THE ERNEST ERNEST KEMPTON K EMPTON ADAMS ADAMS FbND FOR PHYSICAL RESEARCH OF COLUMBIA UNM3R!XI’Y REPRINT SERIES
GENERAL THEORY
OF THREE-DIMENSION THREE-DIMENSIONAL AL
CONSOLIDATION
BY
MAURICE A. BIOT
R e p r i n t e d fr fr o m J O U R N A O F A P P L I E D
E Y S I C S , V o l. l. 1 2 , No. , pp. 155-164,
February, 1941
Reprinted
from
O F A P L I E D P H Y S I CS , Vol.
JOURNAL
Copyright
1941 by th e American
12, No. 2, 155-164, Febru ar y, Institu te of Ph ysic
1941
Pr inted n U. S. A
Genera l Theory of Thr ee-Dimensiona l MAURICE Columbia
Univcrsdy,
(Received The set tlement mechan ism
A. New
October
BIOT
York, New York
2.5, 1940)
of soils un der load is caused by a phen omenon
is known to be in ma ny cases identical
Consolidat ion*
called consolidation,
whose
with the process of squee zing water out of
an elasti c porous medium. The mathematical physical conseque nces of this viewpoint are establ ishe d in the present paper. The number of physical constants necessary to determine the properties of the soil is derived al ong with the general equation s for the prediction of settlements and stresses in three-dimensi onal problems. imple applications are treated as examples. The operational calculus is shown to be a powerful method of solution of consolidation problems. INTRODUCTION
T is well known to engineering pra ctice th at a soil under load does not assume an instantan eous defle tion under t hat load, but settles gradually at a variable rat e. Such settlement is very a pparen t in lays an d san ds satu rat ed with water. The settlement is aused by a gradua ada pta tion of th e soil to th e load var iation. This pr ocess is kn own as soil cons olidat ion. A simp le mechan ism to explain t his phen omenon was first proposed by K. Terzaghi.’ He ass um es th at th grain s or par ticles onstitu ting th e soil ar e more or less boun d together by ert ain mole ular forces and onstitu te a porous ma teria l with elastic propert ies. The voids of th e elastic skeleton are filled with water. A good example of such a model is a rubber sponge satu rat ed with water. A load applied to this system will produce a gradua l settlement, depending n the rat e at which t he water is being squeezed out of th voids. Terzaghi applied t hese concepts to th an alysis of th e sett lement of a olum n of soi un der a nsta nt load and prevent ed rom lateral expans ion. The r emar kable success of th is th eory in predicting th e sett lement for ma ny t ypes o soils ha s been one of th e str ongest incent ives in th e creat ion of a science of soil mecha nics. Terzaghi’s tr eat men t, however, is rest ricted to th e one-dimensional problem of a olum n un der a const an t load. From th e viewpoint of ma th ema tical physics two genera lizations of th is ar Publi cation assisted by the Ernest Kempton Adams Fund for Physical Research of Columbi a University. 1 K. Terzaghi, Erdbaumechanik au Boden~hysikabischer Grundlage CLeipzig . Deuticke, 1925); “Principle of soil mechanics,” Eng. News Record (1925), a series of articles. VOLUME
12, FEBRUARY,
1941
possible the extensi n to the thr ee-dimensional case, an d th e esta blishm ent of equat ions valid for any arbitrary load variable with time. The theory was first presented by the author in rather abst ra ct form in a previous publicat ion.2 The presen t pa per gives a more r igorous an d complete treatment of the theory which leads to results more general t han those obtained in the previ us paper. The following bas ic pr opert ies of th e soil ar assumed (1) isotr opy of th e ma ter ial, (2) reversibility of str ess-stra in relat ions u nder final equilibrium cond itions, (3) linear ity of st res sstr ain relat ions, (4) sma ll str ains , (5) th e wat er onta ined in th e pores is incompr essible, (6) th wat er may cont ain air bubbles, (7) th e wat er flows th rough th e porous skeleton according to Darcy’s law. Of th ese basic assu mpt ions (2) an d (3) ar most subject to criticism. However, we should keep in mind tha t t hey also nstitu te the basis Terzaghi’s th eory, which ha s been foun d quit sat isfactory for th e pra ctical r equiremen ts of engineering. In fact it an be imagined tha t t he grains composing th e soil ar e held together in certain pattern by surface tension forces and tend to ass um e a onfigura tion of minimu m potential energy. Th is would especially be tr ue for th colloidal pa rt icles cons tit ut ing clay. It seem reasonable to assume that for small strains, when the grain pat tern is not too much disturbed, the as su mp tion of rever sibility will be ap plicable. The assu mpt ion of isotr opy is not essential and M. A. Biot, “Le probkme de la Consolidation Mat&-es argileuses sous une charge,” Ann. Sot. Bruxelles B55, 110-113 (1935).
