design approach for rectangular mudmats under fully...

Design approach for rectangular mudmats under fully three-dimensional loadingFeng, X., Randolph, M., Gourvenec, S., & Wallerand, R. (2014). Design approach for rectangular mudmatsunder fully three-dimensional loading. Géotechnique, 64(1), 51-63. DOI: 10.1680/geot.13.P.051

Published in:Géotechnique

DOI:10.1680/geot.13.P.051

Document VersionPublisher's PDF, also known as Version of record

Link to publication in the UWA Research Repository

General rightsCopyright owners retain the copyright for their material stored in the UWA Research Repository. The University grants no end-userrights beyond those which are provided by the Australian Copyright Act 1968. Users may make use of the material in the Repositoryproviding due attribution is given and the use is in accordance with the Copyright Act 1968.

Take down policyIf you believe this document infringes copyright, raise a complaint by contacting [email protected]. The document will beimmediately withdrawn from public access while the complaint is being investigated.

Download date: 11. Jun. 2018

https://doi.org/10.1680/geot.13.P.051

http://research-repository.uwa.edu.au/en/publications/design-approach-for-rectangular-mudmats-under-fully-threedimensional-loading(64fba3f8-728d-4142-8539-d8ce65ddd76f).html

Feng, X. et al. (2014). Geotechnique 64, No. 1, 51–63 [http://dx.doi.org/10.1680/geot.13.P.051]

51

Design approach for rectangular mudmats under fully three-dimensionalloading

X. FENG�, M. F. RANDOLPH�, S . GOURVENEC� and R. WALLERAND†

Mudmats are used in the offshore oil and gas industry to support subsea infrastructure for pipelineterminations and well manifolds. Expansion and contraction of connected pipelines and jumpers dueto changing thermal and pressure conditions impose fully three-dimensional loading on the founda-tions, dominated by horizontal, moment and torsional loading rather than high vertical loads. Themudmat foundations are rectangular, and include shallow skirts in order to increase capacity,particularly for sliding. Offshore design guidelines for shallow foundations tend to excessiveconservatism; optimisation of mudmat capacity under general loading has thus become critical inorder to keep foundation footprints within the limits of current installation vessels. The paper presentsan alternative design method, based on failure envelopes derived from an extensive programme ofthree-dimensional finite-element analyses, focusing on the sliding and rotational capacity of thefoundation. Starting from expressions that quantify the uniaxial capacity under each of the six degreesof freedom, failure envelope shapes for different biaxial combinations are developed. Ultimately, theallowable capacity under the six-degree-of-freedom loading is expressed in terms of a two-dimensionalfailure envelope for the resultant horizontal and moment loading, after due allowance for the verticaland torsional components of load.

KEYWORDS: bearing capacity; clays; design; failure; footings/foundations; numerical modelling

INTRODUCTIONThe increasing focus on deep-water developments in theoffshore industry has led to renewed interest in the design ofshallow foundations, often referred to as mudmats, or matfoundations. Mudmats, which have historically been used toprovide temporary stability for fixed-jacket platforms in rel-atively shallow water, are now widely employed for deep-water subsea production infrastructure, such as manifolds,end terminations and in-line structures for pipelines. Opera-tional loads can be complex and three-dimensional, owing tothermal expansion and contraction of connecting pipelinesand jumpers, with significant moment and torsional compo-nents of load. By contrast, the vertical loading is static, andtypically small in comparison with the bearing capacity. Themagnitude of the applied loading is such that the size ofmudmats is now reaching the limit that can be installed bymoderate-sized vessels such as used for pipe-laying, andthere is a strong incentive to develop robust but accuratedesign methods in order to optimise the foundation size.

Conventional design approaches, as embodied in industrydesign approaches such as ISO (2003) and API (2011), focuson the vertical bearing capacity but modified to take accountof the applied horizontal and moment loading. The under-lying solutions are those for plane-strain conditions (Prandtl,1921; Davis & Booker, 1973), to which semi-empiricalfactors are applied to account for foundation shape, loadinclination and eccentricity (Meyerhof, 1953; Brinch Hansen,1970). The design approach is based on the conservativeassumption of no tensile stresses being sustainable beneaththe (skirted) foundation, and further conservatism arises for

practical load combinations from the simple linear super-position of separate solutions for load inclination and eccen-tricity (Gourvenec, 2007).

Alternative approaches based on failure envelopes invertical, horizontal and moment (V–H–M) loading spacehave been derived for shallow foundations of various geom-etries, including strip, circular and rectangular, using eitherupper- and lower-bound plasticity analysis (e.g. Bransby &Randolph, 1998, 1999; Houlsby & Puzrin, 1999; Randolph& Puzrin, 2003; Salgado et al., 2004) or finite-elementanalyses (e.g. Taiebat & Carter, 2000, 2002, 2010; Gourve-nec & Randolph, 2003; Gourvenec, 2007, 2008; Bransby &Yun, 2009; Gourvenec & Barnett, 2011). A summary of thescope of these various contributions was provided by Ran-dolph (2012).

Among most of the documented results, a typical assump-tion is that the combined vertical load (V), horizontal load(H) and moment (M) applied to the structure act in a singlevertical plane. In addition, torsional loading (T) has tendedto be ignored, or limited to its effect in reducing either thebearing capacity in the absence of lateral loading (Finnie &Morgan, 2004) or the sliding capacity (Yun et al., 2009;Murff et al., 2010). In practice, mudmats for subsea struc-tures (but also shallow foundations in general) are subjectedto fully three-dimensional, six-degree-of-freedom loading.While the resulting foundation response may be assessed ona case-by-case basis using three-dimensional finite-elementanalysis, initial design studies require much simpler ap-proaches where changes in mudmat geometry or loadingconditions can be investigated efficiently.

The paper describes such an approach, leading ultimatelyto algebraic expressions for a failure envelope in the two-dimensional loading space of resultant moment and resultanthorizontal load, for different magnitudes of vertical load andtorsion. Although the approach may be applied more gener-ally, detailed results are restricted here to the conditionsshown in Fig. 1, where a rectangular skirted foundation restson a deep clay layer where the strength may be idealised as

Manuscript received 4 April 2013; revised manuscript accepted 23July 2013. Published online ahead of print 23 September 2013.Discussion on this paper closes on 1 June 2014, for further details seep. ii.� Centre for Offshore Foundation Systems, The University of WesternAustralia.† Subsea 7, Paris, France.

increasing linearly with depth. The mudmat is acted on byfully three-dimensional loading, involving vertical dead load(self-weight) V, biaxial live horizontal loads Hx, Hy,moments My, Mx, and torque loading T (referred to here asV–H2 –M2 –T loading). The sign conventions follow thosesuggested by Butterfield et al. (1997). The mudmat geometryis focused on those typically adopted for subsea systems,with aspect ratio, B/L, of about 0.5 and skirt depth ratios,d/B, up to 0.2. The load reference point is taken at thecoordinate origin, at the surface of the soil, as typicallyassumed during design.

OVERALL STRATEGYThe overall strategy adopted for simplifying the problem

was influenced by typical design conditions, whereby thevertical load is essentially static, and failure is generallygoverned by sliding (including torsion) or overturning. Themain steps in the approach, which are discussed sequentiallyin the following sections, are summarised in Table 1. Theoverall size of the failure envelope is dictated by the extremepoints, represented by the ‘uniaxial’ capacities where asingle component of load or moment acts on the foundation.Two-dimensional slices of the failure envelope are thenconsidered, with the uniaxial capacities due to horizontal,moment and torsional loading reduced according to themagnitude of the applied vertical load. Maximum resultanthorizontal and moment capacities are evaluated, according tothe angle (in plan view) of the resultant loading (see Fig. 2),leading to a two-dimensional failure envelope in (resultant)H–M space. The final failure envelope is then obtained byadjusting the maximum values, Hmax and Mmax, according tothe magnitude of torsion to be sustained by the foundation.Note that although My will generally be negative, accordingto the right-hand axis convention, for ease of comparisons inthe results presented later it is taken as a positive quantity,reversing its sign.

