Undrained bearing capacity factors for conical footings on clay

TLDR: In this paper, the bearing capacity of shallow circular foundations on undrained clay is investigated, and the results have widespread application, particularly in the offshore industry, where the footing is not placed at the ground surface and it is important to take into account the depth of embedment.

Abstract: INTRODUCTION The bearing capacity of circular foundations on undrained clay is of fundamental importance in many geotechnical problems. In particular there are a number of designs of offshore foundations where the foundation can be treated approximately as a large circular footing, for instance some gravity bases, the spudcan foundations of jack-up units, and the more recently developed caisson foundations. In most cases the footing is not placed at the ground surface, and it is important to take into account the depth of embedment. Furthermore, the base of a spudcan is generally not flat, but approximates a shallow cone. For foundations on soft clays, the effect of the increase of strength of the soil with depth needs to be taken into account, and this is particularly important for large foundations. The purpose of this note is to present calculations of bearing capacity factors for shallow circular foundations, accounting for embedment, cone angle, rate of increase of strength with depth, and surface roughness of the foundation. The results have widespread application, particularly in the offshore industry. The soil is assumed to be rigid-plastic, with yield determined by the Tresca condition with an undrained strength su. The method of characteristics is used for the bearing capacity calculation, as described by Shield (1955), Eason & Shield (1960), Houlsby (1982) and Houlsby & Wroth (1982a) for application to undrained axisymmetric problems. Some previous results have been published for this problem using similar numerical techniques (e.g. Houlsby & Wroth, 1982b; Salencon & Matar, 1982; Houlsby & Wroth, 1983; Tani & Craig, 1995; Martin, 2001), but the study presented here involves a much more comprehensive coverage of the parameters. Where comparisons can be made with the previous solutions, the factors differ by up to about 0·5%, which gives some indication of the level of accuracy attainable with this numerical technique. Exceptionally, the rough footing results given by Tani & Craig (1995) are higher by up to about 5%, but this may be due to a problem with their numerical integration procedures (see Martin & Randolph, 2001).

Figures

Table 5. Values of Nc0 V= R2su0 for 1508

Table 1. Values of Nc0 V= R2su0 for 308

Fig. 1. (a) Outline of footing; (b) variation of undrained strength with depth

Table 2. Values of Nc0 V= R2su0 for 608

Fig. 3. Comparison with Skempton’s formula and field data

Table 3. Values of Nc0 V= R2su0 for 908

Full text

Houlsby, G. T. & Martin, C. M. (2003). Ge
´
otechnique 53, No. 5, 513–520
513
TECHNICAL NOTE
Undrained bearing capacity factors for conical footings on clay
G. T. HOULSBY

