PART 4 SHEAR MODULUS
Terms:
GP Pressuremeter shear modulus
Gs Secant shear modulus
Gt Tangential shear modulus
G100 Secant shear modulus at the maximum elastic shear strain
GHH, GVH Shear moduli for transversely isotropic material
EH, EV Young’s modulus in
the horizontal and vertical direction
n HH,
n HV Poisson’s ratios for
transversely isotropic material
n Ratio of horizontal to vertical Young’s modulus
EH/EV
Ko Ratio of horizontal to vertical effective insitu stress
t Shear stress
pc Pressure measured at the cavity wall
ec Circumferential strain
measured at the borehole wall
g Shear strain
gc Shear strain measured
at the borehole wall
gs Invariant shear
strain
h Radial stress intercept
b Elastic exponent
a Shear stress intercept
4.1 Background
Values of stiffness in real soils however measured are strain level and stress
level dependent. Pressuremeter stiffness is affected by the additional factor
of cross anisotropy. The pressuremeter used conventionally gives shear modulus
parameters of type GHH, where the first suffix shows the direction
of loading and the second suffix the direction of particle movement. Most
design calculations that require a value for shear modulus mean in practice
the independent shear modulus GVH. Translating between pressuremeter
values and alternative expressions for modulus is complex but worth pursuing
because of the high quality of the pressuremeter measure. What follows is
a brief outline of a possible approach.
Cycles of unloading and reloading loops are the primary source for shear modulus data. If the material was linear elastic then a simple construction can be used to derive the shear modulus. Figure 16 shows a typical example of one such cycle. A cursor has been placed by eye to bisect the loop. The slope of the cursor is approximately twice the shear modulus. This value is quoted in the top left hand corner of the plot together with an indication of the size of the loop expressed as the change of pressure and strain, and the co-ordinate of the centre of the loop. The equation used is:
G = [1+ec][D pc/2D ec] ... [Equ.33]
Implicit in this equation is the assumption that D pc is equivalent to D tc, that is to say the material has linear elastic characteristics.
In addition, the program carries out a regression analysis of the data points that are part of the reload loop. If the loop is good, that is symmetrical and without indications of scatter, then the two values of modulus obtained will be the same. However the regression analysis is sensitive to misplaced data points, which the visual technique can ignore. The value obtained by regression is quoted in the bottom right hand corner of the plot.
Fig.16 Expanded view of an Unload/Reload cycle
It is important that the effects of creep (for whatever cause) be minimised before starting the cycle, and in Figure 16 ‘deleted’ points before the start of the unloading show where the pressure in the probe was held for a short period of time.
4.2 Non-linear stiffness/strain response
In recent times it has become widely acknowledged that the stiffness/strain
relationship is not linear. The unload/reload cycle can be made to give a
comprehensive description of this non-linear relationship by looking at smaller
steps of pressure/strain other than the points at the extreme ends of the
cycle.
For reasons explained in Whittle et al (1992) it is preferable to examine one half of the rebound cycle only, that which follows the reversal of stress in a loop. The lowest recorded value of stress and strain then becomes the origin for subsequent data points until the original loading path is rejoined.
In Figure 17, once a new origin is defined then every data point on the reloading part of the loop (A, B, C etc.) can be used to give a value for shear modulus. This value can then be plotted against the associated strain increment as measured from the new strain origin.
Fig.17
The procedure for deciding the origin is not ideal - even better results for very small strains could be obtained if the origin were decided by inspection. The procedure suggested here is readily implemented on a spreadsheet, however, and means that any person handling the data will obtain identical results.
It follows that it is not necessary to take loops of small strain amplitude in order to obtain small strain stiffness parameters. Indeed it is better to make the cycles as large as possible in order to obtain parameters for as wide a strain range as possible.
It is often stated as a caution that unload/reload loops should have a pressure amplitude no greater than twice the mobilised shear stress (Figure 1 shows why). Strictly speaking this is true, if one wants to use the whole loop to derive a single modulus parameter as in figure 16.
However the response is still elastic immediately following the turnover point in the loop, so the data is by no means useless if an incremental approach is used. The real penalty for a loop that exceeds the elastic range of the material is a permanent and irrecoverable shift in the strain origin; the loading curve following such a loop is not a continuation of the loading path prior to the loop.
Provided the loops were taken at the same effective stress then the data from all will plot the same trend. Conversely, if the loops plot one above the other then this indicates different effective stress conditions which in a clay test would prove that the expansion was not undrained.
Using the local origin for each cycle the reloading data can also be plotted on log axes of Dpc versus Dgc. Figure 18 is an example. Two reloading events are plotted. The gradient of the best fit straight line to the data points is used within the Bolton & Whittle analysis as the non-linear elastic exponent b . The correspondence to a straight line is excellent.
Fig. 18 The non-linear elastic response
The linear relationship between pressure and shear strain on log scales expands
to a power law with the general form
pc = h
g b
where p is the change in pressure measured at the borehole
wall, g is the corresponding shear
strain and h and
b are the intercept and gradient of the
log log relationship.
4.3 From pressuremeter modulus to secant and tangential
modulus
As shown in figure 17, the variation of stiffness with strain seen in a
pressuremeter rebound cycle and in other soil tests can be expressed as a
power law (Bolton & Whittle, 1999). Specifically, while the soil is
responding elastically, pressure measured at the borehole wall is given by
pc = h gb ... [Equ.33]
At first sight it would seem that the power law expression for secant shear modulus will be Gp = h g b -1 but this is not so. The Palmer result given by equation [31] still applies, therefore substituting for pc using the right hand side of equation [33] allows the differential equation to be solved giving
tc = h bgb ... [Equ.34]
h b is equivalent to a , giving equation [11], the initial assumption of the Bolton & Whittle analysis.
