Discussion: A non-linear elastic/perfectly plastic analysis for plane strain undrained expansion tests Free

In their interesting Technical Note, in which they offer one possible alternative method for the interpretation of self boring pressuremeter test (SBPT) in clays, the authors have made certain statements which merit some additional comments. They state that in SBPT interpretation, ‘it is common practice to reduce such data to fundamental strength and stiffness properties using analyses that assume the ground response to loading is simple elastic/perfectly plastic’. This is a very strange statement, because it is well known that, since 1972, most investigators have been using one of the methods based on the discretization of the stress–strain curve for pressuremeter test interpretation in clays (e.g. Ladanyi 1972; Palmer, 1972). The discussers have been using with success the Ladanyi method since that time. They did not find it to be ‘awkward’, as stated by the authors; it made it possible to follow closely the whole stress–strain curve of the soil, and to detect the tensile cracking point of the test (e.g. Ladanyi & Johnston, 1973), all without making any a priori assumptions concerning the stress–strain behaviour of the soil.

For SBPT interpretation purposes, the authors propose, as an alternative, the use of a stress–strain model for clay, composed of a power law and a straight line, as shown in their Fig. 2, and then they develop the complete solution of cylindrical cavity expansion based on that law. In this connection, it is interesting to note that complete solutions for expansion of both spherical and cylindrical cavities, based on exactly the same assumption, were published about 25 years ago for ϕ = 0 and ϕ > 0 materials by Ladanyi & Johnston (1974) and Ladanyi (1975). For example, equation (14) in the Technical Note is found to be identical to equation (35) in Ladanyi (1975).

However, in the two aforementioned papers, the non-linear elastic/perfectly plastic solution was used only in connection with creep and creep rupture of frozen soils. It was not used in connection with pressuremeter test interpretation in unfrozen clays, first because it was considered that there is no sense in imposing any particular stress–strain law on the soil, and second because it is difficult to accept that the soil stiffness tends to infinity when shear strain tends to zero, as implied by the power law. In that respect, a better assumption may be that proposed by Denby & Clough (1980), who use a hyperbolic stress–strain law as a basis for their pressuremeter test interpretation.

We welcome the opportunity to recognize the achievements of the senior discusser, not least in the field of cavity expansion analysis. The independent analyses in 1972 of both Ladanyi and Palmer, in performing an elegant transformation from pressuremeter measurements to shear stress–strain data for isotropic soil, are well known to us, as they should be to every research worker in this field.

It is possible to share the discussers' exasperation that original research fails to penetrate engineering practice, but it would be unwise to ignore the facts. Linear elastic–perfectly plastic solutions based on Gibson and Anderson are widely used to derive parameters for engineering consultants who demand them. These engineers presumably do not know of the errors that may be generated by this unnecessary restriction. In our view, the neglect of Professor Ladanyi's more fundamental approach is due principally to its main achievement—displaying shear data without the need for parameters. Engineers want parameters. The modest aims of the Technical Note were:

• to demonstrate that real data from overconsolidated soils fit a power curve for the stress–strain law prior to full mobilization of shear strength

• to show that interpretations based on the false assumption of a linear–elastic perfectly plastic soil model are significantly misleading.

The discussers are obviously disconcerted with our presentation of the theory for power law soils in a fashion that might appeal to practising engineers. We admit we were unaware of the similar power law solution published by the senior discusser, 24 years ago, in a paper on the bearing capacity of strip footings on frozen soils. However, we were surprised that the discussers were not as pleased as we were with the excellent fit obtained by analysing field data in stiff clays using power curves. Recent work on clastic compression at Cambridge (McDowell & Bolton 1998) has improved our understanding of how and why soils may adopt limited fractal structures during initial normal consolidation, and it is well known that power laws inevitably emerge from fractal structures.

Of course, we realize that different parameters must apply to the three different ranges of soil behaviour corresponding respectively to elasticity, rearrangement and crushing of the grains. We concentrated on the broad middle range of particle rearrangement, during which the tangent shear modulus continuously reduces as shear strain increases (the misnamed S-curve). We selected a power law for this range of behaviour, and did not quote stiffness parameters for shear strains smaller than 10⁻⁴. Nor did we attempt to resolve the impact of volumetric hardening due to grain crushing, which might make the fitting of power curves to the large-strain data of normally consolidated soils rather less accurate.

Non-linearity of soils is both manifest and important in the strain range 10⁻⁴ to 10⁻², which covers most geotechnical designs. The practical consequences of non-linearity for the behaviour of engineering structures are well documented in research publications, but possibly not so widely known by practising engineers. Where unload–reload data from pressuremeter tests happen to fit a power law for these small-to-medium strains, simple power law expressions are available to represent the deformation of soils below foundations, or adjacent to walls or piles (Bolton 1993). Our Note enables engineers to obtain the index β and yield strain γ_y that best describe the degree of non-linearity, together with the undrained shear strength c_u, so that they can use these parameters directly in their designs.

