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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.