Stability of Offshore Gravity Structure Foundations Using the Upper Bound Method
Authors: Murff J.D. , Miller T.W. 
Published: December 2007

The upper bound method of theoretical plasticity is described in detail and is contrasted with more conventional methods used in soil mechanics. This method is simple to apply, even for very complex foundation conditions. The method is more consistent than conventional techniques. Several examples are presented that apply when determining the stability of gravity structure foundations.

Introduction
The offshore gravity structure presents several unique foundation problems. Among the more important are investigations of stability against collapse. These stability analyses are complicated by complex site conditions, loadings, and base geometries.An appropriate idealization for this analysis is to neglect small elastic deformations by treating the structure as a rigid body and the soil as a rigid, perfectly plastic material. The equations of equilibrium, an plastic material. The equations of equilibrium, an appropriate yield condition, and boundary conditions provide a mathematical model of the system described provide a mathematical model of the system described by a rather cumbersome set of partial differential equations. In general, these equations are difficult to deal with; however, they have been solved for a few special cases. Such special solutions provide the basis for bearing capacity theory and have been extended to different loadings and geometries using empirical techniques. In some cases, other approaches (e.g., limit equilibrium methods) have been used to investigate effects of nonhomogeneous soil properties as well as unusual loading conditions. Both approaches are weakened by extending their empirical or experience basis since they rely heavily on subjective reasoning.Another alternative is to approximate the solution of the partial differential equations by using the bounding methods of plasticity. Ideally, lower bound methods should be used in stability analysis, but generally lower bounds near the exact solution are difficult to obtain. Here, we concentrate on applications of the upper bound method since it is adapted more readily to the complex loading conditions typical for offshore gravity structures. In addition, this method provides a systematic basis for studying the effects of variable soil conditions, loading conditions, and base geometries on collapse loads. The following sections discuss the use of the upper bound method in detail, contrast this method with other approximate methods, and show examples of various applications of the method.

The Upper Bound Method
The upper bound method is given in one form by Calladine's upper bound theorem of plasticity: "If an estimate of the plastic collapse load of a body is made by equating internal rate of dissipation of energy to the rate at which external forces do work in any postulated [kinematically admissable] mechanism of deformation of the body the estimate will be either high or correct." To apply this theorem,assume a collapse mechanism that is kinematically admissable (kinematic admissability will be discussed later),equate the rate of energy dissipation in the plastic material to the rate at which external forces (one of which is unknown) do work, andsolve for the unknown force. This is equivalent to the virtual work method, where virtual velocities are used instead of virtual displacements.
Each item is essential to the upper bound method.
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