RowE, R. K. & DAVIS,E. H. (1982). Geotechnique 32, No. 1,9-23

87

The behaviour of anchor plates III clay

R. K. ROWE* and E. H. DAVISt

The undrained behaviour of anchor plates with a verticalor horizontal axis, resting in a saturated clay, isexamined. Theoretical consideration is given to the effectsof anchor embedment, layer depth, overburden pressureand breakaway condition as well as anchor roughness,thickness and shape. The influence of these parameterson the failure mechanism and the anchor capacity isdiscussed. It is shown that in many cases ultimatecollapse is preceded by significant anchor displacementand a definition of failure which allows reasonabledisplacement predictions to be made at working loads isproposed. Model tests for anchors with a vertical axis arereported. A comparison of these results and other pub-lished data with the theoretical solutions indicates en-couraging agreement. The results of this study arepresented in the form of charts which may be useddirectly in hand calculations for estimating the undrainedfailure load for anchor plates.

L'article decrit Ie comportement non-draine des plaquesd'ancrage a axe vertical ou horizontal qui reposent dansI'argile saturee et decrit du point de vue theorique leseffets de I'encastrement de I'ancrage, de la profondeurdes couches, de la pression du terrain de couverture de larugosite de I'epaisseur et de la forme de I'ancrage.L'influence exercee par ces parametres sur Ie mecanismede rupture et la charge utile de I'ancrage sont aussidiscutees. On demontre que dans beaucoup de cas larupture finale est precedee d'un deplacement significatifde I'ancrage et on propose une definition de la rupturequi facilitera la prevision raisonnable des deplacementssous charges de service. Un compte-rendu est donne detests-modeJes pour les ancrages a axe vertical. U necomparaison de ces resultats et d'autres donnees dejapubliees avec les solutions theoriques indique unecorrespondance encourageante. Les resultats de cetteetude sont presentes sous la forme d'abaques qui peuventetre employes de fa~on directe dans des calculs manuelspour evaluer la charge de rupture des plaques d'ancragedans des conditions.

INTRODUCTIONThe solution of many civil engineering problemsrequires a prediction of the behaviour of buriedstructures. Frequently these buried structures maybe idealized as an anchor plate. Such structuresinclude anchors or buried footings used to supporttransmission towers, retaining walls, bridges and

Discussion on this Paper closes I June 1982. For furtherdetails see inside back cover.

* University of Western Ontario.

t Formerly University of Sydney, now deceased.

tension roofs, as well as submerged pipelines sub-ject to uplift pressures.

The prediction of anchor plate behaviour isusually restricted to the limiting conditions ofelastic displacement (e.g. Fox, 1948; Douglas &Davis, 1964; Rowe & Booker, 1979a, 1979b) orultimate capacity (e.g. Meyerhof & Adams, 1968;Vesic, 1971). Elastic solutions may be convenientlyused for estimating displacements provided thatthe load-detlexion response within the workingload range is quasi-linear. However, this conditionwill be satisfied only if there is limited local yieldwithin the material. In general, the extent of localyield will depend on material properties, the initialstress state, the boundary conditions at the anchorinterface and the load level relative to the collapseload.

Many investigators have proposed approximatetechniques for determining the collapse load foranchor plates. Most approaches involve the use ofeither limit equilibrium concepts or the method ofcharacteristics, frequently combined withempirical corrections (e.g. Meyerhof & Adams,1968; Balla, 1961; Vesic, 1971). Others (e.g.Ladanyi & Johnston, 1974; Vesic, 1971) haveproposed different uses of cavity expansiontheories for predicting anchor capacity. None ofthese approaches provides a rigorous solution tothe general problem of predicting the ultimatecapacity of anchor plates (although a number ofthe approaches have been successfully used forspecific cases).

The finite element method provides a convenientmeans of analysing the load-detlexion behaviour ofanchor plates up to collapse, and allows considera-tion of many factors excluded from other analyses.Several authors (e.g. Ashbee, 1969; Davie & Suther-land, 1977) have performed finite element analysesfor circular anchor plates but no general studyappears to have been attempted.

In this Paper a finite element study of theundrained behaviour of anchor plates in homo-geneous, isotropic saturated clay is reported; theresults are compared with the Authors' model testsand other available experimental data. In a com-panion paper (Rowe & Davis, 1982) considerationis given to anchor plates in sand as well as anchorsin a cohesive-frictional soil. In both papers empha-sis is placed on the effect of local yield on the load-detlexion response and in some cases the adoption

9

SB

ROWE AND DAVIS 88

ss ISOIL SURFACE T

h

~rr ~SB

~AXIS OF SYMMETRY

if)

jRIGID BASE

WITH A VERTICAL AX ISANCHOR

--1 ~T

SOIL SURFACE I

~ r

~lRIGID BASE

ANCHOR WITH A HORIZONTAL AXIS

Fig. I. Problem analysed

of a displacement-related practical failure load(which may be less than or equal to the collapseload) is advocated.

Attention is largely directed towards predictingthe behaviour of strip anchor plates of width B,buried to a depth h, with both vertical and hori-zontal loading as indicated in Fig. 1. Considera-tion is then given to the effect of anchor thicknessand shape. The results are presented in the form ofcharts which may be used in hand calculations fordetermining design failure loads.

NUMERICAL ANALYSISThe numerical solutions presented in this Paper

were obtained from an elasto-plastic finite elementanalysis using the soil-structure interaction theorydescribed by Rowe, Booker & Balaam (1978). Thissubstructure approach allows consideration ofplastic failure within the soil, anchor breakawayfrom the soil behind the anchor, and shear failureat a frictional dilatant soil-structure interface with-out the introduction of special joint or interfaceelements.

For the purposes of this study, the anchor wasassumed to be thin and perfectly rigid. The mainanalysis was for plane strain conditions, so that theanchor is considered to be an infinite strip. Alimited number of analyses were performed foraxisymmetric conditions. The soil was assumed tohave a Mohr-Coulomb failure criterion.

