8/20/2019 bearing capacity of foundation

1/2

APPENDIX I. REFERENCES

Bowles, J. E. (1996).

Foundation analysis and design 5th Ed.

McGraw

Hill, Inc., New York, N Y

Clough, W G., Smith, E. M., and Sweeney, B P (1989). Movement

control

of

excavation support systems by iterative design.

Foundation

engineering current practices and principles. ASCE Geotech. Spec.

Pub/. No. 22,

F

H.

Kulhawy, ed., ASCE, New York, N.Y., 869-884.

Mana, A. I. , and Clough,

W.

G. (1981). Prediction

of

movements for

braced cuts in

clay. Geotech. Div.

ASCE, 107(6),

756-777.

National Coal Board. (1975). Subsidence engineers handbook. National

Coal Board Production Department, London, England.

Peck, R B. (1984). State

of

the art: Soft-ground tunnelling.

Tunnelling

in soil  n rock K

Y Lo, ed., ASCE, New York, N.Y.,

1-11.

APPENDIX II. NOTATION

The following symbols are used in this paper:

Dr =

distance from edge of excavation relative

to

excavation

depth;

D

ri

=

relative distance at

any

point

at

distance

i

from

edge of

excavation;

D

rm

=

maximum relative distance from edge of excavation;

Sr =

settlement relative

to

excavation depth;

Sri

= relative settlement

at

any point at distance

i

from edge

of excavation; and

  rm X

=

maximum relative settlement at edge of excavation.

NUMERICAL

STUDIES

OF

BEARING-CAPACITY FACTOR

N  

Discussion by Abdul-Hamid Soubra/

Christelle Bay,4

and

Jean-Georges SiefTert

5

The bearing capacity problem is a matter

of

interest to the

discussers. The authors presented the bearing capacity factor

Ny based on both the finite difference program FLAC and the

finite element program OXFEM. The authors then compared

their results to those given

by

Garber and Baker (1977) and

those given by Bolton and Lau (1993).

The aim of this discussion is:  I) to comment upon the

 January 1997, Vol. 123, No.

I

by Sam Frydman and Harvey J Burd

(Paper 11594).

'Lect., ENSAIS, 24 Bid de la victoire, 67084 Strasbourg cedex, France.

'Doctoral Student, ENSAIS, 24 Bid de la victoire, Strasbourg.

'Prof., ENSAIS,

24

Bid de la victoire, Strasbourg.

comparison made by the authors with other authors ' results;

and (2) to present some results obtained by the discussers.

Comparison

of

the authors' results with those of the upper

bound solution given by Garber and Baker shows significant

differences between the results. The difference exceeds 100

for = 45°. The solution given by Garber and Baker (1977)

is based on a variat ional l imit equil ibrium method, which is

equivalent to an upper-bound method

of

the limit analysis the

ory for a rotational log spiral failure mechanism. On the other

hand, Chen (1975) considered three symmetrical failure mech

anisms, referred to as Prandtl l, Prandtl2, and Prandt13, and

gave rigorous upper-bound solutions for the three mechanisms

in the framework

of

the limit analysis theory. Prandt1l is com

posed

of

a triangular active wedge under the footing, two ra

dial log-spiral shear zones, and two triangular passive wedges.

Prandtl2 differs from Prandtll only in that an additional rigid

body zone has been introduced. Prandtl3 closely resembles the

Prandtl l mechanism; however, each Prandt13 shear zone is

bounded by a circular arc. Finally, Soubra (1997a) considered

translational nonsymmetrical log-sandwich and arc sandwich

mechanisms and gave upper-bound solutions of the bearing

capacity factors

Ny N

c

 

and

N

q

•

Another translational failure

mechanism has been recently investigated by Soubra (1997b):

a general translational failure mechanism composed

of

several

triangular rigid blocks that allows the rupture surface to de

velop more freely.

As is well-known in the framework

of

the limit analysis

theory, the exact solution

of

the bearing capacity problem can

be bracketed by the minimal upper-bound solution and the

maximal lower-bound solution. Therefore, one must consider

the minimum values obtained by the available different mech

anisms. Table 7 presents the bearing capacity factor Ny given

by these upper-bound solutions (Garber and Baker, 1977;

Chen 1975; Soubra 1997a,b). It is clear that the solution by

Garber and Baker (1977) gives the greatest upper-bound so

lution and that Soubra's (1997b) solution gives the lowest up

per-bound solution. Therefore, better comparison

of

the au

thors' results may be made with the upper-bound solution

given by Soubra (1997b). Comparison

of

the authors' finite

difference results with those

of

Soubra (1997b) shows that the

difference is smaller than 23 for

=

45°.

On the other hand, the finite element method has been used

by the discussers to compute the bearing capacity factor Ny

for rigid rough strip footings. The calculation has been made

using the finite element program CESAR, developed at the

LCPC in Paris. The hypothesis used in the present finite ele

ment analysis assumes that

B =

0.5 m, G

=

100 MPa,

 y =

18

kN/m

3

,

v

=0.2, and

c

=O

The present mesh is composed

of

98 eight-noded elements,

or 337 nodes in total (Fig. 16). This mesh is constructed in a

manner that permits study

of

three-dimensional problems with

complex loading. An elastic perfectly plastic model, based on

the Mohr-Coulomb and the Drucker-Prager criterion was used

to model the soil.

