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18-20 The Civil Engineering Handbook, Second Edition

The ﬂow medium in Fig. 18.20(a) consists of two horizontal layers of thickness d

and d

, underlain

by an impervious base. The coefﬁcient of permeability of the upper layer is k

and of the lower layer k

The coefﬁcients of permeability are related to a dimensionless parameter e by the expression

(18.39)

(p is in radian measure). Thus, as the ratio of the permeabilities varies from 0 to •, e ranges between 0

and 1/2. We ﬁrst investigate the structures shown in Fig. 18.20(a) for some special values of e.

1. e = 0. If k

= 0, from Eq. (18.39) we have e = 0. This is equivalent to having the impervious base

at depth d

. Hence, for this case the ﬂow region is reduced to that of a single homogeneous layer

for which the discharge can be obtained directly from Fig. 18.11.

2. e = 1/4. If k

= k

, the sections are reduced to a single homogeneous layer, of thickness d

+ d

for which Fig. 18.11 is again applicable.

3. e = 1/2. If k

= •, e = 1/2. This represents a condition where there is no resistance to ﬂow in the

bottom layer. Hence, the discharge through the total section under steady state conditions is

inﬁnite, or Q/k

h = •. However, of greater signiﬁcance is the fact that the inverse of this ratio

equals zero: k

h/Q = 0. It can be shown [Polubarinova-Kochina, 1962] that for k

Æ •,

(18.40)

Thus, we see that for the special values of e = 0, e = 1/4, and e = 1/2, measures of the ﬂow quantities

can be easily obtained. The essence of the method then is to plot these values, on a plot of k

h/Q versus

e, and connect the points with a smooth curve, from which intermediate values can be had.

Example 18.7

In Fig. 18.20(a), s = 10 ft, d

= 15 ft, d

= 20 ft, k

= 4k

= 1 • 10

–3

ft/min, h = 6 ft. Estimate Q/k

Solution. For e = 0, from Fig. 18.11 with s/T = s/d

= 2/3, b/T = 0, Q/k

h = 0.39, k

h/Q = 2.56.

For e = 1/4, from Fig. 18.11 with s/T = s/(d

+ d

) = 2/7, Q/k

h = 0.67, k

h/Q = 1.49.

For e = 1/2, k

h/Q = 0.

The three points are plotted in Fig. 18.21, and the required discharge, for e = 1/p tan

–1

. (1/4)

1/2

= 0.15,

is k

h/Q = 1.92 and Q/k

h = 0.52; whence

In combination with the method of fragments, approximate solutions can be had for very complicated

structures.

Example 18.8

Estimate (a) the discharge through the section shown in Fig. 18.22(a), k

= 4k

= 1 ¥ 10

–3

ft/day and

(b) the pressure in the water at point P.

Solution. The ﬂow region is shown divided into four fragments. However, the form factors for regions 1

and 4 are the same. In Fig. 18.22(b) are given the resulting form factors for the listed conditions. In

Fig. 18.22(c) is given the plot of k

h/Q versus e. For the required condition (k

= 4k

), e = 0.35, k

h/Q =

1.6 and hence Q = (1/1.6) ¥ 0.25 ¥ 10

–3

¥ 10 = 1.6 ¥ 10

–3

/(day) (ft).

The total head at point P is given in Fig. 18.22(b) as Dh for region 4; for e = 0, h

= 2.57 ft and for

e = 1/4, h

= 3.00. We require h

for k

= •; theoretically, there is assumed to be no resistance to the ﬂow

tan

----=

--------

----=

Q 0.52 1 10

3–

6¥¥¥=

3.1 1¥ 0

3–

min()§ ft()=

Groundwater and Seepage 18-21

in the bottom layer for this condition. Hence the boundary between the two layers (AB) is an equipotential

line with an expected value of h/2. Thus h

= 10/2 = 5 for e = 1/2. In Fig. 18.22(d) is given the plot of

versus e. For e = 0.35, h

= 2.75 ft and the pressure in the water at point P is (3.75 + 5)g

= 8.75g

The above procedure may be extended to systems with more than two layers.