AND C. M. MARTIN

KEYWORDS: bearing capacity; clays; plasticity; theoretical
analysis
INTRODUCTION
The bearing capacity of circular foundations on undrained
clay is of fundamental importance in many geotechnical
problems. In particular there are a number of designs of
offshore foundations where the foundation can be treated
approximately as a large circular footing, for instance some
gravity bases, the spudcan foundations of jack-up units, and
the more recently developed caisson foundations. In most
cases the footing is not placed at the ground surface, and it
is important to take into account the depth of embedment.
Furthermore, the base of a spudcan is generally not flat, but
approximates a shallow cone. For foundations on soft clays,
the effect of the increase of strength of the soil with depth
needs to be taken into account, and this is particularly
important for large foundations.
The purpose of this note is to present calculations of
bearing capacity factors for shallow circular foundations,
accounting for embedment, cone angle, rate of increase of
strength with depth, and surface roughness of the foundation.
The results have widespread application, particularly in the
offshore industry.
The soil is assumed to be rigid-plastic, with yield deter-
mined by the Tresca condition with an undrained strength
s
u
. The method of characteristics is used for the bearing
capacity calculation, as described by Shield (1955), Eason &
Shield (1960), Houlsby (1982) and Houlsby & Wroth
(1982a) for application to undrained axisymmetric problems.
Some previous results have been published for this problem
using similar numerical techniques (e.g. Houlsby & Wroth,
1982b; Salenc¸on & Matar, 1982; Houlsby & Wroth, 1983;
Tani & Craig, 1995; Martin, 2001), but the study presented
here involves a much more comprehensive coverage of the
parameters. Where comparisons can be made with the
previous solutions, the factors differ by up to about 0·5%,
which gives some indication of the level of accuracy attain-
able with this numerical technique. Exceptionally, the rough
footing results given by Tani & Craig (1995) are higher by
up to about 5%, but this may be due to a problem with their
numerical integration procedures (see Martin & Randolph,
2001).
CALCULATIONS
The soil is taken to be isotropic but non-homogeneous,
with the undrained strength defined as varying linearly with
depth:
s
u
¼ s
um
þ rz (1)
where s
um
is the undrained strength at the ground surface, z
is the depth below the surface, and r is the rate of increase
of strength with depth. It is convenient to define the strength
at the level of the footing as s
u0
¼ s
um
+ rh, as shown in
Fig. 1(b). The average bearing pressure q (for weightless
soil) is then expressed in terms of this strength:
q ¼ N
c0
s
u0
(2)
The remainder of this note is concerned with the value of
the dimensionless factor N
c0
, which is a function of the cone
angle, cone roughness, depth of embedment and the rate of
increase of strength with depth of the clay. Each of these
variables is expressed through a dimensionless parameter,
and a parametric study has been made of the problem in
which the following cases were examined (see Fig. 1):
(a) six values of the cone angle ( ¼ 308,608,908, 1208,
1508 and 1808)
(b) six values of the cone roughness factor Æ ¼ a
u
/s
u
,
where a
u
is the maximum shear stress that can be
mobilised at the cone surface and s
u
is the local value
of the undrained shear strength (Æ ¼ 0·0, 0·2, 0·4, 0·6,
0·8 and 1·0)
(c) six values of the dimensionless depth of embedment
(h/2R ¼ 0·0, 0·1, 0·25, 0·5, 1·0 and 2·5)
(d) six values of the dimensionless rate of increase of
strength with depth (2Rr/s
um
¼ 0·0, 1·0, 2·0, 3·0, 4·0
and 5·0).
The bearing capacity factor is expressed as N
c0
¼ N
c0
(, Æ,
h/2R,2Rr/s
um
). In order to explore all the above cases 1296
analyses were required in total. The values calculated are
given in Tables 1–6.
In all analyses the soil was assumed to be weightless, as it
can be shown that the values of N
c0
are independent of the
soil unit weight, ª. Note, however, that when using these
bearing capacity factors in practice, the ‘cohesive’ bearing
capacity, q, given by equation (2) should be augmented by a
surcharge term ªh (or ª9h for a submerged footing on the
seabed). If there is complete backfilling of the hole above
the foundation, as is often the case with a deeply penetrated
spudcan, then equation (2) can be used to give the net
available bearing capacity directly.
As shown in Fig. 1(a), it has been assumed for the
purposes of analysis that the space above the footing is
occupied by a rigid, smooth-sided shaft. The results are thus
not strictly applicable to cases where there is an unsupported
sidewall above footing level. For cases where backfill soil or
a caisson-type structure is present, however, the stress fields
obtained from the method of characteristics (Fig. 2) are
statically admissible. Calculations to demonstrate the exten-
sibility of these ‘partial’ stress fields (and thus confirm their
status as strict lower bound solutions) were not undertaken
as part of this exercise, but Martin & Randolph (2001) have
shown that acceptable extension fields can be constructed for
many combinations of the parameters examined here. It
therefore seems reasonable to adopt the N
c0
factors in Tables
1–6 as lower bound collapse loads.
Manuscript received 6 November 2002; revised manuscript accepted
22 January 2003.
Discussion on this paper closes 1 December 2003; for further
details see p. ii.

Department of Engineering Science, Oxford University, UK.

