THE MECHANICS OF INFLATABLE ANCHORS IN COHESIONLESS SOIL

409

i) Associate Professor, Geotechnical Research Centre, Department of Civil and Environmental Engineering, The University of Western Ontar-io, Canada.

ii) Associate Professor, ditto (shinchberger＠eng.uwo.ca).iii) Ph.D Student, ditto.

The manuscript for this paper was received for review on August 30, 2007; approved on March 26, 2009.Written discussions on this paper should be submitted before January 1, 2010 to the Japanese Geotechnical Society, 4-38-2, Sengoku,Bunkyo-ku, Tokyo 112-0011, Japan. Upon request the closing date may be extended one month.

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SOILS AND FOUNDATIONS Vol. 49, No. 3, 409–420, June 2009Japanese Geotechnical Society

THE MECHANICS OF INFLATABLE ANCHORSIN COHESIONLESS SOIL

TIM NEWSONi), SEAN HINCHBERGERii) and YI LIANGiii)

ABSTRACT

This paper describes an investigation into the performance and pullout capacity of an in‰atable anchor system. Theanchor system comprises a hydraulically in‰ated rubber membrane or packer that may be bored into place and then in-‰ated to provide pullout resistance. A series of scaled physical model tests were used to study the anchor performanceand pullout capacity. The model tests were done in a calibration chamber using cohesionless sand and anchors of vari-ous length, diameter, embedment depth and in‰ation pressure. The anchor behaviour during pullout is interpreted us-ing ˆnite element analysis that accounts for non-linear soil behaviour, in‰ation and subsequent deformation of the in-‰atable membrane, and anchor-soil interaction. The scaled model tests and interpretations assist with identifying thedominant mechanisms aŠecting the pullout capacity of in‰atable anchor systems.

Key words: Finite Element Method, in‰atable anchor, pullout capacity, sand (IGC: E4)

INTRODUCTION

Many remote operated vehicles (ROVs) are neutrally orpositively buoyant. Consequently, utilizing an ROV foroŠshore activities such as in situ testing or maintainingpipelines may require anchorage of the ROV to control orlimit its movement. Previous attempts to use `classical'anchor systems such as helical screws, and suction, duck-bill or plate anchors have had variable success. Many ofthese anchor systems are di‹cult to deploy and they takeaway from the payload capacity of the ROV. Conse-quently, an in‰atable anchor system that is inexpensive,reusable and has several uses would be desirable foroŠshore applications.

This paper describes an investigation into the mechan-ics of in‰atable anchor systems. A series of scaled physi-cal model pullout tests were done in a calibration cham-ber using dry cohesionless sand and anchors of variouslength, diameters, embedment depth and in‰ation pres-sures. The anchor behaviour during the pullout tests is in-terpreted using the ˆnite element (FE) program PLAXIS.The FE model accounts for non-linear soil behaviour, an-chor in‰ation, deformation of the in‰atable membraneand soil-anchor interaction. The scaled physical modeltests in conjunction with the FE analyses help to identifythe factors aŠecting the pullout capacity of this type ofanchor system. It is shown that, for a given soil condi-tion, the anchor pullout capacity is a function of embed-ment, eŠective length and the degree of cavity expansion

during anchor in‰ation. The improved understanding ob-tained from this study should assist with the developmentof anchor capacity diagrams for use in oŠshore applica-tions.

EXPERIMENTAL STUDY

EquipmentReduced scale physical model testing was carried out in

a large cylindrical steel test chamber of 700 mm internaldiameter and 1200 mm height (Newson et al., 2003).Figure 1 shows the test apparatus and setup.

A typical test consisted of an initially ``wished-in-place'' in‰atable anchor in sand followed by in‰ation ofthe anchor and a pullout tests. A three-phase motor at-tached to a gearbox (200:1 gearing ratio) and screw jackwas used to pull out each anchor. The motor was com-puter controlled permitting diŠerent pullout rates to beapplied. The pullout rate was held constant during eachtest. Anchor head displacement was measured using alinear voltage displacement transducer (LVDT) mountedon the same load frame as the screw jack ( see Fig. 1) andresting on the anchor head.

Axial load was measured using a 2.5 kN load cell con-nected to the screw jack and anchor head. The pulloutload and anchor head displacement were both recordedusing a computer controlled data acquisition system. Theanchor was in‰ated with water using two screw pumpsequipped with LVDTs to measure the pump displacement

410

Fig. 1. Model test chamber (from Newson et al., 2003)

Fig. 2. Anchor details

410 NEWSON ET AL.

and output, and a pressure transducer to measure the in-‰ation pressure. For some of the tests, a motor with anRS232 interface was added to permit computer control ofone of the pumps.