The overall strategy will be illustrated in an example at the

end of the paper. However, the main focus of the paper is todocument the results of extensive three-dimensional finite-element analyses undertaken in order to derive generic valuesfor the uniaxial capacities and the various two-dimensionalfailure envelopes required to verify the full six-degree-of-freedom interaction. The results are strictly applicable only tothe shallowly embedded rectangular foundation geometry andlinearly varying soil strength profile considered, although theunderlying methodology has wider applicability.

FINITE-ELEMENT ANALYSESGeometry and meshes

All of the finite-element analyses were conducted usingthe software ABAQUS (Dassault Systemes, 2010). Theproblem modelled is as shown in Fig. 1. Rectangular foun-dations of breadth B, length L and embedment depth d wereconsidered. The foundation was modelled as a solid rigidplug, on the assumption that sufficient internal skirts wouldbe provided (in reality) that the confined soil plug woulddisplace rigidly. The majority of the analyses targeted atypical aspect ratio of B/L ¼ 0.5, although sensitivity studieswere undertaken within the range 0.33 < B/L < 1. The skirt

Hy

y

V

Hx

Mx

My

LRP

B

T

z

x

L

Mudline

z

susum

su0d

k

Fig. 1. General loading for a rectangular mudmat and linearly varying soil strength

Table 1. Design steps for mat foundation

Step Details

1 For given foundation geometry evaluate su0 and non-dimensional quantities B/L, d/B and k2 Evaluate uniaxial capacities for vertical, horizontal, moment and torsional loading3 Reduce ultimate horizontal, moment and torsional capacities to maximum values available, according to mobilised (design) vertical capacity,

v ¼ V/Vult

4 For given angle of resultant horizontal load, evaluate corresponding ultimate horizontal capacity, and similarly for ultimate moment capacity5 Evaluate reduced ultimate horizontal and moment capacities due to normalised torsional loading6 Evaluate extent to which applied (design) loading falls within H–M failure envelope, and thus safety factors on self-weight V, live loading H,

M, T or material strength su0

y

M Mres max�H Hres max�

θm

θ

Mx

My Hy

Hx

x

Fig. 2. Resultant horizontal and moment loading

52 FENG, RANDOLPH, GOURVENEC AND WALLERAND

depth ratio d/B was varied from 0 to 0.2, again encompass-ing typical skirt depths, which in many cases are less than0.1. Actual mudmat dimensions are generally restricted towidths of 5–10 m, and lengths of up to 20 m, depending onvessel configurations, but results are presented here in non-dimensional form.

The finite-element meshes were extended 3B from theedges of the mudmat and 3B beneath it, with fully con-strained nodes at the base, and horizontally constrainednodes at the sides, sufficiently remote that the failure loadswere not affected by the boundary conditions. Various differ-ent mesh densities were investigated to achieve a time-efficient model without compromising the accuracy. Thetypical example shown in Fig. 3 comprised approximately28 000 eight-node linear brick, hybrid elements.

Material properties and interface conditionsThe material response under undrained soil conditions was

represented with a linear elastic perfectly plastic constitutivelaw according to a Tresca failure criterion. Only undrainedconditions were considered (with Poisson’s ratio set to 0.49),and the soil shear strength was assumed to vary linearly withdepth z according to su ¼ sum + kz, where sum is the shearstrength at the mudline (z ¼ 0), and k is the shear strengthgradient (Fig. 1). The degree of strength heterogeneity isexpressed by the ratio k ¼ kB/su0, where su0 is the shearstrength at skirt-tip level, considering values ranging from 0(uniform) to 10. A constant rigidity index of G/su ¼ 333 wasadopted, ensuring failure at small deformations and thusminimising mesh distortion. With this choice of rigidityindex the displacements when the corresponding ultimatecapacities were approached were observed to be less than0.03B for vertical loading and 0.01B for horizontal loading.The submerged unit weight of the soil was taken asª9 ¼ 6 kN/m3, so that ª9d/su,avg was maintained equal to 1.2(for all k values) for the deepest skirt considered of d ¼ 1 m.

The mudmat was modelled as a discrete rigid body,embedded to a depth d, with the underside of the topcapaligned with mudline level, and the interface between themudmat base and the subsoil was fully bonded – that is,rough in shear with no detachment permitted. The contactproperties between the vertical mudmat sides and the sur-rounding soil were taken as frictionless with separationallowed (zero tension condition). This choice was governed bythe wish to obtain conservative values of capacity for em-bedded foundations, notwithstanding the assumption of tensilestresses allowed at the foundation base. Further discussion of

the influence of the contact properties is provided later thepaper. Depending on particular design loads and durations,this would need to be checked, but typical unfactored loadingconditions do not lead to tensile stresses on the foundationbase (e.g. Gaudin et al., 2012; Dimmock et al., 2013).

METHODOLOGYFailure envelopes under combined loading conditions were

evaluated using two separate techniques:

(a) displacement-controlled ‘swipe’ loading paths, as intro-duced in physical model tests by Tan (1990), and whichhave been shown to follow the failure envelopes closelyfor surface foundations in V–H and V–M load spaces(Gourvenec & Randolph, 2003; Gourvenec, 2008)

(b) fixed-ratio probe tests, both displacement-controlled andload-controlled (e.g. Bransby & Randolph, 1998; Taiebat& Carter, 2002; Gourvenec & Randolph, 2003; Bienen etal., 2006; Yun & Bransby, 2007; Gourvenec, 2008).

Swipe tests can provide extended sections of the failureenvelope from a single analysis, but tend to undercut thetrue failure envelopes when foundations are embedded(Gourvenec, 2008). Probe tests are more intensive computa-tionally, since they provide only a single point on the failureenvelope, particularly under load control, where very smallload increments are necessary as failure is approached inorder not to overshoot the failure envelope. Using bothapproaches in combination provided robust estimates of thefailure envelopes.

RESULTSPure uniaxial capacityUniaxial vertical capacity. Effects of soil heterogeneity onundrained vertical bearing capacity have been investigated forstrip and circular foundations (Davis & Booker, 1973;Houlsby & Wroth, 1983), but there are no published solutionsfor rectangular foundations embedded in heterogeneous soil.For surface foundations on homogeneous soil the undrainedvertical bearing capacity factor reduces gradually as theaspect ratio, B/L, reduces, with the value for B/L ¼ 0.5 beingclose to midway between those for circular and stripfoundations (Salgado et al., 2004; Gourvenec et al., 2006).However, as the degree of strength heterogeneity increases,the shape factor relative to a strip foundation reduces,becoming less than unity for values of k less than about 2.For B/L ¼ 0.5 it was found that purely vertical capacities forembedded rectangular foundations may be expressed as

V ult

Asu0

¼ 5:7 1þ 0:234 tanh 4:78d

B

� ��

3 (1þ 0:2k� 0:012k2 þ 0:0004k3)

(1)

The quality of fit to the finite-element results is shown inFig. 4. Some analyses (not shown) were conducted forslightly greater embedment, up to d/B ¼ 0.3, and indicatedthat the depth factor remained accurate, but this has notbeen explored fully, or for deeper embedment.

Uniaxial horizontal capacity. The horizontal capacity isdominated by the translational sliding resistance at skirt-tiplevel, which is the product of the base area, A, and the shearstrength at that level, su0: For an embedded foundation thereis an additional net contribution from the active and passiveearth pressures on the mudmat skirts, which can be estimatedas a bearing factor Np times the shear strength at the mid-Fig. 3. Finite-element mesh for surface mudmat

DESIGN APPROACH FOR RECTANGULAR MUDMATS UNDER FULLY THREE-DIMENSIONAL LOADING 53

depth of the skirts (Bransby & Randolph, 1999). If the soilremains attached to (or collapses against) the active (rear)edge of the foundation, there is no effect of the soil self-weight, since work done against the self-weight balances outon the passive (front) and active (rear) skirts. Whendetachment at the active side occurs (for low ratios of self-weight stress to shear strength) the mechanism becomesasymmetric, with a single soil wedge pushed up at the passiveside. The soil-self-weight then affects the Np factor.

