746 McMahon, B. T. et al. (2013). Ge ´otechnique 63, No. 9, 746–752 [http://dx.doi.org/10.1680/geot.12.P.061] Cavity expansion model for the bearing capacity and settlement of circular shallow foundations on clay B. T. McMAHON  , S. K. HAIGH  and M. D. BOLTON  The use of an ellipsoidal cavity expansion model to estimate the bearing capacity and settlement of circular shallow foundations on clay is presented. The model uses an upper-bound energy approach with contours of constant soil displacement taken to be ellipsoidal within a hemispherical outer boundary. The elastic and plastic work done within the soil are equated to the footing work, with yield being defined using the von Mises’ yield criterion. It is shown that, for two different soil rigidities, results are consistent with those obtained from finite-element analyses available in the literature. A relationship between the bearing stress on a circular shallow foundation and its normal- ised settlement is developed, with an expression provided for the linear working range. KEYWORDS: bearing capacity; clays; plasticity; settlement INTRODUCTION The design of a shallow foundation requires consideration of the ultimate bearing capacity, and of the settlement at the working load. Settlements are often critical, owing to the vulnerability of typical structures to differential settlements. There are two approaches that can be adopted in deter- mining the bearing capacity of foundations. For shallow foundations, a local mechanism of indentation, shear and heaving is invoked, which leads to Terzaghi’s bearing capa- city equation (Terzaghi, 1943). For the particular case of a shallow foundation on the surface of a purely cohesive soil with undrained shear strength c u , the bearing capacity, q ult , is determined using q ult ¼ c u N c s c (1) where N c and s c are the bearing capacity and shape factors respectively. Prandtl (1921) used plasticity theory to determine for a strip footing on the surface that N c ¼ 2+ ð ¼ 5 . 14. Refinements have been made to the equa- tion since, including research on different-shaped footings and the effects of embedment, footing roughness and soil profile. Cox et al. (1961) determined that N c ¼ 5 . 69 for a smooth circular footing, and Eason & Shield (1960) found that N c ¼ 6 . 05 for a rough circular footing. The program Analysis of Bearing Capacity (ABC) (Martin, 2003) supports these values, with an example of the characteristic net being shown in Fig. 1(a). Concerns about the use of a rigid-plastic material to represent soil led to the second approach for determining foundation stiffness and ultimate bearing capacity: a cavity expansion idealisation. This method has been used primarily for deep foundations such as piles, where capacity is pro- vided by the resistance that the soil offers to the expansion of a cavity corresponding to the volume indented by the pile. Using spherical cavity expansion theory combined with plasticity, the bearing capacity factor has been determined to be a function of both the strength and the stiffness of the soil. This arises from the recognition that soil in the far field must remain elastic if the indentation of the foundation Manuscript received 3 May 2012; revised manuscript accepted 12 October 2012. Published online ahead of print 31 January 2013. Discussion on this paper closes on 1 December 2013, for further details see p. ii.  Department of Engineering, University of Cambridge, UK. (a) C L 2·5 m 0·2 m (b) Fig. 1. Mechanisms beneath shallow foundations: (a) stress characteristic method from ABC (Martin, 2003) for Prandtl mechanism; (b) undrained mechanism at prototype scale from centrifuge testing