The in‰atable anchor comprised a hollow steel anchorrod with two layers of 1.5 mm thick rubber tubing ˆxedto the rod using pipe clamps. The anchor is shown in Fig.2. Tests were done using two diŠerent outer membranes:(i) one with sand glued to the membrane to improve its in-terface strength, and (ii) an untreated rubber membranein contact with the sand.

Materials and PropertiesCongleton silica sand was used for each test. This sand,

which was studied by Gallacher (2000), has a uniformgradation with D50＝0.3 mm, and a speciˆc gravity, Gs,of 2.65. The constant volume friction angle is 329fordensities varying from 1.52 t/m3 to 1.79 t/m3. In addi-tion, interface strength testing was also performed byGallacher (2000) using a direct shear apparatus for mem-brane-to-sand interfaces both with and without sandbonded to the membrane. The engineering properties ofCongleton sand and the various interface properties are

411

Table 1. The engineering properties of Congleton sand and sand-membrane interface (from Gallacher, 2000)

Sand PropertiesVoid ratio (e) Density r (t/m3)

0.48 (min.) 1.79 (max.)0.75 (max.) 1.52 (min.)

Interface PropertiesInterface Friction Angle

Rubber/rubber 19.19Sand/rubber 26.59

Sand/Roughened membrane 349

Table 2. Summary of the conducted tests

SeriesNo.

TestNo. Parameters Varied

A 1–3 Anchor length (L) varied from 100 to 300 mmB 4–6 In‰ation pressure varied from (P) 80 kPa to 200 kPaC 7–9 Density of sand (r) variedD 10–11 Surface roughness of membrane varied

Fig. 3. Typical pullout load-displacement response for in‰atable an-chors in sand Fig. 4. Calculated and measured unconstrained membrane expansion

411MECHANICS OF INFLATABLE ANCHORS

summarized in Table 1.

Test SequenceEach pullout test comprised ˆlling the test chamber

with Congleton sand. Sand beds were constructed by airpluviation using a spot-pouring container at a controlledrate and a ˆxed height from the soil surface. Higher sanddensities were achieved by layered compaction and the in-place density was measured by small density `pots'(Ueno, 1998). The in‰atable anchor was embedded in thesand during sand placement. Consequently, the in‰uenceof anchor installation on factors such as soil stiŠness andstress state has been neglected; although as detailed belowthe in‰ation pressures are very high and thought todominate any disturbance eŠects. After ˆlling the testchamber, the reaction beam was bolted to the test cham-ber ( see Fig. 1), the screw jack was mounted to the reac-

tion beam and connected to the drive motor, the load cellwas connected to the anchor head and screw jack, and theLVDT was mounted to the reaction beam and set to reston the top of the anchor. The anchor was subsequentlypulled out of the sand using a constant rate of displace-ment while simultaneously measuring axial load and dis-placement and maintaining constant anchor in‰ationpressure. Table 2 summarizes the conducted tests. Figure3 shows typical pullout force-displacement curves.

NUMERICAL VALIDATIONS

The ˆnite element program PLAXIS (Brinkgreve et al.,2002) was used to numerically simulate each of the in‰at-able anchor tests to evaluate the main factors aŠecting thepullout response. For this purpose, the ideal model wouldbe one that reproduced the exact experimental sequence.As such, the FE model was formulated such that it ac-counted for the elastic behaviour of the membrane, cavi-ty expansion of the membrane in the sand and pullout ofthe in‰ated anchor. The following is a detailed descrip-tion of the model and its veriˆcation.

Single Membrane ModelThe anchor membrane was modeled as a linear elastic

material even though elastomers are known to exhibitnonlinear elastic response. Such an idealization was con-sidered to be adequate given the limited strain-range thatthe conˆned membrane underwent during each test andsince emphasis was placed on predicting the limit capacityof each anchor. An unconstrained in‰ation test was per-formed on the rubber membrane to deduce its elasticproperties. The results of this test are shown in Fig. 4(a).Figure 4(b) shows the axisymmetric FE model used tosimulate the unconstrained test.

The FE model for unconstrained in‰ation comprised amembrane of height 300 mm and thickness t＝3 mm cor-responding to the exact geometry of the anchor mem-brane. The membrane was discretized using 15-nodedlinear strain triangles as shown in Fig. 4(b). Rigid ˆxedboundary conditions were adopted at both ends of themembrane and expansion of the membrane was simulat-

412

Fig. 5. FE model for cavity expansion in sand

Fig. 6. Comparison of theoretical (Yu and Houlsby, 1991), calculated(FE Method) and measured cavity expansion behavior in sand

Table 3. Summary of the properties of the interfaces for FE model

Interface Model

ParametersFriction angle betweenthe interfaces (Degree)g?

