Cross-sectional Warping Research Articles

A coupled model to analyse dynamic behaviour of non-prismatic rotating anisotropic thin-walled beams

The current research presents a dynamically coupled mathematical model considering the energy method and employing the classical Ritz-approximation method for a thin-walled anisotropic tapered rotating box beam. The novelty of this work lies in presenting the effect of circumferentially uniform stiffness (CUS) ply layup to investigate the reverberations of not considering bending-shear couplings toward vibration behavior. The current model is formulated by considering transverse shear and cross-sectional warping of order primary and secondary. The repercussions of not including non-classical effects like Coriolis-force, centrifugal-acceleration and warping, inertial-warping is explored. Mode switching arising for change in geometry and centrifugal stiffening is presented for various ply angles and rotational speeds.

Free vibration of nonlocal radially graded nanotubes with cross-sectional warping

Free vibration of nonlocal radially graded nanotubes with cross-sectional warping

Cross-sectional warping and precision of the first-order shear deformation theory for vibrations of transversely functionally graded curved beams

Cross-sectional warping and precision of the first-order shear deformation theory for vibrations of transversely functionally graded curved beams

Dynamic stability of rectangular and circular cantilevers on Winkler-Pasternak elastic foundation by a higher-order beam theory

The structural stability of clamped-free beam-columns resting on a two-parameter foundation under a nonconservative force is investigated. The higher-order beam theory is developed by considering the warping of the cross-section, rotary inertia, and axial force’s direction. The characteristic equation of dynamic stability of rectangular/circular cantilevers on a Winkler-Pasternak foundation is obtained for a generalized follower force. Critical loads for divergence instability and flutter instability are evaluated. The influences of foundation stiffness, cross-sectional warping shape, the tangency coefficient on the critical loads are analyzed. The load-frequency interaction-curves are given for different directions of the follower force. The normal/shear stiffness of foundation on divergence and flutter loads has different effects. The normal stiffness increases the first divergence load but decreases the flutter load, reducing to the Herrmann–Smith paradox for Euler columns. The rise of shear stiffness enlarges the first divergence load and lowers the flutter loads. The effects are more sensitive for short beam-columns. Different warping shapes slightly affect the critical loads and the natural frequencies. The exponential warping shape gives the smallest critical divergence loads and the fundamental frequencies. The results of Haringx hypothesis are larger than those of Engesser hypothesis. For shorter columns, the classical results are overestimated and the higher-order beam theory is necessary.

Free vibration and buckling of functional graded box beams and the effect of the cross-sectional warping shape

Box beams have been used widely in engineering fields. This paper proposes a higher-order shear deformation beam theory to investigate the transverse vibration and stability of box beams in pure bending, especially free vibration and buckling of functionally graded box beams. A warping shape function is constructed for the first time to take shear deformation and rotary inertia of the hollow-core rectangular cross section into account simultaneously during bending. The governing equation and boundary conditions are derived. The frequency equations for typical end supports are given. The characteristic equations for buckling and critical loads are determined. Numerical results are evaluated. Besides its simplicity and easy-to-implement, the proposed theory has sufficient accuracy on the mechanical behavior of the box beams. By comparing the obtained results with the simulation results based on three-dimensional finite element method, the accuracy of the numerical results obtained is verified. The effects of parameters such as aspect ratio, gradient index, and wall thickness on the natural frequencies and critical loads are explored in detail. When box beams become rectangular beams, our model reduces to the well-known Levinson beam model.

FREE VIBRATION ANALYSIS OF COMPOSITE LAMINATED PLATES USING HOST 12

This paper presents an application of a Higher Order Shear Deformation Theory (HOST 12) to problemof free vibration of simply supported symmetric and antisymmetric angle-ply composite laminated plates.The theoretical model HOST12 presented incorporates laminate deformations which account for the effectsof transverse shear deformation, transverse normal strain/stress and a nonlinear variation of in-planedisplacements with respect to the thickness coordinate – thus modeling the warping of transverse crosssections more accurately and eliminating the need for shear correction coefficients. Solutions are obtained inclosed-form using Navier’s technique by solving the eigenvalue equation. Plates with varying number oflayers, degrees of anisotropy and slenderness ratios are considered for analysis. The results compared withthose from exact analysis and various theories from references

Warping included mixed finite elements for bending and stresses of functionally graded exact curved beams

