7.2 Application of the EPPE contour approach
7.2.2 Variation of stress consideration
7.2 Application of the EPPE contour approach
rotation small excess pore pressures develop. In the upper part no excess pore pressure accumulates, because of the free drainage boundary condition. The trend over time is almost constant due to the used design dissipation approach. The general excess pore pressure accumulation is not very pronounced due to the homogenous sandy soil layering (without any cohesive layers) and especially because of the comparatively large hydraulic conductivity.
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 6.74 13.47 20.21 26.94 33.68 40.41 47.15 53.89 60.62 67.36 74.09 80.83
(a) (b)
Figure 7.7: Final excess pore pressure ratio field Ru (a) and results of consolidation analysis in the form of excess pore pressure ∆u (b).
Figure 7.8 shows the monotonic and post-cyclic load-displacement curves. However, the cyclic curve should only be analysed regarding a ULS proof, since additional effects such as accumulation of plastic strains and, hence, a softer response in the initial part of the curve are not taken into account in this assessment. For the 0.1D criterion, a degradation of bearing capacity of 53% is expected.
Figure 7.8: Monotonic and cyclic load-displacement curves for reference system.
field, the final excess pore pressure ratio field Ru as well as the resulting post-cyclic bear-ing capacity. Although the procedure is modular and dependbear-ing on the input some results may change, the overall response and influences of variations are expected to be similar.
Additionally, it should be kept in mind that the presented problem is highly non-linear since several non-linear effects are sequentially considered.
Figure 7.9 shows the resulting bearing capacity for a deformation criterion at mudline of 0.1D and the approach according to Manoliu et al. (1985) normalized by the post-cyclic value for the reference model. The approach according to Manoliu et al. (1985) will result in slightly larger capacities than the 0.1D criterion. For the reference model a value of 1 is obtained due to normalization. The monotonic capacity is 1.89 times the capacity of the cyclic one. Different model variations are depicted on the x-axis and will be explained briefly in the following. Figure 7.9 depicts the influence of the general input.
The bearing capacities are 18.7 MN for the cyclic and 35.4 MN for the monotonic case for the 0.1D criterion. The monotonic results agree well with the results from the analytic calculation.
Figure 7.9: Comparison of different modelling approaches in terms of normalized post-cyclic capacity (normalized to reference system).
Application of cyclic load amplitude in an undrained manner
The global cyclic load amplitude can be applied in an undrained or drained manner (cf.
Section 6.1.1). For an undrained application, a larger stiffness modulus and a constant volume boundary arise. In the reference model, the global load amplitude is applied in a drained way by means of a simplification to avoid a coupled hydraulic-mechanical model.
In this coupled model, the mean load is applied over a long simulation period to avoid the build-up of excess pore pressure and allow for volumetric strains. The global cyclic amplitude is applied much faster in the finite element model. The difference in capacity is
7.2 Application of the EPPE contour approach
an increase of roughly +8% based on the stiffer response (Figure 7.9). The CSR field is de-picted in Figure 7.10 (a). The CSR values are generally smaller compared to the reference system, which results in a smaller excess pore pressure degradation field (Figure 7.10 (b)).
The reference approach very well estimates the bearing behaviour conservatively and does furthermore bear advantages regarding the derivation of an equivalent number of cycles (see Appendix B).
0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.20 0.23 0.26 0.29 0.32 0.35
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 7.10: Spatial distribution of CSR (a) and resulting excess pore pressure fieldRu (b) for an undrained application of load amplitude.
Calculation of cyclic stress ratio
The equivalent shear stress builds-up non-linearly over the course of the global load ap-plication. The CSR is based on the shear stress amplitude within one element. The amplitude can be calculated in the following two ways (Equation 7.2 and see Section 6.1.2). In the reference procedure, the stresses at mean and maximum global load are used. Alternately, the CSR can be based on half of the loading span from minimum to maximum global load.
0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.20 0.23 0.26 0.29 0.32 0.35
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 7.11: Spatial distribution of CSR (a) and resulting excess pore pressure fieldRu (b) for CSR based on half of the deviatoric stress span.
Figure 7.11 shows the resulting smaller CSR field compared to the reference system as well as the damage field (Figure 7.11 (b)). The change in bearing capacity is herein +14%
compared to the reference case (Figure 7.9). In the reference procedure the stress ampli-tude is evaluated in order to evaluate the non-linear stress at the global load ampliampli-tude.
This procedure is moreover conservative.
CSR= σeq,F′
mean−σeq,F′
max
2σoct,F′ mean alternatively : CSR= 0.5(σ′eq,F
min −σeq,F′
max)
2σ′oct,Fmean (7.2)
0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.20 0.23 0.26 0.29 0.32 0.35
0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.20 0.23 0.26 0.29 0.32 0.35
(a) (b)
Figure 7.12: Field of calculated CSR based on octahedral stress at maximum global load (a) and based on the stress at initial conditions (b).
