Variational Asymptotic Beam Sectional Analysis Research Articles

Prediction of sectional collapse of thin-walled structure under pure bending by nonlinear composite beam theory

Brazier (1927) found that when one dimension of the beam cross-section was relatively smaller than the others, large in-plane displacements over the cross-section might occur, even though the strains could remain very small. Under this circumstance, the so-called Brazier effect refers to the cross-sectional ovalization, which leads to nonlinear bending buckling and collapses. This paper studies the Brazier effect by the nonlinear Variational Asymptotic Beam Sectional Analysis (VABS) theory which considers finite cross-sectional deformations. Nonlinear VABS reduces three-dimensional (3D) continuum to a one-dimensional (1D) beam analysis and a two-dimensional (2D) cross-sectional analysis featuring both geometric and material nonlinearities without unnecessary kinematic assumptions. The present theory is implemented using the finite element method (FEM) in the VABS code, a general-purpose beam cross-sectional analysis tool. An iterative method is applied to solve the finite warping field for the classical-type model using the Euler–Bernoulli beam theory. The deformation gradient tensor is directly used to deal with finite deformation, various strain definitions, and several types of material laws. Numerical examples demonstrate the capabilities of VABS to predict the sectional collapse of thin-walled structures under pure bending.

Geometrically exact aeroelastic stability analysis of helicopter composite rotor blades in forward flight

In this paper, a sophisticated method for analyzing the geometrically exact aeroelastic stability of composite rotor blades in forward flight has been proposed. The unique advantage of the proposed method is to use the geometrically exact beam model with the updated Variational Asymptotic Beam Sectional analysis (VABS) for structural modeling to not only calculate the blade cross-sectional properties and blade spanwise deformations more accurately, but also take into account the effect of transverse shear deformation and blade initial twist and curvatures sophisticatedly. The Peters finite state airloads theory and the Peters-He finite state dynamic inflow model are coupled to calculate the three-dimensional unsteady airloads applied on the blade. The auto-pilot trim scheme is adopted to determine the blade pitch controls. The finite element spatial discretization method, the Newmark numerical integration method, the Newton-Raphson method and the moving-block analysis method are combined to solve the established geometrically exact aeroelastic equations and obtain the geometrically exact aeroelastic stability of composite rotor blades. The proposed method is validated by correlation with existing analytical results. The investigations show that the effect of transverse shear deformation on the aeroelastic stability of composite hingeless rotors in forward flight is non-negligible. Blade initial curvatures have significant effect on the aeroelastic stability of composite hingeless rotors in forward flight, which is not only related to the blade elastic couplings, but also to the advance ratio.

Geometrically Exact Beam-based Aeroelastic Modeling and Solution Of Composite Rotor Blades in Forward Flight

In this paper, the geometrically exact beam model and aeroelastic solution methods for composite rotor blades in forward flight by the latest variational asymptotic beam sectional analysis (VABS) have been employed. The geometrically exact beam equations of motion in the mixed variational form and the latest VABS are used to deal with one-dimensional blade analysis and the structural property of blade cross section, respectively. The methods can be used for the aeroelastic solution of composite rotor blades with arbitrary cross-sectional shape and material distribution, large deflections and significant nonclassical effects such as cross-sectional warping, transverse shear deformation, and elastic couplings caused by anisotropic material properties. The Peters–He finite state dynamic inflow model and the Peters finite state airloads theory are used to calculate the induced velocity and blade airloads, respectively. An auto-pilot trim scheme is used for calculating the blade pitch controls to meet the trim requirements. The convergence issue encountered when solving the geometrically exact, mixed variational aeroelastic equations in time domain has been successfully addressed. The values of the empirical parameters in the auto-pilot trim scheme for the presented aeroelastic model have been properly selected. The accuracy of the presented aeroelastic modeling and solution methods has been verified against the SA349/2 flight-test data. The influence of transverse shear deformation on the aeroelastic response of composite rotor blades was also investigated, indicating that this effect has a nonnegligible influence on the aeroelastic response of the five different kinds of elastically coupled hingeless composite rotors investigated in this paper.

