Computers and Geotechnics 1 (1985) 207-220

F I N I ~ K I ~ N T ANALYSIS OF EXCAVATION

P.T. Brown and J.R. Booker School of Civil and Mining Engineering

University of Sydney Sydney, N.S.W., 2006

Australia

ABSTRACT

The finite element method has often been used to simulate excavation. When the soll is linearly elastic, the results of excavation should be independent of the number of stages in the excavation process, and lack of such independence indicates an incorrect procedure. The simple direct method described in this paper provides the required independence In the case of linearly elastic materials, and hence can be used for multl-stage excavation in non-linear problems without excessive errors. However methods whose errors increase with the number of stages of excavation are quite unsuitable for non-llnear problems. Alternative methods of analysis, errors arising from the inability of the elements to model adequately the stress gradients near the toe of the excavation and excavation adjacent to a diaphragm wall are discussed.

INTROML~FION

The finite element method provides geotechnlcal engineers with a

potentially powerful tool for simulation of the excavation process.

However, care must be taken in the use of this tool if significant errors

are to be avoided.

For example Christian and Wong in reference i reported some serious

errors discovered while developing a program to simulate construction of

braced excavations. They concluded that excavations should be simulated in

as few stages as possible, because trouble arises when points of stress

concentration are excavated.

While it may be satisfactory to simulate an excavation in linearly

207 Computers and Geotechnics 0266-352X/85/$03.30 © Elsevier Applied Science Publishers Ltd, England, 1985. Printed in Great Britain

208

elastic material by excavation in one stage, this form of simulation is

unlikely to be satisfactory when the material behaviour is non-linear.

Hence the authors of this paper sought a method which did not introduce the

errors referred to above. This method could be checked for the linearly

elastic case as the results should be independent of the number of stages

shown in reference [2]. Such a method is presented here, and as is

intended as the basis for simulation of excavation in non-linear

materials.

EXCAVATION SIMULATION"

The basic aspects of simulation of excavation by the finite element

method are summarised as follows. Fig.l(a) shows a body of soil from which

the shaded portion A is to be excavated, leaving the unshaded portion B.

The behaviour of B will be identical if material A is removed and replaced

by the tractions (T) which were previously internal stresses in the soil

mass, as indicated in Fig.l(b). Then the behaviour of B due to excavation

will be the behaviour of B when the tractions T are removed, for example by

applying equal and opposite tractions, as shown in Fig.l(c).

Simulation of a stage of excavation thus involves determination of

the tractions T at the new portion of the soil boundary, determination of

the stiffness of the soil mass B, and application of tractions -~ to the

new portion of the boundary.

Finite element implementation of this process involves determination

(a) (b) (¢)

FIGURE i. Simulation of Excavation

209

of the noda l f o r c e s which a r e e q u i v a l e n t to the t r a c t i o n s shown i n

Fig. l (c).

This may be carried out by a variety of methods. Christian and Wong

considered methods involving direct determination of tractions and equlva-

lent nodal forces from known values of stress. These gave results which

were dependent on the number of stages of excavation, except when a high

order polynomial was used for stress extrapolatlon. The implementation of

such methods appears laborious, and as Christian and Nong say, the method

may not be applicable to all types of excavation problem.

Clough and Mana in reference [3] report that blana in reference [4]

proposed to determine the excavation boundary forces as

M f = ~ f sTo dV ~ m=l v

in which M is the number of elements which have a common boundary with

unexcavated elements, B is the displacement strain matrix, o is the stress

vector, and f is the vector of nodal forces. However, Desal and Sargand

in reference [5] report that this method is unable to produce correct

results for a simple problem involving excavation off the top of a single

column of elements. Desal and Sargand propose a method using a hybrid

stress model, which is also based on integration of stresses, but includes

integration of surface tractions on the boundary of the domain.

An alternative approach, which is the basis of this paper, is to start

from the virtual work equation including the terms due to stresses, body

forces and external tractions. Then as is shown in the next section of

this paper, the nodal forces can be found by appropriate numerical integra-

tion of stresses, body forces and external tractions throughout the soil

mass, in such a way that total equillbrit~n is approximately maintained at

each stage of excavation. While methods based on extrapolation of known

stresses are conceptually simpler, the recommended method is easier to

210

implement ~Ithout loss of accuracy. Other integration methods, which

ignore part or all of the contribution of body forces to the total strain

energy, will in general give incorrect results, although they may provide

adequate solutions to some problems.

