Indian Geotechnical Conference – 2010, GEOtrendz
December 16–18, 2010
IGS Mumbai Chapter & IIT Bombay
Simplified Non-Linear Theory of Vertical Consolidation of
Thick Clay Layers
1
Khan, P. Ayub
Madhav, M. R.1
Reddy, E. Saibaba1
Associate Professor
e-mail: [email protected]
Professor Emeritus
e-mail: [email protected]
Professor
e-mail: [email protected]
Department of Civil Engineering, ACE Engineering College, Hyderabad
Department of Civil Engineering, J. N. T. U. Hyderabad, College of Engineering, Hyderabad
ABSTRACT
The conventional one-dimensional consolidation theory developed by Terzaghi is based on linear void ratioeffective stress relationship for thin layers only. This paper proposes a simplified theory of non-linear consolidation
for one-dimensional consolidation of a thick clay deposit considering linear void ratio-log effective stress
relationship, variation of initial in-situ effective stress with depth but neglecting the slight variation of initial void
ratio with depth. The governing differential equation of consolidation is solved numerically by the finite difference
method and the results compared with the conventional linear theory. The results indicate that the conventional
thin layer theory under-estimates the degree of consolidation and over-estimates the degree of dissipation of
excess pore pressures. The degree of settlement and degree of dissipation of pore pressures are sensitive to the
magnitude of applied load unlike in the conventional thin layer theory.
1. INTRODUCTION
The conventional one-dimensional consolidation theory
developed by Terzaghi neglects the effect of self weight of
the soil and assumes linear relationship of void ratio and
effective stress and infinitesimal strain. Richart (1957)
reviewed the theory of consolidation and concluded that
the effect of considering void ratio as a variable did not
significantly change the consolidation-time characteristics
of consolidation by vertical flow. The non-linear theory of
one-dimensional consolidation developed by Davis and
Raymond (1965) considering linear void ratio-log effective
stress relationship, is valid only for a thin layer of clay.
Gibson et al. (1981) presented a finite strain non-linear
one-dimensional consolidation theory for thick
homogeneous clays by considering the self weight of soil,
variation of void ratio with depth and Lagrangian and
convective coordinate system. Analytical solution proposed
by Xie et al. (2001) for non-linear one-dimensional
consolidation of double layered soil is based on constant
effective stress with depth. Explicit analytical solutions of
one-dimensional consolidation of clay are derived by Xie
& Leo (2004) for both thin and thick clay layers with
mechanics of large strain.
A simple approximate theory of non-linear
consolidation is proposed herein for a thick clay layer
considering void ratio-log effective stress relationship but
assuming coefficient of consolidation is constant (the
coefficient of permeability and the coefficient of volume
change are inversely proportional to effective stress),
constant thickness of clay layer and constant initial void
ratio along the depth but accounting for the variation of
initial effective stress with depth. The proposed theory for
a thick clay layer is an extension of the non-linear theory
of consolidation developed by Davis and Raymond (1965)
for a thin layer.
2. PROBLEM STATEMENT AND FORMULATION
A homogeneous saturated clay layer of thickness, H, which
is initially fully consolidated under its self weight, (Fig.
1), consolidates under a uniformly distributed load, q,
applied on the top.
Surcharge, q
Pervious
z
Clay Layer
Pervious
or Impervious
Fig. 1: Problem Statement
922
P. Ayub Khan, M.R. Madhav and E. Saibaba Reddy
Considering linear e-log σ΄ relationship, assuming
(1+e) constant during consolidation and constant coefficient
of consolidation, cv, the following equation of consolidation
can be derived following Davis and Raymond (1965) as
 1 ∂ 2 u  1  2 ∂u ∂σ ′ 
1 ∂σ ′

− 
= −c v 
σ ′ ∂t
 σ ′ ∂z 2  σ ′  ∂z ∂z 
For t>0; z=H; for pervious bottom, u(H, t) =0 then
 γ ′H + q 
 or
w( H , t ) = log 10 
 γ ′H 
w( H , t ) = log 10 (1 + q *)
(1)
where σ΄ is the effective vertical stress; t - time of
consolidation; u - the excess pore water pressure and zdepth from surface.
 σ′ 
(2)

w = log 10 
 γ ′.H 
where σ = (γ΄z+q-u) and γ΄ is the submerged unit weight
of the soil,
For t → ∞ ; 0 ≤ z ≤ H ; u ( z , ∞) = 0 ;
 γ ′z + q 

w( z , ∞) = log 10 
 γ ′H 
Average degree of Settlement, Us, is
Let
 ∂ 2 w γ ′ ∂w 
∂w
= cv 
+