This study presents a warping included mixed finite element formulation for the static and stress analyses of functionally graded (FG) exact curved beams. Another distinctive feature of this study is to present the range of geometric and material parameters where the first-order shear deformation theory lacks necessary precision. The constitutive equations are derived from the 3D elasticity theory of an elastic continuum. The couple effects are considered in addition to the variations of FG material constituents through the transverse/axial directions. The mixed finite element formulation is enhanced by the cross-sectional warping deformations over a displacement-type finite element formulation. The two-noded curved mixed finite elements with 12 degrees of freedom at per node are derived over exact curvature and length. Satisfactory results are obtained for the static responses and stresses of axially FG-sandwich and transversely FG exact curved beams with power-law dependence compared to the 3D behavior of brick finite elements. Finally, the influences of ellipticity, width to thickness ratio, FG material constituents, and material gradient index on the static response of FG exact curved beams are investigated.

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.

Analytical Solution and Exact Effective Thickness of Multilayered Laminated Glass Cantilevered Balustrades

Abstract Cantilevered laminated glass balustrades present design challenges. The cross-sectional warping allowed by the end constraint induces such strong asymmetrical deformations that traditional methods, defining the effective thickness of a monolith with equivalent bending properties, cannot be accurate. By using a recently proposed refined zig-zag theory for laminates, we analytically solve the representative problem of a short simply supported laminated beam with a long cantilevered overhang. The variables are the beam displacement and the mean sectional shear angle, defining the zig-zag warping of the cross section. Geometric and natural boundary conditions, as well as the matching condition at the intermediate roller constraint, necessary to solve the governing differential problem, are found variationally. The analytical solutions under concentrated and distributed loads exactly determine the effective thickness of the laminate. Comparisons are made with other approaches that that, however, apply only to three-layered packages. The expressions proposed here can be directly used in the design practice.

Advances on kinematically exact rod models for thin-walled open-section members: Consistent warping function and nonlinear constitutive equation

This work proposes two advances on kinematically exact rod models for thin-walled open section members: (i) an explicit analytical expression for the cross-section warping function, including both primary and secondary warping for arbitrary section geometries, and (ii) a nonlinear elastic constitutive equation with all higher-order strain terms included, suitable for truly finite strains. By not neglecting any of the strain terms at the constitutive level, full bending, compression and torsional strain couplings are enabled. This improves the model performance in capturing buckling phenomena and describing the developed post-critical configurations. Since the stress-resultants end up with elaborate expressions, numerical rather than analytical integration over the cross-section is preferred for their computation, which explains the need for knowing a priori the warping function (and its directional derivatives) in the entire cross-section, including in the thickness direction. The advances are implemented in a 7-DOF (3 displacements, 3 rotations and one warping intensity) rod model available in an in-house finite element program. Validation is performed using existing results from the literature as well as solutions obtained with shell models from a commercial software. The proposed advances proved to fix pathological behavior of previous rod models in both torsional and lateral–torsional buckling problems, as well as numerical convergence issues in developed post-critical configurations.

Mathematical Approach for Mechanical Behaviour Analysis of FGM Plates on Elastic Foundation

This paper presents the flexural analysis of functionally graded plates resting on elastic foundations using new two-dimensional (2D) and quasi-three-dimensional (quasi-3D) higher order shear deformation theories. The main interesting feature of this theory is that it proposes a new displacement field with undetermined integral variables which involves only five unknown functions, unlike other shear and normal deformation theories, hence making it easier to use. A parabolic transverse shear deformation shape function satisfying the zero shear stress conditions on the plate outer surfaces is considered. The elastic foundation follows the Pasternak mathematical model. The material properties change continuously across the thickness of the FG plate using different distributions: power law, exponential, and Mori–Tanaka models. The governing equations of FG plates subjected to sinusoidal and uniformly distributed loads are established through the principle of virtual works and then solved via Navier’s procedure. In this work, a detailed discussion on the influence of material composition, geometric parameters, stretching effect, and foundation parameters on the deflection, axial displacements, and stresses is given, and the obtained results are compared with those published in previous works to demonstrate the accuracy and the simplicity of the present formulations. The different obtained results were found to be in good agreement with the available solutions of other higher-order theories. The proposed model is able to represent the cross section warping in the deformed shape and to demonstrate the validity and efficiency of the approach, the findings reported herein prove that this theory is capable of predicting displacements and stresses more accurately than other theories, as its results are closer when compared to numerical methods reported in other literatures.

