Extending the strain path method analogy for modelling penetrometer installation

*Correspondence to: B. M. Lehane, Department of Civil Engineering, Trinity College, University of Dublin, Ireland

Received 8 October 1998Copyright ( 2000 John Wiley & Sons, Ltd. Revised 5 September 1999

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS

Int. J. Numer. Anal. Meth. Geomech., 2000; 24:477}489

Extending the strain path method analogyfor modelling penetrometer installation

D. R. Gill and B. M. Lehane*Department of Civil Engineering, Trinity College, University of Dublin, Ireland

SUMMARY

The Strain Path Method (SPM) is an approximate framework for simulating the disturbance caused by pilesor penetrometers in soil. The key conceptual assumption of the SPM is that the deformation and strain "eldscaused during these penetration processes are strongly kinematically constrained (especially during un-drained penetration of clays) and can be estimated independently from the actual constitutive properties ofthe surrounding soil. Previous applications of SPM have estimated strain "elds for a variety of penetrometergeometries using velocity "elds of ideal inviscid #uids. This paper re"nes the strain "eld for penetrometerswith 603 conical tips using numerically computed velocity "elds in viscous #uids with a variety of boundaryconditions imposed on the penetrometer shaft. Following a parametric study, a set of #ow conditions isselected which provides a best "t between computed soil deformations and physical displacement measure-ments made in three separate experiments. The approach is simple and rapid and, while highlighting some ofthe inaccuracies associated with the existing SPM solution, may also be used for comparative purposes toassist the development of other approaches to the deep penetration problem. Copyright ( 2000 John Wiley& Sons, Ltd.

KEY WORDS: penetrometer; installation; #ow anology; displacement/strain "elds

1. INTRODUCTION

In geotechnical problems involving retaining walls, slopes or shallow foundations, the stresses inthe ground can be related directly to the depth of overburden and the applied loads. However, thekinematic constraints present at depth during the advance of a penetrometer or driven pile aresuch that the strains induced in the ground are relatively insensitive to the precise shearingcharacteristics of the soil. Such insensitivity was observed in a series of laboratory simulations ofpenetrometer installation performed by Baligh,1 who subsequently developed what is now knownas the Strain Path Method (SPM).2 Previous applications of the SPM have estimated that duringundrained penetration the relative position of soil particles to a penetrometer are the same as thestreamlines followed by an inviscid incompressible #uid as it #ows around a "xed penetrometer.Baligh2 predicted the path of such streamlines using potential #uid #ow theory and modelledpenetrometer geometries through a combination of sources and sinks located within a uniform#ow.

Teh3 adopted an alternative numerical procedure to potential theory and used a "nitedi!erence method to predict the velocity "eld of an inviscid #uid #owing around a penetrometer

with a 603 conical tip. The procedures employed by both Baligh2 and Teh3 do not involvespeci"cation of a constraint to the #ow tangential to the penetrometer boundary, but deduce byapplication of elasto}plastic soil models to the computed strains that their solutions approximatethose of a smooth boundary.

The SPM is just one of the approaches available for the numerical analysis of penetrationproblems. Promising new methods under development use a variety of large strain "nite elementformulations to cope with the rapidly varying boundary conditions (e.g., see References 4}6) oremploy re-meshing procedures following small incremental penetrations.7 These approachesdemand signi"cant computer resources and are appreciably more complex than the SPM.

The approach adopted here explores the possibility of using a standard Finite Element (FE)package to predict realistic strain "elds around penetrometers by a simple extension of the SPM.The SPM analogy is maintained but two additional parameters are varied, namely the penet-rometer boundary condition and the #uid viscosity. Simple interpolation and integration tech-niques are applied to the FE predictions of the #uid velocity "eld to derive the soil strains anddisplacements. These displacements are then compared with experimental measurements ofdisplacements around penetrometers to ascertain whether a di!erent set of #ow assumptions tothose adopted by the SPM can yield improved predictions of displacement and strain "elds.

