Topic7 -Time+rate+of+consolidation

TABLE OF CONTENTS

PREVIEW .................................................................................................................... 7.2 Introduction ............................................................................................................. 7.2 Objectives ................................................................................................................. 7.2

PREFACE .................................................................................................................... 7.2

TERZAGHI’S ONE-DIMENSIONAL CONSOLIDATION EQUATION. ........... 7.2

DETERMINATION OF C V ........................................................................................ 7.5 Displacement vs log(time) – Casagrande (1938) ................................................... 7.5 Displacement vs squareroot (time) – Taylor (1948) ............................................. 7.5

SECONDARY COMPRESSION ............................................................................... 7.6

REVIEW QUESTIONS .............................................................................................. 7.7

PREVIEW

Introduction

In Topic 7, the time dependent behaviour of consolidation was introduced. The oedometer test was also introduced as one method to determine the likely volume changes that resulted from the consolidation process. The oedometer test can also tell us how long it will take for consolidation to occur. This is an important practical problem, because it is essential that we know how fast a structure will settle. Objectives

?To understand Terzaghi’s one-dimensional consolidation theory and be able to apply it to practical problems

?To be able to interpret oedometer tests to determine properties for estimating the rate at which consolidation occurs

?To differentiate between primary and secondary consolidation and be able to estimate secondary consolidation settlement.

PREFACE

Consolidation results from the dissipation of excess pore water pressures which generates movement of pore water within the soil. The amount of water that is squeezed out is directly proportional to the amount of excess pore water pressure that is dissipated. It follows then, that the rate of consolidation is directly related to the rate of excess pore pressure dissipation.

TERZAGHI’S ONE-DIMENSIONAL CONSOLIDATION EQUATION. As introduced in Topic 2 and will be dealt with in the unit CIV3248 Groundwater and Environmental Engineering in Semester 2, the movement of pore water is governed by Darcy’s law. Darcy’s law tells us that the quantity of flow depends on the hydraulic gradient and the permeability of the soil. By equating the volume change of the soil due to water egress with the volume change of the soil due to change in effective stress, it is possible to derive the following differential equation which governs

one dimensional consolidation. Here u is the pore water pressure, c v is

the coefficient of consolidation (a soil “property”), t is time and z

denotes the position where u is determined - u is a function of both z

and t.

This equation was first suggested as a model of consolidation by Terzaghi in 1925. It can also be derived in three dimensions, but for most practical applications the one-dimensional equation is usually assumed.

Note that the rate of consolidation is governed by the coefficient of consolidation not the permeability. c v depends on the permeability, k,

and compressibility, m v, of the soil. This implies that for a soil and rock

with the same permeability, the pore pressures will dissipate faster in the rock because it has lower compressibility (or greater stiffness)

v

w

v m

k c

γ

=

t

u

z

u

c

2

2

v?

?

=

?

?

Terzaghi’s consolidation equation can be solved using analytical or nume rical

techniques. The solution obtained depends on the boundary conditions. For the oedometer test, with a sample of height, 2H, the boundary conditions are :

1. Complete drainage at top and bottom of sample; i.e. u = 0 at z = 0 and z = 2H

2. The initial excess pore water pressure ?u = u I is equal to the applied stress increment ?σ; i.e. when t = 0, ?u = u I = ?σ = (σ'2-σ'1).

Note the length of the longest drainage path is half the sample height, H , since drainage can occur to both the top and bottom of the sample.

The solution is obtained as a Fourier series which can be conveniently expressed in the following form :

Where U z is the degree of consolidation at time t at depth z, and T is a non-dimensional time factor. U z and T are given by :

where H dr is the length of the longest drainage path. The solution is shown graphically in Figure 1.

Figure 1 : Time rate of consolidation curves for two way drainage

This diagram can be used to determine the degree of consolidation throughout the layer. For example, the degree of consolidation, U z , at mid height of an oedometer test for T=0.2 (Point A) is approximately 23%. However at the same time (and time factor) at other locations, U z is different. At z/H=0.5, U z = 44% and at z/H=0.1, U z = 86%.

C o n s o l i d a t i o n r a t i o , U z

()

T f H z f 1U 20n 1z ∑∞

=??

?

??-=i 1

21211z u u

1e e e e U -

=σ'-σ'σ'-σ'=--=

2dr

v

H t

c T =

Note that other boundary conditions have different solutions. These can be obtained from most reference books that deal with consolidation.

