Thin-walled Beam Model Research Articles

A data-driven procedure for one-dimensional dynamic analysis of thin-walled beams with arbitrary cross-sections

This paper develops a one-dimensional dynamic model for thin-walled beams with arbitrary complex cross-sections in a data-driven way. In order to consider complicated deformations in the framework of a beam theory, a universal node system is first created to fit the deformed shape of a thin-walled cross-section, whether prismatic or curved, with or without symmetry. This helps to reduce the three-dimensional displacement field of the thin-walled beam to one dimension, and results in a preliminary higher-order beam model with limited precision loss. For the purpose of largely condense DOFs of the new model, a data-driven approach is proposed to identify cross-section deformation modes through the principal component analysis of free vibration deformation data of an unconstrained thin-walled beam. As a result, a compact set of core deformation modes are obtained and selected to update the preliminary model, leading to the refined higher-order beam model of high accuracy and efficiency. Examples are presented to illustrate the concrete implementation and check the effects of data-related controlling parameters of the proposed procedure. We also verify through numerical examples that the proposed model agrees well with plate/shell and other beam theories, and has remarkable advantages in physical interpretation, modeling simplicity and computation efficiency.

One-dimensional viscoelastic von Kármán theories derived from nonlinear thin-walled beams

We derive an effective one-dimensional limit from a three-dimensional Kelvin–Voigt model for viscoelastic thin-walled beams, in which the elastic and the viscous stress tensor comply with a frame-indifference principle. The limiting system of equations comprises stretching, bending, and twisting both in the elastic and the viscous stress. It coincides with the model already identified via Friedrich and Kružík (Arch Ration Mech Anal 238:489–540, 2020) and Friedrich and Machill (Nonlinear Differ Equ Appl NoDEA 29, Article number: 11, 2022) by a successive dimension reduction, first from 3D to a 2D theory for von Kármán plates and then from 2D to a 1D theory for ribbons. In the present paper, we complement the previous analysis by showing that the limit can also be obtained by sending the height and width of the beam to zero simultaneously. Our arguments rely on the static Gamma -convergence in Freddi et al. (Math Models Methods Appl Sci 23:743–775, 2013), on the abstract theory of metric gradient flows (Ambrosio et al. in Gradient flows in metric spaces and in the space of probability measures. Lectures mathematics, ETH Zürich, Birkhäuser, Basel, 2005), and on evolutionary Gamma -convergence (Sandier and Serfaty in Commun Pure Appl Math 57:1627–1672, 2004).

Effective Stiffness of Thin-Walled Beams with Local Imperfections.

Thin-walled beams are increasingly used in light engineering structures. They are economical, easy to manufacture and to install, and their load capacity-to-weight ratio is very favorable. However, their walls are prone to local buckling, which leads to a reduction of compressive, as well as flexural and torsional, stiffness. Such imperfections can be included in such components in various ways, e.g., by reducing the cross-sectional area. This article presents a method based on the numerical homogenization of a thin-walled beam model that includes geometric imperfections. The homogenization procedure uses a numerical 3D model of a selected piece of a thin-walled beam section, the so-called representative volume element (RVE). Although the model is based on the finite element method (FEM), no formal analysis is performed. The FE model is only used to build the full stiffness matrix of the model with geometric imperfections. The stiffness matrix is then condensed to the outer nodes of the RVE, and the effective stiffness of the cross-section is calculated by using the principle of the elastic equilibrium of the strain energy. It is clear from the conducted analyses that the introduced imperfections cause the decreases in the calculated stiffnesses in comparison to the model without imperfections.

Thermoviscoelastic dynamics of composite thin-walled booms on spacecrafts subjected to solar radiation

Viscoelastic behavior of polymer-matrix composites depends very strongly on temperature. This paper presents a thermoviscoelastic dynamics modelling for fiber-reinforced polymeric composite structures with time-varying and nonuniform temperature field. It is used to investigate the thermally induced vibrations of spacecraft booms, for which a new anisotropic thin-walled beam model based on the generalized beam theory is also proposed. Furthermore, efficient evaluation of the thermoviscoelastic constitutive equations in hereditary integral form is performed by means of the derived recursion formula. A harmonic balance procedure in the circumferential direction of tubular booms is devised for both thermal and structural analyses. By comparing the results of viscoelastic and elastic models, the influence of thermoviscoelastic effects on dynamical responses of composite booms under thermal shock is illustrated. The results show that viscoelasticity may play a significant role in vibration suppression of on-orbit flexible structures, especially at high temperature. The work not only provides a novel model for thermal-structural dynamic analysis, but also develops thermoviscoelasticity theory oriented toward engineering and numerical applications.