Porosity-dependent free vibration and transient responses of functionally graded composite plates employing higher order thickness stretching model

Tailoring the mechanical properties in hybrid materials, like functionally graded materials (FGMs), has become more pragmatic with the recent progress in material manufacturing and design. The porosities in FGMs, which develop during the manufacturing process, may become detrimental to armored plates or bulletproof designs. On the other hand, gradient-porosity distributions may have broad advantages in the design of lightweight and variable-stiffness aircraft components. The present study investigates the free vibration and transient responses of porosity-gradient FGM plates. A higher order theory, which considers the transverse shear and normal deformation, warping of the transverse cross-section, and higher order rotary inertia, is utilized for the first time to model the deformation of the porous FGM plate. The advantages of this model are (i) higher order kinematic terms for membrane and bending deformations due to the coupled membrane-bending behavior of FGMs in addition to the thickness-stretching effects, and (ii) the flexibility of applying external loads at different points along the thickness direction of the FGM plate. An analytical solution technique, popularly known as Navier’s approach, is adopted for the spatial solutions, and Newmark’s average acceleration method is utilized for the temporal solutions. The influence of the geometrical and porosity parameters on the structure’s vibration response is studied. Natural frequencies are affected significantly with uniform porosity distribution, while graded porosities can tweak vibration response in a small range but with better control. In the absence of the elasticity solutions, the present results may be considered the possible benchmark solutions for comparison of the other two-dimensional (2D) kinematic models.

Analysis of geometrically exact beams with torsion-warping deformation by weak form quadrature elements

Geometrically exact beams with torsion-warping deformation are studied. With three-dimensional finite strains, the Wagner effects as well as initial curvature and twist are taken into account in a weak form quadrature element formulation. The cross-section warping function of arbitrary cross-sections is obtained by a two-dimensional weak form quadrature element analysis, enhancing the overall accuracy of the beam formulation. Typical examples with different cross-sections and loading cases are presented to illustrate the effectiveness and robustness of the formulation.

Development of a cross-sectional finite element for the analysis of thin-walled composite beams like wind turbine blades

A method for structural analysis of thin-walled composite beams like wind turbine blades is presented. This method is based on the Nonhomogeneous Anisotropic Beam Section Analysis (NABSA) which consists in discretizing the beam cross section using finite elements. The proposed implementation uses 3-node line cross-sectional finite elements with nodes having rotational degrees of freedom to describe the cross-sectional warping displacements. Solutions obtained using this approach were verified against the corresponding analytical or numerical solutions. Agreement was very good to excellent for the computation of cross-sectional properties and distribution of stresses, strains and warping displacements for a broad range of possible composite beam behaviors including geometric and material couplings, open sections, multicell sections, and arbitrary laminates. For thin-walled layered structures, the proposed method provides models with fewer degrees of freedom than equivalent models based on a two-dimensional discretization of cross sections using triangular or quadrilateral elements such as conventional NABSA or VABS which suggests that computation time could be reduced.

Stress concentration effect on deflection and stress fields of a master leaf spring through domain decomposition and geometry updation technique

Abstract Theoretical and experimental large deflection and stress analysis of a master leaf spring considering stress concentration effect of clamping is reported. The non-uniformly curved master leaf spring under three point bending subjected to moving boundaries is modeled. Geometrically nonlinear strain-displacement relations, as necessary for the theoretical analysis, are derived through visualization of physics behind the large deformation problem. An embedded curvilinear coordinate system is considered, to study the combined effects of non-uniform curvature, bending, stretching and shear deformation including cross-sectional warping. Governing equation of the non-uniformly curved beam system is derived in variational form using energy method, based on linear material constitutive relations and the derived nonlinear kinematic relations. An iterative solution scheme through successive geometry updation is developed and executed in MATLAB® software to solve the governing equation involving strong geometric nonlinearity together with complicating moving boundary effect. Experimental deflection profiles under static loading are obtained through manual image processing technique using AutoCAD® software. Whereas, strain measurements are performed using strain gauges with data acquisition system (HBM-MX840B). Comparison between the theoretical and experimental results lead towards observation on stress concentration effect due to presence of geometric discontinuity in form of a small hole in the physical system. A modified formulation is proposed using domain decomposition method incorporating effect of geometric discontinuity through an equivalent curved beam geometry of variable cross-section. The modified theoretical model is validated successfully with the experimental results, and observations on stress characteristics and effect of hole diameter to beam width ratio are made.