18.6 Piping

By virtue of the viscous friction exerted on a liquid ﬂowing through a porous medium, an energy transfer

is effected between the liquid and the solid particles. The measure of this energy is the head loss h between

the points under consideration. The force corresponding to this energy transfer is called the seepage force.

It is this seepage force that is responsible for the phenomenon known as quicksand, and its assessment

is of vital importance in stability analyses of earth structures subject to the action of ﬂowing water

(seepage).

The ﬁrst rational approach to the determination of the effects of seepage forces was presented by

Terzaghi [1922] and forms the basis for all subsequent studies. Consider all the forces acting on a volume

of particulate matter through which a liquid ﬂows.

1. The total weight per unit volume, the mass unit weight, is

where e is the void ratio, G is the speciﬁc gravity of solids, and g

is the unit weight of the liquid.

2. Invoking Archimedes’ principle of buoyancy (a body submerged in a liquid is buoyed up by force

equal to the weight of the liquid displaced), the effective unit weight of a volume of soil, called

the submerged unit weight, is

(18.41)

To gain a better understanding of the meaning of the submerged unit weight consider the ﬂow

condition shown in Fig. 18.23(a). If the water column (AB) is held at the same elevation as the

discharge face CD (h = 0), the soil will be in a submerged state and the downward force acting

on the screen will be

(18.42)

FIGURE 18.21 Example 18.7.

0.15 0.25 0.50



1.92

tan p =



Ge+()

1 e+

----------------------=

–

G 1–()

1 e+

-----------------------==

FØ

LA=

18-22 The Civil Engineering Handbook, Second Edition

where g¢

= g

– g

. Now, if the water column is slowly raised (shown dotted to A¢B¢), water will

ﬂow up through the soil. By virtue of this upward ﬂow, work will be done to the soil and the force

acting on the screen will be reduced.

3. The change in force through the soil is due to the increased pressure acting over the area, or

Hence, the change in force, granted steady state conditions, is

(18.43)

FIGURE 18.22 Example 18.8.

13 ft

15 ft

1234

10 ft

5 ft

1.6

0.350.25

3.75

0.5

0.50

(

)



(

)

(

)

(

)

Region

1II 0.740.96 2.57 3.00

1.63

3.00

2.14

2.57

0.40

0.74

2.46

0.80

0.96

3.74

ΣΦ

2.362.730.581.02V2

Type

Check:

10.0

Check:

10.0

 = 0

∆

 = 0

∆

 =

= ΣΦ

15 ft

LA h

A–=

Groundwater and Seepage 18-23

Dividing by the volume AL, the resultant force per unit volume acting at a point within the ﬂow region

(18.44)

where i is the hydraulic gradient. The quantity i g

is the seepage force (force per unit volume). In general,

Eq. (18.44) is a vector equation, with the seepage force acting in the direction of ﬂow [Fig. 18.23(b)]. Of

course, for the ﬂow condition shown in Fig. 18.23(a), the seepage force will be directed vertically upward

[Fig. 18.23(c)].

If the head h is increased, the resultant force R in Eq. (18.44) is seen to decrease. Evidently, should h

be increased to the point at which R = 0, the material would be at the point of being washed upward.

Such a state is said to produce a quick (meaning alive) condition. From Eq. (18.44) it is evident that a

quick condition is incipient if

(18.45)

Substituting typical values of G = 2.65 (quartz sand) and e = 0.65 (for sand, 0.57 £ e £ 0.95), we see

that as an average value the critical gradient can be taken as

(18.46)

When information is lacking as to the speciﬁc gravity and void ratio of the material, the critical gradient

is generally taken as unity.

At the critical gradient, there is no interparticle contact (R = 0); the medium possesses no intrinsic

strength, and will exhibit the properties of liquid of unit weight

(18.47)

Substituting the above values for G, e, and

= 124.8 lb/ft

. Hence, contrary to popular belief,

a person caught in quicksand would not be sucked down but would ﬁnd it almost impossible to avoid

ﬂoating.

