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Eur. J. Mech., A/So/ids, 14, n 3, 377-396, 1995
Ultimate bearing capacity of shallow foundations onder inclined
and eccentric loads.
Part II: purely cohesive soil without tensile strength
J, SALENON *and A. PECKER **
ABSTRJ\Cf. - The problem of determining the bearing capacity of
a strip footing resting on the surface of a homogeneous half space
and subjectcd to an inclined, eccentrie Joad, is solved within the
framework of the yield design theory assuming thal the soi! is
purcly cohesive without tensile strength according to Tresca's
strength eriterion with a tension eut-off. The soi! foundation
interface is also purely cohesive, in tenns of the homologous
strength eriterion with a tension eut-off.
As in a eompanion paper [Salenon & Peeker, 1995). both the
static and the kinematie approaches of the yicld design theory are
used. New stress fields are eonstrueted, in order to comply with
the condition of a tension eut-off within the soi! medium, and new
lower bounds are determined as substitutes to those given in the
eornpanion paper. Veloeity fields taking advantage of the tension
eut-off contribution in the expression of the maximum resisting
work arc also implcmented, giving new lower bounds.
As may be expceted from eommon sense and from the general
rcsults of the theory, it appears thal the tension eut-off
condition within the soi! medium results in lower values of the
bcaring capaeity of the foundation, and that the grnvity forces
acting in the soi! mass have a stabilizing effect.
1. Introduction
The problem solved in the cm-rent study, relates to the ultimate
beming capacity of a strip footing resting on the surface of a
homogeneous half space and subjected to an inclined, eccentric
Joad. The foundation , with width B, is assumed to be rigid and to
have an infinite length. It is subjected to uniformly distributed
external loads along the direction Z (Fig. 1). The evaluation of
the ultimate bearing capacity is obtained within the framework of
the yield design the01y [Salenon, 1983, 1993 ], as detai led in the
companion paper [Salenon & Pecker, 1995].
In the companion paper, the problem is studied assuming the soit
to be plllely cohesive, in accordance with Tresca's strength
condition, white the interface between the soit and the foundation
is also purely cohesive, with a similar cohesion and no normal
tensi le strength. In the present case, a tension eut-off condition
is added to the soi t strength criterion, describing a pmely
cohesive medium with no tensile strength. As in the companion
paper, other interface strength conditions can be easily taken into
account.
* Laboratoire de Mcanique des Solides, cole Polytechnique, 91128
Palaiseau Cedex, France. ** Godynamique et Structure, 157, rue des
Blains, 92220 Bagneux, France.
EUROPEAN JOURNAL OF MECHANlCS. NSOLIDS. VOL 14, N 3, 1995
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378
h
J. SAI.F.NON AND A. PECKER
y
- ------ - - - , \
\ T
B Fig. 1. - Strip foundation undcr inclined , eccentric
load.
x
Under aforementioned assumptions, the problcm can be studied
within the framcwork of the plane s train yield design the01y, as
defined in [Salcnon, 1983, 1993].
Numerous publications (i.e. [Meyerhof, 1953-1963; Hansen,
1961-1970; Tran Vo Nhiem, 197 1; Khosravi , 1983; Swami Saran &
Argawal, 1991]) deal with the problem of load inclination and load
ecccntricity for a purely cohesive soil; however, to the best of
the authors' knowledge, the problcm for a cohesive soil without
!ensile strength has never been solvcd before.
2. Theoretical framework
The notations, identical to those of the companion paper, are
recalled in Figure 1. We denote 'Y as the unit weight of the
soil:
(1) 1 = - "t.y
and
(2) F = - N _u + T __n M = -1\1 _~ arc the force system
resultants, computecl at 0, the mid-point of the foundation Ji' Ji;
__ro .y and .= arc the unit vectors of the cartcsian coorclinate
axes Ox, Oy, Oz.
Let S be the point of application of F on O:r and e the
algcbraic eccentricity which is positive along Oa:; it follows
that:
(3) 111 =Ne, l e i ~ B/2 For pratical situations, the load
eccentricity onto the foundation may arise duc to an elcvatcd point
of application of the horizontal force componcnt T.. = T __";
hence:
(4) 1H =Th,
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The soil is assumed to be purely cohesive without !ensile
strength . The strength criterion using Tresca's criterion with
tension eut-off is:
(5)
where a1 and a2 are the in-plane principal stresses of the
stress tensor g; C designates the soil shear strength and tensile
stresses arc positive.
