Interim summary
The results show that the post-cyclic capacity is reasonably estimated with the reference EPPE method. The use of the more complex dissipation method will lead to larger capacities – especially for an increasing number of cycles. The effect of both methods is amplified with a sequential analysis, although a mandatory analysis in such a way does not seem to be necessary. A simplified analysis by using a 1D finite differences model will lead to slightly larger capacities but neglects spatial influences. There is no need to use the equivalent number of cycles for the input of the dissipation model.
7.3 Comparison with different estimation approaches
Figure 7.25: Comparison of different modelling approaches by means of total capacity with dif-ferent variations.
Regarding the contour input, not only a deviation in contour approximation is possible, but also that only symmetric two-way loading DSS test results are present. Hence, Fig-ure 7.26 (b) shows the resulting excess pore pressFig-ure ratio field for this condition. The influence on the ULS design proof is very small compared to the reference case (Fig-ure 7.25). The deviation can be explained with the fact that for LTR = 0 a smaller degradation is expected compared to LTR > 0.
0.00 0.08 0.16 0.23 0.31 0.39 0.47 0.54 0.62 0.70 0.78 0.85 0.93
0.00 0.08 0.17 0.25 0.33 0.42 0.50 0.58 0.67 0.75 0.83 0.92 1.00
(a) (b)
Figure 7.26: Resulting excess pore pressure fieldRu for scaled contour plot (a) and excess pore pressure field Ru for a global symmetric one-way loading, but contour plot only based on symmetric two-way loading as input (b).
Consideration of cyclic triaxial test results
There is a different soil response for different laboratory devices regarding the anisotropy of the soil specimen and also due to a rotation of principal axis. This is especially the case for triaxial and direct simple shear tests. The differences in cyclic soil response have been presented and implementation for the framework described in Chapter 6. The contour
plot was presented in Section 5.3.2. The influence of this distinction shall be shown on the reference system. Since only results for symmetric two-way loading are present for triaxial conditions, the interpolation is only done for this case. The influence on the bearing capacity when changing the contour plot in the mentioned way has already been explained in the last subsection.
The distribution of the Lode angle is shown in Figure 7.27 (b). The region in which triaxial conditions are present is represented by 0◦ and 60◦ and DSS conditions for 30◦. Since no cyclic extension tests have been performed, the compression contour plots are scaled ver-tically by a factor of 0.5 in order to qualitatively consider this effect. Hence, a more severe soil response is assumed for triaxial extension conditions. Andersen (2015) shows that this is the case for sandy material for a medium relative density, but for the very dense state, as in the presented study, a factor between DSS and triaxial test extension results of 1 may be assumed. This implies that a factor of 1.0 would have also been reasonable. How-ever, the region of triaxial extension conditions is not very pronounced (Figure 7.27 (b)) which makes the factor choice almost indifferent. Figure 7.27 (a) shows the different excess pore pressure ratio field as well as the influence on the load-displacement curve.
The application of cyclic constant-volume DSS tests is faster and leads to a slightly more conservative design. There is only a small effect associated with the change of the contour input regarding the DSS case. Even though compression conditions are present in front of the pile, the general degradation is smaller which leads to larger capacities as shown in Figure 7.25.
0.00 0.08 0.17 0.25 0.33 0.42 0.50 0.58 0.67 0.75 0.83 0.92 1.00
0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00
(a) (b)
Figure 7.27: Consideration of cyclic triaxial results with excess pore pressure ratio Ru after N = 1 (a) and Lode angle Θ based on monotonic reference calculation (b).
