Cross-sectional Stiffness Matrix Research Articles

Cross-sectional analysis of initially curved composite beams with rectangular and circular sections using an asymptotic meshless method

An analytical meshless cross-sectional dimensional reduction method is developed to perform structural analysis of initially curved composite beams. By introducing the Pascal polynomials in the Cartesian coordinates system for beams with rectangular cross-sections and in the polar coordinates system for the beams with circular cross-sections to the warping field, a three-dimensional beam problem can be split into a two-dimensional cross-sectional analysis and a one-dimensional along the reference line of the beam. The obtained cross-sectional stiffness matrix based on higher-order of the strain energy can be incorporated into a one-dimensional finite element solution for initially curved composite beams, for the purpose of strain and stress analysis. Finally, the three-dimensional strain through the thickness of the beam can be achieved by the recovery analysis. The present method takes advantage of Pascal polynomials so that the solution procedure will be more simplified and computationally more efficient compared to three-dimensional finite element (3D FE) method. The proposed method for different isotropic and anisotropic beams is examined and compared with the literature, 3D FE, and Variational Asymptotic Beam Sectional (VABS) package which is a finite element based cross-sectional analysis tool for composite beams.

Development of an anisotropic co-rotational beam model including variable cross-section

The aim of this article is to expand the general approach of the flexible beam model to consider tapered geometry and anisotropic properties by updating the cross-sectional stiffness matrix. The advantage of this approach is that continuously variable cross-section, as well as irregular axes on the cross-sections, are considered simultaneously by the co-rotational method. Anisotropic and isotropic cantilevered beam cases are simulated. A static force or moment is applied as the external load. NREL 5 MW wind turbine blade is analyzed as a practical example. The results are compared against the existing literature and ABAQUS model, and they show excellent agreement.

Modal properties and stability of bend–twist coupled wind turbine blades

Abstract. Coupling between bending and twist has a significant influence on the aeroelastic response of wind turbine blades. The coupling can arise from the blade geometry (e.g. sweep, prebending, or deflection under load) or from the anisotropic properties of the blade material. Bend–twist coupling can be utilized to reduce the fatigue loads of wind turbine blades. In this study the effects of material-based coupling on the aeroelastic modal properties and stability limits of the DTU 10 MW Reference Wind Turbine are investigated. The modal properties are determined by means of eigenvalue analysis around a steady-state equilibrium using the aero-servo-elastic tool HAWCStab2 which has been extended by a beam element that allows for fully coupled cross-sectional properties. Bend–twist coupling is introduced in the cross-sectional stiffness matrix by means of coupling coefficients that introduce twist for flapwise (flap–twist coupling) or edgewise (edge–twist coupling) bending. Edge–twist coupling can increase or decrease the damping of the edgewise mode relative to the reference blade, depending on the operational condition of the turbine. Edge–twist to feather coupling for edgewise deflection towards the leading edge reduces the inflow speed at which the blade becomes unstable. Flap–twist to feather coupling for flapwise deflections towards the suction side increase the frequency and reduce damping of the flapwise mode. Flap–twist to stall reduces frequency and increases damping. The reduction of blade root flapwise and tower bottom fore–aft moments due to variations in mean wind speed of a flap–twist to feather blade are confirmed by frequency response functions.

Asymptotic Approach to Oblique Cross-Sectional Analysis of Beams

Structural and aeroelastic analyses using beam theories by default choose a cross section that is perpendicular to the reference line. In several cases, such as swept wings with high AR, a beam theory that allows for the choice of a cross section that is oblique to the reference line may be more convenient. This work uses the variational asymptotic method (VAM) to develop such a beam theory. The problems addressed are the planar deformation of a strip and the full 3D deformation of a solid, prismatic, right-circular cylinder, both made of homogeneous, isotropic material. The motivation for choosing these problems is primarily the existence of 3D elasticity solutions, which comprise a complete validation set for all possible deformations and which are shown to be accurately captured by the current analysis. A secondary motivation was that the development and final results of the beam theory, i.e., the cross-sectional stiffness matrix and stress-strain-displacement recovery relations, are obtainable as closed-form analytical expressions. These results, coupled with the VAM-based beam analysis being devoid of ad hoc assumptions, culminate in what is expected to be of significance when formulating a general oblique cross-sectional analysis for beams with anisotropic material and initial curvature/twist, the detailed treatment of which will be alluded to in a later paper.

A consistent total Lagrangian finite element for composite closed section thin walled beams

This work presents a consistent geometrically exact finite element formulation of the thin-walled anisotropic beam theory. The present formulation is thought to address problems of composite beams with nonlinear behavior. The constitutive formulation is based on the relations of composite laminates and thus the cross sectional stiffness matrix is obtained analytically. The variational formulation is written in terms of generalized strains, which are parametrized with the director field and its derivatives. The generalized strains and generalized beam forces are obtained by introducing a transformation that maps generalized components into physical components. A consistent tangent stiffness matrix is obtained by parametrizing the finite rotations with the total rotation vector; its derivation is greatly simplified by obtention of the derivatives of the director field via interpolation of nodal triads. Several numerical examples are presented to show the accuracy of the formulation and also its frame invariance and path independence.