The strength cri teri on for the interface il' A, y = 0, k1:l
< n /2 is written as:
(6) f (a.1 y, ayy) = Sup (ia.ty l - C', ayy) :S 0 The halfspace
is subjected to the foll owing boundary conditions:
zero di splacement at infinity
(7) y:=:; 0, lxi --+ oo : U = 0 stress free boundary surface
outside the foundation A' A
(8) y:=:; 0, 1.'1:1 > B /2 : rJ.I'.I' = rJ,/'!} = O.
The external load , in addition to the gravity field , is
applied on the upper face of the interface A' A by the rigicl
founclation which enforces the following bounclary condition for y
= o+, l:t:l :=:; lJ /2: the velocity field U must be a rigid body
motion in the plane 0:1:y delined by its components in 0: U 0 and ~
= - w ~= .
The loading parameters of the problcm namely N (g), T (g), lvi
(g) and 'Y for any statically admissible stress field, together
with the associated kinematic ones for any kinematically admissible
velocity field have been detailed in the companion paper.
The founclation bearing capacity is given by the boundary of the
surface which, in the space (lV, T, 1\1), given 'Y and C',
delincates the set of ali values for the parameters N, T, 1\1, for
which the equil ibrium of the foundation is ensurecl without
violating the strength criteria (5) and (6).
In thi s papcr, both the static approach from "insicle" and the
kinematic approach are usccl to determine bounds for the foundation
bcaring capacity. The use of the static approach has been described
in detail in the companion paper together with thal of the
kinematic approach, which requires the introduction of the concept
of maximum resisting work.
Tt is recalled that the former approach yields lower bouncl
estimates for the bearing capacity, the latter, upper bouncl
estimates.
The ex pression for the maximum resisting work in the soil
medium presented in the companion paper must be changed. Referring
to [Salenon, 1983, 1993], the expressions for the corresponding
densities of maximum resisting work, 1r (~) and 1r (!!:, [un. for
EUROPEAN JOURNAL 01' MECHANICS. MSOLII>S. VOL. 14, N 3, 1995
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380 J. SALENON AND A. PECKER
any plane strain velocity fi eld, in the case of a purcly
cohesive sail without tensile strength , are:
{ w(d) = +oo if trd < 0 1f (g) = C Id Il + ldzl - tr g) if
trd ~ 0 (9)
(10) { 1r (!.!:, [UD = +oo if [U] !.!: < 0 1r (!.!:, [U]) = C
(I[U]I- [U] !.!:) if [U] 1J: ~ 0
The kinematic approach then states that, if U is a kinematically
admissible velocity field with the kinematic data !.f..o and w, the
inequality
( 11) -N(Uo)v+ T (Uo).r+Mw~ { 1r(g)dD+ { 7r(1J:, [U])clE ln l
r:. + ; ?T ( [ U]) eix + [' "Y Uv rln
A'A ~ yields an upper bou nd for the foundation bearing capacity
(N, T , 111).
3. Fundamcntal results
3.1. STAIJILIZING EFFECT OF THE GRAYITY FORCES ON THE FOUNDATION
BEARING CAPACITY
In the companion paper, it was recalled thal the bearing
capacity of the considered foundation did not depend on the unit
weight of the soil medium. This resu lt refers to a classical proof
based upon the fact that Tresca's strength criterion is a function
of the stress deviator only. In the present case, where the
strength criterion of the soil exhibits a tension eut-off, the same
reasoning can be followed, but the conclusion will be
different.
Let q_0 be a stress field in equilibrium with zero gravity
forces in the soil medium ("Y = 6) and complying with the strength
criteria (5) and (6), and consider the stress field q_ defined
by:
( 12) Vy ~ O, Vx; g1 (x , y)= g0 (x , y)+ "Y YJ,, "Y~ O. It is
clcar, from Eq. (12), that g is in equilibrium with the gravity
forces 1 = - 1 ~v Si nee "Y y ~ 0 and sz:0 complies with the
strength criterion (5) in the soi! medium, so does sz:A' . Since q_
(;, 0) = q_0 (x, 0), q_ obviously satisfies the strength criterion
(6) in the irrterface; mK g' and g0-equilibrate the same values of
the strength parameters then:
(13)
It follows, from the static approach, that the bearing capacity
in the case of non zero gravity forces is greater than or at !east
cqual to the bearing capacity for zero gravity forces.
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Contrary to the case of the companion paper, Eq. ( 12) does not
provide an exhaustive construction of ali the stress fields g which
comply with the strength criteria (5) and (6) and are in
equilibrium with the gravit y forces 1.. = - "Y ~~r It means that
the beming capacity on a purely cohesive soil without tensile
strength may, under certain circumstances, depend upon the soi l
unit weight which acts as a stabilizing factor.
