The evaluation of neutral pressures in soil mechanics problems is a fundamental step to evaluate deformations in soils. In this paper, we present some results obtained by using the boundary element method for plane problems, describing the undrained situation as well as the consolidation problem

Alarcón Álvarez, Enrique

García-Suárez, C.

Reverter, A.

English

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EFFECTIVE AND NEUTRAL STRESSES IN SOILS USING 
BOUNDARY ELEMENT METHODS 
E. Alarcdn, C. Garcia—Su&rez, A. Reverter 
Escuela T£cnica Superior de Ingenieros Industriales 
Polytechni cal Uni versi ty of Madri d. Spai n 
The evaluation of neutral pressures in soil mecha 
nics problems is a fundamental step to evaluate de 
formations in soils. In this paper we present some 
some results obtained by using the boundary element 
method for plane problems, describing the undrained 
situation as well as the consolidation problem. 
UNDRAINED CASE 
Since long ago (Scott,1965), it has been.recognized 
that, for loads normal to the boundary, the undrai 
ned case can be reduced to a potential problem. 
This has produced several analytical solutions, 
(P.N.Sundaram, 1980;J.Bielak, 1982.),but only re 
cently (Alarcdn et al.1983),it has been clear where 
Are the limitations of that approach. 
In the hypothesis of working in a 1inear media, 
the Beltrami equations for zero body forces produce 
y*1=0 or y2e=0 ... (1) 
where I is the first stress invariant and 6 the vo 
lumetric deformation.Taking into account the princi 
pie of effective stresses it is possible to write 
3 Y 2 U + V 2 I ? = 0 ...(2) 
or 
7*u=-( 72l>)/3 ...(3) 
where u is the pore pressure and I* is the first in 
variant of effective stresses. 
For continuty it is necessary now to stablish the 
compatibility of deformations between the soil ske 
leton and the water filling completely the voids in 
the case of a fully saturated soi1.The volumetric 
soil skeleton change is 
&V=V(I*/K*) ...(4) 
where V is the volume under study and K'the effecti 
ve bulk modulus of the soil skeleton. 
In the water case 
AV =(nV/K )u ... (5) 
w w 
where n is the porosity and K the bulk modulus of 
w 
the water.As 
I'=I - 3 u . (6) 
equating eqs.(4)&(5),and inserting eq.(6) it is po 
ssible to obtain the following expression for u: 
I 
u 
W 
(7) 
We have now in Equation (7) the fundamental relati 
on we were looking for.It expresses the neutral pre 
ssure as a function of soil properties as well as 
the total pressures.Inserting eq.(1) into eq.(7) it 
is possible to state the proposed problem as the pa 
ir of the following equations: 
y2u=0 in £1 
u=eq. (7) in Sit 
. . . (8) 
where Si is the domain and Sii its boundary. 
In the special case in which it is possible to assu 
me K =« , that is, full incompressibi 1 i ty 
u=I/3 in Sit ... (9) 
or,when there Ar& only normal pressures acting on 
the boundary 
u=p in £« .. (10) 
which is Scott's equation. 
From what has been said we see that the Laplace's 
equation approach is only useful in the fully incom 
pressible case,or when it is possible to define the 
first total stress invariant in the boundary.In the 
case it is better to work directly with eq 
(7) after having solved an elasticity problem.lt is 
possi ble then, to use the enormous amount o-f analyti 
cal experience stored in books like that from 
Poulos(1974). 
As an example we show in fig.l the total effective 
and neutral pressures obtai ned,by usi ng known analy 
tical expressions in the classical elasticity theo 
ry,under the corner of a shallow foundation when 
there is a uniform distribution of tangential stre 
sses acting at the interface area.Fig.la.represents 
an aspect ratio of 0.5,while fig.l.b is for a squa 
re foundation. 
On the contrary,if as is usually the case,one assu 
mes full incompressibi1ity and forces acting ortho 
gonally to the boundary,the pore pressure distribu 
tion can be obtained throuhg the solution of a po 
tential problem .In this case the Boundary Element 
Method is the cheapest solution.In what follows we 
present some results obtained for an embedded strip 
foundation.The computer program was prepared for 
an IBM-PC,using several features to improve the efe 
ctiveness and speed of the BEM procedure as descri 
bed elsewhere(G£*-SuArez et al 1983).Fig.2a. shows 
the discretization used at the boundary,as well as 
the cell's mesh utilized in the domain to interpola 
te the isolines.Those are shown in fig.2b. for a 
regular spacing of neutral pressures. 
In order to produce results with practical applica 
tions we ran several cases which have been summari 
zed in figure 3.There,the evolution of the neutral 
pressures under the center of the footing is collee 
ted versus the depth ratio z/H ,where H refers to 
an impervious boundary at the bottom of a permeable 
stratum.Fig. 3a.represents the influence of differ 
ent degrees of embeddment,while Fig.3b. shows the 
dependence on the stratum's depth. 
CONSOLIDATION PROBLEMS 
When there is the possibility of continuos draina 
ge i n the soi1 the prob1em i s cal1ed Consoli dati on 
The excess pore-pressure is dissipated slowly and 
a continuous transfer of load is produced between 