The authors present a useful solution to the undrained expansion of a cylindrical cavity in a non-linear elastic/perfectly plastic material. The most interesting find may be the determination of non-linear stiffness from the unload–reload loops of a pressuremeter test. The difference between the non-linear analysis and linear analysis in the estimation of the in situ horizontal stress may not be significant.

It is well known that the unloading curve of a pressuremeter test is significantly affected by creep and consolidation in a clayey layer. The authors suggested using the reloading curve of determine α and β. In the interpretation of reload curves, the authors make an important assumption that the power law function can be written as following in the non-linear elastic stage:

where γ_r and τ_r are the shear strain and shear stress prior to reloading, respectively. However, the values of γ_r and τ_r may affect the determination of α and β. Further study on the laboratory tests is needed.

For a cylindrical cavity expansion in an isotropically consolidated soil, the yield strain γ_y of soil during cavity expansion can be calculated from equation (27) with γ_r = 0 and τ_r = 0 when the values of c_u, α and β are obtained. The authors suggested using equation (19) which is wrongly printed. Determination of γ_y by equation (19) requires the cavity reference pressure p₀ of in situ horizontal stress σ_ho. The values of r_y and G_y for the SBP tests in London clay and Singapore marine clay calculated by using equations (17) and (27) are presented in Table 2. Based on the linear elastic analysis, the shear modulus G_ur can be obtained from the unload–reload loops. The values of G_ur are larger than those of G_y, as shown in Table 2. Correspondingly, the values of γ_y, obtained from the linear analysis are smaller than those from the non-linear analysis.

Having obtained γ_y, P_Limit, c_u and β, p₀ can be calculated by using equation 14. The values of p₀ are shown in Table 2. It is interesting to note that the values of p₀ obtained from the non-linear analysis and linear analysis fall in the same range. This is due to the non-linear analysis with β < 1 and larger γ_y, and the linear analysis, on the other hand, with β = 1 and smaller γ_y. It is also noted that the values of p₀ estimated by non-linear analysis and linear analysis are close to those estimated by the lift-off method. The lift-off pressure is 449 kPa in London clay and 480 kPa in Singapore marine clay, respectively.

It is realized that the lift-off pressure is affected by the excess pore water pressure Δu_i caused by the insertion of the pressuremeter. Fig. 11 shows that the lift-off pressure for the testing in London clay is close to the initial pore water pressure u_i. Testing in Singapore marine clay also shows that the lift-off pressure is larger than u_i, which is much larger than the static water pressure u₀. This means that u_i consists of u₀ and Δu_i for an SBP test in clayey layer. Thus, the lift-off pressure obtained from the SBP test in a clay layer is generally larger than σ_h0.

The value of p₀ estimated by the non-linear or linear analysis is also affected by Δu_i because of P_Limit, which is estimated from the total pressure, including the effect of Δu_i. Therefore, without the value of Δu_i, the correct estimation of σ_ho based on the non-linear or linear total stress analysis is impossible.

For the SBP test in Singapore marine clay, Δu_i is 176 kPa and u₀ is 239 kPa. Assuming that u_i does not change during the SBP test, the effective in situ horizontal stress σ′_ho can be estimated by using σ′_ho = p₀ − Δu_i − u₀. Table 4 provides the values of σ′_ho and the earth pressure at rest K₀. The value of K₀ ranges from 0·56 to 0·74, which appears reasonable for the light overconsolidated marine clay with an overconsolidation ratio OCR of 1·5. Based on the expression suggested by Mayne & Kulhawy (1982), K₀ = (1 – sin ϕ′)OCR^sinϕ′, where ϕ′ is the effective friction angle, the value of K₀ is 0·71 for the marine clay.

The authors thank Cao for his interest. In particular we agree that equation (19) in the Note should have been set so that the exponent −1 on the right-hand side of the equation applied to the calculated exponential result, not (apparently) to its argument.

With regard to the origin for stress and strain, alluded to in the discusser's equation (27), we note that this was not broached in the paper. However, a longer discussion about the variation in the initial state of the soil following insertion of the instrument can be found in Whittle 1999. The question is then raised of lift-off pressure in relation to possible excess pore pressures following insertion. Excess pore pressures are, as we showed, a natural concomitant of non-linear stress–strain laws. Pore pressures induced by over-cutting or under-cutting on insertion may, therefore, be treatable as a byproduct of our analysis requiring no further corrective action. We agree that the power curve analyses of the data introduced by the discusser seem reasonable.

The further use of the power law fitting to interpret the effects of disturbance during installation on the results of self-boring PMTs, leading to better estimates of p₀, can be found in Whittle (1999).