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Exact bearing capacity calculations using the method of characteristics

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This paper discusses the use of the method of characteristics (commonly referred to as the slip-line method) to solve the classic geotechnical bearing capacity problem of a vertically loaded, rigid strip footing resting on a cohesive-frictional halfspace. It would appear that, contrary to popular belief, the method of characteristics can be used to establish the exact plastic collapse load for any combination of the parameters c, φ, γ, B and q - including the infamous 'Nγ problem'. This applies to footings of arbitrary roughness, though only the extreme cases (smooth and fully rough) are considered in detail here. Many analytical and numerical techniques can be used to calculate the vertical bearing capacity of a rigid strip footing. These include the method of characteristics, upper bound and limit equilibrium calculations based on assumed mechanisms, finite element formulations of the lower and upper bound theorems, and conventional finite element or finite difference analyses employing displacement elements. (Note that here and throughout the paper, all references to lower bound, upper bound and exact plasticity solutions imply the assumption of an associated flow rule.) If the self-weight of the soil is neglected, the method of characteristi cs is the preferred technique because it allows the construction of a simple lower bound stress field from which the bearing capacity can be determined in closed algebraic form. The resulting expressions for Nc and Nq are universally adopted in design when applying Terzaghi's equation: 1 uu 2 cq qQB c Nq NBN γ = =+ +γ
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Exact bearing capacity calculations using the method of
characteristics
C.M. Martin
Department of Engineering Science, University of Oxford
Keywords: bearing capacity, shallow foundation, cohesive-frictional, limit analysis
ABSTRACT: This paper discusses the use of the method of characteristics (commonly referred to
as the slip-line method) to solve the classic geotechnical bearing capacity problem of a vertically
loaded, rigid strip footing resting on a cohesive-frictional halfspace. It would appear that, contrary
to popular belief, the method of characteristics can be used to establish the exact plastic collapse
load for any combination of the parameters c, φ, γ, B and q including the infamous ‘Nγ problem’.
This applies to footings of arbitrary roughness, though only the extreme cases (smooth and fully
rough) are considered in detail here.
1 Introduction
Many analytical and numerical techniques can be used to calculate the vertical bearing capacity of
a rigid strip footing. These include the method of characteristics, upper bound and limit equilibrium
calculations based on assumed mechanisms, finite element formulations of the lower and upper
bound theorems, and conventional finite element or finite difference analyses employing
displacement elements. (Note that here and throughout the paper, all references to lower bound,
upper bound and exact plasticity solutions imply the assumption of an associated flow rule.) If the
self-weight of the soil is neglected, the method of characteristics is the preferred technique
because it allows the construction of a simple lower bound stress field from which the bearing
capacity can be determined in closed algebraic form. The resulting expressions for Nc and Nq are
universally adopted in design when applying Terzaghi’s equation:
1
uu 2
cq
qQBcNqNBN
γ
==+ (1)
There are several ways of showing that the stress fields for Nc and Nq (which have identical
meshes of characteristics) can be extended throughout the semi-infinite soil mass without violating
equilibrium or yield. For weightless soil it is also straightforward to derive a rigorous upper bound
on the bearing capacity that coincides precisely with the lower bound, and furthermore it can be
shown that the roughness of the footing has no effect on the collapse load. When the soil has self-
weight, however, the status of (numerical) solutions obtained using the method of characteristics is
far less certain; there is also debate over how footings of different roughness should be handled.
These issues are frequently used as a justification for adopting the other calculation techniques
mentioned above. Indeed, it would be fair to say that the method of characteristics has acquired
something of a credibility problem in recent years. The view of Frydman & Burd (1997) is typical:
“Evaluation of the third coefficient, Nγ, requires that the self-weight of the soil be considered. In this
case, stress characteristics fields may be obtained numerically, but it is not generally clear whether
the resulting slip mechanisms (which include curved slip lines) are also kinematically admissible,
or whether the stress field can be extended outside the identified plastic zone. Consequently, it is
not clear that the solutions obtained using this technique … are exact, or even lower bounds to the
exact solution … A further complication exists in the use of the method of characteristics when the
base of the footing is assumed to be fully rough. In this case … the precise nature of the boundary
conditions that should be applied at the base of the footing is not clear”. Similar observations have
been made by a number of other authors (see e.g. Ukritchon et al., 2004; Hjiaj et al., 2005).
It is true that, when the soil has self-weight, previous studies of bearing capacity using the method
of characteristics have often lacked theoretical rigour. Sometimes this has been freely
acknowledged: Cox (1962) admits that his analyses “are concerned solely with the calculation of a
region of the stress field sufficiently extensive to determine the stresses on the indenter, it being
assumed that the rigorous mathematical justification of the results obtained is merely a matter of
additional computation”, while Salençon & Matar (1982) acknowledge that “To be quite rigorous …
it would be necessary to complete the stress field in the entire soil layer … and to build a velocity
field associated to this stress field by using the mathematic rule of normality … for reason of
simplicity this question will be left aside here and afterwards”. In many other studies, the important
questions of stress field extensibility and kinematic admissibility have only been mentioned in
passing, if at all (Lundgren & Mortensen, 1953; Sokolovskii, 1965; Larkin, 1968 and discussers;
Graham & Stuart, 1971; Ko & Scott, 1973; Bolton & Lau, 1993; Kumar, 2003). The paper by Davis
& Booker (1971) is a notable exception in that both issues are acknowledged, and backed up with
appropriate calculations (or at least reports thereof). Although their work represents perhaps the
most rigorous treatment of the strip footing problem to date, it is unsatisfactory in several respects.
First, they concentrate on the problem of a fully rough footing, on the grounds that “adaptation of
the analysis to the smooth case is relatively straightforward”. They also appear to be under the
impression that the smooth footing solutions of Cox (1962) are rigorous, when in fact they are not
(see above). Second, their discussion and illustration of the velocity field is somewhat obscured by
the consideration of a general non-associated flow rule, ψ φ. Third, although they claim “In all
cases it was then found that the stresses satisfied equilibrium and did not exceed the strength of
the material”, a completed stress field is only depicted for one (unspecified) case when φ = 30°.