The finite element boundary conditions areshown in Fig. 1. The case of an anchor at infinite

S

P

~liMIT SOLUTIONS FROM

hPLASTICITY THEORY

1 UPPER BOUND

I- S -I LOWER SOUND

NO TENSION PERMITTED __~

AT INTERFACE __ --

6PSo

4

I/J.0

2

his 2 3

Fig. 2. Limit solutions for a shallow anchor; no interfacetension

depth was analysed by specifying rigid boundariesat a distance of eight anchor widths in eachdirection. The finite element mesh consisted of670-1200 constant strain elements (depending ongeometry) arranged in a crossed triangular con-figuration as advocated by Nagtegaal, Parks &Rice (1974).

To ensure that the behaviour of the singularityat the anchor tip was modelled as accurately aspossible, a technique of introducing potential rup-ture lines (Rowe & Davis, 1977) near the edge ofthe anchor was adopted. This approach attemptsto overcome the inhibition of free plastic flowinherent in the usual stiffness formulation of thefinite element method by permitting the formationof velocity discontinuities in the regions of highstress and velocity gradient near the tip of theanchor plate. A description and justification of thistechnique, and numerical checks on incrementalprocedure, load step size and convergence aregiven by Rowe (1978).

The finite element results obtained from thisstudy were compared with available benchmarksolutions from elasticity (e.g. Douglas & Davis,1964; Rowe & Booker, 1979a, 1979b) and plasticitytheory and were found to be in reasonable agree-ment. For example, in Fig. 2 the finite elementcollapse load is compared with the best availableupper and lower bounds for an anchor in aweightless, purely cohesive soil for the condition ofno tension at the anchor interface. Although theexact collapse load is not known for this problem,there is good agreement with the limit solutions inregions of close bounding.

The finite element collapse loads obtained forthe limiting cases of a surface footing and a smoothvertical retaining wall in a purely cohesive materialwere 4'3% and 2'5% above the analytical collapseloads of 5.14Bc and 2Bc respectively. The numeri-cal collapse load for a fully bonded anchor atinfinite depth exceeded the best available upperbound of 11'42Bc by 4'6%. The load-deflexion

BEHAVIOUR OF ANCHOR PLATES IN CLAY

12

UPPER BOUNDh/B

'ao

PBe

B

PRANDTLSOLUTIONh/B. 0

4

5.0 10.0 15.0 20.0ANCHOR DISPLACEMENT E I Be

Fig. 3. Load-displacement curves for two rigid anchors;

fully bonded

curves for the limiting cases of a fully bondedanchor with a vertical axis are shown in Fig. 3. Theclose agreement between the numerical andanalytical loads for these cases suggests that thenumerical collapse loads for intermediate embed-ment ratios could be expected to be within 5% ofthe actual collapse load. In fact, if the numericalcollapse loads for the fully bonded anchor are allreduced by 5% then the finite element results liebetween the best available upper and lowerbound solutions (Rowe, 1978), as shown in Fig. 4.

THEORETICAL RESULTSAnchor capacity

The average applied pressure qu required tocause undrained failure of an anchor plate in asaturated clay with cohesion c and cPu= 0 may beexpressed in the form

qu=cFc' (1)

where Fe' is the lower value given by

Fe' = Fe+ sq,jc (2a)or

where Fe is the dimensionless anchor capacityfactor corresponding to the case where the soil isinitially stress-free and the interface between theback of the anchor and the soil is incapable ofsustaining tension, i.e. is unbonded. Under theseconditions there will be immediate breakaway ofthe soil from the back of the anchor as soon as loadis applied. Fe* is the dimensionless anchor capacityfactor for an anchor which is fully bonded to thesurrounding soil. By definition, there can be nobreakaway between the anchor and the soil. Thissituation would arise if the interface could sustaintension due to suction or adhesion or if the initialstresses were sufficiently large to ensure that the

89

--

BOUND

4COLLAPSE LOADS FROMFINITE ELEMENT ANALYSIS(REDUCED BY 5 % )

(2b)

If; , o.

00 2 3 4trB

Fig. 4. Limit solutions for a fully bonded anchor with avertical axis

stresses behind the anchor were compressive for allanchor loads up to and including the failure load.qh is the overburden pressure at depth hand s is acoefficient for the effect of overburden pressure onanchor capacity.

For intermediate levels of initial stress, break-away will occur when the compressive stressbehind the anchor is reduced to zero. Under theseconditions, the value of Fe' will be between thelimiting values of Fe and Fe* and will depend onthe initial overburden pressure qh and, for anchorswith a horizontal axis, the initial stresses actingnormal to the anchor plate Ko qh' The transitionbetween Fe and Fe* can be given in terms of s inequation (2a).

Definition of failure: the k4 failure conceptFinite element analyses were performed to

obtain the anchor capacity factors for a range ofembedment ratios. These analyses indicated thatalthough clearly defined collapse loads could beobtained, in many cases the deformation due tocontained plastic flow before collapse was so greatthat, for practical purposes, failure would bedeemed to have occurred at a load well below thecollapse load.

The effect of local yield (contained plasticity) onsurface footing behaviour has long been recog-nized. Terzaghi (1943), for example, defined localshear failure in terms of the load at which the load-deflexion curve passes into a steep straight tangent;Terzaghi & Peck (1967) recommend values of cand tan cPreduced to two thirds of their measuredvalues in bearing capacity theory when dealingwith loose or soft soils which exhibit this type ofbehaviour. D'Appolonia, Poulos & Ladd (1971)have used the finite element to show the import-ance of local yield on the displacement of founda-tions in soft clay.

The results of model and field tests on anchorplates suggest the presence of significant containedplastic deformation before collapse and failure is

12

12P

8C

B

8E

4

ROWE AND DAVIS

COLLAPSE LOAD

FOR'k =/

/ I/J ~ 0/

)' ~ 0/ «We ~ 0

// ELASTIC 'VB' =

/ 38E

PRACTICAL FAILURELOAD (k4)

COLLAPSE LOAD

FOR tyB= I

oo 5 10 15 20

ANCHOR DISPLACEMENT E / Be

Fig. 5. Definition of failure

often defined according to an arbitrary rule, suchas load at a specified displacement, or according toTerzaghi's (1943) local yield criterion. A loadingpath finite element analysis provides a convenientmeans of identifying this effect more clearly. Forexample, load-<leflexion curves obtained for twoanchors with a vertical axis are shown in Fig. 5 forthe immediate breakaway condition. The collapseload for a deep anchor (cP= 0) is independent ofthe initial stress conditions; however, the deforma-tion before collapse varies considerably. In the caseof immediate breakaway, there is significant localyield at low load levels (PIBc> 3) and the load-deflexion curve passes into a steep, relativelystraight tangent as the plastic region graduallyexpands until collapse is reached after a very largedisplacement. For such cases, the use of the truecollapse load in conjunction with typical factors ofsafety of 2'5-3 would give working loads in thenon-linear range of behaviour and would result indisplacements much larger than would be pre-dicted from elastic analyses and probably largerthan practica1\y acceptable.