des S ci.
15.5
an isotr opy can easily be intr oduced as a refinemen t. Another refinement which might be of pra ctical importa nce is th e influen e, u pon th str ess distribution and the settlement, f the state of initial stress in the soil before application of the load. It was shown by the present author3 th at th is influence is great er for m at erials of lo elastic modulus. Both refinement s will be left out of th e presen t t heory in order t o avoid un due heaviness of presentation. The first an d second sections deal mainly with th e ma th emat ical ormulat ion of th e physical propert ies of th e soil an d th e nu mber of onsta nt necessary to describe t hese propert ies. Th nu mber of th ese onsta nt s including Dar y’ per mea bility coefficient is foun d equ al t o five in th e most general ase. Section 3 gives a discussion of th e physical int erpr etat ion of th ese various nst ant s. In Se tions 4 and 5 are established the un dament al equations r the consolidat ion an d an applicat ion is ma de to th one-dimensional problem corresponding to sta nda rd soil test . Section 6 gives th e simplified th eory for the case most import an t in pra tice of a soil ompletely sat ur at ed with wat er. The equat ions for th is case oincide with th ose of th previous publicat ion.2 In the last section is shown how the mat hemat ical tool known as the operational calculus can be applied most conveniently for the calculat ion of th e sett lement without ha ving to alculate an y str ess or wat er press ur e distribut ion inside th e soil. This m eth od of at ta ck const itut es a ma jor simplificat ion an proves to be of high value in th e solution of th more omplex two- an d th ree-dimens iona l problems. In the present paper applications are rest ricted to one-dimensional examples. A series of ap plicat ions to pr actical cases of two-dimensional consolidation will be th e object of su bsequent papers. 1. SOIL STRESSES Consider
a sma ll
ubic element
of th e con-
solidating soil, its sides being par allel with t he co rdin at e axes. This element is ta ken to be large enough compa red to th e size of th e pores so th at it may be treated as homogeneous, and at the *M. A. Biot, “Nonlinear th eory of elast icity an d th linearized ase for a body under initial stress.” 15
sam e time small enough, ompar ed to the scale of th e ma croscopic phen omena in which we ar interested, so tha t it may be nsidered as infinitesimal in the mat hemat ical treat ment. The average str ess ondition in th e soil is th en repr esent ed by forces distr ibuted un iformly on the faces of this cubic element. The corresponding str ess mponents are denoted by
They mu st satisfy the well-known cond itions of a st res s field.
equilibrium
(1.2)
Ph ysically we ma y th ink of th ese str esses as comp osed of two pa rt ne wh ich is au sed by th e hydrosta tic press ur e of th e wat er filling th pores, the other caused by the average stress in the skeleton. In t his sense the st resses in the soi are said to be carried partly by the water and par tly by th e solid constitu ent 2. STRAIN RE LATED TO STRESS AND WATER PRESSURE We now all ur at tent ion to th e str ain in th soil. Den oting by u, V, zet he comp onen ts of th displacement f the soil an d assuming the stra in to be small, the values of the strain components
ez=-
ax’ av
e,=--,
ay
aw ez=-
a~
av
ay
a2
r.T -+-_,
a24 aw -YII=-+-, a2
ax
av
a24
ax
ay
-yz=-+-.
