# D.3 Pressuremeter creep pressure

If there are at least two sets of readings both in the second and in the third group, the creep pressure *p*_{fM} shall be estimated, using the following two graphical analyses.

- A graphical analysis of the (
*p*, Δ*V*_{60/30}) diagram: 2 straight lines shall be drawn on the (*p*, Δ*V*_{60/30}) graph, one involving the data points in the second group, the second one involving the data points in the third group, as illustrated on Figure D.2; the abscissa of the intersection of the 2 straight lines shall give a first value for*p*_{fM}: call it*p*_{fMi}. - A graphical analysis of the (
*p, V*_{60}) diagram: the borderline between the second group of readings (pseudo-elastic phase) of the pressuremeter curve and the third group of readings (large strains) shall be determined: call*p*_{2i}its abscissa.

The creep pressure value shall lay between *p*_{fMi} and *p*_{2i}. The closer *p*_{fMi} and *p*_{2i} are, the better is the quality of the test.

This value shall be confirmed during the final check (see D.6) when considering the values of *p*_{LM} and *E*_{M} obtained in the next sections.

## D.4 Pressuremeter limit pressure

### D.4.1 Definition

Since the pressuremeter limit pressure is obtained when the volume of the central measuring cell, which is also called the volume of the pocket, is doubled and since the volume readings do not involve the original volume *V*_{c} of the central measuring cell (see B.4.2.1), the limit pressure shall be the corrected pressure for which the corrected injected volume in the probe central cell is such that (see Figure D.2):

### D.4.2 Direct solution

When, during a test, the injected volume is such that the pressuremeter central cell volume becomes bigger than *V*_{c} + 2*V*_{1}, the limit pressure shall be then obtained by linear interpolation.

### D.4.3 Extrapolation methods

D.4.3.1 General

When; during an expansion test, the injected liquid volume is smaller than *V*_{c} + 2*V*_{1} it is impossible to use the direct method. Therefore, the limit pressure shall be extrapolated.

Each of the two extrapolation methods described in D.4.3.2 and D.4.3.3 shall be applied to test results The final value of the limit pressure which is to be reported shall be determined using the method given in D.4.4.

For these methods, extrapolation is only permitted when the number of pressure holds applied beyond pressure *p*_{fM} is at least two (see D.6).

If the limit pressure is not attained either by the direct method or by extrapolation methods, the limit pressure value shall be reported as *p*_{LM} > *p*, *p* being the last corrected pressure applied.

D.4.3.2 Reciprocal (1/*V*) method

The (*p*, *V*) pairs of readings shall be transformed into (*p*, 1/*V*) values and plotted. A linear regression shall then be performed using the last three readings.

This extrapolation shall be obtained by the following transformation:

*Y = Ap + B*

with

*Y = V*^{–1}

where

*A* and *B* are coefficients obtained by a least square regression of *Y* on *p*. The limit pressure shall be determined by the following equation:

D.4.3.3 Double hyperbolic method

The pressuremeter curve shall be approximated by a straight line tangential to two hyperbolic segments as defined by the following equation:

The coefficients *A*_{5} and *A*_{6} are the abscissae of the vertical asymptotes to each hyperbola.

The matrix of four coefficients [*A*] = [*A*_{1}, *A*_{2}, *A*_{3}, *A*_{4} ] shall be obtained for values of the asymptotic limits *A*_{5} and *A*_{6}, by the following matrix transformation.

[*A*] = [*X*^{t} × *X*]^{–1} × [*X*^{t} × *V*]

where

,

*A*_{5} and *A*_{6} are found by a least square analysis on *V* based on the Gauss/Newton method.