This suggests the need for a practical definitionof failure for problems in which the full ultimatecapacity is obtained only after extensive containedplastic deformation. Although many such de-finitions have been used, a definition that isconvenient and rational in the present context isthat the failure load is considered to have beenreached when the displacement is a selectedmultiple of that which would have been reachedhad conditions remained entirely elastic. Thisdefinition is arbitrary in terms of the choice ofmultiple to be used; however, it is not dependenton scale or modulus and it gives a calculable limiton the displacement before failure, provided theload path to failure is monotonic. In this Paper, 4is chosen as the multiple and the failure load isdenoted as the k4 failure load; it corresponds to anapparent stiffness of one quarter of the elasticstiffness. (Similarly k2 and k3 are loads correspond-

90

25

ing to an apparent stiffness of one half and one thirdrespectively of the elastic stiffness.) The adoption ofa multiple of 4 in conjunction with a typical factorof safety of 2'5-3 will generally ensure that theworking load is close to the linear range and hencethe displacement may be estimated from elasticsolutions. Creep effects increase with increasingcontained plasticity within the soil and hence theadoption of the k4 practical failure load willminimize the contained plasticity and creep atworking loads.

Not a1\ anchors exhibit large deformationsbefore co1\apse. In particular, for shallow anchorswith a vertical axis and for fully bonded anchors,the k4 practical failure load is identical with theultimate collapse load. This can be seen forhlB = 1 in Fig. 5.

Limiting cases: immediate and no breakawayconditions-smooth anchor

Dimensionless anchor capacity factors Fe andFe* determined for the limiting cases of immediatebreakaway and no breakaway are shown in Figs 6and 7 for anchor plates with vertical and hori-zontal axes respectively. The anchor capacity fac-tors determined from the actual collapse load areindicated by a full line; those determined using apractical (k4) definition of failure are denoted bylong dashed lines. In these cases where containedplastic deformation governs the anchor response,the dimensionless loads corresponding to two,three and five times the elastic displacement areshown by short dashed lines, to indicate thesensitivity of the anchor capacity factor to thedefinition of the practical failure load.

Anchors may be classified as shallow or deep,depending on the nature of the anchor response. Adeep anchor is not appreciably affected by theproximity of the soil surface and any increase inembedment beyond the critical embedment atwhich the anchor is first classified as being deepwill not have a significant effect on the anchorcapacity. Anchors with a vertical axis exhibit deepanchor behaviour for embedment ratios greaterthan about 4 for the immediate breakaway casesand about 3 for the no breakaway case. Anchorswith a horizontal axis have a critical embedmentratio of 3 for both breakaway conditions.

The effect of embedment depth and breakawaycondition on anchor behaviour is evident from theplastic region and velocity fields at failure shown inFigs 8-12. In these figures, failure corresponds tothe k4 failure load, where applicable, as indicatedby the dashed lines in Figs 6 and 7.

Figure 8 indicates that for the immediate break-away case, failure of a shallow anchor with avertical axis is associated with the development ofa limited shear zone near the edge of the anchor

BEHAVIOUR OF ANCHOR PLATES IN CLAY

IMMEDIATE~ B ~ BREAKAWAY

k~ Fe ----------;'::::---T~- - ~-= =-~=- =~=-=-=-~---=---

~==::::::= - n-- -- -- - -- ---/-v/ IMMEDIATE BREAKAWAY LOAD PIBe

CORRESPONDING TO 2.3,4 AND 5 TIMESTHE ELASTIC DISPLACEMENT

and an almost rigid upward movement of a blockof soil directly above the anchor. Here, the failureinvolves complete collapse. Under similar condi-tions, failure of a very deep anchor is due toextensive contained plastic deformation andsignificant additional plastic failure would be re-quired to achieve complete collapse. Fig. 9 showsthe failure mechanisms for the corresponding nobreakaway case. Here k4 failure coincides withultimate collapse for both shallow and deepanchors. Plastic flow extends to the soil surface forthe shallow anchor and from front to back for thevery deep anchor.

The failure mechanism for an intermediate case(hlB = 3), corresponding to the transition fromshallow to deep anchor behaviour, is shown in Fig.10. Although the behaviour of this anchor isinfluenced by the presence of the free surface, thefailure load is very close to that of an infinitelydeep anchor. In particular, the velocity field for theno breakaway case shows that collapse is pre-dominantly associated with plastic flow from thefront to the back of the anchor.

12

NO BREAKAWAY (FULLY BONDED)

Fe'

- COLLAPSE LOAD

- - k4 PRACTICALFAILURE LOAD

Fe'8

ORFeORPBC4

4 0.16

12

NO BREAKAWAY (FULLY BONDED)F..e

oI 5 0.16

91

The behaviourof an anchor with a horizontalaxis is in many respects similar (and for a very deepanchor is identical) to that of anchors with avertical axis. The transition from shallow to deepanchor behaviour occurs at an embedment ratio ofabout 3 as indicated by Figs 11 and 12. Shallowanchor (hlB = 1) failure is characterized by plasticflow to the soil surface. For a deeper anchor(hlB = 3) failure is more contained, and in the nobreakaway case involves plastic flow from front toback.

