86584_04b

ARTICLE 25 THEORY OF CONSOLIDATION 227

oped for computing the rate of consolidation for processes such as those illustrated in Fig. 25.3. The continuity equation for one-dimensional flow in the vertical direction (Eq. 23.1) is

3 dx dy dz = -at a (n dx dy dz) az

The Darcy equation in terms of excess porewater pressure is

k, a d Y w az

v, =

Assuming that the coefficient of permeability k, is the same at every point in the consolidating layer and for every stage of consolidation, and expressing the porosity n in terms of void ratio e, we obtain

a ( e d x d y dz) dx dy dz = at l+e k, a2Ui

Y w az --

If we assume that the changes in void ratio during consolidation are small, (1 + e) can be approximated by (1 + eo), where eo is the initial void ratio of the consolidating element and e is the current void ratio during consolidation. Then, (dx dy dz)/(1 + eo), which is equal to the volume of the solids in the element, is independent of time, and we have

k, a 2 d Yw az at l + e

ae dx dy dz -- d x d y d z = - -

or

(25.3)

Equation 25.3 is the hydrodynamic equation of one- dimensional consolidation based on the assumptions that the coefficient of permeability is constant and the strains are small during consolidation.

If we assume that the time lag of the compression is caused exclusively by the finite permeability of the soil, so that in Eq. 16.1 a,, = 0, and if we assume further that in Eq. 16.1 uvs is equal to -a, , which is the same at every point in the layer and for every stage of consolidation, then Eq. 16.1 becomes

de - dah

If the total vertical stress u, and the reference porewater pressure us remain unchanged during consolidation, then dahldt = -du’/dt, and Eq. 25.3 becomes

- - a v z dt

k, d2u’ - a, a d yw az2 1 + e tit

In terms of m, = a,/ (1 + e), where m, = AE,/Au: and E, is vertical strain, we have

1 k, a 2 d a d y w m v az2 at ---=-

By introducing the coefficient of consolidation e, defined as

1 k v e , = - -

Y w mv

we obtain

(25.4)

(25.5)

The dependent variable u’ is a function of the independent variables z and t. In the partial differential Eq. 25.5, u’ is differentiated twice with respect to z and once with respect to t. Consequently, the solution of Eq. 25.5 requires two boundary conditions in terms of z and an initial condition in terms of t. These conditions depend on the drainage boundaries and the loading as shown in the diagrams in Fig. 25.3. The boundary conditions that determine the consolidation of a half-closed layer and a uniform consolidation pressure increment in Fig. 25.4 may serve as an example. The initial condition is

At t = 0 and at any distance z from the impermeable surface, the excess porewater pressure is equal to Ao,; that is u’ (z, 0) = ha,.

The boundary conditions are

At any time t other than zero at the drainage surface z = H, the excess porewater pressure is zero; that is u’ (H, t ) = 0.

At any time t other than zero at the impermeable surface z = 0, the hydraulic gradient is zero; that is dur (0, t ) l& = 0.

The differential Eq. 25.5 can be solved subject to any set of initial and boundary conditions to obtain an expres-

Freely Draining Boundary

u ’a t z =O

dp, 4 Llrnperrneoble Boundory

Figure 25.4 uniform initial excess porewater pressure distribution.

Isochrone for half-closed layer subjected to a

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228 HYDRAULICS OF SOILS

sion for the excess porewater pressure. A solution using the Fourier expansion method leads to

u’(z, t ) = ha, exp(-M2T,) (25.6) m=O M

where M = 7~ (2m + 1)/2,

CVt T, = - H 2

(25.7)

is a pure number called the time factor, and H is the maximum drainage distance.

The degree of compression of a sublayer during consolidation is

(25.8) eo - e U(Z, t ) = -

eo - ep

where ep is the void ratio when the excess porewater pressure becomes zero. Because a linear time-independent relationship between void ratio and effective vertical stress is assumed in formulating the theory of consolidation represented by Eq. 25.5, the degree of compression defined by Eq. 25.8 is identical with the degree of effective vertical stress increase. This in turn is equal to the degree of excess porewater pressure dissipation

u; - U ’ U(Z, t) = -

Ui’

The degree of consolidation of the H is

S u = - SP

layer of thickness

(25.9)

where sp is the settlement of the layer when the excess porewater pressure becomes zero throughout the thickness H. The expression for U as a function of the time factor T, is obtained by integrating with respect to z the degree of excess porewater pressure dissipation of the sublayers. For example, from Eq. 25.6 we obtain

n L U = 1 - -exp(-M2T,) (25.10) m=O M2

Equation 25.5 can be solved to obtain excess porewater pressure isochrones and U =f(T,) for any set of initial and boundary conditions such as those in Fig. 25.3. Several U - T, curves are shown in Fig. 25.5. For an open layer (thickness 2H) the relationship between U and T, is determined by Eq. 25.10 and the curve C1 for all cases in which the initial excess porewater pressure varies linearly with z. Therefore, the curve C, represents the solution for all the initial and boundary conditions represented by Fig. 25.3 a, b, c, and e. If the initial excess porewater pressure is uniform throughout the consolidating layer,

T h e Fucfor

nmc furtor 7;

Figure 25.5 Relation between degree of consolidation and time factor. In (a) the time factor is plotted to an arithmetic and in (b) to a logarithmic scale. The curves C,, C,, and C, correspond to different conditions of loading and drainage, represented by a, d, and J respectively, in Fig. 25.3 (after Terzaghi and Frohlich 1936).

curve C1 also represents the rate of consolidation for a half-closed layer with thickness H.