Aeroelasticity of Flying-Wing Aircraft Subject to Morphing: A Stability Study

In this work, we present a study on the effects of modifications on wing geometric configurations via morphing. Specifically, we investigate the resulting impacts on aeroelastic stability. In the primary analysis presented, we assess a wall-mounted configuration for an airfoil section of a two-dimensional wing. This morphing-wing system comprises a torsional spring at the trailing-edge flap hinge, which induces the morphing phenomenon. The behavior of this system is studied within the context of divergence and reversal. The analysis following this examines a highly flexible flying-wing configuration, subject to a transient morphing scheme. The geometries corresponding to the unmorphed and morphed configurations are acquired from a physical morphing trailing-edge wing prototype, and the structural modeling and cross-sectional properties are obtained using the Gmsh tool and variational asymptotic beam sectional analysis. Furthermore, we evaluate the effects of the geometric variations on the aeroelastic stability through leveraging the computer program, NATASHA (which stands for nonlinear aeroelastic trim and stability analysis of high-altitude long-endurance aircraft). The results obtained from this analysis insinuate the favorable impact on the aeroelastic stability of the system, namely, in terms of flutter characteristics.

Evolutionary Aeroelastic Design of Flying-Wing Cross Section

The uniform distribution of stresses in flying-wing aircraft improves the aeroelastic flight envelope. In this paper, we document the effect of wing cross-section configuration on the stress distribution and flutter characteristics of a flying-wing aircraft. We determined the flow of stresses through the wing structure, and changed the structure to avoid stress strangulations. The emerging structure is more stable. We used the computer programs Gmsh, Variational Asymptotic Beam Sectional Analysis, and Nonlinear Aeroelastic Trim and Stability of High Altitude Long Endurance Aircraft. The wing structure was evolved by holding fixed the flight condition, mass per unit length, and material type. The results indicate that particular configurations of wing cross sections favor a uniform stress distribution, and therefore aeroelastic stability. The configuration with higher flutter speed is associated with the smoother flow of stresses through the wing structure.

Stress analysis of elliptical composite tubular beams using a meshless dimensional reduction method

In the present paper, a meshless dimensional reduction method is presented for the efficient stress analysis of composite tubular beams with elliptical cross-section. A polynomial based method in polar coordinates in conjunction with an asymptotic variational method were used to carry out cross-sectional analysis. The two-dimensional cross-section strain energy is perturbed by perturbing the warping field to obtain unknown warping of the cross-section. The transverse shear effect is taken into consideration using transformation to Timoshenko-Like beam model. Since the present approach does not use finite element mesh in two-dimensional analysis of the cross-section (such as that used in the software Variational Asymptotic Beam Sectional Analysis (VABS)), it is termed meshless method. The one-dimensional finite element analysis through the length is carried out to determine the transverse deflections and rotations of the beam. This procedure is much faster computationally as compared to the three-dimensional finite element method. It is simple-input, and needs less pre-processing time to prepare the model. The obtained results are compared with literature, strength of materials and VABS tool. In addition, the effect of ellipse diameter ratios and lay-up sequence on the mechanical behavior of elliptical tubular beams are investigated.

Structural analysis of composite tubes using a meshless analytical dimensional reduction method

AbstractA polynomial‐based model in conjunction with dimensional reduction method are presented to perform cross‐sectional analysis and to determine strain distribution in composite tubes under bending loading. For beam structures with tubular cross‐section, the Variational Asymptotic Method (VAM) has been employed to decompose a three‐dimensional (3D) elasticity problem into a two‐dimensional cross‐sectional analysis and a one‐dimensional analysis along the length. This greatly reduces computational time as compared to 3D Finite Element Method (FEM). There also exists publically available Variational Asymptotic Beam Sectional Analysis (VABS), a FEM cross‐sectional analysis tool based on VAM. For VABS, FEM mesh for the beam section needs to be generated. For the case of composite tubes with many layers, the mesh generation consumes efforts and is unnecessary. We introduce a new meshless dimensional reduction method for the analysis of composite tubes. This method utilizes Pascal polynomials in polar coordinates to model the warping functions. Using this, one can obtain stiffness constants and 3D strains of the composite tube. This method is straightforward, meshless, with similar computation time as VABS, and is much more efficient than conventional 3D FEM. The accuracy of the proposed method is examined by comparing the obtained results with 3D FE (ANSYS), VABS, literature, and experiment.