2. FLOW MODELLING

The "rst step in modelling the penetration process is to solve the #uid #ow problem of anaxisymmetric "xed body located within a uniform #ow "eld. The ANSYS8 "nite element #uiddynamics package was used for this purpose. The use of "nite elements allows the modelling ofdi!erent #ow types and penetrometer shapes to be achieved far more simply than with potentialtheory, and is more suited to solving problems with complex boundaries than "nite di!erencemodels.9

The #uid #ow solutions presented in this paper are steady-state incompressible Newtonian#ow analyses produced by the FLOTRAN computational #uid dynamics solver within ANSYS.8A two-dimensional axisymmetric co-ordinates system is used with the penetrometer located onthe axis of symmetry. The stream function values (t) are derived from the calculated velocitiesand stored on a nodal basis at the end of each iteration. The number of iterations required toachieve a converged solution depends on the grid meshing and stability of the problem. All the#ow problems were meshed with four-noded quadrilateral elements. The analyses conducted for#uids with high viscosities produced smooth, ordered velocity "elds and converged easily.Convergence of #ows with very low viscosities was more di$cult and, following the recommenda-tions of the ANSYS8 authors', involved a number of analyses, each of which began iterations fromthe results of a preceding analysis which had been performed for a higher viscosity.

The boundary conditions adopted for modelling the #ow distribution around a cone penet-rometer, of radius, R, are shown in Figure 1. For all solutions, a uniform vertical velocity (;) isapplied at the inlet AF (upstream) and far "eld boundary (FE). On the axis of symmetry (r"0axis), the velocity normal to the #ow (the r-direction) is set to zero and but is unconstrained in thez-direction. A zero normal velocity is applied on the penetrometer face and, unlike the SPM, thetangential velocity (v

1%/5) is assigned a value between zero (termed no-slip) and the #ow "eld

velocity, ; (full slip). A zero relative pressure is speci"ed at the outlet (DE) for all solutions.Figure 2 shows the vertical and radial velocity pro"les close to the penetrometer at a height of"ve radii above the tip (z/R"5) for &full slip' #ow and for two meshes: Mesh &A' (with 4080

478 D. R. GILL AND B. M. LEHANE

Copyright ( 2000 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech., 2000; 24:477}489

Figure 1. Flow model: (a) boundary conditions; (b) nodal grid

Figure 2. Velocities for the 603 cone at z/R"20 for two FE meshes

MODELLING OF PENETROMETER INSTALLATION BY STRAIN PATH METHOD 479

Copyright ( 2000 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech., 2000: 24:477}489

elements) and Mesh &B' (with 11,140 elements). Signi"cant di!erences between the predictedvelocities of the two meshes only occur very close to the penetrometer and these are relativelysmall. Given that the interpolation procedure discussed below resulted in smoother pro"les,Mesh &A' was selected as the best compromise between accuracy and computational/dataprocessing time and is used to produce all the solutions presented in this paper.

3. INTERPOLATNG FLOW PATHS

The SPM derives strains in the soil by integration of strain rates along streamlines (or #ow paths)followed by the #uid #ow. Each streamline is de"ned by a constant value of t, termed the streamfunction (which is given directly by potential theory). The standard ANSYS8 package does not,however, predict the location of streamlines but instead outputs the t value at each node. Thelocation of streamlines therefore has to be found by interpolating for a constant stream functionthrough the nodal grid. A polynomial interpolation function that could be integrated ordi!erentiated without di$culty was used for this purpose.