The limit pressure *p*_{LMDH} shall be determined for *V*_{l} = *V*_{c} + 2*V*_{1} as derived from the double hyperbolic equation above, using the analytical expression given by the unique positive solution such as 0 < *p*_{LMDH} < *A*_{6}, in the third degree equation:

*–A*^{2} × *p*^{3}_{LMDH}* *+ [*V – A*_{1}* + A*_{2}(*A*_{5}* + A*_{6})] × *p*^{2}_{LMDH}* *+ [(*A*_{1} – *V*)(*A*_{5} *+ A*_{6}) – *A*_{5 }*× A*_{6} × *A*_{2 }*+ A*_{3 }*+ A*_{4}] × *p*_{LMDH}* + *[(*V – A*_{1}) × *A*_{5}*× A*_{6} – *A*_{3}*× A*_{6}* – A*_{4}* × A*_{5}] = 0

NOTE Reference for the mathematical modelling can be found in references [2]-[4] in the Bibliography.

### D.4.4 Limit pressure by extrapolation, final step

The sum of the errors Σ* _{i}* |

*V*

_{calculated}–

*V*

_{measured}| for each extrapolated curve obtained by the two methods described in D.4.3.2 and D.4.3.3.shall be calculated and divided by the number of data points used. The limit pressure

*p*

_{LM}retained shall be the one obtained by the method giving the lowest mean error.

## D.5 Obtaining the Ménard pressuremeter modulus

### D.5.1 Choice of the pseudo-elastic range

The analysis of a corrected pressuremeter curve shall begin by calculating the slope *m _{i}* of each linear segment between two adjacent data points (see Figure 5).

with

*p _{i}*,

*V*the coordinates of the beginning of segment No.

_{i}*i*(

*i*≥ 1).

The lowest *m _{i}* value, always positive, is called

*m*

_{E}. The coordinates of the origin of this segment (

*p*

_{E},

*V*

_{e}) and of its end (

*p*'

_{e},

*V*'

_{E}) shall be used to calculate a coefficient β as follows:

where δ*V* is a tolerance for *V* taken as 3 cm^{3} initially.

In a first approach, the pseudo-elastic range along which the pressuremeter modulus shall be obtained by including all the consecutive segments which exhibit a slope less than or equal to β times the lowest non-zero *m*_{E} gradient. This range shall then extend in both directions from the origin of the first such segment to the end of the latest segment. The coordinates of the origin of the pseudo-elastic range shall be denoted (*p*_{1}, *V*_{1})and those of its end (*p*_{2}, *V*_{2}). If the number *n* of intervals becomes too low (for example *n* < 3) the tolerance interval δ*V* shall be increased. Engineering judgment shall be exercised, for example by considering *p*_{2} closer to or equal to *p*_{fMi}.

NOTE At any time of the test reading and test reporting, quick approximation of the pseudo-elastic range boundaries (*p*_{1}, *V*_{1}), (*p*_{2}, *V*_{2}), can be obtained by an analysis of the variation of *DV/DP* between pressure holds.

### D.5.2 Ménard pressuremeter modulus EM

D.5.2.1 General

According to the type of probe cover, the pressuremeter modulus shall be obtained by using the corresponding equations given in D.5.2.2 or D.5.2.3.

D.5.2.2 Flexible cover

where

*v* is the Poisson's ratio, conventionally taken as 0,33.

The *E*_{M} modulus shall be given in MPa.

D.5.2.3 Slotted tube

When using the slotted tube, *E*_{M} shall be obtained either from the equation given in D.5.2.2, or from the

following equation:

where

is the volume of the central measuring cell after calibration;

is the volume of the central measuring cell, including the slotted tube;

*V*_{m} = (*V*_{1} *+ V*_{2})/2.

NOTE For further information on the equation, see reference [4] in the Bibliography. The corresponding equation according to either D.5.2.2 or D.5.2.3 used shall be reported.

## D.6 Final check on pressuremeter parameters

Before finalizing the interpretation of a pressuremeter test, the *p*_{1}, *p*_{2}, *p*_{fM} and *p*_{LM} values shall be marked on the horizontal axis of the pressuremeter test curve (Figures 5 and D.2) and the fit checked with the corrected curve so as to detect any error or incorrect extrapolation and to check the choice of boundaries for the trio of results [*p*_{fM}, *p*_{lm}*, E*_{m}].

When the limit pressure is obtained by extrapolation, the limit pressure *p*_{LM} stated in the test report shall not be smaller than the last corrected pressure hold applied to the ground.