PART 2                  ANALYSES FOR INSITU LATERAL STRESS

2.1 Overview

The expansion pressuremeter test is a sequence of measured co-ordinates of pressure and displacement of the cavity wall (once suitable corrections have been made to compensate for the response of the elastic membrane).

In order to solve the boundary problem, an origin for the expansion has to be determined. For insertion methods that imply stress relief, the origin is taken to be the point where insitu conditions are restored to the cavity. This means that an estimate of the insitu lateral stress has to be made, and the measured radius of the cavity at the point where the insitu lateral stress is restored is used to convert subsequent displacements to strain.

For an SBPM it is possible to recognise the insitu lateral stress by inspection, the so-called lift-off method. It is also possible to recognise by inspection the shear stress limit (the point marked p_f in figure 1) as this is indicated by the onset of a markedly non-linear response. An iterative procedure first suggested by Marsland & Randolph (1977) allows the insitu lateral stress to be inferred.

Both methods are outlined by Mair & Wood (1987). Note that these methods amount to obtaining a value for the cavity reference pressure, p_o. It is impossible to measure the insitu lateral stress s _ho because the act of placing instrumentation must result in some disturbance, no matter how small. The methods above are indirect indicators for determining s_ho. It is open to question whether the reference stress is equivalent to the insitu lateral stress, and it is usual to bring a range of evidence to bear in order to decide if a particular value for p_o is also a plausible value for s_ho . External evidence might take the form of using the derived reference stress within a K_O calculation, or checking that the derived vertical/horizontal anisotropy can be supported by the material shear strength i.e.

s_ho - s_vo < 2c_u                                                            . _...[Equ.1]

The software is so arranged that the analyst must make an explicit choice of s_ho the result being the 'best estimate' that is quoted on the results summary sheet. Having decided on the best estimate the displacement offset interpolated from this stress is treated as the origin for the expansion.

For an ideal LCPM test the external stress from the ground is balanced by an internal pressure so keeping the load cells in a zero stress condition. Several hours and sometimes days are allowed to elapse until a state of equilibrium around the probe is achieved. Reading the internal pressure gives the external stress directly.

In practice equilibrium around the probe is reached with the load cells in a partially stressed condition. The derived external stress is therefore a combination of the internal stress and load cell transducer outputs.

2.2 Lift-off
This method is applicable only to the SBPM. In principle it is a straightforward procedure. The instrument is assumed to be bored into the ground with insignificant disturbance caused to the surrounding material. If the insitu conditions around the instrument remain unchanged by the insertion process then the pressure at which the membrane first moves and the cavity begins to expand is p_o. The corresponding cavity diameter will be the same as the at rest diameter of the instrument. Because the initial part of a SBP test is very stiff the choice is made from an enlarged view of the first 0.2mm (0.5% strain) of the expansion.

Difficulties arise because the instrument has a finite stiffness and hence there is instrument compliance to be separated from the expansion of the cavity. In addition the instrument is being externally loaded by the lateral stress when the test is started. This external stress tends to deflect the arms of the instrument and reveals any imperfections in the seating of the arms. The imperfections, in effect small movements, are revealed when the pressure differential across the membrane is equalised, i.e. exactly at the point where the cavity reference pressure is reached.

In a simplistic approach these arm 'signatures' could be considered as positive indications of the reference pressure. However in the ground it is not possible to have displacements without an associated change in stress, which add to or subtract from the reference pressure.

As a result of finite instrument stiffness and small movements from the displacement sensors applying the lift-off analysis means that there is much uncertainty attached to identifying a plausible reference pressure.

Conventional practice for coping with this uncertainty is to relax the definition of 'lift-off' to mean something more like 'significant movement'.

Fig.3  An example of lift-off

If the strict definition of 'lift-off' could be applied then no assumptions concerning soil response are required. Accepting that some movement takes place prior to ‘lift-off’ implies that assumptions be made about the mode of deformation. In the less rigorous application of 'lift-off' it is important that the analyst identifies the onset of plastic behaviour as a guide to deciding that some conspicuous change of form in the loading curve at a lesser stress is likely to be p_o. Our plots would still refer to such a break point as 'lift-off' but clearly it is something else, p_o by inspection perhaps.