(2.1)
In order to describe completely th e macroscopi ondition of th e soil we mu st consider an addiOURNAL OF APPLIED PHYSICS
tional variable giving the amount of water in the pores. We therefore denote by 0 the increment of water volume per unit volume of soil and call this quantity the zJariation in water content. The incremen t’of wa ter pressure will be denoted by u. Let us consider a cubic element of soil. The water pressure in the pores may be considered as uniform throughout, provided either the size of the element is small enough or, if this is not the case, provided the changes occur at sufficiently slow rate to render the pressure differences negligible. It is clear that if we assume the changes in the soil to occur by reversible processes the macroscopic condition of the soil must be a definite function of the stresses and the water pressure i.e., the seven variables e,
ey e,
the shear modulus and Poisson’s solid skeleton. There are only constants because of the relation G=-.
ratio for the two distinct
(2.3)
v+v1
Suppose now that the effect of the water pressure is introduced. First it cannot produce any shearing strain by reason of the assumed isotropy of the soil; second for the same reason its effect must be the same on all three components of strain e, e, e,. Hence taking into account the influence of u relations (2.2) become e.=~-~(uv+u,,+i
3H
Yx YU YE.
must be definite functions of the variables: uz
Qu
UE
7,
i-g
7,
CT.
Furthermore if we assume the strains and the variations in water content to be small quantities, the relation between these two sets of variables may be taken as linear in first approximation. We first consider these functional relations for the particular case where u= 0. The six componentsof strain are then functions only of the six stress components cI uy uZ rZ ry 7,. Assuming the soil to have isotropic properties these relations must reduce to the well-known expressions of Hooke’s law for an isotropic elastic body in the theory of eIasticity have
~z
i-z/G,
Y?/ =
r,lG,
yz= TJG,
where H is an additional physical constant. These relations express the six strain components of the soil as a function of the stresses in the soil and the pressure of the water in the pores. We still have to consider the dependence of the increment of water content 0 on these same variables. The most general relation is 6 = aiuo+a2uy+a3ur+a4r2 +a6ry+a67,+a7u.
e,=~-~(uz+uu),
~z
(2.2)
rz/G,
mu=7,/G,
Now because of the isotropy of the material change in sign of r2 ry rZ cannot affect the water content, therefore a4=aL=a6=0 and the effect of the shear stress components on 0 vanishes. Furthermore all three directions x, y, z must have equivalent properties al = a2 = a3. Therefore relation (2.5) may be written in the form
s=-&-(u~+u,+uz)+~, 2.6)
yz= rz/G.
In these relations the constants E, G, v may be interpreted, respectively, as Young’s modulus, VOLUME
12, EBRUARY, 1941
(2.5)
where
and
are two physical constants. 157
Relations (2.4) and (2.6) contain five distinct physical constants. We are now going to prove that this number may be reduced to four; in fact that H=Hl if we introduce the assumption of the existence of a potential energy of the soil This assumption means that if the changes occur at an infinitely slow rate, the work done to bring the soil from the initial condition to its final state of strain and water content, is independent of the way by which the final state is reached and is definite function of the six strain components and the water content. This assumption follows quite naturally from that of reversibility introduced above, since the absence of a potential energy would then imply that an indefinite amount of energy could be drawn out of the soil by loading and unloading along a closed cycle. The potential energy of the soil per unit volume is U=~(uoez+uy~y+u~~o.+7z~z +ry~y+r.yz+~@.
(2.7)
In order to prove that H=Hl let us consider particular condition of stress such that
sidered as a function of the two variables Now we must have au
u 1,
Hence
E, B.
u.
aal au de
or
de
1 Ha==
We have thus proved that H=Hl write
and we may (2.10)
Relations (2.4) and (2.10) are the fundamental relations describing completely in first approximation the properties of the soil, for strain and water content, under equilibrium conditions. They contain four distinct physical constants G, v, H and R. For further use it is convenient to express the stresses as functions of the strain and the water pressure u. Solving Eq. (2.4) with respect to the stresses we find
Then the potential energy becomes lJ=$(ure+u0)
with
E=e,+ey+er
u,=ZG( g+fi)
and Eqs. (2.4) and (2.6) t=
3(1-2~)
u
uI+~,
e=u~/H~+ulR.