(kN/m3)Eref

(kPa)n

c?(kPa)

q?(Degree)

c?(Degree)

AF/FE/FC Mohr-Coulomb 0 6000 0.35 0.5 329 09 329

412 NEWSON ET AL.

ed by imposing a uniform horizontal pressure on the in-side of the membrane that was increased from 0 to 200kPa in 40 steps. An approximate geometrically non-linearanalysis was done using mesh updating.

The calculated relationship between in‰ation pressure(P ) and radial expansion (dr/r0) was investigated and it iscompared with the measured behaviour in Fig. 4(a). Thenonlinear response of the rubber membrane is evidentfrom the laboratory results presented in Fig. 4(a).However, there is reasonable agreement between the cal-culated and measured radial expansion versus in‰ationpressure for radial expansion (dr/r0) up to 30z and mem-brane properties of E＝1500 kPa and n＝0.35 ( see Fig.4(a)). Consequently, these properties were adopted insubsequent ˆnite element analyses. It should be notedthat although the radial expansion was in the order of 40z for the unconstrained membrane tests, the radial ex-pansion rarely exceeded 30z during the constrained cavi-ty expansion that occurred during the in‰ation phase ofeach pullout test.

Cavity Expansion in SandCavity expansion of the in‰atable membrane in sand,

similar to that of a pressuremeter test (Hughes et al.,1977), was experimentally measured and then modeledusing PLAXIS. This was done to gain conˆdence in thecomputational model and to obtain elastic properties forthe sand used in the anchor pullout tests. Figure 5 showsthe FE model geometry and Fig. 6 compares experimen-tal and calculated (PLAXIS) results with the Yu andHoulsby (1991) cavity expansion solution for dilatantsoils.

Referring ˆrst to Fig. 5, the FE model geometry tookinto account the axisymmetric geometry. In order tomaintain an isotropic initial stress state identical to thatassumed in the cavity expansion theory (Yu and Houlsby,1991), the overburden pressure and horizontal groundpressure at rest were simulated by applying a uniformpressure of 4.35 kPa, along lines A–B and B–C–D of theFE model. This pressure corresponds to the overburdenpressure at the midpoint of the in‰atable portion of theanchor during the constrained anchor in‰ation test. Theanchor in‰ation was simulated by incrementally applyinga uniform normal stress along boundary EF in Fig. 5 in15 steps. An elastic perfectly plastic constitutive modelbased on the Mohr Coulomb failure criterion was adopt-ed for the sand material assuming a unit weight g? ofzero. Strength parameters of q?＝329, c?＝09and c?＝0.5 kPa were adopted in the analysis for sand, which cor-responds to the constant volume parameters of Congle-

ton sand obtained from laboratory direct shear tests. Theelastic properties of the sand were adjusted in the FEanalysis to match the measured pressure versus volumechange response of the anchor. Table 3 summarizes thematerial properties adopted in the FE analysis.

To obtain a FE solution to the constrained cavity ex-pansion, an approach similar to that described in Yu andHoulsby (1991) was used. Interface elements were in-troduced between the vertical anchor shaft-to-sand (A–F)interface and membrane-to-sand (E–F) interface and ahorizontal rigid perfectly plastic interface was introducedin the sand layer at the top of the anchor ( see F–C in Fig.5). Although the interface F–C does not physically exist,it is required in the FE mesh (e.g., see Schanz et al., 1999;Brinkgreve et al., 2002) to improve numerical stability byeliminating the local stress concentration that occurs atpoint F due to the sudden change in boundary constraint

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Fig. 7. Geometry of the FE model for embedded in‰atable anchor oflength L＝300 mm

Table 4. Summary of the parameters used in the FE analysis

Parameter Anchor Shaft(Steel) Membrane Sand Unit

Material model Linear Elastic Linear Elastic Mohr-Coulomb —Behavior Non porous Non porous Drained —Unit weight, g? 24 15.6 14.9¿15.6 kN/m3

Young's modulus, Eref 2×108 1500 6000 kN/m2

Poisson's ratio, n 0.3 0.35 0.35 —Cohesion, c? — — 0.5 kN/m2

Friction angle, q? — — 329¿369 DegreeDilatancy angle, C? — — 0¿3.29 DegreeInterface strength, d(for each material in contact with sand)

3¿49 26.5¿349(For Le)0.59(Below Le)a

Equal to q?for A-B, C-D

Degree

a Le＝aL eŠective length and a is in the range of 0¿1.0

413MECHANICS OF INFLATABLE ANCHORS

from free along E–F to ˆxed along F–A. The strengthperameters of interface F–C were identical to that of thesoil. The full set of interface parameters used in the anal-ysis is summarized in Table 3. Referring to Fig. 5, it isnoted that the FE mesh was locally very ˆne adjacent tothe anchor and becoming coarser with distance from theanchor.