Figure 4 is a good illustration of the problems involved with identifying lift-off. Here the individual arms from EP1 Test 6 are plotted together, Figure 3 being the average output of these arms.

Marked on the plot are the lift-off points corresponding to a rigorous interpretation of what is implied by the term. However for each arm there are alternative points of inflexion – many analysts would pick about 780kPa for arm 1 for example. Interestingly the combined data given in figure 3 has cancelled out some of the compliance – this is not always the case, the usual tendency being for the average lift-off to be similar to that obtained from the first arm to move.

Fig.4  Lift-off and separate arms

Although the ‘zoomed’ data shown in Figure 4 looks horrifying it is important to bear in mind the scale. All the lift-off information is concentrated into the first 50 microns of the expansion or about 0.12% cavity strain. In this test the elastic strain range is about 0.5% cavity strain. Because the movements are well within the elastic range of the material the analyst is justified in attributing significance to the output of the separate arms. In this case the arithmetic mean of the separate lift-off points is often a more useful parameter than lift-off derived from averaged arm displacement data.

The correct use of the Strain Control Unit can sometimes assist in this process of identifying significant movement. Initially the unit is set to input pressure into the probe at a rate that is too low to allow expansion at a constant rate of strain. Hence this part of the expansion is by default stress controlled and the initial elastic response from the soil is indicated by equally spaced data points along the displacement and pressure axes. As the soil response becomes plastic the SCU is gradually able to initiate constant strain rate expansion. There is a transition period where the pressure rate remains regular but each increment of pressure has an increasing effect on the displacement until eventually the displacement rate becomes constant. Thereafter the rate of pressure advance steadily reduces as the plastic zone around the pressuremeter increases. This changeover phase shows up clearly in a plot of total pressure versus time, and can be helpful for identifying yield.

The six arm SBP allows a fuller appreciation of the initial stress state around the instrument. The early part of the test, when the circumferential stress still exceeds the radial stress, is the only part of the test where the behaviour of the separate arms is more informative than the averaged output. The six arms allow average, pairs and individual arms to be inspected, and it is clear that in over-consolidated material the observed variations in behaviour of the separate arms invalidate the assumption of minimal disturbance. It is usually the case that when one arm shows a very high lift-off stress, its opposite partner shows a very low level. This indicates insertion disturbance, due either to the drilling process itself or to imperfect vertical alignment of the probe. Axes that read similar on both sides are accorded greater significance in the analysis than axes that read very different.

2.3 Marsland & Randolph (1977) Analysis
Marsland & Randolph analysis relies on being able to identify the onset of plastic behaviour, the yield stress p_f. The argument is as follows:

• that in the vicinity of the insitu lateral stress the soil behaves elastically and therefore the pressure/strain plot will be linear
• that this elastic behaviour will cease when the undrained shear strength of the soil is reached in the wall of the cavity, and hence the pressure /strain plot will begin to curve (see Figure 1).

This can be expressed as:        p_f = p_o + c_u                                                                               ...[Equ.2]

From this it follows that p_o can be deduced by a process of iteration. Initially a guess is made of a value for p_o; using this guess to define a temporary strain origin a total pressure:log volumetric strain plot is then generated in order to derive a value for c_u. The sum of these two parameters is compared with the selected value of p_f. The choice of p_o is then adjusted and the process repeated until a match is found. It is a straightforward matter to carry out this procedure on the computer.

The modified method in current use is a response to the difficulty that the Gibson & Anderson model is too simple for use in real soils and yield may occur at a shear stress that is different from the large strain shear strength. Hawkins et al (1990) suggested that the most appropriate choice was that value of shear stress pertaining at the apparent onset of plasticity, so equation [2] now becomes:

p_f = p_o + t_f                                                                                                ... [Equ.3]

t_f  can be obtained from a total pressure:log volumetric strain plot by selecting the slope at the pressure and strain corresponding to the choice of p_f .  In practice, this is using the Palmer (1972) argument to identify the mobilised shear stress at failure.