Referring to the kinematic approach, it can also be stated that
any upper bound estimate for the bearing capacity on a purely
cohesive soil still remains valid for a soit with the same shear
strength but without tensile strength. It follows that such an
upper bound estimate is valid whatever the gravity forces in the
soi! medium.
The derivation of additional (and better) upper bounds for the
bearing capacity on a soi! without !ensile strength will proceed
from the implementation of velocity fields in equality (11) which
take ad van tage of the new expressions (9) and (1 0) for the
maximum resisting work within the soi!. Such velocity fields
exhibit dilation ( tr d > 0) and/or uplift velocity jumps ([U~
TI: > 0): consequently, the work of the gravity forces within
the soi! medium is always negative (the result is established using
Stokes' formula) which proves that the unit weight of the soit acts
as a stabili zing factor for the upper bound estimates (except for
the case when the velocity field only exhibits velocity jumps at
the soi! boundary below the interface A' A and is non dilatant
anywhere else).
As a consequence of the preceding considerations, the problem
will be studied assuming zero gravity forces, both for the static
and the kinematic approaches. Due to the fonn of the implementee!
velocity fields, the computed upper bouncls will prove to be valid,
even without this assumption, while the lower bounds might be
conservative.
3.2. CONVEXITY AND METHOD OF THE REDUCED WIDTH FOUNDATION
The general results regarding the convexity of the lower bound
estimates of the bearing capacity, and the application of the
methocl of the reduced width foundation presented in the companion
paper, remain valid without any alteration.
4. Foundation bearing capacity for an inclined cccentric Joad on
a pmely cohesive soi! without tensile strength
Since the strength domain defined by Eq. (5) for a purely
cohesive soi! without tensile strength is contained within the
strength domain for a classical purely cohesive soit (as defined in
the companion paper), the following general statement holds
[Salenon, 1983]:
-for a given sai l unit weight "f, the foundation bearing
capacity (N, T, M) for a purely cohesive soit with tension eut-off
is lower than the founclation bearing capacity for a classical
purely cohesive soi! (the proof is straigthforward and stems from
the static approach from inside).
Moreover, as previously stated, for a soil without !ensile
strength, the bearing capacity may depend upon the soi! unit weight
acting as a stabilizing factor: for increasing values of "f, the
bearing capacity increases but remains bounded by the beming
capacity of the foundation for a classical purely cohesive
soi!.
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382 J. SALENON AND A. PECKER
The object of the following study is twofold. The kinematic
mechanisms described for the purely cohesive soil are reanalyzed
to
introduce slight modifications in orcier to take aclvantage of
ex pressions (9) and ( 1 0) for the maximum resisting work and
therefore yield new, better, upper bounds for the bearing capacity.
New mechanisms will subsequently be introduced.
Thereafter, the stress fields of the static approach used for
the classical purely cohesive soi! will be checked with respect to
the strength criterion with tension eut-off (5). New stress fields,
compatible with the new criterion, are constructed.
4.1. KlNE~IATIC APPROACH
As previously stated, the upper bound estimates described in the
companion paper which were derived from the kinematic mechanisms
and do not depend upon the soi! unit weight, rernain valid for the
soi! without tensile strength.
The associated vclocity fields do not imply any volume change
within the soi! medium:
(14) { tr g = 0 [U] rr = o
The interface A' 1l incorporates, for the mechanisms A and C, an
inclined (not tangential) velocity discontinuity [U] which
corresponds to a separation betwecn the two sides, that is, the
soi! and the foundation: the discontinuity takes place within the
interface. The analysis focuses on this specifie aspect.
The new mechanisms Ao and Co are definecl below, given that: -
the velocity is identical to thal of the corresponding mechanisms A
or C within
the soit medium; - the velocity discontinuity, equal to the
velocity discontinuity in the interface A' A
for the A or C mechanisms, takes place within the soi/
immediate/y below the inteJ:face. Consequently, when the new
mechanisms llo and Co , are compared with the original
mechanisms A or C', it is found thal - the work of the externat
loads remains unchanged and independent of the soif unit
weight, - the maximum resisting work is alterecl using Eq. ( 1
0) to re flect the contribution of
the velocity discontinuity along I A (or a A). The maximum
resisting work is smaller than or equal to the values derivccl for
the purely cohesive soi!. The llo and Co mechanisms consequently
yield better upper bouncls for the beming capacity than the A or C'
mechanisms.
Mechonism Ao Referring to Figure 2, the velocity discontinuity
[U] develops along A' I in the
soi! immediately below the interface. Therefore, the maximu m
resisting work in that mechanism ineludes a contribution along a
discontinuity tine within the soif (whose normal is ~y) insteacl of
a contri bution within the interface.