Table 1. Values of N
c0
V = R
2
s
u0
for  308
2Rr/s
um
h/2R Roughness factor, Æ
0·0 0·2 0·4 0·6 0·8 1·0
0·0 0·0 4·608 5·513 6·383 7·224 8·027 8·787
0·1 4·795 5·697 6·561 7·398 8·199 8·954
0·25 5·051 5·941 6·803 7·631 8·427 9·178
0·5 5·405 6·290 7·144 7·967 8·757 9·501
1·0 5·982 6·853 7·695 8·508 9·290 10·027
2·5 7·124 7·981 8·810 9·612 10·376 11·102
1·0 0·0 7·531 9·020 10·456 11·843 13·190 14·467
0·1 7·451 8·891 10·270 11·606 12·897 14·127
0·25 7·376 8·731 10·046 11·318 12·550 13·722
0·5 7·276 8·550 9·784 10·981 12·137 13·239
1·0 7·205 8·378 9·514 10·614 11·676 12·682
2·5 7·344 8·386 9·394 10·369 11·300 12·192
2·0 0·0 10·448 12·510 14·511 16·442 18·308 20·097
0·1 9·653 11·532 13·334 15·083 16·785 18·399
0·25 8·891 10·577 12·191 13·757 15·271 16·716
0·5 8·197 9·665 11·089 12·469 13·805 15·079
1·0 7·598 8·871 10·104 11·298 12·451 13·542
2·5 7·365 8·438 9·476 10·479 11·437 12·354
3·0 0·0 13·360 15·980 18·561 21·032 23·419 25·706
0·1 11·512 13·765 15·925 18·022 20·052 21·998
0·25 9·979 11·891 13·718 15·492 17·206 18·849
0·5 8·745 10·329 11·866 13·357 14·799 16·178
1·0 7·792 9·114 10·395 11·636 12·834 13·977
2·5 7·396 8·457 9·507 10·522 11·491 12·423
4·0 0·0 16·269 19·460 22·574 25·617 28·524 31·318
0·1 13·102 15·676 18·142 20·537 22·860 25·079
0·25 10·827 12·874 14·862 16·791 18·657 20·441
0·5 9·109 10·770 12·383 13·961 15·470 16·906
1·0 7·906 9·258 10·568 11·838 13·063 14·231
2·5 7·399 8·467 9·523 10·545 11·523 12·457
5·0 0·0 19·177 22·938 26·610 30·200 33·632 36·921
0·1 14·480 17·331 20·063 22·715 25·289 27·748
0·25 11·461 13·637 15·749 17·800 19·783 21·680
0·5 9·368 11·085 12·768 14·382 15·942 17·434
1·0 7·982 9·354 10·683 11·972 13·214 14·402
2·5 7·400 8·473 9·533 10·559 11·548 12·481
h
V
R
β
τ ⫽ 0
τ
max
⫽ a
u
⫽ αs
u
z
(a) (b)
1
ρ
s
u0
⫽ s
um
⫹ ρh
s
um
s
u
Fig. 1. (a) Outline of footing; (b) variation of undrained strength with depth
514 HOULSBY AND MARTIN

CURVE FITTING
It is convenient to develop an algebraic expression that
fits the calculated bearing capacity factors, and this is done
in the following way. First it can be noted that a substantial
part of the effect of cone roughness is accounted for by the
vertical component of the shear force developed on the
inclined surface of the cone, so that we can write:
N
c0
¼ N
c0Æ
þ
Æ
tan(=2)
1 þ
1
6 tan(=2)
2Rr
s
u0

(3)
where N
c0Æ
¼ N
c0Æ
(, Æ, h/2R,2Rr/s
u0
) is the contribution
of the normal stresses on the cone face only, and the second
term is the contribution of the shear stresses, which for fully
developed roughness can be expressed analytically as above.
It is then found empirically that N
c0Æ
can be expressed in
the following form, where N
c00
¼ N
c00
(, h/2R,2Rr/s
um
)is
the value of N
c0
for a smooth footing:
N
c0Æ
¼ N
c00
1 þ f
1
Æ þ f
2
Æ
2