FIGURE 18.23 Piping.

A′ B′

Area,

′

(

)

(

)

(

)

Screen

Water

column

Material

–=

------

G 1–

1 e+

------------==

1ª

Ge+

1 e+

------------

Ë¯

Êˆ

18-24 The Civil Engineering Handbook, Second Edition

Many hydraulic structures, founded on soils, have failed as a result of the initiation of a local quick

condition which, in a chainlike manner, led to severe internal erosion called piping. This condition occurs

when erosion starts at the exit point of a ﬂow line and progresses backward into the ﬂow region forming

pipe-shaped watercourses which may extend to appreciable depths under a structure. It is characteristic

of piping that it needs to occur only locally and that once begun it proceeds rapidly, and is often not

apparent until structural failure is imminent.

Equations (18.45) and (18.46) provide the basis for assessing the safety of hydraulic structures with

respect to failure by piping. In essence, the procedure requires the determination of the maximum

hydraulic gradient along a discharge boundary, called the exit gradient, which will yield the minimum

resultant force (R

min

) at this boundary. This can be done analytically, as will be demonstrated below, or

from ﬂow nets, after a method proposed by Harza [1935].

In the graphical method, the gradients along the discharge boundary are taken as the macrogradient

across the contiguous squares of the ﬂow net. As the gradients vary inversely with the distance between

adjacent equipotential lines, it is evident that the maximum (exit) gradient is located where the vertical

projection of this distance is a minimum, such as at the toe of the structure (point C) in Fig. 18.11. For

example, the head lost in the ﬁnal square of Fig. 18.11 is one-sixteenth of the total head loss of 16 ft, or

1 ft, and as this loss occurs in a vertical distance of approximately 4 ft, the exit gradient at point C is

approximately 0.25. Once the magnitude of the exit gradient has been found, Harza recommended that

the factor of safety be ascertained by comparing this gradient with the critical gradient of Eqs. (18.45)

and (18.46). For example, the factor of safety with respect to piping for the ﬂow conditions of Fig. 18.11

is 1.0/0.25, or 4.0. Factors of safety of 4 to 5 are generally considered reasonable for the graphical method

of analysis.

The analytical method for determining the exit gradient is based on determining the exit gradient for

the type II fragment, at point E in Fig. 18.14. The required value can be obtained directly from Fig. 18.24

with h

being the head loss in the fragment.

Example 18.9

Find the exit gradient for the section shown in Fig. 18.17(a).

Solution. From the results of Example 18.4, the head loss in fragment 3 is h

= 5.9 ft. With s/T = 1/3,

from Fig. 18.24 we have I

s/h

= 0.62; hence,

FIGURE 18.24 Exit gradient.

0.2 0.4 0.6 0.8 1.0

1.0

0.8

0.6

0.4

0.2

Groundwater and Seepage 18-25

To account for the deviations and uncertainties in nature, Khosla et al. [1954] recommend that the

following factors of safety be applied as critical values of exit gradients: gravel, 4 to 5; coarse sand, 5 to

6; and ﬁne sand, 6 to 7.

The use of reverse ﬁlters on the downstream surface or where required serves to prevent erosion and

decrease the probability of piping failures. In this regard, the Earth Manual of the U.S. Bureau of

Reclamation, Washington [1974] is particularly recommended.

Deﬁning Terms

Coefﬁcient of permeability — Coefﬁcient of proportionality between Darcy’s velocity and the hydrau-

lic gradient.

Flow net — Trial-and-error graphical procedure for solving seepage problems.

Hydraulic gradient — Space rate of energy dissipation.

Method of fragments — Approximate analytical method for solving seepage problems.

Piping — Development of a “pipe” within soil by virtue of internal erosion.

Quick condition — Condition when soil “liquiﬁes.”

References

Bear, J. 1972. Dynamics of Fluids in Porous Media. American Elsevier, New York.

Bear, J., Zaslavsky, D., and Irmay, S. 1966. Physical Principles of Water Percolation and Seepage. Te c hnion,

Israel.