Intermediate break way cases: smooth anchorsFor an anchor where the interface between the

back of the anchor and the soil is incapable ofsustaining tension (i.e. unbonded), the immediatebreakaway and no breakaway conditions alreadydiscussed correspond to the limiting cases wherethe initial overburden pressure qh is zero and verylarge respectively. Finite element analyses indicatethat the overburden pressure required to ensure ano breakaway response for a homogeneous elasto-plastic material is approximately 6c for anchors

Fig. 6 (top left). Variation of anchor capa-city factors with embedment ratio

Fig. 7 (bottom left). Variation of anchorcapacity factors with embedment ratio

Fig. 8 (below). Plastic regions and velocityfields at collapse; immediate breakaway

3.3Bo ~ SURFACE I

Q~3B

3B

'N,'~.

,.........II.,..., .

:<,;:,':.'

--.iM.:.:.:

o

3B

1-

3B

trB ~00

3

h/B~I

~3h/B ~3

SURFACE

iJj

ROWE AND DAVIS 92

SURFACE~~~,

3B 3B

- -

3B3B

f ~ ... _

t", .

tt;f~';,~\' . .-"1.'I'. ~ # t

":..."..

I' ~ _ ..

I"'..,

,t .", ."f", ...

"'"I, .

II" I' .

".'II.

t.,.,.''1'1'"

,1,1,',',.

,

~~.:-

I'".,I'......t,'f'4'_, t

!...!...!.;',',1\,_", ...'

". -'".. ..'"

..41 ..

IMMEDIATE BREAKAWAY NO BREAKAWAY

Fig. 9 (below left). Plastic regions and velocity fields atfailure; no breakaway (fully bonded)

Fig. 10 (below right). Effect of breakaway on plasticregions and velocity fields at failure; hiB = 3

I .'" ]

.fl 4B 4B(0) PLASTIC REGION

. . ,

Fig. 11. Plastic regions and velocity fields at failure; immediatebreakaway.

. , . ,'''::.' .

,'

:::: ::::..~

.:::::. . -- .. .... - ... .. -- -. .... .. ... .. .. . .. ., .. ... ...." .,

.-"

Fig. 12 (right). Behaviour of an anchor at collapse; hlB = 3, nobreakaway (b) VELOCITY FIELD

12

8F.'c

4

012

8

F.'c

4

.00 2 4 6 8q%

Fig. 13. Variation in anchor capacity with overburdenpressure q,j c

BEHAVIOUR OF ANCHOR PLATES IN CLAY93

with a vertical axis and typically ranges between4c/Ko and 6c/Ko for anchors with a horizontalaxis. Results (Ladd & Edgers, 1972) for a numberof clays suggest that the dimensionless overburdenpressure q,jc will often lie in the range 6~qh/C~ 12for normally consolidated clays down to1~ q,jc ~ 2.5 for highly overconsolidated clays(OCR = 10). Thus, situations will commonly arisein which the initial overburden pressure lies be-tween the limiting values required to achieve eitheran immediate or no breakaway response. For theseintermediate cases, the anchor response willinitially be as if it were fully bonded. However,breakaway from the soil behind the anchor willoccur some time during the loading sequence.

The increase in anchor capacity with over-burden pressure is shown in Fig. 13. Anchorcapacity may be considered to increase linearlywith overburden pressure between the limitsimposed by the immediate and no breakawaycases. Thus, the anchor capacity factor can begiven by equation (2a), where the factor s corre-sponds to the rate of increase in anchor capacitywith overburden pressure. For anchor systemswith a vertical axis, s may be taken to be unity andat least approximately independent of Ko.

For anchors with a horizontal axis and hydro-static initial stress conditions (Ko = 1), s varieswith embedment ratio from s = 0.5 for h/B = 1 tos = 0.96 for h/B = 3. The value of s forintermediate embedments may be obtained bylinear interpolation. For deep anchors (h/B> 3) smay be taken to be unity. For anchors with ahorizontal axis and non-hydrostatic initial stressconditions, the value of s is approximately equal toKo times the value obtained for hydrostatic condi-tions.

Full ultimate collapse load for deep anchors in apurely cohesive soil is independent of the initialstress state within the soil mass, and hence isindependent of the breakaway condition. How-ever, the extent of plastic deformation beforecollapse is highly dependent on the initial over-burden stress, and the practical failure load in-creases in direct relation to the overburden pres-sure up to the limiting no breakaway condition asshown in Fig. 13. Although this variation is notentirely linear, to sufficient accuracy, it may beapproximated by a straight line with a slope sequal to unity as previously indicated. (The slightdifference between the results for deep anchorswith horizontal and vertical axes arises from thedifference in the refinement of the finite elementmeshes used in the different analyses.)

The load-deflexion curves obtained for theintermediate breakaway cases are identical to theno breakaway curves, until breakaway occurs. Fullbreakaway is accompanied by extensive plasticdeformation and the practical failure load is gener-

APPROXIMATEo RELATIONSHIP

h

}S'I.O: Vs' ~ FINITE ELEMENT

. coRESULTS (k4)

//",v t II =co ./............-

h B /"

ns1IBl //U

/-APPROXIMATE

$= 1.0 "",. 3 RELATIONSHIP

.hiS'~}FINITE ELEMENT: =

RESULTS (k4)

ally obtained shortly after full breakaway. Thus,working loads deduced by applying a factor ofsafety of 2.5-3 to the anchor capacity given byequations (I) and (2) typically correspond to the nobreakaway portion of the load-deflexion response.Thus, for a homogeneous soil and an overburdenpressure q,jc greater than unity, it would generallybe appropriate to estimate the working loaddeflexions from elastic solutions for a fully bondedanchor.

Adhesion and/or suctionSuction or adhesion between the back of the

anchor and the soil will give rise to a no breakawayresponse until cavitation occurs or the adhesivestrength of the interface bond is exceeded. Oncebreakaway occurs, the stress redistribution arisingfrom the loss of adhesion or suction will lead to anappreciable increase in displacement. The failureload for this case may be estimated from equation(1) where Fe'= Fc+qJc and q. is the availableadhesion or suction. However, because of the effectof stress redistribution after breakaway and theuncertainty as to the actual magnitude of adhesionand suction that may be mobilized, it is suggestedthat particular caution should be adopted in anyanchor design which relies on suction or adhesion.Estimation of the magnitude of q. can be muchmore important in interpreting tests on a modelscale than for field scale design.