Fourth, they simply declare that “the velocity field was evaluated for each problem and the rate of
plastic work calculated at the nodes of the characteristic net. In every case this rate was nowhere
negative thus confirming the kinematic validity of the solutions”. No further details of these
calculations are given, and no actual upper bounds are evaluated, so their claims to have
established exactness are not totally convincing.
The other major area of concern relates to the boundary conditions that should be applied when
using the method of characteristics to solve problems involving rough (or semi-rough) footings. It is
certainly true that there is confusion in the literature, dating back to Caquot & Kerisel (1953). Other
questionable analyses of rough footings have been made by Graham & Stuart (1971), Bolton &
Lau (1993) and Kumar (2003). These authors make various a priori assumptions about the stress
field under the footing, often involving a wedge of non-plastic soil having some predetermined
shape. Such assumptions are unnecessary, and invariably lead to erroneous results when the soil
has self-weight. In fact, the shape of this wedge or ‘false head’, which may or may not span the full
width of the footing, emerges naturally during the iterative process needed to determine the extent
of the mesh of characteristics for a particular combination of parameters. The correct procedure
has been outlined (and used) by Lundgren & Mortensen (1953) and Davis & Booker (1971), though
admittedly the clarity of their explanations and figures could have been better.
Within the limited space available, this paper attempts to address some key questions concerning
bearing capacity calculations using the method of characteristics:
Is the collapse load obtained from the incomplete stress field exact, even when γ > 0?
What makes the Nγ problem so difficult to solve? Can exact values of Nγ be obtained?
As mentioned in the Abstract, only the extreme cases of smooth and fully rough footings will be
considered in the example problems. Some values of Nγ for semi-rough footings will be given, but
their derivation will not be discussed here (generally speaking, intermediate roughness is no more
complicated than full roughness). All analyses are restricted to plane strain geometry and are
based on the classical assumptions of rigidperfectly plastic soil behaviour, i.e. a linear Mohr-
Coulomb yield envelope with an associated flow rule (ψ = φ). The latter assumption means that the
stress and velocity characteristics coincide; they indicate the planes on which the Mohr’s circle of
stress touches the yield envelope, and the directions in which the extensional strain is zero.
2 Example problems
Stress and velocity calculations for vertically loaded footings using the method of characteristics
have been implemented in the publicly available computer program ABC (Martin, 2004). The user
manual gives full details of the relevant theory, the numerical methods employed, and the
extensive collections of test problems that have been used for validation. The examples in Table 1
will be used to illustrate the process of establishing the exact bearing capacity. All three problems
have been studied before the first by Cox (1962) and the other two by Salençon & Matar (1982)
but these previous analyses were confined to calculation of the stress field (without completion).
Figure 1 shows the stress and velocity fields for the three problems, using coarse meshes of
characteristics for clarity. In Figure 1(a) the footing is smooth, so every α characteristic proceeds
from the soil surface to the underside of the footing. An iterative process is used to determine the
extent of the mesh, such that the final (i.e. outermost) α characteristic intersects the footing on the
axis of symmetry. In the associated or ‘consistent’ velocity field, there are discontinuities along the
underside of the footing and along the outside of the mesh (the exterior soil being assumed rigid).
In Figure 1(b) the footing is fully rough, but for this combination of parameters, the need to satisfy
the symmetry condition on the z axis means that no α characteristics can proceed to the footing.
Instead they all terminate in mid-soil, and there is never any need to apply the rough footing
boundary condition. The iterative adjustment process now involves two variables, namely, the
extent of the mesh along the soil surface, and the aperture of the fan. In the consistent velocity
field, the blank false head region moves down with the footing as a rigid body, giving rise to a
velocity discontinuity along the innermost β characteristic. In Figure 1(c) the footing is again rough,
but now the problem parameters are such that the inner α characteristics can proceed to the
footing. The β characteristics that emerge from the endpoints are tangential to the footing, and on
this part of the interface the full shear strength of the soil is mobilised. The remaining α
characteristics terminate in mid-soil, resembling those of the second example. Construction of this
type of mesh requires the iterative adjustment of two distances along the soil surface, delineating
the α characteristics that proceed to the footing from those that do not. In the consistent velocity
field the blank false head region again moves down with the footing; there are velocity jumps along
the innermost β characteristic and along the part of the footing where full roughness is mobilised.
Having constructed the stress and velocity fields, two separate calculations of the bearing capacity
can be performed: a ‘stress calculation’ involving integration of the boundary tractions under the
footing (see Figure 1), and a ‘velocity calculation’ involving evaluation of the internal and external
work rates. It is important to realise that these two calculations are approximate they incorporate
errors introduced by the use of finite difference (rather than exact) equations when constructing the
stress and velocity fields, as well as errors resulting from the use of numerical integration in the
calculations themselves. It is of course possible to perform a sequence of increasingly accurate
analyses, each one involving a finer mesh of characteristics, until the results converge to the
desired precision. Until this has been done, however, it is not appropriate to refer to the stress and
velocity calculations as lower and upper bound calculations (e.g. as a mesh is refined, the result of
the ‘upper bound’ calculation may converge from below). Table 2 shows the results obtained.
Despite the coarseness of the initial meshes, the stress and velocity calculations for each problem
converge very quickly as the subdivision counts are doubled, quadrupled, etc. The original results
reported by Cox (1962) and Salençon & Matar (1982) were 37.8, 1.61×103 and 44.9 kPa, so the
level of agreement is encouraging. Even more encouraging is the fact that, in all three problems,
the stress and velocity calculations converge to identical values.
Table 1. Example problems.