Brahma, S. P., and Harr, M. E. 1962. Transient development of the free surface in a homogeneous earth

dam. Geotechnique. 12(4).

Casagrande, A. 1940. Seepage Through Dams. In Contributions to Soil Mechanics 1925–1940. Boston

Society of Civil Engineering, Boston.

Cedergren, H. R. 1967. Seepage, Drainage and Flow Nets. John Wiley & Sons, New York.

Dachler, R. 1934. Ueber den Strömungsvorgang bei Hanquellen. Die Wasserwirtschaft, no. 5.

Darcy, H. 1856. Les Fontaines publique de la ville de Dijon. Paris.

Domenico, P. A., and Schwartz, F. W. 1990. Physical and Chemical Hydrogeology. John Wiley & Sons, New

Yo r k .

Dvinoff, A. H., and Harr, M. E. 1971. Phreatic surface location after drawdown. J. Soil Mech. Found.

ASCE. January.

Earth Manual. 1974. Water Resources Technology Publication, 2nd ed. U.S. Department of Interior, Bureau

of Reclamation, Washington, D.C.

Forchheimer, P. 1930. Hydraulik. Te ubner Verlagsgesellschaft, Stuttgart.

Freeze, R. A., and Cherry, J. A. 1979. Groundwater. Prentice-Hall, Englewood Cliffs, NJ.

Harr, M. E. 1962. Groundwater and Seepage. McGraw-Hill, New York.

Harr, M. E., and Lewis, K. H. 1965. Seepage around cutoff walls. RILEM, Bulletin 29, December.

Harr, M. E. 1977. Mechanics of Particulate Media: A Probabilistic Approach. McGraw-Hill, New York.

Harza, L. F. 1935. Uplift and seepage under dams on sand. Trans. ASCE, vol. 100.

Khosla, R. B. A. N., Bose, N. K., and Taylor, E. M. 1954. Design of Weirs on Permeable Foundations. Central

Board of Irrigation, New Delhi, India.

Lambe, H. 1945. Hydrodynamics. Dover, New York.

Muskat, M. 1937. The Flow of Homogeneous Fluids through Porous Media. McGraw-Hill, New York.

Reprinted by J. W. Edwards, Ann Arbor, 1946.

Pavlovsky, N. N. 1956. Collected Works. Akad. Nauk USSR, Leningrad.

Polubarinova-Kochina, P. Ya. 1941. Concerning seepage in heterogeneous (two-layered) media. Inzhenernii

Sbornik. 1(2).

0.62 5.9¥

----------------------- 0.41 or FS

0.41

---------- 2.44== ==

18-26 The Civil Engineering Handbook, Second Edition

Polubarinova-Kochina, P. Ya. 1962. Theory of the Motion of Ground Water. Tr anslated by J. M. R. De Wiest.

Princeton University Press, Princeton, New Jersey. (Original work published 1952.)

Prasil, F. 1913. Technische Hydrodynamik. Springer-Verlag, Berlin.

Scheidegger, A. E. 1957. The Physics of Flow through Porous Media. Macmillan, New York.

Te rzaghi, K. 1922. Der Grundbruch and Stauwerken und seine Verhütung. Die Wasserkraft, p. 445.

Further Information

This chapter dealt with the conventional analysis of groundwater and seepage problems. Beginning in

the mid-1970s another facet was added, motivated by federal environmental laws. These were concerned

with transport processes, where chemical masses, generally toxic, are moved within the groundwater

regime. This aspect of groundwater analysis gained increased emphasis with the passage of the Compre-

hensive Environmental Response Act of 1980, the superfund. Geotechnical engineers’ involvement in

these problems is likely to overshadow conventional problems. Several of the above references provide

speciﬁc sources of information in this regard (cf. Domenico and Schwartz, Freeze and Cherry). The

following are of additional interest:

Bear, J., and Verruijt, A. 1987. Modeling Groundwater Flow and Pollution. Dordrecht, Boston: D. Reidel

Pub. Co., Norwell, MA.