Effect of a “Slight” Curvature of the Axis of a Thin-Walled Nonprismatic Beam on its Free Vibration

This paper presents an analysis of the effect of a “slight” curvature of the axis of a thin-walled beam on its eigenfrequencies and forms. By “slight” we understand such a beam curvature at which no theory of curved beams (arches, horizontally curved girders), but classical beam theory is applied. The subject of the analysis is the open nonprismatic thin-walled beams with any geometrical parameters, and prismatic beams. The thin-walled beam model used in the analysis was derived on the basis of the momentless theory of plates, using the Vlasov assumptions. The free vibrations of the beams were analysed. Calculations were performed for four groups of beams. Each of the groups comprised two beams with the same configuration of flanges and identical geometrical cross section characteristics, but differing in web geometry. One of the beams had a symmetrical web (a beam with a rectilinear axis in the web plane), whereas the web of the other beam was asymmetrical (a beam with its axis curved in the web plane). The algorithm based on Chebyshev polynomials was used to solve the variable coefficient equations describing the model’s vibration. According to this algorithm, solutions are sought in the form of Chebyshev series and the closed analytical recurrence formulas generated by the algorithm are used to calculate the coefficients of the equations. It has been shown that even a “slight” curvature of the beam’s axis has a significant effect on its eigenfrequencies and forms, as confirmed by a comparison of the results with the ones obtained using Finite Element Method (FEM).

Joint modelling in advanced thin-walled beam models

A methodology for thin-walled frame analysis using displacement modes of beams at joint interfaces has recently been proposed. This paper introduces an implementation that uses a specific advanced beam element based on semi-analytical beam mode solutions of a generalised beam theory. The end cross-sections of the beam elements consist of wall elements with displacement degrees of freedom that can be joined to conventional finite shell element models of the joints. However, before joining the beam element and joint model at the interface, both beam and joint model are transformed into cross-section displacement mode degrees of freedom. This transformation enables the use of a reduced mode space at the interface as well as in the beam. The quadratic nature of the beam mode equations of the beam element leads to a non-unique number of beam displacement modes. However, this paper introduces an exact mode selection technique that leads to a unique set of modes. Necessary details of the beam element formulation are included in order to make a self-contained description of the mode selection technique. Mode reduction through a length scale dependent choice of the number of exponentially decaying distortional modes is investigated and discussed for three types of joint design in four examples. The three joint design types investigated are based on in-plane membrane action, flexural out-of-plane action and a combination of both membrane and flexural action.

An Analysis of the Effect of a Change in the Support Point Location on the Vibration of Thin-Walled Beams

This paper deals with an analysis of the free vibration of nonprismatic thin-walled beams, with a special focus on the effect of a change in the support point location on the eigenfrequencies of the systems. A change in the support point location is understood here as occurring within the same fixed cross-section of the beam where the latter is supported. The original elements of this study are a thin-walled beam model and a method of solving differential equations, not previously used by other authors. The equations describing the model used in this paper were derived using the momentless theory of shells and the Vlasov theory assumptions. The displacement equations were derived relative to an arbitrary rectilinear reference axis. In most works known to the authors the equations describing the vibration of the nonprismatic thin-walled beam do not take into account the effects due to the curvature of the axis formed by the shear centers. The recursive algorithm was used to solve the obtained differential equations with variable coefficients. The algorithm enables one to solve the analyzed displacement equations in the form of series relative to the orthogonal Gegenbauer polynomials. For special values of the parameter defining the order of the polynomials the Gegenbauer polynomials became Chebyshev or Lagrange polynomials. In the provided numerical example, the effect of a change in the support point location within the fixed cross-section is examined. It is also analyzed which of the approximation polynomials (Chebyshev or Lagrange polynomials) yield more precise results for a small approximation base. In order to verify the model and the effectiveness of the adopted solution method, the results obtained using this method are compared with the results yielded by FEM. The results obtained showed a significant effect of the position of the support point, within a fixed section, on the eigenfrequencies values of the thin-walled systems.

