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consol1aa.sm

NUMERICAL SOLUTION OF 1-D CONSOLIDATION EQUATION

BY FINITE DIFFERENCES - Homogeneous Soil

This program solves the 1D second order partial differential equation of consolidation for a homogeneous soil layer. Finite diferences are used.

δu δt / Cv δ 2^u * δz 2^/ * :

where:

u=pore pressure

z depth :

t time :

Cv = coefficient of consolidation = k/(γw mv)

Reference: "Advanced Soil Mechanics" by B. M. Das. McGraw-Hill 1983

INPUT DATA

Cv 1510 * mm 2^sec / * :

Coefficient of consolidation:

Thickness of soil layer

h 5 m * :

Number of soil subdivisions

nH 40 :

Final integration time

tf 19 hr * :

Boundary conditions

enter key = 1 for both ends free draining enter key = 2 for bottom end impervious, top free draining

key 1 :

Initial pore pressure vs depth

Bottom depth is 0.0

depth

initial pore pressure

depth 0 m * h 21 mat :

uinit 808021 mat kPa * :

Finite difference solution

CALCULATIONS

tunit 1 sec * :

lunit 1 m * :

punit 1 Pa * :

Δtmax 0.5 h nH / (2^* Cv / :

Maximum size of Dt: (from Das page 273)

nt1 tf Δtmax / :

Number of time steps

nt nt1 trunc 1 + :

H h :

Cv1 Cv tunit lunit 2^/ * :

H1 H lunit / :

non-dimensionalize quantities

Δz H1 nH / :

Δt tf nt tunit * / :

Const Cv1 Δt * Δz 2^/ :

depth depth lunit / :

uinit uinit punit / :

Interpolate initial pore pressure vs depth and place resulting vector as the first row of matrix u(i,j) Rows of matrix u(i,j)are pore pressures at time step i

j 1 nH range zz j 1 - (Δz * : u 1 j el depth uinit zz linterp : 21 line for

key 2 ≡ u 11 el uinit 1 el : u 11 el 0.0 : if

u 1 nH 1 + el 0 :

Compute pore pressures vs time by finite differences

t 1 nt range z 2 nH (range u t 1 + z el u t z el Const u t z 1 - el 2 u t z el * - u t z 1 + el + (* + (eval : for key 2 ≡ u t 1 + 1 el u t 1 el Const 2 u t 2 el * 2 u t 1 el * - (* + eval : continue if 21 line for

RESULTS

Number of time steps nt 1314

enter several time step numbers to plot u vs z:

tp 1100400 nt 41 mat :

Solution time increment Δt tunit * 52.0548 s *

c9 punit kPa / :

uu u transpose c9 * :

generate z depth vector

i 1 nH 1 + (range d i el i 1 - (Δz * : for

i 1 nH 1 + (range u1 i 2 el d i el : u1 i 1 el uu i tp 1 el el : 21 line for

i 1 nH 1 + (range u2 i 2 el d i el : u2 i 1 el uu i tp 2 el el : 21 line for

i 1 nH 1 + (range u3 i 2 el d i el : u3 i 1 el uu i tp 3 el el : 21 line for

i 1 nH 1 + (range u4 i 2 el d i el : u4 i 1 el uu i tp 4 el el : 21 line for

u1 u2 u3 u4 4 1 sys

Degree of Consolidation U(t):

area of initial excess pore pressure:

Ai u 1 i el 0.5 u 11 el u 1 nH 1 + el + (* + i 1 nH sum (Δz * :

area of dissipated pore pressure at each time step

t 1 nt range Ad t el u t i el 0.5 u t 1 el u t nH 1 + el + (* + i 1 nH sum (Δz * eval : U t el Ai Ad t el - Ai / : 21 line for

generate time vector

i 1 nt range tt i el i (Δt * : for

UT tt hr / U 100 * augment :