In most settlement calculations we are interested in the average degree of consolidation of the entire layer rather than the point values given above. This can be calculated by finding the area under the appropriate T curve in the above diagram. For problems in which initial excess pore pressure is constant throughout the consolidating layer, the following values apply.

Some useful approximations (Cassagrande, 1938; Taylor, 1948) are:

U av > 60% T = 1.78 - 0.933log(100-U av %)

2

2100%44

%60??

?

??=

=

av U U T U ππ

In terms of settlements :

where s(t) is the settlement at any time, t , and s c is the total primary consolidation settlement at t 100.

c av s )t (s U =

DETERMINATION OF C V

The coefficient of consolidation can be determined from the oedometer displacement and time data or from the consolidation phase of a triaxial test. Only the oedometer test will be dealt with. The displacement and time data need to be plotted in the form of displacement vs square root (time) or displacement vs log(time).

Displacement vs log(time) – Casagrande (1938)

1.Determine end of primary consolidation t p (or t100) and d100 by plotting tangents to

curve- see below.

2.Determine true origin of test; i.e. d0 and t0 by the following process : Choose any

two times t1 and t2 where t2= 4 t1. Determine d2-d1, the settlement increment over the period t2-t1. d0 is located distance d2-d1 above d1.

3.d50 (or U=50%) is located midway between d0and d100 . t50 can be read directly of

the log(time) axis.

4.Determine c v from

2

dr

50

2

dr

v

H

T

T H

c=

=

line back to zero time to establish d 0.

2. Draw second line from d 0 with abscissas 1.15 times larger than that of the first line.

3. The intersection of this second line and the laboratory curve defines d 90 and t 90.

4. Determine c v from

Note :

1. c v determined from root time method is usually slightly greater than that determined from the log time method.

2. c v is not constant, but depends on stress level.

3. For pressures less than the preconsolidation pressure, consolidation occurs quite rapidly, and c v can be relatively high. However, interpretation can be quite difficult, as the displacement time graphs do not have the “classical” shapes shown earlier.

Typical values of c v for soft clays range from 0.1 to 0.5 m 2/year and gradually increase with OCR.

Due to soil fabric, roots, small sample size, etc; c v measured in the laboratory tend to be less than those measured in the field.

SECONDARY COMPRESSION

Secondary compression or creep occurs at a much slower rate than primary consolidation and differs from primary consolidation in that it occurs at a constant effective stress. Nevertheless, creep settlements can be very large.

One method of estimating creep settlement is through the secondary compression index C α.

90

2dr

902dr v t H T t T H c =

=t

e

C log ??=

α

where ?e is the change in void ratio between times t a and t b , and ?log t =log t b - log t a . C α is usually estimated over one log cycle of time; e.g. from t a = 100 to t b =1000 (or from 1000 to 10000 etc), then ?log t = 1 and C α = ?e . See Figure 2.

The modified secondary compression index , C αε, , giving strain rather than void ratio change, can also be defined as

Where e p is the void ratio at the start of the linear portion of the e vs log t curve. (e 0 is often used with little loss of accuracy). Both C α and C αε can be determined from the slope of the straight line portion of the displacement vs log(time) curve following the end of primary consolidation.

Typical values of C α /C c :

organic soft clays : 0.05 +/- 0.1 inorganic soft clays : 0.04 +/- 0.1 sands : 0.015 – 0.03

REVIEW QUESTIONS

1. Which properties of a soil govern the rate at which excess pore water pressures dissipate ?

2. For the following construction activities, plot on the same graph the expected variation of pore water pressure (on the vertical axis) with time (on the horizontal axis) for a point in the soil immediately beneath the construction activity. Assume that the initial pore water pressure is the same in all cases and that all activities cause the same local magnitude change in pore water of ?u. ?u can be positive or negative depending on the activity.

(a) Addition of extensive fill overlying a deep deposit of clay (b) Addition of extensive fill overlying a deep deposit of sand

(c) Addition of extensive fill overlying a shallow layer of clay overlying sand (d) Footing on deep deposit of clay (e) Excavation in deep deposit of clay

p

e C

C +=1α

αε

(f)Lowering of ground water table in clay

(g)Raising of ground water table in clay

3.Holtz and Kovacs : from page 423 on; Problems 9-1 to 9-36