Aeroelastic vibration analysis of wind turbine blade with Gurney flap

In this paper, based on the parametric design of 3 D blades including typical cross-section and natural frequency calculation of the equivalent model, an integral method for aerodynamic performance and aeroelastic vibration analysis of blade with Gurney flap is proposed. Parametric design and unstructured grid are used to pre-process the blades with/without the Gurney flap. The discrete aerodynamic performance parameters distribution including the lift, drag, and torsion coefficients calculated by Fluent is fitted by the nonlinear least square method based on the trust-region algorithm, and the natural frequencies of the rotating blade are accurately solved by the equivalent thin-walled beam model and Green’s functions. Based on the aerodynamic performance coefficients and natural frequencies obtained by the accurate calculation above, the aeroelastic response equation of typical cross-section considering local aerodynamic damping matrix is established, and the vibration response of blade in flap and torsion direction is further described. From the analysis results, it can be seen that the Gurney flap structure can not only bring higher lift performance to the blade, but also can reduce the amplitude and vibration range of aeroelastic vibration, improve the aeroelastic stability of the blade, and prove the effectiveness of Gurney flap.

A thin-walled composite beam model for light-weighted structures interacting with fluids

A thin-walled beam model is proposed for structures of variable cross-section, which can be either open or closed and includes multicellular cross-sections with either isotropic or orthotropic materials. The proposed model does not require any priori definition of cross-sectional warping which instead results from the solution of the problem. To achieve that a special deformation pattern is superimposed on the bending deformation described by Euler–Bernoulli beam theory. All sectional properties are automatically incorporated in the analysis as a result of the usual variational formulation of the system of equations. The proposed model is specifically designed to simulate the dynamics of wind/hydrokinetic turbine blade with low computational cost, especially in fluid–structure interaction (FSI) simulation. A number of test cases have been carried out to validate the proposed structural model which show good agreement between the results obtained her e and the solutions available in literature. Finally, FSI simulation of a hydrokinetic blade under field condition is carried out to illustrate the capability of the current thin-walled beam model in practice.

Data-driven approach for a one-dimensional thin-walled beam analysis

This study presents a data-driven approach for developing a one-dimensional thin-walled beam model. In order to analyze complicated deformations occurring in a thin-walled beam by a beam theory, it is important to identify core cross-sectional deformations that are the bases for the one-dimensional beam analysis. In this study, we derive core cross-sectional deformations through data processing using the shell-based static analysis results of a thin-walled beam. We perform a principal component analysis for the data processing, in which the desired core cross-sectional deformations are obtained without specific assumptions pertaining to the behavior of the beam’s sectional deformations. Then, the core cross-sectional deformations can be obtained in explicit functional forms (shape functions) to facilitate the subsequent one-dimensional higher-order beam analysis. Important issues related to the establishment of a well-defined dataset are addressed. Using the data-driven shape functions, a series of static, vibration and buckling analyses are performed for various thin-walled beams. We demonstrate by numerical examples that the present data-driven results agree well with those obtained by shell analysis results. While the shape functions are obtained only from the static analysis results, the vibration and buckling analyses using them were also found to be sufficiently accurate.

A thin-walled beam element based on semi-analytical solution modes

Using energy principles, a thin-walled beam element is introduced for the analysis of beams with deformable cross-sections that are prone to distortion. The beam element is based on previously attained semi-analytical displacement solution modes of an advanced thin-walled beam model. The first-order beam element for linear analysis handles shear deformations related to both Timoshenko and Mindlin-Reissner type deformations, warping effects of torsion, cross-section distortion including associated warping effects, as well as the transverse displacement effect from normal stress. The formulation can handle both open and closed cross-sections without special attention. The formulation of the displacement solution modes and the stiffness integration of the products of the advanced displacement modes using the Hadamard product are described. The paper also presents the transformations between modal degrees of freedom and element displacement degrees of freedom. Four examples show the beam element capabilities and good agreement with results obtained using the shell and solid elements of a commercial finite element program. The kinematic assumptions that the thin-walled beam model accommodates leads to local shear stress transfer at corners. This transfer of shear stresses is not normally seen in thin-walled beam formulations or shell models. However, the shear transfer is verified through examination of a finite element model using solid elements.

Displacement modes of a thin-walled beam model with deformable cross sections

A novel one dimensional beam model for analysis of prismatic thin-walled beams with deformable cross sections is introduced and a novel cross section mode determination procedure, which leads to the three dimensional beam displacement modes, is derived. The first order beam model for linear analysis includes: shear deformations related to both Timoshenko and Mindlin-Reissner type shear deformations, the warping effects of torsion, cross section distortion with related warping effects, as well as the Poisson effect with transverse displacements due to normal stress. The generality of the model allows it to handle open, closed and multi-cell cross sections with branched walls. The cross section analysis procedure leads to two types of beam displacement modes referred to as distortional beam modes and fundamental beam modes, with exponential and polynomial variations along the beam axis, respectively. It turns out that each of the beam deformation modes consists of a sum of one to four cross section displacement fields each with an individual axial variation. The displacement modes can facilitate the formulation of an advanced thin-walled beam element. The beam displacement modes will be illustrated for an open and a closed cross section.