Two finite elements for general composite beams with piezoelectric actuators and sensors

Two finite elements for the static analysis of smart beams with piezoelectric sensors/actuators are presented: ad hoc smart beam element (ADSBE) and variational asymptotic smart beam element (VASBE). Both elements rely on the computation of the cross-sectional matrices associated with the electromechanical properties of the beam cross-section. ADSBE uses the Timoshenko cross-sectional stiffness matrix computed by the VABS program, and the electric field is assumed constant across the thickness of each piezoelectric layer. Taking advantage of the cross-section discretization of the beam, all the matrices related to the electric field are also computed by performing a numerical integration using the VABS program that was extended to account for these new quantities. VASBE is based on the fully coupled Timoshenko theory for smart beams constructed using the variational asymptotic method. This theory decouples the original three-dimensional electromechanical problem to a two-dimensional electromechanical cross-sectional analysis and a one-dimensional beam analysis. The cross-sectional analysis provides a one-dimensional constitutive model for the beam analysis without a priori assumptions regarding the geometry of the cross-section, the electric field distribution, and the location of smart materials. Several examples available in the literature are used to validate the accuracy of these two new elements. The numerical results obtained using ADSBE and VASBE correlated well with other published results. For structures that are out of the limits in which one structure may be modeled as a beam, the ADSBE showed considerable errors and, therefore, should not be used. Nevertheless, VASBE was able to predict the 3D results available in the literature with an error smaller than 8%.

Aeroelastic Response of Composite Helicopter Rotor with Random Material Properties

This study investigates the effect of uncertainty in composite material properties on the cross-sectional stiffness properties, natural frequencies, and aeroelastic responses of a composite helicopter rotor blade. The elastic moduli and Poisson’s ratio of the composite material are considered as random variables with a coefficient of variation of around 4%, which was taken from published experimental work. An analytical box beam model is used for evaluating blade cross-sectional properties. Aeroelastic analysis based on finite elements in space and time is used to evaluate the helicopter rotor blade response in forward flight. The stochastic cross-sectional and aeroelastic analyses are carried out with Monte Carlo simulations. It is found that the blade cross-sectional stiffness matrix elements show a coefficient of variation of about 6%. The nonrotating rotor blade natural frequencies show a coefficient of variation of around 3%. The impact of material uncertainty on rotating natural frequencies varies from that on nonrotating blade frequencies because of centrifugal stiffening. The propagation of material uncertainty into aeroelastic response causes large deviations, particularly in the higher-harmonic components that are critical for the accurate prediction of helicopter blade loads and vibration. The numerical results clearly show the need to consider randomness of composite material properties in the helicopter aeroelastic analysis.

Obliqueness effects in asymptotic cross-sectional analysis of composite beams

Cross-sectional stiffness properties for general composite beams, along with the interior solution for warping, are derived in such a way as to provide the extra freedom to choose a cross-sectional plane that is not perpendicular to the local beam reference line. The present development treats prismatic as well as initially twisted and curved beams. The main purpose of allowing such a choice of coordinate systems is for the convenience of the analyst; one, of course, should not expect the final 3D results to change. The 3D strain field is derived for the corresponding nonorthogonal curvilinear set of coordinates which describe the undeformed and the deformed geometry of the beam. The analysis is carried out based on the variational-asymptotic method which is used to determine the warping field. For the development of the solution, it is shown that there is a fundamental difference between the solutions for a prismatic beam and a beam with initial twist and curvature, which in turn produces a limitation of the angle of obliqueness: the effect of obliqueness is treated exactly for a prismatic beam, while for the initially twisted and curved beam the obliqueness is regarded as a small parameter. The ultimate goal of the analysis is the determination of the cross-sectional stiffness matrix, which can then be used as input for the 1D problem. Then, using the recovering relations, one can recover the strain field over the entire cross section, once the previous problem has been solved. The validity of the method is demonstrated by recovery of accurate cross-sectional stiffnesses associated with a normal cross-sectional plane from those associated with an oblique cross-sectional plane, using a simple rotation-of-axes transformation.

Theory of Anisotropic Thin-Walled Beams

Asymptotically correct, linear theory is presented for thin-walled prismatic beams made of generally anisotropic materials. Consistent use of small parameters that are intrinsic to the problem permits a natural description of all thin-walled beams within a common framework, regardless of whether cross-sectional geometry is open, closed, or strip-like. Four “classical” one-dimensional variables associated with extension, twist, and bending in two orthogonal directions are employed. Analytical formulas are obtained for the resulting 4×4 cross-sectional stiffness matrix (which, in general, is fully populated and includes all elastic couplings) as well as for the strain field. Prior to this work no analytical theories for beams with closed cross sections were able to consistently include shell bending strain measures. Corrections stemming from those measures are shown to be important for certain cases. Contrary to widespread belief, it is demonstrated that for such “classical” theories, a cross section is not rigid in its own plane. Vlasov’s correction is shown to be unimportant for closed sections, while for open cross sections asymptotically correct formulas for this effect are provided. The latter result is an extension to a general contour of a result for I-beams previously published by the authors. [S0021-8936(00)03003-8]