Continuous ongoing efforts to better predict the mechanical behaviour of complex beamlike structures, such as wind turbine blades, are motivated by the need to improve their performance and reduce the costs. However, new design approaches and the increasing flexibility of such structures make their aeroelastic modelling ever more challenging. For the structural part of this modelling, the best compromise between computational efficiency and accuracy can be obtained via schematizations based on suitable beamlike elements. This paper addresses the modelling of the mechanical behaviour of beamlike structures which are curved, twisted and tapered in their unstressed state and undergo large displacements, in- and out-of-plane cross-sectional warping, and small strains. A suitable model for the problem at hand is proposed. Analytical and numerical results obtained by its application are presented and compared with results from 3D FEM analyses.

New methods are continuously being sought to improve the performance and efficiency of horizontal-axis wind turbines (HAWTs). Specifically, such improvements aim to increase their energy capture capacity, develop more reliable structures and lower overall energy costs (Wiser et al., 2016). Such goals can be achieved through the use of advanced materials, the optimization of the aerodynamic and structural behaviour of the blades, and the exploitation of load control techniques (see, for example, Ashwill et al., 2010; Bottasso, 2012; Stäblein, 2017). However, new design strategies and the increasing flexibility of those structures make modelling their aeroelastic behaviour ever more challenging. For the structural part of this modelling, schematizing the blades through suitable beamlike elements may represent the best compromise between computational efficiency and accuracy. Modern blades are however very complex beamlike structures. They may be curved, twisted and also tapered in their unstressed state. Even ignoring the complexities related to the materials and loading conditions, their shape alone is sufficient to make mathematical description of their mechanical behaviour a very challenging task. This work addresses the modelling of the mechanical behaviour of structures of this kind, with a particular focus on their main geometrical characteristics, such as the twist and taper of the transversal cross sections, as well as the in- and out-of-plane cross-sectional warping and the large deflections of their reference centre line.

Over the years several theories have been developed for beamlike structures (see, for example, Love, 1944; Antman, 1966; Rubin, 2000) for applications in different fields, from helicopter rotor blades in aerospace engineering to bridge components in civil engineering and surgical tools in medicine. Nevertheless, due to the continuous need for ever more rigorous and application-oriented models, interest in advanced theories for complex beamlike structures has led to even further research in recent years. The focus of this paper is on the effects of important geometrical characteristics of those structures, such as the curvature of their centre line as well as the twist and the taper of their cross sections. After an introduction to modelling approaches for structures of this kind (Sect. 2), a suitable model is proposed for the problem at hand (Sect. 3). Finally, analytical results and numerical examples obtained by applying the proposed modelling approach to reference beamlike structures are presented and compared with results from 3D FEM analyses (Sects. 4 and 5).

Modelling the mechanical behaviour of modern blades can be performed via different approaches. See, for example, the reviews on aeroelastic modelling approaches for wind turbine blades of Hansen et al. (2006) and L. Wang et al. (2016), which discuss and compare aerodynamic and structural models used in research and industrial applications. For the structural modelling, two main choices are based on 3D FEM and beam models. In general, 3D FEM approaches can be very accurate and flexible, but they can be computationally demanding for analyses of complex systems, especially if they are coupled with CFD methods for aerodynamic analyses. The overall computational cost can be reduced using faster aerodynamic models, such as those based on the blade element momentum theory (see, for example, Hansen et al., 2006). However, this may not yet be sufficient in the case of multi-objective optimization tasks, in which the optimization of several aspects (e.g. aerodynamic performance, structural characteristics and control systems) has to be addressed at the same time (see also Bottasso et al., 2012). Therefore, faster structural models may be needed as well, such as suitable beam models, which may provide accurate information on the deflection of the structure's centre line as well as the strain and stress fields in the three-dimensional solid. The use of fast aerodynamic models along with suitable beam models may then represent the best compromise between computational efficiency and accuracy. In this work, the focus is only on the structural modelling. In particular, a mathematical model is proposed to simulate the behaviour of non-prismatic beamlike structures, which may be curved, twisted and tapered in their unstressed reference state and undergo large deflections, in- and out-of-plane cross-sectional warping, and small strains (such a model is referred to here as a beamlike model or BLM).