Analysis of thin-walled beams with variable monosymmetric cross section by means of Legendre polynomials

Abstract This article deals with the vibrations of a nonprismatic thin-walled beam with an open cross section and any geometrical parameters. The thin-walled beam model presented in this article was described using the membrane shell theory, whilst the equations were derived based on the Vlasov theory assumptions. The model is a generalisation of the model presented by Wilde (1968) in ‘The torsion of thin-walled bars with variable cross-section’, Archives of Mechanics, 4, 20, pp. 431–443. The Hamilton principle was used to derive equations describing the vibrations of the beam. The equations were derived relative to an arbitrary rectilinear reference axis, taking into account the curving of the beam axis and the axis formed by the shear centres of the beam cross sections. In most works known to the present authors, the equations describing the nonprismatic thin-walled beam vibration problem do not take into account the effects stemming from the curving (the inclination of the walls of the thin-walledcross section towards to the beam axis) of the analysed systems. The recurrence algorithm described in Lewanowicz’s work (1976) ‘Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series’, Applicationes Mathematicae, XV(3), pp. 345–396, was used to solve the derived equations with variable coefficients. The obtained solutions of the equations have the form of series relative to Legendre polynomials. A numerical example dealing with the free vibrations of the beam was solved to verify the model and the effectiveness of the presented solution method. The results were compared with the results yielded by finite elements method (FEM).

Investigation of coupled instability for shear flexible FG sandwich I-beams subjected to variable axial force

The general thin-walled beam model is presented to study the flexural-torsional coupled buckling behavior of shear flexible sandwich I-beams made of functionally graded materials based on an orthogonal Cartesian coordinate system. The derived beam model includes the transverse shear, the restrained warping induced shear deformations, the structural coupling coming from the material anisotropy, and the variable axial force effect. Material properties of the beam such as Young’s and shear moduli are assumed to be graded across the wall thickness. The strain energy and the potential energy due to the variable axial force are introduced. The seven coupled instability equations are derived from the energy principle. To solve the instability problem, three types of finite beam elements, namely, linear, quadratic, and cubic elements are employed with the scope to discretize the governing equations. In order to verify the accuracy and efficiency of the beam model presented by this study, numerical results are presented and compared with results from other researchers. By numerical examples, the bisymmetric and mono-symmetric I-beams with two types of material distributions are considered to investigate the effects of shear deformation, variable axial force, gradient index, thickness ratio of ceramic, boundary conditions, material ratio, and span-to-height ratio on the buckling behavior of FG sandwich I-beams. Particularly, the crossover phenomenon in buckling modes with changes in gradient index and thickness ratio of ceramic in flanges is investigated.

Improved formulation for spatial free vibration of thin-walled Al/Al2O3 FG sandwich beams with non-symmetric open, single- and double-cell sections

In this study, an improved formulation is presented for the spatially coupled free vibration analysis of Al/Al2O3 thin-walled functionally graded (FG) sandwich beams with non-symmetric open, single- and double-cell sections. The thin-walled beam model is based on the Euler-Bernoulli beam theory for bending and the Vlasov theory for torsion. The FG beam consists of ceramic (Al2O3) and metal (Al) phases varying through the wall thickness. Three types of material distribution, which leads to the material coupling, are considered. The strain and kinetic energies are derived by introducing displacement parameters defined at the arbitrary chosen axis considering the effects of axial-flexural-torsional coupling and restrained warping. For finite element analysis, cubic polynomials are utilized as the shape functions of two-noded Hermitian beam element. Elastic stiffness and mass matrices for the non-symmetric cross-sections are precisely evaluated. In order to illustrate the accuracy of this formulation, the present finite element solutions of thin-walled sandwich beams with non-symmetric open, single- and double-cell sections are presented and compared with available results. Especially, the effects of various structural parameters of thin-walled FG sandwich beams such as gradient index, thickness ratios of ceramic in flanges and webs, and non-symmetricity of cross-section on the warping to torsion ratio of cross-section and the spatially coupled natural frequencies are parametrically investigated.