Problem no. c
[kPa] φ
[°] γ
[kN/m3] Footing type B
[m] q
[kPa]
1 1 20 10 Smooth 2 0
2 16 30 18 Rough 4 18
3 1 + 2.5z 10 16 Rough 4 0
Figure 1. Stress and velocity fields for example problems.
Table 2. Convergence of qu (kPa) in example problems.
Mesh Problem 1
stress calc. Problem 1
velocity calc. Problem 2
stress calc. Problem 2
velocity calc. Problem 3
stress calc. Problem 3
velocity calc.
Initial (Fig. 1) 37.75 37.77 1.627×103 1.627×103 44.97 44.99
× 2 37.76 37.77 1.626×103 1.626×103 44.99 44.99
× 4 37.76 37.76 1.626×103 1.626×103 44.99 44.99
× 8 37.76 37.76 1.626×103 1.626×103 44.99 44.99
× 16 37.76 37.76 1.626×103 1.626×103 44.99 44.99
× 32 37.76 37.76 1.626×103 1.626×103 44.99 44.99
(a)
(b)
(c)
x
z
β
α
Figure 2. Completed stress fields for example problems.
Once converged, the velocity calculations in Table 2 constitute strict upper bound solutions. To
confirm that the converged stress calculations are strict lower bounds (thereby establishing
exactness) the stress fields of Figure 1 must be completed. This can be done by following Davis &
Booker (1971) and using an ingenious extension strategy originally developed by Cox et al. (1961)
for weightless soil. The basic idea, as shown in Figure 2, is to extend the stress field downwards
and outwards by ‘bouncing’ characteristics off the z axis, continuing as far as the minor principal
stress trajectory originating from the extremity of the original mesh. The stresses in the extension
need not be fully plastic (except at the boundary with the original mesh), but this is a convenient
assumption since the governing equations and finite difference routines then remain unchanged.
On the other side of the minor principal stress trajectory, the stress field is comprised of columns or
‘spokes’ of uniaxial stress, superimposed on a hydrostatic stress field. To be more precise, if the
principal stresses (compression positive) at a point (x0, z0) on the interior of the trajectory are σ1
and σ3, then the principal stresses at any point (x, z) along the supporting spoke are given by
(
)
1,ext103,ext
;
zzzq
σ=σ+γσ=γ+
(2)
In the limiting case of an infinite number of spokes, the stress field outside the minor principal
stress trajectory satisfies equilibrium (Cox et al. 1961). It is also easy to show that in a given spoke,
the yield criterion will be satisfied everywhere provided it is satisfied at the starting point (x0, z0).
The verification of the extension therefore reduces to a check on the yield criterion at the start of
each spoke. For this it is convenient to define a utilisation factor that compares the size of the
Mohr’s circle of stress with the maximum allowable size at the same level of mean stress:
( )
10
10
Utilisation
2cossin
zq
czq
σγ−
=
φ+γ
(3)
If all the utilisation factors are less than or equal to unity, the extension is safe, and the bearing
capacity calculated from the incomplete stress field is confirmed as a strict lower bound. Note that
the yield criterion is checked at all points on the trajectory, not just those where labels are printed
in Figure 2. For rough footings it is also necessary to extend the stress field into the false head
region, whether it spans all or part of the footing. This is a straightforward operation that involves
bouncing characteristics off the z axis, followed by truncation of the stress field along the underside
of the footing. The completed stress fields in Figure 2 are all statically admissible, and remain so
during refinement, so the converged results in Table 2 are formally established as exact.
3 Evaluation of Nγ
The bearing capacity factor Nγ depends on the friction angle of the soil and the roughness of the
footing (it also depends on the dilation angle of the soil, but the aim here is to obtain definitive
solutions for the case ψ = φ). Many graphs and tables of recommended Nγ values have been
published, but the range covered by these solutions is large, especially for high friction angles and
rough footings. It is clearly desirable to resolve this issue, at least for the case of an associated
flow rule, when the uniqueness theorem guarantees that there is an unequivocal, exact value of Nγ
for each combination of friction angle and roughness. While there is no need for the bearing
capacity factors used in design to be particularly accurate, if the exact values of Nγ can be found,
then it would surely make sense to use them (just as the exact values of Nc and Nq are used).
When employing the method of characteristics, it is numerically convenient (see e.g. Larkin, 1968;
Graham & Stuart, 1971; Davis & Booker, 1971; Bolton & Lau, 1993; Martin, 2004) to consider the
Nγ problem as a limiting case of bearing capacity on cohesionless soil:
uu
00
2
uu
lim =lim
with 0
lim2=lim2
qBqBq
BqBq
NqqQqB c
NqBQB
γγ→
γγγ→∞
=
=
=γγ
(4)
It is also possible to consider a cohesive-frictional soil with no surcharge, letting the cohesion tend
to zero, though the approach in equation (4) is usually preferred because of its transparent
physical interpretation (q/γB corresponds directly to the familiar embedment ratio D/B).
Table 3. Determining Nγ from analyses in which c = 0 and γB/q (see equation (4)).