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Ul:l'li\IATE BEARING CAPACITY OF SHALLOW FOUNDATTONS. PART Il
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A' x L
1
0 t __ ....
.....
Fig. 2.- Mcchanism Ao.
Using the notation presented in the companion paper, the
corresponding upper bound estimate for the bearing capacity is
given by:
{ ~ + ~ + ( ~ - et) / cos2 a }
_N_ < 2 ___..o. __ +_t_a_n_2
_o:_ _[ l..,..---,-t_a_n_2 _(--'~':-------;::_~ --,)
,--_L_n:-:(=-t-a_n_2 ..,...(-=~=----=-~-)_)_] _/4-''----
CB - (1 +tan 8 tan n) + ef B - 1/2 ( 15)
in which the angle e is clefined by:
(16) tane =(A' Tjrtl) = (1 - )/tan cr,
The right hand side of inequality ( 15) is minimized, for a
fixed 6, with respect to the parameters ,\ and Cl' under the same
constraints that were appliecl to mcchanisms il:
( 17) (1/2- efD)/ (1 +tanO' tan )< ::::; 1
Meclwnism Co Referring to Figure 3, the velocity discontinuity
[U] for the Co mechanism, devclops
along A' a within the soi/ immediately below the interface. The
application of the same modifications used in mechanisms A leads to
the inequ ality
{ ~ + [Ln 1 tan(~- :_)/tan(~ - ~) } sm2 (l' 4 2 4 2
+Arctan (sin e)- Ar
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384 J. SALENON AND A. PECKER
y F
x
Fig. 3. - Mechanism Co.
where the ang le t: is defined, using the parameters . and a,
by:
(19) tan t: = (A' I /01) = (1- .>.)j.>. tana, 0 < <
7r/2 The right hand side is minimized, for a fixed 8, with respect
to . and a under the same constraints which were applied to
mechanisms C
(20) { 0 < Ct< 7r/2 0 < .< 1/2,
Mechanism Fo
(1/2 - e/B)/(1- tano/tana) < .
With one exception, the other mechanisms implemented in the
companion paper, namely the B and D mechanisms, do not involve a
velocity discontinuity along the interface A' A. Therefore, they
are not affected by the poss ibility of locating this discontinuity
within the soi! medium and consequently of improving the
corresponding upper bounds. The exception concerns the limiting
case of the "unilateral mechanism" examined in the companion paper,
which reduced to a slip in the interface A' A when the velocity V
was tange ntial to the surface (x = 1r /2) and to an uplift
velocity jump in the interface when 7r/2 < x < 1r.
In the present case, the homologous mechanism can be developed,
assuming that the velocity jump takes place within the soil medium
immediately below the interface A' A (Fig. 4):
y
v
A' x
Fig. 4. - Mcchanism 1'0.
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- the foundation follows a translation rigid body motion with
the velocity V defined by its inclination x;
- the velocity is continuons across the interface A' il; - a
constant velocity discontinuity [U] = V develops within the soi!
immediately
below the interface A' A; - the sail medium (y < 0) is
motionless. The maximum resisting work in the mechanism is obtained
through Eq. ( 1 0). For any
prescribed value of x. the corresponding upper bound for the
bearing capacity is given by:
(21) { 7r/2
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386 J. SALENON AND J\. PECKER
N/CB
T/CB
Fig. 5. - lnclined, cccenlric Joad: kinemalic approaches from
mechanisms Ao, /J , Co, D , Fo and synunctric ones for various ej B
values.
comments, suggcsted by a comparison with the results in Figure
16 of the companion paper, apply:
- e/B = 0: for 0 ~ lc51 ~ 1/4, the best results are obtained
from the unilateral mechanisms of the companion paper (i.e. a
degenerated mechanism B with a = {3); for 1 / '1 ~ lc51 ~ 1r /2,
the mechanisms Fo and F~ prevail.
- e/B = 0.1: for c5 ~ 0, increasing from the vertical loading
(c5 = 0), the same estimate as for the classical purely cohesive
soil is achieved through the mechanisms B (with {3 = 1 /2); the
associated curve is then extended by an arc obtained from the
mechanisms Ao, which is closely concident with the upper bound
obtained by the mechanisms Fo; for c5 ~ 0, decreasing from the
vertical loading, the same evaluation as for the purely cohesive
soil is obtained through the mechanisms D, extcnded to ti = - 1r /2
by the upper bounds from the mechanisms F~.