The horizontal capacities parallel to the x- and y-axes maybe expressed as

Hxult

Asu0

¼ 1þ d

BNp þ 2Æskirt

B

L

� �1� k

2

d

B

� �

Hyult

Asu0

¼ 1þ d

BNp

B

Lþ 2Æskirt

� �1� k

2

d

B

� � (2)

Friction on the side skirts has been allowed for in theabove expression, through the friction ratio of Æskirt, eventhough this was taken as zero in the finite-element analyses.The finite-element results (allowing for up to 9% discrep-ancy) are fitted reasonably using a skirt bearing factor ofNp ¼ 2.2 + ª9d/2su,avg < 4.4 for smooth skirts and Np ¼ 4.4for fully rough skirts (see Fig. 5), where su,avg is the shearstrength at the mid-depth of the skirts.

Uniaxial moment capacity. Exact solutions for the momentcapacity of surface strip (with width B) and circularfoundations (with diameter D) are Mult ¼ 0.69B2su0 andMult ¼ 0.67ADsu0 respectively (Randolph & Puzrin, 2003).Shape factors for surface rectangular foundations on homo-geneous soil, assuming a fully bonded soil interface (notensile failure), have been estimated based on finite-elementanalysis (Gourvenec, 2007), leading to

Mult,rectangular

Mult,strip

¼ 1þ 0:22B

L(3)

This would give Myult/ABsu0 ¼ 0.77 and Mxult/ALsu0 ¼0.99 for B/L ¼ 0.5.

Finite-element analysis of a surface strip foundation onhomogeneous soil gave Mult ¼ 0.72B2su0 for the meshes usedin the present work, suggesting a numerical error of about4%. For the rectangular mudmat (with B/L ¼ 0.5), thecorresponding finite-element results were Myult/ABsu0 ¼ 0.84and Mxult/ALsu0 ¼ 1.04. Reducing these by 4% or 5% wouldthen give reasonable agreement with the values quoted abovebased on the moment capacity shape factor.

Following a similar pattern as for Vult, the numericalvalues of uniaxial moment capacity were fitted by

0

2

4

6

8

10

12

14

0 2 4 6 8 10

Ver

tical

cap

acity

,/

VA

sul

tu0

Heterogeneity factor, /κ � kB su0

d B/ 0�

d B/ � 0·05

d B/ � 0·10

d B/ � 0·20

Fits

Fig. 4. Finite-element results and fit for pure vertical bearingcapacity factor

0

0·3

0·6

0·9

1·2

1·5

1·8

0 2 4 6 8 10

Hor

izon

tal c

apa

city

,/

HA

sxu

ltu0

Heterogeneity factor, /(a)

κ � kB su0

Heterogeneity factor, /(c)

� kB su0κ

Heterogeneity factor, /(b)

� kB su0κ

Heterogeneity factor, /(d)

� kB su0κ

d B/ 0�

d B/ � 0·05d B/ � 0·10d B/ � 0·20Fits

0

0·3

0·6

0·9

1·2

1·5

1·8

2·1

2·4

2·7

0 2 4 6 8 10

Hor

izon

tal c

apa

city

,/

HA

sxu

ltu0

d B/ � 0d B/ � 0·05d B/ � 0·10d B/ � 0·20Fits

0

0·3

0·6

0·9

1·2

1·5

1·8

0 2 4 6 8 10

Hor

izon

tal c

apa

city

,/

HA

syu

ltu0

d B/ � 0d B/ � 0·05d B/ � 0·10d B/ � 0·20Fits

0

0·3

0·6

0·9

1·2

1·5

1·8

2·1

2·4

0 2 4 6 8 10

Hor

izon

tal c

apac

ity,

/H

As

yult

u0

d B/ � 0d B/ � 0·05d B/ � 0·10d B/ � 0·20Fits

Fig. 5. Finite-element results and fit for pure horizontal capacity factor: (a) parallel to x-axis, one-sided mechanism,smooth skirts; (b) parallel to x-axis, two-sided mechanism, rough skirts; (c) parallel to y-axis, one-sided mechanism,smooth skirts; (d) parallel to y-axis, two-sided mechanism, rough skirts


Myult

ABsu0

¼ 0:84 1þ 0:254 tanh 4:51d

B

� ��

3 (1þ 0:2k� 0:01k2 þ 0:0004k3)

Mxult

ALsu0

¼ 1:04 1þ 0:124 tanh 8:31d

B

� ��

3 (1þ 0:3k� 0:028k2 þ 0:00134k3)

(4)

These provide a good fit at each embedment depth ratio,as indicated in Fig. 6. For application, it is suggested thatthe leading coefficients be reduced to 0.79 and 0.99 respec-tively. As for vertical loading, the expressions for the depthfactor appear accurate up to at least d/B ¼ 0.3, but shouldbe used with caution for deeper embedment.

Uniaxial torsional capacity. Pure torsional resistance of asurface foundation can be calculated by integrating the shearstrength across the base area. An exact solution has beenpresented by Murff et al. (2010) and an approximaterelationship as a function of the aspect ratio, B/L, by Finnie& Morgan (2004), expressed respectively as

T ult

ALsu0

¼ 4

AL

ðB=2

0

ðL=2

0

(x2 þ y2)1=2

dxdy

¼ L

6B

sinŁ0

2 cos2 Ł0

þ 0:5 ln tan�

4þ Ł0

2

� ��

þ B2

6L2

cosŁ0

2 sin Ł0

� 0:5 ln tanŁ0

2

� ��

� 0:25þ 0:04B

Lþ 0:09

B

L

� �2

(5)

where Ł0 ¼ arctan(B/L). For a mudmat with aspect ratioB/L ¼ 0.5, the torsional capacity can be calculated asTult/ALsu0 ¼ 0.297 (or 0.29 from the approximate relation-ship). The three-dimensional finite-element analysis gaveTult/ALsu0 ¼ 0.317, an error of just under 7%.

The additional torsional capacity from the skirts may beestimated in much the same way as that of horizontalcapacity, assuming a net passive resistance factor of Np fromskirts on opposite sides of the foundation, to give

˜T ult

ALsu0

¼ d

B

N p

41þ B

L

� �2" #

þ 1:4Æskirt

B

L

( )1� k

2

d

B

� �

(6)

A reasonable fit to the finite-element results is obtainedusing a value of Np ¼ 2.5 + ª9d/2su,avg < 5.0 for smoothskirts and Np ¼ 5.0 for fully rough skirts, as shown in Fig.7. The slight divergence for the deepest embedment ratio ismost probably a result of increasing numerical error.

