Derivation procedure

In order to derive a contour plot, the shear strain and excess pore pressure are plotted over the number of cycles and predefined isolines are interpolated. These can then be connected with an appropriate equation. When deriving the contour plots from laboratory results, it is important to choose the boundary conditions of cyclic laboratory tests carefully (e.g.

CSR), because on the one hand the asymptotic value (stable range) should be derived realistically with smaller CSR values, but on the other hand the most influencing excess pore pressures are derived for a small number of cycles and large CSR values. If an insufficient amount of test results is present for this region, a small deviation at the beginning of the liquefaction curve (N < 10) within the regression analysis may have considerable effects later on. For a first estimation the maximum MSR and CSR are needed. For sandy material, the mean load is applied in a drained manner. Hence, for a CSR = 0, the maximum MSR value is tan(φ^′). Regarding the CSR, the failure line and the phase transformation line can help to estimate the maximum CSR. For the reference soil, the undrained dilatant behaviour can be estimated with the PTL position at roughly CSR = 0.2. The testing grid of the reference soil is depicted in Figure 5.20. Most of the points are concentrated for MSR and CSR values smaller than 0.2. If only a small number of tests shall be performed, these should be placed at symmetric two-way and one-way load, since it is possible to use these positions for deriving contour plots and scale existing data.

Exemplary results of these cyclic tests have already been shown. After all data is present, a mathematical regression framework is required. Initially a high degradation rate is needed

5.3 Contour plots for reference sand

Figure 5.20: Combinations of CSR and MSR within the laboratory program for the reference soil at a relative density of 0.85.

and for larger number of cycles the rate needs to decrease. The power law function given in Equation 5.2 is often used in literature and practice. The equation is used to derive one particular R_u isocurve and hence this has to be done for several different excess pore pressure ratios values.

CSR_Ru,i =a N^b+c (5.2)

Herein, Zorzi et al. (2019) keep the parameter b constant at -0.35, similar to Boukpeti et al. (2014) for carbonate silt sediments. Both fitted shear strain plots instead of excess pore pressure ratios. Zografou et al. (2019b) used the equation with values forb from 0.00 to -0.35 and c from 0.25 to 0.61. However, they dealt with kaolin clay with failure shear strain assumed at 5%. Since this approach is widely used it shall be investigated in the following.

A different approach is presented by Ronold (1993) (also in a compacted form in DNV-RP-C212). The values a and b in Equation 5.3 are empirical regression parameters and constant for different R_u values.

CSR_Ru,i = 0.0001R_u

(a N +b) (5.3)

Equation 5.4 uses a regression for the trend of excess pore pressure over the CSR for N = 1 and adds the degradation over the number of cycles with the second term.

CSR_N=1 =tanh(a R^b_u) CSR_Ru,i =a CSR^b_N=1(1 + 10N)^−c (5.4) Wichtmann and Triantafyllidis (2011) report a logarithmic approach for the description of volumetric strain in cyclic triaxial tests. The approach fits well for poorly graded fine gravel, but the function depends on test material. Hence the following equation was

derived in order to fit CSR with the related N and R_u. Equation 5.5 uses the shape of a quadratic form and an apex at the position of N = 1000. The asymptotic CSR value equals the b value within the equation. It can mathematically only be used up to 1000 cycles and needs two regression parameters. This applies to the DSS tests, because mainly 1000 cycles for each test were performed within the test program.

CSR_Ru,i =a∗(ln(1000)−ln(N))²+b (5.5) After the possible equations have been presented, they shall be applied to the reference soil (D_r = 0.85). Figure 5.21 to Figure 5.24 show the fitted contour plots for the reference soil for MSR = 0. Figure 5.21 shows the regression based on Equation 5.2. This equation is also known from practical projects. The regression works well for predefined ranges of the regression parameters ( 0 ≤ a ≤ 1, −0.5 ≤ b ≤ −0.2, 0 ≤ c ≤ 0.5) and the asymptotic CSR value can actively be controlled. In order to have a mathematical framework, the regression parameters are furthermore fitted over the normalized excess pore pressure with a hyperbolic tangent function (Equation 5.6 with parameter definition of a and b ) (cf. Figure 5.25).

a=tanh(a₁∗R^a_u²) b =tanh(b₁∗R^b_u²) (5.6) This is done for all the following approaches, based on preliminary analysis. Hence, Equation 5.2 needs in total six fitting values and is the most complex one in terms of fitting.

Figure 5.21: Contour plot based on Equation 5.2 for MSR = 0 and the reference relative density from cyclic direct simple shear tests.

Figure 5.22 shows the approach in accordance to the DNV (Equation 5.3). This regression seems to be quite good, but is not easy to control regarding the physical meaning of regression parameters. Figure 5.23 shows the fitting of the values with Equation 5.4.

5.3 Contour plots for reference sand

Figure 5.22: Contour plot based on Equation 5.3 for MSR = 0 and the reference relative density from cyclic direct simple shear tests.

Figure 5.23: Contour plot based on Equation 5.4 for MSR = 0 and the reference relative density from cyclic direct simple shear tests.

Figure 5.24: Contour plot based on Equation 5.5 for MSR = 0 and the reference relative density from cyclic direct simple shear tests.

Equation 5.4 cannot be fitted to the measured data straightforward since first the curve for N = 1 is needed. It does not give a large decrease over the number of cycles which may become difficult for a larger degradation within the first cycles.

The resulting larger excess pore pressure values are not conservative in comparison with the other approaches. The asymptotic value cannot be fitted separately. Furthermore, the curves are not parallel for 1000 cycles and may intersect for larger numbers of cycles.

Figure 5.24 shows the regression based on Equation 5.5. This approach fits the cyclic data very well and the regression parameters can also be fitted easily (Figure 5.25). The regression of the fitting values a and b is done over the excess pore pressure ratios.

Figure 5.25: Regression of fitting parameter over normalized excess pore pressure ratio for a (a) and b (b) for MSR = 0 for Equation 5.5.

Table 5.2: Final regression parameters for reference soil for excess pore pressure ratio at a relative density of 0.85.

MSR a₁ a₂ b₁ b₂

[1] [1] [1] [1] [1]

0.00 0.0205 0.3328 0.0804 0.6601 0.05 0.0201 0.7823 0.0580 0.3353 0.10 0.0150 0.8000 0.0476 0.4265 0.15 0.0050 0.9000 0.0378 0.2744 0.35 0.0041 0.9000 0.0237 0.1624

All test results were subsequently fitted with Equation 5.5 to derive plots for different MSR values. The observed laboratory test results, in terms of the excess pore pressure R_u and the cyclic shear accumulationγ over the cyclic load number N, are described by a set of best fit parameters. Table 5.2 gives an overview of the used regression parameters for all presented contour plots. The final excess pore pressure ratio contour plot is depicted in Figure 5.26 with CSR and MSR over a number of cycles for excess pore pressure ratios R_u of 0.01, 0.05, 0.10, 0.20, 0.50, 0.95. The blue lines show the slices of type 3 plots and the red lines the type 2 representations.

5.3 Contour plots for reference sand

Figure 5.26: Excess pore pressure ratio R_u (from bottom to top: 0.01, 0.05, 0.10, 0.20, 0.50, 0.95) over CSR, MSR and number of cycles N.