sma11 - Numerical Solution of the 1-D Consolidation...

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Numerical Solution of the 1-D Consolidation Equation
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The 1-D Consolidation Equation c u z u t q t v 2 2 = - For homogeneous soil the equation governing one dimensional consolidation is given by Analytical solutions are only available for very simple initial and boundary conditions. Approximate numerical solutions are required for other cases.
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q 1 2 3 4 Setting up the numerical solution grid Soil Layer
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z t t z t = 0 t = t 1 t = t 2 q 1 2 3 4 1 2 3 4 z z t Setting up the numerical solution grid Soil Layer
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A B C z z = + ∆ z z = -∆ t = t n Assumed excess pore pressure variation u = u A u = u B u = u C z u
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Assume Finite difference formulae u a a z a z 1 = + + 2 3 2
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Assume Then Finite difference formulae u a a z a z u a u a a z a z A 1 B C 1 = - + = = + + 2 3 2 1 2 3 2 u a a z a z 1 = + + 2 3 2
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Assume Then Hence Finite difference formulae u a a z a z u a u a a z a z A 1 B C 1 = - + = = + + 2 3 2 1 2 3 2 u a a z a z 1 = + + 2 3 2 a u a u u z a u u u z B C A A C B 1 2 3 2 2 2 2 = = - = + -
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Finite difference formulae u a a z a z 1 = + + 2 3 2 u z u u z u z u u u z B C A B A C B = - = + - 2 2 2 2 2 a u a u u z a u u u z B C A A C B 1 2 3 2 2 2 2 = = - = + - + (1 d)
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Finite difference form of consolidation equation
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+ Finite difference form of consolidation equation
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u t q t c u u u z B B v A C B - = + - 2 2 + Finite difference form of consolidation equation
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u t q t c u u u z B B v A C B - = + - 2 2 where Finite difference form of consolidation equation u q c z u u u dt B B v A C B t t t = + + - + 2 2 [ ]
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F t dt t t t F t t ( ) ( ) + To proceed an approximate numerical solution of the integral is needed. This is as follows Finite difference form of consolidation equation
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F t dt t t t F t t ( ) ( ) + t F(t) To proceed an approximate numerical solution of the integral is needed. This is as follows Error in approximation t t + t Finite difference form of consolidation equation Area = F(t) t
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Finite difference form of consolidation equation u q c z u u u dt B B v A C B t t t = + + - + 2 2 [ ] where u q u t u t u t B B A C B = + + - β [ ( ) ( ) ( )] 2 c t z v = β 2 (2d)
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This note was uploaded on 08/30/2011 for the course CIVL 2410 taught by Professor Dairey during the Three '11 term at University of Sydney.

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sma11 - Numerical Solution of the 1-D Consolidation...

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