2
∂t
σ ′ ∂z 
 ∂z
(4)
Equation (4) is re-written in non-dimensional form as
∂w  ∂ 2 w
1 ∂w 
+
=

2
∂T  ∂Z
10 w. ∂Z 
(5)
where Z and T are the normalized parameters; Z=z/H and
T=t/tR; reference time, tR=H2/cv.
Boundary and Initial Conditions:
Two boundary conditions, viz., (i) pervious top impervious
base (PTIB) and (ii) pervious top pervious base (PTPB) are
considered.
Initial Condition : t=0; 0 ≤ z ≤ H ; u(z,0) = q;
H
(6d)
σ′
) dz
σ o′
σ ′f
∫ log(
Us =
0
H
∫ log(
0
σ o′
(7)
) dz
where σ΄o and σ΄f (=σ΄o+q) are the initial and final effective
stresses respectively. Average degree of dissipation of excess
pore pressure, Up, is
H
u ( z, t ) 
dz
U p = ∫ 1 −
u o 
0
(8)
where uo = q the initial excess pore pressure.
2. RESULTS AND DISCUSSION
Eqn. (5) is solved numerically using the finite difference
approach. The layer was divided in to hundred sub-layers
and convergence ensured. The average degree of settlement
for the entire thickness, Us, by the proposed non-linear
theory, is presented in Figs. 2 and 3 for PTPB and PTIB
cases respectively and compared with the conventional onedimensional consolidation theory by Terzaghi. The degree
0
10
20
30
q*=1
10
100
10000
40
Us (%)
∂ 2σ ′
∂ 2u
∂σ ′
∂u
=−
=γ′−
and
(3)
∂z
∂z
∂z 2
∂z 2
Substituting the above equations into Eqn. (1), the following
differential equation of consolidation can be obtained for a
thick clay layer.
(6c)
50
60
 γ ′.z 