The analysis is implemented graphically, using a number of rulers to identify significant points on the curve.

Fig.5  An example of the Marsland & Randolph analysis

There are a number of problems, both theoretical and practical:-

• There can a choice of slopes for G_i, giving alternatives for p_f . In practice the first slope encountered is usually too stiff to make a credible choice for G_i and is probably an indication of insertion disturbance.
  
• The assumption of simple elastic response - in practice soils exhibit marked non-linear elastic characteristics. One important consequence of non-linearity is that changes in radial stress applied by the pressuremeter to the borehole wall no longer have a 1:1 correspondence with changes in the mobilised shear stress. This point is developed later.

• The original analysis was developed as an aid to the interpretation of pre-bored pressuremeter tests where the process of forming the pocket results in the complete unloading of the cavity prior to the test commencing. It is certain therefore that the soil has seen stress relief. It is arguable whether in these circumstances that the yield point remains unchanged, as more than elastic unloading has taken place. However the form of such tests does tend to give an unambiguous choice for the onset of plasticity.

• In a self bored pressuremeter test the situation is not so clear cut. The very factors that make the test desirable also results in more realistic behaviour being seen in the form of the early part of the test, with non-linear elasticity being a feature. Hence a choice of p_f is by no means easy. Often the better the test the harder such a choice becomes. However it is probable that in a good test the lift off pressure would be a credible choice so that in the wider context it is not a serious problem.

A disturbed SBP test does not necessarily imply stress relief. Typically disturbance arises out of damage to the shoe cutting edge; if the shoe is enlarged then stress relief will result. However if the shoe is damaged in such a way that it cuts undersize then stress increase will take place and plasticity will be masked by a rise in the pore water pressures around the instrument. In this event the analysis can contribute nothing – forcing such data to fit the assumptions of the analysis will result an over-estimate of the insitu lateral stress.

2.4 Earth Pressure – The Load Cell Pressuremeter

2.4.1 Introduction
The procedures now described are specific to the LCPM. The probe is essentially a smooth brass & steel cylinder of similar proportion to the SBPM. Spaced equally around the circumference are six compartments sealed with a flush mounted cap. Each compartment contains a load cell fixed to the cap, an internal pressure transducer and a pore water pressure transducer. The load cell and hence the cap can move in and out a tiny amount, a maximum of 14 microns in each direction. 14 microns deflection is caused by a differential stress across the cell of 2MPa. Each compartment is separately linked to a gas source on the surface.

The LCPM is self bored into the ground and is left in place for many hours while the consequences of any drilling disturbance are allowed to minimise. The ideal mode of operation is an active system whereby throughout the drilling and settling, deflections of a load cell trigger the gas control system so maintaining the load cells at a null stress position. The gas pressure required to do this is measured by the internal pressure transducer giving a direct reading of the external stress acting on the cell cap.

In practice the system reaches equilibrium with the load cell in a mildly stressed state. The earth pressure at which this occurs is obtained by combining the output of the internal pressure transducer with that of the load cell.

Provided the external stress does not exceed 2MPa it is possible to run the system in an entirely ‘passive’ mode whereby gas control is not used. The ground bears on the load cell and the output of the cell alone gives the earth pressure. The flaw in this arrangement is the movement of the cell and the consequent stress relief caused to the surrounding material. This may be small and in principle calculable using estimates of the soil stiffness.

The probe can be switched from the passive to active condition and visa versa at will. Unfortunately the rate at which the controlling system makes adjustments to the internal gas pressure seems to be too rapid. The consequent movements of the load cell cap cause large changes in pore water pressure suggesting the cells are carrying out a one dimensional compression test on the soil. Analysing such data is too complex for the present report although we have some suggestions of how to approach the problem.

2.4.2 Inspecting the data
Figure 6 is an example of the output of the load cell pressuremeter. The data from one of the six measuring positions are displayed, and the outputs of all three transducers in the cell are shown.