Stability analysis of an origami helical antenna using geometrically exact fully intrinsic nonlinear composite beam theory

A reconfigurable structure of antenna improves electromagnetic (EM) performance and its capability compared with other designs. The design satisfies both EM and mechanical needs simultaneously. This paper studies the stability analysis of a foldable helical antenna with the Kresling origami pattern, which is subjected to tip follower force. The structure of the antenna is simulated using the geometrically-exact, fully intrinsic nonlinear composite beam theory of Hodges. The cross-sectional properties of the antenna’s structure are obtained using the Variational Asymptotic Beam Sectional (VABS) analysis. While the results of the stability analysis are in agreement with commercial finite element tools, the numerical platform developed in this paper offers significant computational advantages. In a case study, the stability analysis based on the Hopf-bifurcation method is presented for a foldable helical antenna made from Kapton and fabric. The behavior of eigenvalues and mode shapes for this antenna are also studied. Furthermore, we investigated the effects of several geometrical parameters on the stability of the antenna, such as the number of polygon sides, height, and the element length ratio. The results indicate the antenna stability changes significantly by increasing the element length ratio and the number of polygon sides.

Structural Design, Analysis, and Testing of a 10 kW Fabric-Covered Wind Turbine Blade

Reducing the weight of a wind turbine blade is a major issue. Wind turbines have become larger in size to increase power generating efficiency. The blade has also grown in length to take more wind energy. A fabric-based wind turbine blade, introduced by General Electric Co., reduced the blade weight. In this study, a small fabric-covered blade for a 10 kW wind turbine was developed to verify structural ability. The blade was designed on the cross-section using variational asymptotic beam sectional analysis (VABS), structural analysis was carried out using MSC.Nastran for the design loads. A modal analysis was performed to compare the modal frequency and mode shapes. Static structural testing and modal testing were fulfilled. The analysis results were compared with the testing results. The fabric-covered structure was confirmed to reduce the blade mass with sufficient strength.

Constructal Design of Flying Wing Aircraft: Curved and Swept Configurations

Constructal law considers the design as a physics principle and uses energy flow configurations (e.g., the flow of stresses through the structure) to explain and predict the time arrow of change and evolutionary design in nature. Here, constructal law is used to explain why swept flying wings achieve higher flutter speed compared with curved wings. This investigation of the aeroelastic instability through the lens of constructal law is a new concept that envisions the evolutionary design of aircraft. Nonlinear Aeroelastic Trim And Stability of HALE Aircraft (NATASHA) performs the aeroelastic stability analyses. NATASHA benefits from geometrically exact, fully intrinsic, composite beam formulation of Hodges combined with Peters aerodynamic model. Variational Asymptotic Beam Sectional Analysis (VABS) conducts stress analysis. The results shed light on how changing the geometry would affect the flow of stresses and stability. The findings reveal an analogy between the instability and stress flow strangulation in the structure. Stress strangulation is observed in both configurations, whereas the curved configuration has lower flutter speed and stress strangulation with larger magnitude. The swept wing configuration provides easier access to the flow of stresses with less stress strangulation and leads to a more stable configuration.

