Thin-walled Composite Box Beams Research Articles

Numerical/Experimental Validation of Thin-Walled Composite Box Beam Optimal Design

Thin-walled composite box beam structural configuration is representative of a specific high aspect ratio wing structure. The optimal design procedure and lay-up definition including appropriate coupling necessary for aerospace applications has been identified by means of “ad hoc” analytical formulation and by application of commercial code. The overall equivalent bending, torsional and coupled stiffness are derived and the accuracy of the simplified beam model is demonstrated by the application of Altair Optistruct. A simple case of a coupled cantilevered beam with load at one end is introduced to demonstrate that stiffness and torsion angle distribution does not always correspond to the trends that one would intuitively expect. The maximum of torsional stiffness is not obtained with fibers arranged at 45° and, at the maximum torsional stiffness, there is no minimum rotation angle. This observation becomes essential in any design process of composite structures where the constraints impose structural couplings. Furthermore, the presented theory is also extended to cases in which it is necessary to include composite/stiffened hybrid configurations. Good agreement has been found between the theoretical simplified beam model and numerical analysis. Finally, the selected composite configuration was compared to an experimental test case. The numerical and experimental validation is presented and discussed. A good correlation was found confirming the validity of the overall optimization for the optimal lay-up selection and structural configuration.

Experimental and numerical studies of a cracked thin-walled box-beams

This paper discusses the problem of detection and location of cracks in a cantilever beam. To solve this problem, a damage index based on the modal displacement method is applied in both experimental and numerical studies. Thin-walled composite box-beams with rectangular cross-section are tested. Two types of beam configuration are examined: circumferentially asymmetric stiffness (CAS) and circumferentially uniform stiffness (CUS). The CAS configuration is characterized by the coupling between bending in a flexible direction and torsion, while the CUS configuration is characterized by coupled bending in two perpendicular directions (flexible and stiff). Numerical models of the tested beams are made by the finite element method. A series of simulations are performed for different damage locations and ply orientations. The Lanczos method is used to solve the eigenvalue problem. Natural frequencies and free vibration modes of the beams are determined. To validate the numerical models, a test stand is build. The test specimens are made of unidirectional carbon-epoxy prepreg by the autoclave technique. The damage of the reinforcement fibres is induced by making a crack in the beam using a circular saw. Selected numerical results are validated with two different experimental methods. First, the experimental test is conducted using a non-contact laser vibrometer provided with a scanning head and a data acquisition-visualization unit. Next, a modal analysis is performed by the contact method using the LMS analyzer with a modal hammer and an acceleration sensor. All numerical and experimental results of the beams with cracks are compared with the findings obtained for the specimens without any damage.

Exact solutions for static analysis of laminated, box and sandwich beams by refined layer-wise theory

Abstract In the present work, a close-form solution based on a unified one-dimensional model is proposed and then applied to static response analyses of cross-ply laminated and sandwich beams subjected to simply supported boundary conditions. The hierarchical beam model is derived within the framework of the Carrera Unified Formulation (CUF), which makes use of Lagrange polynomials to express the three-dimensional (3D) displacement field via arbitrary order approximation of pure displacement variables at each layer over the cross section, in a Layer-Wise (LW) sense. The governing equations are derived via the principle of virtual work and a Navier-type close-form solution is employed to solve the resulting boundary value problem. Four benchmark numerical examples are carried out to demonstrate the efficiency of this novel method, including compact multi-layered cross-ply laminated beams, a thin-walled composite box beam and a composite sandwich-box beam. The results show that accurate displacement and stress components can be obtained as the order of the expansion increases, accompanied by a significant reduction in computational costs in comparison with the 3D finite element solutions. Besides, numerical cases in this research may be taken as benchmarks for future assessments in this field.