Mitchell, J. K. 1993. Fundamentals of Soil Behavior. John Wiley & Sons, New York.

Nyer, E. K. 1992. Groundwater Treatment Technology. Van Nostrand Reinhold, New York.

Sara, M. N. 1994. Standard Handbook for Solid and Hazardous Waste Facility Assessments. Lewis Publishers,

Boca Raton, FL.

Consolidation and

Settlement Analysis

19.1 Components of Total Settlement

19.2 Immediate Settlement

19.3 Consolidation Settlement

Total Consolidation Settlement

•

Rate of Consolidation

Settlement

19.4 Secondary Compression Settlement

In order to evaluate the suitability of a foundation or earth structure, it is necessary to design against

both bearing capacity failure and excessive settlement. For foundations on cohesive soils, the principal

design criterion is typically the latter

the control of expected settlements within the limits considered

tolerable for the structure. As a result, once allowable foundation displacements have been established,

the estimate of total settlement over the service life of the structure is a major factor in the choice of

foundation design.

The purpose of this chapter is to present the fundamental concepts regarding settlement analysis for

saturated, inorganic, cohesive soils. In addition, the recommended procedure for estimation of founda-

tion settlements is described. Much of this chapter is based on Leonards [1968], Perloff [1975], and Holtz

[1991]. Readers may refer to these works for additional information on consolidation and settlement

analysis.

19.1 Components of Total Settlement

During construction, surface loads from foundations or earth structures are transmitted to the underlying

soil proﬁle. As a result, stresses increase within the soil mass and the structure undergoes a time-dependent

vertical settlement. In general, this time–settlement curve can be represented conceptually as shown in

Fig. 19.1. The

total settlement

, is calculated as the sum of the following three components:

(19.1)

where

is the

immediate settlement

is the

consolidation settlement

, and

is the

secondary

compression settlement

Immediate settlement is time-independent and results from shear strains that occur at constant volume

as the load is applied to the soil. Although this settlement component is not elastic, it is generally calculated

using elastic theory for cohesive soils such as clays. Both consolidation and secondary compression

settlement components are time-dependent and result from a reduction of void ratio and concurrent

expulsion of water from the voids of the soil skeleton. For consolidation settlement, the rate of void ratio

++=

Patrick J. Fox

Purdue University

-2

The Civil Engineering Handbook, Second Edition

reduction is controlled by the rate at which water can escape from the soil. Therefore, during consoli-

dation, pore water pressure exceeds the steady state condition throughout the depth of the layer. Over

time, the rate of consolidation settlement continuously decreases as effective stresses increase to approach

their equilibrium values. Once the consolidation process is completed at time

, settlement continues

in the form of secondary compression. During secondary compression, the rate of void ratio reduction

is controlled by the rate of compression of the soil skeleton itself. As such, it is essentially a creep

phenomenon that occurs at constant vertical effective stress and without sensible excess pressure in the

pore water.

The time–settlement relationship shown in Fig. 19.1 is conceptually valid for all soil types. However,

large differences exist in the magnitude of the components and the rate at which they occur for different

soils. For granular soils, such as sands, the hydraulic conductivity is sufﬁciently large that consolidation

occurs nearly instantaneously with the applied load. In addition, although granular soils do exhibit creep

effects, secondary compression is generally insigniﬁcant. For cohesive soils, such as clays, hydraulic

conductivity is very small and the consolidation of a thick deposit may require years or even decades to

complete. Secondary compression can be substantial for cohesive soils. Different from both sands and

clays, peats and organic soils generally undergo rapid consolidation and extensive, long-term secondary

compression.

The ﬁrst step in a settlement analysis is a careful study of the changes in applied loads and the selection

of appropriate fractions of live load pertinent to each of the three total settlement components. Often,

insufﬁcient attention is given to this aspect of the problem. In general, immediate settlements should be

computed using 100% of both live and dead loads of the structure. Consolidation and secondary com-

pression settlements should be calculated using 100% of the dead load and permanent live load, but only

a reasonable fraction of the transient live load. The proper estimate of this fraction should be made in

consultation with the structural engineer on the project [Leonards, 1992].