If the consolidation pressure increment for a half-closed layer decreases from some value havt at the top to zero at the bottom, as shown in Fig. 25.3d, the relation between U and T, is given by the curve C,. If it increases from zero at the top to havb at the bottom, as in Fig. 25.3J the relation is given by curve C,. Figure 25.53 shows the curves C , to C3 plotted to a semilogarithmic scale from which small values of U can be obtained more accurately. In the arithmetic plot, Fig. 25.5a, the initial part of the curve CI has a parabolic shape. In fact, up to a degree of consolidation of 60% the relation between U and T, is accurately defined by U = 2 m.

The coefficient of consolidation c, at any a: is either computed by using Eq. 25.4 after directly measuring k, and taking m, (Eq. 2 5 . 3 ~ ) from the EOP e vs oh curve, or it is determined from the definition of the time factor by Eq. 25.7. In the latter procedure it is usual to compute


c, = 0.2 @/t50, where the time factor 0.2 from curve CI in Fig. 25.5 (which is applicable.to the commonly used double-drained incremental loading oedometer test) corresponds to the elapsed time t5o required to complete 50% primary consolidation (Casagrande and Fadum 1940; Taylor 1948). Alternative graphical interpretations of the theory of consolidation for estimating c, from settlement vs time data have also been used (e.g., Asaoka 1978). Values of c, computed using Eq. 25.4 together with direct measurements of k, are shown in Fig. 25.6. The magnitude of c, in the recompression range is generally larger than that in the compression range. The abrupt decrease in c, near the preconsolidation pressure in some clays reflects the change in m, as the soil passes from the recompression to the compression range. The ratio of c, in recompression to that in compression ranges from about 100 for a highly structured soft clay from eastern Canada to about 2 for a Boston Blue clay overconsolidated by desiccation. For typical soft clays, however, the ratio is in the range of 5 to 10. The value of c, is practically constant in the recompression range. In the compression range, it either remains constant or increases moderately with increasing a:. Data on c, in the compression range for a large number

Mexico City W I =%I%

o./

of clays are plotted in Fig. 25.7. Although in a general way c, decreases with increasing liquid limit, for clays with a given liquid limit c, varies widely.

25.4 Other Initial and Boundary Conditions It is obvious that the predictions of the time rate of settlement are not even approximately correct unless the assumed hydraulic boundary conditions are in accordance with the drainage conditions in the field. Every continuous sand or silt seam located within a bed of clay acts like a drainage layer and accelerates consolidation of the clay, whereas discontinuous lenses of sand and silt have no such effect. If test boring records indicate that a bed of clay contains partings of sand or silt, the engineer is rarely able to find out whether the partings are continuous. In such instances the theory of consolidation can be used to determine only upper and lower limiting values for the rate of settlement. The real rate remains unknown until it is observed.

In Figs. 25.1,25.3, and 25.4 the upper and lower boundaries of the consolidating layer are assumed to be either freely draining (u’ = 0) or impermeable (du’ /az = 0). In the field the behavior of a boundary layer may lie somewhere between these extremes. This is illustrated in Fig. 25.8 which represents a consolidating layer of thickness H and permeability k , separated from freely draining upper and lower surfaces by incompressible layers of thickness H , and Hb and finite permeability k,, and kvb, respectively. Equation 25.5 has been solved subject to these boundary conditions and a uniform initial excess porewater pressure distribution with depth (Mesri 1973). The drainage capacity of the adjacent incompressible layers is characterized by the drainage factors

and

Son Froncisco 8 M

w, :33%

10 QY - 4

Figure 25.6 Coefficient of consolidation of various soils as a function of the consolidation pressure ai.

The solutions indicate that values of R equal to 0.1 and 100 closely approximate impermeable and freely draining boundaries, respectively. For example, the U vs T, curve in Fig. 25.8 that corresponds to Rh = 0.1 and R, = 100 is very close to curve C1 in Fig. 25.5.

The U vs T, curves in Figs. 25.5 and 25.8 correspond to solutions of Eq. 25.5 in which it is assumed that the consolidation pressure increment is applied instantane- ously at t = 0. In the field, construction operations that may lead to consolidation take time. Under these loading conditions the consolidation pressure increment, and therefore the initial excess porewater pressure, reach their final values after weeks, months, or sometimes years. Equation 25.5 can be solved readily by assuming a linear


Computed using Eq. 25.4 ond Direct k ,. Measurements

o Computed using Eq. 25.7 and Settlement vs Time Datu

0 e o O d 0

! G .Mexico city Cloy Sample

t 0

a/ I I I I I I l l l l I I I I I I I I IO loo loo0

wj l%/

Figure 25.7 Comparison of coefficients of consolidation of various clays computed by use of direct measurements of k, and computed from settlement-time data.

R,, = O i 3 40 t \Y 8 I L

Figure 25.8 Relation between degree of consolidation and time factor for impeded drainage boundaries.

increase with time of AD, and therefore of u;, up to the end-of-construction time t,. Thereafter, Au, is assumed to remain constant. The U vs T, relationship in Fig. 25.9 is a function of the construction time factor T,, = c,t,/H?-. The combinations of c, H, and t, that produce values of T,, less than 0.0 1 closely approximate instantaneous loading; the U vs T, curve corresponding to 0.01 is very close to curve C, in Fig. 25.5.

