This paper presents a closed form solution for undrained cavity expansion and contraction. The shear stress/shear strain response is modelled as non-linear elastic/perfectly plastic with a power law describing the reduction of soil stiffness with strain. The constants of the power law are obtained from pressuremeter unload/reload cycles. The solution can be used to generate a pressuremeter curve for comparison with field data and an iterative procedure with one degree of freedom identifies p. The analysis is applied to a number of self boring pressuremeter tests in Gault clay at a site in Cambridgeshire.
The standard reference for analysing an undrained self boring pressuremeter (SBP) test is Windle & Wroth (1977). This assumes a linear elastic/perfectly plastic shear stress/shear strain response and is adapted from the classic Gibson & Anderson 1961 analysis. The solution is expressed in terms of fundamental soil properties and results can be compared against other types of soil test. For deriving shear strength parameters (in effect using the perfectly plastic part of the solution) it remains useful. Using the complete solution to predict verifiable elastic parameters fails, because outside of a finite element mesh soil does not deform as a linear elastic medium up to the point of failure. This paper publicises a recently developed non-linear elastic/perfectly plastic solution better able to describe the distribution of stress and strain in the pressuremeter test. To apply the solution data are required for the non-linear stiffness characteristics of the soil and these are obtained from pressuremeter unload/reload cycles.
True elasticity in soils is confined to a strain range below the resolution of the SBP and too small to be practicable for design purposes. 'Non-linear elasticity' is used here to describe the ground response prior to the full shear strength of the material being mobilised. Deformations in this range are substantially but not completely recoverable.
It has been possible for some time to obtain reasonable data for the stiffness:strain response of soils from pressuremeter tests. Jardine (1992) gives empirically derived expressions for converting SBP unload/reload data to curves for the degradation of stiffness with strain. Muir Wood (1990) applies the Palmer (1972) analysis using empirical relationships proposed in Jardine et al (1986) to solve Palmer's differential equation for undrained cavity expansion.
Neither procedure is suitable for introducing the variation of stiffness with strain into classic boundary problems such as the pressuremeter test. However Gunn (1992) suggests the reduction of soil stiffness with strain can be approximated satisfactorily by a simple power law. Bolton & Whittle (1999) is a closed form solution for the undrained plane strain expansion of a non-linear elastic/perfectly plastic soil where a power law defines the soil response prior to failure. This paper extends the solution to cover the case of cavity contraction.
Using contraction data is not common practice but ought to be. Pressuremeter loading data are affected by the process of inserting the instrument. Even the 'minimal disturbance' attributed to the SBP is too great for confidence in the initial data. The importance of the SBP test is the high probability that disturbance is within the elastic range of the material and potentially, can be allowed for in the subsequent analysis. A small expansion of the cavity is enough to erase the evidence of this disturbance. Unloading from this new state gives a curve defined by the same parameters as the loading but with less uncertainty about the origin for the contraction event.
The complete solution allows pressuremeter curves to be generated from a coupled set of soil stiffness and strength parameters for direct comparison to the measured data. With the exception of cavity reference pressure p all parameters are defined by the analysis so the procedure gives a sensitive test of p and by inference, the insitu horizontal stress. The results obtained from applying this method to a number of SBP tests in Gault clay are given.
The non-linear elastic/perfectly plastic model
Figure 1 shows the proposed shear stress/shear strain relationship for the non-linear elastic/perfectly plastic soil. It is assumed that prior to failure the shear stress: shear strain response is given by:
where is shear stress, is shear strain, is the exponent of non-linearity and is a modulus, here called the shear stress constant. It will be shown later that  gives an excellent fit to field data.
Failure on first expansion occurs after mobilising the undrained shear strength cu for a shear strain ye. During contraction, failure occurs after a shear stress reduction 2cu at a shear strain yc. Because of non- linearity, yc/ye = 21/. G is secant shear modulus given by the quotient of  and the pertinent shear strain . As indicated, non-linearity has a disproportionate effect on the contraction phase of the test.
Deriving the power law parameters from unload/reload cycles
Figure 2 shows an SBP field curve. The axes are total pressure and radial displacement. If displacements were normalised by the initial radius of the instrument then the axes become total radial stress and hoop strain at the cavity wall.