19.2 Immediate Settlement

For saturated or nearly saturated cohesive soils, a linear elastic model is generally used for the calculation

of immediate settlement. Although clays do not behave as linear elastic materials, the rationale for the

use of elastic theory has been the availability of solutions for a wide variety of boundary conditions

representative of foundation engineering problems. In general, the elastic approximation performs rea-

sonably well in the case of saturated clays under monotonic loading conditions not approaching failure.

In addition, for these same conditions, the elastic parameters can generally be assumed as approximately

constant throughout an otherwise homogeneous soil mass [Perloff, 1975].

For cohesionless soils, in which the equivalent elastic modulus depends markedly on conﬁnement, the

use of linear elastic theory coupled with the assumption of material homogeneity is inappropriate.

FIGURE 19.1

Time–settlement curve showing total settlement components.

Settlement, S

Time, t

Consolidation and Settlement Analysis

-3

Immediate settlement on granular soils is most often estimated using the procedure of Schmertmann

[1970]. Holtz [1991] reviews this and other available methods in detail.

For those cases in which a linear elastic model is acceptable, solutions for stress distribution and surface

deﬂection under a variety of ﬂexible and rigid surface loading conﬁgurations can be found in Harr [1966],

Perloff [1975], Poulos and Davis [1974], and Holtz [1991]. One particularly useful relationship is provided

herein for the immediate settlement of a circular or rectangular footing at the surface of a deep isotropic

stratum. In this case, the immediate vertical displacement is given by

(19.2)

where

= immediate settlement of a point on the surface

= shape and rigidity factor

= equivalent uniform stress on the footing (total load/footing area)

= characteristic dimension of the footing

= Poisson’s ratio

= undrained elastic modulus (Young’s modulus)

The coefﬁcient

is a function of the shape and rigidity of the loaded area and the point on the footing

for which the immediate settlement estimate is desired. Thus, Eq. (19.2) can be used for both rigid and

ﬂexible footings, with the appropriate values of

given in Table 19.1. The characteristic footing dimen-

sion,

, is taken by convention as the diameter of a circular footing or the short side length of a rectangular

footing. For saturated cohesive soils, constant volume strain is usually assumed and Poisson’s ratio,

, is

set equal to 0.5. For soils that are nearly saturated,

will be less than 0.5. However, using

= 0.5 is

generally acceptable since the magnitude of computed immediate settlement is not especially sensitive

to small changes in

Reliable evaluation of the remaining soil parameter, the undrained elastic modulus,

, is critical for

a good estimate of immediate settlement. In general,

is the slope of the undrained stress–strain curve

for a stress path representative of the ﬁeld condition. Figure 19.2 illustrates the measurement of

from

a plot of principal stress difference,

, versus axial strain,

, as typically obtained from an undrained

triaxial test. Principal stress difference is deﬁned as

–

, where

and

are the major and minor

TA BLE 19.1

Values of

Shape and Rigidity Center Corner Edge/Middle of Long Side Average

Circle (ﬂexible) 1.00 0.64 0.85

Circle (rigid) 0.79 0.79 0.79

Square (ﬂexible) 1.12 0.56 0.76 0.95

Square (rigid) 0.82 0.82 0.82 0.82

Rectangle (ﬂexible):

length/width

2 1.53 0.76 1.12 1.30

5 2.10 1.05 1.68 1.82

10 2.56 1.28 2.10 2.24

Rectangle (rigid):

length/width

2 1.12 1.12 1.12 1.12

5 1.6 1.6 1.6 1.6

10 2.0 2.0 2.0 2.0

Source:

Holtz, R. D. 1991. Stress distribution and settlement of shallow foundations.

Foundation Engineering Handbook,

2nd ed., ed. H.-Y. Fang. Van Nostrand Reinhold,

New York.

1 v

–

-------------

Ë¯

Êˆ