Aeroelastic Response Analysis of Composite Blades Based on Geometrically Exact Beam Theory

The geometrically exact nonlinear beam theory consisting of the latest version of two-dimensional variational asymptotic beam sectional analysis (VABS) and one-dimensional geometrically exact beam theory (GEBT) has been widely used for the structural analysis of composite beam structures. The theory can be used for establishing the aeroelastic model of composite blades undergoing large deflections to improve computational accuracy and efficiency. In this paper, the theory has been extended from structural analysis to aeroelastic analysis of blade, and an accurate and efficient method for aeroelastic response analysis of composite blades has been presented based on the theory and unsteady aerodynamic model. The geometrically exact nonlinear equations of motion and the latest VABS are used to deal with one-dimensional beam analysis and the structural property of blade cross section, respectively. The Peters–He finite state dynamic wake model and the Peters finite state airloads theory are used to calculate the induced velocity and blade airloads, respectively. The presented method has been used to analyze the aeroelastic responses of composite blades, and its accuracy has been verified by experimental data. The influence of transverse shear deformation on the aeroelastic response of composite blades was also investigated, indicating that the transverse shear deformation has a nonnegligible effect on aeroelastic response analysis of hingeless composite rotors in hover.

The effect of a movable mass on the aeroelastic stability of composite hingeless rotor blades in hover

In this paper, the aeroelastic stability of a composite hingeless rotor blade with a chordwise movable mass is investigated. The point mass is located near the tip of the blade and its chordwise location is variable with respect to the elastic axis and can be moved during the flight. This movable mass is added to the blade to actuate the blade twist during flight. By actuating the mass in the chord direction of the blade during the flight, a bending moment which is the result of the centrifugal force of the mass and its offset is induced on the blade. This bending moment induces twist in the blade, due to bend-twist coupling in the composite lamination. The blade is modelled by using the geometrically exact fully intrinsic beam equations along with the variational asymptotic beam sectional analysis. The aerodynamic loads are simulated by using the two-dimensional strip theory combined with a uniform inflow. The nonlinear partial differential aeroelastic equations are discretized by a time-space scheme, and the converged results are compared with those reported in the literature and a very good match is observed. The results show that by positioning the mass near the tip of the blade, and also by using the ply angle of about 30 degree in this configuration, the highest possible twist change is achieved when the mass moves from the leading edge to the trailing edge of the blade. Moreover, the spanwise location of the mass slightly changes the stability boundaries, while the chordwise movement significantly affects the aeroelastic instability.

Simplified Formulation of Mechanics of Structure Genome

This paper presents a simplified formulation of the mechanics of a structure genome (MSG). The MSG is a unified theory for multiscale constitutive modeling for all types of composite structures. It is generalized from the previous research works based on the variational asymptotic method (VAM), including the variational asymptotic beam sectional analysis, the variational asymptotic plate and shell analysis, and the variational asymptotic method for unit cell homogenization. In this paper, the original complex formulation of the MSG is simplified and the principle of minimum information loss underpinning the MSG is formalized. The simplified formulation of the MSG is applied to construct the linear elastic Euler–Bernoulli beam model, the Kirchhoff plate model, and three-dimensional Cauchy continuum model. The connections of the MSG to previous VAM-based theories are clearly pointed out.

Structural Dynamic Analysis of a Tidal Current Turbine Using Geometrically Exact Beam Theory

This paper presents a numerical study of the dynamic performance of a vertical axis tidal current turbine. First, we introduce the geometrically exact beam theory along with its numerical implementation the geometric exact beam theory (GEBT), which are used for structural modeling. We also briefly review the variational-asymptotic beam sectional analysis (VABS) theory and discrete vortex method with free-wake structure (DVM-UBC), which provide the one-dimensional (1D) constitutive model for the beam structures and the hydrodynamic forces, respectively. Then, we validate the current model with results obtained by ANSYS using three-dimensional (3D) solid elements and good agreements are observed. We investigate the dynamic performance of the tidal current turbine including modal behavior and transient dynamic performance under hydrodynamic loads. Finally, based on the results in the global dynamic analysis, we study the local stress distributions at the joint between blade and arm by VABS. It is concluded that the proposed analysis method is accurate and efficient for tidal current turbine and has a potential for future applications to those made of composite materials.