In two dimensions, a complete nth-order polynomial in the r}z plane may be written as (e.g. seeReference 8):

Pn(r, z)"

T (2)n+k/1

akrizi, i#j)n (1)

where the number of terms in the polynomial is

¹ (2)n

"(n#1) (n#2)/2

and the coe$cients (ak) control the magnitude of the distribution of the two-dimensional "eld

variable. With the local co-ordinates (m, g) and in a nine-node local mesh system (also used byTeh3) the polynomial used to describe the variation of t becomes

t (m, g)"a0#a

1m#a

2g#a

3m2#a

4mg#a

5g2#a

6m2g#a

7mg2#a

8m2g2 (2)

The corresponding matrix formulation for (2) is

MtN"[M]MaN (3)

where [M] is a square (9]9) matrix. MaN is then found from

MaN"[M]~1MtN (4)

This set of simultaneous equations gives MaN for the central node in the local mesh and is appliedto every node in the grid (except the boundary nodes). Interpolation of streamline paths (ofconstant stream function, t) through the nodal grid can then be achieved by simply solving form at each horizontal grid row (when the local y-direction co-ordinate, g, is zero) using equation (2),which is reduced to the following quadratic equation.

t"a0#a

1m#a

3m2 (5)

4. CALCULATION ON STRAINS AND DISPLACEMENTS

The calculation of strains from the ANSYS8 output "les is an adaptation of the method presentedby Teh3 for "nite di!erence solutions. Following the derivation of the streamline location, as



outlined, the determination of strains involves calculation of the velocities and strain rates at eachinterpolated point along the #ow path. The accumulated strain can then be calculated along thestreamline by integration of the strain rates at each interpolated point.

The velocities and strain rates are obtained using the expressions given in Appendix Iwhich are di!erential forms of equation (2). The strain accumulated along a streamline is thencalculated by dividing the streamline along its length into linear segments, each of which is anapproximation of the path length (ds) between the established points. Each strain increment iscalculated at the mid-point between the streamline points and the integration moves from thebottom grid (boundary AF in Figure 1) upwards in the direction of #ow. The cumulative strain isgiven by

eij"

n+k/1

eRij

ds

vN4

(6)

where vN4is the average velocity in the direction of the streamline between points i and j and eR

ijis

the strain rate between these points.The radial displacement at any node n (dr

n) is simply the radial shift of the streamline

(equation (7)). The increment of vertical displacement is given by the di!erence between thevertical and free "eld #ow velocity multiplied by the time over which these velocities act.The total vertical displacement (dz

n) at each streamline point is the sum of these increments

(equation (8)).

drn"r

n!r

0(7)

dzn"

n+k/1

(v3%-

, t)k"

n+k/1A(vz!;)

dz

vzBk

(8)

An example demonstrating the accuracy of the interpolation technique is shown in Figure 2,which shows that velocities derived from the interpolated stream functions compare very wellwith those outputted directly from the Mesh &A' solution. In fact, it is evident that the interpola-tion process smoothens small irregularities associated with the density of the FE mesh.

5. STRAIN PATH PREDICTIONS

The in#uence of #uid viscosity and shear stresses on the penetrometer boundary (as a conse-quence of v

1%/5/; being less than 1) are now investigated for the 603 cone by examining the strain

paths predicted in eight separate ANSYS8 analyses. The relevant details of each analysis, desig-nated #ows &A'}&G', are summarized in Table I. To facilitate comparison of the analyses, the strainpath predictions were performed for a streamline located at an initial radial distance of onepenetrometer radius from the centreline, i.e. r

0/R"1.

The strain paths are depicted in Figures 3}5 in terms of variations with z/R of the three &E'strain invariants proposed by Baligh:2

E1"e

zz, E

2"(e

rr!ehh)/J3, E

3"2e

rz/J3 (9)

where ezz

, err

and ehh are the direct vertical, radial & circumferential strains and erz

is thecomplimentary shear strain in the r}z plane. E

1, E

2and E

3correspond to strains induced under

triaxial, cylindrical cavity expansion and direct simple shear loading.



