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Method of Andrus and Stokoe (1998)

The method of Andrus e Stokoe  is based on measurements from seismic refraction tests, (Vs).

The velocity of the shear waves is corrected by the overpressure, using the equation (Robertson et al., 1992):

where

VS1 shear waves velocity corrected by the overpressure

VS shear waves velocity measured in situ

Pa atmospheric pressure (about 100 kPa)

σ’vo effective initial pressure in the same units of measure as Pa

For the calculation of the resistance to liquefaction, Andrus and Stokoe have proposed the following relationship:

Where the presence of fine content FC (%) intervenes in the calculation model by the following specifications:

CSR is calculated using the formula mentioned in the Introduction chapter of this Guide, for different magnitude must be inserted the corrective factor MSF (Magnitude Scaling Factor) as recommended by NCEER (see Table 1 - Seed Idriss method).

The critical value VS1c and the VS1/VS1c margin

The Andrus and Stokoe relationship is defined only for \(V_{S1} < V_{S1c}\): its second term contains \(1/(V_{S1c} - V_{S1})\), so as \(V_{S1}\) approaches \(V_{S1c}\) the resistance CRR diverges. \(V_{S1c}\) — written \((V_{S1})_{cs}\) in the formulas above and Vs1c in the results table — is the limiting upper value of the normalized velocity beyond which cyclic liquefaction is not observed.

For the intervals where \(V_{S1} \geq V_{S1c}\), Liquiter:

  • classifies the layer as Soil not susceptible to liquefaction;
  • still reports \(V_{S1}\), \(V_{S1c}\), the reduction coefficient \(r_d\) and the CSR, which remain defined;
  • shows CRR and Fs as --, because in that range they are not defined. This is not missing data: extending the formula beyond \(V_{S1c}\) would return negative values with no physical meaning.

Note

For these intervals the liquefaction index is 0.00 and the risk is "Very low": the layer does not contribute to the liquefaction potential of the vertical.

The VS1/VS1c column

To quantify by how much an interval lies above or below the threshold, the results table reports the margin \(V_{S1}/V_{S1c}\):

  • greater than 1 — the interval is outside the liquefaction threshold, and the value measures its relative distance: 1.30 means that \(V_{S1}\) exceeds the critical value by 30%;
  • lower than 1 — the interval falls within the range of validity of the method, and CRR and Fs are computed for it;
  • equal to 1 — limit condition.

Unlike Fs, the margin is defined along the whole vertical. Where the Fs diagram breaks off, the margin diagram carries on.

In the chart the margin is drawn as a diagram next to the Fs one, with the red reference line at 1.00 and green (not susceptible) or red (susceptible) markers: the same convention used for the safety factor against the code Fs.

Liquefaction potential and risk

Below the results table, both on screen and in the report, the summary of the vertical is printed, for example:

Liquefaction potential Liquefaction risk in 50 years: Moderate PLS=7.76%

The PLS value is the depth-weighted average of the liquefaction indices \(PL_i\) of the individual intervals:

\[PLS = \frac{\sum_i PL_i \, z_i \, w_i}{\sum_i z_i \, w_i} \qquad w_i = \frac{10 - 0.5 \, z_i}{100}\]

The weight decreases with depth and vanishes at 20 m. The risk classes are:

PLS (%) Liquefaction risk
≤ 2 Very low
2 < PLS ≤ 5 Low
5 < PLS ≤ 15 Moderate
15 < PLS ≤ 25 High
> 25 Very high

The liquefaction potential of the vertical computed after Iwasaki and Sonmez is described in Liquefaction potential index LPI.