Damage analysis by physically nonlinear composite beam theory

Damaging constitutive models are incorporated into the variational asymptotic beam sectional analysis method (VABS). Under the governing of the virtual work principle, Newton’s method is applied to solve for the finite cross-sectional warping field and the damage variables according to continuum damage mechanics models from the literature. These solutions result in mesh-insensitive predictions of the damaged beam stiffness and three-dimensional stress and strain fields in the damaged state. The present theory is implemented using the finite element method to capture the high-fidelity elastic-to-damaged beam constitutive behaviors by the cross-sectional analyses. The predicted beam constitutive relations and local fields are compared with results from finite element analyses (FEA) using solid elements and plate elements, and experiments.

Effect of Inertial and Constitutive Properties on Body-Freedom Flutter for Flying Wings

The class of high-altitude long-endurance aircraft often has very slender wings with minimal structure. This leads to large trim deflections and coupling between the vehicle flight dynamics and wing vibrations. When this coupled behavior becomes unstable, it is referred to as body-freedom flutter. Body-freedom flutter behavior is dependent on the inertial and constitutive properties of the wing, as well as the fuselage. This relationship is explored for an aircraft representative of the Horten IV flying wing, using an efficient yet rigorous analysis that relies on geometrically exact, fully intrinsic beam equations and a finite-state induced flow model, which is implemented in the computer code NATASHA (which stands for nonlinear aeroelastic trim and stability of high-altitude long-endurance aircraft). The wing inertial and stiffness properties were calculated using a realistic representative section using the section analysis tool called variational asymptotic beam section analysis. Trade studies on the body-freedom flutter behavior were performed by varying the fuselage properties and the internal wing structure, as well as examining the effects on the flutter speed, flutter frequency, and flutter mode shape.

Nonlinear Variational Asymptotic Sectional Analysis of Hyperelastic Beams

A nonlinear sectional analysis is developed for modeling beams made of hyperelastic materials and subject to finite deformation. Exploiting the fact that the cross-sectional dimension is much smaller than the wavelength of the global beam deformation, the three-dimensional hyperelasticity is rigorously reduced to a one-dimensional nonlinear beam analysis and a two-dimensional cross-sectional analysis featuring both geometric and material nonlinearities. The deformation gradient is directly used to enable the capability of dealing with various types of hyperelastic material models defined by the strain energy function. This new theory can predict phenomena from both material and geometric nonlinearities of beam-like structures, such as the trapeze and Poynting effects. The present theory is implemented using the finite element method in variational asymptotic beam sectional analysis, a general-purpose cross-sectional analysis tool that deals with composite beams. Examples are used to validate the theory and the companion code.

A variational asymptotic approach for thermoelastic analysis of composite beams

A variational asymptotic composite beam model has been developed for thermoelastic analysis. Composite beams, including sandwich structure and laminates, under different boundary conditions are examined. Previously developed beam model, which is based on variational-asymptotic method, is extended to incorporate temperature-dependent materials experiencing large temperature changes. The recovery relations have been derived so that the temperatures, heat fluxes, stresses, and strains can be recovered over the cross-section. The present theory is implemented into the computer program VABS (Variational Asymptotic Beam Sectional analysis). Numerical results are compared with the 3D analysis for the purpose of demonstrating advantages of the present theory and use of VABS.

Discussion: “Vibration Analysis of Composite Beams With End Effects Via the Formal Asymptotic Method” (Kim, J.-S., and Wang, K. W., 2010, ASME J. Vib. Acoust., 132(4), p. 041003)

