![]() In an earlier post we discussed consolidation settlement. We trust that this presentation has been enlightening and informative. We have covered the basic, classic case of consolidation settlement in this post and its predecessors From Elasticity to Consolidation Settlement: Resolving the Issue of Jean-Louis Briaud’s “Pet Peeve” and Deriving and Solving the Equations of Consolidation. The distinction between the drainage length h ( in the graph above) and the layer thickness H is clear. The above graph also gives approximating equations it is easy to see that, for, the equation given is simply the first two terms of Equation (14). We use the variable to emphasise that we are dealing with the “standard” case. ![]() In any case the usual method for determining –and by extension the degree of consolidation–is generally done either using a graph or a table, as is shown in the graph at the start of the post (repeated below:) Degree of Consolidation for Instantaneous Uniform Loading and One-Dimensional Flow. For certain cases, it is possible to use the first two or three terms. It is regrettable that, in defining, the value was not included, as using Equation (14) would be much simpler. That being the case, if we integration Equation (9) with Equation (11), we obtainĪs was the case with Equation (9), only the odd values of n are considered the even ones result in zero terms. Let us further define the dimensionless time constant In this case the result is divided by the uniform pressure p and the height h. For the specific case (governing equations, initial equations and boundary conditions) at hand, the degree of consolidation–the ratio of settlement at a given point in time to total settlement–can be determined as follows: Where is the settlement at any time before complete settlement. We start by defining the degree of consolidation thus: We know what the ultimate deflection will be based on Equation (3) above (or more complicated equations when preconsolidation is taken into consideration.) But how does the settlement progress in time? One thing that our theory presentation demonstrated was the interrelationship between pore pressure, stress and deflection. If we invoke the effective stress equation and solve this for the boundary and initial conditions described, we have a solution Where $latex u(x,t) is the pore water pressure. DefiningĪnd making some assumptions about the physics, we can determine the equation for consolidation as Where is the permeability of the soil and is the unit weight of water. The flow of water is regulated by the overall permeability of the soil, and all of this can be combined to yield Where is the flow of water out of the pores and is the porosity as a function of position and time. The change in porosity, for a saturated soil whose voids are filled with an incompressible fluid (hopefully water) induces water flow, Turning to the post Deriving and Solving the Equations of Consolidation, we first determined that the change in porosity could, for small deflections, be equated to the change in strain. Where is the compression index, is the initial void ratio of the layer, is the change in pressure induced from the surface, and is the average effective stress in the layer. The problem is that is not constant, and the settlement more accurately obeys the law Where is the settlement of the layer, is the thickness of the layer and is the uniaxial stress on the layer. We also showed that, for a homogeneous layer, Where E is the modulus of elasticity and is a factor based on Poisson’s Ratio and includes the effects of confinement, be that in an odeometer or in a semi-infinite soil mass. We were able to start with the theory of elasticity and, considering the effects of lateral confinement, define the coefficient of volume compression by In the post From Elasticity to Consolidation Settlement: Resolving the Issue of Jean-Louis Briaud’s “Pet Peeve”, we discussed the issue of how much soils (especially cohesive ones) settle through the rearrangement of particles. ![]() Note: the material for this derivation and those that preceded it have come from Tsytovich with some assistance from Verruijt. We need to start by a brief summary of what has gone before. This is the last (hopefully) post in a series on consolidation settlement. ![]()
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