γB/q φ = 10°
Smooth φ = 10°
Rough φ = 30°
Smooth φ = 30°
Rough φ = 50°
Smooth φ = 50°
Rough
103 0.2938 0.4495 7.780 14.91 374.4 745.5
104 0.2826 0.4354 7.670 14.77 372.3 743.2
105 0.2811 0.4334 7.655 14.76 372.0 742.9
106 0.2809 0.4332 7.653 14.75 372.0 742.9
109 0.2809 0.4332 7.653 14.75 372.0 742.9
1012 0.2809 0.4332 7.653 14.75 372.0 742.9
If highly accurate values of Nγ are sought obviously anything more than 2-digit precision is purely
for academic interest then γB/q needs to be surprisingly large. For example, Table 3 shows that
γB/q must be at least 106 in order to obtain Nγ factors that are correct to 4 digits over the range φ =
10° to 50°. Note that these results are specific to program ABC, which uses a particular adaptive
subdivision strategy to maintain the accuracy of the stress field calculation near the edge of the
footing. This can be seen in Figure 3 (drawn for γB/q = 109) where the light-coloured characteristics
are those resulting from an adaptive subdivision. Many researchers have noted that as γB/q
becomes large, the usual fan zone becomes severely distorted, and degenerates into a single β
characteristic as γB/q (see arrows in Figure 3). For accurate results, an extremely fine mesh
is needed in the vicinity of the footing edge, and adaptive subdivision (Larkin, 1968; Graham &
Stuart, 1971; Martin, 2004) can achieve this far more efficiently than a fine but uniform spacing
(Bolton & Lau, 1993; Kumar, 2003). Another peculiarity of the Nγ problem is the consistent velocity
field: as γB/q , the velocities emerging close to the edge of the footing also approach infinity. If
the entire velocity field is plotted to scale, as in Figure 3, the picture is not very informative.
Figure 3. Stress and velocity fields for Nγ problem (c = 0, φ = 30°, γB/q = 109, smooth and rough).
(a)
(b)
Each bearing capacity in Table 3 is a converged solution obtained by systematic refinement of the
relevant mesh of characteristics. Each value is also exact for its combination of γB/q, φ and
roughness (in all cases it is found that the converged stress and velocity calculations agree, and
the stress field is extensible). For the case φ = 30° and γB/q = 109, details of the convergence
behaviour are given in Table 4. The stress results converge quickly, but the velocity calculations
now require several refinements before the desired 4-digit precision is achieved (cf. Table 2). In
view of the comments on the velocity field in the previous paragraph, this is not surprising. Having
determined that the converged stress and velocity calculations agree, completion of the stress field
is the final formality needed to confirm exactness. This is illustrated in Figure 4, again for the case
φ = 30° and γB/q = 109. The utilisations are acceptable, and remain so with further refinement.
The parametric study in Table 3 suggests that, when using the approach of equation (4), a γB/q
ratio of 109 is comfortably large enough to evaluate Nγ to 4-digit precision. This has been confirmed
for friction angles in the range 1° to 60°, for both smooth and rough footings. A selection of Nγ
values is given in Table 5, and these have all been confirmed as exact solutions in the manner of
Table 4 and Figure 4 (readers wishing to verify this are welcome to download ABC and run it with
the appropriate parameters). A fuller version of Table 5, covering the range φ = 1° to 60° in 1°
intervals and giving Nγ to 6-digit precision, will appear in a forthcoming journal paper; it can also be
It is worth pointing out that numerical values of Nγ for any friction angle and roughness can be
obtained much more quickly without using the method of characteristics. When c = q = 0 exactly,
the equations governing the stress field reduce to a pair of ordinary differential equations that can
be solved by Runge-Kutta or similar techniques (Lundgren & Mortensen, 1953; Sokolovskii, 1965;
Booker, 1970; Salençon, 1977). This approach allows an Nγ factor to be calculated to very high
precision in a fraction of the time that it takes to construct (and then systematically refine) a mesh
of characteristics for a large value of γB/q. It is therefore ideal for compiling a large table of values,
and the calculations for the above-mentioned spreadsheet were performed in this way. Although it
gives the same result for Nγ as taking the limit in equation (4), the drawback of the alternative
approach is that it effectively amounts to the calculation of an incomplete stress field. If proof of
exactness is required, it is necessary to revert to the method of characteristics, at least to construct
the velocity field (it may well be possible to complete the stress field by solving ODEs only).
Table 4. Convergence in Nγ problem (c = 0, φ = 30°, γB/q = 109).
Mesh Smooth
stress calc. Smooth
velocity calc. Rough
stress calc. Rough
velocity calc.
Initial (Fig. 3) 7.645 7.682 14.72 14.82
× 2 7.651 7.660 14.75 14.77
× 4 7.653 7.655 14.75 14.76
× 8 7.653 7.653 14.75 14.76
× 16 7.653 7.653 14.75 14.75
× 32 7.653 7.653 14.75 14.75
Table 5. Selected exact values of Nγ.
φ
[°] Smooth δ/φ = 1/3 δ/φ = 1/2 δ/φ = 2/3 Rough
10 0.2809 0.3404 0.3678 0.3929 0.4332
20 1.579 2.167 2.411 2.606 2.839
30 7.653 11.75 13.14 14.03 14.75
40 43.19 73.55 80.62 83.89 85.57
50 372.0 690.8 728.9 739.8 742.9
Figure 4. Completed stress fields for Nγ problem (c = 0, φ = 30°, γB/q = 109, smooth and rough).
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50 60
φ [°]
Nγ / Nγ,rough
Smooth
/
φ
= 1/3
/
φ
= 1/2
/
φ
= 2/3
Figure 5. Influence of roughness on Nγ.
Figure 5 shows the effect of roughness on Nγ. Rather surprisingly, footings with interface friction
angles as small as φ/3 still provide at least 75% of the fully rough capacity, regardless of φ. The
smooth footing also exhibits intriguing behaviour at large friction angles, its bearing capacity
approaches exactly half of the rough capacity. The mathematical reason for this is by no means
obvious, but the circumstantial evidence is compelling (the capacity ratio for 60° is 0.500043).
One final remark on the Nγ problem: for rough or semi-rough footings, the false head never spans
the full width of the footing, i.e. the solutions are always of the type shown in Figures 1(c) and 3(b).
Since the fan is degenerate, it is obviously impossible to have a solution of the type in Figure 1(b).
4 Conclusions
The method of characteristics is not a versatile technique, but the strip footing bearing capacity
problem is one that it can solve, and solve exactly when ψ = φ. Apart from the examples presented
here, several hundred cases covering the full spectrum of soil and footing parameters have been
analysed, and the results obtained have always been confirmed as exact plasticity solutions
(though sometimes a small modification to the basic stress field extension strategy is required).