- efB = 0.2: for c5 ~ 0, the entire curve comes from the
mechanisms Ao; for c5 ~ 0, the curve is first derived from the
mechanisms D then extended by an arc following from the mechanisms
Co, which, for c5 values close to 1r /2, al most concides with the
upper bound yielded by the mcchanisms F~ .
- e/B = 0.3 and 0.4: for ~ 0, the curves are generated from the
mechanisms Ao; for ti ~ 0, from the mechanisms Co.
Comparing thcse results with the diagrams of Figure 16 in the
companion paper clearly shows a significant decrease in the bearing
capacity for the soil without !ensile strength. This decrease is
more pronounced when the Joad inclination and/or the Joad
eccentricity increases. The curves will then follow from the
mechanisms Ao, Co and Fo
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ULTI~ IATE REAKING CAPACITY OF SHALLOW FOUNDAT IONS. PART Il
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which take advantage of the decrease in the maximum resisting
work for the sa il without tensile strength. From thi s point of
view, the evolution of the leading mechanisms with increas ing Joad
eccentricity and Joad inclination is in good agreement with what
could be intuitively expectcd.
4.2. STATIC APPROACH FROM INSIDE FOR i\ CENTERED LOAD
Tt is worth comparing the upper bounds g iven in Figure 5 for
the bearing capacity of the foundation on a cohcsive sail with a
tension eut-off, with the lower bounds presentee! in Figure 17 of
the companion paper corresponcling to a classical pure ly cohesive
soil. lt appears thal, for high values of Joad eccentricity or of
Joad inclination, the bearing capacity of the soil with a tension
eut-off is (signi ficantly) smaller than the bearing capacity of
the classical purely cohesive so il. This proves that, in those
cases, the stress fields constructecl for the classical purely
cohesive so i! are definitely not compatible with the strength
criterion (5), due to the presence of "unconfined" zones where at
!east one of the principal stresses is (positive) !ensile.
Referring to the comments regarding the influence of gravity
forces, it must be recalled that those stress fields are in
equilibrium with zero gravity forces within the sail medium and may
be substituted for g0 in Eq. (12). Assuming that such a stress
field, defined for y ::; 0, incorporai es unconfined zones, Eq. (
12) shows thal, provided that the minimum depth of those zones is
non zero and the tens ion remains fini te, the incorporation of the
sail unit weight 'Y > 0 in 'YY in the expression for q_l may
balance the tractions in the lleld g0 : the stress field g' will
then bccome compaible with the strength criterion (5) for the sai l
with a tensile strength eut-off.
This is the stabilizing effcct of the g ravit y forces stated in
Paragraph 3.1, the application of which requires thal the stress
fields for the classical pure! y cohesive sail be reexamined in o
rcier to cletect the presence of unconfined zones. The s tress
fields in equilibrium with centered loads will be checkecl fi rsl,
before the application of the method of the recluced width
founclation and the use of convexity propertics.
Stress field in equilibriu11t with 011 oxiol /oad The stress
field [Ls ( 1r /2) in Figure 6 is composee! of the Pra nd tl stress
field in the a rea
located above the tine D' C' BC' D and of the Shielcl extension
belo w.
D' x
D
Fig. 6. - Prand!l 's stress field and Shicld's extension.
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388 J. SAI.ENON AND A. PECKER
The formulation of the firs t stress field is straightforward
and shows that the principal stresses arc compressive
everywhere.
The Shield stress field is detailed in the original publication
[Shield, 1954] and in [Philips, 1956] (see also lSalcnon, 1969,
1973]). It is composed, in the areas located left of B1t' and right
of Bu, of the samc stress fields as in a reas A' DG' and A' C' D',
and ABC and AC'D respectively: the principal stresses are again
eithcr positive or nil everywhere. In the area located in between
the !ines of discontinuity Dt/ and Btt, the principal directions
are Ox and Oy; a .r.r (resp. ayy) docs not depend upon :l: (resp.
y); the !ines of discontinuity Bu' and Bu and the stresses a .r .1
and a !1!1 are detennined from the conti nuit y of the stress
vcctor and from the adclitional condition a !I!J - a .r.r = 2 C' on
Du' and fl1t .
Eqs. (26), with o: = 1r /2, are derivcd and indicate thal the
stresses a.1-.r and a !J!I are either compressive or zero
everywhere. Consequently, the stress field in Figure 6 which is in
equilibrium with the axial load
NfC'D=1r+2 , T/CB = 0, M = O and complies with the tension
eut-off strength criterion (5).
lt follows thal this Joad is also the exact value for the
bearing capacity of an axially loaded soi! with a tcnsile strcngth
eut-off.