Unloading and reloading data are plotted separately on log scales in Figure 3. The axes are now change of pressure and change of shear strain from a local origin. For the unloading data it is the point where the direction of straining reverses, for reloading data it is the turnaround point in the cycle.
Both unloading and reloading data plot similar trends but it can be hard to identify with sufficient precision the origin for the unloading event. The start of unloading will be masked by an overshoot, primarily a rate effect, unless preventative steps are taken. Here the operator has closed off the gas supply before taking a loop. The borehole continues to expand but at a slowing rate, the bigger volume resulting in a slight pressure reduction.
Once expansion has stopped the pressure is vented in regular steps to force the direction of straining to reverse and in this test there are clear indications of the start of unloading. It is always easier to identify the commencement of reloading and such data are preferred.
For all practicable purposes the two loops give the same results. The quoted correlation coefficients confirm that the elastic characteristics of the soil can be obtained from a rebound cycle anywhere along the loading path and a power law is adequate for describing this response. The exponential form of the results in Figure 3 is:
p = 
where p and are total pressure and shear strain at the cavity wall. is the elastic exponent and the intercept.
For undrained plane strain loading, Palmer (1972) shows the shear stress t at any point on the pressuremeter curve is given by:
Substituting the result of equation  for p and differentiating gives:
(-1) = 
Comparing  and  indicates . From  and  it can be inferred that:
The quotient of  and shear strain gives secant shear modulus Gs:
Gs = -1 = -1 
As an aside, tangential modulus Gt is obtained as follows (Muir Wood 1990):
Gt = Gs + [dGs]
the solution from the power law being:
Gt = -1 = -1 
The solution for the non-linear elastic/perfectly plastic undrained cavity expansion
Equation  applies until the mobilised shear stress reaches the undrained shear strength cu. At the cavity wall, the criteria for failure is a total pressure pf where:
pf - po = cu/
po is cavity reference pressure, ideally the horizontal lateral stress ho. Below failure total pressure and shear strain are defined by the power law, so:
p - po =  
Combining  and  gives the shear strain at failure ye:
ye [Cu]1/ 
Were the material response to be linear elastic then would become 1 and would be the unique value of secant shear modulus G. In clays is nearer 0.5, suggesting:
pf po + 2cu.
Perfect plasticity and constant volume deformation mean the complete solution is merely the integration from the cavity wall to the elastic/plastic boundary of the failure condition. The mathematical steps necessary to do this are given by Gibson & Anderson (1961) for a linear elastic/perfectly plastic loading, the solution being:
p=po+cu[1+ln(e/ye)] for e ye
where e is the current shear strain at the borehole wall during expansion. At first failure e ye, and  reduces to the linear elastic failure stress condition:
Substituting the non-linear elastic failure condition of  into  gives the non-linear elastic/perfectly plastic solution:
p = po+cu [(1/+ln(e/ye)] 
Equation  is the result given by Bolton & Whittle  albeit by a slightly different argument. For indefinite expansion e=1 and p then becomes limit pressure pL.
Extending the non linear elastic/perfectly plastic solution to the contraction phase
Jefferies (1988) gives the solution for the stresses and strains around an expanded cavity unloading in undrained conditions. A simple elastic/ perfectly plastic response is assumed and the solution can be written:
p = pmax - 2cu [1ln (c/yc)] for c yc 
where c is the current shear strain at the borehole wall during contraction. The similarity to the loading expression is apparent. po is replaced by the new stress origin pmax, the maximum radial stress at the end of the expansion phase. The signs change and the deviator stress is now 2cu. At first failure in contraction the log term disappears leaving the linear elastic contraction failure definition:
p = pmax - 2cu 
Due to the non-linearity of the soil, failure occurs when the radial stress at the cavity wall is:
p = pmax - 2cu/
Substituting this result into  gives the non-linear elastic/perfectly plastic solution for undrained contraction:
p = pmax _ 2cu [(1/ln(c/yc)]
While the soil is contracting elastically, pressures and strains are connected by:
p - pmax = [
being a restatement of equation  with a new stress origin. The shear strain at failure in contraction yc is given by:
yc = [2cu ]1/ = ye (21/)
A note on shear strain
For clarity shear strain has been quoted in a schematic form in the preceding equations. The shear strain used throughout is current cavity shear strain, usually written V/V meaning constant volume ratio. For devices measuring radial displacement where plane strain expansion is assumed V/V is given by:
e = 1 _ (ro )2
where ro is the initial radius and rc the current radius of the cavity. For contraction, V/V (after Jefferies, 1988) is obtained from:
c = (rmax) _ (rc ) 
where rmax is the radius of the cavity at the end of the expansion phase.