Proponents of other techniques would do well to examine the accuracy of their own solutions for
this fundamental benchmark problem, especially with regard to the bearing capacity factor Nγ (e.g.
the rough footing results of Frydman & Burd (1997) are all high by 40 to 80%, and about one-third
of the supposedly rigorous lower bounds of Hjiaj et al. (2005) lie slightly above the exact values).
5 Acknowledgements
This paper was written while the author was visiting the Centre for Offshore Foundation Systems in
Perth. The financial support arranged by Prof. Mark Randolph is gratefully acknowledged.
6 References
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J., 30(6), 1024-1033.
Booker J.R. 1970. Applications of theories of plasticity to cohesive frictional soils. PhD thesis, University of Sydney.
Caquot A., Kerisel J. 1953. Sur le terme de surface dans le calcul des fondations en milieu pulvérulent. Proc. 3rd Int. Conf. on
Soil Mech. and Found. Eng., Zurich (Switzerland), 1, 336-337.
Cox A.D. 1962. Axially-symmetric plastic deformation in soilsII. Indentation of ponderable soils. Int. J. Mech. Sci., 4, 371-380.
Cox A.D., Eason G., Hopkins H.G. 1961. Axially symmetric plastic deformation in soils. Proc. R. Soc. London (Ser. A), 254, 1-45.
Davis E.H., Booker J.R. 1971. The bearing capacity of strip footings from the standpoint of plasticity theory. Proc. 1st Australia-
New Zealand Conf. on Geomech., Melbourne (Australia), 276-282.
Frydman S., Burd H.J. 1997. Numerical studies of bearing-capacity factor Nγ. J. Geotech. Geoenv. Eng., 123(1), 20-29.
Graham J., Stuart J.G. 1971. Scale and boundary effects in foundation analysis. J. Soil Mech. Found. Div., 97(11), 1533-1548.
Hjiaj M., Lyamin A.V., Sloan S.W. 2005. Numerical limit analysis solutions for the bearing capacity factor Nγ. Int. J. Sol. Struct.,
42(5-6), 1681-1704.
Ko H-Y., Scott R.F. 1973. Bearing capacities by plasticity theory. J. Soil Mech. Found. Div., 99(1), 25-43.
Kumar J. 2003. Nγ for rough strip footing using the method of characteristics. Can. Geotech. J., 40(3), 669-674.
Larkin L.A. 1968. Theoretical bearing capacity of very shallow footings. J. Soil Mech. Found. Div., 94(6), 1347-1357.
Lundgren H., Mortensen K. 1953. Determination by the theory of plasticity of the bearing capacity of continuous footings on sand.
Proc. 3rd Int. Conf. on Soil Mech. and Found. Eng., Zurich (Switzerland), 1, 409-412.
Martin C.M. 2004. ABC Analysis of Bearing Capacity. Available online from www-civil.eng.ox.ac.uk/people/cmm/software/abc.
Salençon J. 1977. Applications of the theory of plasticity in soil mechanics, Wiley, New York (USA).
Salençon J., Matar M. 1982. Bearing capacity of circular shallow foundations. In Foundation engineering (ed. G. Pilot), 159-168,
Presses de l’ENPC, Paris (France).
Sokolovskii V.V. 1965. Statics of granular media, Pergamon, Oxford (UK).
Ukritchon B., Whittle A.J., Klangvijit C. 2004. Reply to discussion of “Calculations of bearing capacity factor Nγ using numerical
limit analyses”. J. Geotech. Geoenv. Eng., 130(10), 1107-1108.
... For instance, Terzaghi (1943) proposed the superposition method by assuming the shear strength of soil as a linear Mohr-Coulomb (MC) failure envelope. Sokolovskii (1966), Kumar (2003), Martin (2005), and Han et al. (2016) determined the collapsed load by utilizing the method of characteristics. Unlike existing theories used to compute the bearing capacity factors of rough strip footing, Kumar (2003) performed the analysis by considering a curved non-plastic wedge under the footing base bounded by curved slip lines being tangential to the base of the footing at both its edge and inclined at an angle π=4 − ϕ=4 with a vertical axisymmetric axis. ...
... It is observed that the current solutions provide a value slightly greater than some solutions available in the literature. Comparisons were performed with the solutions generated by Hansen et al. (1969), Booker (1969), and Martin (2005) utilizing the stress characteristics method, Bolton and Lau (1993) and Han et al. (2016) using method of characteristics, Ukritchon et al. (2003) using limit analysis, Frydman and Burd (1997) employing FDM, and Manoharan and Dasgupta (1995) using FEM. However, for ϕ > 40°, Frydman and Burd (1997) and Ukritchon et al. (2003) give significantly larger values than the current analysis. ...
... It is found that the differences are increased with increasing ϕ. For instance, when ϕ ¼ 40°, it is observed that the discrepancies reach 43%, −29%, −30%, 18%, −27%, −1%, −10%, −36%, −10%, −29%, and −29% for Sokolovskii (1966), Hansen et al. (1969), Booker (1969), Chen (1975), Bolton and Lau (1993), Michalowski (1997), Frydman and Burd (1997), Ukritchon et al. (2003) (lower-bound analysis), Ukritchon et al. (2003) (upperbound analysis), Martin (2005), and Han et al. (2016), respectively. In addition, when ϕ ¼ 35°, it is found that Manoharan and Dasgupta (1995) provided differences of 35% and −20% for associated and nonassociated flows, respectively. ...