Failure envelopes for biaxial loading planes including VVertical and horizontal loading plane. The shape of failureenvelopes in vertical and horizontal load space (M ¼ T ¼ 0)generally follows the plane-strain plasticity solution proposedby Green (1954). Previous studies on surface strip andcircular foundations have shown that, when normalised by theultimate loads, all the failure envelopes fall in a very tightband for deposits with any linearly varying shear strengthdistribution, that is, different k value (Gourvenec &Randolph, 2003). For the rectangular foundation consideredhere, normalised failure envelopes, with V and H normalisedby their ultimate values as v ¼ V/Vult, h ¼ Hx/Hxult (for

0

0·5

1·0

1·5

2·0

2·5

0 2 4 6 8 10

Mom

ent c

apac

ity,

/s

MA

Byu

ltu0


κ � kB su0


� kB su0κ

d B/ 0�

d B/ � 0·05

d B/ � 0·10

d B/ � 0·20

Fits

0

0·5

1·0

1·5

2·0

2·5

3·0

3·5

0 2 4 6 8 10

Mom

ent c

apac

ity,

/M

ALs

xult

u0

d B/ � 0

d B/ � 0·05

d B/ � 0·10

d B/ � 0·20

Fits

Fig. 6. Finite-element results and fit for moment capacity factor:(a) about y-axis; (b) about x-axis

0

0·1

0·2

0·3

0·4

0·5

0·6

0·7

0·8

0·9

0 2 4 6 8 10

Tors

iona

l cap

acity

,/

TA

Lsul

tu0


κ � kB su0


� kB su0κ

d B/ � 0

d B/ � 0·05

d B/ � 0·10

d B/ � 0·20

Fits

0

0·1

0·2

0·3

0·4

0·5

0·6

0·7

0·8

0·9

0 2 4 6 8 10

Tors

iona

l cap

acity

,/

TA

Lsul

tu0

d B/ 0�

d B/ � 0·05d B/ � 0·10d B/ � 0·20Fits

Fig. 7. Finite-element results and fit for torsional capacity factor:(a) one-sided mechanism, smooth skirts; (b) two-sided mechan-ism, rough skirts


loading angle Ł ¼ 0) or h ¼ Hy/Hyult (for Ł ¼ 908), may beapproximated as follows.

For v < 0.5:

h ¼ 1 (7a)

For h , 1:

v ¼ 0:5þ 0:5(1� h2:5�cos2 Ł)0:5

(7b)

A comparison of this expression with the finite-elementresults is shown in Fig. 8. Some data points fall a littleinside the envelope, particularly as d/B increases, and thismay be compensated for by slight adjustment of the coeffi-cients to give the following failure envelope.

For v < 0.4:

h ¼ 1 (8a)

For h , 1:

v ¼ 0:4� 0:6(1� h2:5�cos2 Ł)0:5

(8b)

Vertical and moment loading plane. The failure envelopes innormalised V–M space (H ¼ T ¼ 0) derived from the finite-element results may be fitted reasonably accurately by asimple power-law curve of the form (Gourvenec & Randolph,2003)

v ¼ (1� m)p (9)

The exponent p was found to range from 0.23 to 0.43 fora strip foundation as k increased from 0 to 10. The value ofp changes with the shape of foundation, and a simplequadratic function for a rectangular foundation of B/L hasbeen proposed previously for homogeneous soil (Gourvenec,2007). The value of p also depends on the degree of strength

heterogeneity, and again this can be expressed as a poly-nomial function of k.

For moment loading about the long axis (i.e. the V–My

plane), close fits were obtained for an exponent given by

p ¼ 0:23(1þ 0:19k� 0:02k2

þ 0:001k3) 1þ 0:4B

L� 0:1

B

L

� �2" #

(10)

essentially independent of the embedment ratio. For momentloading about the short axis (V–Mx plane) the same expres-sion can be used, but replacing B/L by L/B.

A synthesis of all data was plotted, irrespective of thestrength heterogeneity, by taking the (1/p)th root of equation(9), and then plotting the linear relationship of v1= p againstnormalised moment, m. These fits are shown for the V–My

and V–Mx planes in Fig. 9. In both cases, the maximumerror is extremely small, and only data for surface founda-tions (d/B ¼ 0) fall inside the proposed failure envelopes.

Vertical and torsional loading plane. Under combinedvertical and torsional loading (H ¼M ¼ 0), the finite-elementresults fit a failure envelope similar to that for V–H loading.A single expression provides an adequate (and mostlyconservative) fit, as follows.

For v < 0.5:

t ¼ 1 (11a)

For t , 1:

v ¼ 0:5þ 0:5(1� t2:5)0:5

(11b)

The quality of the fit is illustrated in Fig. 10.

Resultant horizontal and moment capacitiesThe ultimate horizontal capacity in any direction, denoted

Hult, can be explored on a plot of Hx against Hy by carryingout analyses for different ratios of Hx/Hy: The failureenvelope expands and changes shape slightly with increasingembedment ratio (as shown in Fig. 11), but the shape is notaffected significantly by the degree of soil heterogeneity. Ageneral expression for the failure envelope, taking accountof the embedment ratio, can be expressed as

Hx

Hxult

� �2þ6(d=B)

þ Hy

Hyult

� �2

¼ 1 (12)

for 0 < k < 10; V/Vult < 0.5; M ¼ T ¼ 0.For a surface mudmat, the limiting horizontal load in any

direction theoretically satisfies Hult ¼ Asu0, and the failureenvelope in the Hx –Hy plane should be circular, whichequation (12) reverts to. The expression is essentially accu-rate for low levels of vertical load (V/Vult < 0.5), where thefailure mechanism for combined vertical and horizontalloading is one of sliding. An example normalised Hx –Hy

failure envelope for a surface foundation is shown in Fig. 12to verify the applicability of equation (12) within the pro-posed range of vertical load. However, within the overallmethodology, it is also applied for larger vertical loads, withthe values of Hxult and Hyult adjusted according to equation(8). The maximum resultant horizontal load, Hult, may bededuced from that equation for any given angle of loading(or ratio, Hx/Hy).

A similar generalised ellipse may also be used to fit thefailure envelopes in the My –Mx plane. For moment loadingthe shape of the ellipse was found to be essentially indepen-

d B/ 0; 0� �κd B/ ; 0� �κ0·05d B/ ; 0� �κ0·10d B/ ; 0� �κ0·20d B/ ; 10� �κ0d B/ ; 10� �κ0·05d B/ ; 10� �κ0·10d B/ ; 5� �κ0·20Green (1954)Fit: 0°θ �

1·00·80·60·40·2

d B/ 0; 0� �κd B/ ; 0� �κ0·05d B/ ; 0� �κ0·10d B/ ; 0� �κ0·20d B/ ; 10� �κ0d B/ ; 10� �κ0·05d B/ ; 10� �κ0·10d B/ ; 5� �κ0·20Green (1954)Fit: 90°θ �

Compensated fit: 90°θ �

1·00·80·60·40·20Normalised vertical load, /

(b)v V V� ult

0

0·2

0·4

0·6

0·8

1·0

0

Nor

mal

ised

hor

izon

tal l

oad,

/h

HH

�x

xult

Normalised vertical load, /(a)

v V V� ult

Compensated fit: 0°θ �

0

0·2

0·4

0·6

0·8

1·0

Nor

mal

ised

hor

izon

tal l

oad,

h�

HH

yy

/ul

t

Fig. 8. Finite-element results and fit for V–H failure envelopes:(a) V–H envelope in x–z plane (parallel to short edge); (b) V–Henvelope in y–z plane (parallel to long edge)


dent of the embedment ratio, but contracted inwards as thedegree of soil heterogeneity increased. A generalised formmay be expressed as (Figs 13 and 14)

My

Myult

� �1:5

þ Mx

Mxult

� �3�0:1k

¼ 1 (13)

for V ¼ H ¼ T ¼ 0.