Table I. ANSYS #ow solutions for 603 cone

Flow Viscosity, k(N s m~2)

Velocity onpenetrometer

boundary, v1%/5

/;

Reynolds number,Re"

Flow-type description

A! 1]10~60 1 1)2]1062 Very low viscosity full slip #owB 1)2 0 2000 Viscous non-slip #owC 1)2 1 2000 Viscous full slip #owD 100 1 24 High-viscosity full slip #owE 1]104 1 0)24 High-viscosity full slip #owF 1)2 0)975 2000 Viscous partial slip #owG 1)2 0)75 2000 Viscous partial slip #owH 4]10~3 0)97 6]105 Low-viscosity partial slip #ow

!Fluid density, o"1 kg/m3 and uniform #ow "eld velcotiy, ;"2 m/s. For all other solutionso"2000 kg/m3 and ;"0)02 m/s."For internal duct #ows; Re"o<D

h/k, where D

his the hydraulic diameter and < is the #uid velocity.

5.1. Comparison with SPM

Teh3 presents the strain path for a streamline with r0/R"1 for an inviscid #ow around a 603

cone. This is reproduced in Figure 3 and may be considered as a benchmark representing theSPM solution for this streamline. Flow &A', which adopted a very low viscosity and a v

1%/5/; ratio

of unity, closely approximates the assumptions of the SPM. A comparison of the strain pathspredicted by this #ow (see Figure 4) indicates very good agreement with the SPM solution apartfrom the fact that E

3experiences a reversal at z/R"3 and increases subsequently as z/R

increases. A separate ANSYS8 analysis performed with an unspeci"ed tangential velocity on thepenetrometer boundary revealed that the E

3response predicted at large z/R values arises because

of the viscosity of the #uid.

5.2. The ewects of yuid viscosity (k)

The predictions plotted in Figure 4 illustrate the e!ect of #uid viscosity (k) on &full slip' #ow(v

1%/5/;"1) for the 603 cone. It is evident that high viscosity #ows with low Reynolds numbers

(Re), i.e., Flows D & E, produce somewhat di!erent strain paths to #ows with lower viscosity(A & C). Further points of note include:

(1) High k results in greater compressive strains (note that E1

is negative when compressive) asthe penetrometer tip approaches. The elements for Flows D & E are seen in Figure 4 to havea net vertical compression throughout the penetrative process.

(2) E2

and E3

are generally lower for high k #ows with &full slip'. However, the other analysesindicated that friction on the penetrometer can lead to higher E

2and E

3values for high

k #ows.(3) The strain paths produced are most sensitive to changes in viscosity over the range from 1 to

100 Ns/m2 (compare Flows C and D).

In general, all analyses performed (including those involving partial slippage on the penet-rometer surface) indicated that the overall e!ect of increasing the #uid viscosity is to increase the



Figure 3. Teh's3 solution for SPM: strain path at r0/R"1 for 603 cone

range of in#uence of the penetrometer and its speci"ed boundary conditions on the surroundingelements.

5.3. Relative velocity of penetrometer

The predictions plotted in Figure 5 examine the in#uence of the slip condition on the boundaryof the penetrometer for a #uid with a viscosity approximately 1000 times that of water. The strainpaths for #ows with &full slip' (C), &no-slip' (B), and two &partial slip' conditions (F & G) arepresented. Reducing the velocity of the penetrometer increases the friction on its surface and thisresults in;

(1) large increases in the shear strain (E3) once the cone shoulder has passed (particularly evident

in Flows B & G)



Figure 4. Deviatoric strain paths at r0/R"1 during penetration of a 603 cone for &full slip' #ow

(2) increased vertical compressive strain (E1

remains compressive throughout the penetrativeprocess in Flows B & G)

(3) greater peaks in E2

(the cylindrical expansion strain) at the cone shoulder, but also greaterreductions in E

2as the distance above the tip (z/R) increases.

6. COMPARISON WITH EXPERIMENTAL OBSERVATIONS

Experimental observations of the displacement "elds around penetrometers have been made insmall scale model tests by Francescon10 and Gue,11 while a limited set of data from the "eld have



Figure 5. Deviatoric strain paths at r0/R"1 during penetration of a 603 cone for di!erent boundary conditions with

constant viscosity

been presented by Cooke & Price.12 These measurements used in conjunction with the ANSYS8

analyses for Flows A}G may be used to estimate the most appropriate values of k and v1%/5

/; forthe modi"ed SPM approach described.