Stress field in equilibrium with an inclined centered load The
derivation of the stress field in Figure 7 is basee! on the
solution for the bearing
X'
/ / t / d~ - o2
x
fig. 7. - Bearing capacity under inclincd Joad: static
approach.
capacity on an infi nite trapezodal wedge whose stress field gs
(a:) is depicted in Figure 8. 1t is composed of the Prandtl stress
field whose principal stresses are always either compressive or nil
and of ils extension by Shicld's method , whosc description has
becn given abovc. With the notations of Figure 8, the following
expressions are dcrived on
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u'! '. u 1 1
Fig. 8. - Infinite trapezoida! wedge with apex angle 2 o:
Prandtl's stress field and Shicld's extension.
bu' for u.r.1 and uyy (Eq. 26) which determine the stress field
g.s (a) in the area Ill between bn and bn'.
(26) { u.r.r = 2 C [cos (a - 0)- 0- 1] uyy = 2 C [cos (a- 0) -
0]
l n the extended area, the principal st resses abovc the !ines
of discontinuities bu and bu' are cither compress ive or nil
everywhere. In between these lines, Eqs. (26) show thal u.1.r is
always compressive and that:
- if 0 < a < 1, Uyy is everywhere bounded and tensile (
< 2 C), - if 1 ~ a < 1r /2, rr uu is compressive within a
"column" centered on the Oy axis
which spreads out when o: increases until it occupies the entire
area in between bu and bn' when a = 1r /2. Outside this column ,
ayy is bounded and tensile ( < 2 C').
It follows thal the stress field g.s (a), for 0
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390 J. Si\LENON AND A PECKER
-for 0 < loi :::; 7r/4, although the stress fi elds of Figure
7 arc no longer compati ble with the strcngth criterion, it is not
possible to make a conclusion rcgarcling the instability of the
corresponding loacls for they might be balanccd by other stress
fields which comply with the criterion. Moreover, it has becn
proven that thesc stress fields arc stabili zed by the soit unit
weight. Jn the fotlowing, attempts are made to construct new stress
fields making the application of the Jowcr bound approach possible
with the strength criterion (5).
For loi > 1r /4, the kinematic approach of Paragraph 4. 1
proves thal the diagram is no longer valid.
New stress .field in equilibrium with an inclined centered load
Khosravi and Salenon [Khosravi, 19831 considered the discontinuous
stress field with
three zones, presented in Figure 9.
--
\ '\... \!13 \ \
\ \
t' \
\ \ \ \ t
x
Fig. 9. - Stress Jicld in cquilibrium with a ccntcred.
incline
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ULTI~IATF. BEARING CAPACITY OF SIIALLOW FOUNDATIONS. PART Il
391
whose inclination is 6 = ( 1r /2 - (3), i.e. the force F is
parai lei to the li nes of discontinuity A' t' and At.
ln view of the constraints set on C1 and Cz, these fields
obviously comply with the strength criterion for the purcly
cohesive soil.
Using the static approach with thesc stress fields, which depend
upon thrce parameters, leads to the determination of the convex
envelope of the loads (30) while the parameters (3 , Cl/C, Cz / C
vary under the constraints alrcady listed:
0 :=:; CJ/C :=:; 1,
This envelope is defi ned by the equations for the arcs il, B
and C (see Fig. 1 0): Arc A
(3 1)
Arc JJ
(32)
Arc C
(33)
{
0 :=:; (J :=:; 7r/4 hence 1rj2 ~ 6 ~ 1rj1l Cl/C = 0 (a= 0) ,
Cz/C = 1 N / C B = ( 1 - cos 2 /3) = 1 + cos 2 6 TfC JJ = sin 2/3
=sin 28
{
7r/'1 :=:; /3 :=:; 37r/8 hence 7r/4 ~ 8 ~ 1rj8 CI/C = - 1/tan
2/3 (a= /3 = -1r /4) , Cz / C = 1 N ;en= tan (J T/CB = 1
{
37r/8 :=:; (3 :=:; 1rj2 !tence 1rj8 ~ 6 ~ 0 CI/C = 1 (a = 1rj2 -
(3) , Cz/C = 1 N 1 c n = -2 cos 2 f3 ( 1 - cos 2 /3) = 2 cos 2 8 (
1 + cos 2 8) T/CB = - sin 11/3 = sin 46
When thesc stress fields arc analysed with regards to the
strength criterion (5) for a soil with a tension eut-off, it is
found thal the most sensitive stress, in area 2, is an which
vanishes on the arc ;1 and is compressive on the arcs B and C. In
contras!, a y y is always nil in arcas 1 and 3. It follows that the
arcs A, Band C in Figure 10 effectively yield a lower boum! for the
bearing capacity of an inclined centric Joad on a soi! with a
tension eut-off.