1  f
3
h
2R þ h

(4)
Suitable values of the empirical constants are f
1
¼ 0·212, f
2
¼ –0·097 and f
3
¼ 0·53.
Furthermore, N
c00
can be expressed approximately as a
linear expression in 2Rr/s
u0
:
Table 2. Values of N
c0
V = R
2
s
u0
for  608
2Rr/s
um
h/2R Roughness factor, Æ
0·0 0·2 0·4 0·6 0·8 1·0
0·0 0·0 4·446 4·964 5·450 5·897 6·315 6·687
0·1 4·677 5·193 5·672 6·119 6·531 6·898
0·25 4·981 5·495 5·960 6·404 6·813 7·177
0·5 5·414 5·900 6·370 6·809 7·208 7·566
1·0 6·066 6·545 7·010 7·434 7·838 8·181
2·5 7·327 7·813 8·245 8·659 9·049 9·391
1·0 0·0 5·808 6·510 7·150 7·768 8·343 8·870
0·1 5·916 6·593 7·228 7·827 8·378 8·885
0·25 6·040 6·698 7·299 7·880 8·420 8·910
0·5 6·199 6·835 7·406 7·964 8·475 8·943
1·0 6·431 7·046 7·582 8·122 8·590 9·033
2·5 6·974 7·546 8·079 8·539 8·979 9·387
2·0 0·0 7·139 8·017 8·840 9·600 10·324 10·988
0·1 6·916 7·729 8·493 9·212 9·884 10·504
0·25 6·741 7·496 8·177 8·845 9·461 10·028
0·5 6·591 7·290 7·912 8·526 9·086 9·608
1·0 6·548 7·198 7·763 8·334 8·827 9·299
2·5 6·986 7·492 8·033 8·504 8·955 9·370
3·0 0·0 8·486 9·537 10·495 11·423 12·292 13·099
0·1 7·774 8·697 9·565 10·383 11·145 11·847
0·25 7·239 8·025 8·799 9·528 10·198 10·819
0·5 6·822 7·559 8·211 8·860 9·448 10·002
1·0 6·605 7·271 7·850 8·436 8·943 9·429
2·5 6·989 7·467 8·011 8·488 8·943 9·360
4·0 0·0 9·830 11·018 12·161 13·242 14·259 15·181
0·1 8·507 9·524 10·480 11·382 12·219 12·996
0·25 7·611 8·442 9·264 10·038 10·748 11·410
0·5 6·974 7·736 8·410 9·080 9·688 10·262
1·0 6·637 7·312 7·902 8·495 9·011 9·506
2·5 6·864 7·453 8·000 8·478 8·936 9·354
5·0 0·0 11·174 12·522 13·827 15·062 16·197 17·264
0·1 9·142 10·232 11·263 12·249 13·149 13·991
0·25 7·899 8·784 9·631 10·432 11·174 11·868
0·5 7·083 7·845 8·550 9·237 9·859 10·446
1·0 6·658 7·323 7·936 8·534 9·056 9·557
2·5 6·854 7·444 7·992 8·472 8·931 9·350
UNDRAINED BEARING CAPACITY FACTORS FOR CONICAL FOOTINGS ON CLAY 515

N
c00
¼ N
1
þ N
2
2Rr
s
u0
(5)
where the coefficients are expressed in the form N
1
¼ N
1
(,
h/2R), N
2
¼ N
2
(, h/2R). The values of N
1
and N
2
are given
in Tables 7 and 8. If curve-fitted expressions are required for
these values then the following procedure can be used, with
only slight loss of accuracy. The value of N
2
is well
approximated by
N
2
¼ f
4
þ f
5
1
tan =
ð
2Þ

f
6
þ f
7
h
2R

2
(6)
with the empirical constants f
4
¼ 0·5, f
5
¼ 0·36, f
6
¼ 1·5
and f
7
¼0
:
4.
The factor N
1
gives the bearing capacity of smooth cones
in homogeneous soil. It is less easy to fit with a simple
expression than the other variables described above, but it
can be reasonably approximated by
N
1
¼ N
0
1  f
8
cos =
ð
2
½Þ
1 þ
h
2R

