Deformation of a loose medium. I. Plane model

1981

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1. An understanding of the lateral-thrust coefficient is basic to the development of approximate schemes for computation of the pressures developed by a loose material. If the pressure problem is solved in the strict statement, there is no need to introduce a lateral-thrust parameter. Only a mathematical model of the deformation, the characteristics of the medium, and the edge conditions are given In this case. Relationships between stresses, which can be interpreted as the lateral-thrust coefficient, can be derived even after solution of the specific problem-Applied problems are usually solved using an approximate schemes, one of the premises of which is the introduction of an additional characteristic of the medium -the lateral-thrust coefficient -and a number of a priori assumptions on stress distribution [1]. The basic assumptions are contained in the hypothesis that the lateral-thrust coefficient k is constant.This hypothesis contaIns a whole series of assumptions on the fact that the lateral-thrust coefficient is independent of the yield of the enclosing surfaces, the character of the stress distribution, the material' s load history, etc. It is known, however, that the quantity k depends on these and many other factors. All hypotheses contaIning the assumption k = const are therefore only specific approximations. We can expect that depending on the specific situation, sone of these hypotheses will be proven rather accurate, others less accurate, and a third will be unsatisfactory. Of major Interest is the theoretical evaluation of various assumptions, which can be made from the solution of problems posed strictly.2. For this purpose, it is necessary to select a mathematical strain model, the load range, and the proper edge conditions. Let us restrict ourselves to the case of plane strain within the framework of the model described by Revuzhenko [2], which describes the basic properties of the medium and does not assume limitingstate conditions. In approximate computational schemes, the hypothesis k = const is introcklced as a generalization of Pascal's law; In this case, the magnitude of the coefficient itself characterizes the capacity of thin layers of material to transmit a load distributed over their entire length in the direction of the lateral enclosing surfaces. We will therefore treat the thIn layer as a loadingregion. The conditions on the boundaries of the layer depend on the method of loading and the entire strain region from which this layer is isolated. To evaluate the hypothesis, however, it is sufficient to restrict ourselves to certain characteristic cases when the condltious on the layer boundaries are given directly, and are not determined by the complete problem.Let us examine the case where there are no tangential stresses on the boundaries, and the lateral surfaces are orthogooal surfaces AB (Fig. 1). Let us assign normal-stress or displacement distributions to the surface AB, Wlnkler yield conditions to the lateral surfaces, and the condition of no vertical displacements to the surfac...

Section: Fh~mentioning

confidence: 97%

Section: Fh~mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Numerical experiments to determine the lateral-thrust coefficient of a loose medium

1981

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“…Let us consider this problem in the context of the mathematLcal model in [7]. We will limit ourselves to the case of plane deformation.…”

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confidence: 99%

Janssen's problem

1981

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1. Consider a vessel filled with liquid. The problem of finding the pressure of the liquid on the bottom and sides of the vessel has an elementary solution. This solution follows from the equations of equilibrium, Pascal's law, and the absence of tangential stresses at the boundary. Now suppose that the vessel is filled with an ideally loose material consisting of separate solid particles. In this case the problem of determining the pressure is much more difficult. This is because there is no analog of Pascal's law for loose materials, and because, in contrast with liquids, the pressure of a loose material depends not only on the depth of filling of the vessel but also on the method of filling.Determining the pressure of a loose material on the bottom and sides of a vessel is a classical problem in mechanics. Interest in it is prompted by the need to analyze various bunkers for storing loose and powdered materials, chemical reactors filled with granulated components, and so on.The first solution to this problem for vertical side walls was derived by Janssen [1]. His solution is based on two hypotheses: 1) The coefficient of lateral thrust k, equal to the ratio of the normal pressure on the wall to the mean normal pressure in a horizontal cross section, is constant (an analog of Pascal's law); 2) the tangential stress on the walls is completely developed and is proportional to the normal pressure.Subsequent investigations of this problem were essentLally based on these hypotheses [2]. The constructional norms and specifications at present in force are also based on Janssen's solution, to which a number of empirical correction coefficients are added [3].To make use of Janssen' s solution we must have data on the coefficient of lateral thrust. The existing recommendednorms for k range over very wide intervals. For example, for grain stores the USSR norm s give k ~ 0.4 [3]; the USA norms give k ~ 0.6; in West Germany the value k ~ 0.3 was used for manyyears, then the value k ~ 0.5 was intrvduced; these values have been described as "purely conjectural~ [4]. Thus in Janssen's scheme the question of the value of the coefficient of lateral thrust is still open. Moreover, while adhering to this scheme it is impossible to estimate its error. Experimental estimates of the error encounter various theoretical difficulties associated with the dependence of the pressure on the yield of the transducers and vessel walls and on other factors.In this connection solving the problem by the methods of mechanics of a continuous medium without involving hypotheses (1) and (2) is of consLderable interest. Solving the problem by exact methods enables us to find the distributions of all the stresses and to make a quantitative estLmate of the influence of various factors. With this approach, in contrast with Janssen's scheme, generalization to the case of inclined or curved side walls presents no new difficulties. From the known solution we can calculate the coefficient of lateral thrust and estimate the error of the engineering solution b...

Pressure of friable materials on pliable protective structures

1981

The class of problems dealing with mechanisms of soil and friable material reduces to the determination of the pressures on protective structures. In certain formulations the pliability of the structures is not important. A classical example is the problem dealing with the active and passive pressures of the ground on bulkheads.In this and other cases, the loading is determined for boundary cases of quite large wall displacements, such that the displacements themselves do not affect the loading. However, for the analysis of actual situations, these formulations as a rule are inadequate. In the first place, this is because the active and passive pressures differ considerably between one another. Therefore, as an estimate of the actual loads (and even more so, the pressure and moment diagrams) they give a very rough approximation. Because of this, the necessity arises for investigating the problem with more rigorous formulations.It is well known that in actual situations the pressures on the protective structures depend significantly on their pliability [i, 2]. Therefore, in a strict formulation the problems of pressure calculations must be set as statically indeterminate.For these formulations, it is necessary to formulate a closed system of equations describing the deformation of the medium and the correct boundary conditions. No less important also is the question of the adequacy of these conditions in the actual conditions at the boundary. The following classification can be made for the inherent boundary conditions; the following are assigned at the boundary: I) the stress or displacement vector; 2) the individual components of the displacement and stress vectors; 3) the relation between the components of the stresses (e.g., the condition of evolved dry friction); 4) limitation nn the stress components in the form of an inequality (e.g., dry friction or adhesion on unknown sections of the boundary); 5) a functional relation between the component of the stress acting at a defined point of the boundary and the component of the displacement of this point; 6) the component of the displacement at a function of the stress distribution at a defined section of the boundary, etc. For plane deformation, it is necessary in the general case to assign two boundary conditions to the whole closed contour bounding the region of deformation. Problems with boundary conditions of type 3-5 were investigated in [3, 4]. Taking account of the pliability of the protective structures leads to more complex boundary conditions of type 6.Let us consider the method of implementing these conditions by themethod of finite elements. We will assume that the component of the displacement u~ depends on the stresses a~1 acting on the rectilinear section of the boundary AB (x~ = const, xl and x2 are Cartesian coordinates) : B u I (z,) = .~ G (z,, z) a11 (~) dz, z, ~ AB.(i) A Here, it is assumed that the protective structure functions elastically. The function G is determined by the parameters of the structure. The second condiKion at the boundary...