(2.8)
7,
e #J1=----RA
HA’
A= The potential 158
3(1-2~) ER
--.
(2.9)
1 HH1
energy in this case may be con-
(2.11)
GYP,
it, = Grv, with
--E +3(1 -2V)e u=&A EA
-cyu,
ez+fi)
u,=2G(
The quantity t represents the volume increase of the soil per unit initial volume. Solving for ul and u
--au,
TZ=GYZ
2(I+v) CY= 3(1--2~)
G 5’
In the same way we may express the variation in water content as e=ffe+u/Q, where
1 1 -=---. QR JOURNAL
(2.12)
a!
OF APPLIED
PHYSICS
3. PHYSICAL INTERPRETATION OF THE SOIL CONSTANTS
, The constants and v have t he same meaning as Young’s modulus the shear modulus an d th e Poisson ra tio in th e theory of elasticity provided time h as been allowed for th e excess
water to squeeze out. These quantities may be consider ed as th e avera ge elastic const an ts of th sohd skeleton. Ther e ar e nly t wo distinct such const an ts since th ey mu st satisfy relat ion (2.3) Assum e, for example, th at a column of soil supports a n axial load po= -ur while allowed to expand freely lat era lly. If th e load h as been applied long enough so th at a fina l sta te of sett lement is rea ched, i.e., all th e excess water ha been squeezed ut and u = 0 then the axial str ain is, ccordin g to (2.4), ez=
--
PO
VP0
-vez.
(3.2)
The coef icient v mea sur es t he rat io of the lat eral bulging to the vertical strain under final equilibrium conditions. To interpret the nst an ts consider a sam ple of soil enclosed in a thin ru bber bag so th at th e st ress es applied to the soil be zero. Let us drain the water rom this soil thr ugh a thin tube passing through the walls of the bag. If a negative pressure u is applied to th e tube a certain amount of water will be sucked out. This am ount is given by (2.10)
fy=
2(l+v)
Cl!=
3(1-22)
1 1 -_=---_
(3.3)
(3.5)
CY (3.6)
QR
is a measu re f the amount f water which an be forced into the soil under pressure while the volume of th e soil is kept const an t. It is quite obvious th at th e consta nt s (Y an d Q will be o significan ce for a soil not completely sat ur at ed with water a nd c nta ining air bubbles. In tha ase the c nst ant s a and Q an take values dependin g on th e degree of sat ur at ion of th e soil. The standard soil test suggests the derivation of addit iona l cons ta nt s. A column of soil su pport a load p. = - ul an d is confined latera lly in a rigi sheath so that no lateral expansion can occur. The wat er is allowed to escape for insta nce by applying th e load t hr ough a porous slab. When all the excess water has been squeezed out the axial str ain is given by r elations (2.11) in which we put u=O. We write er= -
-“.
E
According to (2.12) it mea sur es th e ra tio of th wat er volume squeezed out to th e volume ha nge of th e soil if th e latter is compr essed while allowing th e wat er t o .escape (a=O). The coefficient l/Q defined as
(3.1)
and the lateral strain ez=ey=-=
sur e. The t wo elastic c nst an ts and th e nst ant are the ur distinct onstan ts whic un der ur assu mption define mpletely the physical proportions of an isotr opic soil in th equilibrium conditions. Other constants have been derived from these four . F or in st an ce Q! s a coefficient defined as
pea.
(3.7)
The coefficient l-2v
The corr esponding given by (2.4)
volume
change of th e soil is
(3.4) The coefficient l/H is a measure of the compressibility of th e soil for a cha nge in water pressure, while measu res th e han ge in /R water ntent r a given han ge in water presVOLUME
12, EBRUARY, 1941
l7,=
2G(l-
v)
(3.8)
will be called th e fina l comp res sibility. If we measure the axial strain just after the load has been applied so that the water has not had time to ilow out, we must put B=O in relat ion (2.12). We dedu ce t he value of th e wat er pressure u = - aQe&.