In their paper [1], Kim and Wang presented the vibration analysis of composite beams by using a finite element-based formal asymptotic expansion method. After a detailed mathematical treatment of the formal asymptotic method-based beam analysis (FAMBA), they considered various examples of composite beams with solid and closed thin-walled cross sections for verification purposes and the numerical results were compared with those from 3D FEM and other methods, such as the variational-asymptotic beam section analysis (VABS) [2].One particular case, the 15 deg CUS thin-walled composite box beam (whose ply material properties are listed in Table 3 of [1]), was analyzed with VABS by taking the stiffness model given in [2] and applying it to a typical Rankine–Timoshenko beam model. The normalized first and second eigenvalues computed by 3D FEM, FAMBA, VABS, etc. were plotted against the beam length-to-height ratio in Figs. 6 and 7. While it is clear that FAMBA-second predictions are reasonably close to those of 3D FEM, the authors (Kim and Wang) also show, and hence claimed, that results of VABS deviate significantly from those of 3D FEM, even qualitatively so for the second eigenvalue.The normalized first, second, and eighth mode shapes in Figs. 8 and 9 and 10 in [1] also show VABS as being unable to match 3D FEM predictions, with the deviation most obvious in Figs. 8(b), 9(a), and 10(b) in [1]. This is solely attributed by the authors to VABS inaccurately capturing the bending-shear coupling of the box beam. These claims call for the following comments:1. The stiffness model given in [2] (as calculated by versions of VABS till v.3.1), when used in tandem with a 1D beam analysis code based on a geometrically exact intrinsic beam theory (GEBT), provide results that are both qualitatively and quantitatively in close agreement to 3D FEM unlike the results reported in [1]. This can be seen in Figs. 1–5 which have now been recalculated by the authors (Kim and Wang). We can safely assume that the fault lies neither with the Rankine–Timoshenko model that the authors used nor the geometrically exact nonlinear beam theory (since the former seemed to have worked with other beam models and the latter is designed to capture all the geometrical nonlinearities obtainable by a beam model). Therefore, it seems logical to conclude that the source of discrepancy was an error in applying the given stiffness model correctly to the macroscopic analysis.2. The discrepancy has been examined by the authors (Kim and Wang). It is found that the sign convention used in VABS [2] was not consistently applied to the calculation of VABS results in [1]. The vibration analysis results from VABS and others have been corrected and replotted, as shown in Figs. 1–5.In conclusion, we find that although there are minor differences in the prediction of the bending-shear coupling for the CUS box beam between VABS [2] and FAMBA [1], both stiffness models yield very acceptable results in terms of vibration analysis. Specifically, VABS does not significantly deviate, qualitatively or quantitatively, from 3D FEM predictions. Additionally, the authors (Kovvali, Hodges, and Yu) would like to note that since the publication of [1], VABS has been updated and now provides results that are still further improved for some problems; see [3].

External fixator configurations in tibia fractures: 1D optimization and 3D analysis comparison

The use of external fixation devices in orthopedic surgery is very common in open tibial fractures. A properly applied fixator may improve the healing process while one improperly applied might delay the healing process. The several external fixator systems used in clinical today, can be categorized into uniplanar–unilateral, uniplanar–bilateral, biplanar and multiplanar. The stability on the fracture focus and, therefore, the fracture healing process, is related with the type of external fixator configuration that is selected. The aim of this study is to discuss the principles for the successful application of unilateral–uniplanar external fixation, the assembly of its components, for the case of a transverse fractures using computational models. In this context, the fixation stiffness characteristics are evaluated using a simplified 1D finite element model for the tibia and external fixator. The beams are modeled with realistic cross-sectional geometry and material properties instead of a simplified model. The VABS (the Variational Asymptotic Beam Section analysis) methodology is used to compute the cross-sectional model for the generalized Timoshenko model, which was embedded in the finite element solver FEAP. The use of Timoshenko beam theory allows accounting for several kinds of loads, including torsion moments. Optimal design is performed with respect to the assembly of fixator components using a genetic algorithm. The optimization procedure is based on the evaluation of an objective function, which is dependent on the displacement at the fracture focus. The initial and optimal results are compared by performing a 3D analysis, for which different three-dimensional finite element models are created. The geometrical model of a tibia is created on the basis of data acquired by CAT scan, made for a healthy tibia of a 22 year old male. The 3D comparison of the 1D optimal results show a clear improvement on the objective function for the several load cases and, therefore, it is shown that appropriate selection of the external fixator geometrical features can lead to an improvement on the stability of the external fixator. The results obtained show that the optimal position of the side beam and the first pin should be as close as possible to the bone interface and as close as possible to the fracture focus, respectively. Concerning the second pin, it should be placed away from the first pin in case of flexion loads, to axial and torsion loads the second pin should be placed near the first pin.