Terzaghi's Consolidation Equation - Finite Difference Solution

Title

Page 1

By: Murray FredlundSoilVision Systems Ltd.

2109 McKinnon Ave. S, Saskatoon, Sask., S7J 1N3Voice: (306) 477-3324Fax: (306) 955-4575

Email: [email protected]

Features:Handles varying boundary conditions

1-Dimensional

Terzaghi's Consolidation Equation - Finite Difference Solution

Title

Page 2

By: Murray FredlundSoilVision Systems Ltd.

2109 McKinnon Ave. S, Saskatoon, Sask., S7J 1N3Voice: (306) 477-3324Fax: (306) 955-4575

Email: [email protected]

Features:Handles varying boundary conditions

1-Dimensional

Terzaghi's Consolidation Equation - Finite Difference Solution

Help

Page 3

HelpThe following help is provided to aid in the successful operation of the spreadsheet.

Only change cell values marked yellowChanging non-yellow cells could disrupt normal calculation processProcedure

Press New Problem buttonEnter values for yellow cells and proceed through sheets by pressing NextPress the Generate button where availablePress Initial button on Consolidation sheet to initializePress Calculate button to begin calculationsPress the Stop button to stop calculationsView results on the GraphPWP sheet

Nodes

Page 4

Node GenerationThe number of nodes to be used in the solution must be input. A greater number of nodes willyield a more accurate solution but will slow computations. A minimum of 11 nodes and a maximum of 680 nodes are allowed.Stresses are calculated at the nodes while soil properties are based on the space between the nodes.

25 : Number of Nodes

Valid number of nodes

Elements are regions between nodes1

2

3

Nodes

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The number of nodes to be used in the solution must be input. A greater number of nodes willyield a more accurate solution but will slow computations. A minimum of 11 nodes and a maximum of 680 nodes are allowed.Stresses are calculated at the nodes while soil properties are based on the space between the nodes.

InitialPWP

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Pore-Water PressuresThis defines the initial pore-water pressure distribution.

Node # Height (m)25 12.00 100.0024 11.50 100.0023 11.00 100.0022 10.50 100.0021 10.00 100.00 25 : Expected # of nodes20 9.50 100.00 25 : Actual # of nodes19 9.00 100.00 OK18 8.50 100.0017 8.00 100.00 0.5 : Height increment (m)16 7.50 100.00 9.81 : Unit Weight of Water (kN/m^3)15 7.00 100.0014 6.50 100.0013 6.00 100.0012 5.50 100.0011 5.00 100.0010 4.50 100.00

9 4.00 100.008 3.50 100.007 3.00 100.006 2.50 100.005 2.00 100.004 1.50 100.003 1.00 100.002 0.50 100.001 0.00 100.00

Initial PWP's (kPa)

Boundaries

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Boundary ConditionsThe boundary conditions for the top and bottom of the soil layer must be specified.

0.00 : Pressure Head (m)

0.00 : Pressure Head (m)

25 : Expected # of nodes25 : Actual # of nodes

OK

Top Boundary

Bottom Boundary

Boundaries

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: Expected # of nodes: Actual # of nodes

Boundaries

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Pressure head = constantflux = zero

1

Pressure head = constantflux = zero

1

document.xls

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On/Off 1Height Increment 0.5 mNumber of nodes 25

Max Time Inc. 100,000,000 s 1,157.4 days 3.17 yrsCalc Time Inc. 18,111 s 0.2 days 0.00 yrsAct. Time Inc. Err:523 s Err:523 days Err:523 yrs

Unit Wt. of Water 9.81 kN/m^3Initial Void Ratio 0.30 0.23 porosity, n

Coeff. of Compressibility, av 1.00E-04 1/kPaCoeff. of Permeability, k 5.00E-09 m/s

Coeff. of Consolidation, cv 6.63E-06 m^2/s Calculated - if the manually input cv cell is blank, this one will be usedCoeff. of Consolidation, cv m^2/s Manually input - to use calculated, clear contents of this cellCoeff. of Consolidation, cv 6.63E-06 m^2/s This is the Coefficient of Consolidation that will be used for calculations

Beta 0.48 < 0.5 SafeMax Time 157,784,630 s 1,826.2 days 5.00 yrs

Current Time Err:523 s Err:523 days Err:523 yrsCounter Err:523

Max Iterations 1Iteration Err:523

Update Values (1/0) Err:523

Water Flow PastHeight Initial Pressure (kPa) Pressure (kPa) Node Cutoffs Err:523 : total flow (m^3)

(m) Pore-water Pressures t t+dt (+=up) m^3 2.769 : water available for flow (m^3)0.00

12.000 100.00 0.00 0.00 Err:52311.500 100.00 0.93 Err:523 Err:52311.000 100.00 Err:523 Err:523 Err:523

Finite Difference Solution to Terzaghi's Consolidation Equation

document.xls

Page 11

10.500 100.00 Err:523 Err:523 Err:52310.000 100.00 Err:523 Err:523 Err:5239.500 100.00 Err:523 Err:523 Err:5239.000 100.00 Err:523 Err:523 Err:5238.500 100.00 Err:523 Err:523 Err:5238.000 100.00 Err:523 Err:523 Err:5237.500 100.00 Err:523 Err:523 Err:5237.000 100.00 Err:523 Err:523 Err:5236.500 100.00 Err:523 Err:523 Err:5236.000 100.00 Err:523 Err:523 Err:5235.500 100.00 Err:523 Err:523 Err:5235.000 100.00 Err:523 Err:523 Err:5234.500 100.00 Err:523 Err:523 Err:5234.000 100.00 Err:523 Err:523 Err:5233.500 100.00 Err:523 Err:523 Err:5233.000 100.00 Err:523 Err:523 Err:5232.500 100.00 Err:523 Err:523 Err:5232.000 100.00 Err:523 Err:523 Err:5231.500 100.00 Err:523 Err:523 Err:5231.000 100.00 Err:523 Err:523 Err:5230.500 100.00 Err:523 Err:523 Err:5230.000 100.00 0.00 0.00 Err:523

0.00

document.xls

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Equation solved:

Calculated - if the manually input cv cell is blank, this one will be usedManually input - to use calculated, clear contents of this cellThis is the Coefficient of Consolidation that will be used for calculations

Note: Hit 'Esc' twice after starting Excel iterating (Ignore error and press enter)

: total flow (m^3): water available for flow (m^3)

#########

Finite Difference Solution to Terzaghi's Consolidation

¶u¶t

=c v¶2u¶ x2

GraphPWP

Page 13

0.00 20.00 40.00 60.00 80.00 100.00 120.000.000

2.000

4.000

6.000

8.000

10.000

12.000

14.000

Pore-Water Pressure Distributions

Current

Initial

PWP (kPa)

Hei

gh

t (m

)