Over the years many approaches have been developed for beamlike structures, from classical beam models (Love, 1944) for extension, twisting and bending to the formulation of Reissner (1981), which also accounts for transverse shear deformation, to geometrically exact and asymptotic approaches, involving the research efforts of many investigators (such as Antman, 1966; Giavotto et al., 1983; Simo, 1985; Ibrahimbegovic, 1995; Ruta et al., 2006; Pai, 2011; Yu et al., 2012; Hodges, 2018). The available theories may be broadly grouped into engineering theories and mathematical ones. The former are usually based on ad hoc corrections to simpler theories (e.g. Rosen and Friemann, 1978) or exploit geometrically exact approaches (such as Q. Wang et al., 2016); the latter are generally based on the directed continuum (see, for example, Rubin, 2000) or exploit asymptotic methods (e.g. Yu et al., 2012). Reviews on beam theories, which summarize modelling approaches and complicating effects, are also available in the literature. For example, many theories have been developed for helicopter rotor blades with an initial twist (Hodges, 1990). In this regard, a wide-ranging review on pre-twisted rods is by Rosen (1991), which covers several aspects of the problem, from the response to static loads to dynamics and stability issues. Kunz (1994) also provided an overview on modelling methods for rotating beams, discussing how engineering theories for rotor blades have evolved over the years, from the recognition of the importance of bending flexibility to the development of linear equations for bending and torsion to the introduction of non-linear terms to such equations. More recently, Rafiee et al. (2017) reviewed the vibrations control issues in rotating beams, summarizing beam theories and complicating effects, such as non-uniform cross sections, initial curvature, twist and sweep. In general, it seems that, unlike for the case of pre-twisted rods, the results published for curved rotating beams with initial taper and sweep are quite scarce, although all these geometrical characteristics may play an important role. This is particularly true for modern wind turbine blades, which are ever more flexible and longer than in the past, pre-bent and swept, and, in addition, characterized by significant chord and twist variations.

To date many research efforts have been devoted to developing powerful theories for beamlike structures. However, complex non-prismatic cases still require further investigation. In general, the geometry of the reference and current states of the structure must be appropriately described, as the curvature, twist and taper are important geometrical design features and should be explicitly included in the model. Moreover, the analysis should not be restricted to small displacements. The model should provide the stress and strain fields in the three-dimensional solid, be rigorous and application-oriented, and provide classical results when applied to prismatic cases. Following these guidelines, a mathematical model to simulate the mechanical behaviour of the mentioned non-prismatic beamlike structures is proposed hereafter.

Here we are concerned with developing a mathematical model to describe the mechanical behaviour of beamlike structures which are curved, twisted and tapered in their reference state and undergo large displacements. One of the main issues with such a task is how to describe the motion of the structure. See, among others, the works of Simo (1985), Ruta et al. (2006) and Pai (2014) for some examples of different approaches. Here, we consider a non-prismatic beamlike structure as a collection of deformable plane figures (i.e. the reference cross sections) along a suitable three-dimensional curve (i.e. the reference centre line). We assume that each point of each cross section in the reference state moves to its position in the current state through a global rigid motion on which a local general (warping) motion is superimposed. In this manner, the cross-sectional deformation can be examined independently of the global motion of the centre line. It is thus possible to consider the global motion to be large, while the local motion and the strain may be small. An analytical description of how the motion of the considered structure is modelled in this work is presented and further discussed in the following section.

We begin by introducing two local triads of orthogonal unit vectors. The
first is the local triad,

Schematic of the reference and current states, centre lines, cross sections, and local frames.

We continue by introducing the kinematic variables we use to describe the
motion of the considered structure. To this end, the orientation of frames

In a similar manner, we introduce the skew tensor field

Function

Now, we start modelling the motion of the cross section's points. In
particular, we introduce two mapping functions,

It is worth noting that

Example curved, twisted and tapered beamlike structure and local
frame

The positions of the points in the current state are defined in a similar
manner by means of the (current) mapping function:

By using maps (Eqs. 9 and 12), we can determine the 3D tensor,

Given the strain tensor,

We are now in a position to define the cross-sectional stress resultants,
namely the force

To complete the formulation, we conclude with considerations on the principle of expended power and the balance equations for the considered structure. For hyper-elastic bodies (Gurtin, 1981), we write the principle of expended power in the form

For small warpings, small strains and small local rotations, if the power
expended by surface and body loads on the warping velocities is neglected,
the external power,

The cross-sectional warpings may be important in calculating the 3D energy
function and cannot be neglected in the internal power,