ROWE AND DAVIS 94

Effect of anchor roughnessThe load-deflexion response of a deep anchor,

or an anchor with a vertical axis, is not noticeablyaffected by anchor roughness because the sym-metry of the problem prevents the development ofsignificant shear stresses at the anchor-soil inter-face. However, the failure mechanism associatedwith shallow anchors having a horizontal axis isnon-symmetric and high shear stresses may bedeveloped at a rough anchor-soil interface. Theroughness of these shallow anchors (hiB ~ 2) willlead to an appreciable increase in anchor capacityand the values given in Fig. 7 for a smooth anchorrepresent a lower limit to the available capacity ofan anchor. For example, the anchor capacity of arough anchor with a horizontal axis and hlB = 1.5is 30% above the anchor capacity of a similar, butsmooth, anchor. For embedment ratios greaterthan 3, roughness ceases to have a significant effect.

Effect of layer depthThe finite element analyses used to determine

the anchor capacity factors given in Figs 6 and 7were performed for a layer of depth D equal toeight plate widths B, i.e. DIB = 8. However, it canbe shown (Rowe, 1978) that the entire elasto-plasticbehaviour up to collapse may be normalized withrespect to the apparent elastic stiffness of theanchor, so that it is largely invariant for areasonably wide range oflayer depths, i.e. DIB ~ 5.If the elastic stiffness of the anchor for a particularlayer depth is k, then the load corresponding to adisplacement which is twice, three times, etc. theelastic displacement is obtained by constructinglines with slope k12, k13, etc., and then finding theintersection of these lines with the load displace-ment curve. These loads correspond to the k2, k3and k4 values already discussed; these values arerelatively insensitive to change in layer depth. Forlayer depths in the range 5~DIB~ 18, the

14

1.0

0.6

o 0.4t

B

04B

t

0.8 0.8

variation in .the k2, k3, etc. loads is typically lessthan 3% of the values given in Figs 6 and 7.

Anchor inclinationThe results presented in this Paper are for

anchor systems with vertical and horizontal axes.However, on the basis of these results and theelastic solutions of Rowe & Booker (1979a, 1979b,1980), some tentative suggestions can be maderegarding the estimation of anchor capacity forintermediate inclinations.

It is suggested that the anchor capacity forshallow anchors with hlB = 3 and axes at 60° or lessto the vertical may be approximately determinedusing the results presented in Fig. 6; the capacityfor anchors with axes at more than 60° to thevertical may be estimated from Fig. 7. For shallowinclined anchors, judgement may have to beexercised in deciding whether the embedded depthh should be measured to the point on the inclinedanchor furthest from the soil surface or to thecentre.

At embedment ratios greater than 3, the anchorcapacities for the immediate and no breakawayconditions are both independent of anchororientation. For intermediate breakawayconditions, the value of s in equation (2a) willrange between Ko and unity depending on theanchor inclination.

Anchor thickness

The anchor capacity factors given in Figs 6 and7 were determined for an anchor plate of negligiblethickness. The effect of anchor thickness on thecapacity of a deep, fully bonded anchor may beestimated from a limit analysis solution. Althoughthis solution is for a diamond-shaped section itshould be approximately applicable to othershapes. The results of this analysis are presented inFig. 14 in the form of a correction factor R, which

Fig. 14 (left). Effect of anchor thickness on anchorcapacity; fully honded

Fig. 15 (below). Ratio of anchor capacity factorsF, for a circle and strip

1.8Fe (CIRCLE)

Fe (STRIP)

C/J= o'IMMEDIATEBREAKAWAY

14

~"" "-

"-"-

"-"-

",-

'- <k~ :LUES

1.02 3

0.330.17

Bh

oah

B

BEHAVIOUR OF ANCHOR PLATES IN CLAY 95

indicates the variation in anchor capacity withthickness t.

The ultimate capacity of a perfectly smoothanchor decreases appreciably with increasingthickness tl B of the anchor. However, anchors willgenerally be closer to being perfectly rough thanperfectly smooth and the ultimate capacity of arough anchor is relatively insensitive to thicknessfor practical ranges of anchor thickness. Theseresults suggest that the theoretical anchor capacityfactors already presented for a plate anchor arealso relevant to rough anchors of finite thickness.

Anchor shapeFinite element elasto-plastic analyses may be

performed for rectangular or circular anchors atdifferent inclinations. However, the cost of suchnon-linear three-dimensional analyses isprohibitive. The determination of the effect ofshape on anchor capacity will, therefore, berestricted to the special case of circular anchorswith a vertical axis, under immediate breakawayconditions.

Circular anchors may be considered to be deepat an embedment ratio of 2.5 as compared with 4for a strip anchor. As in the plane strain analyses,extensive deformation due to contained plasticflow was observed, necessitating the adoption of apractical k4 failure load. In the case of deepanchors, this load is considerably less than the finalcollapse load.

The effect of shape on the failure loads is shown

v'0.5

8 - '18 MhO- E

1.0

3

STRIP

o

STRIP

2 4 0.18

Ii"

0.2h

B ( b)

Fig. 16. (a) Influence factor MhD for a fully bondedancbor; (b) correction factor Ro for immediate breakaway

in Fig. 15 in terms of the ratio of anchor capacityfor a circle divided by the anchor capacity of astrip. For a homogeneous, isotropic elastic-plasticmaterial, the anchor capacity may differ by up to afactor of 2 for very shallow anchors. However, theeffect of shape decreases rapidly with increasingembedment and for moderately deep anchors(hiB > 3) the use of anchor capacity factors for astrip would underestimate the capacity of acircular anchor by less than 25%.

Displacements at working loadsThe anchor capacity factors given in this Paper

may be used in conjunction with an appropriatefactor of safety to estimate a working load foranchor plates in clay under undrained conditions.Similarly, the working load may be estimated fordrained conditions in either clay or sand from theanchor capacity factors given in a companionpaper (Rowe & Davis, 1982). The working loaddeflexions may then be estimated from an elasticanalysis or elastic solutions.