The initial excess porewater pressure distributions in Fig. 25.3, b, c, and d, could result from dewatering operations that decrease the porewater pressure at the top or bottom boundaries of the consolidating layer, or at both. Consolidation resulting from such operations closely approximates one-dimensional compression and one- dimensional flow in the vertical direction. The initial conditions in Fig. 25.3 b, c, and d, may also result from the distribution of Am, with depth that occurs when the consolidating layer is thick with respect to the width of the loaded area. Under these conditions, consolidation is not likely to be one-dimensional. Usually, loading and deformation boundary conditions are such that the compression is more or less one-dimensional, and one-dimensional settlement analysis provides reliable estimates of the EOP settlement. Nevertheless, in these situations, the flow of water is likely to be three-dimensional (two- dimensional for plane-strain loading conditions), and the horizontal flow is likely to accelerate the process of consolidation.

A three-dimensional theory of consolidation was devel- oped by Biot in 1941. With respect to the fundamental assumptions, it is practically identical with the one- dimensional Terzaghi theory except that it includes the flow of water in all directions. The Biot theory, or simpler versions of it (Rendulic 1937, Gibson and Lumb 1953), may be used to evaluate the contribution of horizontal flow to the rate of consolidation (Gibson and McNamee 1957, Gibson 1961, Christian et al. 1972, Davis and Poulos 1972). Two sets of U vs T, curves from Davis and Poulos (1972) are shown in Figs. 25.10 and 25.11 to illustrate this contribution. Figure 25.10 corresponds

ARTICLE 25 THEORY OF CONSOLIDATION 23 1

3

Figure 25.9 Relations between degree of consolidation and time factor for time-dependent increase in consolidation pressure.

0, 1 1 1 1 1 , 1 , , I , , , , . , , , , 1 , , 1 , 1 1 , , , , , , , , . , ,

Figure 25.10 Consolidation under a uniform pressure over a circular area on the assumption of three-dimensional flow of the water (after Davis and Poulos 1972).

to a uniform pressure Ap applied over a circular area of radius a to the upper surface of a half-closed consolidating layer of thickness H. A similar condition is shown in Fig. 25.1 1 , except that the uniform pressure Ap is applied over a long strip of width 2b. The U vs T,] curves corresponding to H / a and H / b values of 0.5 differ only slightly from curve C , (Fig. 25.5). However, for values of H l a or H l b greater than 1, horizontal flow makes a substantial contribution to the rate of consolidation. The U vs T, curves in Figs. 25.10 and 25.1 1 correspond to the center of the loaded area and were obtained by assuming kh = k,. In the field, if kh is greater than k , the contribution of horizontal flow may be

.$ r 6 \\ \\\

CY I T,= - H'

Figure 25.11 Consolidation under a uniform pressure over a long strip on the assumption of two-dimensional flow of the water (after Davis and Poulos 1972).

even more significant, especially beneath the boundaries of the loaded area.

25.5 Consolidation with Vertical Drains The primary consolidation of fairly thick soft clay layers subjected to permanent reclamation fills, embankments, or temporary preloads may require a considerable time if the excess water flows out of the clay in only the vertical direction. For example, the primary consolidation


of a soft clay layer with a maximum drainage distance of 5 m may require more than 10 years. To accelerate primary consolidation, vertical drains may be installed (Article 44.3.6). They are especially effective in stratified soils such as varved clays, in which the permeability is larger in the horizontal than in the vertical direction.

In the absence of a permeable granular layer above the consolidating layer, the drains are connected to a drainage blanket placed on the ground surface. Vertical drains often penetrate fully through the consolidating layer and into a permeable lower boundary. Such a drain is open at both top and bottom, and its length is denoted by 2 1, where 1, is the maximum drainage length. If the drain partially penetrates the consolidating layer, it is open only at the top, and its length is denoted by 1,

The radius r, of a circular sand drain is typically in the range of 80 to 300 mm. The equivalent radius rw of wick drains with a rectangular cross-section is computed as rw = (a + b ) h , where a and b are the thickness and width of the drain, respectively (Hansbo 1979). Typical values of a and b of 3.2 to 4.0 mm and 93 to 100 mm, respectively, lead to values of r, in the range of 31 to 33 mm.

Vertical drains are usually installed in a triangular pattern at a spacing DS in the range of 1 to 5 m (Figs. 25.12 and 25.13). The radius re of the soil cylinder discharging water into a vertical drain is 0.525 DS.

The function of a vertical drain is to accept radial flow of water from the consolidating ground, transport it in the vertical direction, and discharge it into a top or bottom drainage layer, or both, with as little hydraulic resistance as possible. However, most vertical drains have a finite discharge capacity, qw = T .", k , where k , is the permeability of the drain. Therefore, the horizontal permeability of the consolidating soil and the maximum drainage length 1, of the drain determine whether a vertical drain

0 I o I o Figure 25.12 Triangular pattern of installation of vertical drains.

I I I

I T I

I I I I

I 'I I I

I -----+ I I I

I I I I I I I I I

I

I - I I I I I

I I -

Drainage rs-!-- f Blanket

I I

I I T

I I I I I FlowLIne I - I I I

I I

I I I

I I I

I I I I

I I

T I

~ ! . , n w / * \ " / n \ \ ,

Impermeable

Figure 25.13 Vertical drain of radius r, smear zone of radius r,, and consolidating soil cylinder of radius re. Flow lines based on assumption of no vertical flow of water. Isochrone at depth z drawn on assumption of well resistance.

may drain freely or may display a well resistance. The discharge factor D is defined as