Vibration analysis of rotating composite beams using polynomial based dimensional reduction method

Free vibration analysis of rotating composite beams with arbitrary cross section is presented. The analysis is based on dimensional reduction method. Contrary to most of the previous studies, which finite element is used to analyze the cross section, the present study examines a simple polynomial based sectional analysis. The three-dimensional (3D) elasticity problem of the composite beam is decomposed into a two-dimensional (2D) cross section analysis and a one-dimensional (1D) beam analysis. In 2D cross-sectional problem in-plane and out-of-plane warpings are calculated by minimizing the energy functional with respect to warping functions. Simple polynomial series are employed to derive the necessary warping functions analytically. Cross sectional properties are determined as a fully populated 4×4 stiffness matrix. Based on a 1D beam potential and kinetic energy, natural frequencies of rotating and non-rotating beams are obtained. The presented approach is applied to isotropic, laminated composite beams as well as thin-walled composite box beams. The effects of fiber angle and rotational speed are investigated for flap and lag bending vibration. The accuracy of the results is validated in comparison with other theories, 3D FEM and experimental data. This procedure, which gives accurate predictions, eliminates the costly use of 3D finite element analysis.

Investigation of the Effects of Constitutive Equations on the Free Vibration Behavior of Single-Celled Thin-Walled Composite Beams

An investigation of the free vibration behavior of thin-walled composite box-beams is carried out by considering different assumptions in the constitutive equations. Within the present model some non-classical effects, such as restrained warping and transverse shear, are incorporated. Free vibration results are validated against experimental and numerical results, which are available in the literature. The natural frequencies obtained based on the different assumptions of constitutive equations are compared, and it is revealed that these assumptions play an important role in the proper treatment of the free vibration behavior of torsion-bending coupled composite beams. The results obtained based on the proposed constitutive equations are demonstrated to have a good agreement with the finite element results as far as the lower natural frequencies of the beams are concerned.

Axial–torsional vibrations of rotating pretwisted thin walled composite beams

Axial–torsional vibrations of rotating pretwisted thin-walled composite box beams exhibiting primary and secondary warping are investigated. Considering the nonlinear strain–displacement relations, the coupled nonlinear axial–torsional equations of motion are derived using Hamilton's principle. Ignoring the axial inertia term leads to differential equation of motion in terms of elastic torsion in the case of axially immovable beams. Centrifugal load in the presence of material anisotropy and pretwist angle leads to an induced static torque. The nonlinear equation should be linearized about the corresponding equilibrium state to obtain the linear differential equation of motion. Extended Galerkin's method is utilized to achieve the proper eigenvalue problem. The results obtained in this paper seek to clarify the individual and collective effects of axial loading, pretwist, stagger and fiber angles on the torsional behavior of the non-uniform thin-walled composite blades. The results are compared to available analytical and experimental results in the literature which reveals excellent agreements. The outcomes of this study are expected to offer better predictions of the dynamic behavior of this kind of structures in general, and in design of rotor blades of turbo-machinery, in particular.

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].

Validation of the thin-walled composite box beams using FEM

Validation of the thin-walled composite box beams using FEM

Experimental Investigations of Shear Lag Effect in a Simply Supported [0o ∕±45o2 ∕ 0o]T Laminated Box Beam

With the features of high efficiency, low consumption and good mechanical characteristic, thin-walled composite box beams have been broadly adopted in structural engineering, and its mechanical behavior has became an active research area. As shear lag effect can bring an uneven normal stress distribution on the flanges, it would remarkably affect the strength design of thin-walled beams. This paper focuses on the experimental investigations of shear lag effects in [0o∕±45o2∕ 0o]T laminated box beam under concentrated loads, and test results indicates that the shear lag effect in this composite box beam can be simulated by the two parabola.

An improved model for naturally curved and twisted composite beams with closed thin-walled sections

This paper presents an improved model for naturally curved and twisted anisotropic beams with closed thin-walled cross-sections. By introducing eigenwarping functions and expanding axial displacements in series of eigenwarpings, the differential equation involving the generalized warping coordinate and the expression for eigenvalues can be derived using the principle of minimum potential energy. In the model the effects of some factors such as the initial curvature, torsion of the beams as well as torsion-related warping, transverse shear deformations and elastic coupling are incorporated. As an application, the present model is adopted to do an analysis for closed thin-walled composite box beams. Comparison with the existing experimental observation and numerical results shows that the proposed model is valid for analyzing such naturally curved and twisted beams.

