Cyclic accumulation model

A method for predicting the load-bearing behaviour of foundation structures under cyclic loading that is conceptually applicable to a variety of general cyclic problems was pre-sented by Jostad and Andresen (2009) and is based on ideas according to Andersen (1976) and Andersen et al. (1978). They combine the information from site-specific cyclic direct simple shear and cyclic undrained triaxial tests in the form of contour plots as input for the numerical simulations (cf. Chapter 3). The general approach is based on the stress path philosophy (Lambe, 1967; Lambe and Marr, 1979; Bjerrum, 1973). Herein, represen-tative tests are chosen for represenrepresen-tative stress states of elements. It implies that different laboratory test results can be seen as representative and be used to describe the soil re-sponse (Wood, 1990). In general, this requires true triaxial tests; however, since these are not feasible for practical application, triaxial tests and direct simple shear tests are used.

The problem is mainly that the stress path does not only depend on the soil element it-self, but that stresses are redistributing. The model assumption hence is that the element follows exactly this stress path. By using the contour plots for different representative load types over the number of cycles, no separate constitutive framework is needed due to the additional inter- and extrapolation of laboratory results. The general method can be used for both undrained (UCDAM) and partially drained (PDCAM) analyses (Jostad et al., 1997, 2014). An approximate implementation is given in Appendix F.2.

Undrained conditions

The undrained cyclic strain accumulation model (UDCAM) analyses the undrained be-haviour under mean and cyclic load as a non-linear stress-strain response with anisotropic features (Andersen, 2009). It is solely based on cyclic laboratory results and uses total stresses. This means that the non-linear stress - strain curve is theoretically extracted without any derivation of e.g. a stiffness modulus or undrained cohesion of the soil from laboratory tests required for a constitutive model. However, in order to have a mathemat-ical representation some values are still derived in reality. For clarification, this undrained method is explained prior to the partially drained model. The UDCAM method can only

4.2 Explicit numerical methods

be used for completely undrained conditions, which occur mainly with cohesive materials.

A first implementation was presented in Jostad and Andresen (2009) and an extension to three-dimensional application in Jostad et al. (2014). The model uses the strain accumu-lation procedure (Andersen, 1976) (explanation in Appendix C) and accounts for cyclic soil degradation in the form of reduced stiffness due to accumulated strains in the inte-gration points of the numerical model for a calculated equivalent number of cycles. The laboratory results are not fitted to advanced constitutive models but are rather directly used (Andresen et al., 2011) – either as a look-up table or as a mathematical framework.

It uses results of cyclic triaxial compression and extension tests for the estimation of the cyclic response dependent on the individual stress state; and for all other stress states cyclic direct simple shear test results.

UDCAM-S

A simplified version of the UDCAM model is implemented in the finite element code PLAXIS (Bentley Systems, 2022) and termed UDCAM-S (Jostad et al., 2014). The sim-plified model is easier to handle, but has some limitations. It is still mainly intended for the application of undrained soil behaviour of clay or silt with low permeabilities. This model is being explained prior to the full UDCAM model since it is more comprehensi-ble.

The UDCAM-S model in PLAXIS is strictly speaking a pre-processor for the input val-ues of the elasto-plastic anisotropic shear strength (NGI-ADP) constitutive model with anisotropic shear strength and anisotropic hardening function (Grimstad et al., 2012).

The NGI-ADP was developed for monotonic anisotropic soil response. Cyclic loading is considered by means of contour plots. Hence, the main input are the contour plots for different boundary conditions. The NGI-ADP is an improved model which is based on plane strain conditions (model of Davis and Christian (1971)). The model requires the shear strain at failure for these three different stress types (direct simple shear, triaxial compression and triaxial extension), as well as the shear strength values for the different states normalized to the static shear strengths_u as input parameters. It uses an elliptical interpolation between the three stress states. The design cases FLS, SLS and ULS can be investigated by extracting the associated values from the contour plots (see Section 3.4).

