Incipient failure of a circular cylinder under gravity
Introduction
The yield stress of a slurry is an important parameter used in industrial processes. In mining engineering, the yield stress is a control parameter governing the pumping and disposal of mine tailings [1], whereas in the construction industry, the yield stress of fresh concrete affects its mixing, transportation, placing and consolidation behaviour [2], [3]. Laboratory instruments such as the vane rheometer [4] can be used to measure the yield stress of these materials. However, in the field the “slump test” presents a simpler approach [1]. In this test a frustum of a cylinder or a cone is placed on a flat base and filled with slurry. Lifting the frustum causes the slurry to slump down to a lower height, which can then be related to the yield stress. Rather than considering this flow problem, we examine the height of incipient failure, i.e., the height of material required to just initiate flow. This height is of fundamental importance to the understanding and operation of the slump test, since it must be exceeded for flow to occur. Also, calculation of the point of incipient failure can be used to develop a method for determining the yield stress, as we shall discuss. Since cylindrical geometry is used typically in practice, we extend previous work on a plane-strain rectangular block [5] to the case of an axisymmetric cylinder.
A simplistic approach commonly used to analyse the slump test is to assume that all stresses are vertical, and uniform across any horizontal plane [1], [3], [6]. This model can also be used directly to calculate the height of incipient failure H, and giveswhere σy is the uniaxial yield stress, ρ is the density and g is the acceleration due to gravity. This shall henceforth be referred to as the “uniform stress model”. It predicts that the radius of the cylinder and friction conditions at the base do not affect the height of incipient failure, and that the material yields only along the base.
In this paper, we rigorously analyse the height of incipient failure of a cylinder. We assume that the slurry is a perfectly rigid-plastic material. A finite-differencing implementation of the slip-line field method is used [7], [8], [9], incorporating the Haar–Karman hypothesis. We allow for frictional effects at the base, which are modelled as Coulombic in nature. In so doing, we show that there exists a critical coefficient of friction, where increasing the coefficient of friction above this critical value does not change the height of incipient failure. The predictions of the uniform stress model are assessed using these results. Finally, we compare the slip-line field cylinder solutions with analogous results for the plane-strain problem.
In the next section, we describe the governing equations and the underlying assumptions in the model. This will be followed in Section 3 by a discussion of how to construct slip-line fields for the axisymmetric circular cylinder, with the details relegated to the appendices. Example slip-line field diagrams are given for several cases of base friction and radius. In Section 4, the results of the slip-line field calculations are discussed with reference to (i) predicting the height of incipient failure, (ii) measuring the yield stress, (iii) comparing the results to those for a rectangular block, and (iv) discussing the implications of the Haar–Karman hypothesis. Conclusions are given in Section 5.
Section snippets
Model
A schematic illustration of the circular cylinder and the coordinate system used is given in Fig. 1. The angular coordinate in this cylindrical coordinate system is denoted by θ. We assume that the stress field and the incipient strain field are symmetric about the vertical axis of the cylinder, with no incipient displacement in the θ direction. The strain components εθr and εθz are therefore zero. We also assume that the corresponding stress components Σrθ and Σzθ are zero, although these
Construction of plastic field
The axisymmetric slip-line field solutions are constructed using an approach similar to that used for the analogous plane-strain problem [5]. The slip-line field method applied here is based on propagating information along two sets of characteristics known as α- and β-lines. Fig. 2 shows an α-line CA, a β-line CB and the slip-line angle φ between the r-axis and the α-line. Further details on how the stress field at C is calculated from the stress field at A and B are given in Appendix A. This
Results and discussion
The height of incipient failure for a circular cylinder loaded by gravity can be calculated from the stress fields constructed in the previous section, using a method based on that used for the plane-strain analysis [5]. The slip-line field calculation gives the stress tensor along a line from the top of the plastic part of the vertical free surface at height hp to the centre of the cylinder (e.g., TBCD in Fig. 10). This line defines a surface of revolution in the axisymmetric geometry. The
Conclusions
We have investigated the incipient failure of a cylinder under gravitational loading using slip-line field theory. The cylinder was assumed to behave as a perfectly rigid-plastic material and satisfy the Haar–Karman hypothesis. The friction at the base was modelled using a modified Coulomb law. The stress field in the deformable plastic region of the cylinder at the point of incipient failure has been calculated. This gives a prediction of the height of incipient failure of a cylinder as a
Acknowledgements
We acknowledge the support of the Particulate Fluids Processing Centre (PFPC), an Australian Research Council Special Research Centre.
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