Introduction to Flexible Components
In Bladed, the support structure is modelled as a single linear flexible component (body). The blade is also simulated as a flexible component, with the option to be subdivided into multiple linear flexible components in order to accurately model large deflections.
The fundamental finite element model assumes that the flexible components are linear space beams or more general space frames that comprise assemblies of multiple members modelled by Timoshenko beam elements. As the model is linear, deflections are assumed to be small within each body.
For the blades, the finite element model degrees of freedom can be used directly as the generalised freedoms. For the blades and support structure, modal reduction can be performed to reduce the number of generalised freedoms. Using the modal approach, the deformation is represented by a linear combination of some pre-calculated mode shapes. The scalars of this linear combination are the modal amplitudes Clough and Penzien, 1993, that represent the generalised strains and hence the degrees of freedom of the component. It is important to note that the mode shape functions are constant in time.
The applied beam element may be considered as an extension to the standard three-dimensional engineering Timoshenko beam element Przemieniecki, 1968 with two nodes or stations located at the two ends. This element has twelve fundamental degrees of freedom, i.e. three translational and three rotational degrees of freedom at both stations. The deflection at all intermediate points is calculated via interpolation functions that are derived from a set of homogenous equilibrium equations valid for prismatic beam elements. It is important to note that this beam element includes the effect of shear deformation that may be important for some support structures, in particular complicated offshore foundations. The magnitude of the shear deformations relative to bending deformations for an element may be evaluated by the element shear parameter conveniently defined as
where
\(\bscalar{E}\bscalar{I}^e\) is the bending stiffness,
\(\bscalar{G}\bscalar{A}^e\) is the corresponding shear stiffness,
\(\bscalar{l}^e\) is the element length.
The beam element supports an arbitrary spatial variation of the stiffness along the beam element, but in the present implementation it is assumed that the bending, torsional, axial and shear stiffness vary linearly. The orientation of the element is defined by the position of the two ends as well as the orientation of the principal axis around the neutral axis (elastic centre). The effect of possible coupling between bending and torsion is included in terms of the position of the shear centre (torsion centre), and a transformation between displacements and forces relating to the shear centre and the neutral axis is included. The resulting stiffness matrix is constant and calculated by numerical integration. For prismatic elements, where the shear centre is located at the neutral axis, the stiffness matrix is identical to the standard engineering Timoshenko beam element Przemieniecki, 1968.
An important feature of the derived method is that some fundamental degrees of freedom may be constrained, which is particularly useful in cases where the effect of elongation and/or torsion can be neglected. In order to enable the description of rigid connection the deflection of a beam element may also be constrained completely. The constraints are modelled in terms of a constant constraint matrix together with Lagrange’s method Cook, Malkus and Plesha, 1989.
Second-order effects of the internal axial forces are included in terms of a geometric stiffness matrix (stress stiffening) that is calculated from the derivatives of the interpolation functions Clough and Penzien, 1993. For the blades the dominating part of the axial force is caused by centrifugal forces for which reason the term centrifugal stiffness is traditionally used in this case. A similar effect occasionally referred to as geometric destiffening can be observed in the support structure due to gravity loading.
Further second-order effects of the internal shear forces are accounted for by applying extra external loads based on a method by Krenk, 2009. This model can particularly enhance the prediction of torsional deflection in blades with a torsional degree of freedom.
Inertia forces acting on the element and the proximal node are derived as described in the multibody dynamics approach from the fundamental displacement hypothesis using the principle of virtual work. An important feature of the derived method is that the inertia forces are expressed directly in terms of the modal amplitudes, i.e. the strains and corresponding derivatives as originally suggested in Shabana, 1998. In principle this means that the time for calculating the accelerations scales with the number of mode shapes rather than the number of beam elements of the flexible component.