Equivalent linear pseudostatic and dynamic modelling of vertically vibrating monopile

Highlights

• A general equivalent linear dynamic frequency domain FE-tool.

• Empirical or laboratory soil shear modulus and damping curves.

• Validated with experiments of vibro-compaction and pile vibration.

• Load amplitude dependent vertical foundation stiffness and damping for monopile.

• Which parts of soil contributes to foundation damping.

Abstract

To optimize offshore wind turbine (OWT) design, an engineering tool has been developed allowing for a detailed investigation of the effects of nonlinear soil stiffness and damping on foundation dynamics. We have studied the response of a vertically oscillating offshore wind monopile foundation in a realistic soil profile subjected to loads between 1 and 200 MN in the frequency range 0–10 Hz with pseudo-static and equivalent linear dynamic model. The non-linear soil behaviour is modelled with an equivalent linear method with shear modulus reduction and damping curves as input. The tool is verified and validated by comparison with elasto-dynamic model and experiments. With increasing load amplitudes foundation stiffness increases and damping decreases. For large load amplitudes the lower part of the pile foundation contributes more to foundation damping. The results indicate the nonlinear foundation stiffness and damping can be modelled rationally by combining stiffness and hysteretic damping from nonlinear static tools with apparent mass and radiation damping from elasto-dynamic analysis. The tool can be used to compute soil springs and dampers based on laboratory-based soil stiffness and damping.

Introduction

Dynamic response at resonance often governs the design loads for foundation and substructures of offshore structures, thus their stiffness and damping properties are important. Design optimization of offshore wind turbine (OWT) structures, life extension and load increase for existing structures often require detailed studies with due consideration to cyclic soil behaviour for accurate calculation of foundation stiffness and damping ([3,55]). This is especially important for OWT structures which are designed to have their lowest resonance frequencies in a very narrow frequency band.

Offshore foundation stiffness and damping can be computed with static finite element analysis with stress-strain and damping-strain curves based on a combination of static and cyclic laboratory tests. NGI has developed a software for computing hysteretic foundation damping by integrating hysteretic energy loss in the soil around the foundation ([21], [22], [27]). It has recently been applied to the NREL OWT to compute foundation damping for emergency shutdowns [7] which compare very well with measurements [10] and is also used for estimating foundation stiffness and damping for jack-up rigs [53]. However, the software does not account soil dynamic effects such as inertia and radiation damping, which are necessary for e.g. earthquake loading and vibrating machine foundations.

To account for soil and foundation dynamic effects in the design of machine foundations and earthquake resistant structures, empirical formulas for dynamic frequency dependent foundation stiffness and damping have been established ([14]) based on linear elasto-dynamic models e.g. Green's function approach (e.g Refs. [32,63]). A good overview of the early history of elasto-dynamic soil structure interaction is given by Kausel [28]. The effects of soil non-uniformity and reduced soil stiffness in the vicinity of foundations have been studied by several researchers since the 1980s with linear elasto-dynamic theory by including zones of lower stiffness near the foundation to approximately account for soil nonlinearity. Novak and Sheta [45] developed a model validated against experimental data [11], which was further developed and described by Han [18,20]. Angelides and Roesset [4] developed a tool for nonlinear foundation analysis based on an equivalent linear FE model with an absorbent boundary formulation [4] extending the work in [6]. Lysmer et al. also developed equivalent linear dynamic modelling tools [38,39]. More recent works are reported in Ref. [44,59]. For the application to pile and sheet pile installation, Holeyman et al. [25] extended the work of Michaelides et al. ([40,42]). In current performance based seismic design there is considerable research on how to account for various forms of soil and geometric nonlinearity, including gapping between the soil and the foundation, in the computation of the dynamic foundations stiffness and damping (see e.g. Refs. [1,9,15,26,45,48,49,52,56,57], and [63]).

Recently, an equivalent linear dynamic procedure has been developed [12] in the FE software COMSOL Multiphysics (COMSOL 2018) to analyse sheet pile vibratory installation. This frequency-domain-based steady state analysis tool can handle soil non-linearity around the foundation in an equivalent linear manner. The stiffness and damping in each element are determined from the maximum shear strain amplitude in a cycle. The new feature in COMSOL is tested and verified with the code PILES [32] for frequencies up to 10 Hz. For validation of the nonlinear capability we compare the results with field measurements of vibratory roller compaction [60] and pile vibration experiments ([18,45]).

Vertical loading of monopiles is not a critical issue in foundation design for storm loads. However the seismic response of OWTs should be considered in many seismic regions such as Taiwan, Japan, and U.S. West Coast. Particularly the vertical seismic response of an OWT can become large due to coincidence of vertical resonance frequencies of the OWT with earthquake loading frequencies. Thus the vertical foundation stiffness and especially the radiation damping is of importance [26], [65]. Therefore the effect of soil nonlinearity on dynamic stiffness and damping of a vertically vibrating monopile foundation has been studied further here. Moreover, the dynamic response of jacketed offshore wind turbines and offshore structures under horizontal environmental and earthquake loading is dominated by the vertical response of the piles. The findings of the present study can therefore be extended to these piles. As demonstrated in this paper, the results of the analyses can be presented as frequency- and load-dependent impedances at the pile head which can directly be included in the dynamic model of the turbine substructure for integrated dynamic analyses of the structure for optimized structural design.

Furthermore, recently vibratory installation of monopiles has been performed at several windfarms [8], [16], [66] and the developed numerical method can allow for better understanding of the important parameters affecting such vibratory installation. The developed tool is general and can be used for the lateral equivalent linear static and dynamic response of different types of foundations. We also compare foundation stiffness and damping with equivalent linear and the non-linear static approach for a non-linear hyperbolic soil model.

We give a brief description of the main features of frequency dependent stiffness and damping for monopiles in section 2 in the framework of linear elasto-dynamic modelling. The finite element model is described in section 3 and the input data are given in section 4. The results are presented in section 5, and finally conclusions and recommendations are given in section 6.