Shear strength: Because the analysis assumes perfect plasticity, deriving shear strength is identical to the widely used procedure of plotting total pressure against log shear strain (Figure 4). An inspection of  indicates that a plot of total pressure against log shear strain has a slope cu and a pressure intercept L. The contraction solution of  can be presented in a similar way to give a plot whose slope is _2cu. A difference between expansion and contraction values probably indicates insertion disturbance. This point is developed below.
Shear modulus: Equations  and  allow secant and tangential shear modulus to be quoted for any elastic shear strain. In practice there is a small strain threshold below the resolution of the equipment where G is a maximum value and is constant. Deciding this threshold has to be an arbitrary choice - it is probably not sensible to use the method to provide shear modulus parameters for shear strains smaller than 10-4.
Cavity reference pressure: The cavity reference pressure po is undefined by the analysis and is chosen initially from inspection of the first part of the loading curve for a point of inflexion. For example the inset in Figure 2 suggests po is about 350kPa. Deciding whether this is a good choice is done subsequently by comparing the field curve to a curve formed from the derived parameters. What follows is a description of a method currently being used on a spreadsheet.
1. Obtain an estimate of po by inspection.
2. Convert the measured displacements to shear strain using the at rest radius of the probe as the strain origin ro in .
3. Produce a semi-log plot of the loading and so derive the shear strength.
4. Also obtain an estimate of the shear strength from the contraction curve.
5. Adjust the strain origin to ensure loading and contraction shear strengths agree.
6. Plot the reloading part of the unload/reload cycles on log scales using the turn-around point in the loop as an origin. Hence obtain the non-linearity exponent and the shear stress constant .
7. Derive the yielding shear strain values ye and yc.
8. Obtain pmax and rmax from inspection of the pressure versus displacement plot.
9. Using equations , ,  and  draw the pressuremeter curve.
10. Adjust po for the best fit.
The results for the test shown in Figure 2 are given in Figure 5. The measured data have been stripped of unload/reload cycles and displacements are now written as cavity strain. Three calculated data curves are plotted, showing the best fit and the effect of changing the estimates of po by +/-10%.
The consequences of insertion disturbance
Cambridge Insitu has the use of a field where occasional demonstration and experimental SBP tests are carried out. All tests are logged and the remainder of this paper presents some of these results, collected over about two years. All tests have been interpreted using the non-linear elastic model. Several instruments were used, an assortment of operators were involved (including trainees) and a number of boring methods tried. Normally papers of this kind stress the care taken to ensure the best quality data; because of commercial imperatives it is not true of these tests.
The soil is Gault clay overlain by glacial till and the site is described in Dalton & Hawkins (1982). The water table is about 1m below the surface. Tests were taken in a cluster of boreholes spaced at about 5m intervals. Each borehole was cased down to 3m with 150mm diameter water well casing hammered into the ground by a percussion rig. The casing provided an excellent anchorage for the driving part of the self boring system.
Tests can be divided into two classes - those where the expansion and contraction shear strengths are the same and those where they differ. It is a requirement of the methodology described here that they are the same. The question is, what change will make the shear strengths agree?
As described above, the origin for the contraction event is fixed, so no change can be made to the estimate of shear strength derived from that part of the field curve. However the origin for the loading depends on the insertion disturbance - it can be greater or less than the initial diameter of the SBP depending on whether the material has been over or under drilled. Small adjustments to the strain origin can be used to make the loading and unloading shear strengths agree. 'Small' means within the elastic range of the material. If an elastic adjustment is sufficient then the assumption of drilling disturbance seems the most likely explanation for the difference. A loading cu less than the contraction cu means the probe was underdrilled (the usual case) and the strain origin must be made more negative. The converse is true. Moving the origin rotates the plot obtained from applying  until a match for the contraction value is obtained.
Figure 6 gives the limit pressure and shear strength parameters derived from the tests. With the exception of the second test in borehole 7 there is a reasonable trend of increasing strength with depth. This test is discussed below.