(3.9) 159
substituting
this value in (2.11) we write er= -parri.
(3.10)
The coefficient aa=
(3.11)
1 +a2aQ
will be called the instantaneous compressibility. The physical constants considered above refer to the properties of the soil for the state of equilibrium when the water pressure is uniform throughout. We shall see hereafter that in order to study the transient state we must add to the four distinct constants above the so-called coeficient of permeability of the soil. 4. GENERAL EQUATIONS GOVERNING CONSOLIDATION We now proceed to establish the differential equations for the transient phenomenon of consolidation, i.e., those equations governing the distribution of stress, water content, and settlement as a function of time in a soil under given loads. Substituting expression (2.11) for the stresses into the equilibrium conditions (1.2) we find G
GV2~+--l-2vax
de --
aa (Y--o, ax
ae
V,= -k;,
V,= -$,
V,= -Pt.
(4.2)
The physical constant is called the coeficient of permeability of the soil. On the other hand, if we assume the water to be incompressible the rate of water content of an element of soil must be equal to the volume of water entering per second through the surface of the element, hence
a v, de -=-----7 at ‘ ax
a v,
av
ay
a
(4.3)
Combining Eqs. (2.2) (4.2) and (4.3) we obtain
G de da ff--0. Gv%~-l----l--2vay dy GV2wS -.--L,_=O, l-2vdz
There are three equations with four unknowns u, v, w, 6. In order to have a complete system we need one more equation. This is done by introducing Darcy’s law governing the flow of water in a porous medium. We consider again an elementary cube of soil and call V, the volume of water flowing per second and unit area through the face of this cube perpendicular to the x axis. In the same way we define V, and Vl. According to Darcy’s law these three components of the rate of flow are related to the water pressure by the relations
(4.1)
da
de
1 an
at
Q at
kV2a = a~---+-
-.
(4.4)
The four differential Eqs. (4.1) and (4.4) are the basic equations satisfied by the four unknowns u, 21,w, c.
dz
v2 a 2 / a X2 + d 2 a y 2 + a 2 / a 2 2 .
5. APPLICATION TO A STANDARD SOIL TEST Let us examine the particular case of a column of soil supporting a load PO= - cz and confined laterally in a rigid sheath so that no lateral expansion can occur. It is assumed also that no water can escape laterally or through the bottom while it is free to escape at the upper surface by applying the load through a very porous slab. Take the z axis positive downward; the only component of displacement in this case will be w. Both w and the water pressure u will depend only on the coordinate z and the time t. The differential Eqs. (4.1) and (4.4) become
2w
a22
=0, a2
k a 2 u - C a 2 W ’ au a . 2 2 azat Qat’ 160
(5.1)
(5.2)
JOURNAL OF APPLIED PHYSICS
wher e a is th e final compr essibility const an t.
defined by (3.8). Th e st ress (TVhr oughout
Fr om (2.11) we ha ve
1 ---+ci a
PO = -az= and from (2.12)
th e loaded column is
(5.3)
w e=a-+--.
Note that Eq. (5.3) implies (5.1) and that
1 PW
da
adzat=%. This relation carried into (5.2) gives
with
Pa -=--, dz2
*
1 aa
(5.4)
c at
1 -=a”+-_.
(5.5)
Qk
The const an t c is alled th consolidation constant. Equat ion 5.4) shows the import ant result th at the wat er pres su re sat isfies th e well-known equat ion of hea t condu ction. This equation along with the bounda ry an d the initial condit ions leads to a complete solut ion of th e problem of consolidation. Taking th e height of th e soil column to be and z=O at the top we have th e bounda ry nditions a=0 -=0
for z=O, (5.6)
for z=h.
The irst c ndition expresses th at the pr essure of the water un der the load is zero be aus e the permeability of th e slab thr ough which t he load is app lied is as sum ed t o be large with resp ect t o th at of th soil. The second condition expresses t ha t no wat er escapes th rough th e bottom. The initial condit ion is th at th e change of wat er cont ent is zero when th e load is app lied becau se th wat er m ust escape with a finite velocity. Hen ce from (2.12) ~=cP--+--0
for t=O.