The accuracy of the lower boum! is improved by taking advantagc
of convexi ty properties using the exact value of the bearing
capacity for an axial Joad, yielding the arc D in Figure 1 O.
EUROPEAN JOURNAl. 0 1' ~IECIIANICS, MSOI.IDS. VOL. 14, N 3,
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392 J. SALENON AND A. PECKER
N/ CB
B
T/ CB
Fig. 10. - Ccntered, inclined Joad: convex envelopc of the lowcr
bound estimates.
Synthesis of the results for a centered load Comparing these
results with thosc using the kinematic approaches presentee! in
Figure 5, il appears that for ef B = 0, with the exception of
the case of the axial Joad (8 = 0) which has already been
mentioned, both approaches give identical results for 1r /8 ~ 181 ~
1r /4 in the fonn of the vertical segments:
n/4 ~ N/C'B ~ 1 + ../2, ITI/C'B = 1 and for 1r / 4 ~ 181 ~ 1r /2
where they both yie ld the sa me quarter o f a circle who se
equation is given by (24) or (31) and its symmetric. The bearing
capacity is computed exactly.
The identity of the two results reveals that the mechanisms Fo
consideree! in the kinematic approach and the stress fields which
defi ne the arcs A and B in Figure 10, are associated with each
other as shown in Figure 11 . For 1r /4 ~ 181 ~ 1r /2, the
foundation rests on the " soit column" t'A' At (area 2) which is in
unconfined compression pmallel to the force F. The tension eut-off
in the strength critcrion makes a velocity discontinuity
A' 1
- 0- ' 1 ' '
A x 1
' - 0---....
' 1 0 ........_-2C ' ,. 0 '
' t' , t' ,
\ \ ' 0
'.
t' \
\ 0 \
\ \ t \
x
Fig. Il . - Vclocity field F0 and associated slress field.
EUROPEAN JOURNAL OF ~IECHANICS. NSOLIDS. VOL. 14, N 3, 1995
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ULTIMATE BEARING CAPACITY OF SHALLOW FOUNDATIONS. PART Il
393
[U] = V possible in the soi! medium immediately below the
interface A' A, acting at an inclination x = 2 5.
For n /8 ::::; 151 ::::; nf 4, the soi! column parallel to the
force F in a rea 2 is laterally confined by areas 1 and 3; the
principal stress directions are inclined at "Tn /4 on Ox , and the
velocity field is a slipping mechanism within the soit below the
interface A' A, and parallel to A' A.
Referring to Fig me 10 and recalling the stabilizing effccl of
the gravity forces within the soil medium on the stress fields of
Figure 7, it also follows that, when 'Y D f C is large enough, the
lower bou nd estimate of the beming capacity for 0 < 5 < n /8
is improved: the lower bound obtained for the classical purely
cohesive soit can then be retained for the cohesive soit wilh a
tension eut-off.
4.3 . STATIC APPROACHES FOR AN ECCENTRIC LOAD
The lower bounds for eccentric loads are obtained using the
reduced width foundation melhod and taking advantage of the
convexity properties in the (JV, T, M) space stated in Paragraph
3.2.
The results are presented in Figure 12 where it appears that the
convexity property, which has been numerically implemented, does
improve the lower bound estimate: this improvement is particularly
noticeable for the maximum value of Tf C B, for a fixed ef D .
N/ CB
T/ CB Fig. 12. - Dearing capacity undcr inclined eccentric
Joad.
Solid !ines: upper bound estimatcs. Dotted !ines: lower bound
estimates.
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394 J. SALENON AND A. PECKER
4.4. BEARING CAPi\CITY FOR AN ECCENTRIC INCLINED LOAD ON A
PURELY COHESIVE SOIL \VITH A 'l'ENSILE STRENGTH CUT-OFF
Figure 12 compares the results of both approaches showing the
upper and lower bound estimates for the beming capacity for each
indicated value of e/ B. This diagram may be compared with the
corresponding one for the classical purely cohesive soi!.
As statecl previously, for a centered Joad ( The comparison with
the results obtained for the classical purely cohesive sail for the
centered Joad ( ej B = 0) shows that, for 6 = 0 and 1r /8 < 6
< 1r /4, the beming capacities are equal to each other,
regardless of the strength criterion , they do not depend upon the
soil unit weight; for 0 < 6 < 1r /8, the difference, if any,
between both values, cloes not exceed 7 to 8% for 1 B /C' = O. The
most significant difference appears for 1r /4 < 161 < 1r /2.