FI]IITE ELEMENT IMPI..EMlmT&TXON

Let us consider a multi-stage excavation for which the inital stresses

are no' and the initial strains are ~o~ and are zero as shown in Fig.2.

The soil mass is subject to body forces ~ and surface tractions t outside

the excavation, and after the i-th stage of excavation the stresses and

strains are ~i and ~i respectively. V i is the volume of the soll mass

mass after the i-th stage of excavation has taken place, and it is required

to determine the nodal forces which must be applied to this volume in order

to simulate satisfactorily the process of the i-th stage of excavation.

t t i i l i i / I I

Vo---

~ : .o - o

.~o: O

Initk~l s t o r e J

V~

g=.~l

-.~o = D .~1

End of f i r s t s tage

t t i 1 t

_~ = .~2

O'2- O1 = D (e2-el)

End of second s tage

FIGURE 2. Definition of stresses and strains at each stage.

211

The procedure adopted for determination of these nodal forces is to

impose an arbitrary (virtual) nodal displacement at the end of the i-th

stage of excavation. Then by equating the internal and external work

performed during that displacement, the nodal forces required to simulate

the i-th stage of excavation can be determined, based on total not

incremental equilibrium.

Let the virtual displacement be 6u and the corresponding strains and ~

nodal displacements be 6e and ~a. Equating the internal work done by the

stresses and the external work done by the body forces ~ and the tractions

f(6~T~i)dV = f (~uT!)dV + f (~T!)dS V i V i S i

(1)

where S i is the surface of the soll mass at the end of the i-th stage of

excavation.

Now writing

A~i ffi ~i - ~i-I and taking ~o = 0

A~i ffi ~i - Zi-1

Ao i~ = DAe i~ where D is the incremental stress-strain matrix

equation (i) can be re-written as

f (~cT(Aoi + Zi_l))dV = f (~uT~) dv + f(6~T~)dS Vi ~ ~ Vi ~ ~ Si

f(6~TDA~i)dV = - f (6Toi_ l )dV + f(6uT~)dV + f(eTt)dS V i V i V i ~ ~ Si

(2)

f(~aTBTDBAai)dV ffi - f(~aTBToi_l)dV + f ( 6a~T ~) dV + f(~aTNT~)d$ V i V i V i S i

where N is the mat r ix of shape f u n c t i o n s .

212

Since the above relationship holds for arbitrary displacements

(~(BTDB)dV)Aa = -f(BToI_I)dV+f(NTy)dV + f(NTt)dS

Vi ~ Vi - _ V I ~ Si ~

or in finite element form

K'A~i = [ i , i - 1 + ~i + ~i (3)

where ~i'i-l' ~i' ~i are the vectors of nodal forces arising from integra-

tion of ~i-l' ~ and ~ respectively and the first subscript indicates the

the body over which the integration takes place, and the second subscript

(if any) indicates the stress state which is integrated.

The sum of the vectors f, g and h ~ represents the nodal forces which

need to be applied to retain total equilibrium in carrying out the i-th

stage of excavation.

Restricting attention now to the linearly elastic case, it is of

interest to rewrite equation (2) in the form

f(6~ TD ~i)dV ffi -f6eT(Oi_l - D~i_l)dV + f(6uT~f)dV~ ~ + f(6uTt)dS~ ~

V i V i V i S i

(4)

From Fig 2 it will be seen that

~i-I - D ~i-I = ~o° (5)

and making this substitution in equation (4) and converting to finite

element form

K'~i ffi ~i,O + ~i + ~i (6)

Equation (6) corresponds to a single stage excavation, and is

precisely equivalent to repeated application of equation (3) for the

linearly elastic case, provided that the sample points used to evaluate the

stiffness matrix are also used to evaluate the nodal forces corresponding

to each 'initial stress'. Thus while there will be errors in the results

213

obtained due to inadequate representation of the stress distribution by the

finite elements, no additional errors will be caused by increasing the

number of excavation stages, if this method is used. Hence this method

achieves the theoretically correct equality between single-stage and

multi-stage excavation.