In general, a 3D non-linear elasticity problem can be formulated as a
variational problem. However, if we try to solve the variational problem
directly, the difficulties encountered in solving the elasticity problem
remain. For beamlike structures whose transversal dimensions are much
smaller than the longitudinal one, assumptions based on the shape of the
structure and the smallness of the warping and strain fields can lead to
useful simplifications. In particular, solving the 3D non-linear elasticity
problem can be reduced to solutions of two main problems. See, for example,
Berdichevsky (1981), who seems to be the first in the literature to plainly
state this for elastic rods. One of the two problems governs the local
distortion of the cross sections and is referred to here as the cross-section problem. The other governs the global deformation of the
centre line and is referred to here as the centre-line problem. Hereafter,
we consider the following variational statement to determine the warping
fields which are responsible for deformation of the cross sections:

Note that to determine the current state of the structure, we also need the displacements of its centre-line points. They can be determined by solving the centre-line problem, which is a non-linear problem governed by the set of kinematic, constitutive and balance equations introduced in Sect. 3 (in particular, we are referring to the constitutive model in Sect. 3.2, which relates stress resultants and strain measures, and the balance equations for the stress resultants in Sect. 3.3).

In the next sections we show some analytical solutions (Sect. 4) and numerical results (Sect. 5) that can be obtained by applying the proposed modelling approach to some reference beamlike structures.

In this section we consider the case of a beamlike structure with bi-tapered elliptical cross sections. For this case we can obtain analytical solutions in terms of warping fields, while for generic shapes (e.g. the aerodynamic profiles used in wind and steam turbine blades as well as helicopter rotor blades) the problem corresponding to Eq. (29) can be solved using numerical methods. However, this is not surprising, since analytical solutions are available only for a limited number of cases even in the classical linear theory of prismatic beams (see, for example, Love, 1944).

As discussed in Sect. 3, we are assuming that the warpings, strains and local rotations are small. Moreover, hereafter we choose that the current local frames be tangent to the current centre line and include possible shear deformations within the warping fields. In addition, with the aim of showing a first analytical solution for bi-tapered cross sections, here we neglect the effects of the initial cross-sectional twist. Then, we write the
Euler–Lagrange equations corresponding to Eq. (29), whose unknown functions are the warping fields,

In this section we present the results of simulations conducted using the modelling approach discussed in Sect. 3, which we have implemented into a numerical code in MATLAB language. The results are also compared with those obtainable via 3D FEM simulations with the commercial software ANSYS.

Global deformation with 3D BLM for increasing

In particular, we show a first set of test cases in which a beamlike structure with rectangular cross sections undergoes large displacements while fixed at one end and loaded at the other by a force of progressively increasing magnitude. The second set of test cases addresses a more complex geometry, that is, a beamlike structure with elliptical cross sections, which is curved, twisted and tapered in its reference configuration and under the same loading condition as in the first set of test cases. Finally, the third set of test cases regards four different beamlike structures under the same loading conditions. In particular, we consider a first prismatic structure with elliptical cross sections. The second structure is a modification of the first, on which the same cross section is maintained at 18 m from the root, while taper is added according to the taper coefficients in Fig. 2. Starting with this latter structure, we then consider a third structure which includes twisting of the cross sections, assuming the twist law in Fig. 2. The fourth and final case is a curved, twisted and tapered structure obtained from the third (tapered and twisted) by adding a centre-line curvature. Once the simulations have been completed, we compare the results obtained to highlight the effects of their different geometries on their mechanical behaviour.

In all cases, the displacements of the reference centre-line points are
calculated by solving the centre-line non-linear problem through the
previously mentioned numerical procedure we have implemented in MATLAB,
which is based on the kinematic, constitutive and balance equations
introduced in Sect. 3. In particular, the constitutive model introduced in
Sect. 3.2 is used to relate stress resultants and strain measures. We
define the local frames orientation using Euler angles and simulate
orientation changes in terms of the derivatives of those angles over the
arch length,

In this set of test cases we consider a rectangular cross-sectioned beamlike structure which undergoes large displacements while clamped at one end (i.e. the root) and loaded at the other (i.e. the tip) by a force,

The simulations are run for different values of the tip force. The model we have implemented in MATLAB for solving the non-linear problem renders results on the structure's deformed configuration (e.g. Fig. 3a) within a few seconds. In all cases, the simulation time is less than 2.4 s, which is significantly less than that required for the corresponding non-linear 3D FEM simulations carried out on the same computer, while the accuracy of the results is almost the same. A summary of the results obtained, in terms of global displacements and simulation times, is shown in Figs. 3 and 4.