The foregoing theoretical results show that thegeometric factors that mainly determine the failureload are the anchor orientation and theembedment ratio hiB. However, the displacementsat working load are also significantly affected bythe anchor shape (LIB for rectangular anchors)and the depth D of the layer in which the anchor isburied. Many elastic solutions are available andthese make provision for the effects of anchorshape (e.g. Fox, 1948; Selvadurai, 1976; Rowe &

2.588eloslic

2.0

1.5

1.0

2.588elaslic

Ih

-IT81'18 1

2.0

1.5

o 1.0o 08

Fig. 17. Increase in anchor displacement due to localyield: immediate breakaway

ROWE AND DAVIS 96

Booker, 1979a, 1979b), inclination (e.g. Douglas &Davis, 1964; Rowe & Booker, 1979b; 1980), andnon-homogeneity and anisotropy (Rowe &Booker, 1981). Typical solutions for a bonded stripplate anchor with a vertical axis are shown in Fig.16(a) and a correction factor for the effect ofimmediate breakaway is shown in Fig. 16(b).

The anchor capacity factors were selected suchthat they provide some limit on the displacementwhich will occur before failure. However, even withthe application of a factor of safety of 2.5 or 3, localyield may occur which will increase the displace-ment above that predicted from elastic theory. Themagnitude of this increase in displacement willdepend on the breakaway condition, the initialstress state and the load level.

Local yield effects are of greatest practicalsignificance for the immediate breakaway case.For this case, Fig. 17 shows the increase indisplacement due to local yield for the initial stressstate Ko = 1. Even for a factor of safety of 3(q/qu = 0.33) the displacement may be increasedby up to 20% because of local yield.

EXPERIMENTAL RESULTSDescription of tests

A test program was designed to study the upliftbehaviour of model rectangular anchors for thecase of immediate breakaway. The model anchorswere made from 6 mm thick brass bar with width Bin the range 13-38 mm, length L in the range 64-190mm and apsect ratio L/B between 3 and 8.

Thirty uplift tests were performed on anchorsburied to a depth h of up to 180 mm in kaolin(LL = 45%, PL = 33%, W= 51%) which wascontained in a large (590mm dia. by 480mm deep)pressure vessel. The clay was consolidated under apressure of 200 kPa until primary consolidationwas complete. The soil was then unloaded giving asoft to firm overconsolidated clay, which typically

200

P.LB.k Po

100

125mm x 125mm ANCHOR)

~. 4.63

k4 FAILURE k4 FAILURE

TEST 14 (19mm x 96mm ANCHOR)

~. 3.6

TEST 28

TEST 17 !38mm x 190mm ANCHOR)

~'1.67

o 5 ~ ~ 20 25ANCHOR DISPLACEMENT: mm

Fig. 18. Load-displacement curves for anchor plates inclay LIB = 5

had a moisture content of 35% and bulk density1.85 t/m3. The cohesion and elastic modulidetermined from undrained triaxial tests onsamples removed from the pressure vessel aftereach test were in the ranges 43.7-55.5 kPa andI850-2410 kPa respectively, with average values of50 kPa and 2060 kPa and typical coefficients ofvariation of 0.1 and 0.25. The dimensionless over-burden pressure qJc was always less than 0.06, andfor shallow anchors was less than 0.02.

A technique of underlaying the anchor withfilter paper and gauze was adopted to preventadhesion between the underside of the anchorplate and the soil. Hollow anchor rods were usedto prevent the development of suction below theanchor. This procedure was similar to that used byAdams & Hayes (1967).

In an attempt to achieve undrained conditionson a model scale, the anchors were loaded quicklyusing a lever system and a constant rate ofpenetration machine (Rowe, 1978). Each test took10-30s. The load applied to the anchor and thedisplacement was recorded using a stiff provingring and electronic displacement transducers.

ResultsThe observed anchor behaviour may be broadly

divided into two categories: shallow anchorbehaviour (h/B~4.5) and deep anchor behaviour(h/B>4.5). The behaviour of all anchors withembedment ratios of less than 2.5 (at the beginningof the test) was characterized by a clearly definedcollapse load (see test 17, Fig. 18) and the forma-tion of a tension crack, which began to develop atrelatively low displacements. The behaviour ofshallow anchors with embedments greater than 2.5(i.e. 2.5 ~ h/B ~4.5) was characterized by clearly-defined collapse without the development oftension cracks. A typicalload-deftexion curve (test14) is shown in Fig. 18. From this it can be seenthat, although collapse was clearly defined in thesecases, considerable deformation had occurredbefore collapse and the adoption of a k4 practicalfailure load could be justified. In most cases, thegeometry changes before collapse are significant,with the anchor generally having displacedthrough almost one anchor width. For example, intest 14 the initial embedment ratio was 3.6; at thek4 failure load the effective embedment ratio hadreduced to 3.3 and at collapse the effective embed-ment ratio was only 2.5. Thus, although the initialembedment ratio is also reasonably applicable atthe k4 failure load, there is some doubt as to whatembedment ratio (i.e. initial, final or somewherebetween) should be associated with actual collapse.

Deep anchor behaviour was generally observedfor anchors with embedment ratios greater than4.5. In these cases, failure was not clearly defined

0 0.1

h8 = 3

P 04 LBc = 0

BEHAVIOUR OF ANCHOR PLATES IN CLAY

97

and the load-deflexion curve was still rising whenthe test was terminated after a displacement ofbetween one and two anchor widths. There was noindication of tension crack formation in any ofthese tests, and in most cases there was noapparent surface indication that the anchor hadbeen loaded, despite the fact that it had beendisplaced 25-45 mm. A practical definition offailure such as the k4 definition seems essential forthis type of behaviour.

After each test the soil was removed in such away as to allow assessment to be made of thefailure mode for each anchor. In each case failurewas associated with indentation of the anchor intothe clay, and involved rupture between the soilabove the anchor and the soil at the sides of theanchor. For very shallow anchors, the tensioncrack was along the longitudinal axis of the anchorplate and extended from the soil surface to the topof the anchor. In these cases the crack was up to5 mm wide at the soil surface and reduced to ahairline crack at the anchor level. It appeared thatcrack propagation had been encouraged by thepresence of the anchor rods.

For all tests on shallow anchors there was nosignificant indication of plastic flow around theanchor and the void below the anchor had almostthe same cross-sectional area as the anchor plateitself. However, there was noticeable flow arounddeeper anchors and the cross-sectional area ofthe void below deep anchors varied fromapproximately 60% to 80% of the plate area.