(25.10)

where kh is the horizontal permeability of the consolidating layer and q, and 1, pertain to the vertical drain. Analyses of field performance of wick drains (Article 44.3.6) in soft clay deposits indicate that well resistance is negligible when D is greater than 5 (Mesri and Lo 1991). That is, the minimum discharge capacity qw (min) of vertical drains, required for negligible well resistance, is qw (min) = 5 kho l;. The most typical values of kho and 1, for soft clay and vertical drain installations lead to values of qw (min) in the range of 2 to 80 m3/year. However, these magnitudes of discharge capacity are required only at the beginning of consolidation. As kh decreases during consolidation, less water enters the drain during a given time and, therefore, a smaller 4, (min) is required to discharge the water with negligible hydraulic


resistance. The value of D = 5 together with kho specify the upper limit for 4, (min) which decreases approximately according to Eq. 25.10 as kh decreases. Therefore, if the initial 4, (min) required for negligible well resistance is 10 m3/year, and kh decreases by a factor of 2 by the time 50% primary consolidation is completed, then the minimum discharge capacity required beyond 50% consolidation is less than 5 m3/year.

Installation of vertical drains creates a cylinder of dis- turbed soil or smear zone of external radius r, around the drain. In this zone the permeability and preconsolidation pressure are reduced and the compressibility is increased. The decrease in horizontal permeability is most significant for laminated soils such as varved clays, and the increase in compressibility is most significant for highly structured clays.

For one-dimensional vertical compression together with vertical and radial flow, Eq. 25.5 becomes

where ch = kh/ rw m, The excess porewater pressure u’ is a function of time as well as of z and I: Carillo (1942) showed that the solution of Eq. 25.11 can be obtained by combining separate solutions for vertical compression by vertical flow and vertical compression by radial flow. The excess porewater pressure and degree of consolidation, at any time, were found to be

(25.12)

and

u = 1 - (1 - U,)(l - U,) (25.13)

where u: and u: are the excess porewater pressures for vertical flow only and for radial flow only, respectively, and U, and U, are the corresponding degrees of consolidation. Relying on Carillo’s contribution, most formulations of the rate of consolidation with vertical drains have considered only the radial flow of water through the consolidating layer and have ignored the vertical flow. The contribution of vertical flow is taken into account using the solution of Eq. 25.5 together with the Carillo equations. Vertical flow through the consolidating layer is likely to be significant when His small and DS is large.

Barron (1944) formulated the rate of consolidation of a layer with fully penetrating vertical drains on the basis of the same assumptions regarding the permeability and compressibility of the soil as those that were made in developing Eq. 25.5. Assuming no smear zone, no well resistance, and equal vertical strain at any depth throughout the consolidation process, he found

u = 1 - e x p [ x ] -2T, (25.14)

where U is the degree of consolidation for radial flow only, n = re / r , F, = In (n) - 3 /4 for values of n greater than 10, and

Later expressions for vertical compression with radial flow into a vertical drain include the effect of a smear zone of radius r, = s rw with permeability and compressibility different from those of the undisturbed soil, and the effect of a vertical drain having a finite permeability k, or discharge capacity qw (Barron 1948, Hansbo 1979, 1981, Zeng and Xie 1989). For example, the formulations by Hansbo (198 1) and Zeng and Xie (1989) include a smear zone with permeability k, different from kh of the undisturbed layer but with the same compressibility as that of the undisturbed soil. They can be used to illustrate, for a value of k, = kh/2 , the effect of the magnitudes of n, 4w and s on the U vs T, relationship (Fig. 25.14). Figure 25 .14~ shows that n has a relatively small effect on the U vs T, relation; however, it has a significant effect on the rate of consolidation because of its influence on re = n r, in the time factor T, (Eq. 25.15).

A value of s = 1 means that there is no smear zone. For sand drains r, = r,,,, and the value of s is unlikely to exceed 4. However, for wick drains rw is significantly smaller than the radius r,,, of the mandrel used to install them. Therefore, even a value of rJr, = 2, for typical values of r,,, = 80 mm and r, = 32 mm, leads to s = 5.

Analyses of field performance of wick drains suggest that in most soft clays the mobilized discharge capacity of prefabricated drains of high quality is comparable to the minimum discharge capacity required for negligible well resistance (Mesri and Lo 1991). Drains of poor quality or drains that are damaged during installation are represented by a value of equal to 4w (min)/lO in Fig. 25.14~.

25.6 Limitations of the Theory of Consolidation The theory of consolidation that has been formulated in this article and solved for various initial and boundary conditions, including three-dimensional flow of water and consolidation with vertical drains, is based on a number of important assumptions that limit its validity. It has been assumed that:

1. The coefficient of permeability k, or kh remains constant during consolidation.

2. The relation between void ratio or vertical strain and effective vertical stress is linear; that is, the coefficient of compressibility a, or m, remains constant during consolidation.


e the progress of settlements that are expected to be large. Such settlements are associated with values of the final effective vertical stress, ab, = a:, + Au,, in the compression range; that is, beyond the preconsolidation pressure ai (Fig. 25.15). The consolidation associated with an increase in effective vertical stress from a:, to a:,-involves a decrease in void ratio from eo to ep and thus a substantial decrease in permeability. The permeability of each sublayer is initially k,,, kho and decreases during consolidation. Because the consolidation of a layer proceeds from

e k YO

/ - e ; p -~ ~

r, =

Or I

r ; o r p

e;\

Figure 25.14 Relations between degree of consolidation and time factor for radial flow into a vertical drain. Loading is assumed to be instantaneous and excess porewater pressure constant with depth. (a) different drain spacing; (b) different mobilized discharge capacity; (c) different size of smear zone.


homogeneous layer characterized by k,, and kho, k, and kh vary with both time and location until the end of primary consolidation is reached.

