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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

==++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 rigid–perfectly 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+γ−σ=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

σ−γ−

=

φ+σ+γ+φ

(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

γ→γ→

γγ→∞γ→∞

=

=

=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

downloaded from the author’s website in spreadsheet form.

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

δ

/

φ

= 1/3

δ

/

φ

= 1/2

δ

/

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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