Torsion effects on VMH failure envelopesTorsion will effectively reduce the maximum horizontal

load or moment that can be sustained by the soil. Torsionaleffects on horizontal capacity were explored by conducting aseries of load-controlled finite-element analyses. Combined

1·00·80·60·40·2

(a)

d B/ 0; 0� �κ

d B/ ; 2� �κ0

d B/ ; 5� �κ0

d B/ ; 10� �κ0

d B/ ; 0� �κ0·05

d B/ ; 2� �κ0·05

d B/ ; 5� �κ0·05

d B/ ; 10� �κ0·05

d B/ ; 0� �κ0·10

d B/ ; 2� �κ0·10

d B/ ; 5� �κ0·10

d B/ ; 10� �κ0·10

d B/ ; 0� �κ0·20

d B/ ; 5� �κ0·20

Fit

1·00·80·60·40·2

(b)

0

Normalised vertical load, ( / )v V V1/ult

1/p p�

0

0·2

0·4

0·6

0·8

1·0

0

Nor

mal

ised

mom

ent c

apa

city

,/

mM

My

yy

�ul

t

Normalised vertical load, ( / )v V V1/ult

1/p p�

d B/ 0; 0� �κ

d B/ ; 2� �κ0

d B/ ; 5� �κ0

d B/ ; 10� �κ0

d B/ ; 0� �κ0·05

d B/ ; 2� �κ0·05

d B/ ; 5� �κ0·05

d B/ ; 10� �κ0·05

d B/ ; 0� �κ0·10

d B/ ; 2� �κ0·10

d B/ ; 5� �κ0·10

d B/ ; 10� �κ0·10

d B/ ; 0� �κ0·20

d B/ ; 5� �κ0·20

Fit

0

0·2

0·4

0·6

0·8

1·0

Nor

mal

ised

mom

ent c

apa

city

,/

mM

Mx

xx

�ul

t

Fig. 9. Finite-element results and fit for linearised V–M failure envelopes: (a) V–My plane(rotation about y-axis); (b) V–Mx plane (rotation about x-axis)

1·00·80·60·40·20

0·2

0·4

0·6

0·8

1·0

0

Nor

mal

ised

tors

iona

l cap

acity

,/

tT

T�

ult

Normalised vertical capacity, /v V V� ult

d B/ 0; 0� �κ

d B/ ; 0� �κ0·05

d B/ ; 0� �κ0·10

d B/ ; 0� �κ0·20

d B/ ; 10� �κ0

d B/ ; 10� �κ0·05

d B/ ; 10� �κ0·10

d B/ ; 5� �κ0·20

Fit

Fig. 10. Finite-element results and fit for V–T failure envelope

1·00·80·60·40·20

0·2

0·4

0·6

0·8

1·0

0

Nor

mal

ised

hor

izon

tal l

oad,

/h

HH

yy

y�

ult

Normalised horizontal load, /h H Hx x x� ult

d B/ 0�

d B/ � 0·05

d B/ � 0·10

d B/ � 0·20

κ 0, 5, 10�

κ 0, 5�

Fig. 11. Normalised failure envelopes for biaxial horizontalloading of mudmats with varying embedment ratio


horizontal–torsional (H–T) action was realised by applyingthe load at an eccentricity eT from the centreline of thefoundation, and the direction of the resultant horizontal loadwas controlled by the ratio Hx/Hy: The failure envelopes areplotted in Fig. 15. Different eccentricities for the horizontalcomponents (i.e. eTx 6¼ eT y) were also investigated, but theeffect on the H–T failure envelopes was found to be insignif-icant. The main reason is that the resultant torsion is always

along the z-axis, meaning that verifying eccentricities for thehorizontal components just changes the gradient of the H–Tload path for a given Hx/Hy: Therefore the H–T loadcombination at failure should fall on the same failure envel-ope as that produced by the given Hx, Hy with identical eT:

It can be observed from Fig. 15 that the normalisedfailure envelope is smallest when H is applied along theshort axis, and it extends outwards gradually as the horizon-tal loading direction changes from 08 to 908. Hence, for agiven magnitude of H/Hult, higher torsional resistance isavailable when the horizontal load acts parallel to the longside of the mudmat.

The failure envelopes from finite-element analysis werefitted by the general proposed relationship between torsionand horizontal load (Finnie & Morgan, 2004), expressed as

H

Hult

� �m

þ T

T ult

� �n

¼ 1 (14)

for V ¼M ¼ 0.For a rectangular foundation, the power m ranges from 1

to 2, and n from 2 to 2.5. Contrary to Finnie & Morgan(2004), the investigation by Yun et al. (2009) on the basis offinite-element results demonstrated that m varies from 1.9 to2.02, and n from 0.95 to 1.84. In this study, generalexpressions have been derived for the powers m and n, as

m ¼ 1:85

n ¼ 1:25þ 0:75(sin Ł)2:5(15)

The resulting envelopes are compared with the finite-element results in Fig. 15. The power for the torque termchanges from n ¼ 1.25, for horizontal loading parallel to thelatitude axis of the foundation, to n ¼ 2, for loading parallelto the longitude axis: these values are reasonably close tothose suggested by Yun et al. (2009). It was found that thefailure envelopes were essentially insensitive to either thesoil heterogeneity factor, k, or the embedment ratio, d/B, asillustrated in Fig. 16. For loading parallel to the long axis ofthe foundation (Ł ¼ 908), the expression for the failureenvelope becomes increasingly conservative for low levels oftorsion as the embedment depth increases. More accurateenvelopes can be obtained by keeping m unchanged butadjusting n to

n ¼ 1:25þ 0:75þ 2:5d

B

� �(sin Ł)2:5(1þ5d=B) (16)

Normalised moment–torsion, M–T, failure envelopes (Fig.17) were also explored for surface and shallowly embeddedmudmats, and for soil with different degrees of shearstrength heterogeneity. A general equation was developed to

1·00·80·60·40·20

0·2

0·4

0·6

0·8

1·0

0

Nor

mal

ised

hor

izon

tal l

oad,

/h

HH

yy

y�

ult

Normalised horizontal load, /h H Hx x x� ult

V V/ 0ult �

V V/ ult � 0·10

V V/ ult � 0·25

V V/ ult � 0·50

Fit: / 0·5V Vult �

Fig. 12. Failure envelopes for biaxial horizontal loading of asurface mudmat with varying vertical load

1·00·80·60·40·20

0·2

0·4

0·6

0·8

1·0

0

Nor

mal

ised

mom

ent,

/m

MM

xx

x�

ult

Normalised moment, /m M My y y� ult

0, 0·5, 1, 2, 5, 10κ↑ �

Fig. 13. Effects of soil heterogeneity on combined My –Mx failureenvelopes

1·00·80·60·40·20

0·2

0·4

0·6

0·8

1·0

0

Nor

mal

ised

mom

ent,

/m

MM

xx

x�

ult

Nomalised moment, /m M My y y� ult

d B/ 0�

d B/ � 0·05

d B/ � 0·10

d B/ � 0·20

κ↑ � 0, 5, 10

Fig. 14. Effects of embedment ratio on combined My –Mx failureenvelopes

0

0·2

0·4

0·6

0·8

1·0

0

Nor

mal

ised

tors

ion,

/t

T T

�ul

t

Normalised horizontal load, /h H H� ult

FE results

FitB

L x (0°)

30°

45°60°

y (90 )°0 , 30 , 45 , 60 , 90θ↑ � ° ° ° ° °

H

1·00·80·60·40·2

Fig. 15. Normalised H–T failure envelopes for surface mudmat


estimate the M–T failure envelopes, which were found to beindependent of moment direction (Łm), foundation embed-ment depth and soil heterogeneity factor. The failure envel-ope can be expressed as

M

Mult

� �6

þ T

Tult

� �2

¼ 1 (17)

Combined M–H failure envelopesThe yield envelopes expressed by equations (14) to (17)

are used to deduce the maximum resultant horizontal load,Hmax, and moment, Mmax, corresponding to a given mobilisa-tion of torsional capacity. The final stage is then to considerthe failure envelope in moment–horizontal load space.

Although from a design perspective it is convenient totake the load reference point (LRP) at the mudline, it wasfound that the M–H failure envelopes become effectivelyindependent of the embedment ratio if the LRP is taken atskirt-tip level – that is, by substituting a moment ofM� ¼M + Hd, but without changing Hmax (which would alsobe affected by the position of the LRP). An example of theeffect of this transformation on the shape of failure enve-lopes is illustrated in Fig. 18. The failure envelopes are,however, still affected by the degree of soil heterogeneity(Fig. 19) and the direction of the resultant horizontal load(Fig. 20).

