Three-Dimensional Consolidation

Three-Dimensional Consolidation J. A. DE WET, Research Officer, National Building Research Institute, South African Council for Scientific and Industrial Research, Pretoria The one-dimensional theory of Terzaghi can be extended to three dimensions if the initial distribution of pore water pressures is known. This theory is similar to that of Biot with the exception that the normal strains are not defined during consolidation other than by the general conditions of compatibility of strain. Only the volumetric strain is pre• scribed. Experimental evidence is presented in favor of the former approach, and a complete solution, including stability analysis, is made of a consolidating rigid circular footing using the generalized Terzaghi theory. A study is also made of secondary compression which is shown to be due to variations in the coefficient of per• meability. The bulk modulus or coefficient of compression is shown to be very nearly constant, in spite of variation of the Poisson ratio during consolidation, and is proportional to the average permeability. • THIS PAPER presents experimental evidence that the space distribution of pore water pressures, caused by an applied load, is independent of the strain field and is very nearly equal to the normal stress on octahedral planes. Small pore pressures are caused by the reorientation of particles which follows after shearing strain, but these are shown to be negligible in the field. If Darcy's law is valid with a constant coefficient of per• meability, then the dissipation of pore pressure is governed by the classical equation like that for the diffusion of heat in solids, and therefore, the pore pressure history can be traced if the initial distribution is known. This is given by the theory of elasticity which is shown to be applicable to soils if the shear stress-strain curve is linearized. The bulk modulus relating a normal strain to a normal stress is shown experimentally to be constant over a reasonable stress increment, although Poisson's ratio varies during the consolidation of a saturated clay—from 0. 5 to approximately 0. 4. The strain in the vertical direction (i. e., settlement) can be split into a component due to shearing stress only, which wUl be almost instantaneous for practical footings, and a component due to normal stress only, which will be time dependent and for a saturated soil will have to await the expulsion of the pore water (coincident with dissi• pation of pore pressure). Only the latter is defined as consolidation and the resulting normal strains are independent of the conditions of drainage in full accord with the theory of elasticity. But both settlements can be evaluated if the average shear modulus and the bulk modulus are known. As an illustration, the complete solution is given for a cylindrical footing on a semi-infinite layer for both a constant load and a variable load, with an analysis of the stability of the footing. It is apparent that it is unnecessary to consider consolidation as an elastic problem with a variable body force as done by Biot {!) in 1941, when he derived equations iden• tical to the elastic equations used to compute secondary effects due to a nonuniform tem• perature distribution. Cryer, in a note as yet unpublished, has shown that, for small values of the Poisson's ratio v , these equations can lead to a pore pressure CT at the center of a clay sphere, equal to 157 percent of a hydrostatic pressure round the sphere (there being no shearing stresses). No such increases have been observed during tests conducted by the author of the present paper. The pore pressure is always equal to the hydrostatic pressure as predicted by the classical theory of Terzaghi (2). 152 153 This case corresponds to a solution of Blot's equations for Poisson's ratio v = 0. 50, when the solutions of Blot are also solutions of the generalized Terzaghi Theory. This method outlined for the treatment of three-dimensional consolidation is also identical to that of Terzaghi and is considerably simpler mathematically than that of Blot. The reason why Biot's equations predict a possible pressure increase is because his fundamental stress-strain relations are of the form V 'x = E- • T ^^y * ^z^ ^ ^^^^ ^z = i - 1 ^ V ^ in which ax, cry, and normal stresses on the soil element, a is the pore water pressure, and p, v, E are elastic constants. These equations imply that a change in the pure water pressure affects each normal strain in the same way or, conversely, a strain must cause a pore pressure change in a unique manner. That such is not the case is shown by experimental results described later and illustrated by Figure 2. The distribution of pore pressures at the center of a consolidating sample, with volume change AV as shown by the curve 2, was found to be unaffected whether there was a lateral strain equal to the vertical unit strain cy. or no lateral strain whatever, whereas the normal stresses cj^ = ^^'^ CTX = ^y = were nearly the same in both cases. Apparently the only certainty about a consolidating soil is that there is a reasonably constant bullc modulus (i.e., that there will be a volumetric strain with a pore pressiure change), but the individual normal strains cannot be prescribed other than by the rather wide conditions of compatibility. This point is further illustrated by Figure 3, which shows that the volumetric strain Is virtually the same for a confined and unconflned sample, although In the former case the strain is necessarily all In the vertical direction, whereas the latter was characterized by isotropic unit strains (being the same in the three coordinate directions). The main error in the extended Terzaghi approach rests In the so-called phenomenon of "secondary compression," defined as consolidation that does not follow the time behavior predicted by the simple diffusion equation with constant coefficients. This phenomenon is shown experimentally to be due to variations in the coefficient of per• meability which are present at the small pressure gradients In the latter stages of con• solidation; this is in direct confirmation of an hypothesis by Schiffman (3). The average coefficient of consolidation Cy was found for intervals In an interrupted consolidation test and these values were used in a numerical solution of one-dimensional consolida• tion by a method also mentioned by Schiffman (3). This numerical solution provides a far better fit to laboratory consolidation curves and, using this fit. It was possible to show that the coefficient of consolidation was practically independent of the loading sequence in the virgin range. If the soil Is partially saturated, there will still be consolidation, but the pore water pressure cr will only be some fraction a of the octahedral stress CTQ on a soil element at the moment of application of the load, and, according to a thermodynamic relation• ship derived by Coleman and Croney (4), the volume change £N of the soli element will be the same fraction a of AV^, the quantity of water expelled. Therefore, the settle• ment due to consolidation can stlU be determined if ce and the relation between a change In the water pressure and a corresponding change In AVm are known. The latter Is given numerically by the slope of the pF moisture content curve, if the pF Is converted to a natural scale of pressure. APPARATUS AND SAMPLES Most of the experimental evidence was obtained from a large triaxial machine which was adapted as a three-dimensional consolldometer without side friction and without any 154 shear stresses (see Fig. 1). Provision was also made for measuring the pore pressures in the center of the sample by means of a pore pilot, which consisted of a flat, hollow brass tube roughly 1 ^/z in. long by ^/i. in. wide, with a smaller aperture at the end cov• ered by 200-mesh wire. The press remeasuring device was extremely simple, consist• ing merely of the movement of a slug of mercury against air entrapped at the end of a sealed glass tube (0.4-mm bore). The air volume changes were very small (of the order of 10"* cu in.) and so were measured by micrometer. The system could be calibrated by exerting pressure on the reservoir connected to the water lines. By using flexible Saran tubing it was possible to insert the pore pilot while water was dripping through its end in order to ensure both a thorough de-airing as well as a direct contact between the soil water and the water in the measuring assembly. Loads could be applied to the cylindrical sample (height and diameter 3 in.) either by a cell pressure or a vertical load, or both. In the former case, it was necessary to ensure contact of the loading bar with the top of the sample by observing the "response" of an ammeter connected to a measuring bridge which was in balance with a linear variable differential transformer attached to a diaphragm on the base of the cell. Deformations could therefore be registered either by the vertical dial gauge d or by volume changes dV of castor oil in the right hand tube connected to the cell. The sample was insulated from the oil by means of a thin rubber membrane provided with a nipple Pf88. Lin« Cell Pr«s8 =er3- Vtrt Dial d Looding Bar Cell Vol Change dV Pore Preee Wattr Vol Chongt dV^ Drainage Line Figure 1. Modified triaxial compression apparatus. 155 for the pore pilot insertion. These volume changes could be correlated with the quan• tity of water dV^ expelled from the sample. If dVm = dV, the sample was defined as saturated.

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