... The bearing capacity factor N q for a frictional, weightless soil has an exact solution that depends on the friction angle ϕ of the soil (Lyamin et al. 2007;Reissner 1924;Smith 2005). The expression for the bearing capacity factor N γ , based on work done by Martin (2005) using the method of characteristics, is accurate for soils that are assumed to be perfectly plastic and follow an associated flow rule (in a material following a model with the Mohr-Coulomb yield criterion, dilatancy angle ψ equal to ϕ), although it is well known that, for real sands, ψ is considerably less than ϕ (Loukidis and Salgado 2011;Salgado 2020;Tatsuoka 1987). However, limited guidance on the estimation of a representative friction angle for calculation of bearing capacity factors N q and N γ is available in the literature. ...
... Eq. (8) closely approximates the exact values of N γ obtained by Martin (2005) for an associated flow rule using the method of characteristics. 6. Determine the depth factor d q from the limit unit bearing capacity q bL j D>0 of a strip footing embedded at a given depth using ...
Article
Bearing capacity calculation is an important part of shallow foundation design. The expressions for the shape and depth factors available in the literature for bearing capacity calculation are mostly empirical and are based on results obtained using limit analysis or the method of characteristics assuming a soil that is perfectly plastic following an associated flow rule. This paper presents the results of an experimental program in which load tests were performed on model strip and square footings in silica sand prepared inside a half-cylindrical calibration chamber with a transparent visualization window. The results obtained from the model footing load tests show a significant dependence of footing penetration resistance on embedment depth. The load test results were subsequently used to determine experimentally the shape and depth factors for model strip and square footings in sand. To obtain the displacement and strain fields in the sand domain, the digital image correlation (DIC) technique was used to analyze the digital images collected at different stages during loading of the model footing. The DIC results provide insights into the magnitude and extent of the vertical and horizontal displacement and maximum shear strain contours below and around the footing base during penetration.
... To evaluate the bearing capacity of foundations on different types of soils, most of the geotechnical researchers employ the Mohr-Coulomb (MC) failure criterion, in which case, the material yield parameters do not depend on stress levels [21,27,28,32,36,39,47,49,51,54]. Although it is quite evident from a number of experimental studies that most of the soils obey the nonlinear failure criterion in which case the yield parameters depend on stress level especially in lower stress regions with the mean effective stress lesser than about 100 kPa [3], there are still only a few studies reported in the literature for the evaluation of the bearing capacity of foundations by using the nonlinear yield failure criterion [2, 3, 8, 19, 38, 44-46, 52, 53, 55]. ...
... In the early research pertaining to this area, the shape of the non-plastic wedge was assumed to be a triangular in shape [5,9,35,51]. However, later it was found that this nonplastic wedge is actually curvilinear which results in a lower magnitude of the collapse loads [21,23,25,26,30,32]. In the present investigation, since the footing is assumed to be completely rough, it is assumed that there will be formation of a non-plastic ...
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Experimental studies indicate that the yield parameters for soils remain generally stress dependent, in which case, the internal friction angle reduces continuously with an increase in the normal stress. Such a yield behaviour cannot be modelled correctly by using a linear failure envelope for which case the friction angle does not depend on the stress level such as the Mohr–Coulomb failure criterion. In the current manuscript, a nonlinear yield criterion, considering pure-frictional as well as cohesive-frictional power-type failure envelope, has been employed to compute the bearing capacity of a rough strip footing placed horizontally on sloping ground surface in the presence of pseudo-static seismic inertial forces. The analysis has been performed by using the method of stress characteristics approach. The variation of the seismic bearing capacity factor Nσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N}_{\sigma }$$\end{document}, with an increase in horizontal earthquake acceleration coefficient (αh)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha}_{\mathrm{h}})$$\end{document} for various ground inclinations (β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}), has been provided. The obtained solutions have been compared with different available numerical and experimental results. The factor Nσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N}_{\sigma }$$\end{document} reduces continuously with increases in earthquake acceleration as well as slope inclination. It has been clearly noted that the power-type failure criterion provides a much better prediction of the bearing capacity as compared to the conventional Mohr–Coulomb yield criterion.
... The limit state of the shallow footing settlements and the approximation of the ultimate load and the consequent displacement field represent a principal issue of geomechanics. The previous scientific publications comprise of investigations of the problem from deterministic [1][2][3][4][5][6][7][8][9][10][11][12] and stochastic [9,[13][14][15][16][17][18][19][20][21][22][23][24][25] perspectives. In the deterministic perspective, the failure mechanism is obtained. ...
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In this article, a set of neural networks for the prediction of the stresses and the correspond�ing strains at failure of cohesive soils when subjected to a load of a shallow foundation are presented. The data are acquired via Monte Carlo analyses for different types of loadings and stochastic input material variabilities, and by adopting the clayey soil domain and modified Cam Clay material yield function. The mathematical functions for the estimation of the failure stresses and strains are computed with the feed forward neural network method (FNN). It is demonstrated that the accuracy of the derived relations is in the order of a maximum relative error of 10−5 in all monitored output variables. In addition, the number of training epochs required for convergence is relatively low and this means that the computational and data costs for the construction of the FNN are low. The critical input variable for the estimation of the most unfavorable situations is the Karhunen Loeve series expansion for porous analyses, while for non-porous analyses the constant distribution over depth is the one that provides more critical estimations for the monitored output variables of stresses and strains at failure. This set of functions can estimate the aforementioned variables of the footing settlement in clays with high accuracy; consequently, it can be an important tool for geotechnical engineering design, especially in providing the largest stress allowed from the foundation
... The difficulties in predicting the actual behaviour of foundation soil upon loading are due to non-uniformity of soil properties which mainly occur due to heterogeneous nature and geographical variability of the soil at different locations (Ma et al. 2015;Kuo et al. 2008;Kalantari and Prasad 2014). These challenges have promptly lead to a proper and robust analysis at construction sites which would serve as a prerequisite for measuring the resistance level of the soil resulting from pressures caused by imposed loads (Martin 2005). ...