DISCUSSION OF RXSULI'S

Finite element analyses were carried out using 9 Gauss points per

element and the meshes shown in Fig.3, which consist of 8-noded

smoot h ~

(a)

smooth~

(b)

smooth~__

(c)

P,| Im

• 0 1"0 1-0 1"0 1'0 1"0 1'0 1"0

1"0

1'0

1.0

1"0 fixed

0.1k~7 1'0 1.0 1-0 1.0 1-0

o.1 0.2

not to scale

Y/) IIII I I I I

" ' I ; ' , ;~1111 ' , I I I I I I I I

1.0

0.9 0.1 ~--------0.1 " ~ ' 0 . 2

1.0 " ' f i x e d

0"72

~ 0 < ) 6

. . . . . ,,,, ~ 0 " 0 6

III111111 1.0 " 0 . 7 2 0 1.0 1.0 1-0 1.0 " ' f i x e d

b o6cb(b(b66

FIGURE 3. Finite element meshes used.

214

isoparametric rectangles, and where the material was homogeneous isotropic

and linearly elastic. The excavated elements have been cross-hatched, and

the excavation was carried out in one-stage, or in several stages each

consisting of one horizontal row of elements. In each case the nodal

displacements obtained at the completion of excavation were identical,

whether the excavation was carried out in one or several stages. For all

the analyses discussed in this paper Young's modulus (E) was I0,000 kPa,

Poisson's ratio (v) was 0.47, K o = 0.5 and the unit weight of the soil

was 20 kN/m 3 .

Analyses using the mesh shown in Fig. 3(a), were also carried out

using 4 Gauss points per element, in which the nodal forces were determined

by linear extrapolation of stresses from the Gauss points in the elements

next to be excavated. This was followed by fitting a straight line to the

resulting values of stress on the excavation boundary for each element, and

/l/'/~'2' Proposed Method , / / ~ Stress Extrapolation:

/:~ / 2 Stage. 0.Sm Elements o o o J J I 2 Stage, ~Om Elements . . . . . " k ~ 4 Stage, C)-Srn Elements . . . . . . .

1 " - ~ I I 4 3 2 1 0

Horizontal Displacement (mm)

FIGURE 4. Displacements of side of cut using stress extrapolation.

215

calculation of the equivalent nodal forces. Two similar analyses were

carried out with the two upper rows of elements in Fig. 3(a) subdivided

into elements each of height 0.5 m, in order that the excavation could be

carried out in 4 stages and also in 2 stages of two rows of elements. The

horizontal displacements of the side of the cut resulting from use of the

recommended method, and the analyses based on stress extrapolation are

shown in Fig. 4. Excavation in one stage produced the same results as the

recommended method, but progressively greater errors are seen to occur,

using the stress extrapolation method, when the excavation is carried out

in 2 stages and 4 stages. Similar increases in error with an increasing

number of excavation stages were reported by Christian and Wong when

evaluating nodal forces only from elements near the excavation (not using a

high order polynomial for extrapolation). However it is of interest to

note that at most depths there is a smaller error for 2 stage excavation

with the finer mesh than with the coarser mast.

Nodal forces were found to arise only on the excavated boundary of the

current stage when the recommended method was used. When the contribution

to the nodal forces arising from the body forces is ignored, vertical

forces arise at every node in the soil mass, and produce grossly erroneous

results. However if one also ignores the nodal forces except at the nodes

on the current excavated boundary, a reasonably satisfactory solution is

obtained. Table i shows the vertical nodal forces on the boundary to be

excavated in the first stage of excavation using the mesh shown in

Fig. 3(a), and 9 Gauss points per element. The contributions arising from

integration of stresses and body forces are shown separately, as well as

the total nodal forces for each of the relevant nodes, whose numbers and

locations are shown in Fig. 5.

216

TABLE i Vertical Nodal Forces (kN)

Node fBTo dV fNT~ dV Total

7 13 21 27 35 36 37

1.67 20.00 3.33

20.00 -1.67 6.67

-1.67

1.67 -6 • 67 3.33

-6.67 5.00

-6.67 1.67

3.33 13.33

6.67 13.33

3.33 0 0

Total 48.33 -8.33 40.00

0

.•0.5

0

o

--Node Numbers ( ~ o 1 2 3 4 5 Cram) / ~

L ® ® ®

1.5 I (>5 0

Distance f rom Side of Cut (m)

FIGURE 5. Displacements at end of excavation.