Car rying th is into (5.3) we derive th e initial value of th e wat er pr essu re az*,/(A+Ck!)
fort =0
or
a=yP 0,
(5.7)
wher e a, an d a ar e th e inst an ta neous an d final compr essibility coeffi ients defined by (3.8) and (3.11) The solution of th e differen tial equa tion (5.4) with th e boun dar y condit ions (5.6) an d th e initial ondition (5.7) ma y be writt en in th e form of a series 4 a-ai a=----~0~exp[-(~)z~t]sin~+~exp[-(~)2~t]sin~+~~~). .X ffa The sett lement
ma y be foun d from relat ion (5.3). We ha ve -=&a-ape.
VOLUME
(5.8)
12, EBRUARY,
1941
(5.9) 161
The total settlement is wo= -
Im media tely
-ak
az
=
--$(u-ui)hpo 0 (2n+1)2
aft er loadin g (t=O),
th e deflection
exp
+Uhpo. -[(Znzl’n]z t
(5.10)
is
Taking int o acc un t th at WC a,hp,,
(5.11)
which checks with th e res ult (3.10) a bove. The fina l deflection for t = 00 is wc.,
(5.12)
ahpo.
It is of inter est to find a simplified expression for th law of sett lement in th e period of time immediat ely after loadin g. To do this we first elimina te th e initia l deflection wi by considerin (5.13) This expresses t ha t par t of th e deflection which is caused by c nsolidat ion. rate of settlement. dw, 2c( -ai) p0 exp ( -[(2”i1)r-J &}
We th en consider th
(5.14)
or t =0 th is series does not c nverge; which m ean s tha t at th e first insta nt f loading th e ra te of sett lement is infinite. H ence th e curve repr esentin g th e sett lement w, as a function f time sta rt with a vertical slope an d ten ds asympt tically toward th e value (a-ud)hpl as shown in Fig. 1 (curve 1). It is bvi us th at dur ing t he initial per iod f sett lement th e height h of th e column cannot ha ve any influence on the phenomenon because the water pressure at th e depth z=h has not yet had time to cha nge. Therefore in rder to find the na tu re f the settlemen t curve in th e vi inity f t = 0 it is enough to consider the case where h = w. In this case we put n/h=
&
l/h=Al
and write (5.14) as dw z= ,,for h= 60,. The ra te
f sett lement
exp [ bec mes th e integra
dw,
00
-=2c(u-uai)po
exp (- r2E2ct)dE=
dt
The value of th e sett lement
-a2(E+$A~ )2ct]A,$
c(a-UJPO (Tct);
(5.15) .
is bta ined by int egrat io w*=
w,
-dt=2(u-u&o
(5.16)
o dt
It follows a pa ra bolic curve as a function of tim e (curve 2 in Fig. 1). 16
JOURNAL
OF APPLI ED
PHYSICS
6. SIMPLIF IED THE ORY F OR A SATURATED CLAY
For a mpletely satu rat ed lay the sta ndar test shows th at t he initial compressibility ai ma be ta ken equ al t o zero compa red to th e fina ompressibility a, an d th at th e volume change of th e soil is equal to the amoun t of wat er squeezed out. Accord ing to (2.12) an d (3.11) this implies Q=w,
cr=l.
(6.1)
This redu ces th e nu mber of physical consta nt s of the soil to the two elastic constants and the perm eability. Fr om relations (3.5) an d (3.6) we deduce
2GO+v)
H=R=
(6.2)
3(1--2~)
and from (5.5) the value of the consolidation nsta nt takes th e simple rm c = k/a.
(6.9)
The sett lement r th e stan dard test of a lumn f clay of height h under th e load ~0 is given by (5.13) by putting ai=O. w,=-ahp, 7r2
5
(2nfl)Z
X[ l-e~p [-((2fl~1)n )2&]}. Fr om (5.16) the sett lement column is
6= e.
(6.4) equations
G de GV2u+-------=Q, 1-2va~ Gvzv+---
Va = 0.