The results obtained in the case

of

an associated flow rule

TABLE

7

Bearing Capacity Factor

Ny

for

15°

q

<

45°

Chen 1975

Garber and Baker

min Soubra

Soubra

 >

1977

Prandtl 1 M1 Prandtl 2 M2 Prandtl 3 M3 M1, M2, M3 1977a

1977b

1   2 3

(4)

 5

6 7

8

15

-

2.7 2.3

2 1 2 1 2 1 1.9

20

-

5.9

5.2

4.6 4.6 4.8 4.5

25 16.5 12.4

11.4 10.9

10.9 11 1 9.8

30

38.1 26.7

25.0

31.5 25.0 25.0 21.5

35 92.5 60.2

57.0 138.0

57.0 57.1 49.0

40

243.9

147.0 141.0

1,803.0 141.0

140.5

119.8

45 536

- -

-

- -

326.6

JOURNAL OF GEOTECHNICAL

AND

GEOENVIRONMENTAL ENGINEERING /

MAY

1998 465

J. Geotech. Geoenviron. Eng. 1998.124:465-466.

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FIG. 16. Finite-ElementMesh fo r CESAR-LCPCRuns

TABLE 8. OXFEM and CESAR-LCPC Finite-Element Results

fo r

a Rough Strip Footing

1/1

= 35°)

Concerning the authors' results obtained by the finite dif

ference analysis (FLAC), Fig. 17 shows that for

<

35°, there

is good agreement with the discussers' results using both the

finite element method and the limit analysis theory. The dif

ference increases with

The maximum percent difference is

about 16 with the discussers' finite element solution and

does not exceeds 23 with the discussers' limit analysis so

lution.

Calculations performed by means

of

the finite-element pro

gram CESAR become more difficult for great

values or

when the difference between and   increases. The number

of

increments must be large for a cohesionless soil (c

=

0).

The discussers' conclusions conform to the authors' results.

A P P E N ~ X REFERENCES

Chen,

W

(1975).

Limit analysis and soil plasticity.

Elsevier Scientific,

London, U.K.

Soubra, A. H. (1997a).

 Seismic

bearing capacity of shallow strip footing

in seismic conditions.

Proc. Instn.

Civ

Engrs. Geotech. Engrg.

Soubra, A H. (1997b). Upper-bound solutions of the bearing capacity

of

strip footings. Internal report.

The authors thank the discussers for their contribution. The

authors are aware

of

the upper-bound nature

of

Garber and

Baker's variational approach and note with interest the alter

native solutions given in Table 7. Soubra's 1997a solutions

have   y values similar to those reported by Chen (1975).

Soubra's 1997b solutions, however, are somewhat lower, par

ticularly for cases where the friction angle is large.

 

these

solutions satisfy the necessary conditions to be kinematically

admissible, then they represent a significant improvement over

the other solutions given in Table 7. Unfortunately, the pro

cedures used to obtain these solutions have not yet been pub

lished, so it is not possible to comment on their admissibility.

The authors note that Soubra's 1997b solutions are signifi

cantly lower than the .finite difference results obtained for the

case

of

full association, particularly for the higher friction an

gles. This unexpected trend deserves further investigation.

The limit theorems may be applied rigorously only to as

sociated materials. Solutions obtained using limit-state ap

proaches should therefore be applied with some caution to

non-associated materials, such as sands. In such cases, it

cannot be proven that kinematically admissible solutions are

necessarily upper bounds. The comparison given in Fig. 17,

then, is potentially misleading. The authors' numerical results

were obtained using a non-associated flow rule, whereas Soub

ra's 1997b results were obtained on the assumption

of

nor

mality. Fig. 17 also shows additional finite element results ob

tained by the discussers. The discussion does not make it clear

whether these were based on an associated flow rule (to com

pare directly with the limit state solutions) or on a non-asso

ciated flow rule.

 Prof., Facu.

of

Civ. Engrg.,

Technion-Israel

Inst.

of

Technol., Haifa,

Israel.

7LeCt. Dept.

of

Engrg. Sci., Univ.

of

Oxford, England.

Closure by Sam Frydman

6

and

Harvey

J.

Burd

45

Ny

 3)

48.4

47.4

40

5,000

5,000

35

Number of steps

 2)

0'4'-----+-----+-------1

30

Finite element

program

 1

)

500 +-------1f-----+-------F-I

200   - - - - - - - 1 r - - ~ - - r - - - ~ ~ _ _ _ I

,(deg)

. . . . .

Garber   Baker

......

discussers' results (Limit Analysis)

. . . . -

discussers' results (Finite Elements)

  *

authors' results (Finite Difference)

FIG. 17. Var iat ion

of

Ny with for Rough Strip Footing

400

  r I

Z

300 + - - - - - - - 1 f - - - - - + - / - - - - - - , I ~

6

r ..... ... r

100   -----:::zJ ;:IF'-----1

OXFEM

CESAR

Coulomb material

 1\1 =

= 35°) are presented in Table

8.

There is good agreement with the authors' results using

OXFEM: The difference does not exceed 2 , although the

mesh used by the discussers is less refined. The

Ny

values

obtained by the discussers using the Drucker-Prager soil be

havior model are identical to those obtained using the Mohr

Coulomb model.

466 JOURNAL OF GEOTECHNICALAND GEOENVIRONMENTAL ENGINEERING / MAY 1998

J. Geotech. Geoenviron. Eng. 1998.124:465-466.