It can be seen that ignoring the body force contributions, leads to

significant error in the vertical nodal forces. Similar errors occur in

the second stage of excavation. Refinement of the mesh by subdividing the

upper two rows of elements into elements of height 0.5 m, had no

significant effect on the final displacements. Despite these errors in

217

vertical nodal forces, the final displacements of the boundary of the

excavation resulting from use of the recommended method, and stress

integration ignoring body forces and non-boundary forces, produce similar

displacements as shown in Fig. 5. The recommended method gives the correct

solution for the selected mesh, since it is precisely the same as the

solution for one-stage excavation using this mesh. However this stress

integration method provides a solution which is reasonably similar except

for the vertical displacements near the top of the cut.

mesh in f c / / a

i i

4 3 2

Horizontal deflection (mm)

I I O

FIGURE 6. Horizontal deflections of side of excavation when using 9 Gauss points.

A very coarse mesh such as that shown in Fig.3(a) cannot be expected

to model the stress concentrations at the bottom corner of the excavation.

Better representation of the stress distribution in this region, by the use

of a finer mesh near the corner, should produce more accurate values of

displacements. This aspect was investigated by means of additional finite

element analyses using the meshes shown in Fig.3(b) and (c). The resulting

218

horizontal nodal displacements along the vertical excavated face for the

case of 9 Gauss points are shown in Fig.6. The first mesh refinement leads

to an increase in deflection of the top corner of the excavation of 15%,

however the second mesh refinement only leads to an additional increase in

deflection of 3%.

It is also of interest to examine the effects of changes in element

size when the excavation is carried out adjacent to a diaphragm wall. For

this purpose finite element analyses were carried out on the meshes shown

in Fig.7 which include a concrete diaphragm wall. The wall deflections

above the base of the excavation are sbown in Fig.8.

/ srnooth

~, i 1 '0

L ~ f i x e d 1.0 1.0 0.5 1.0 1.0 1.0 1.0 1.0 1.0

f smooth.

0 ~ 0 1-0 1.0 1.0 1.0 1-0

• 1 0.5

1.O

1.0 ....~ 0-1 ~ 0 . 2 0.7

1.0 " ' f i x e d

FIGURE 7. Finite Element Meshes used for diaphragm wall.

The effect of refining the mesh is smaller than when no wall was

present, as the increase in deflection at the top corner of the excavation

is only 10%.

219

Finer mesh

/ 4 Gauss points / / 9 Gcuss points

I I I I 4 3 2 1 0

Well deflection (ram)

FIGURE 8. Deflections of diaphragm wall.

CONCLUSIONS

A theoretically sound basis has been established for a simple correct

method of finite element simulation of excavation. This produces results

which are independent of the number of stages in which the excavation is

carried out for the case of a linearly elastic medium. The implementation

of this method appears to be more economical to program than the high order

polynomial fitting proposed in reference [I], aud is more accurate than

other suggested methods. The method described is therefore recommended as

the basis for simulation of excavation in non-llnear materials.

It has been demonstrated that for a linearly elastic material, very

little refinement of the mesh is required in order to obtain horizontal

deflections of engineering accuracy.

220

ACKIIOIIL~IqTS

The work described in this paper was carried out with the aid of a

grant from the Australian Research Grants Committee.

REIzgRENCKS

i.

2.

3.

4.

5.

Christian, J.T., and Wong, I.H., Errors in simulating excavation in elastic media by finite elements. Soils Fdns. (Japan). 13 (1973) i-i0.

Ishihara, K., Relations between process of cutting and uniqueness of solutions. Soils Fdns. (Japan). IO (1970) 50-65.

Clough, G.W., and Mana, A.I., Lessons learned in finite element analysis of temporary excavations. Proc. 2nd Int. Conf. Num. Meth. Geomech., A.S.C.E. I (1976) 496-510.

Mana, A.I., Finite element analysis of deep excavation behaviour. Ph.D. Thesisp Stanford University, Stanford, Calif. (1976).

Desai, C.S., and Sargand, S., Hybrid FE procedure for soil-structure interaction. J. Geotech. Ensg. , A.S.C.E. iio (1984) 473-486.