Shear modulus Gs is given by
tc/gc ... [Equ.35]
so the expression for secant shear modulus is given by
Gs = a gcb-1 ... [Equ.36]
This gives a means of determining the secant shear modulus at any elastic shear strain, although an arbitrary cut-off strain must be assumed below which the modulus will be constant and a maximum – this strain is below the resolution of the SBPM.
When comparing triaxial results with pressuremeter results, triaxial invariant shear strain gT is given by:
gT = (2/Ö3) gc ... [Equ.37]
Tangential shear modulus Gt is given by Gt = Gs + ec[dGs/dec] ... [Equ.38]
Hence from the power law
Gt = a b gcb-1 ... [Equ.39]
For the purpose of finding the single value of secant shear stiffness governing the pressuremeter response seen in the measured loading curve, G100 is required. This is the secant modulus at the maximum elastic shear strain, sometimes termed Gmin or Gyield. It is probably too conservative a value for design purposes.
There is an alternative way of deriving Gs and Gt from pressuremeter unload/reload cycles, what might be described as the transformed strain approach. If the data points of an unload/reload cycle are used to derive a pressuremeter modulus GP (in effect D pc/D g c) curve then Jardine (1991) gives two empirically derived expressions for Gs and Gt. The expressions are
g c/g s = 1.2 + 0.8 log10(g c/10-5) for converting GP to Gs ... [Equ.40]
and
g c/g s = 4.5 + 2.65 log10(g c/10-5) for converting GP to Gt ... [Equ.41]
The effect of applying equations [40] and [41] is to re-calculate the strain at which a given value of pressuremeter modulus applies.
Figure 19 shows all these possible ways of quoting modulus applied to a single unload/reload cycle from a pressuremeter test in clay. There is good agreement between the empirically derived Jardine transformations and the rigorous derivatives from the power law expression.
Fig.19 Alternative ways of plotting stiffness-strain curves
4.4 Stress level
For modulus parameters derived from undrained expansion tests the mean effective
stress remains unchanged throughout the expansion and all stiffness:strain
data will plot the same trend. Conversely, failure to plot the same trend
implies changes in the mean effective stress. This is true of tests affected
by consolidation, but is also true of a heavily disturbed loading where the
effects of the pressuremeter installation method have yet to be overcome.
For such data it is reasonable to take modulus parameters from as late in
the loading as possible. Division of the modulus values by a normalising
stress such as the effective insitu lateral stress or yield stress gives
a dimensionless parameter for modelling purposes.
4.5 Cross hole anisotropy
The pressuremeter test gives values for GHH, the shearing stiffness
in the horizontal plane. This is directly applicable to the analysis of radial
consolidation or cylindrical cavity expansion due to pile insertion.
GVH is applicable all shearing which has an element of deformation
in the vertical plane, such as under a footing or round an axially loaded
pile.
To convert from GHH to GVH some relationship between the two must be assumed. Wroth et al (1979) suggest that anisotropy arises from two causes:
The second cause implies the stiffness in any direction will be a function
of the effective insitu stress in that direction, ie a function of
KO.
It can be shown
GHH = EH/[2(1+nHH)] ... [Equ.42]
For undrained expansion
nHH = 1-n/2 ... [Equ.43]
and
n = EH/EV = Ko ... [Equ.44]
From this it follows
EH = (4-n)GHH ... [Equ.45]
EV = (4-n)GHH/n ... [Equ.46]
This is as far as argument from first principles can go, because of the additional contribution of the manner in which the material is deposited. Ko is likely to lie between 0.5 and 2, so from equation [45] EH/GHH lies between 2 and 3.5. From equation [46] EV/GHH lies between 1 and 1.75.
It is likely that GVH will be linked to EV by Poisson’s ratio in a relationship of the form of equation [42]. Plausible values of EV/GVH would seem to be 2.4 to 3. Hence in a material with Ko of 2, GVH could be as low as GHH/3. Simpson et al (1996) come to the same conclusion, but find in practice heavily over-consolidated London clay gives relationships of the order of GVH @ 0.65GHH. The influence of the strain range is not separately considered in these studies, and it is quite possible that the G100 values would be similar in all planes.
Lee & Rowe (1989) give details of the anisotropy characteristics of many clays varying from lightly overconsolidated to heavily overconsolidated. The general conclusion is EV/GVH lies between 4 and 5, rather more than the isotropic relationship of 3. However their paper was concerned with the impact of anisotropic stiffness properties on surface settlement. Deriving GVH from EV is therefore unsatisfactory, because although GVH is insensitive to the direction of loading, EV is not.
In material with a Ko of 1 it is likely that GVH will be the same as GHH. For values of Ko smaller than 1 then the vertical shear modulus GVH may even be greater than the horizontal value.
4.5.1 Recommendations for manipulating pressuremeter unload/reload data
4.6 Shear modulus from other parts of the pressuremeter curve.
The initial part of the loading will give a value for secant shear modulus, usually referred to as Gi. Provided the insertion disturbance is low this will be a plausible value but affected by the same considerations of stress level and strain range as other parts of the curve. The calculation is that given by Equation [33] and figure 5 is an example.
The first part of the unloading can in principle give a similar parameter but by the time the pressuremeter unloads the creep strains due to consolidation and rate effects will be large, so there will be a tendency for the initial unloading to be too stiff. However provided some allowance is made for this then reasonable estimates of the shear modulus will be obtained. Equation [33] can be used.
Analyses such as Bolton & Whittle also imply a value for the secant shear modulus at yield – it will be cu/gye, called Gye in figure 12. Although this is not likely to be the best way of deriving shear modulus data it is important justification for using the analysis that it can predict this independently measurable stiffness.