Analytical determination of the influence of geometric and material design parameters on the stress and strain fields in non-prismatic components of wind turbines

Abstract

Non-prismatic beamlike elements have long been used in engineering applications to optimize the performance of structures such as wind turbines, aircraft, and civil bridges, just to mention some examples. Unfortunately, engineering methods and formulas commonly used to analytically evaluate stresses and strains in prismatic beams do not hold and provide incorrect results for non-prismatic geometries. Large displacements and non-uniform material properties further complicate the analytical prediction. In order to determine the state of stress and strain in non-prismatic beamlike elements and its dependence on important design factors (such as taper parameters), a suitable mapping of the shape of such elements and a variational approach are used. The resulting field equations are exploited to derive application-oriented stress-strain formulas. Examples and comparisons with results of non-linear 3D-FEM analyses confirm the effectiveness of the modelling approach and of the new formulas proposed in this work.

Stress and strain fields in non-prismatic inhomogeneous beams

“…As is apparent, equation ( 12) extends the well-known Jourawski's solution [8]: the mean shear stress over a chord parallel to the cross-section width 2h2 consists of a first Jourawski-like term (proportional to the s-derivative of the bending curvature k2), plus additional terms proportional to the s-derivative of taper and material functions, which are needed to accurately predict the shear stresses in non-prismatic inhomogeneous cases, as is also confirmed by numerical analyses for the present beam shape (not reported here for brevity), as well as from the analyses performed on other beam shapes in recent works (e.g., [3,5,15,16]).…”

“…On the contrary, coefficients Zi account for the effects of taper, pre-twist, and material inhomogeneity, which are absent in prismatic homogeneous cases and are unpredictable via usual linear theories of prismatic homogenous beams. Specifically, the shear formula ( 8) is a generalization of results derivable from the classical Saint-Venant's theory (such as Jourawski's formula), as well as of formulas presented in recent works [5,15]. Moreover, it is worth noting that if the s-derivative of the material and taper functions vanish, together with the pre-twist function (i.e., in the prismatic homogenous case), the cross-sectional shear flow q will depend only on the s-derivative of the bending curvatures kα via the area moments Sα.…”

“…In the study of helicopter rotor blades, for instance, one needs to properly account for non-trivial couplings among bending, torsion, and traction associated with the cross-sectional pretwist [1][2][3]. In addition to this, further complexities are present in wind turbine blades, which are also tapered [4][5]. Not to mention the complexities associated with large displacements and complex material properties.…”

“…Literature works usually address tapered beams undergoing small displacements, assume Navier's formula for the cross-sectional normal stresses, and derive formulas for the corresponding shear stresses by the static equilibrium of a beam slice in its reference (undeformed) state. Migliaccio's works [5,15,16] do not rely on such assumptions, account for the effects of large displacements, and is the basis of the present study.…”

Analytical Solutions of Partial Differential Equations Modeling the Mechanical Behavior of Non-Prismatic Slender Continua

“…Non-prismatic slender elements are widely employed in engineering applications for their structural efficiency compared to prismatic ones, but their peculiar shape makes it complex to predict their state of stress and strain via analytical methods. Examples are components of aircraft, wind turbines, and civil structures [1][2][3][4][5]. The prototypical model for elements of this kind is that of the non-prismatic slender elastic continuum, i.e., a three-dimensional elastic body with one dimension, say, the longitudinal one, much larger than the other two, which are variable along the longitudinal dimension.…”