7.3.1 Displacement-controlled equation approach
Some authors state that due to stress redistributions monopiles are loaded in a controlled manner (Andersen et al., 1978; Cai et al., 2014). The advantages of displacement-controlled tests have already been explained. Most of the cyclic investigations deal with load-controlled tests, however, no in-depth analysis regarding the most representative load type currently exists to the knowledge of the author. A strain-approach for excess pore pressure estimation was established and is presented in Appendix D (see also Saathoff and Achmus (2021)). The herein used approach bases solely on displacement-controlled test results. Instead of the derivation of an equivalent shear stress, an equivalent shear strain
7.3 Comparison with different estimation approaches
is used. The equation was presented in Chapter 6 (Equation 6.16). Due to the different estimation equation (by usingγeq), the general spatial excess pore pressure distribution for N = 1 is altered. Figure 7.25 shows the resulting bearing capacity. Compared to contour based approaches, the bearing capacity is +17% larger and hence slightly less damage is induced. The value is based on the excess pore pressure ratio field in Figure 7.28 (a) which is derived with the equivalent shear strain distribution depicted in Figure 7.28 (b). The used constitutive model is elastic-ideal plastic and hence, the shear strain distribution may change when using a material law with hardening. Nevertheless, the larger strains by using the Mohr-Coulomb model can be interpreted as conservative in comparison to smaller strains derived with a more sophisticated one.
0.00 0.08 0.17 0.25 0.33 0.42 0.50 0.58 0.67 0.75 0.83 0.92 1.00
0.00 0.00 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.04
(a) (b)
Figure 7.28: Final excess pore pressure ratio fieldRu (a) and equivalent shear strain amplitude γeq,cyc (b).
7.3.2 Alternative load-controlled equation approach according to Seed et al. (1975)
A more simple approach is the one according to Seed et al. (1975b). The CSR curve for different MSR values was shown in Section 5.2.2. The normalized bearing capacity is shown in Figure 7.25. The excess pore pressure field Ru for N = 1 and the Ru field after dissipation is depicted in Figure 7.29. This approach is much simpler and yields a bearing capacity larger to the contour approach. The liquefaction curve is similar, but the trend over the (normalized) number of cycles is different. There is a small initial increase for the case of the semi-empirical equation. This trend is used in the analytical superposition and influences the final excess pore pressure value. Hence, smaller excess pore pressures arise which lead to a larger bearing capacity. An estimation of the capacity degradation can be obtained by using this simplified approach, however, the use of the reference EPPE approach is recommended.
7.3.3 Iterative calculation
A sequential dissipation was already presented and it was concluded that the dissipation effect is slightly amplified, but that this procedure may not be necessary. The next step is to not only calculate the excess pore pressure sequentially within the dissipation model, but the stresses can be calculated sequentially, too. So far, only one calculation was done and the CSR values for each integration point derived. However, due to excess pore
0.00 0.08 0.17 0.25 0.33 0.42 0.50 0.58 0.67 0.75 0.83 0.92 1.00
0.00 0.08 0.17 0.25 0.33 0.42 0.50 0.58 0.67 0.75 0.83 0.92 1.00
(a) (b)
Figure 7.29: Final excess pore pressure ratio fieldRu after N = 1 (a) andRu after superposition (b) with equation approach.
pressures, stress redistributions occur and the local stress values may change. A sequential calculation should be done successively in the finite element model for each cycle due to stress redistribution and, hence, different excess pore pressure responses. The dissipation analysis is performed in each run once. However, the iterative calculation is accompanied with high computational efforts and will not be taken into account within the reference procedure in order to keep the procedure simple. The influence of this model assumption is analysed in the following.
For a sequential calculation, the soil gets softer in the upper part, but the same bedding resistance is required, therefore larger stresses arise in the lower part. There is a gradual decrease in strength and stiffness due to excess pore pressure. In this way other areas are loaded and excess pore pressure is generated in a more distinct manner; a redistribution occurs. The complete behaviour and the areas where the soil softens and how redistri-butions take place is mainly influenced by the drainage paths. Figure 7.30 shows the bearing capacity over five iterations. Herein, the post-cyclic model was used in order to calculate the input values by means of CSR and LTR to derive the accumulated excess pore pressure ratios. The bearing capacity converges to a value which is slightly larger to the first estimation.