(6.10)
(6.3)
Relation (2.12) becomes
The general differential (4.4) are simplified,
Equa tions 6.5) and 6.8) are the funda menta equa tions govern ing th e cons olidat ion of a completely sa tu ra ted lay. Because of (6.4) the initial condition 8=0 becomes e=O, i.e., at th e inst an of load ing no volum e chan ge of th e soil occurs This condition intr oduced in Eq. (6.7) shows th at at the instant of loading the water pressure in the pores also sat isfies Laplace’s equat ion.
(4.1)
and
&s ax
G
ae au ---=0, i-2vay ay
G ae GV2w+------=O, 1-2v a2
(6.5)
au
w,=2apo
for an infinitely high
ct
-
(6.11)
It is easy to ima gine a mechan ical model h aving th e propel-ties implied in th ese equa tions. Consider a system made f a great num ber f small rigid particles held together by tiny helical sprin gs. This syst em will be elastically deform able an d will possess avera ge elast ic consta nt s. If we ill completely with wat er th e voids between th
a2
By adding th e derivatives with r espect t o x, y, of qs. (6.5), re sp ectively, we find Ve2= aV$,
(6.7)
wher e a is th e fina l comp res sibility given by (3.8). Fr om (6.6) an d (6.7) we der ive
Vc2=! at’
(6.8)
Hen ce th e volume change of th e soil sat isfies th equat ion of hea t conduction. VOLUME 12, FE BRUARY, 1941
FIG. 1. Settlemen t caused by onsolidation a s a functio of time. Cur ve 1 represen ts-the settlemen t of a column of height h under a load ~0. Curve 2 represents the settlement for a n infinitely high column
16
particles, we shall have a model of a completely saturated clay. Obvi usly such a system is incompr essible if no wat er is allowed to be squeezed out (this corresponds to the ndition Q= a) an d the han ge of volume is equal to the volume of wat er squeezed out (this corr esponds to th e condit io LY 1). If th e system s cont ained air bubbles th is would not be the case and we would have to consider th e genera l case where Q is finite a nd a#l. Whether th is model represent s schematically th e actua l constitu tion of soils is un ert ain. It i
Eqs. (5.1) become
ak aa --=-, a.22
tion pr oblems. We cons ider th e case of a clay column infinitely high and t ake as before th e top to be th e origin of th e vertical coordinat e z. For a completely sat ur at ed clay = 1, Q = 00 an d with th e operational nota tions, r eplacing a/at by p,
16
Cle-“‘P/“’
a
The bounda ry
Hence
condit ions
ar e for z=O
gz=-l=--, a a c
-
a=O.
,
C2=l.
The sett lement w. at th e top (z = 0) caused by the su dden a pplicat ion of a un it load is
The calculation f settlement un der a suddenly app lied load leads na tu ra lly t o th e application of operational met hods, developed by Hea viside for the analysis of transients in electric circuits. As an illust ra tion of th e power an d simplicity intr oduced by th e operat iona l calculus in th tr eat men t of consolidation problem we sha ll derive by th is procedur e th e sett lement of completely sat ur at ed clay column alrea dy calculated in the previous section. In subsequent ar ticles th e operational met hod will be us ed exten sively for th e solut ion of var ious consolida-
(7.2)
P -
a=&--
Cl=a TO
(7.1)
c,p(Plc)t
WC
model. 7. OPERATIONALCALCULUS APPLIE CONSOLIDATION
aw
A solution of these equations which vanishes at infinity is
quite possible, however, th at t he soil par ticles ar prett y mu h the same way a s the springs of th
a%
kaz,=pdz.
c w,=a
The
m eanin g
of
(t).
-
th is
symbolic
derived from th e operational
expression
is
equa tion4 (7.3)
The sett lement as a load PO s th erefore
unction
time under the
ct -
(7.4)
w*=2apo
This coincides with t he value (6.11) above. 4 . Bush, Operational
New York,
1929), p. 192.
Circuit
Analysis
(John
Wiley,
JOURNAL OF APPLIED PHYSICS