Figure 16.1 shows that when a consolidation pressure increment spans the preconsolidation pressure, a, is initially small in the recompression range, increases abruptly as a6 is exceeded, and then decreases continuously as u; increases in the compression range. Thus, in most practical situations involving the consolidation of soft clays, a, or m, does not remain constant during consolidation.

It is possible to reformulate the theory of consolidation by making more realistic assumptions with respect to the permeability and compressibility of the soil as consolidation proceeds. For example, the decrease in k , or kh may be expressed in terms of the Ck between the initial void ratio eo and the EOP void ratio ep (Article 14.4). The relation between void ratio and effective vertical stress may be approximated by a constant recompression index C, in the recompression range from a:, to ai, and by a constant compression index C, in the compression range from a; to ah, (Mesri and Rokhsar 1974). Alternatively the e vs log k , e vs log kh, and e vs log a: curves may be digitized and directly input into a numerical solution of the consolidation equation by use of a computer (Mesri and Choi 1985a, Mesri and Lo 1989). For some soils and for consolidation pressure increments that are completely in the compression range, Ck /C, = 1. Under these conditions the decrease in m, compensates for the decrease in k , or kh, and c, or c h remains practically constant during consolidation. Therefore, in some situations, the assumption of a constant c, or c h during consolidation may be reasonable and may lead to realistic predictions of the rate of settlement. However, the dissipation of porewater pressure at any point in the consolidating layer is highly dependent on the relation between void ratio and effective vertical stress. Reliable settlement analysis, especially excess porewater pressure prediction, requires realistic assumptions regarding the permeability and compressibility of the consolidating layer (Mesri and Choi 1979).

It has been observed repeatedly in both laboratory tests and in the field that when the excess porewater pressure approaches zero, the settlement does not come to a stop. Secondary compression continues beyond the end of primary consolidation. However, the assumption that a,, = 0 excludes secondary compression. Consequently, the U vs T curves in Figs. 25.5, 25.8, 25.9, 25.10, 25.11, and 25.14 and therefore the s vs t curves of the theory of consolidation level off at the end of primary consolidation.

The Ca/Cc law of compressibility (Article 16.7) pre- dicts and explains secondary settlement as a function of time for any value of When the EOP e vs log a: relation is nearly linear and thus C, is constant near aif, the progress of secondary settlement can be computed

by Eq. 16.12. If the EOP e vs log a: curve is highly nonlinear near akand thus C, changes with a:, a graphical construction can be used to compute the secondary settlement as a function of time (Mesri and Godlewski 1977, 1979, Mesri and Shahien 1993). The progress of post- surcharge secondary settlement is determined by Eq. 13.13. Information on the duration rp of the primary consolidation stage after loading or on the duration rpr of the primary rebound stage after the removal of surcharge, in addition to the compressibility parameters, is required for computing the rate of secondary settlement. The value of rp is obtained by using a theory of consolidation, and rpr is determined with a theory of expansion. Most often, rp or rps is approximated by the time required to complete 95% primary consolidation or primary expansion. For example, by using the Terzaghi theory of consolidation and the curve C, in Fig. 25.5, tp is computed by means of Eq. 25.7, and T, = 1.13. When vertical drains are used, then rp is computed by using Eq. 25.15. From the Barron Eq. 25.14 and, for example, n = 25, T, is found to be 3.7 for U = 95%.

The assumption of a,, = 0 ignores the contribution of compressibility with time during the primary consolidation stage. The compression of a sublayer that occurs during the time that an increment of effective vertical stress acts, however, consists of the contributions of both avs and a,,,. Whereas a,, during the secondary compression stage is well defined by direct measurements, the evalua- tion of a, and a,, during primary consolidation is not practicable. In the absence of observed data, the assumption that avr = 0 has been replaced by alternative assumptions (e.g., Berry and Poskitt 1972, Mesri and Rokhsar 1974). Mesri and Choi (19854, for example, assume that the EOP e vs log a: curve is independent of the duration of primary consolidation. The EOP e vs log a: curve from the oedometer test is used with a constant Ca/C, to estimate avs and a,, for increments of time during the primary consolidation stage. This approach has no effect on the magnitude of the EOP settlement and leads to predictions of rate of settlement that are in good agreement with direct observations (Mesri et al. 1994).

The hydrodynamic equation of consolidation can also be reformulated without assuming small strains during consolidation (Gibson et al. 1967). However, for most consolidation problems involving soft clay deposits, the small strain approximation is not a serious limitation (Mesri and Rokhsar 1974). On the other hand, a finite strain or large strain formulation may be more appropriate for consolidation of clay dredgings and slurries.