Analyses were carried out for horizontal load directions of08 (i.e. parallel to B), 308, 458, 608 and 908 (i.e. parallel toL), and for soil heterogeneity factors of k ¼ 0, 0.5, 1, 2, 5

and 10. These led to approximate failure envelopes ex-pressed as

M�

Mmax

M�

M�j j

� �q

1� ÆH

Hmax

M�

M�j j þ �H

Hmax

� �2" #

þ H

Hmax

� �2

¼ 1

(18)

where the power, q, and coefficients Æ and � are functionsof the direction of the resultant horizontal loading, expressedas an angle Ł from the x-axis, and of the soil strengthheterogeneity, k. They may be approximated as

q ¼ 3 1þ k10

� �

Æ ¼ 0:3 3þ k10

(cos2 Ł� 2 sin2 Ł)

� �

� ¼ 0:09k

(19)

Comparisons of the failure envelope fits provided by theseexpressions with the results from finite-element analysis areshown in Fig. 21 for various values of soil strength hetero-geneity factor, k, and embedment ratio, d/B, and zero torque(so Hmax ¼ Hult and Mmax ¼ Mult in equation (18)).

Finally, Fig. 22 shows comparisons of the failure envelopefits for different levels of torsion. The results are for surfacemudmats, but for various combinations of soil strengthheterogeneity factor, k, and loading direction, Ł. Theproposed failure envelopes from equations (18) and (19)

1·00·80·60·40·2

1·00·80·60·40·2

0

0·2

0·4

0·6

0·8

1·0

0

Nor

mal

ised

tors

ion,

/t

T T

�ul

t

Normalised horizontal load, / ( 0°)(a)

h H H� �ult θ

d B/ 0; 0� �κd B/ ; 0� �κ0·05d B/ ; 0� �κ0·10d B/ ; 0� �κ0·20d B/ ; 10� �κ0d B/ ; 10� �κ0·05d B/ ; 10� �κ0·10d B/ ; 5� �κ0·20Fit

0

0·2

0·4

0·6

0·8

1·0

0

Nor

mal

ised

tors

ion,

/t

T T

�ul

t

Normalised horizontal load, / ( 90°)(b)

h H H� �ult θ

d B/ 0; 0� �κd B/ ; 0� �κ0·05d B/ ; 0� �κ0·10d B/ ; 0� �κ0·20d B/ ; 10� �κ0d B/ ; 10� �κ0·05d B/ ; 10� �κ0·10d B/ ; 5� �κ0·20Fit: 0d B/ �

Fit: 0·05d B/ �

Fit: 0·10d B/ �

Fit: 0·20d B/ �

Fig. 16. Effects of soil heterogeneity and foundation embedmentratio on normalised H–T failure envelopes: (a) combined Hx –Tloading capacity; (b) combined Hy –T loading capacity

d B/ 0; 0� �κd B/ ; 0� �κ0·05

d B/ ; 0� �κ0·10

d B/ ; 0� �κ0·20

d B/ ; 10� �κ0

d B/ ; 10� �κ0·05

d B/ ; 10� �κ0·10d B/ ; 5� �κ0·20Fit

1·00·80·60·40·2

d B/ 0; 0� �κd B/ ; 0� �κ0·05

d B/ ; 0� �κ0·10

d B/ ; 0� �κ0·20

d B/ ; 10� �κ0

d B/ ; 10� �κ0·05

d B/ ; 10� �κ0·10d B/ ; 5� �κ0·20Fit

1·00·80·60·40·20

Nor

mal

ised

mom

ent,

/(

90°)

mM

M�

�ul

tmθ

Normalised torsion, /t T T� ult

(b)

0

0·2

0·4

0·6

0·8

1·0

0

Nor

mal

ised

mom

ent,

/(

0°)

mM

M�

�ul

tmθ

Normalised torsion, /t T T� ult

(a)

0

0·2

0·4

0·6

0·8

1·0

Fig. 17. Effect of soil heterogeneity and foundation embedmentratio on normalised M–T failure envelopes: (a) resultant momentangle Łm 08; (b) resultant moment angle Łm 908


provide reasonable, and slightly conservative, fits to thefinite-element results.

EXAMPLE APPLICATIONAn example application is considered here, showing how

the proposed method may be used in design. The input datafor the problem, summarised in Table 2, comprise founda-tion geometry, shear strength profile and (factored) designloads. In a design problem, the size of the mudmat would be

varied in order to optimise the design, with the requiredreserve strength ratio, or material factor applied to the soilshear strength. Here, the material factor is evaluated for thegiven mudmat size. The calculations were performed follow-ing the design steps summarised in Table 1, as detailedbelow including the relevant equations and figures.

Step 1: From input parameters in Table 2, values ofsu0 ¼ sum + kd and non-dimensional quantities B/L, d/B andk ¼ kB/su0 are evaluated (see Fig. 1).

d B/ 0; 0� �κ

d B/ ; 0� �κ0·05

d B/ ; 0� �κ0·10

d B/ ; 0� �κ0·20

1·00·80·60·40·20�0·2�0·4�0·6�0·8

d B/ 0; 0� �κ

d B/ ; 0� �κ0·05

d B/ ; 0� �κ0·10

d B/ ; 0� �κ0·20

1·00·80·60·40·20�0·2�0·4�0·6�0·8�1·0

Nor

mal

ised

mom

ent,

/(

90°)

mM

* M

��

ult

mθ


h H H� �ult θ

0

0·2

0·4

0·6

0·8

1·0

1·2

�1·0

Nor

mal

ised

mom

ent,

/(

90°)

mM

M�

�ul

tmθ


h H H� �ult θ

0

0·2

0·4

0·6

0·8

1·0

1·2

Fig. 18. Example showing effect of position of LRP on normalisedfailure envelopes for Hx–My loading: (a) LRP at z 0; (b) LRP atz d

1·00·80·60·40·20�0·2�0·4�0·6�0·80

0·2

0·4

0·6

0·8

1·0

1·2

�1·0

Nor

mal

ised

mom

ent,

/(

90°)

mM

M�

�ul

tmθ

Normalised horizontal load, / ( 0°)h H H� �ult θ

0, 0·5, 1, 2, 5, 10κ↑ �

0, 0·5, 1, 2, 5, 10κ↑�

Fig. 19. Effects of soil heterogeneity on normalised Hx–My failureenvelopes for surface mudmat

0, 5, 10κ↑ �

0, 5, 10κ↑ �

Fit

0, 5, 10κ↑ �

0, 5, 10κ↑ �

Fit

d B/ 0�

d B/ � 0·05

d B/ � 0·10

d B/ � 0·20

d B/ 0�

d B/ � 0·05

d B/ � 0·10

d B/ � 0·20

1·00·80·60·40·20�0·2�0·4�0·6�0·8�1·0

Nor

mal

ised

mom

ent,

/(

0°)

mM

* M

��

ult

mθ


h H H� �ult θ

0

0·2

0·4

0·6

0·8

1·0

1·2

1·00·80·60·40·20�0·2�0·4�0·6�0·8�1·0

Nor

mal

ised

mom

ent,

/(

90°)

mM

* M

��

ult

mθ


h H H� �ult θ

0

0·2

0·4

0·6

0·8

1·0

1·2

Fig. 21. Effects of embedment ratio and soil heterogeneity onnormalised H–M failure envelopes (LRP at z d): (a) normalisedHx –My failure envelopes; (b) normalised Hy –Mx failure envelopes

1·00·80·60·40·20�0·2�0·4�0·6�0·80

0·2

0·4

0·6

0·8

1·0

1·2

�1·0

Nor

mal

ised

mom

ent,

/m

M M

�ul

t

Normalised horizontal load, /h H H� ult

0°, 30 , 45 , 60 , 90θ↑ � ° ° ° °

Fig. 20. Effects of horizontal loading direction on combined H–Mcapacity for surface mudmat on homogeneous soil


