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L11-numerical_solution_for_consolidation

L11-numerical_solution_for_consolidation - Soil Mechanics...

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Soil Mechanics CIVL2410 11. Numerical Sol n to 1D Consolidation 1 11. NUMERICAL SOLUTION OF THE 1-D CONSOLIDATION EQUATION Contents 1. Introduction. ............................................................................................................................ 1 2. Finite Difference Formulae ..................................................................................................... 1 3. Finite Difference Approximation of Consolidation Equation .................................................... 3 4. Stability ................................................................................................................................... 4 5. Boundary Conditions .............................................................................................................. 4 a) Fully Permeable Boundary .................................................................................................. 5 b) Impermeable Boundary ....................................................................................................... 5 1. Introduction. The 1-D equation of consolidation cannot be solved analytically except for some very simple situations. For more difficult cases it is necessary to use approximate numerical techniques. One numerical technique that can be used for consolidation problems is the finite difference approach. In this method the solution is evaluated at a number of points at different times as indicated on Figure 1 below. Figure 1: Grid showing points at which solution calculated 2. Finite Difference Formulae The 1-D consolidation equation and the boundary conditions are approximated by finite difference formulae. These can be derived by referring to below and taking local axes at B:
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Soil Mechanics CIVL2410 11. Numerical Sol n to 1D Consolidation 2 Figure 2: Excess pore water pressure variation at time t . Suppose that the excess pore pressure at any time t can be approximated by a parabola u a a z a z = + + 1 2 3 2 (1a) The constants in this equation can be related to the values of the excess pore pressures at points A, B, C. Taking B as the origin for z gives: ) 1 ( 2 3 2 1 1 2 3 2 1 b z a z a a u a u z a z a a u C B A + + = = + - = so that ) 1 ( 2 2 2 2 3 2 1 c z u u u a z u u a u a B C A A C B - + = - = = thus evaluating the slope and curvature of u at the point B ( z = 0 ) it is found:
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Soil Mechanics CIVL2410 11. Numerical Sol n to 1D Consolidation 3 ) 1 ( 2 2 2 2 2 d z u u u z u z u u z u B C A B A C B - + = - = 3. Finite Difference Approximation of Consolidation Equation The equation of consolidation is: c u z u t q t v 2 2 = - (2a) where q is the change in total stress, due to applied loads, from the initial equilibrium situation when the excess pore pressures were zero. When this equation is evaluated at any point in the soil it is equivalent to evaluating the equation at point B, and hence the finite difference formulae developed above can be introduced so the equation becomes: u t q t c u u u z B B v A C B - = + - 2 2 (2b) if the above equation is now integrated from times t to t+ Δ t it is found that: u q c z u u u dt B B v A C B t t t = + + - + 2 2 [ ] (2c) Where: u u t t u t and q q t t q t B B B B B B = + - = + - ( ) ( ) ( ) ( ) Figure 3: Approximate integral evaluation.
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Soil Mechanics CIVL2410 11. Numerical Sol n to 1D Consolidation 4 If the integral appearing in equation (2c) is now approximated as indicated in Figure 3, it is found that: u q u t u t u t B B A C B = + + - β [ ( ) ( ) ( )] 2 (2d)
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