Calculated and measured net pressure versus radial ex-pansion, dr/r0, is compared with cavity expansion theoryin Fig. 6. In this ˆgure, the net pressure is the anchor in-‰ation pressure corrected for membrane eŠects (e.g., sub-tracting contributions due to the membrane, see Fig.4(a)). From Fig. 6, it can be seen that a limit pressure(PL) of 70 kPa was obtained from the FE analysis at dr/r0

of 32z. A limit pressure of 64 kPa was obtained fromcavity expansion theory (Yu and Houlsby, 1991) at amaximum dr/r0 of about 46z. The limit pressures fromboth FE analysis and cavity expansion theory are com-parable and they agree well with the actual measured be-haviour of the anchor during constrained cavity expan-sion. The analysis and discussions presented above serveto demonstrate the adequacy of the FE model used in thestudy. From Fig. 6, it can be seen that the FE analysisand measured response are within 10z for net pressuresup to about 50 kPa, corresponding to an internal in‰a-tion pressure in the order of 100 kPa.

FE MODEL FOR INFLATABLE ANCHOR IN SAND

Model Conˆguration and Materials PropertiesFigure 7 shows the FE model used to investigate the in-

‰atable anchor behaviour and Table 4 summarizes thematerial properties. The FE model consists of the anchorshaft, rubber membrane and sand around the anchor.Axisymmetric geometry was modeled using 15-noded cu-bic strain elements to discretize the sand and rubber mem-brane. Each ˆnite element analysis comprised: (i) ˆrst set-ting up the initial stresses in the sand; (ii) simulating in‰a-tion of the membrane inducing cavity expansion; and (iii)ˆnally simulating the pullout response. The model con-ˆguration, material properties and other details of the FEcalculations are described below. Finite element discreti-zation of the membrane and soil did not deviate sig-

niˆcantly from that described in the veriˆcation problemsdiscussed above.

Anchor ShaftThe 35 mm diameter anchor shaft was modeled using

continuum elements assuming linear-elastic material(steel) behaviour. The Young's modulus and Poisson'sratio were taken as 200 MPa and 0.3, respectively. Thecover depth (H ) and the length of the in‰atable anchor(L) were variable corresponding to each of the test cases( see Table 2). Figure 7 shows the geometry corre-sponding to H＝140 mm and L＝300 mm.

MembraneBased on the unconstrained membrane tests described

above, the membrane was modeled as a linear-elasticmaterial with a thickness of t＝3 mm, and Young'smodulus of 1500 kPa and Poisson's ratio of 0.35. Thelength of the membrane corresponds to the length of thein‰atable anchor for the speciˆc test considered (Fig. 7).

Sand LayersThe sand was modeled as an elastic-perfectly plastic

material with failure governed by the Mohr Coulomb

414414 NEWSON ET AL.

failure criterion. The sand properties are listed in Table4.

The sand was modeled to the bottom of the anchoronly based on studies that have shown the soil below theanchor base has a negligible eŠect on the anchor behav-iour for drained conditions due to breakaway (e.g.,Kanakapura et al., 1994). From back-calculation of theconstrained cavity expansion test described above, theYoung's modulus and Poisson's ratio of the sand were6000 kPa and 0.35, respectively. The eŠective friction an-gle was varied between 329and 369depending on thesand density and taking into account the results oflaboratory direct shear tests on the Congleton sand (Gal-lacher, 2000). For loose sand (g?＝14.9 kN/m3), the con-stant volume friction angle of q?cv＝329and c?＝09wereused in the FE model. For the more dense sand, peak fric-tion angles, qp?, of 349and 369were used for sands withg?＝15.2 kN/m3 and g?＝15.6 kN/m3, respectively, andthe corresponding dilation angle c? was deduced usingthe relationship c?＝0.8(qp?－q?cv) from Bolton (1986).The coe‹cient of lateral earth pressure at rest was as-sumed to be K0?＝0.5 to calculate the initial stresses.

EŠective Length of the In‰atable Anchor (Le)The main purpose of the FE analyses reported in this

paper was to interpret the ultimate pullout load (pulloutcapacity) of the in‰atable anchors. However, during theinitial back analysis of the anchor tests, it was found thatthe pullout capacity could not be interpreted for all an-chors using an eŠective friction angle, q?, for the sandthat agreed with laboratory direct shear tests (Gallacher,2000). In addition, in many cases it was necessary to varyq? for anchors of diŠerent length but installed in sandwith the same density. As a result, the test results werereinterpreted by modeling: (i) an upper eŠective zone,denoted Le for eŠective length, with interface propertiescorresponding to those measured for the sand-to-rubberinterface and (ii) a lower zone, which was assumed to beineŠective and assigned a very small interface friction an-gle to limit the shear strength to approximately zero.