f
9
(7)
where N
0
¼ 5·69 is the bearing capacity factor for a smooth
flat footing at the surface of a homogeneous soil, and the
remaining empirical factors are f
8
¼ 0·21 and f
9
¼ 0·34.
Using the values for N
0
and f
1
to f
9
given above, all 1296
Table 3. Values of N
c0
V = R
2
s
u0
for  908
2Rr/s
um
h/2R Roughness factor, Æ
0·0 0·2 0·4 0·6 0·8 1·0
0·0 0·0 4·643 5·022 5·364 5·672 5·946 6·172
0·1 4·904 5·277 5·609 5·913 6·182 6·405
0·25 5·223 5·594 5·927 6·226 6·490 6·710
0·5 5·680 6·033 6·363 6·657 6·915 7·138
1·0 6·372 6·714 7·047 7·324 7·581 7·787
2·5 7·649 8·028 8·320 8·604 8·859 9·049
1·0 0·0 5·568 6·046 6·470 6·867 7·222 7·535
0·1 5·741 6·206 6·619 7·005 7·356 7·653
0·25 5·938 6·377 6·788 7·162 7·496 7·794
0·5 6·164 6·605 6·993 7·355 7·679 7·972
1·0 6·499 6·931 7·298 7·644 7·948 8·208
2·5 7·246 7·575 7·941 8·248 8·535 8·776
2·0 0·0 6·463 7·028 7·539 8·013 8·445 8·824
0·1 6·410 6·944 7·434 7·878 8·281 8·648
0·25 6·409 6·883 7·345 7·756 8·144 8·645
0·5 6·401 6·879 7·293 7·692 8·031 8·347
1·0 6·539 6·991 7·372 7·735 8·057 8·328
2·5 7·157 7·494 7·863 8·176 8·469 8·715
3·0 0·0 7·359 7·995 8·592 9·142 9·648 10·083
0·1 6·993 7·573 8·104 8·599 9·055 9·447
0·25 6·699 7·239 7·726 8·174 8·587 8·939
0·5 6·540 7·040 7·469 7·880 8·240 8·568
1·0 6·557 7·018 7·407 7·778 8·108 8·386
2·5 7·118 7·458 7·828 8·145 8·441 8·686
4·0 0·0 8·223 8·964 9·644 10·254 10·820 11·334
0·1 7·489 8·111 8·678 9·222 9·704 10·139
0·25 6·940 7·505 8·013 8·485 8·917 9·293
0·5 6·632 7·145 7·584 8·006 8·378 8·715
1·0 6·567 7·032 7·427 7·803 8·137 8·420
2·5 7·048 7·437 7·809 8·127 8·425 8·670
5·0 0·0 9·110 9·934 10·664 11·354 11·998 12·564
0·1 7·874 8·550 9·174 9·744 10·261 10·746
0·25 7·124 7·710 8·236 8·722 9·171 9·567
0·5 6·696 7·219 7·666 8·088 8·475 8·819
1·0 6·573 7·042 7·440 7·820 8·156 8·442
2·5 7·034 7·425 7·797 8·116 8·415 8·660
516 HOULSBY AND MARTIN

