Consolidation Theories of Soils

General Consolidation is gradual process involving drainage, compression & stress transfer. In geology, it refers to the hardening of soil to a rock like condition. In geotechnical engineering , it refers to adjustment of loads to applied loading. It may require a long time for a soil formation to come to an equilibrium under load. When equilibrium is reached, the soil is considered as a fully consolidated.

Cont’d…

Cont’d… The rate at which the volume change (consolidation) occurs is directly related to the permeability since the permeability controls the speed at which the pore pressure can escape. In pervious soil ( clean sand)- consolidation is almost instantaneous. In soil of low permeability ( clay)- consolidation will be quite slow.

Mechanics of Consolidation Consolidation can be explained with the help of a piston and spring mechanical analogy shown below. The spring represents the soil grains and the water represents the moisture in the soil.

Cont’d…

cot……….

General outline of Terzaghi-Froehlich’s one-dimensional Theory of Consolidation Can be explained with the help of Fig. below Consider a clay layer 2H laying between to pervious sand layers and is subjected to a surface unit load equal to . The water is squeezed out the two pervious boundaries – double drainage

Cont’d… At the instant the pressure is applied (t=t0), it is entirely carried by the pore pressure. That means, A few instant later, water will start escaping into the sand, so that the pore pressure at both boundaries will equal to zero. At time, t = ∞, consolidation will be completed and excess pore pressure will be zero. At any time, the area between the curve pertaining to that time and the initial hydrostatic excess pressure diagram gives the load transferred to the soil grains up to that time.

Cont’d… Progress of consolidation process at a given point in the soil is indicated by where The average percent of consolidation of the entire layer at any time is numerically equal to the percentage change in thickness or settlement. Then to estimate the rate of settlement, it is necessary to establish the variation of U with time. The relationship is established based on the assumptions stipulated below.

Cont’d… a) Assumptions used in Terzaghi-Froehlich’s theory 1) Homogeneous soil 2) Complete saturation 3) Negligible compressibility of soil grains and water 4) One dimensional compression 5) One dimensional flow 6) The validity of Darcy’s Law 7) The Coefficient of permeability, k, is the same everywhere within the layer and remains constant during consolidation.

Cont’d… b) Coefficient of Consolidation, Cv - Indicates the combined effects of permeability and compression for a given void ratio range. where cV - coef. of consolidation k - coef. of permeability aV - coef. of compressibility γω - unit weight of water .

Cont’d… c) Longest Drainage Path, H - represents the longest distance travelled by drop of water in reaching an outlet The path is longer in a soil stratum with single drainage as shown in Figure below.

Cont’d… - thickness of clay layer d) Time-rate of settlement:- depends on the - thickness of clay layer - number of drainage faces - permeability of the soil - magnitude of the consolidating pressure acting on the layer

Mathematical Formulation of Terzaghi-Froehlich’s Theory of Consolidation From the flow in earth mass, the time rate of change of volume is expressed as From one dimensional flow, and the case Refer Teferra /Leikun (1999) for the step-by-step derivation of consolidation equation , pp 168-177

Cont’d… Generally, the one-dimensional consolidation equation is given as where

Rigorous Solution Using Fourier Series As stipulated above, the z coordinate distance is measured downward from the surface of the clay sample. For analysis, the thickness of the sample is designated by 2h, the distance h being the length of the drainage path ( see Figure on the next slide). The boundary condition are: a) Double drainage b) Single drainage

A. Double drainage Rectangular initial stress

Cont’d… Triangular initial stress

Cont’d… Trapezoidal initial stress

Cont’d…

B. Special triangular initial stress Type-A

cont………….

cont…………

C. Single drainage Rectangular initial stress

Cont’d… Triangular initial stress

Cont’d… Trapezoidal initial stress

Cont’d…

D. Special triangular initial stress

Approximate Solution By replacing the isochrones with parabolic curves, approximate solutions have been provided. Tables for degree of consolidation, U, and time factor, T, for different initial stresses and boundary condition are prepared. The table is shown in the next slide.

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