Thus, it was assumed that each anchor has an eŠectivelength Le that is less than the physical length of the an-chor and the FE analysis was undertaken to back calcu-late Le for each anchor using properties that were consis-tent with Congleton sand and its corresponding state.Additionally, it is noted that the concept of eŠectivelength is very common in the analysis of anchors and pilesduring pullout (see Wald et al., 2008; Degil', 1991; In-draranta, 1991; Iskander et al., 2002; Bhattacharya et al.,2003; Ahmad et al., 2005; Nihar et al., 2006). In the fol-lowing sections, the eŠective length (Le) is oftenrepresented by aL in ˆgures, where a is a coe‹cientvarying from 0 to 1.0, and L is the anchor (membrane)length.

Interface in SandTwo horizontal interfaces denoted A–B and C–D in

Fig. 7 were introduced to solve the cavity expansion or in-‰ation stage of each test. These interfaces were rigid per-

fectly plastic interfaces with shear strength parametersidentical to that adopted for the sand ( see Table 4).

Three vertical interfaces were introduced along thelength of each anchor (I–J) as shown in Fig. 7. In PLAX-IS, the friction angle of the interface, d, is deˆned by theequation tan d＝Rint tan q?, where Rint is a reduction fac-tor and q? is the eŠective friction angle of soil adjacent tothe interface. Based on the laboratory results summarizedin Table 1, the friction angles of the sand-to-sand andsand-to-rubber interfaces are 349and 26.59, respectively.The corresponding reduction factors used in the analysiswere Rint＝0.1 (d＝3–49) on the anchor shaft-to-soil inter-face, Rint＝0.74–1.0 (d＝26.5–349) on the membrane-to-soil interface in the eŠective length zone (Le), and Rint＝0.01 (d＝0.59) on the membrane-to-soil interface belowthe eŠective length zone. A small friction angle was re-quired below the eŠective zone to ensure numerical stabil-ity ( see also Kanakapura et al., 1994). The materialparameters used in the FE analysis are summarized inTable 5.

Boundary ConditionsThe lateral FE boundary extends 350 mm from the an-

chor axis, which is 10 times the anchor diameter and iden-tical to the dimensions of the test chamber. Fixed x-displacement and free y-displacement boundaries wereadopted on both sides of the model, and ˆxed y-displace-ment and free x-displacement boundaries were given tothe base if the sand layer ( see Fig. 7).

Solution SequenceThe FE solution sequence involved three stages. (i)

First, the initial geostatic stresses in the sand were estab-lished assuming K0?＝0.5. (ii) Then, in‰ation of the an-chor (cavity expansion) was simulated numerically usingmesh updating. Thus, the cavity expansion solution in-volved approximate geometric non-linearity. (iii) Finally,anchor pullout was simulated but without mesh updatingas discussed below.

For the in‰ation stage, a physical gap was introducedin the FE mesh between the anchor shaft and the mem-brane elements ( see Physical gap 1 in Fig. 7). This physi-cal gap was required to apply the in‰ation pressures (seeFig. 7), which were speciˆed as a uniform stress (Load A)incrementally from 0 to the in‰ation pressure reportedduring each model test (in kPa) along the inner surface ofthe membrane.

After simulating the anchor expansion, pullout wassimulated by prescribed displacements at the top of theanchor. A second physical gap (Physical gap 2 in Fig. 7)was introduced in the model at the bottom of the anchorto avoid tension along this interface. Unlike the anchorin‰ation stage, the pullout stage was analyzed withoutmesh updating because the mesh updating caused numer-ical instability during the pullout simulations. The sourceof instability is thought to arise from the use of interfaceelements along line C–J in Fig. 7, which can tolerate onlya limited amount of shear displacement during mesh up-dating before becoming ill-conditioned (e.g., the length

415

Table 5. Summary of the lab tests (Newson et al., 2003) and the corresponding FE analyses

ParameterLab Test/FE Analysis

Unit1b/Ac 2/B 3/C 4/D 5/E 6/F 7/G 8/H 9/I 10/J 11/K

Anchor length1 L 100 200 300 300 300 300 300 300 300 300 300 mm

Embedment depth1 H 140 140 140 140 140 140 140 140 140 140 140 mm

In‰ation pressure1 P 150 150 150 80 150 200 150 150 150 150 150 kPa

Unit weight1 g? 15.6 15.6 15.6 14.9 14.9 14.9 14.9 15.2 15.6 14.9 14.9 kN/m3

EŠective Length coe‹cient3 a 0.9 0.5 0.45 0.15 0.45 0.45 0.45 0.45 0.45 0.45 0.45