Fig.6  Typical output of LCPM, cell array 4

Selecting a value for the external stress p_o from the long passive phase is straightforward. The value after 20 hours read directly from the load cell. By combining the output of the load cell with the pore water pressure transducer it is possible to quote a value for the effective external stress, p'_o.

The active value is obtained by combining the internal pressure cell and load cell. In principle the load cell should indicate zero stress but as seen in figure 6 the cell reaches equilibrium slightly depressed into the body of the probe and so indicates a small positive stress. This is added to the internal pressure cell, giving the black trace visible in figure 6. It is also apparent that the settling period for the active condition is not long enough, so this value will be higher than the ultimate state.

For tests run entirely in the active state the view of the test is slightly different. Figure 7 is an example from a test where lowering down the borehole, drilling in and waiting were all carried out with the probe in an ‘active’ condition. The changes of stress during drilling are too rapid for the reaction time of the system but once drilling has stopped control is quickly established. As a consequence the output of the load cell falls close to zero and the output of the internal pressure transducer gives most of the external stress p_o. In this example the pore water pressure settles quickly to a value close to the expected ambient PWP. Although the data are plotted for 20 hours it is apparent that the external stress is still falling gently at almost a constant slope.

Fig.7   Typical active output of LCPM, cell array 5

The instrumentation is complex and from time to time problems arise. Figure 8 shows an example. For the drilling and settling the test is run in passive mode. The control system flattens the battery after 17 hours and as a consequence the system starts to oscillate. This is not the only difficulty.

Fig.8   An LCPM test with problems, cell array 4

Load cell 4 has started to drift and is spoiling the derived value of external stress. Note that if the system had been in active mode this would have caused the control system to put gas into the probe to correct for the apparent movement. Because the movement is not real the consequence would have been to push the cell cap out into the soil and to raise the local pore water pressures. This would eventually have affected all the other 5 cell arrays. It is an implicit requirement of the active system that all transducers be in fully working order.

2.4.3 Anisotropic insitu lateral stress
For all LCPM tests there is a spread of values for the external earth pressure from the six cell arrays. Where structures such as tunnels have been driven through the material an anisotropic lateral stress distribution is to be expected. One reason for using a six cell array LCPM is the potential this gives for resolving the stress distribution, the magnitude of the major and minor stress and their orientation. The stress circle can be resolved given only three stress parameters at 120 degree spacing, so the output of the odd numbered cell arrays and even numbered cells arrays are separately analysed together with the average of opposing cells. This offers a means of assessing the data for consistency.

Fig.9  An example of lateral stress anisotropy

Figure 10 gives the geometry of the problem. b and a are the major and minor stresses respectively, r₁, r₂ and r₃ are three stress readings at 120°.

r₁ = ½(b+a) + ½(b-a)cos2q                           ...[Equ.4]
r₂ = ½(b+a) + ½(b-a)cos2(q+2p /3) ...[Equ.5]
r₃ = ½(b+a) + ½(b-a)cos2(q+4p /3) ...[Equ.6]

These relationships can be used to show the minor stress a is given by:

a = ¹/₃(r₁+r₂+r₃) - ²/₃(r₁²+r₂²+r₃²-r₁r₂-r₁r₃-r₂r₃)^½                                                               ...[Equ.7]

and the major stress b by:

b = ¹/₃(r₁+r₂+r₃) + ²/₃(r₁²+r₂²+r₃²-r₁r₂-r₁r₃-r₂r₃)^½                                                ...[Equ.8]

The angle q can be obtained from:

sin2q = 2(r₂-r₃)/Ö 3(b-a)                                                                    ...[Equ.9]

and also from:

cos2q = 2(2r₁-r₂-r₃)/3(b-a)                                                               ...[Equ.10]

Fig.10  Mohr's circle for stress

The angle q is quoted as a clockwise rotation from cell array 1, so knowing the orientation of cell array 1 is important. The difference between the major and minor stress has to be supported by the shear strength of the soil, giving a means of identifying implausible values.

There are many other reasons why the soil may appear to have anisotropic properties:

• The probe may not be inserted perfectly vertical.
• The instrument may have a defect, such as being bent.
• The self boring process may have caused irrecoverable disturbance to the ground.

