Fig. 2. Distortions caused by probes of different shape.

Fig. 3. Shear strain caused by various in situ tests.

**IMPORTANCE OF THE STRESS HISTORY PARAMETER K**_{D} TO PREDICT SETTLEMENTS

The dilatometer modulus *E*_{D} should not to be used, as such, for predicting settlements, because *E*_{D} is only slightly sensitive to Stress History, while the operative modulus increases considerably with Stress History.* E*_{D} must first be corrected for Stress History :

*M*_{DMT} = E_{D} *×* *R*_{M} (*K*_{D}*, I*_{D}) (2)

where *E*_{D} is the dilatometer modulus and *R*_{M} is a correction factor applied to *E*_{D} depending primarily on the stress history index *K*_{D} (Marchetti 1980). As to the operative Young’s modulus *E*, if required, it can be estimated from Mdmt using the theory of elasticity (*E* ≈ 0.8-0.9 *M*_{DMT}).

Incorporating stress history information in the settlement calculation, using *K*_{D}, is not a refinement, but a necessity. The availability of the parameter *K*_{D} is important. There are not many alternatives to *K*_{D} for obtaining in situ information on stress history. On the other hand if the investigation is carried out with probes of modest sensitivity to stress history, the benefits of stress history are ignored, leading to a settlement overprediction and a more expensive design.

**NOTES**

- Not rarely
* M *estimated in situ are higher than *M* estimated in the laboratory, due to sample disturbance. E.g. Schmertmann (1988) compared *M* obtained by different methods at Sunshine Skyway Bridge, Tampa Bay in Florida. On the average, *M* estimated by DMT was 200 MPa, by laboratory oedometers 50 MPa, back calculated from observed settlements 240 MPa. The laboratory oedometers were in this case 4 times too soft, possibly due to sample disturbance and stress relief. On the other hand already Terzaghi and Peck, in their 1967 book, had warned that even a good oedometer of OC clay may 2 to 5 times softer than in situ. In sands in situ estimates of *M* are even more useful, due to the difficulty of recovering undisturbed samples in sand.
- Eq. (1), based on linear elasticity, provides a settlement proportional to the load, and is unable to provide a non linear prediction. The predicted settlements is meant to be the
*settlement in working conditions*
- Immediately after a DMT is completed, the predicted settlements may give an idea of the proper type of foundation. E.g. in case of buildings, very roughly, if the predicted settlement is < 3 cm (or possibly 4 or 5 cm), then a shallow foundation can be adopted, otherwise a deep foundation has to be adopted. In general the entity of settlements has a significant economical consequence. Accurate estimates may permit a more economical design.
*M* can also be predicted as *M* = a *Q*_{c}. The problem is that a depends on stress history or OCR, a missing information if only *Q*_{c} is available. The range of variation of a is quite wide (a = 2 to 20).

**APPENDIX**.

1 **Legitimacy of using M as a constant**

It is well known that oedometer moduli *M* are not constant, but vary with the applied vertical load. In particular the oedometer modulus increases up to the maximum past pressure *p*_{c}. At *p*_{c }, break point in the e-log p curve, the modulus decreases, to increase again at higher loads. Therefore the average modulus to be used for predicting settlement should in principle be chosen as the average modulus in the interval between the initial and the final vertical load. This can be done if the e-log p curve from the oedometer is available, but cannot be done if only the constrained modulus at geostatic stress is available. Since the target of DMT is specifically the 1-D modulus at vertical geostatic stress, and since DMT does not provide information on modulus at stresses higher than geostatic, predicting settlements using *M*_{DMT} involves approximation. Fig. 4 shows schematically two typical e-log p oedometer curves, and the values of the moduli *M* at various applied vertical load *p*. In many natural soils, with the exception of highly structured clays, where the break is sharp, the variation of the modulus across *p*_{c} is moderate. Hence the error in assuming *M*≈constant is often relatively acceptable for practical purposes. This assertion is supported by the large number of case histories in the recent decades indicating good agreement between observed and DMT-predicted settlements. On the other hand moduli estimated by alternative methods are not rarely affected by errors (e.g. disturbed samples) much larger than the mentioned approximation.

It is reminded that *M*_{DMT} provides an estimate of the operative modulus during the consolidation. Hence the predicted settlement is the primary settlement, and does not include the secondary settlement.

Fig. 4. Schematic variation of the oedometer moduli with applied load.

**2. Deriving M drained from an undrained test**

In clay, the expansion of the membrane occurs in undrained conditions. Therefore the dilatometer modulus *E*_{D} is an undrained modulus. Thus, according to logic, the correlation to be investigated should be between *E*_{D} and the undrained modulus *E*_{u}. Attempts of this kind were carried out in the early days of the DMT development. However a big obstacle, precluding such possibility, was the high variability of the undrained moduli provided by different laboratories, at least in part due to the high sensitivity of *E*_{u} to the disturbance. Hence, as a second attempt, the correlation *E*_{D} – *M* was investigated. This correlation involves many soil properties, including material type, anisotropy, pore pressure parameters etc. Hence no unique *E*_{D }– *M *correlation can be expected. On the other hand the DMT provides, in addition to *E*_{D}, also the parameters *I*_{D} and *K*_{D} , containing at least some information on material type and stress history. This availability provides some basis to expect at least some degree of correlation *E*_{D} –* M* , using *I*_{D} and *K*_{D} as parameters. Moreover, while the correlation *E*_{D} –* M* is, at least in principle, weaker than *E*_{D} – *E*_{u }, at least *E*_{D} – *M* can be tested, because *M* by different laboratory have much less variability than *E*_{u}.

Obviously the final word goes to real world observations. Large number of case histories have generally proved the favorable comparisons between observed and DMT-predicted primary settlements, thereby supporting the use of *M*_{DMT }as operative constrained modulus.

Note also Lambe et al. (1977 Jnl Asce GE, 106, GT3): “Drained moduli of saturated clays are typically about *one-third* to* one-fourth* the undrained values”. Hence a broad connection drained-undrained stiffness has already been invoked in the past.

**3. Using The 1-D Settlement Terzaghi Formula even in 3-D Situations**

Settlements calculations are generally carried out using the 1-D elasticity formula in 1-D problems, say *large* rafts, or the 3-D elasticity formula in 3-D problems, say *small* isolated footings. The well known formulae are respectively:

(3)

(4)

However the general recommendation is to use in all cases the 1-D formula, for the following reasons (Marchetti, 1991):

- The 3-D formula (unlike the 1-D) involves Poisson’s ratio n (unknown) and makes use of the horizontal stresses s
_{h} that “may be grossly over-/underestimated by theory of elasticity”, while the vertical stresses “are surprisingly well predicted by simple elastic theory”
- In most cases the 1-D formula gives settlements that are within 10% of the 3-D calculated settlements, because of the following compensation: M in Eq. 10 is higher than E in Eq. 11, but Eq. 11 contains a negative term
- Errors due to the formulae are small compared with errors in deformation parameters. As noted by Poulos: What is important is the modulus, not the formula.

Since the above two formulae give similar answers, it appears preferable to use the 1-D formula, simpler and avoiding elusive n or σ_{h.}