Coupled stability analysis of thin-walled composite beams with closed cross-section

For the coupled stability analysis of thin-walled composite beam with closed cross-section subjected to various forces such as eccentric constant axial force, end moments, and linearly varying axial force, the efficient numerical method to evaluate the element stiffness matrix is newly presented based on the homogeneous form of simultaneous ordinary differential equations. The general bifurcation type of buckling theory for thin-walled composite box beam is developed based on the energy functional corresponding to semitangential rotations and semitangential moments. The coupled stability equations including variable coefficients and the force–displacement relationships are derived from the energy principle and explicit expressions for displacement functions are presented based on power series expansions of displacement components. The element stiffness matrix is evaluated by applying member force–displacement relationships to these displacement functions. In addition, the finite element model based on the cubic Hermitian interpolation polynomial is presented. In order to verify the accuracy and validity of this study, numerical solutions are presented and compared with the finite element solutions using the Hermitian beam elements and the available results from other researchers. Particularly, the influence of the eccentricity and the force ratio of axial forces, the fiber orientation, and the boundary conditions on the buckling behavior of composite box beam are parametrically investigated. Also the emphasis is given in showing the phenomenon of buckling mode change.

Interaction curves for vibration and buckling of thin-walled composite box beams under axial loads and end moments

Interaction curves for vibration and buckling of thin-walled composite box beams with arbitrary lay-ups under constant axial loads and equal end moments are presented. This model is based on the classical lamination theory, and accounts for all the structural coupling coming from material anisotropy. The governing differential equations are derived from the Hamilton’s principle. The resulting coupling is referred to as triply flexural–torsional coupled vibration and buckling. A displacement-based one-dimensional finite element model with seven degrees of freedoms per node is developed to solve the problem. Numerical results are obtained for thin-walled composite box beams to investigate the effects of axial force, bending moment, fiber orientation on the buckling loads, buckling moments, natural frequencies and corresponding vibration mode shapes as well as axial-moment–frequency interaction curves.

Geometrically nonlinear theory of thin-walled composite box beams using shear-deformable beam theory

A general geometrically nonlinear model for thin-walled composite space beams with arbitrary lay-ups under various types of loadings is presented. This model is based on the first-order shear deformable beam theory, and accounts for all the structural coupling coming from both material anisotropy and geometric nonlinearity. The nonlinear governing equations are derived and solved by means of an incremental Newton–Raphson method. A displacement-based one-dimensional finite element model that accounts for the geometric nonlinearity in the von Kármán sense is developed. Numerical results are obtained for thin-walled composite box beams under vertical load to investigate the effects of shear deformation, geometric nonlinearity and fiber orientation on axial–flexural–torsional response.

Free vibration of axially loaded thin-walled composite box beams

A general analytical model applicable to flexural–torsional coupled vibration of thin-walled composite box beams with arbitrary lay-ups under a constant axial force has been presented. This model is based on the classical lamination theory and accounts for all the structural coupling coming from the material anisotropy. Equations of motion are derived from the Hamilton’s principle. A displacement-based one-dimensional finite element model is developed to solve the problem. Numerical results are obtained for thin-walled composite box beams to investigate the effects of axial force, fiber orientation and modulus ratio on the natural frequencies, load–frequency interaction curves and corresponding vibration mode shapes.

Torsional analysis of thin-walled composite beams with single- and double-celled sections

The exact solutions for twist angle and fiber stresses of thin-walled composite box beams with single- and double-celled sections subjected to torsional moment are presented by introducing fourteen displacement parameters. For this, a general thin-walled composite box-beam theory including the effects of elastic couplings and restrained warping is developed based on the Vlasov’s assumptions. The equilibrium equations and the force–deformation relations are derived from the energy principle. A system of linear algebraic equations with non-symmetric matrices is constructed by introducing fourteen displacement parameters and by transforming the higher order simultaneous differential equations into first-order ones. This numerical technique determines eigenmodes corresponding to multiple zero and non-zero eigenvalues and derives exact displacement functions for displacement parameters based on the undetermined parameter method. Finally, the exact stiffness matrix is determined using the member force–deformation relations. The theory developed by this study is validated by comparing several torsional responses from the present approach with those from the finite element beam model that uses third-order Hermitian polynomials and detailed two-dimensional analysis results using the shell elements of ABAQUS for coupled composite beams with single- and double-celled sections.