Progressive failure analysis for thin-walled composite beams under fatigue loads

A thin-walled beam (TWB) model for integrated progressive failure analysis in composite materials under fatigue loading is presented for the first time. The model is computationally lean and capable of tracking the spatial distribution and further propagation of fatigue-induced damage in beam-like thin-walled composite structures (TWCS) by accounting for both sudden and gradual degradation of strength and stiffness moduli of material cells located at arbitrary positions. An integrated TWB model of a cylindrical symmetrical 8-layer test structure was studied to obtain insights into damage progression patterns and interactions between different damage mechanisms. It was shown that it is possible to localize and quantify the extent of damage at individual layers while providing explanations for damage initiation and propagation in engineering terms. The proposed model provides a computationally efficient platform for performing fatigue-induced progressive failure analysis in TWCS such as rotating blades of wind turbines or helicopters, allowing the assessment of a number of what-if scenarios with current off-the-shelve computing power.

The role of nonlinear torsional contributions on the stability of flexural–torsional oscillations of open-cross section beams

An open-cross section thin-walled beam model, already developed by the authors, has been conveniently simplified while maintaining the capacity of accounting for the significant nonlinear warping effects. For a technical range of geometrical and mechanical characteristics of the beam, the response is characterized by the torsional curvature prevailing over the flexural ones. A Galerkin discretization is performed by using a suitable expansion of displacements based on shape functions. The attention is focused on the dynamic response of the beam to a harmonic force, applied at the free end of the cantilever beam. The excitation is directed along the symmetry axis of the beam section. The stability of the one-component oscillations has been investigated using the analytical model, showing the importance of the internal resonances due to the nonlinear warping coupling terms. Comparison with the results provided by a computational finite element model has been performed. The good agreement among the results of the analytical and the computational models confirms the effectiveness of the simplified model of a nonlinear open-cross section thin-walled beam and overall the important role of the warping and of the torsional elongation in the study of the one-component dynamic oscillations and their stability.

A higher order beam model for thin-walled structures with in-plane rigid cross-sections

A higher order thin-walled beam model for the analysis of thin-walled structures considering the cross-section warping and shear deformation is presented in this paper. The formulation is derived from the corresponding elasticity governing equations by adopting an approximation of the beam displacement field over the cross-section by a set of linearly independent basis functions, which are refined according to the accuracy required for the representation of the three-dimensional behaviour of the structure. A set of beam-like equations is obtained through the integration over the cross-section of the corresponding elasticity equations, properly weighted by the cross-section approximation functions. A set of uncoupled beam deformation modes is obtained from a non-linear eigenvalue problem that stems directly from the homogeneous solution of the beam differential equilibrium equations, being reduced to a generalized eigenvalue problem for a transversely rigid cross-section. The criteria put forward for uncoupling the beam deformation modes is mathematically consistent and allows the interpretation of the involved structural phenomena. Two kinds of deformation modes are obtained: (i) classic deformation modes, being associated with a null eigenvalue, requiring an adequate computation of a Jordan chain and (ii) higher order modes corresponding to the non-null eigenvalues, allowing to measure the mode decay along the beam axis. Some examples are presented in order to verify the capability of the model to simulate the non classic effects associated with the higher order deformation modes of thin-walled structures.

Definition of warping modes within the context of a higher order thin-walled beam model

A procedure for the definition of uncoupled warping modes within the framework of a higher order beam model is presented. An approximation of the displacement field over the cross-section by a set of linearly independent basis functions is considered in the models formulation so as to capture 3D structural phenomena. By considering the cross-section in-plane rigid, a linear eigenvalue problem stemming from the models governing equations is derived, allowing to retrieve classic solutions and to derive a set of hierarchical warping modes. Numerical examples are presented in order to verify the ability of the model to simulate the warping of thin-walled structures.

A non-linear rod model for folded elastic strips

We consider the equilibrium shapes of a thin, annular strip cut out in an elastic sheet. When a central fold is formed by creasing beyond the elastic limit, the strip has been observed to buckle out-of-plane. Starting from the theory of elastic plates, we derive a Kirchhoff rod model for the folded strip. A non-linear effective constitutive law incorporating the underlying geometrical constraints is derived, in which the angle the ridge appears as an internal degree of freedom. By contrast with traditional thin-walled beam models, this constitutive law captures large, non-rigid deformations of the cross-sections, including finite variations of the dihedral angle at the ridge. Using this effective rod theory, we identify a buckling instability that produces the out-of-plane configurations of the folded strip, and show that the strip behaves as an elastic ring having one frozen mode of curvature. In addition, we point out two novel buckling patterns: one where the centerline remains planar and the ridge angle is modulated; another one where the bending deformation is localized. These patterns are observed experimentally, explained based on stability analyses, and reproduced in simulations of the post-buckled configurations.