As a matter of fact, within this range of 6, the beming capacity
diagram is nothing but the representation, using axes (N/C'B , T/C'
fl ), of the comprehensive criterion for the interface .111 Jl
fSalenon , 1983], i. e. of the critcrion expressed in tenns of ayy
and a.,.y which simultaneously takes into account the limitations
imposee! by the interface itself and by the constituent materials
on either side of the interface (in the present case, only the soi!
is consideree! , since the foundation is assumed to be rigicl).
This is not merely a coincidence since the mechanisms corresponding
to the arcs on the beming capacity diagram (unilateral mechanism
with x = 1r /2, and mechani sm Fo) imply a velocity cliscontinuity
across the interface.
Tt must be 110tcd that the most important parameter is the
tension eut-off in the soil, since in both cases, the interface
does not present any resistance in tension.
Referring to Figure 9 in the companion paper, where the diagram
obtained from Meyerhof [Meyerhof, 1963], "parabolic" formula is
drawn, it appears thal for 1r /3 < 161 < 1r /2, the values
obtained using this formula slightly overest imate the bearing
capaeity.
As the Joad eccentricity increases, the difference between both
bounds inereases signiflcantly, despite the slight improvement
achieved by utili zing the convexity propertics in the results
obtained from the recluced width founclation mcthod. Comparing
these results with the results for the purely cohesive soi!, it
appears thal:
- the beming capacitics are considerably lower for large Joad
inclinations, - for 1r / 12 s; 161 s; 1r /6, the lower bounds
computecl for the same cf B values are
comparable. Furthermore, the symmetry of the lower bound
diagrams for e/ fl f. 0 is a result of
their construction methods; on the othcr hanc! , it may be
observed thal the upper bound cliagrams appear to be more
symmetrical in Figure 12 than they did for the classical purely
cohesive soil ; however, a definite conclusion on thal point would
be risky.
Finally, it wi ll aga in be recalled, as explainecl in the
companion paper, th at in the case of the interface which exhibits
a strength criterion which cliffers from (6), ali the
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ULTI~IATE REARING Ci\Pi\CITY OF SHAU .OW FOUNO,\TIONS. PART Il
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cliagrams in Fig me 12 should be truncatcd by the corresponding
curvcs which express the interface strength criterion in tenns of N
j C'B and 'J 'jC n insteacl of ayy and a.,.y: e.g. 1'1'1/ C 13 ::=;
N tan (Pl/ C n for a Coulomb strengh cri teri on with a fri ction
angle c/J t. Obviously, this is in line with what has just bcen
stnted about the comprehensive strength criterion of the
interface.
5. Conclusions
Within the same framework of the yielcl design thcmy which was
adoptcd for the case of a classical purely cohesive soi! in the
companion paper [Salenon & Pecker, 1995], the ultimate bcaring
capncity of n founclation resting on a cohesivc soil with no
!ensile strength has been studied for an inclined eccentric
load.
Due to the strength criterion of the soil, the bearing capacity
in the general case of inclination and eccentric ity can no longer
be proven to be independent from the gravity forces, whose effcct
will nlways be a stabilizing one, and is governcd by the
adimensional factor 1 B / C. However, for the case of a centered
load with a small incl ination, indcpcndence has been proven or may
be anticipated.
As a general comment, the effect of the tension eut-off
condition can be perceivccl with in the soit medium and in the
interface. Therefore, new stress fields were constructcd in orcier
to comply with the additional tension eut-oiT condition. New
velocity fields were then implementee! which could take advantagc
of the expression of the maximum resisting work associatecl with
the comprehensive strength criterion of the interface.
The foundation bearing capacity has bccn computed, exactly or,
at least, with a very high accuracy, for a centered Joad. For an
eccentric Joad the bcaring capacity has bccn brackctecl between
lower and upper bounds, as a function of the Joad eccentricity and
Joad inclination. From a practical standpoint, in vicw of the
safety factors uscd in the design of foundations under vertical,
centered, loads, the implicati ons of the inclinat ions and
eccentricities are usually small.
The results prcsented herein have bccn used for the seismic
analyses of foundations and arc rcported in LPecker & Salcnon,
199 1 J.
Sincc both cases studied in this paper and in the companion one,
represent extrcme conditions for the tensilc resistance for the
purely cohcsive soil, it can be statccl thal the results bracket
the variations of the bearing capacity as a function of the soil
!ensile strength.
Acl
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396 J. SALENON AND A. PECKER
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(Manuscript rcccived Septcmbcr 6, 1993; rcviscd Marcl1 23, 1994;
accepted May 16, 1994.)
EUROPEAN JOURNAL 01' MECHAN!CS, A/SOLIDS, VOL. 14, N 3, 1995