PART 3.             ANALYSES FOR UNDRAINED SHEAR STRENGTH (c_u)

Once the origin is known, the expansion and contraction phases of the test can be used to determine the material shear strength. Two methods of approach have been applied:

1. Assume the shape of the shear strength:shear strain curve and hence derive a closed form solution for the pressuremeter curve. For the SBPM tests on this contract a non-linear elastic/perfectly plastic soil response has been assumed. This has been solved (Bolton & Whittle 1999) for the case where the non-linear elastic characteristics are given by a power law. Strictly, the form of the elastic phase is of no consequence once perfect plasticity is assumed, and the classic procedure developed by Gibson & Anderson (1961) could be used. For both solutions the slope of the pressure /strain curve plotted on semilog axes gives the shear strength directly and an estimate of the ultimate limit pressure. However the terminology of the non-linear elastic solution is different from the linear elastic model and this avoids some conceptual problems.
2. Make no assumptions about the shape of the shear stress:shear strain curve but differentiate the measured pressuremeter curve directly to give the shear stress response. Palmer (1972) gives the differential equation used to describe the complete shear stress:shear strain response of a material deforming under undrained conditions. The equation can be solved graphically or numerically by taking the current tangent of the total pressure/cavity strain plot, but the success of the method depends on the smoothness of the measured data.
   

3.1 Bolton & Whittle (1999)
Figure 12 gives the shear stress: shear strain response of a non-linear elastic/perfectly plastic soil. For the sake of completeness both expansion and contraction are included.

Fig.12  Non-linear elastic/perfectly plastic shear stress:shear strain curve

Because this solution is not widely known it is given in greater depth here than is strictly necessary. Assume that the non-linear elastic response of soils can be fitted with a power law of the form

t = ag^b                                                                            ...[Equ. 11]

This assumption will be justified later by inspecting unload/reload cycles.

Around the pressuremeter, assume that the soil is deformed under conditions of axial symmetry and the expansion is undrained. The following relationships apply (see figure 2 for an explanation of the symbols used):

Axial strain e_a = 0
Circumferential strain e_q = -r/r
The expansion is undrained so radial strain e _r = -e_q = r/r
Shear strain g = e_q + e_r = 2r/r = dA/A

The equation of radial equilibrium applies throughout the expansion:

                                                                  ... [Equ. 12]

where s_r is radial stress and s_q is circumferential stress.

Using t to represent the maximum shear stress, equation [12] becomes:

                                                                           ... [Equ.13]

Now using the constitutive relationship t = ag^b and writing the current area in terms of radius:

                                                                  ... [Equ. 14]

Noting that (1/r)(1/r²)^b = r^{-(2b +1)} :

                                                        ... [Equ. 15]

and integrating between the reference state, and the pressure and radius at the cavity wall:

                                                     ... [Equ. 16]

so

                                     ... [Equ. 17]

The right hand side of this result is the shear stress mobilised at the cavity wall and can be written as t_C/b .
Note that if b = 1, the condition for linear elastic response, the right hand side of equation [17] reverts to the following familiar expression where a is shear modulus G:

                                                                  ... [Equ. 18]

The end of the elastic phase is reached when t_C = c_u for the expansion, hence

p - p_o = c_u/b                                                                  ... [Equ. 19]

Thereafter, there is a plastic zone confined by the limiting elastic radial stress of c_u/b .

Equation 12 still applies, so

                                                        ... [Equ. 20]

This gives

                                                         ... [Equ.21]

Integrating between the radii of the cavity wall and of the elastic-plastic transition:

Hence

                                                  ... [Equ.22]

In a soil being sheared at constant specific volume (Gibson & Anderson 1961) the ratio of the shear strain required to initiate plasticity during expansion g_ye to the shear strain at the cavity wall during expansion g_ce. This leads to

                                               ... [Equ.23]

This result resembles the simple elastic/perfectly plastic solution proposed by Gibson & Anderson. For the special case of a simple elastic response when b =1 the two solutions are identical. Indefinite expansion of the borehole is predicted by:

                                                      ... [Equ. 24]

and substituting this into equation [23] gives

                                                           ... [Equ. 25]

showing the undrained shear strength and limit pressure can be obtained from the gradient and intercept of a plot of total pressure at the cavity wall versus the natural log of the current cavity shear strain (Figure 13).