Step 2: Uniaxial capacities for vertical, horizontal, momentand torsional loading are calculated from equations (1), (2),(4), (5) and (6), and Figs 4, 5, 6 and 7. The uniaxial capacitiesare tabulated in Table 3. The critical modes of loading are intorsion and sliding.Step 3: The ultimate horizontal and moment values areadjusted to account for the vertical load mobilisation throughequations (7) and (9), and Figs 8 and 9.Step 4: For the given angle of resultant horizontal load(Ł ¼ arctan(Hy/Hx)) and that of resultant moment(Łm ¼ arctan(My/Mx)), the corresponding ultimate horizontalcapacity is evaluated from equation (12) and Fig. 11, and the

ultimate moment capacity through equation (13) and Figs 13and 14. This leads to maximum allowable values of resultanthorizontal load and moment of Hmax ¼ 372 kN andMmax ¼ 2373 kN m for this example. The resulting H–Mfailure envelope expressed by equations (18) and (19) is thenas shown by the outer envelope in Fig. 23(a), together with thedesign loading.Step 5: Reduced ultimate horizontal and moment capacitiesdue to normalised torsion loading are evaluated throughequations (14) and (17), and Figs 16 and 17. The effect of theapplied torsion in this example is to further reduce Hmax andMmax to 312 kN and 2282 kN m respectively, as indicated bythe inner failure envelope in Fig. 23(a).Step 6: The extent to which the applied (design) loading fallswithin the H–M failure envelope, and thus the safety factor onmaterial strength, su0, is then evaluated. For this example, thefactored resultant horizontal load (156 kN) is about 50% of themaximum allowable value. However, the reserve strength ratio

FE results Estimation(a)

1·00·80·60·40·20�0·2�0·4�0·6�0·8�1·0

FE results Estimation(b)

1·00·80·60·40·20�0·2�0·4�0·6�0·8�1·0Nor

mal

ised

mom

ent,

*/(

30°)

mM

M�

�ul

tmθ


T T/ 0, 0·25,0·5, 0·75, 0·9

ult �

FE results Estimation(c)

1·00·80·60·40·20�0·2�0·4�0·6�0·8�1·0

FE results Estimation(d)

1·00·80·60·40·20�0·2�0·4�0·6�0·8�1·0Nor

mal

ised

mom

ent,

*/(

0°)

mM

M�

�ul

tmθ


T T/ 0, 0·25,0·5, 0·75, 0·9

ult �

Nor

mal

ised

mom

ent,

*/(

90°)

mM

M�

�ul

tmθ


T T/ 0, 0·25,0·5, 0·75, 0·9

ult �

0

0·2

0·4

0·6

0·8

1·0

1·2N

orm

alis

ed m

omen

t,*/

(60

°)m

MM

��

ult

mθ


T T/ 0, 0·25,0·5, 0·75, 0·9

ult �

0

0·2

0·4

0·6

0·8

1·0

1·2

0

0·2

0·4

0·6

0·8

1·0

1·2

0

0·2

0·4

0·6

0·8

1·0

1·2

Fig. 22. Normalised failure envelopes for H–M loading for varying torsion: (a) kk 0, Ł 308; (b) kk 0, Ł 608; (c) kk 5,Ł 08; (d) kk 10, Ł 908

Table 2. Input data for example mudmat design

Parameter Value

Width, B: m 6Length, L: m 12Skirt, d: m 0.5Mudline strength, sum: kPa 3.4Strength gradient, k: kPa/m 1.5Skirt friction ratio, Æ 0.5Soil unit weight, ª9: kN/m3 6Vertical load, V: kN 700Horizontal load, Hx: kN 100Horizontal load, Hy: kN 120Moment, Mx: kN m 200Moment, My: kN m �360Torsion, T: kN m 640

Table 3. Uniaxial capacities for example mudmat design

Parameter Value Mobilisation ratios

Vertical, Vult: kN 2561 V/Vult 0.273Horizontal, Hxult: kN 369 Hx/Hxult 0.271Horizontal, Hyult: kN 351 Hy/Hyult 0.342Moment, Mxult: kN m 5845 Mx/Mxult 0.034Moment, Myult: kN m 2149 My/Myult 0.167Torsion, Tult: kN m 1405 T/Tult 0.456


is much less than 2, since as the material factor on soil shearstrength is increased, the mobilisation ratios for vertical andtorsional loading increase, resulting in disproportionate de-creases in Hmax and Mmax: This is illustrated in Fig. 23(b),which shows conditions corresponding to the ‘failure’ materialfactor, which turns out to be 1.53. Hmax and Mmax are nowrespectively 157 kN and 1301 kN m.

CONCLUSIONSThe overall aim of the work presented in this paper was

to investigate the response of shallowly embedded rectangu-lar mudmats to fully three-dimensional, six-degree-of-freedom, loading conditions, V–H2 –M2 –T. The specificapplication for the work is the type of foundation used indeep water by the offshore oil and gas industry for pipelineterminations and well manifolds. Finite-element analysesconsidered a rectangular mudmat with aspect ratio B/L of0.5 and embedment ratios d/B up to 0.2 in soil with linearlyincreasing shear strength with depth, 0 < k ¼ kB/su0 < 10.

A design method is proposed whereby the six-degree-of-freedom V–H2 –M2 –T interaction is reduced to a two-dimensional failure envelope in resultant H–M space, ad-justed for the effects of vertical and torsional loading.Algebraic expressions are proposed for prediction of uniaxialand combined loading capacities. The expressions presentedin the paper are accurate for interpolating mudmat capacitywithin the range of conditions considered in this study. Themethod is considered applicable for a wider range of condi-tions, but the coefficients of the fitting expressions – and insome cases the fitting expressions themselves – may needredefining for foundation aspect ratios, B/L, much less thanthe value of 0.5 considered, or for embedment ratios sig-nificantly greater than 0.2.

An example application was presented, illustrating how

the method may be applied to evaluate the reservestrength capacity for a given mudmat geometry andfactored loads. In principle, the approach allows themudmat size to be optimised for given required loadingconditions, soil shear strength profile, and design loadand material factors.

ACKNOWLEDGEMENTSThe research presented here derives from a research col-

laboration between the Centre for Offshore Foundation Sys-tems (COFS) and Subsea 7. Subsea 7 is the owner of theintellectual property resulting from the collaboration, andsome features contained in this work are covered by pendingpatent. COFS is currently supported as a node of theAustralian Research Council Centre of Excellence for Geo-technical Science and Engineering and in partnership withThe Lloyd’s Register Foundation.

NOTATIONA area of rectangular mudmat foundationB mudmat breadthd mudmat embedment

eTx, eT y horizontal eccentricity relative to x- and y-axesHmax maximum allowable horizontal load

Hx, Hy horizontal load along x- and y-axesHxult, Hyult ultimate horizontal load along x- and y-axes

h normalised horizontal load (H/Hult)k shear strength gradientL mudmat length

Mmax maximum allowable momentMx, My moment along x- and y-axes

Mxult, Myult ultimate moment about x- and y-axesm normalised moment (M/Mult)

Np bearing factorp exponent

su undrained shear strengthsu,avg undrained shear strength at mid-depth of skirts

sum undrained strength at mudline levelsu0 undrained strength at level of skirt tip

T torsionTult ultimate torsional capacity

t normalised torsion (T/Tult)V vertical load

Vult ultimate vertical loadv normalised vertical load (V/Vult)z depth

Æskirt friction ratio of skirtsª9 effective unit weightŁ direction of resultant horizontal load

Łm direction of resultant momentk soil heterogeneity factor (kB/su0)

REFERENCESAPI (2011). Recommended practice 2GEO geotechnical and foun-

dation design considerations, 1st edn. Washington, DC, USA:American Petroleum Institute.

Bienen, B., Byrne, B. W., Houlsby, G. T. & Cassidy, M. J. (2006).Investigating six-degree-of-freedom loading of shallow founda-tions on sand. Geotechnique 56, No. 6, 367–379, http://dx.doi.org/10.1680/geot.2006.56.6.367.

Bransby, M. F. & Randolph, M. F. (1998). Combined loading ofskirted foundations. Geotechnique 48, No. 5, 637–655, http://dx.doi.org/10.1680/geot.1998.48.5.637.