For isotropic soil conditions, only the soil response from DSS tests is assessed. The result of the analysis are load-deformation curves for cyclic (FLS), average (SLS) or total loads (ULS). The default contour plots are for Drammen clay with a specific w_l, P I and I_P. But also contour plots for sandy materials exist. In order to perform an analysis on other soil, this plot can be scaled vertically according to site specific data for stronger or weaker soil responses. The dominant influence is the CSR, which is the vertical axis. Scaling is done to match a reference contour plot to cyclic test data for the soil being calibrated.

The calibration procedure is mainly done by the “cyclic accumulation tool”. The procedure shall briefly be described. First an equivalent number of cycles N_eq is derived with the strain accumulation procedure under symmetric two-way loading (cf. Appendix C). The result is a type 2 contour plot for this N_eq (Figure 4.2 (a)). A stress path is needed which defines the relationship between normalized cyclic and mean shear stress (Figure 4.2 (b)).

The stress path is one of a few relevant parameters and the value must be chosen carefully.

If the ratio is small, the behaviour will be dominated by the mean response, if the ratio is high, the behaviour will be dominated by the cyclic response. For a practical value, the stress path should be derived from the cyclic and the mean loading of the structure.

The path starts at zero normalized cyclic and mean shear strain. For structures under dead-weight, the starting point should be the mean dead-weight (MSR), since this shear stress is already mobilized. Since this is done before the numerical calculations, no exact stress conditions are considered. Furthermore, no stress redistributions are considered.

After Neq and a stress path inclination have been chosen, the next step is to evaluate the type 2 contour plots. The sum of mean and cyclic shear strain and shear stress (termed cyclic shear strength) is used for ULS case. In case of dynamic analysis, only the cyclic shear strain component is evaluated. The NGI-ADP model is fitted to the stress-strain path by means of a simple particle swarm optimisation.

0.5 0.4

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Shear strain [%]

toct/σ[1]DSS

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(a) (b)

Figure 4.2: Schematic contour plot (a) and resulting total shear-stress - shear-strain relation for LTR = 2 (b) (modified after Andersen (2015)).

UDCAM

The UDCAM overcomes some of the limitations of the simplified UDCAM-S. The UD-CAM evaluates the stress states in each integration point. This is especially relevant for large shear strains and hence elements at failure. However, if the stress path is chosen correctly, the UDCAM-S simplification may save a lot of computational effort and hence calculation time. Nevertheless, the chosen stress-path would be an approximation and be not correct for all integration-points in the model. One additional advantage of the UDCAM is that it directly incorporates the strain-accumulation procedure and hence equivalent number of cyclesN_eq are calculated for each integration point after each storm bin.

The general procedure of the UDCAM is as follows. The soil is undrained during one cycle and the cyclic degradation is estimated based on the soil response in the integration points.

In a finite element model the global mean load F_mean is applied. In a different model the global cyclic load amplitudeFcycis applied. Within the contour plots the memory variable

4.2 Explicit numerical methods

e.g. pore pressure or shear strain can be followed through different boundary conditions such asN,τcycorτmean. For a given storm load consisting of different bins, the procedure can be used in order to find a N_eq. The material for mean and cyclic soil is updated between each phase which is related to a specific N of the storm for each integration point up to this point within the storm. Iterations are performed to update the mean shear stress based on the results for the cyclic model. The output values are γ_cyc and γ_mean. For plane strain conditions γ_cyc,tri = ε_x −ε_z and γ_cyc,DSS = γ_xz are used (Jostad and Andresen, 2009). For the model with Fmean the output is a function of γcyc and for the F_cyc model the output is function of γ_mean. Afterwards a new N is calculated from the ∆N of the storm bin and the N_eq,old (see Appendix C). Finally, the materials are updated. The stress-strain responses from laboratory results are elliptically interpolated between different stress states and are used within the finite element framework (for a derived N_eq which is representative for a specific storm load bins and integration point).