Note that equation [23] makes no direct reference to shear modulus.

3.2 Analysing pressuremeter undrained contraction data
The expansion phase ends at some value of pressure and cavity strain at the borehole wall p_max and e _cmax. This is the origin for the contraction event. During contraction, the end of the elastic phase is reached when t _C = -2 c_u, hence

p_max - p = 2c_u/b                                                                     ... [Equ. 26]

Fig.13   Deriving undrained shear strength

Jefferies (1988) gives the simple elastic solution for the relationship between pressures and strains at the cavity wall once reverse plastic failure is initiated:

p = p_max – 2c_u[1+Ln(g_cc)-Ln(2g_ye) ]                                               ... [Equ. 27]

This is not quite as his solution is written - g_cc is the shear strain at the cavity wall during contraction (see equation 29 below) and g_ye is the shear strain required to initiate yielding when expanding the cavity. From equation [23] it follows the non-linear elastic version of equation [27] is given by

p = p_max - 2c_u[(1/b ) +Ln(g_cc)-Ln(2g_ye/b )]                                      ... [Equ. 28]

Shear strain g_cc is obtained from conventional cavity strain e_c by:

[(1+e_cmax)/(1+e_c)] - [(1+e_c)/(1+e_cmax)]                                               ... [Equ. 29]

An inspection of equation [28] indicates that a plot of the natural log of the contraction shear strain against total pressure at the cavity wall gives a curve whose ultimate gradient is –2c_u. Figure 14 gives an example for the contraction phase of the same test plotted in figure 13.

As before, if b = 1, the condition for simple elastic response, all non-linear elastic equations given above revert to published solutions for simple elastic/perfectly plastic material.

One important reason for using contraction data to discover the shear strength is the certainty of knowing the origin for the contraction event. The expansion event is relatively insensitive to the choice of initial conditions but noticeable changes to the derived shear strength can be made by adjusting the origin for strain. All things being equal, therefore, a comparison between loading and unloading values may indicate insertion disturbance and give a means for correcting for it.

Fig.14  Using the contraction curve to derive c_u

 3.3   Palmer (1972)
The Palmer analysis is an example of more information being obtained from the pressuremeter test if fewer assumptions are made. The analysis shows that the pressure:strain graph is the integrated shear stress:shear strain curve. Taking the slope of the pressure:strain graph at any point gives the mobilised shear stress directly, and allows the complete shear stress:shear strain curve to be plotted. In terms of cavity strain the shear stress t is:

t = ½e _c(1+e_c)(2+e_c)dp/de_c                                                             ... [Equ.30]

More conveniently, perhaps, equation [30] can also be written in terms of volumetric strain as:

t = dp/d[ln(DA/A)]                                                                       ... [Equ.31]

This implies that the gradient at any point on the semilog plot used for the perfectly plastic analysis gives the mobilised shear stress directly. The example below uses the same data as Figure 13:

Fig.15  An example of the Palmer (1972) analysis

The analysis is awkward to implement on the computer because the differentiation process highlights any irregularities in the data. This is especially irritating because the stress strain response must be a smooth curve. Possible strategies involve curve fitting the measured data prior to applying the solution, but this is a mistake. Minor changes of gradient on the loading path are usually not random, but a response to some event such as the taking of an unload/reload cycle.

If the material does deform perfectly plastically then the two analyses give identical answers. Included on the plot is a horizontal ruler marking the value of shear strength obtained from applying the previous Bolton & Whittle analysis. If there are clear indications of peak and residual shear strength then additional horizontal rulers are available to mark these values. However the true value of the plot is that it is a 'map' of the shear stress, and it is the form of the complete curve which is of interest. The analysis is very sensitive to insertion disturbance - in particular insufficient allowance for stress relief will give an apparent peak in the stress/strain response. It is also possible that the peak is a rate effect – considered from the perspective of the elastic/plastic boundary, the standard strain rate of 1% per minute at the cavity wall gives an equivalent rate of about 70% per minute for this clay immediately following first yield.

The contraction data can also be plotted using equation [31].

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