Bransby, M. F. & Randolph, M. F. (1999). The effect of embedmentdepth on the undrained response of skirted foundations to com-bined loading. Soils Found. 39, No. 4, 19–33.

Bransby, M. F. & Yun, G. (2009). The undrained capacity of skirtedstrip foundations under combined loading. Geotechnique 59, No.2, 115–125, http://dx.doi.org/10.1680/geot.2007.00098.

4003002001000�100�200�300

Zero torque

T 640·0 kN�

Design

Zero torque

T 640·0 kN�

Design

4003002001000�100�200�300�400

Resultant horizontal load, : kN(a)

H

3002001000�100�200�300Resultant horizontal load, : kN

(b)H

0

500

1000

1500

2000

2500

3000

�400

Res

ulta

nt m

omen

t,: k

N m

M

Resultant horizontal load, : kN(a)

H

0

200

400

600

800

1000

1200

1400

1600

Res

ulta

nt m

omen

t,: k

N m

M

Fig. 23. Failure envelopes and design loading for exampleapplication (V 700 kN): (a) unfactored shear strength data;(b) factored shear strength data


http://dx.doi.org/10.1680/geot.2006.56.6.367




http://dx.doi.org/10.1680/geot.2007.00098

Brinch Hansen, J. (1970). A revised and extended formula forbearing capacity. Danish Geotech. Inst. Bull. No. 28, 5–11.

Butterfield, R., Houlsby, G. T. & Gottardi, G. (1997). Standardizedsign conventions and notation for generally loaded foundations.Geotechnique 47, No. 5, 1051–1054, http://dx.doi.org/10.1680/geot.1997.47.5.1051.

Dassault Systemes (2010). Abaqus analysis users’ manual. Provi-dence, RI, USA: Simula Corp.

Davis, E. H. & Booker, J. R. (1973). The effect of increasingstrength with depth on the bearing capacity of clays. Geotech-nique 23, No. 4, 551–563, http://dx.doi.org/10.1680/geot.1973.23.4.551.

Dimmock, P., Clukey, E. C., Randolph, M. F., Gaudin, C. &Murff, J. D. (2013). Hybrid subsea foundations for subseaequipment. J. Geotech. Geoenviron. Engng, http://dx.doi.org/10.1061/(ASCE)GT.1943-5606.0000944.

Finnie, I. M. S. & Morgan, N. (2004). Torsional loading ofsubsea structures. Proceedings of the 4th international offshoreand polar engineering conference, Toulon, France, pp. 326–333.

Gaudin, C., Randolph, M. F., Feng, X., Clukey, E. C. & Dimmock,P. (2012). Centrifuge modelling of a hybrid foundation forsubsea equipment. Proceedings of the 7th international confer-ence on offshore site investigation and geotechnics, London,UK, pp. 411–420.

Gourvenec, S. (2007). Shape effects on the capacity of rectangularfootings under general loading. Geotechnique 57, No. 8, 637–646,http://dx.doi.org/10.1680/geot.2007.57.8.637.

Gourvenec, S. (2008). Undrained bearing capacity of embeddedfootings under general loading. Geotechnique 58, No. 3, 177–185,http://dx.doi.org/10.1680/geot.2008.58.3.177.

Gourvenec, S. & Barnett, S. (2011). Undrained failure envelope forskirted foundations under general loading. Geotechnique 61, No.3, 263–270, http://dx.doi.org/10.1680/geot.9.T.027.

Gourvenec, S. & Randolph, M. F. (2003). Effect of strengthnon-homogeneity on the shape and failure envelopes forcombined loading of strip and circular foundations on clay.Geotechnique 53, No. 6, 575–586, http://dx.doi.org/10.1680/geot.2003.53.6.575.

Gourvenec, S., Randolph, M. & Kingsnorth, O. (2006). Undrainedbearing capacity of square and rectangular footings. Int. J.Geomech. 6, No. 3, 147–157.

Green, A. P. (1954). The plastic yielding of metal junctions due tocombined shear and pressure. J. Mech. Phys. Solids 2, No. 3,197–211.

Houlsby, G. T. & Puzrin, A. M. (1999). The bearing capacity of astrip footing on clay under combined loading. Proc. R. Soc.London, Ser. A 455, No. 1983, 893–916.

Houlsby, G. T. & Wroth, C. P. (1983). Calculation of stresses onshallow penetrometers and footings. Proceedings of the IUTAM/

IUGG symposium on seabed mechanics, Newcastle, UK, pp.107–112.

ISO (2003). ISO 19901-4: Petroleum and natural gas industries –Specific requirements for offshore structures – Part 4: Geotech-nical and foundation design considerations, 1st edn. Geneva,Switzerland: International Organisation for Standardisation.

Meyerhof, G. G. (1953). The bearing capacity of foundations undereccentric and inclined loads. Proceedings of the 3rd interna-tional conference on soil mechanics and foundation engineering,Zurich, Switzerland, vol. 1, pp. 440–445.

Murff, J. D., Aubeny, C. P. & Yang, M. (2010). The effect oftorsion on the sliding resistance of rectangular foundations.Proceedings of the 2nd international symposium on frontiers inoffshore geotechnics, Perth, Australia, pp. 439–443.

Prandtl, L. (1921). Eindringungsfestigkeit und festigkeit von schnei-den. Angew. Math. U. Mech 1, No. 1, 15–20 (in German).

Randolph, M. F. (2012). Offshore geotechnics: the challenges ofdeepwater soft sediments. In Geotechnical engineering state ofthe art and practice: Keynote lectures from GeoCongress 2012(eds K. Rollins and D. Zekkos), Geotechnical Special Publica-tion No. 226, pp. 241–271. Reston, VA, USA: ASCE.

Randolph, M. F. & Puzrin, A. M. (2003). Upper bound limitanalysis of circular foundations on clay under general loading.Geotechnique 53, No. 9, 785–796, http://dx.doi.org/10.1680/geot.2003.53.9.785.

Salgado, R., Lyamin, A. V., Sloan, S. W. & Yu, H. S. (2004). Two-and three-dimensional bearing capacity of foundations in clay.Geotechnique 54, No. 5, 297–306, http://dx.doi.org/10.1680/geot.2004.54.5.297.

Taiebat, H. A. & Carter, J. P. (2000). Numerical studies of thebearing capacity of shallow foundations on cohesive soilsubjected to combined loading. Geotechnique 50, No. 4, 409–418, http://dx.doi.org/10.1680/geot.2000.50.4.409.

Taiebat, H. A. & Carter, J. P. (2002). Bearing capacity of strip andcircular foundations on undrained clay subjected to eccentricloads. Geotechnique 52, No. 1, 61–64, http://dx.doi.org/10.1680/geot.2002.52.1.61.

Taiebat, H. A. & Carter, J. P. (2010). A failure surface for circularfootings on cohesive soils. Geotechnique 60, No. 4, 265–273,http://dx.doi.org/10.1680/geot.7.00062.

Tan, F. S. (1990). Centrifuge and theoretical modelling of conicalfootings on sand. PhD thesis, University of Cambridge, UK.

Yun, G. & Bransby, M. F. (2007). The horizontal-moment capacityof embedded foundations in undrained soil. Can. Geotech. J. 44,No. 4, 409–424.

Yun, G., Maconochie, A., Oliphant, J. & Bransby, M. F. (2009).Undrained capacity of surface footings subjected to combinedV–H–T loading. Proceedings of the 9th internationalnoffshore and polar engineering conference, Osaka, Japan,pp. 9–14.






http://dx.doi.org/10.1061/(ASCE)GT.1943-5606.0000944

http://dx.doi.org/10.1061/(ASCE)GT.1943-5606.0000944



http://dx.doi.org/10.1680/geot.9.T.027










http://dx.doi.org/10.1680/geot.7.00062

design approach for rectangular mudmats under fully...

Documents