Values for cavity reference pressure po and coefficient of earth pressure at rest K0 are given in Figure 7. The saturated unit weight of the clay is assumed to be 20kN/m3.
Figure 8 shows shear modulus and stiffness ratio for these tests. For a non-linear elastic model the strain range pertinent to these parameters must be quoted. Gye is secant shear modulus at yield strain ye. It is too conservative a value for most design purposes but  allows secant shear modulus parameters to be derived for any elastic strain. Results for shear strains of 10-2, 10-3 and 10-4 are given in Figure 9.
Stiffness ratio is Gye/po. Shear modulus increases slightly with depth but the stiffness ratio is almost constant below 8m.
Comments on the field tests
The first two tests in borehole 7 are noticeably different. The boring for these tests was done by 'jetting' the SBP into place. In this method material is cut by high pressure water. Successful results have been reported with this method in soft clay (Benoit et al 1990) but the increased disturbance would seem to invalidate its use in stiff clays.
The majority of the K0 values lie between 1 and 2. Previous work with the SBP in Gault clay, even at this site (Dalton & Hawkins 1982) has implied K0 values much higher from SBP tests at these depths - between 3 and 4. The results given here seem more plausible and it is encouraging that in both boreholes details of the trend against depth are reproduced. One problem with SBP test analysis in the past has been inappropriate allowance for insertion disturbance - of the tests presented here all but two required a small adjustment for underdrilling.
Underdrilling generates excess pore water pressures and raises the total stress around the cavity. Membrane lift-off will now occur at a higher stress than ho. Waiting for these pore pressures to dissipate before starting to expand the cavity is futile. The material consolidates and there is a net increase in the radial stress acting on the probe surface. Logically, it is better to start putting pressure into the SBP immediately the boring stops, to ensure undrained expansion.
A method of interpreting an undrained self boring pressuremeter test has been presented and tested against a number of field tests in Gault clay. Initially the SBP test is analysed as a number of separate events to provide a set of parameters. These are then used as modelling parameters to produce a calculated curve to match the field data. By adjusting the cavity reference pressure very close matching can be achieved, and this final choice of reference pressure gives reasonable values for the insitu lateral stress. The methodology described can accommodate small amounts of insertion disturbance.
The analysis is constrained by the stiffness versus strain response of unload/reload cycles and defines this response in terms of two parameters and .
Values of quoted show that treating a non-linear elastic response with linear elastic relationships will give seriously misleading results. Only consequences for shear modulus have been mentioned here but in general a linear elastic solution will always underestimate the elastic contribution.
The method has also been used to analyse a large number of SBP tests in London clay, with convincing results. However the method does not work as well in normally consolidated clays - a complete explanation is not yet available but the problem is in the final unloading where the mobilised shear stress does not always settle at a constant value. This may be an artefact due to rotation of principal axes under low ko conditions.
There is no reason why the step-by-step approach could not be eliminated in favour of a purely curve matching process. This is risky, with a temptation to accept approximate fits as adequate. The stress: strain response used for the model is still a highly simplified description of how the ground behaves - one of the more useful aspects of the work reported here is the occasional discovery of materials and stress conditions for which the model is not good enough.
Fieldwork for these tests was carried out by members of Cambridge Insitu at the Rectory Farm site owned by Clive Dalton. Equation  was the solution to an Engineering Tripos question posed by Dr Malcolm Bolton of Cambridge University in 1991.
Use of symbols
Shear stress constant
Exponent of elasticity, where 1 is linear elastic
Intercept on log log plot of p versus e - equal to /
ho Insitu horizontal stress
Shear strain at cavity wall, suffix e and c denoting expansion or contraction
ye Shear strain at failure for initial loading
yc Shear strain at failure for unloading
cu Undrained shear strength
p Total pressure applied to cavity wall
pf Total pressure applied to expanding cavity wall at failure
po Cavity reference pressure, being the insitu horizontal stress for the ideal test.
pL Limit pressure at which indefinite expansion of the cavity occurs
pmax Total pressure applied to cavity wall at the end of the expansion phase
G Shear modulus, suffix s or t denoting secant or tangential.
Gye Shear modulus at shear strain ye
Gyc Shear modulus at shear strain yc
V/V Constant volume ratio, being the current shear strain
r Radius of cavity wall, with suffix o, c, or max denoting initial, current or maximum.
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