25.7 Application of a Consolidation Theory to Field Situations

In the field the consolidating layer rarely consists of a single homogeneous soil with respect to permeability and


compressibility. More often it consists of one or more distinct geologic deposits, and within each deposit such soil properties as eo, k,,, kho, and u; may vary significantly with depth. With the help of the computer, it is no longer necessary to idealize the compressible soil profile into a single homogeneous layer. The compressible soil is divided into a number of consolidating sublayers, each with its own values of eo, k,,, kho, ck, C, C,, uho, 0; (or its own e vs log k, e vs log kh, and EOP e vs log a: curves), as well as and Ca/C,. A numerical scheme is used with a computer to solve the differential equation of consolidation (Mesri and Choi 1985a, Mesri and Lo 1989). At the interface between adjacent sublayers j and j + 1 the following conditions are satisfied with respect to the excess porewater pressure and flow rate:

u; = u;+1

and

Vertical drains may be installed before or during construction of a fill or embankment. The drains n a y fully or partially penetrate the compressible soil profile. The compressibility and permeability of the smear zone sur- rounding the vertical drain differ from those of the undisturbed soil and differ among the consolidating sublayers. Field construction schedules often include several stages of loading intervening with rest periods before the end of construction; therefore, in general A u v does not increase linearly from to to tc. Such drainage and loading conditions are best treated with a numerical solution with the aid of a computer program. The computer program ILLICON (Mesri and Choi 1985a, Mesri and Lo 1989), for example, incorporates all the foregoing factors.

To illustrate the capabilities of such a procedure, field observations of ground surface settlement over a period of 25 years are shown in Fig. 25.16 for test fills of granular materials at Ski-Edeby, Sweden (Hansbo 1960, Hansbo et al. 1981). Within the compressible clay profile 12 m thick including the crust, the values of w,, wl, and uLlu~, are in the ranges, respectively, of 45 to 120%, 60 to 150%, and 1.1 to 2.0 (most typical values 1.2 to 1.3). The diameters of the fills range from 30 to 70 m. At four locations, displacement-type sand drains of 180 mm diameter were installed in a triangular pattern at spacings of 0.9, 1.5, and 2.2 m. At one location no vertical drains were used. The drains, 12 m long, penetrated into a granular till underlying the clay.

For a settlement analysis using the ILLICON computer program, the compressible soil profile was divided into seven consolidating sublayers, each with its own values of eo, u:,, uLf, EOP e vs log oh and kv, (Mesri and Lo 1989, Lo 1991). For the entire compressible profile, con-

stant values of Ck/f?o, Ca/Cc, kho/kv,, k,,lkv,, and rs/rw were used. The discharge capacity of the sand drains was computed to be 450 m3 /year which is more than adequate for negligible well resistance. For the smear zone within each sublayer an EOP e vs log a: relation was used consisting of a straight line joining (u;,, eo) and (uh,, e,,) of the undisturbed soil.

The predictions of ground surface settlements are shown by the continuous curves in Fig. 25.16. The measurements and predictions both show that the sand drains significantly accelerated the rate of primary consolidation and that the magnitude of the increase was a function of the spacing of the drains. For example, with the 0.9-m drain spacing, primary consolidation was completed in about 500 days, whereas without vertical drains it would require more than 30 years. Where the vertical drains were used, primary consolidation was completed within the period of observation and secondary compression was observed thereafter. Because at all sublayers except the crust, ahf was significantly larger than uL, the EOP settlement was large and, in comparison, the secondary settlement during the period of observation was small, especially for the drain spacing of 2.2 m.

25.8 Theory of Expansion Time-dependent expansion or swelling of saturated soils in fundamental respects is completely analogous to consolidation. Excavation of overburden or removal of a temporary surcharge results in a decrease in total vertical stress and in an identical decrease in porewater pressure, whereas the effective vertical stress remains unchanged. The porewater pressure at the permeable boundary layers quickly returns to the equilibrium reference condition. The gradient in excess porewater pressure (excess in a negative sense because the porewater pressure in the expanding layer is less than the reference equilibrium porewater pressure) causes water to flow through the boundary surfaces into the expanding layer as the excess negative porewater pressure dissipates. The increase in porewater pressure, the associated decrease in effective vertical stress, and the expansion of the void volume represent the primary expansion stage. Secondary expansion follows the end of primary expansion.

Examples of swelling of saturated shales subjected to a decrement in expansion pressure are shown in Figs. 25.17 and 25.18 (Cepeda-Diaz 1987). The index properties of the Patapsco shale sample from Washington, D.C. were w,, = 21.6%, wl = 77%, wp = 25%, and w , ~ = 11%. The undisturbed specimen, 15 mm thick, was subjected to a pressure decrement of 144 kPa, from ahi = 192 kPa to ukf = 48 kPa. Drainage took place from the permeable top, and porewater pressure was measured at the impermeable bottom of the specimen. At an elapsed time of 4 min an excess porewater pressure of about 142 kPa was observed, which dissipated to zero in about 6000 min.


0 / 09 0 / 1.5

Sweden

Figure 25.16 Observed and computed settlements of ground surface under embankments on a soft clay deposit at Ska Edeby, Sweden.

Figure 25.17 Observed and computed behavior of Patapsco shale subjected to a decrement in expansion pressure.

Secondary swelling followed at a constant effective vertical stress of 48 kPa. The index properties of the Pierre shale sample from Limon, Colorado were w, = 24.3%, wl = 82%, wp = 30%, and w, = 13%. The undisturbed specimen, 12 mm thick, was subjected to a pressure decrement of 45 kPa, from abi = 58 kPa to ahf = 13 kPa. At an elapsed time of 2 min, an excess porewater pressure of 45 kPa was measured that dissipated to zero in about

5000 min. Thereafter, at constant effective vertical stress, secondary swelling was observed.