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There are challenges associated with determining the actual behaviour of soil foundation due to its heterogeneous nature especially when subjected to an imposed loading. This has led to deployment of various soft-computing approaches as an alternative to field experiments for measuring the resistance level of the soil caused by imposed loading. Therefore, this research work was aimed at modelling the soil bearing capacity with specific consideration of index properties parameters of soil, shear strength parameters and relative varied depths, by employing Terzaghi's equations. To considerably overcome complexities, and spontaneous variabilities associated with natural soil foundation, a relatively larger set (45) of data were sourced and used for the modelling. Multiple linear regression (MLR) was used to develop the model using 30 set of data and natural bearing capacity was determined as output with relatively high level of accuracy. The model developed was validated using remaining 15 set of data by employing normal probability plots, which indicates a low level of variance between experimental, and modelled values. Likewise, the corresponding R 2 values of strip, square, and circular footings were found to be 96.98%, 96.93%, and 96.90%, this indicates a high reliability of the model developed.
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An estimate of the bearing capacity of inherently anisotropic sand deposits has been provided by the finite element and linear programming of the lower bound limit analysis. Both strip and circular foundations are considered. The inherent anisotropy can often be reasonably assumed for sands as they are often deposited into layers rendering them a transversely isotropic material. The anisotropy can be described by an internal measure like the fabric tensor. Although determination of the fabric tensor is often a difficult task, for transversely isotropic materials, it was found that no direct measurement on this tensor is required which makes its applications more appealing. In contrast to previous studies on the same topic by assuming a directional dependence of the friction angle, in the present study such an assumption is no longer made and the direct use of the fabric tensor has been made which seems to be logical and more flexible.
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By using the Mohr-Coulomb (MC) yield criterion, the method of stress characteristics (MOSC) has been employed to compute the bearing capacity factor Nγ in the presence of horizontal pseudo-static earthquake inertial forces for a rough strip footing over sloping ground surface. Two different types of failure mechanisms (M1 and M2) were employed. Due to rigorousness of the adopted failure mechanism, the present analysis overcomes the limitation of the existing MOSC formulation available in literature. The solutions in all the cases have also been determined based on the finite elements limit analysis (FELA). The results from the two different analyses, including the failure mechanisms, were found to be very close with each other.
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This study uses finite element analysis to determine the ultimate bearing capacity of unskirted/skirted ring footing under concentric, vertical loads. The clay cohesion and thickness ratio ranged from 8 to 26 kPa and 0.25 to 1.5, respectively. At the same time, the friction angle of the dense sand layer was kept between 40° and 44°. The circular footing had the largest bearing capacity, followed by ring footings with the double skirt, inner skirt, no skirt, and outer skirt, with a thickness ratio of 0.25. The ring footing with a double skirt had the maximum bearing capacity at a thickness ratio of 0.5, followed by a circular, ring without a skirt, a ring with an inner skirt, and a ring with an outer skirt. At a thickness ratio of 1.5, the unskirted circular and ring footings have the equivalent bearing capacity, while ring footing with the outer skirt has a greater bearing capacity than footing with an inner skirt. Furthermore, the double-skirted ring footing had the lowest bearing capacity at this thickness ratio. The generated displacement contours complemented computational findings. The provision of inner and double skirts significantly reduced soil heave via the ring annulus compared to the outer skirt alone.
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This paper is concerned with determining the bearing capacity of axially loaded circular shallow foundations resting on a soil layer of limited or unlimited thickness with a cohesion linearly increasing with depth. It is a continuation of a study made previously by the authors dealing with the bearing capacity of strip footings under similar conditions. Within the frame of the theory of yield design and by means of the theory of axisymmetrical limit equilibria using the Haar-Karman hypothesis, the exact value of the theoretical bearing capacity was determined by a global calculation taking into account all the parameters of the problem simultaneously. In order to make the practical use of the results obtained easier, they are presented in the form of charts, by relating the bearing capacity of the circular footing to that of the strip footing with the same width under the same conditions by means of a shape factor, a function of the nondimensional terms of the problem.
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The bearing capacity of rough strip footings on sand is analyzed using numerical integration of the plasticity equations. The influence of different boundary conditions below the footing is studied for the zero surcharge case, and bearing capacity coefficients are calculated for various assumed settlements at failure. A pressure dependent solution is described which permits calculation of the variation of Φ with stress level in the failure zone. The results are compared with existing trial failure surface solutions and with experimental values. It is concluded that good agreement can be obtained between theoretical results which assume a trapped elastic wedge beneath the footing, and model results related to average Φ-values from triaxial tests in the normal range of cell pressures. The lack of agreement with field results and the computed variation of Φ in the failure zone imply that this relationship is empirical. A procedure is outlined for relating bearing capacity of a given-sized footing to the initial density of a sand using Φ versus pressure results from plane strain shear tests on sand at the same density.
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Plane-strain, finite-element, and finite-difference computations of the bearing-capacity factor, N-gamma, have been made for smooth and rough footings, for friction angles, phi, in the 30 degrees-45 degrees range, and dilation angles, psi, in the 0 less than or equal to psi less than or equal to phi range. The finite-element analyses were based on a tangent stiffness approach. The commercially available code, fast Lagrangian analysis of continua (FLAC), was used for the finite-difference analyses. It was found that these methods may be used to obtain reliable N-gamma values provided they are used cautiously. To obtain high-quality solutions, however, it is necessary to review the results critically, and, if necessary, alter the control parameters (e.g., mesh dimensions, number of calculation increments) being used. Computational difficulties were found to increase with increasing phi, and with increasing difference between phi and psi, and the significant of these difficulties with regard to the use of the two computational techniques is discussed here. Curves relating N-gamma to phi are presented, and compared to previously published relations. It is shown that commonly used values of N-gamma, which have generally been based on associated plasticity calculations,are unconservative for real soils, for which psi<phi.