COMPARISON OF EXPERIMENTAL ANDTHEORETICAL RESULTSRectangular anchors with a vertical axis

Many of the observations made during the testsregarding the general behaviour of model anchorplates are in accordance with the predictions madein the theory. In particular, the distinction betweenshallow and deep anchor behaviour and the needfor a practical definition of failure are consistentwith the finite element findings. However, as noallowance was made in the analysis for tensioncrack formation, it may be anticipated that thetheoretical predictions for very shallow modelanchors will be rather unconservative. At thismodel scale, the dimensionless overburdenpressure qJc is not representative of fieldsituations. This is advantageous in the sense that itallows immediate breakaway tests to beperformed, but it also means that the initialstresses are almost zero and any incrementaltensile stress will induce actual tension in thematerial. In field applications, the overburdenpressures qJc will generally be greater than unityand may be as high as 12. In these cases, actualtension will occur only when the tensions due to

loading exceed the initial stresses. Consequently,the effect of tension crack formation is less likely tobe significant at field scale and the capacities fromthese model tests on very shallow anchors may beconsidered to represent the worst possible case. Thefinite element results are considered to be morerelevant to field situations for these cases.

The load-displacement curves obtained experi-mentally and theoretically for a shallow anchor(hiB = 3) are shown in Fig. 19(a). The experimentalcurve shown is for test 15 and, although this isclassified as a shallow anchor, failure did not occuruntil there was a displacement of 0'9B, and theload-deflexion curve was still rising sharply at thepractical k4 failure load. The theoretical curveshown in Fig. 19(a) was obtained for the casewhere 4>u= 0°, Vu= 0'48, Eu = 1940 kPa andCu= 43,4 kPa. These parameters correspond to themean values obtained from triaxial tests onsamples removed from the pressure vessel after thetest. An alternative comparison between theexperimental and theoretical curves can beobtained by collocation of the two curves at twopoints and then plotting in a non-dimensionalform as shown in Fig. 19(b). Figs 19(a) and 19(b)

200 PLB

( kPo)

150

100 - --- EXPERIMENTALCURVE ~. 3 TEST 15

- THEORETICAL CURVE ~. 3

USING TRIAXIAL PARAMETERS

Eu . 1940 kPa(! 390 kPa)

Cu "43.4 k Po (:!: 4 k Po

)

(0)'lJu

. 0

Vu'"

0.5

0.50.4

EXPERIMENTAL CURVE

- THEORETICALCURVE

"COllOCATION OF EXPERIMENTAL ANDFINITE ELEMENT CURVES TO GIVE

Eu' 2120 kPa

Cu"

40.2 kPa

10 158EBC

Fig. 19. Comparisons of theoreticalload-detlexion curves

20 25

and experimental

8p

(1ft )c'b

4

0 00

0 . 00

.0 t

~klO. THEORETICAL LOAD 6

12 rtJ ~aD

10.

8P

LBc

6

4./

/./

/2

ROWE AND DAVIS 98

PLBc

4

6

. . iIEXPERIMENTAL'

THEORYk 2 VALUESVALUES

6P

LBc4

.EXPERIMENTAL'

THEORYk4 VALUES

LIB~ 3

}Lis ~ 5 k4 LOADS

LIB' 8

2 hB

4 0.2 0.16 B 00811

o

Fig. 20. Comparison of experimental and theoretical k2 and k4loads

CIRCULAR ANCHOR

<lVO.

.6ADAMS 8 HAYES (1967)

6 SPENCE (1965) 6 LANGLEY (1967 I. All (19681 (STIFF CLAY Io All (1968 I (SOFT CLAY). DAVIE 6 SUTHERLAND (1977) (GL Y8EN)o DAVIE 6 SUTHERLAND (1977) (SILTY CLAY)

4h

BFig. 21. Comparison of experimental collapse loads with predicted k4 failure load

a 2 6 8 10.

NO BREAKAWAY (FULLY BONDED I

.

... ---r-------------- . IMMEDIATE BREAKAWAY (k 4)

. RANJAN AND ARORA (1980Ie' 6 k Pc

MACKENZIE (1955)

SERIES A C' 3.9 kPc

SERIES Be' 21.4 k Pc

Fig. 22.

3 5 0..16 0.0.8h/B B/h

Comparison of available experimental and theoretical k4 loads

a

BEHAVIOUR OF ANCHOR PLATES IN CLAY 99

indicate that, subject to uncertainty about theactual soil parameters, the experimental andtheoretical curves are in acceptable agreement.

The experimentalload-deflexion curves may benormalized and expressed in terms of loadscorresponding to displacement twice and fourtimes the elastic displacement (i.e. k2 and k4 loads)in the same manner as was adopted for thetheoretical results. This allows convenientcomparison of the experimental and theoreticalresults as indicated in Fig. 20. Fig. 20 shows thatfor embedment ratios greater than 2.5 there isencouraging agreement between the experimentaland theoretical load factors. In particular, thedimensionless factors are virtually independent ofembedment ratios for hiB > 3 and mostexperimental results lie within :t 15% of thetheoretical prediction.

A comparison of the test results for anchors withdifferent aspect ratios LIB did not indicate anyconsistent difference between the cases whereLIB = 5 and LIB = 8, and it would appear that toall practical purposes both anchors behave as stripanchors. Although these results for LIB = 3 werelimited, it would appear that the reduction inaspect ratio corresponds to an increase in anchorcapacity and the strip results could beconservatively used to predict the behaviour of thisrectangular anchor.

Circular anchors with a vertical axisIn some model and field pull-out tests on

circular anchors special care has been taken toprevent the development of suction or adhesionbetween the anchor and the soil (e.g. Adams &Hayes, 1967; Davie & Sutherland, 1977). In othercases, test results have been corrected to makeallowance for the estimated suction forces betweenthe anchor and the soil (e.g. Spence, 1965; Langley,1967; Ali, 1968). The reported anchor capacitiesobtained from these tests are plotted againstembedment ratio in Fig. 21. Unfortunately,insufficient data exist for a direct comparison of thepredicted and experimental loads at the various k2,k3, etc. displacements. However, the predicted k4and klO failure loads are shown in Fig. 21 so thatsome assessment of the suitability of failure loadspredicted on this basis can be made.