Secondary swelling at ahf begins at tp, which is the time required to complete primary swelling. The value of C,,/C, is used (Article 16.10), together with C, from the unloading EOP e vs log a: curve. Because C,,/C, and in some cases C, increase with OCR, C,, is expected to increase with time. This increase is most significant


0 4 0

074-

p;,

A- e,

Slope = oys

0 7 0 -

0.68 ' ' I , , ,

/O 20 30 40 50 60

r i (kPd

kvo = 36x 10-'2mm/s

ck = 0175

0 Observed -Computed

50 1 10 lo0 l a x , W W toom

Time fminj

Figure 25.18 Observed and computed behavior of Pierre shale subjected to a decrement in expansion pressure.

at very high values of OCR, and for those expansive clays and shales for which C, dramatically increases with a decrease in a:. However, in many practical situations, where the unloading EOP e vs log ai relation is approximately linear, and for values of t Itp less than 10, a constant C,, can be used to estimate heave resulting from secondary swelling.

The theory of consolidation is applicable to the time rate of expansion of saturated soils (Terzaghi 1931), except that time is related to the time factor in terms of the coefficient of expansion, c,, = k,/y, mvs, where mvs = Ae,/Aa: is determined from the EOP rebound curve. Values of c,, for shales as well as for soft clay deposits are shown as a function of overconsolidation ratio in Fig. 25.19. For unloading from the compression range, a,, or mvs increases significantly during rebound, whereas in comparison the increase in k, is relatively small. There- fore, c,, decreases dramatically during a decrement in expansion pressure. Because mvs starts with a very small value at aii and increases with decrease in a;, the assumption of a constant mVs between oli and significantly underestimates the rate of excess porewater pressure dissipation. That is, observed excess porewater pressures in cut slopes and adjacent to underground excavations dissipate faster than predicted by the theory of expansion. However, the U vs T, curves in Figs. 25.5, 25.8, 25.9, 25.10, and 25.1 1, together with an appropriate value of cVs, provide acceptable estimates of the rate of expansion

and heave. Furthermore, the decrease in c,, with overconsolidation ratio is not so dramatic when the unloading is from the recompression range.

The theory of expansion has been reformulated to take into account the increases in mvs and k, during primary expansion (Mesri et al. 1978). The reformulation also includes, in terms of C,,/Cs, the contribution of a,, for expansion. Comparisons of predicted time rate of heave and dissipation of excess porewater pressure by the reformulated theory are shown in Figs. 25.17 and 25.18. The insets in these figures show the significant increase in uvs during the decrements in expansion pressure. These results, together with measurements and predictions for other shales (Cepeda-Diaz 1987), show that for a reliable prediction of the rate of decrease in effective stress after unloading, the variation of uvs must be included.

Time-dependent expansion and associated heave follow excavation of overburden or any other process of unloading. Time-dependent heave also follows a reduc- tion in preconstruction suction that exists above the water table. A portion of this suction becomes an excess negative porewater pressure with respect to the postconstruc- tion equilibrium condition where the ground surface is covered by an impermeable slab for a building, road, runway, canal, or spillway, and the moisture equilibrium between precipitation-evaporation and the water table is altered. To predict the time rate of heave, a formulation of unsaturated flow in expansive soils must then be used


I

OCR = o-;/u;

Figure 25.19 Coefficient of expansion for various shales and soft clays unloaded from the compression range.

(Sokolov and Amir 1973, Yong 1977). Such a formulation must take into account the significant decrease in permeability that occurs with the increase in degree of saturation and with the associated internal slaking and dispersion (Jayawickrama and Lytton 1992).

Problems

1. A wide fill (3 m thick and y = 19.2 kN/m3) is placed on a site with a soil profile consisting of 1.5 m of sand (y = 19 kN/m3) over 6 m of plastic clay (y = 16 kN/m3, eo = 1.7, C, = 1.2, uL/cr:, = 1.4, CJC, = 0.15) over 10 m of organic silt (y = 16 kN/m3, eo = 1.2, C, = 0.8, uL/cri0 = 1, CJC, = 0.1, C,/C, = 0.06, c, = 0.04 m2/day) over coarse sand. The water table is 1.5 m below the ground surface. Compute the settlement resulting from primary consolidation of the plastic clay. Assuming the plastic clay is impermeable compared with the organic silt, compute settlement resulting from compression of the organic silt after 30 years. Using the same assumption, sketch excess porewater pressure isochrones in the organic silt corresponding to lo%, 50%, and 95% primary consolidation.

2. A soil profile consists of 6 m of sand (y = 17.6 kN/m3) over 3 m of silty clay (y = 16.6 kN/m3, eo = 1.1, C, = 1.5, uL/a:, = 1.2, C,/C, = 0.1, and c, = 6 X mZ/day) over gravel. The water table is 2 m below the ground surface. The plan is to pump water from cased wells that penetrate into the gravel layer at such a rate that the water level in the wells will be 8 m below the ground surface. Compute settlement resulting from the primary consolidation of the silty clay layer when the steady-state seepage condition is established. Compute the time required for 95% primary consolidation of the silty clay layer. Sketch excess porewater pressure isochrones corresponding to average degree of consolidation of 0%, 5%, 50%, and 95%.

Ans. 0.60 m; 0.88 m.

Ans. 0.16 m; 423 days.

