Lecture 8-1 - 29 Time Rate of Consolidation: During the...

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ρ 29 Time Rate of Consolidation: During the consolidation process: s What % ρ c will have occurred at a given time? s How much time is required for given % ρ c to occur? One-Dimensional Consolidation: Theory presented by Karl Terzaghi in 1925 ENR. Assumptions: 1. The soil is fully saturated and homogenous. 2. The pore fluid and soil particles are both incompressible. 3. Fluid flow and volume change occur in only one direction. 4. Darcy’s law is valid, i.e. flow area, Δ h, (1/L) (essentially the conservation of energy and momentum combined). o Consider the flow q through a small prism at a point (x,y,z) and conservation of mass: if volume = V then t V dq z - = if velocity = v then = dz z v dv z z and since dxdy v dA v q z z z = = then dxdydz z v dxdy dz z v dq z z z = = and therefore t V dxdydz z v z - = (1) recall Darcy’s equation - = dA z h k q z or, dividing by dA, - = z h k v z where k is the permeability which is assumed to be constant and since γ = z u 1 z h W where u is the excess pore pressure due to the applied load then γ - = z u k v W z and substituting into (1) gives t V dxdydz z u k 2 2 W = γ (2) % ρ C ρ ρ s ρ C Time ρ i x z y dy dz dx q q = q + dq z
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ρ 30 also t V t e V t V e t ) V eV ( t ) V V ( t V S S S S S S V + + = + = + = and since 0 t V S = and + = + = 0 0 0 S e 1 dxdydz e 1 V V where e 0 = initial void ratio and V 0 = the initial volume then t e e 1 dxdydz t e V t V 0 S + = = and substituting into (2) gives t e e 1 1 z u k 0 2 2 W + = γ (3) the change in void ratio in (3) is caused by an increase in effective stress (with a corresponding decrease in pore water pressure) and for small strains may be assumed linearly related to the void ratio as: u a ' a e v Z v = - = σ (4) substituting (4) into (3) we get t u m t u e 1 a z u k v 0 V 2 2 W = + = where a v = coefficient of compressibility and + = 0 V V e 1 a m = coefficient of volume compressibility finally we define: V W 0 V W V a ) e 1 ( k m k c ion consolidat of t coefficien γ + = γ = = units (L 2 /T) so that: t u z u c 2 2 V = which is Terzaghi’s 1-D consolidation theory o The above partial differential equation has been solved using simple boundary conditions and dimensionless parameters as follows:
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This note was uploaded on 09/12/2011 for the course CEG 4012 taught by Professor Staff during the Fall '08 term at University of Florida.

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Lecture 8-1 - 29 Time Rate of Consolidation: During the...

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