Comparison of tip displacements

In particular, Fig. 3b shows the undeformed shape (for

Global deformation with 3D BLM for increasing

Let us now consider a more complex beamlike structure, specifically, one
with a 90 m curved centre line with constant curvatures, which schematizes a
pre-bent and swept beam whose tip is moved 4 m edgewise and 3 m flapwise, as in Fig. 2. The local frames in the reference state are characterized by a
pre-twist of 20

The simulations are run for different values of tip force,

Comparison of tip displacements

Displacement of the reference centre-line points with 3D BLM for
increasing

Apart from the foregoing results, the model is also able to provide other
meaningful information. In particular, we can obtain the displacement fields of the reference centre-line points (Fig. 7), as well as the change in
curvature of the beamlike structure (Fig. 8a–c) and the corresponding moment stress resultant (Fig. 8d–f). The moment components are in the current local frame,

Changes in curvature

Local frame orientations in terms of Euler angles before (green lines) and after deformation.

The last test cases regard four different beamlike structures, starting
with a prismatic elliptical one, to which there is the step-by-step addition
of the taper; the twist of the cross sections; and, finally, the curvature of
the centre line, as discussed in Sect. 5. Note that the curved–twisted–tapered case considered here coincides with that
discussed in more detail in Sect. 5.2 (see Figs. 5–9,

Prismatic case before and after deformation

Centre-line points displacement of prismatic case (blue) and
non-prismatic cases (other colours) for

Tip displacements with 3D BLM (blue) and 3D FEM (red) for different geometries and increasing

Similar results have also been obtained for larger values of tip force,

We conclude now by examining the results for the 3D strain measure

Max longitudinal strain

As verified by many simulations and shown in the examples, the proposed approach performs well in terms of computational efficiency and accuracy. It can be used to study the mechanical behaviour of beamlike structures, which are curved, twisted and tapered in their unstressed reference state and undergo large global displacements. It can moreover provide information on the deformed configurations of the structures, such as their displacement fields, as well as the corresponding strain and stress measures. It is worth noting that it is suitable for beamlike structures with generic reference cross-sections shapes. However, as already pointed out, for bi-tapered elliptical cross sections, analytical solutions can be obtained in terms of warping fields, while for generic reference cross-section shapes problem, Eq. (29) has to be solved using numerical methods.

Many complex engineering structures, such as the rotor blades of wind turbines and helicopters, are non-prismatic beamlike structures, with one
dimension much larger than the other two and a shape that is curved, twisted
and also tapered in the unstressed reference state. The increasing size and
flexibility of such structures make the prediction of their aeroelastic
behaviour ever more challenging. This paper addressed the structural part of
this modelling and proposed a modelling approach, referred to as 3D BLM,
which is computationally efficient, accurate and explicitly considers the
main geometrical characteristics of the mentioned structures, the large
deflections of their reference centre line, and the in- and out-of-plane
warping of their transversal cross sections. In the mentioned approach, the
warping displacements have been thought of as an additional small motion
superimposed on the global motion of the local frames. The strain tensor has
been calculated analytically in terms of geometrical parameters of the
structure, 1D strain measures and 3D warping fields. A method based on a
variational statement has been used to obtain suitable warping fields. The
proposed approach enables one to obtain analytical results in particular cases
and can be implemented into an efficient numerical code in the general case.
The analytical results obtained, along with numerical examples (obtained by
implementing 3D BLM into a computer code) and comparisons with corresponding
results from 3D FEM simulations, have been presented to show the
effectiveness of the modelling approach and the information it can provide.
In all cases, the simulation times with 3D BLM have been significantly
shorter than those required by 3D FEM simulations, while the accuracy of the
results has been always almost the same. In this paper the analyses have
been limited to the terms of order

All necessary research data have been included in the paper. For further information please contact the authors.

MG was responsible for the conceptualization and methodology and RG, BS and BR for supervision and the methodology.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Wind Energy Science Conference 2019”. It is a result of the Wind Energy Science Conference 2019, Cork, Ireland, 17–20 June 2019.

This paper was edited by Katherine Dykes and reviewed by three anonymous referees.