The first calculation starts the iteration with γcyc = 0 within the mean model. The mean model outputs γ_mean for each integration point as an input for the cyclic model.

More information can be found in Jostad and Andresen (2009), Andersen et al. (1978), Khoa and Jostad (2017) and Jostad et al. (2014).

Summarising, the UDCAM model is not implemented trivially and an implementation is not publicly available. The UDCAM-S model can be used with PLAXIS (Bentley Sys-tems, 2022). The shortcomings are that the strain compatibility procedure is used, but was intended for cohesive material and gravity foundations in which the elements within the failure surface need to have the same shear strain. Such a dominant failure surface does not exist for instance at monopile foundations. Here, a procedure based on excess pore pressure should be used. Within the UDCAM-S procedure the mean shear stress is neglected. Furthermore, the assumed stress path, which is equal for all elements, bases on a global load ratio assumption and is not derived from a numerical model. No ex-cess pore pressure is separately considered since the total soil response in the form of shear strain plots does already indirectly consider this. Furthermore, no dissipation is considered which can lead to very conservative and even uneconomical designs for cohe-sionless soils. However, for application to cohesive soil this is not necessary. Furthermore, only one degradation set is evaluated; for gravity based foundations this seems reason-able due to the limited influenced depth, but for monopiles there is a non-homogenous degradation field which cannot be captured with one simple stress - strain curve. Hence, the UDCAM-S approach seems to lack applicability for monopile foundations. Klinkvort et al. (2020) use the approach in combination with an axisymmetric model (with Fourier transformation) called super-fast monopile design (SUMO); even though the strain com-patibility procedure is not recommended for monopiles and the soil degradation shows a large spatial variation (Skau et al., 2017; Andersen et al., 2013). For the investigated case of cohesionless soils the UDCAM (UDCAM-S and SUMO) model appears not well suited due to the aforementioned reasons.

PDCAM

The UDCAM was mainly developed for the soil response of material with negligible drainage. The soil response of sandy material is more complex due to dilatancy and

drainage effects. Therefore, the partially drained cyclic pore pressure accumulation model (PDCAM) was developed (Andersen et al., 1994; Jostad et al., 1997). PDCAM uses a pore pressure accumulation procedure in which the pore pressure can dissipate between the cycle packages (cf. Appendix C). The model is not intended to calculate the effective stresses for different cycles, but rather to represent the response of excess pore pressure and volumetric strain during the cyclic load history (Jostad et al., 2015a). The model works only with inter- and extrapolation of the stress - strain relation from the contour plots without an elasto-plastic framework. Based on the dissipated excess pore pressure, PDCAM also predicts the volumetric strain after the number of cycles. It can hence estimate the structures’ deformation after the storm – similar to the stiffness degradation method (SDM) by Achmus et al. (2009). The PDCAM model works in a similar way as the UDCAM model, but dissipation is allowed under mean loads (Jostad et al., 2015a; An-dersen et al., 1992, 1994). It uses an effective stress model for the mean load component.

The input parameters for the PDCAM model are the contour plots from undrained cyclic laboratory tests, drained triaxial tests as well as oedometer and permeability test results.

The main disadvantage is that it is not publicly available and that the implementation is not well documented. Hence, no critical evaluation can take place.

A first implementation of the elasto-plastic framework presented in Jostad et al. (1997) was based on a constitutive model from Kavli et al. (1989), which was a two surface soil model with end closing cap, stress-dependent stiffness and mobilized friction angle related to plastic shear strain. The calculation procedure is explained in Jostad et al.

(1997) for axisymmetric conditions (a slightly different implementation is presented in Jostad et al. (2015b)). The description of the procedure over the years is not entirely stringent and at some points lacks details. The global mean load and the global cyclic load are coupled as done in the UDCAM model. In the case of storm load, individual load bins start with a restart command from the individual models for which the coupling is done via a database. For stress-redistributions an iterative calculation is performed.

The calculation of volumetric strains during consolidation is done similar to the concept explained in Section 4.3.2.