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By using the method of characteristics, the bearing capacity factor Nγ was computed for a rough strip footing. The analysis was performed by considering a curved nonplastic wedge under the foundation base bounded by curved slip lines being tangential to the base of the footing at its either edge and inclined at an angle π/4 - /2 with the vertical axis of symmetry. The existing theories in the literature for rough footings, which usually employ a triangular wedge below the footing base, were generally found to provide greater values of Nγ as compared with the results obtained in this contribution.Key words: bearing capacity, foundations, failure, numerical modeling, plasticity.
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The two basic equations of plastic equilibrium in a soil mass are solved by a numerical scheme proposed by Sokolovski, for the case of bearing capacity failure of a strip footing under plane strain conditions. A new scaling technique, first proposed by Sokolovski and amplified in detail in this paper, shows that the bearing capacities of strip footings can be conveniently calculated without using the conventional Terzaghi bearing capacity factors, Nc, Nγ, Nq. The Terzaghi equation for bearing capacities is shown to be non-conservative when the proper strain angle of internal friction is used. Previous conclusions that experimental results showed the Terzaghi prediction values to be conservative are misleading because a triaxial compression angle of internal friction was used in a plane strain formula.
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The bearing capacities of very shallow, perfectly smooth strip and circular footings are obtained by integration of the equations of plastic equilibrium of soils. The equations are integrated numerically by a finite difference approximation based on the method of characteristics. Calculations are carried out for cohesionless soils with angles of internal friction of 30○ and 40○. The results are linearized to obtain bearing capacity factors Nγ and Nq and shape factors for a circular footing. The increase of the calculated bearing capacity with depth of burial is sufficiently large to indicate that the small footing settlement prior to failure is a significant factor in the large discrepancies between theoretical and experimentally observed bearing capacities.
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This paper presents numerical upper- and lower-bound solutions for the well-known bearing capacity factor Nγ of a surface strip footing on a frictional soil. The analyses use linear programming and finite-element spatial discretization to solve limit analysis of perfect plasticity, assuming a linear Mohr-Coulomb failure envelope with associated flow within the soil and along the soil-footing interface. The current analyses are to bound the exact value of Nγ within ±5% increasing to ±30% as the internal friction angle increases from 5° to 45° for both smooth and rough interface conditions. Previous solutions by Hansen and Christensen in 1969 and Booker in 1969 are in excellent agreement with the best estimate of Nγ (average of upper and lower bounds) obtained from the current numerical limit analyses. Other well-known analytical solutions and numerical calculations appear to overestimate Nγ for rough footings. Comparisons of predicted upper-bound failure mechanisms and lower-bound contact pressures help to explain similarities and differences among these solutions.
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The method of characteristics is used to establish consistent factors for the vertical bearing capacity of circular and strip footings on soil which satisfies a linear (c, phi) Mohr-Coulomb strength criterion. This method of solution avoids the assumption of arbitrary slip surfaces, and produces zones within which equilibrium and plastic yield are simultaneously satisfied for given boundary stresses. Although similar solutions have previously been published for circular footings, their application has been hindered by errors and confusions over terminology. These are resolved, and the method of solution is explained. It is confirmed that Terzaghi's approach to the superposition of bearing terms containing N-q, N-y, and N-c is both safe and sufficiently accurate for circular footings, as for strip footings. The values to be adopted are tabulated as functions of phi. Differences between the factors applicable to circular and strip footings far exceed the allowances of the empirical shape factors in common use. Some new shape factors are suggested that better represent the relationship between the limiting equilibrium of circular and strip foundations. Some current shape factors attempt to allow simultaneously for the differences in equilibrium solutions and the differences in axisymmetric (triaxial) and plane strain soil parameters. This cannot succeed, since the relationship between strength parameters depends strongly on relative density. The new bearing factors facilitate a more rational approach in which soil parameters appropriate to the geometry can first be determined and then used to find appropriate bearing capacity factors.
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A theoretical investigation is given of quasi-static axially symmetric plastic deformations in soils. The mechanical behaviour of a natural soil is approximated by that of an ideal soil which obeys Coulomb's yield criterion and associated flow rule, with restriction to rigid, perfectly plastic deformations. There are considerable variations in the structure of the associated stress and velocity field equations for the various plastic regimes, but it is noteworthy that real families of characteristics occur in all non-trivial cases. Attention is focused on those plastic regimes agreeing with the heuristic hypothesis of Haar & von Karman as being seemingly of application to certain classes of problems, in particular to those of indentation. The stress and velocity fields are then hyperbolic with identical families of characteristics, and the stress field is statically determinate under appropriate boundary conditions. In applications of the theoretical analysis, attention is confined to situations involving only the Haar & von Karman plastic regimes. First, possible velocity fields are obtained for the incipient plastic flow of a right circular cylindrical sample of soil subjected to uni-axial compressive stress parallel to its axis. Secondly, a complete solution is obtained for the incipient plastic flow in a semi-infinite region of soil, bounded by a plane surface, due to load applied through a flat-ended, smooth, rigid, circular cylinder; numerical results obtained for this problem include the variation of yield-point load with angle of internal friction of the ideal soil. These applications relate to problems of the mechanical testing of soil samples and of load-bearing capacity in foundation engineering.
Article
Cox, Eason and Hopkins1 have developed a general theory of axially-symmetric plastic deformations in ideal soils. This theory was applied, in particular, to discuss incipient plastic flow in a semi-infinite mass of imponderable soil loaded by a smooth, rigid, flat-ended, circular cylinder (or punch). This situation is here discussed further with duo account of the weight of the soil.Values of the mean yield-point pressure for indentation by a punch have been obtained, using the R.A.R.D.E. digital computer AMOS, for a range of values of a dimensionless soil weight parameter, G, and the angle of internal friction of the soil, Φ. In addition, in view of its close similarity to the axially-symmetric problem, results are also presented for the corresponding plane strain problem of indentation by a smooth, rigid, flat-ended die. The results show clearly, in both cases, the strong dependence of the mean yield-point pressure upon soil weight for frictional soils.