The reference stress within the definition of CSR and MSR can be chosen at initial conditions, at the mean global load (as done by default) or the maximum global load.
The influence on the final CSR field is shown in Figure 7.12. The use of the octahedral stress at initial conditions will result in a field with increased CSR values and the stress at maximum global load to decreased CSR values. The change in bearing capacity is -17% and +4% compared to the reference case, respectively (Figure 7.9). There is a difference of about 20% in bearing capacity between the different model assumptions.
This indicates how important a correct definition of the input values can be. The non-linear correlations can well be seen in this example, because the CSR and MSR values are smaller for a normalisation with the octahedral stress at global maximum load. Also the spatial distribution changes slightly. This leads to a larger final damage field. The assumption within the reference EPPE procedure is conservative but too uneconomical within the design process.
Unloading of numerical model to global minimum load
Instead of deriving the cyclic input values from one monotonic loading, the stress state can very well be described by performing a calculation with a complete half cycle in-stead of only loading to the maximum cyclic load level in order to also consider stress-redistributions and above all the stiffer soil response for an un- and reloading stiffness.
Even if no un- and reloading modulus is present in the used Mohr-Coulomb model, the effect shall still be presented. Hence, the deviations are based on stress redistribution.
The CSR in each integration point is calculated with the difference from global mean load to the global minimum load. The bearing capacity is larger compared to the ref-erence case, because the required shear stress to carry the bedding resistance is already mobilised and only stress redistributions and mainly unloading effects for a symmetric one-way loading case occur. This leads generally to smaller CSR values. The depicted CSR field and excess pore pressure ratio field are depicted in Figure 7.13. The bearing capacity changes by +12% because of the less severe CSR field compared to the reference case (Figure 7.9).
7.2 Application of the EPPE contour approach
0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.20 0.23 0.26 0.29 0.32 0.35
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 7.13: Spatial distribution of CSR (a) and resulting excess pore pressure field (b) based on a monotonic reference calculation with unloading toFmin and derivation of CSR by using the amplitude fromFmean toFmin.
Influence of global mean load
There is a different soil degradation depending on different cyclic load levels and dif-ferent global mean loads. A focus was put on a constant load amplitude and not on a constant maximum global load (Figure 7.14). Therefore, Figure 7.15 shows results for non-symmetrical, symmetrical two-way loading with Fmin = −3.4 MN / Fmax = 10.2 MN and Fmin = −6.8 MN / Fmax = 6.8 MN, respectively; as well as non-symmetrical and symmetrical one-way loading withFmin = 6.8 MN /Fmax = 20.4 MN andFmin = 0 MN / Fmax = 13.6 MN in order to evaluate the influence of different load types. Hence, the load amplitude was kept constant as ofFcyc = 6.8 MN. Due to a constant load amplitude and the presented load cases, not only ζc, but also ζb is varied.
Figure 7.14: Schematic of used load types with constant load amplitude.
Figure 7.15 shows the final excess pore pressure ratio field for different loading scenar-ios. The damage decreases from ζc changing from -1 to 0.3 (Figure 7.16). The overall damage is smaller compared to symmetric one-way loading, but the spatial distribution is similar. The resulting bearing capacities normalized to the reference case are depicted in Figure 7.16. It shows that different load types can be approximated with the EPPE approach and that a one-way loading induces a larger damage compared to a two-way
loading. How pronounced the influence is, of course, also depends on the contour repre-sentation. The degradation is mainly on the right side of the pile, because the pile was only loaded once in this direction. The degradation could also be calculated for the other pile side, but this would only marginally change the bearing capacity but increase the computational time.
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
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) (c)
Figure 7.15: Excess pore pressure ratio fieldRuforζc=Fmin/Fmaxequal to -1 (a), -0.5 (b) and 0.5 (c).
It should be kept in mind, that as a conservative model assumption the excess pore pressure contour plots were not adjusted to the smaller damage for small CSR and large MSR values. Within the cyclic DSS tests, a smaller excess pore pressure build-up was measured regarding non-symmetric one-way loading and hence the result can be seen as conservative. For a different contour input, this curve may get closer to the monotonic response.
Figure 7.16: Comparison of bearing capacities depicted with ζc = Fmin/Fmax for the 0.1D cri-terion.
Interim summary
The calculation results show, that the model assumption regarding the stress definitions of the reference EPPE method are reasonable. There is no need to apply the load amplitude under undrained conditions. The stress ratios should not be derived based on the stress span since the stresses are averaged and the use of an unloading step does not significantly
7.2 Application of the EPPE contour approach
change the results but implies that the constitutive model can be used for this application, which is strictly speaking not the case. The use of the mean effective octahedral stress for the normalisation of the CSR and MSR is surely a model assumption in order to transfer the stress conditions from finite element to DSS conditions, but also reasonable.