Young's modulus1 Eref 6 6 6 6 6 6 6 6 6 6 6 MN/m2

Poisson's ratio2 n 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35

Cohesion1 c? 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 kPa

Friction angle1,4 q? 36 36 36 32 32 32 32 34 36 32 32 Degree

Dilatancy angle2,5 c? 3.2 3.2 3.2 0 0 0 0 1.6 3.2 0 0 Degree

Anchor shaft-soil Interface2 d 4 4 4 3.6 3.6 3.6 3.6 3.3 3.5 3 3 Degree

Membrane-soil interface1 (Le Part) d 26.5 26.5 26.5 26.5 26.5 26.5 26.5 26.5 26.5 26.5 34 Degree

Membrane-soil interface3 (below Le) d 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 Degree

Soil-to-soil interface1 d 36 36 36 32 32 32 32 34 36 32 32 Degree

Notes: b The sequence number 1¿11 denotes the lab test number ( see Figs. 8¿11)c The sequence letter A¿K denotes the FE analysis case corresponding to the lab tests (see Figs. 8¿11)1 Parameters that were measured2 Parameters that were estimated3 Parameters that were deduced or back-calculated4 q?p was interpolated from q?cv and q?p data in Gallacher (2000)5 c? was estimated from Bolton (1986)

Fig. 8. Comparison of measured and calculated pullout force versusdisplacement for various anchor lengths (L)

415MECHANICS OF INFLATABLE ANCHORS

of interface elements on boundary C–J was only 5 mm).This limitation in the analysis will be discussed below af-ter presentation of the results.

RESULTS

In this section, FE calculations are compared with themeasured behaviour of model anchors during the pullouttests. For the ˆnite element analyses, the variables consi-dered were the anchor length (L), in‰ation pressure (P ),density of the soil (r), and membrane surface roughness.For all cases, the primary variable was the eŠectivelength, which was deduced by varying Le to obtain agree-ment between the calculated and measured behaviour.

EŠect of Anchor Length (L)Figure 8 compares the calculated and measured pullout

behaviour of 100 mm, 200 mm and 300 mm long anchorswith 140 mm embedment depth (H ). For these tests, thein‰ation pressure (P ) was 150 kPa and the density of thesand (r) was essentially the same, 1.59 t/m3 for each test.To obtain the calculated behaviour, the eŠective anchorlength (Le) was set at 0.9L, 0.5L and 0.45L for anchors oflength (L) 100 mm, 200 mm and 300 mm, respectively.

From Fig. 8, it can be seen that the maximum pulloutcapacity calculated by FE method varies from 0.14 kN to0.29 kN, which agrees well with the lab tests. Such agree-ment could not be obtained assuming that the full length

of the anchor was eŠective in resisting the pullout loads( see Newson et al., 2007). Although there is good agree-ment for the ultimate capacity of the anchors, the dis-placement mobilized at the peak load is consistently lessthan that measured in each test. This may be due to dis-tortion of the rubber membrane possibly caused by stressconcentrations at the top of the anchor near the mem-brane-to-shaft connection. In addition, as shown in Fig.4, the stress-strain response of the rubber is nonlinearwhich has not been accounted for in this analysis.

416

Fig. 9. Comparison of calculated and measured pullout force versusdisplacement for diŠerent in‰ation pressures (P )

Fig. 10. Comparison of measured and calculated pullout force versusdisplacement for diŠerent sand densities (r)

Fig. 11. Comparision of measured and calculated pullout force versusdisplacement for diŠerent membrane surface treatments

416 NEWSON ET AL.

In‰ation Pressure (P)Figure 9 shows the eŠect of varying the in‰ation pres-

sure on the pullout resistance of the in‰atable anchors.For this series of tests, the embedment depth (H ) was 140mm, the anchor length (L) was 300 mm, the soil density(r) was 1.52 t/m3, and the in‰ation pressure (P ) was va-ried from 80 kPa to 200 kPa. In order to obtain reasona-ble agreement between the measured and calculated pul-lout behaviour, the eŠective length (Le) was 0.15L, 0.45Land 0.45L for anchors in‰ated to pressures of 80 kPa,150 kPa and 200 kPa respectively. Consequently, it is in-ferred on the basis of the actual anchor performance thatthe eŠective length is also proportional to the in‰ationpressure.

From Fig. 9, it can be seen that the calculated peak pul-lout resistance varies from 0.13 kN for P＝80 kPa to 0.28kN for P＝200 kPa. The calculated pullout resistanceagrees well with the test data. The displacement mobi-lized at the peak strength is higher for each of the scaledmodel tests compared with the FE calculations. Again,this may be due to nonlinearity of the rubber membraneand membrane distortions at the upper connection.