3. A soil profile consists of 10 m of sand (y = 17.6 kN/m3) over 3 m of organic clay (y = 16 kN/m3, eo = 1.6, C, = 1.3, uL/u:, = 1.3, C,/C, = 0.2, C,/C, = 0.05, and c, in compression range = 0.02 m*/day) over gravel. The water table is 3 m below the ground surface. Before constructing a structure on the site, the organic clay is precompressed by lowering the water table by 3 m and pumping from the gravel layer at such a rate that the piezometric head in the gravel drops by 6 m. Compute the settlement resulting from 80% primary consolidation of the clay layer. Sketch excess porewater pressure isochrones corresponding to 0%, 80%, and 95% average degree of consolidation. After 95% primary consolidation is reached, pumping is stopped so that the water table in sand and the porewater pressure in the gravel return to the original condition. The structure is constructed; it produces an effective vertical stress increase in the clay barely exceeding that produced by precompression. Compute the settlement of the structure after 30 years resulting from the secondary compression of the organic clay.

Ans. 0.06 m; 0.20 m. 4. A 7-m-thick granular fill (y = 20 kN/m3) is to be placed

over a 20-m-thick deposit of silty clay (y = 16 kN/m3, eo = 1.76, C, = 0.9, ui/a:, = 1.6, CJC, = 0.2, c, = 4 X m2/day, ch/c, = 3, and C,/C, = 0.04) over decomposed frac- tured granite. The water table is near the ground surface. Com- putations show that considerable time would be required for primary consolidation of the silty clay if drainage is in the vertical direction only. Therefore, it is decided to speed up consolidation by using prefabricated vertical drains (rw = 33mm) with a spacing of 1 m, in a triangular pattern, and fully penetrating the silty clay layer. Compute: (a) settlement resulting from primary consolidation of silty clay, (b) time required for 95% primary consolidation, and (c) secondary settlement 50 years after construction of the fill.

3.1 m; assuming no well resistance and no smear effect, 70 days; however, assuming vertical drain discharge

Ans.


capacity of 5 rn3/year, a smear zone with rs/rm = 2, radius of mandrel r, = 100 mm, and k,/k, = 1, we obtain 178 days; using tp = 178 days, 0.52 m.

5. A 7-m-thick gravel fill (y = 19 kN/m3) is to be placed over a soil profile consisting of 3 m of stiff fissured crust (y = 18 kN/m3) over 17 m of gray plastic clay (y = 16 kN/m3, eo = 1.76, C, = 0.90, uL/uLo = 1.4, CJC, = 0.1, C,/C, = 0.04, c, = 1 0 - ~ m21day, and ch = 3 x m2/day) over I O m of silty clay (y = 17 kN/m3, eo = 1.5, C, = 0.75, u~lul, = 2.5, CJC, = 0.1, C,/C, = 0.05, c, (recompression) = 5 X lo-* m2/day, c, (compression) = 5 X m2/day) over coarse sand. The water table is 3 m below the ground surface. In order to speed up the primary consolidation of the plastic clay, vertical band drains (rw = 32 mm, triangular pattern) with a spacing of 1.5 m are installed penetrating to the top of the silty clay. Compute settlement of the ground surface after 1, 3, and 50 years.

Ignoring well resistance, smear effect, compression of the crust, and vertical water flow through gray plastic clay, 1.06 m, 1.43 m, 1.72 m.

Ans.

Selected Reading

Solutions for the consolidation of masses of soil having various boundary conditions may be found in the following references.

Terzaghi, K. and 0. K. Frohlich (1936). Theorie der Setzung von Tonschichten (Theory of settlement of clay layers). Leipzig, Deutike, 166 pp.

Bamon, R. A. (1948). “Consolidation of fine-grained soils by drain wells,” Trans. ASCE, 113, pp. 718-742.

Hansbo, S . (1960). “Consolidation of clay, with special reference to influence of vertical sand drains,” Proc. Swedish Geotechnical Institute, 18, Linkoping.

Gibson, R.E., G. L. England, and M.J.L. Hussey (1967). “The theory of one-dimensional consolidation of saturated clays,” GPot., 17, No. 3, pp. 261-273.

Davis, E.H. and H.G. Poulos (1972). “Rate of settlement under two- and three-dimensional conditions,” Ge‘ot., 22, No. 1,

Mesri, G. and A. Rokhsar (1974). “Theory of consolidation for clays,” J. Geotech. Eng., ASCE, 100, No. 8, pp. 889-904.

Hansbo, S . (1979). “Consolidation of clay by band-shaped pre- fabricted drains,” Ground Eng., 12, No. 5, pp. 16-25.

Hansbo, S. (1981). “Consolidation of fine-grained soils by prefabricated drains,” Proc. 10th Int. Con$ Soil Mech. and Found. Eng., Stockholm, 3, pp. 677-682.

Mesri, G. and Y.K. Choi (1985a). “Settlement analysis of embankments on soft clays,” J. Geotech. Eng., ASCE, 111, No. 4, pp. 441-464.

Mesri, G. and D.O.K. Lo (1989). “Subsoil investigation: the weakest link in the analysis of test fills,” The Art and Science of Geotechnical Engineering At the Dawn of the Twenty-first Century, Prentice-Hall Inc., Englewood Cliffs,

Lo, D.O.K. (1991). Soil Improvement by Vertical Drains, Ph.D. thesis, Univ. of Illinois at Urbana-Champaign.

Mesri, G. and D.O.K. Lo (1991). “Field performance of prefabricated vertical drains,” Proc. Int. Con$ on Geotech. Eng. for Coastal Development-Theory to Practice, Yokohama,

Mesri, G., D.O.K. Lo, and T.W. Feng (1994). “Settlement of embankments on soft clays,” Proc. Settlement ’94, ASCE Specialty Conf. Geotech. Spec. Publ. No. 40, 1, pp. 8-56.

pp. 95-114.

N.J., pp. 309-335.

1, pp. 231-236.

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