Details concerning the experimental techniques and displacement resolution in the three testseries considered are given in Table II, while the respective measurements of normalized radialand vertical movements (dz/R and dr/R) are shown in Figure 6. These dz/R and dr/R values areplotted against the normalized radius of each measurement point from the penetrometer prior to



Table II. Experimental measurements of displacements around penetrometers

Test series Experimental details Penetrometer tip Displacement resolution

Francescon Model piles with R"9)5 mm inkaolin; Calibration chamberradius"13R

453 cone cut o!at the apex

0)005R (using an X-ray tech-nique)

Gue Model piles with R"8 mm inkaolin; Semi-cylindrical specimenwith glass front and radius"28R

#at end 0)012R (measured visually)

Cooke andPrice

Field piles with R"84 mm inLondon Clay.

603 cone 0)0002R (using inclinometers)

its installation, i.e. r0/R. Although precise details concerning the "nal z/R location of each

measurement point are not given, it may be assumed that the data presented correspond to a "nalz/R in excess of 10.

Despite the varying penetrometer tip end conditions, chamber sizes and soil types in the threesets of experiments (see Table II), Figure 6 illustrates relatively consistent patterns of displace-ment. It may be seen that vertical movements are large and downwards close to the penetrometerbut are very small and upwards for streamlines at r

0/R'2. In contrast, radial movements remain

relatively large to signi"cant distances from the penetrometer. The values of dr/R recorded atlarge r

0/R values in Francescon's10 experiments may be assumed to be underestimates due to the

in#uence of boundary e!ects in the small chamber used.Values of k"0)004 N s/m2 (four times that of water) and v

1%/5/;"0)97, which were employed

in a further ANSYS analysis of the modi"ed SPM approach (see Flow H in Table I), provided themost satisfactory "t to the experimental data in Figure 6. Displacement predictions for this #owand for Flow A at z/R"20 are overlain on the data in this "gure. It is evident from thecomparison of the two #ows that predictions of radial movements are not very sensitive to thespeci"ed viscosity and relative velocity while the computed vertical movements are very sensitiveto these parameters. The match between all measurements and the output from Flow H is quiteencouraging but should only be viewed as indicative, given the di!erences in the details of the casehistories considered. Further experimental data to test and re"ne the approach are clearlyrequired.

The strain paths predicted for the &best-"t' #ow (i.e. Flow H) and for Flow A (considered toresemble the SPM solution most closely) along the r

0/R"0)5 and r

0/R"1 streamlines for

a penetrometer with a 603 cone are shown on Figure 7. It is apparent that the strain pathspredicted for both #ows at r

0/R"1 are very similar while the magnitudes of the E

1and E

3strains

for r0/R"0)5 are signi"cantly di!erent. These trends are in keeping with the respective displace-

ment plots shown in Figure 6 which indicates that radial displacement predictions of both #owsare alike but vertical displacement predictions close to the penetrometer are markedly di!erent.

7. CONCLUSIONS

Baligh's Strain Path Method (SPM) postulates that strain "elds induced in the ground duringpenetrometer installation may be approximated by the #ow of an inviscid #uid around thepenetrometer. This paper examines the sensitivity of the predicted strain "elds to the introduction



Figure 6. Radial and vertical displacement "elds at z/R"20 during penetration of a 603 cone for &best "t' and &full slip'#ows

of #uid viscosity and di!erent boundary conditions on the penetrometer. It became evident thatincreasing the viscosity extends the penetrometer's range of in#uence in the soil and accentuatesthe in#uence of the boundary conditions. The e!ect of partial slip on the penetrometer boundaryresults in relatively signi"cant strain changes at large distances above the penetrometer tip;partial slip also leads to lower vertical extension strains but higher cavity expansion and shearstrains.