Density of Soil (r)Figure 10 shows the eŠect of soil density on the pullout

capacity of the in‰atable anchors. For this sequence oftests, the sand density (r) was varied from 1.52 t/m3 to1.59 t/m3. Accordingly, the parameters governing thestrength and dilation of the sand, which are summarizedin Table 5, were varied slightly according to the densityof the sand. For each test and FE simulation, the in‰ationpressure (P ) was 150 kPa, the embedment depth (H ) was140 mm and the anchor length (L) was 300 mm. Giventhe consistent geometry of these anchors, the eŠectivelength (Le) was 0.45L for each analysis.

Referring to Fig. 10, it can be seen that the peak pul-lout resistance from the FE analysis agrees very well withthe test results. The calculated displacement mobilized atthe peak strength varied from 20 mm to 26 mm, which isagain less than that obtained during the lab tests. As withthe other simulations, however, the agreement between

calculated and measured peak load is adequate.

Membrane Surface RoughnessTo conclude, variations in the pullout capacity due to

variations in the membrane treatment (rubber only versusrubber with sand bonded to it) were investigated and theresults are summarized in Fig. 11. For each of these tests,the soil density (r) was 1.52 t/m3, the embedment depth(H ) was 140 mm, the anchor length (L) was 300 mm andthe in‰ation pressure (P ) was 150 kPa. The calculated be-haviour in Fig. 11 is based on an eŠective length (Le) of0.45 for both the untreated and roughened membrane.For the roughened membrane surface, the interface fric-tion angle, d, was 349compared to that of the untreatedmembrane, d＝26.59( see Table 1).

From Fig. 11, it can be seen that the peak pulloutresistance from FE analyses varies from 0.22 kN to 0.42kN, which matches the measured peak loads with againdiscrepancies in the displacement mobilized at the peakpullout load.

417

Fig. 12. Calculated pressure versus radial expansion during conˆnedcavity expansion

Fig. 13. Calculated plastic zone for P＝30 kPa and 150 kPa during in-‰ation for test C

417MECHANICS OF INFLATABLE ANCHORS

DISCUSSION

Anchor MechanicsFigure 12 shows the calculated net pressure on the soil

and the internal in‰ation pressure versus radial expansion(dr/r0) for anchors of length 100 mm, 200 mm and 300mm, inclusive, for anchor test C. In addition, the netpressure causing full yielding in the soil is denoted byPoint A. It is noted that a maximum radial expansion of30z was reached at an in‰ation pressures of P＝150 kPa.From Fig. 12, it can be seen that for P＝150 kPa at dr/r0

of 30z, the net pressure supported by the sand is about70 kPa and the diŠerence (80 kPa) was resisted by themembrane. In addition, the soil is well past its yield pointas discussed further in reference to Fig. 13 below.

Figure 13 shows the plastic zones after cavity expan-sion corresponding to in‰ation pressures of 30 kPa and150 kPa. From Fig. 13, it can be seen that there are exten-sive zones of plasticity in the soil surrounding the in‰atedportion of the anchor and that the size and extent of theyield zone increases with in‰ation pressure from P＝30kPa to 150 kPa. As a consequence, the pullout capacityof this type of in‰ated anchor is strongly in‰uenced bythe response of the embedment soil at its limit pressure asshown by Figs. 12 and 13.

Figure 14 shows a typical velocity ˆeld and the associ-ated zones of plasticity during pullout for anchor test C

( see Table 5). From Fig. 14(a)¿(d), it can be seen thatboth the velocity ˆeld and the plastic zones are progres-sively developed around the upper part of the anchor dur-ing pullout, which is above the deˆned eŠective length(Le). The velocity ˆelds and plastic zones are governed bythe eŠective length assumed in the analyses of the an-chors.

Figure 15 shows contours of the shear strain developedin the sand around anchor C during pullout. It can beseen that, for an anchor head displacement dv of 12.5 mmcorresponding to the displacement ratio dv/r0 of 0.7 ( seeFig. 15(a)), the maximum shear strain developed aroundthe upper part of the anchor is 19.1z and it is concentrat-ed within the relatively narrow zone adjacent to the an-chor (i.e., along the interface). At an anchor head dis-placement of 28 mm reaching the peak pullout load withthe displacement ratio dv/r0 of 1.6 ( see Fig. 15(b)), theshear zone begins to propagate toward the soil surfacealong the boundary of the pullout mechanism depicted inFig. 14(c).

Figure 16 shows the measured and calculated relation-ship between the normalized pullout force and the dis-placement. For this ˆgure, the pullout load (P ) was nor-malized by the peak load (Ppeak) during each test. The ver-tical displacement (dv) was likewise normalized by the dis-placement at the peak load (dpeak). As expected, a uniquenormalized curve was obtained from the FE analyses.The normalized FE curve ( see Fig. 16) agrees well withthe normalized test data, which is likely in‰uenced byvariations in the test conditions.