Anisotropy is apparent in most tests carried out with self boring probes suggesting that instrument factors are a major influence. The results of the anisotropy analysis should be considered with a degree of scepticism. For this reason it is probably not worth trying to ‘improve’ the data. However it is possible to run the analysis twice – the first run attempts to measure the contribution due to ‘disturbance’, and uses the results to adjust the measured stress readings by a constant ‘error’ stress, ½ (b-a). The anisotropy analysis is then applied to the new set of data. For this procedure to be successful it is necessary to assume all disturbance is elastic – this assumption is not safe.

2.4.4  Alternative procedures for predicting the ultimate stress state
It is apparent that even in the longest of LCPM tests on this contract the state of stress reached at the end of the active phase is not the ultimate value. In principle there seems no reason why the ultimate stress state could not be predicting from the rate of change of the measured data. In order to make such a scheme work a good system for calculating the rate of settlement is required.

It seems reasonable to assume that the problem with the ‘active’ state is the excess pore water pressure generated by carrying out a one dimensional compression test. Eventually these pore water pressures will settle to the ambient pore water pressure, a known value. These are sufficient data for calculating the half life of the pore water pressures, assuming an exponential decay. By inference the half life of the ‘active’ pressure decay will be the same, and hence it is possible to derive the unknown value it is settling towards.

Fig.11  LCPM Test with short PPP decay time

Unfortunately as figure 11 indicates the soil response to movements of the cell cap is more complex than this simple description suggests. In this figure the transition from the passive to active phase after 20 hours has generated a large excess pore water pressure. This decays very rapidly, but the total external stress changes much more slowly. Although it is observed for 24 hours it clearly has a long way to go to reach the passive state and probably this condition would not be achieved. The small consolidation test carried out by default may have permanently raised the effective stress state acting on the probe. Because the rate of change of total external stress is not directly dependent on the pore water pressure decay the one cannot be used to predict the other.

An explanation for the behaviour in figure 11 is not yet available. In the active state the cell cap cannot move more than the tiny amount required to trigger the control system. Yet despite negligible displacement and no change in pore water pressure the stress acting on the cell is falling. This implies a change in stiffness, perhaps a consequence of changes in Poisson’s ratio. Note the process is reversible, in that switching from active to passive at about 44 hours causes a sharp fall in pore water pressure and total external stress. The pore water pressure soon recovers and there are indications that the ‘passive’ total external stress value is climbing towards the previously recorded level prior to the change to the ‘active’ state.

2.5   Deriving insitu lateral by synthesis
The doubt concerning the appropriateness of using the measured values for cavity reference pressure p_o as best estimates for the insitu lateral stress s_ho mean that other methods for inferring plausible values are required. Jefferies (1988) is a procedure for deriving insitu lateral stress, stiffness and strength from undrained pressuremeter curves by matching the measured data points with an iteratively selected set of numbers. Some rigour is introduced into the procedure by making the single set of parameters match the contraction as well as the expansion phases of the SBPM test.

For the procedure to work the model used to represent the deformation characteristics of the soil has to be realistic. Jefferies (1988) assumes a simple elastic/perfectly plastic shear stress:shear strain response. Outside of a computer there is no such soil and despite the claims made for it, the procedure fails – in particular it cannot predict the measured field values for stiffness, the one property of the soil pressuremeters can provide without major difficulty.

However the procedure can be used with more realistic soil models, and the SBPM tests on this contract have been back analysed using a non-linear elastic/perfectly plastic shear stress:shear strain characteristic. This uses measured values of stiffness and shear strength so the only variable to be decided is the insitu lateral stress. Both expansion and contraction phases of the test are fitted. Details of the model and the solution are given in Part 5.

In general the values for lateral stress derived using this procedure tend to be lower than those obtained by inspection, and are consistent with a view of the test as slightly under drilled, raising the state of stress around the probe. Note that it is only possible to derive one value for insitu lateral stress using these procedures, as isotropy of soil properties is a fundamental assumption.

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