Boundary Condition for Solution & Settlement Correction of 1D-Consolidation

By Sanchari Halder

Submitted to

Professor Dr. Myoung Soo Won

Contents

Boundary Condition for Solution of 1D-Consolidation.

Correction of Settlement for Construction Period:

We can analyze the strain of a saturated clay layer subjected to a stress increase .Considering the case where a layer of saturated clay of thickness H that is confined between two layers of sand is being subjected to an instantaneous increase of total stress of Δσ. This incremental total stress will be transmitted to the pore water and the soil solids. This means that the total stress, Δσ, will be divided in some proportion between effective stress and pore water pressure.

It follows this equation:

Because clay has a very low hydraulic conductivity and water is incompressible as compared with the soil skeleton, at time t = 0, the entire incremental stress, Δσ, will be carried by water (Δσ= Δ u) at all depths (Figure b). None will be carried by the soil skeleton—that is, incremental effective stress (Δσ´) = 0. After the application of incremental stress, Δσ, to the clay layer, the water in the void spaces will start to be squeezed out and will drain in both directions into the sand layers. By this process, the excess pore water pressure at any depth in the clay layer gradually will decrease, and the stress carried by the soil solids (effective stress) will increase. Thus, at time 0 < t < ∞,

However, the magnitudes of Δσ´ and Δu at various depths will change (Figure c), depending on the minimum distance of the drainage path to either the top or bottom sand layer. Theoretically, at time t= ∞, the entire excess pore water pressure would be dissipated by drainage from all points of the clay layer; thus, Δu=0. Now the total stress increase, Δσ, will be carried by the soil structure (Figure d). Hence, Δσ= Δσ´This gradual process of drainage under an additional load application and the associated transfer of excess pore water pressure to effective stress cause the time-dependent settlement in the clay soil layer.

Figure 1(a) shows a layer of clay of thickness 2Hdr (Note: Hdr length of maximum drainage path) that is located between two highly permeable sand layers.

If the clay layer is subjected to an increased pressure of Δσ, the pore water pressure at any point A in the clay layer will increase. For one-dimensional consolidation, water will be squeezed out in the vertical direction toward the sand layer.

Figure 1(a): Clay layer Undergoing consolidation;

Figure 1(b) shows the flow of water through a prismatic element at A. For the soil element shown,

Rate of outflow of water – Rate of inflow of water = Rate of volume change

……….(1)

……….(2)

1 & 2

……….(3)

;According to Laplace’s Equation:

;

During consolidation, the rate of change in the volume of the soil element is equal to the rate of change in the volume of voids. Thus,

But (assuming that soil solids are incompressible)

and

…….(4)

Substitution for and Vs in Eq. (4) yields, ……… (5)

Where e0 initial void ratio. Combining Eqs. (3) and (5) gives,

……… (6)

The change in the void ratio is caused by the increase of effective stress (i.e., a decrease of excess pore water pressure). Assuming that they are related linearly, we have

………(7)

Combining Eqs. (6) and (7) gives,

Where, mv = coefficient of volume compressibility =

Or, Where, cv = coefficient of consolidation =

Thus,

…(8)

Eq. (8)

is the basic differential equation of Terzaghi’s consolidation theory and can be solved with the following boundary conditions:

Boundary Condition for Solving 1D-Consolidation Equation

1. Initial Condition, at time t = 0 ; u = Δσ

2. Boundary Conditions at any time

where z = 0 ; u = 0

For Double Drainage,

z = 2H ; u = 0

The Initial & Boundary Conditions:

The solution yields, …..(9)

The time factor is a non dimensional number. Because consolidation progresses by the dissipation of excess pore water pressure, the degree of consolidation at a distance z at any time t is

where, uz= excess pore water pressure at time t.

…….. (10)

Equations (9) and (10) can be combined to obtain the degree of consolidation at any depth z. This is shown in Figure 2.

The average degree of consolidation for the entire depth of the clay layer at any time t can be written from Eq. (10) as

Figure2: Variation of Uz

withSubstitution of the expression for excess pore water pressure uz given in Eq. (9) into Eq. (11) gives

……. (11)

The variation in the average degree of consolidation with the non dimensional time factor Tv , is given in Figure 3, which represents the case where u0 is the same for the entire depth of the consolidating layer.The values of the time factor and their corresponding average degrees of consolidation for the case presented in Figure 2 may also be approximated by the following simple relationship:

Figure 3:

Sivaram and Swamee (1977) gave the following equation for U varying from 0 to 100%:

Example Problem 1:

Solution:

Correction of Settlement for Construction Period:

In practice, structural loads are applied to the soil not instantaneously but over a period of time.

Terzaghi proposed an empirical method of correcting the instantaneous time–settlement curve to allow for the construction period.

Load

Settlement

Loading curve

Instanous Curve

Effective Construction Time

Time

SR

QP

½t1

½ tc

tc

Correction for construction period: The net load(P′) is the gross load less the

weight of soil excavated, and the effective construction period(tc) is measured from the time when P′ is zero.

It is assumed that the net load is applied uniformly over the time tc and that the degree of consolidation at time tc is the same as if the load P′ had been acting as a constant load for the period tc/2.

Thus the settlement at any time during the construction period is equal to that occurring for instantaneous loading at half that time.

Load

O

Settlement

Loading curve

Instanous Curve

Currected Curve

½ tc

t1T

Effective Construction Time

P1

P´Time

Sc

Sc

1

Example Problem 2:20 kn/m3

γsd = 17 kn/m3

γsw = 19 kn/m3

γcw=20 kn/m3

γw =9.8 kn/m3

Cv ,=1.26 m2/year

Cc =.32

Example Problem:

The Co-efficient of Consolidation Cv ,=1.26 m2/year

The Compression Index, Cc =.32 Calculate the final settlement of the area due to consolidation of the clay and the settlement after a period of 3 years from the start of dumping.Required Equation:

Settlement,

σ1

20 kn/m3

γsd = 17 kn/m3

γsw = 19 kn/m3

γcw=20 kn/m3

γw =9.8 kn/m3

Cv ,=1.26 m2/year

Cc =.32

Solution:20 kn/m3

γsd = 17 kn/m3

γsw = 19 kn/m3

γcw=20 kn/m3

γw =9.8 kn/m3

3m

Cv ,=1.26 m2/year

Cc =.32

Solution:

Cv ,=1.26 m2/year

Cc =.32

Correction for Settlement:

Settlement for Instantaneous Loading,

Scf =182 mm

Correction of Settlement for constructionperiod,

Sc =61 mm182 mm

61 mm

Thank you all….