EŠective LengthAs noted above, a consistent interpretation of the an-

chor behavior could only be obtained for all tests byvarying the eŠective length (Le) of each anchor. Other in-terpretations were possible but only if the soil parameters(such as q?) were varied signiˆcantly outside the rangemeasured in the laboratory for Congleton sand ( see Gal-lacher, 2000; Newson et al., 2007). However, the eŠectivelength concept permitted the use of material parametersthat were consistent from test to test and that agreed withthe measured parameters.

Figure 17 illustrates a hypothesis that could explain themobilization of an eŠective anchor length for in‰atableanchors. After cavity expansion, the anchor shape is ap-proximately cylindrical based on the FE results but with alarger diameter at the top and smaller diameter towardthe bottom. The solid line in Fig. 17 denotes the anchorshape after in‰ation. During pullout, however, it ishypothesized that the anchor deforms into a `tear drop'shape due to the ‰exible nature of the membrane. In thiscase, the lower part of the membrane-soil interface is lesskinematically favorable for the development of shearbands and may not even slip relative to the soil. Thismechanism would tend to reduce the contribution of thelower portions of the membrane toward the pulloutcapacity.

Lastly, it can be shown that the pullout capacity of anin‰atable anchor is dependent on the following indepen-

418

Fig. 14. Calculated velocity ˆelds and corresponding plastic zones during pullout (Test C)

Fig. 15. Calculated shear strain contours in sand during pullout (TestC)

418 NEWSON ET AL.

dent parameters: (i) eŠective stress at the top of the an-chor, s?v, (ii) the slenderness ratio, L/r, (iii) the degree ofcavity expansion, dr/r, (iv) the interface strength, d, (v)the friction angle of the soil, q?, (vi) the anchor weight,Wa, and (vii) the membrane modulus, Em. For the testsdescribed in this paper, Wa was a small fraction of the ul-timate anchor load, and the parameters s?v, and Em, werenot varied. Thus, for a given d and q?, the followingdimensionless parameter can be derived:

j＝(dr/r)(L/r) (1)

where r is the ˆnal radius after cavity expansion and r＝dr＋r0.

Figure 18 summarizes Le/L versus the dimensionlessparameter j for all of the anchor tests. From the FE ana-

419

Fig. 16. Normalized relationship between force and displacement (allanchors)

Fig. 17. Hypothesized shape change of the membrane during pullout

Fig. 18. EŠective length ratio Le/L versus dimensionless parameter, j＝(dr/r)(L/r)

419MECHANICS OF INFLATABLE ANCHORS

lyses, it appears that there is a good correlation between(Le/L) and j where:

(Le/L)＝0.45＋3.0e－2.0j (2)

Thus, the data suggests a transition from short anchorbehaviour where the full anchor length is eŠective at jÃ1to long anchor behaviour for jÆ2.5 where increasing the

anchor length does not signiˆcantly in‰uence the pulloutcapacity.

CONCLUSIONS

Based on the numerical study of the behaviour of an in-‰atable anchor system in sand, the following conclusionscan be drawn:

1) The PLAXIS FE model can approximately simulatethe behavior of the in‰atable anchors in cohension-less soil. The model was however not able to ac-count for geometric nonlinearity during the anchorpullout stage requiring adoption of an eŠectivelength for each anchor.

2) The pullout capacity of this type of in‰ated anchoris strongly in‰uenced by the response of the embed-ment soil at yield.

3) The FE model consistently under predicted dis-placements mobilized at the ultimate load. This ismost likely due to membrane distortions and mem-brane nonlinearity, which were not accounted for inthe analysis.

4) Adoption of an eŠective length for each anchor per-mitted a consistent interpretation of the pulloutresponse. As such, it is concluded that membranedistortions during pullout and consequent changesin the geometry of the anchor are important.

5) The normalized eŠective length (Le/L) is propor-tional to the dimensionless parameter j. The overallresponse suggests that short anchor behaviour oc-curs for jº1 wherein the full anchor length is activein resisting the pullout loads.

6) There is a transition from short anchor behaviour tolong anchor behaviour wherein additional an-chorage length does not improve the pullout capaci-ty. For the test conditions considered, long anchorbehaviour occurs for jÆ2.5.

7) Lastly, future numerical analysis of in‰atable an-chor behaviour should account for the non-linearbiaxial behavior of elastomers and the shape changeof the membrane during pullout.

ACKNOWLEDGEMENTS

The authors would like to acknowledge the contribu-tions of Stephen Gallacher, Fraser Smith and the staŠ ofthe University of Dundee, and Paul Bruning of AcergyNorthern Europe and Canada during the early stage ofthis project.

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