Application of these "ndings to physical displacement measurements showed that the #owassumptions adopted by the SPM can be improved upon and that adoption of a modi"ed form of



Figure 7. Deviatoric strain paths at r0/R"0)5 & 1 during penetration of a 603 cone for &best "t' and &full slip' #ows

the SPM using a standard FE package can allow the derivation of more realistic strain "eldsaround penetrometers.

APPENDIX I

Radial and vertical velocities (v3and v

z) may be written in terms of the stream function (t) as:

vr"!

1

r

LtLz

, vz"

1

r

LtLr

(10)



The radial, vertical, circumferential and shear strain rates are expressed as

err"

Lvr

Lr, eR

zz"

Lvz

Lz, eR hh"

vrr

, eRrz"

1

2 ALv

rLz

#

Lvz

Lr B . (11)

Then from (2) and converting to the local co-ordinate system (note the local and globalco-ordinates can simply be interchanged in the di!erentiation process as dr"dm and dz"dg):

vr"!

1

r(a

2#a

4m#2a

5g#a

6m2#2a

7mg#2a

8m2g) (12)

vz"

1

r(a

1#2a

3m#a

4g#2a

6mg#a

7g2#2a

8mg2) (13)

eRrr"

1

r2Ma

2#a

4(m!r)#2a

5g#a

6m(m!2r)#2a

7g (m!r)#2a

8mg (m!2r)N (14)

eRzz"

1

r(a

4#2a

6m#2a

7g#4a

8mg) (15)

eR hh"!

1

r2(a

2#a

4e#2a

5g#a

6e2#2a

7mg#2a

8m2g) (16)

eRrz"

1

2r2[!a

1#2a

3(r!mQ )!a

4g!2a

5r#2a

6g(r!m)!a

7(2mr#g2)!2a

8m (mr#g2)] (17)

REFERENCES

1. Baligh MM, Scott RF. Quasi-state deep penetration in clays. Journal of Geotechnical Engineering ASCE, 1975;101:1119}1133.

2. Baligh MM. Strain path method. Journal of Geotechical Engineering ASCE, 1985; 111:1108}1136.3. Teh C-I. An analytical investigation of the cone penetration test. Ph.D. ¹hesis, Department of Engineering Science,

University of Oxford, UK, 1987.4. Kiousis PD, Voyiadjis GZ, Tumay MT. A large strain theory and its application in the analysis of the cone

penetration mechanism. International Journal for Numerical and Analytical Methods in Geomechanics 1988; 12:45}60.5. Van Den Berg P, De Borst R, Huetink H. An Eulerian "nite element model for penetration in layered soil.

International Journal for Numerical and Analytical Methods in Geomechanics 1996; 20:865}886.6. Abu-Farsakh MY, Voyiadjis GZ, Tumay MT. Numerical analysis of the nature piezocone penetration tests (PCPT) in

cohesive soils. International Journal for Numerical and Analytical Methods in Geomechanics 1998; 22:791}818.7. Hu Y, Randolph MF. A practical numerical approach for large deformation problems in soil. International Journal for

Numerical and Analytical Methods in Geomechanics 1998; 22:327}350.8. ANSYS. Help Manual for Revision 5.3, SAS IP, Inc., 1997.9. Huebener KH, Thornton EA, Byrom TG. ¹he Finite Element Method for Engineers. Wiley: New York, 1995.

10. Francescon M. Model pile tests in clay. Stresses and displacements due to installation and axial loading. Ph.D. ¹hesis,University of Cambridge, UK. 1983.

11. Gue SS, Ground heave around driven piles. Ph.D. ¹hesis, University of Oxford, UK. 1984.12. Cooke RW, Price G. Strains and displacements around friction piles. Proceedings of the Eigth International

Conference on Soil Mechanics Foundation Engineering, Moscow, vol 2-1, 1973; 53}60.



Extending the strain path method analogy for modelling penetrometer installation

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