Figure 7.30:Bearing capacity over five iterations normalized to the value of the first run.
7.3 Comparison with different estimation approaches
The changes of the CSR field over the iteration are depicted in Figure 7.31. There is a clear stress redistribution in which the bedding area is mostly reduced. The spatial dimension has been reduced and the large CSR values occur in the area of active bedding.
The distribution depicted in Figure 7.31 (c) would, from a theoretical point of view, be the most accurate. However, because the influence in the bedding reaction of the initial field seems to be already sufficiently calculated in terms of a bearing capacity compared to the one resulting from Figure 7.31 (c).
0.01 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.26 0.29 0.32 0.35
0.01 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.26 0.29 0.32 0.35
0.01 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.26 0.29 0.32 0.35
(a) (b) (c)
Figure 7.31:Spatial distribution of CSR field for first (a), third (b) and fifth iteration (c).
7.3.4 Estimation of volumetric strain due to dissipation
The settlement is composed of shear strains from the co-cyclic undrained cyclic soil re-sponse and post-cyclic volumetric strains due to dissipation of excess pore pressure (cf.
Section 4.3.2). The latter is not considered here. However, with respect to the cyclic accumulation of displacements, the storm may also generate displacements that can only be considered if the volumetric strain is calculated on the basis of the dissipated pore pressure.
Many cycles with full dissipation will – in the current procedure – result in no degradation or accumulation of deformation since the last value of excess pore pressure will be zero. If excess pore pressure accumulates, the soil response gets softer and the shear modulus and angle of friction decrease. The static soil response arises (based on the aforementioned procedures). However, the soil went through N cycles and dissipated the excess pore pressure N times which in turn generated a volumetric strain. With the presented excess pore pressure estimation approaches it is also possible to roughly estimate the drained response of the structure by using undrained cyclic tests. Therefore, it is important to transfer the dissipated excess pore pressure to a volumetric strain, which is then accu-mulated over time. The consideration of displacements are also important, because the investigation of tilting is part of the SLS proof. Due to the dissipation additional tilting of the structure may arise. The volumetric strain is derived with the dissipated excess pore pressure (which is similar to an increase in effective stresses) by multiplying with the recompression modulus (cf. Section 4.3.2 and Section 5.2.5). Figure 7.32 shows the resulting accumulated volumetric strain. Because this value depends on the current stress conditions, the largest strains are calculated at the pile tip with 7% although there are a volumetric strains of roughly 3% in front of the pile. The volumetric strain was calcu-lated in each cycle with the dissipated excess pore pressure ratio times the stress at mean
global load and the recompression modulus of 7×10−41/kPa. The resulting additional plastic deformation of the pile was not estimated since only the general applicability of this aspect shall be presented.
Figure 7.32:Derived volumetric strain field εv after 30 cycles.
7.3.5 Interim summary
Based on the presented results, there are some conclusions which can be drawn at this point:
• A small deviation within the regression analysis of the contour curve has no sig-nificant influence on the final load-bearing capacity. The influence of a simplified contour input, using only contour plots for LTR = 0, is also not very pronounced.
• There is a larger bearing capacity, if an interpolation between DSS and triaxial test results is done. This was expected since cyclic triaxial compression results will yield smaller cyclic accumulations. The use of cyclic DSS results is recommended since they can be performed much easier and will result in conservative bearing capacity estimations.
• It is not clear if load- or displacement-controlled cyclic tests shall be performed.
However, if the latter are used, they lead, in this case, to a slightly larger post-cyclic capacity. The alternative incorporation of cyclic results with the semi-empirical equation according to Seed et al. (1975b) leads to larger degradations.
• An iterative calculation, in which the stresses are used after a degradation in order to calculate the CSR field again, seems not to be necessary, because the results of the first run are accurate enough related to the additional computation effort.
• The volumetric strain can be estimated and the order of magnitude seems reason-able. The incorporation in the framework can easily be done, if required.