Salençon, J. and Pecker EC8 Foundation Bearing Capacity

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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, SALENÇON *and A. PECKER ** ABSTRJ\Cf. - The problem of determining the bearing capac ity of a stri p foot ing resting on the surface of a homogeneous hal f space and su bjec tcd to an incl ined, eccentrie Joad, is solve d within the framework of the yield design theory assumi ng thal the soi! is purcly cohesive without tens il e strength acco rd ing to Tresca's strength eriterion wi th a tension eut-of f. The soi! foundati on inte rface is also purely cohe sive, in tenns of the homologous strength eriterion with a tension eut-off. As in a eompanion paper [Salençon & Peeker, 1995). both the static and th e kinematie approaches of the yicld design theory are used. New stress fields are eonstrueted, in order to comply with the condition of a te nsion eut-off within th e soi! medium, and new lower bounds are determined as substitutes to th ose given in the eornpanion paper. Veloeity fields taking advantage of the tensi on eut-off contribution in the expression of the max imum resisting work arc also impl cmented, giving new lower bound s. As may be expceted from eommon sense and from the ge neral rcsults of the th eory , it appears th al the te nsion eut-off condition within the so i! medium results in lower va lues of th e bcaring capae ity of the fou ndation, and tha t th e grnv ity forces acting in the soi! mass have a stabilizing effect. 1. Introduction The problem solved in the cm-rent study, relates to the ultimate bem·ing 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 th e01·y [Salençon, 1983, 1993 ], as detailed in the compani on paper [Salençon & Pecker, 1995]. In the companion paper, the prob lem is studied assuming the so it to be plll·ely 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 tensile strength. In the present case, a tension eut-off condition is added to the soit strength criterion, describing a pm·ely cohesive medium with no tensile strength. As in the companion paper, other interface strength condi ti ons ca n be easily taken into account. * Laboratoire de Mécanique des Solides, École Polytechnique, 91128 Palaiseau Cedex, France. ** Géodynamique et Structure, 157, rue des Blains, 92220 Bagneux, France. EUROPEAN JOURNAL OF MECHANl CS. NSOLI DS. VOL 14, 3, 1995 0997-7538/95/03/S 4.00 Gauthier-Villars

description

A very good paper showing the derivation of bearing capacity of strip foundations, based on which the EC8 equations are adopted.

Transcript of Salençon, J. and Pecker EC8 Foundation Bearing Capacity

  • 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.

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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

  • 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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    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

  • 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

  • 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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    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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    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

  • 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

  • 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.

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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.

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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 395

    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

  • 396 J. SALENON AND A. PECKER

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    HANSEN B. J. , 1961 , A general formula for bearing capacity, Bull. 8, Danish Geotechnicallnstilllte, Copenhagen. HANSEN B. J. , 1970, A rcvisecl and extemled formula for bearing capacity, Ru//. 28, Danish Gl'otec/micallnstitute,

    Copcnhagcn. KIIOSRAVI Z., 1983, tude thorique et exprimentale de la capacit portante des fondations superficielles. Thse

    Dr. lng., cole Nationale des Ponts et Chausses, Paris. MEYERHOr G. G., 1963, Some recent rcscarch on the bearing capacity of foundations, Canadian Geoteclmical

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    sollicitations dynamiques, C.R. Recherche M.R.E.S., 88-F-0212. PECKER A., SALEI\'ON J., 1991 b, Seismic bearing capacity of shallow strip foundations on clay soils,

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    Mechanics, 25, 4, 643-648. SALENON J. 1983, Calcul la mpture et analyse limite, Presses de I'E.N.P.C. , Paris. SALENON J. 1993, Yield design: a survey of the theory, in Evaluation of the global bearing capacities of

    stmctures, G. Sacchi-Landriani, J. Salenon Ed., Springer Verlag, \Vien , New York. SALENON J ., PECKER A., 1995, Ultimate bearing capacity of shallow foundations un der inclincd and cccentric

    Joad; Part !: pu rely cohesive soil, European Jou mal of Meclumics, 1VSolids, 14, 3, 349-375. SHIEI.D R. T., 1954, Plastic potential theory and Prandtl bearing capacity solution, J. Appl. Mech. , trans. ASME,

    21, 193-194. SWA~Il SARAN, ARGAWAL R. K., 1991 , Bearing capacity of eccentrically oblique) y loaded footing, J. Geoteclmical

    Eng., 117, Il , 1669- 1690. TRAN Vo NHtE~I, 1968, Terme de surface de la force portante limite d' une fondation charge incline excentre

    par la mthode du coin triangulaire minimal, C. R. Acad. Sei. Paris, 267, 137- 140. TRAN Vo NHIEM, 1971 , Force portante limite des fondations superficielles el rsistance maximale l' arrachement

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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