calculated factors are fitted to within better than 5% by the
use of just 10 empirical constants.
DISCUSSION
Apart from the comparisons with other solutions using the
method of characteristics, there are few comparisons that
can be made with other published work. The calculations
presented here do, however, invite comparison with the well-
known curve presented by Skempton (1951) for the effect of
depth on the bearing capacity factor, N
c
, for homogeneous
clays. In Fig. 3 we compare Skempton’s curve with our
results for a rough circular flat footing. These are the figures
in the top right hand corner of Table 6. As Fig. 3 shows, the
results are indistinguishable from Skempton’s curve up to h/
2R of about 2. At greater depths our figures are slightly
higher. It is worth noting that, although over the years
Skempton’s curve has been very widely used and is known
to be of practical use, in his original paper he presented only
five case records that were relevant to the curve for circular
footings. These records are shown on the figure, and in fact
those at high h/2R values fall below his curve. He did,
however, present other data supporting an asymptotic value
of the bearing capacity factor of about 9 at great depth.
Table 4. Values of N
c0
V = R
2
s
u0
for  1208
2Rr/s
um
h/2R Roughness factor, Æ
0·0 0·2 0·4 0·6 0·8 1·0
0·0 0·0 4·959 5·253 5·509 5·732 5·918 6·053
0·1 5·228 5·516 5·769 5·987 6·170 6·298
0·25 5·570 5·852 6·100 6·312 6·489 6·617
0·5 6·037 6·310 6·550 6·756 6·934 7·047
1·0 6·737 7·006 7·243 7·441 7·614 7·718
2·5 8·068 8·322 8·551 8·746 8·899 8·988
1·0 0·0 5·687 6·043 6·362 6·646 6·893 7·092
0·1 5·887 6·237 6·547 6·824 7·065 7·259
0·25 6·117 6·454 6·756 7·022 7·258 7·451
0·5 6·393 6·719 7·010 7·266 7·485 7·659
1·0 6·797 7·097 7·367 7·615 7·816 7·970
2·5 7·521 7·817 8·080 8·294 8·493 8·615
2·0 0·0 6·375 6·787 7·161 7·495 7·795 8·038
0·1 6·413 6·803 7·155 7·473 7·751 7·973
0·25 6·465 6·833 7·167 7·461 7·719 7·935
0·5 6·561 6·909 7·220 7·493 7·741 7·918
1·0 6·805 7·119 7·396 7·654 7·867 8·034
2·5 7·427 7·721 7·989 8·207 8·411 8·534
3·0 0·0 7·043 7·509 7·932 8·312 8·658 8·930
0·1 6·838 7·267 7·653 7·998 8·307 8·570
0·25 6·710 7·094 7·447 7·761 8·046 8·271
0·5 6·658 7·018 7·340 7·625 7·882 8·077
1·0 6·805 7·115 7·407 7·671 7·890 8·062
2·5 7·385 7·680 7·948 8·173 8·376 8·507
4·0 0·0 7·696 8·217 8·685 9·108 9·488 9·814
0·1 7·201 7·657 8·071 8·441 8·771 9·031
0·25 6·876 7·285 7·654 7·985 8·274 8·525
0·5 6·721 7·085 7·416 7·715 7·971 8·178
1·0 6·805 7·117 7·413 7·681 7·902 8·078
2·5 7·385 7·658 7·925 8·152 8·356 8·487
5·0 0·0 8·349 8·911 9·429 9·894 10·310 10·668
0·1 7·521 7·991 8·427 8·819 9·176 9·450
0·25 7·012 7·433 7·814 8·153 8·453 8·718
0·5 6·765 7·134 7·471 7·775 8·034 8·250
1·0 6·804 7·119 7·417 7·687 7·910 8·088
2·5 7·341 7·643 7·911 8·139 8·343 8·475
UNDRAINED BEARING CAPACITY FACTORS FOR CONICAL FOOTINGS ON CLAY 517

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Undrained bearing capacity factors for conical footings on clay" ?

Some previous results have been published for this problem using similar numerical techniques ( e. g. Houlsby & Wroth, 1982b ; Salençon & Matar, 1982 ; Houlsby & Wroth, 1983 ; Tani & Craig, 1995 ; Martin, 2001 ), but the study presented here involves a much more comprehensive coverage of the parameters. 

Q2. What is the purpose of this note?

The purpose of this note is to present calculations of bearing capacity factors for shallow circular foundations, accounting for embedment, cone angle, rate of increase of strength with depth, and surface roughness of the foundation. 

Q3. What are the main characteristics of the foundations?

In particular there are a number of designs of offshore foundations where the foundation can be treated approximately as a large circular footing, for instance some gravity bases, the spudcan foundations of jack-up units, and the more recently developed caisson foundations. 

Q4. How many dimensions can be used to give the net available bearing capacity?

If there is complete backfilling of the hole above the foundation, as is often the case with a deeply penetrated spudcan, then equation (2) can be used to give the net available bearing capacity directly. 

Q5. What is the simplest way to test the extensibility of the stress fields?

Calculations to demonstrate the extensibility of these ‘partial’ stress fields (and thus confirm their status as strict lower bound solutions) were not undertaken as part of this exercise, but Martin & Randolph (2001) have shown that acceptable extension fields can be constructed for many combinations of the parameters examined here. 

Q6. what is the stress factor in soils?

2. Typical stress characteristic field ( 1508, Æ 0·8, h/2R 0·5, 2Rr/sum 5)plasticity problems in soils by the method of characteristics.