The wide scatter of experimental resultsindicates the general uncertainty involved inestimating the uplift capacity of anchors withregard to the definition of failure itself and thedetermination of the relevant soil cohesion c and,for model tests, the determination of the relevantembedment ratio hiB. In cases where details of theanchor behaviour before failure have been given, itwould appear that failure was preceded by

extensive contained plastic deformation, aspredicted.

It is difficult to draw any firm conclusions fromthe comparison of experimental and theoreticalfailure loads given in Fig. 21. However, twoobservations can be made. First, for embedmentratios less than 2, most experimental results liebelow the predicted failure load. In some of thesecases it would appear that this is due to tensilefailure arising from flexure of the soil above theanchor. The effect of geometry change betweeninitial loading and failure may also be significantfor shallow model anchors because the embedmentratio before actual collapse may be considerablyless than the initial embedment ratio which isreported.

Second, for anchors with an embedment ratiogreater than 3, the k4 definition of failure generallyprovides a conservative estimate of the failure loadand the klO loads (corresponding to a displacementten times the elastic value) lie within the scatter ofreported anchor capacities.

Strip anchors with a horizontal axisMacKenzie (1955) and Ranjan & Arora (1980)

have reported results obtained from small-scalemodel tests in soft clay. MacKenzie's model haddimensions of 25 mm x 25 mm x 250 mm. Ranjan& Arora's models were 15-80 mm x 3 mm x 80 mm.The soft soil was mixed to the required consistencyand either placed by hand or dropped around theanchor. The anchors were loaded incrementally attime intervals of 16 minutes and 30 minutes forMacKenzie and Ranjan & Arora respectively, withthe entire test being performed over a time periodof two to five hours. It might be expected that theparameters appropriate to small-scale model testsperformed over such a prolonged period would liebetween the drained and undrained values. How-ever, in both cases only the undrained shearstrength is given. MacKenzie reported averagecohesions of 3,9 kPa and 21.4 kPa for his first andsecond series of tests, where these val ues representthe average of results with a range of up to 80%of the mean value (e.g. for series A,2.3kPa < c < 5,5kPa). These results form the basisof the empirical design approach proposed byTschebotarioff (1973). Ranjan & Arora report anaverage undrained shear strength of 6 kPa.

The breakaway condition was not clearlydefined for these tests, and although the over-burden pressure qJc is relatively small, no attemptappears to have been made either to avoid or tomeasure the adhesion or suction that developsbehind the anchor before breakaway.

The uncertainties concerning the effect ofdissipation of pore pressures on soil strength,suction and adhesion, combined with the

ROWE AND DAVIS100

likelihood that the level of breakaway will bedependent on the depth of the anchor and thenature of the soil, make it difficult to comparedirectly these results with theory. However, somecomparison can be made between the experi-mental k4 loads and limiting theoretical cases ofimmediate and no breakaway, as shown in Fig. 22.In comparing the theoretical and experimentalresults, it should be noted that some tensioncracking was reported for shallow anchors andthat this may be expected to relieve suction as wellas reduce the soil's bearing resistance. For deeperanchors, the overburden pressure appears to havebeen sufficient to prevent tension cracking and theincreasing resistance with depth may be attributedto this overburden effect combined with increasedsuction/adhesion that may be mobilized for thesedeeper anchors.

The comparison of these experimental andtheoretical results suggests that the theoreticalpredictions provide reasonable limits for modelanchor behaviour.

CONCLUSIONSThe undrained behaviour of anchor plates with

a vertical or horizontal axis, resting in a saturatedclay, has been examined. Theoretical considerationhas been given to the effect of anchor embedmentand layer depth, overburden pressure and break-away condition, as well as anchor roughness,thickness and shape.

In many cases, it was found that although thecollapse load was easily identified, significantdeformation occurred before ultimate collapse. Inthese cases it is suggested that the failure loadshould be selected such that reasonable displace-ment predictions could be made at a working load,determined by applying a conventional factor ofsafety to the anchor capacity. In this context it wasfound convenient to define the failure load as theload which would give rise to a displacement fourtimes that predicted by an elastic analysis. Failureloads defined on this basis were largely insensitiveto elastic parameters of the soil and to the depth ofsoil beneath the anchor.

For the limiting conditions of immediate or nobreakaway of the soil behind the anchor, ananchor could be considered deep at an embedmentratio of 3-4; increasing the embedment beyond thishad no appreciable effect on anchor capacity.Anchor capacity for the intermediate case in whichbreakaway occurs during loading, is dependent onoverburden pressure, which is a function of anchordepth. Thus, the anchor capacity may increasewith embedment for embedment ratios greaterthan 4, although a critical depth will be reachedbeyond which the ultimate anchor capacity isindependent of overburden pressure, anchor

orientation, Ko and any adhesion/suctiondeveloped between the anchor and the soil.

Anchor roughness was found to increase thecapacity of shallow anchor plates with a horizontalaxis, but was of little importance for other anchorplates. Anchor thickness did not alter the capacityof deep rough anchors for practical ranges ofthickness. However, the capacity of deep, perfectlysmooth anchors decreased with increasingthickness.

The anchor capacity of circular anchors was upto twice that of a strip for very shallow anchors.The difference between these two cases decreasedappreciably with increasing anchor depth.

The results of the Authors' model tests foranchors with a vertical axis have been reported;they show encouraging agreement with thetheoretical solutions. Likewise, comparisons withother published data for anchor plates suggest thatthe theoretical solutions provide reasonablebounds to the observed behaviour of modelanchors.

The results of this study have been presented incharts which may be used directly in hand calcula-tions for estimating the undrained failure load ofanchor plates. These undrained results representthe limiting case of rapid loading. In situationswhere the rate of loading is known to be slow, adrained analysis using the drained cohesion andfriction angle may be more appropriate. Anchorcapacity factors for purely frictional and cohesive-friction soils are presented in a companion paper(Rowe & Davis, 1982). It is suggested that, withjudgement, the results presented in these twopapers provide a means of estimating the capacityof anchors in clay for a wide range of conditions.

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Terzaghi, K. & Peck, R. B. (1967). Soil mechanics inengineering practice. New York: Wiley.

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