The influence of an initial twisting on tapered beams undergoing large displacements

Abstract

The behaviour of pre-twisted and tapered beams (such as turbine or helicopter blades) is characterized by stress distributions that may be quite different from those of the usual beam theory, yielding couplings among bending, twisting and traction. We propose a physical-mathematical model for tapered beams that accounts for the effects of the pre-twist of the cross-sections along the centreline. The beam centre-line may undergo large displacements, while its cross-sections see small warping both in-and out of their plane. Supposing infinitesimal strain, a variational approach provides the field equations, which are perturbed in terms of a small geometric ratio and shall be solved numerically in general. However, analytical closed-form solutions exist in some cases, such as for isotropic beams with pre-twisted, bi-tapered elliptic cross-sections; they are presented and compared with the results of nonlinear 3D-FEM simulations.

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

“…Simulation campaigns based on 1D-FEM approaches (i.e. based on standard beam elements) could be run to identify the effects of some important design parameters on the mechanical behavior of the aforementioned structures, but such approaches cannot identify a number of effects on the stress and strain fields that depend on taper [5][6]. 3D-FEM approaches (based on solid elements) can be used instead and can be accurate, but they would be computationally more demanding and, in any case, just like any other numerical approach, they would not generally provide the same insight in the physical problem as analytical formulas.…”

“…3D-FEM approaches (based on solid elements) can be used instead and can be accurate, but they would be computationally more demanding and, in any case, just like any other numerical approach, they would not generally provide the same insight in the physical problem as analytical formulas. Unfortunately, formulas derivable from the Saint-Venant's linear theory of prismatic beams (see, e.g., [7][8][9][10]) do not hold in non-prismatic cases and may result in significant errors [6,11,12]. A model and new formulas, which are accurate, efficient and tailored to non-prismatic elements as those mentioned in the foregoing, are proposed in this work.…”

Efficient and Accurate Modeling of Non-Prismatic Beamlike Structures

“…It is worth noting that similar but generally more complex expressions for E 11 , E 21 , E 31 can be obtained for other non-prismatic beams as well (see, e.g., [6,21]). However, we do not discuss here all cases in which we can obtain closed-form solutions to problem ( 16)-( 17), but we present another interesting example (in section 3.2) and then proceed to compare the results of our model with those of nonlinear 3D-FEM simulations to verify its effectiveness in terms of computational efficiency and accuracy (section 4).…”

“…Other examples can also be found in previous works (e.g. [6,8,21]), which address other geometries and load cases.…”

“…At the same time, tackling complex engineering problems continuously leads to develop theories to meet ever more stringent requirements on the predictive models. For example, several theories have been developed over the years for helicopter blades with pre-twist, a geometric feature which induces several couplings in their mechanical response: for pre-twisted beams bending is always three-dimensional, and tension may be significantly coupled to torsion [1][2][3][4][5][6]. Wind turbine blades are another interesting example of pre-twisted structures.…”