods have been proposed and implemented, particularly forming geotechnical stability analysis: limit equilibrium, for slope stability analysis. Indeed, early examples of widely limit analysis, slip-line methods, and the displacement finite- used slope stability methods include those of Janbu (1954, element method. In the following, all these techniques will 1973), Bishop (1955), Morgenstern & Price (1965), Spencer be discussed except the slip-line methods. This family of (1967) and Sarma (1973, 1979). A more recent procedure, procedures is omitted, not because they are considered to be described by Donald & Giam (1989b), is also noteworthy, ineffective, but simply because they are not well suited to since it gives a factor of safety that is a strict upper bound the development of general-purpose software which can deal on the true value. with a wide variety of practical problems. The key principles of the limit-equilibrium approach can be illustrated by considering the classical bearing capacity pro- blem for a smooth strip footing, of width B, resting on a deep layer of undrained clay of strength su, as shown in Fig. 1. Manuscript received 14 September 2012; revised manuscript accepted To begin the analysis, the supposition is made that failure 22 January 2013. occurs along a circular surface whose centre lies at some à ARC Centre of Excellence for Geotechnical Science and Engineer- point directly above the edge of the footing, as shown in ing, University of Newcastle, NSW, Australia. Fig. 2. In addition, undrained failure along this surface is 531 532 SLOAN Bearing capacityϭϭq ? σ . ∂ ij . p ϭ f σ ελij ∂ σij B ϭ f()σij 0 Saturated clay ε σ Undrained shear strength ϭ s ij u (a) (b) Fig. 3. (a) Perfectly plastic material model and (b) associated flow rule Fig. 1. Smooth strip footing on deep layer of undrained clay 0 tions and the yield criterion. For a perfectly plastic material model with an associated flow rule, it can be shown that the θ load supported by a statically admissible stress field is a R lower bound on the true limit load. Although the limit load q for such a material is unique, the optimum stress field is not, and thus it is possible to have a variety of stress fields that furnish the same lower bounds. To illustrate the applica- B tion of lower-bound limit analysis, the smooth rigid footing τ ϭ s u problem shown in Fig. 1 is considered again. The simple stress field shown in Fig. 4, which consists of three distinct zones separated by two vertical stress discontinuities, is σn statically admissible since it satisfies equilibrium, the stress boundary conditions, and the undrained (Tresca) yield criter- ó ó Fig. 2. Limit-equilibrium failure surface for strip footing on clay ion 1 À 3 ¼ 2su everywhere in the domain. Note that each stress discontinuity is statically admissible because the nor- mal and shear stresses are the same on both of its opposing assumed to be governed by the Tresca criterion, with the sides, and that equilibrium is automatically satisfied every- maximum shear strength being fully mobilised at every where in each zone because the unit weight is zero and the point, so that the shear stress is given by ô ¼ su: stress field is constant. Although the normal and shear Taking moments about the centre of the failure surface, stresses must be continuous across an admissible stress O, the following is obtained. discontinuity, the normal stress on a plane orthogonal to the B discontinuity is permitted to jump. This feature can be ðÞqB 3 ¼ ðÞ2RŁ 3 su 3 R exploited in the construction of stress fields to give useful 2 lower bounds, and is shown in Fig. 4. or Since the stress field supports a vertical principal stress of ó 4s Ł 1 ¼ 4su in the zone beneath the footing, this defines a lower q ¼ u (1) bound on the bearing capacity of q ¼ 4s : 2 Ł low u sin In contrast to the lower-bound theorem, the upper-bound The lowest value of q, and hence the geometry of the critical theorem requires the computation of a kinematically admis- surface, can be found by setting dq/dŁ ¼ 0. This leads to the sible velocity field that satisfies the velocity boundary condi- simple non-linear equation tan Ł À 2Ł ¼ 0, which can be tions and the plastic flow rule. For such a field, an upper solved to yield the critical angle Łc ¼ 66.88. Inserting this bound on the collapse load is obtained by equating the critical angle in equation (1) gives the approximate bearing power expended by the external loads to the power dissi- capacity as pated internally by plastic deformation. Note that although the true limit load from such a calculation is unique, the q ¼ 5:52s u actual failure mechanism is not. This implies that multiple which is approximately 7% above the exact solution mechanisms may give the same limit load, and it is neces- q ¼ (2 + ð)su derived by Prandtl (1920). sary to seek the mechanism that gives the lowest upper bound. A simple upper-bound mechanism for this strip footing example, shown in Fig. 5, assumes that failure Limit analysis occurs by the rigid-body rotation of a circular segment, with Limit analysis is based on the plastic bounding theorems developed by Drucker et al. (1951, 1952), and assumes qsϭ 4 small deformations, a perfectly plastic material (Fig. 3(a)), low u and an associated flow rule (Fig. 3(b)). The last assumption, which is often termed the normality rule, implies that the åp ó ϭ σ ϭ 4s σ ϭ 0 plastic strain rates _ij are normal to the yield surface, f( ij), σ3 0 1u 3 åp º_ ó º_ so that _ij ¼ @ f =@ ij, where is a non-negative plastic ϭ ϭ σ ϭ 2s multiplier.