w(z,0) = log10
 γ ′.H 
Boundary Conditions: For t>0; z=0; u(0,t)=0;
 q 
 or
w(0, t) = log10
 γ ′H 
w(0, t ) = log10 (q*)
70
80
(6)
90
100
0.0001
Proposed Theory
Terzaghi's Theory
0.001
0.01
0.1
1
10
Tv
Fig. 2: Degree of Consolidation vs. Time Factor (PTPB)
0
(6a)
10
20
30
q *= 1
10
40
50
100
10000
s
U ( %)
q
where q* = ′
γ .H
60
70
80
For t>0; z=H for impervious bottom,
90
Pr o p o s e d T h e o ry
T e r z a g h i's T h e o r y
100
∂u
∂w
0.434
( H , t ) = 0 then
(H , t) =
∂z
∂Z
10 w
0 .0 0 0 1
0 .0 0 1
0 .0 1
0 .1
1
10
Tv
(6b)
Fig. 3: Degree of Consolidation vs. Time Factor (PTIB)
923
Simplified Non-Linear Theory of Vertical Consolidation of Thick Clay Layers
0
q*=100 00
10
100
20
10
30
1
40
Up (% )
of consolidation for a thick layer decreases with increase
of q* at all times. The results from the proposed theory
agree with the conventional thin layer theory for q* (=q/
(γ’H)) e” 10,000 (for H tending to zero, thin layer). The
degree of consolidation is relatively more in the case of
thick layer of clay compared to thin layer for a given load
intensity. Similar observations are made by Gibson et al.
(1981). The degree of consolidation increases from 51% to
60% (Fig. 2 for PTPB) and from 53% to 71% (Fig. 3 for
PTIB), for q* decreasing from 10,000 to 1, at a time factor,
TV of 0.197. The theory of consolidation for thin layers
thus underestimates the degree of settlement.
The excess pore pressures are computed at different
depths and times and the average degree of dissipation of
pore pressures for the entire thickness of clay layer is shown
in Figs. 4 and 5 for different values of q* for PTPB and
PTIB respectively along with the results from conventional
thin layer theory. The degree of dissipation of pore pressure
from non-linear theory is slower than the degree of
settlement (Davis & Raymond, 1965, Gibson, 1981, and
Xie & Leo, 2004). While the degree of settlement is 51%
the corresponding degree of dissipation of pore pressure is
only 8 % for PTPB (Fig. 4) and 53% and 8.30% respectively
for PTIB (Fig. 5), for q* of 10,000 in thick layers against a
value of 50% from linear theory, at a time factor of 0.197.
The degree of dissipation of pore pressure decreases with
the increase of q* as the degree of dissipation of excess
pore pressure decreases with the increase of the ratio of
final to initial effective stresses as established for the nonlinear theory of consolidation (Raymond & Davis, 1965).
Thus the conventional thin layer theory over-estimates the
degree of dissipation of pore pressures or under-estimates
the residual pore pressures of thick layers.
50
60
70
80
Pro pos ed T heory
Ter z aghi's Theor y
90
100
0.0001
0.001
0.01
0 .1
1
10
Tv
Fig. 5: Average Degree of Dissipation of Pore Pressures vs.
Time Factor (PTIB)
Figs. 6 and 7 depict isochrones of excess pore pressure
for q*=10 for PTPB and PTIB respectively. The excess pore
pressure is relatively large or dissipation of pore pressure
is relatively slow at all times according to non-linear theory
of consolidation compared to the conventional theory.
Interestingly, the isochrones in the case of PTPB are slightly
unsymmetrical about the mid depth in contrast to
symmetrical isochrones in the conventional linear theory
for PTPB boundary conditions. The residual pore pressures
are 82% and 78% of q, at depths of 0.2H and 0.8H
respectively for q* of 10 at a time factor of 0.20. The
corresponding residual pore pressure is 46% of q at the
two depths from linear theory.
u/q
0
1
q* =10000
10
0.2
1
40
Up (% )
0.2
0
0.2
30
0.05
0.2
0.8
50
0.8
0.4
z/H
60
70
100
0 .0 0 0 1
0.4
Tv =0.05
100
20
90
0.6
0
10
80
0.8
0.6
Pro p o s e d Th e o ry
T e rz a g h i's Th e o ry
0 .0 0 1
0 .0 1
0.1
1
10
q*=10
Proposed
Terzaghi
0.8
Tv
Fig. 4: Average Degree of Dissipation of Pore Pressures vs.
Time Factor (PTPB)
1
Fig. 6: Excess Pore Pressure Isochrones (PTPB)
924
P. Ayub Khan, M.R. Madhav and E. Saibaba Reddy
u/q
1
0.8
0.6
0.4
0.2
0
0
Tv=0.05
0.2
z/H
0.4
0.05
0.20
0.20
0.6
0.8
0.8
0.8
q*=10
Proposed
Terzaghi
1
Fig. 7: Excess Pore Pressure Isochrones (PTIB)
3. CONCLUSIONS
A simple and approximate non-linear theory of onedimensional consolidation for thick clay layer is developed.
The conventional thin layer theory under-estimates the
degree of consolidation but over-estimates the degree of
dissipation of excess pore pressures. The isochrones in the
case of a thick layer with PTPB condition are slightly
unsymmetrical about the mid depth in contrast to
symmetrical isochrones in thin layer theory. The degree of
settlement and degree of dissipation of pore pressures are
sensitive to the magnitude of loading in the case of a thick
layer while these are independent of loading in the thin
layer theory.
REFERENCES
Davis, E.H. and Raymond, G.P. (1965). A Non-linear theory
of consolidation. Geotechnique, 15(2), 161-173.
Gibson, R.E., Schiffman, R.L. and Cargill, K.W. (1981).
The theory of one-dimensional consolidation of
saturated clays: II. Finite nonlinear consolidation of
thick homogeneous layers. Canadian Geotechnical
Journal, 18(2), 280-293.
Richart, F.E. (1957). Review of the theories for sand drains,
Transactions of ASCE, 124(2999), 709–736.
Xie, K.H. and Leo, C.J. (2004). Analytical solutions of onedimensional large strain consolidation of saturated and
homogeneous clays. Computers and Geotechnics, 31,
301-314.
Xie, K.H., Xie, X.Y. and Jiang, W. (2002). A study on onedimensional nonlinear consolidation of double-layered
soil. Computers and Geotechnics, 29, 151-168.