Geometrically nonlinear analysis of thin-walled composite box beams

A general geometrically nonlinear model for thin-walled composite space beams with arbitrary lay-ups under various types of loadings has been presented by using variational formulation based on the classical lamination theory. The nonlinear governing equations are derived and solved by means of an incremental Newton–Raphson method. A displacement-based one-dimensional finite element model that accounts for the geometric nonlinearity in the von Kármán sense is developed. Numerical results are obtained for thin-walled composite box beam under vertical load to investigate the effect of geometric nonlinearity and address the effects of the fiber orientation, laminate stacking sequence, load parameter on axial–flexural–torsional response.

On sixfold coupled vibrations of thin-walled composite box beams

This paper presents a general analytical model for free vibration of thin-walled composite beams with arbitrary laminate stacking sequences and studies the effects of shear deformation over the natural frequencies. This model is based on the first-order shear-deformable beam theory and accounts for all the structural coupling coming from the material anisotropy. The seven governing differential equations for coupled flexural–torsional–shearing vibration are derived from the Hamilton’s principle. The resulting coupling is referred to as sixfold coupled vibration. Numerical results are obtained to investigate the effects of fiber angle, span-to-height ratio, modulus ratio, and boundary conditions on the natural frequencies as well as corresponding mode shapes of thin-walled composite box beams.

Flexural–torsional behavior of thin-walled composite box beams using shear-deformable beam theory

This paper presents a flexural–torsional analysis of thin-walled composite box beams. A general analytical model applicable to thin-walled composite box beams subjected to vertical and torsional loads is developed. This model is based on the shear-deformable beam theory, and accounts for the flexural–torsional response of the thin-walled composites for an arbitrary laminate stacking sequence configuration, i.e. unsymmetric as well as symmetric. The governing equations are derived from the principle of the stationary value of total potential energy. Numerical results are obtained for thin-walled composites under vertical loading, addressing the effects of fiber angle and span-to-height ratio of the composite beam.

Flexural–torsional buckling of thin-walled composite box beams

Buckling of an axially loaded thin-walled laminated composite is studied. A general analytical model applicable to the flexural, torsional and flexural–torsional buckling of a thin-walled composite box beam subjected to axial load is developed. This model is based on the classical lamination theory, and accounts for the coupling of flexural and torsional modes for arbitrary laminate stacking sequence configuration, i.e. unsymmetric as well as symmetric, and various boundary conditions. A displacement-based one-dimensional finite element model is developed to predict critical loads and corresponding buckling modes for a thin-walled composite bar. Governing buckling equations are derived from the principle of the stationary value of total potential energy. Numerical results are obtained for axially loaded thin-walled composites addressing the effects of fiber angle, anisotropy and boundary conditions on the critical buckling loads and mode shapes of the composites.

Free vibration of thin-walled composite box beams

Free vibration of a thin-walled laminated composite beam is studied. A general analytical model applicable to the dynamic behavior of a thin-walled composite box section is developed. This model is based on the classical lamination theory, and accounts for the coupling of flexural and torsional modes for arbitrary laminate stacking sequence configuration, i.e. unsymmetric as well as symmetric, and various boundary conditions. A displacement-based one-dimensional finite element model is developed to predict natural frequencies and corresponding vibration modes for a thin-walled composite beam. Equations of motion are derived from the Hamilton’s principle. Numerical results are obtained for thin-walled composites addressing the effects of fiber angle, modulus ratio, and boundary conditions on the vibration frequencies and mode shapes of the composites.