the fluid and the soil skeleton.The phenomenon can 
be described as a coupled problem,in which the neu 
tral pressures are mainly governed by a diffussion 
equation while the soi1—skeleton stresses can be 
described by an elasticity equation.The problem is 
a classical one in soil mechanics,and it is well do 
cumented el sewhere.(See Verruijt 1977, for a fini 
te element approach of it). 
Two approaches were presented to that problem,when 
\*6i m. ta. 
+% *•<». i. k. 
4>=o 
=o 
V = o 
s =o 
HC, a e £- boundary^ 
conditions 
FIG, £ b . EquApoleritiaL (u- ci*h| mcs 
i 
+ t 4 + 
o.Z 03 o« 
n<*. 3 a 
05 0.6 
fr 
og 
<zuir<5l vresmrzs r JifjcmewT £dftrx& wWHk 5 
MS. 3 b - N < ^ 
•to restores -J^r d<'{{eir«:n~f 
0? 
laser's dcprti5 
solved by boundary elements. The -first oneiOnishi e 
al. ) works di recti y with a -fundamental solution that 
includes the time, whi le the second (G*Su£rez et al . ) 
works with the -fundamental solution to a Helmoltz— 
type equation produced a-fter accepting a finitediff 
erenc& aproach in the time domain.The coupling term 
equations is due to the first invariant of 
stresses,that is related to the neutral pressure 
value.After the classical transformations in a dir 
ect B.E.fi. the elasticity equation can be written 
a 
c 
- u + P cr n U - P cr* n u +P 0 U , 
lisii i j j j i j£ft i j j i Jfl j i 
+P X U =0(11) 
h i Ji 
The effective stresses can be obtained by the input 
of eq.(11) into the Lam£?s relationships,getting 
2G SI** SI1 SI* 
s + G ( + )(12) 
i j J£ f i k i j JSR k j i 1 V i j SXft Sxi &<<i 
k 
I = P 0 U , 
k i i . ( 1 3 ) 
The derivatives of eq(13) have to be done carefully 
k 
SI SU 
=p0 - 0 lini pU , r, 
Sx J Sx €~*Q J ki i k ...(14) 
m (fl-B€) m tf(ft-B€) 
where the first integral in the r.h.s. refers to 
the domain except a ball of radius € around the po 
le where the fundamental solution is being applied 
and the second integral is extended over the bounda 
ry.It is important to notice that,in principle,we 
do not need the cr-jvalue, but the octaedral one. It is 
possible then to show that the first integral dissa 
pears (Gil Suarez, 1982) , and then it is only neccesa 
ry to reduce the second integration to 
I = -(0.£ 72-29) 
2 ij 
..(15) 
And 
cr' =- 0/2(l-V) 
oct 
(16) 
Needless to say that the most important part in the 
computations Are the volume integrals.We have tried 
several possibilities that Are described in other 
part of this volume.(G^SuArez et al«).In fig 5 we 
have collected some results for a classical consoli 
INITIAL 
F\A/AL 
SETTLEMENT 
< » > 
t = o . £ 4 x l O 6 sac. 
CO0K€J> THEORY 
c*o 
* M * . « 
24oo 
I So© 
'4O0 
O. C4 v (O 6 
• 
• 
3<5oo 
>B 1— 
3 loo 
3'. 600 
WcotfKEfc THEORY 
— - < O O K . B . D THeo*Y 
f 5 
4 -
C3) 
(t 
*=!<%. S 
dation of a stratum 20 m.depth by 20 m. width loa 
ded in the center of the upper surface by a unifor 
mly distributed pressure 6 m. width of 100000 N/m2-
Fig.4 a. shows the comparison between the initial 
and final settlements,comparing the results of a to 
tal stress analysis with other in effective stresse 
s with an equivalent load vector such 
f 0 U , . . .(17) 
Jft ij i 
Finally, it is possible to say that,apparently,the 
coupled consolidation theory is fastest than the 
one obtained by the Terzaghi—Rendulic's assumption. 
Fig.4b. displays both results for a particular time 
while Figs.5 a & b. show the distribution of neutra 
1 pressures along the axis of symmetry in two diffe 
rent times.The same phenomenon can be observed here. 
REFERENCES 
Alarcon,E.& Marti,P.(1983)Discussion to Bielak?s 
note.To be published.Jour.Geot.Eng.Div.ASCE. 
Bielak,J.(1982) Initial Pore Pressure from vertical 
Surface Loads.Jour«Geot.Di v.ASCE. 
GaSu3rez,C.(1982)The BEM in consolidation problems 
(in Spanish).Anales de Ingenieria MecAnica.Madrid 
G*Su£rez,C.&Alarc6n,E.(1982)Consolidation problems 
in Boundary Element Methods in Engineering.Ed.C.Bre 
bbia.Springer Verlag. 
G*SuAre2,C.&Alarc6n,E.(1983)A new scheme for domain 
integration in Boundary Element problemsThis volume 
Onishi,K.,Kuroki,T.&Tomofco,I.(1982)Boundary element 
methods in Biot*s 1inear consolidation.App.Math.Mod 
elling.Vol2,n.2 
Poulos,H.G.&Davis,E.H.(1974)Elastic solutions for 
rock and soi1 mechanics.J.Wiley 
Scott,R.F.(1965)Principles of soil mechanics.Addis 
son Wesley Pub. 
Sundaram,P.N.(1980)Initial Excess Pore Pressure in 
soils-Jour.Geot.Div.ASCE.Vol.100,n GT4 
Verruijt,A.(1977)Generation and dissipation of pore 
-water pressures.In Finite element in Geomechanics 
Ed.by Gudehus.J.Wiley 


Effective and neutral stresses in soils using boundary element methods

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Alarcon,E.& Marti,P.(1983)Discussion to Bielak ?s note.To be published.Jour.Geot.Eng.Div.ASCE.

Bielak,J.(1982) Initial Pore Pressure from vertical Surface Loads.Jour«Geot.Di v.ASCE.

Sundaram,P.N.(1980)Initial Excess Pore Pressure in soils-Jour.Geot.Div.ASCE.Vol.100,n GT4 Verruijt,A.(1977)Generation and dissipation of pore -water pressures.In Finite element in Geomechanics Ed.by Gudehus.J.Wiley

Effective and neutral stresses in soils using boundary element methods

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