Метод граничных состояний в решении задач кручения трансверсально-изотропных тел вращения

The aim of this work is to develop the method of boundary states for the class of torsion problems as applied to transversely isotropic elastic bodies of revolution. Efforts, displacements, or a combination of both are used as twisting conditions at the border. Proceeding from the general solution to the problem of cross section warping, the basis of the space of internal states is formed. The search for an internal state is reduced to the study of the boundary state isomorphic to it. The solution is a Fourier series. The proposed technique is implemented in solving the first main problem for a body in the form of a truncated cone; the second main problem for a circular cylinder; and the main mixed problem for a non-canonical body of revolution. The solution was verified and the calculation accuracy was assessed. The obtained characteristics of the elastic field have a polynomial form. The elastic field in each problem satisfies the specified boundary conditions in the form of their distribution over the surface and does not satisfy them only in the integral sense.

Computational Enhancement of a Mixed 3D Beam Finite Element with Warping and Damage

This paper describes the computational aspects of the beam Finite Element formulation recently developed by the authors to simulate the nonlinear response of structural members subjected to shear and torsion, accounting for cross-section warping. The paper focuses on an efficient consistent solution algorithm that by-passes the iterative procedure required in force-based and mixed Finite Elements and makes the model easy to be implemented in a standard code. Moreover, it proposes a new non-iterative technique to condense out the stress components derived by the three-dimensional constitutive response and not directly included in the fiber section formulation. The efficiency and accuracy of the proposed numerical model are validated by simulating the response of steel and reinforced concrete structural members.

Buckling analysis of open-section beams with thin-walled functionally graded materials along the contour direction

The buckling and lateral buckling of thin-walled functionally graded (FG) open-section beams with various types of material distributions are studied. The approach is based on assumption that the volume fraction of particles varies through the contour direction according to a power law. The governing buckling equation and a finite element method have been developed to formulate the problem. Warping of cross-section and all the structural coupling coming from anisotropy of material are taken into account in this study. The critical load is obtained for thin-walled FG channel-section with arbitrary distributions of material. The lateral buckling parameter and mechanism are expressed for thin-walled FG mono-symmetric I-section beam under uniformly distributed load and pure bending with several types of material distributions. For the validity of the proposed theory, the number comparisons are compared with those of formerly published work. The effects of the material distribution, load height and span-to-height on the buckling characteristics are examined in detail and highlighted.

ВЛИЯНИЕ НАКЛОННЫХ РЕБЕР ЖЕСТКОСТИ В УЗЛАХ СОЕДИНЕНИЯ ДВУТАВРОВ НА ПЕРЕДАЧУ БИМОМЕНТОВ

Shear-free theory of V.Z. Vlasov remains one of the most reasonable approaches to calculating thin-walled bars taking into account constrained torsion. At the same time, the use of this theory for the analysis of deformations of frame structures still requires research in terms of the conditions for transferring forces in bar connections. As noted in some scientific papers, the balance of bimoments can be significantly broken at the joints of thin-walled bars of an open profile in some designs. This paper deals with this phenomenon for steel I-beam profiles, associated with the presence of inclined stiffeners in joint units. Using shell finite element models, the influence of inclined stiffeners on the appearance of bimoment jumps at the pairwise connection of bars is shown. A dependence is derived that makes it possible to take into account the stiffness of the inclined edge in the bar models from the point of view of the restraint of cross-section warping. On the basis of numerical experiments, it was determined that the introduction of such stiffness into the bar finite element schemes of frame structures allows to reflect the condition of bars interaction in the transferring of bimoments with a sufficiently high accuracy for engineering practice.

Enriched beam finite element models with torsion and shear warping for the analysis of thin-walled structures

This paper presents three beam Finite Element (FE) formulations developed for the analysis of thin-walled structures. These account for out-of-plane cross-section warping by removing the classical rigid body cross-section hypothesis and capture the interaction of axial/bending stress components with shear and torsion.The beam FE models rely on different kinematic assumptions to describe out-of-plane cross-section deformations. Indeed, warping displacement field is interpolated in the element volume according to different approaches, with increasing level of accuracy and detail. First two models adopt a coarse warping description, where warping displacement field is defined as the linear combination of assumed warping profiles and unknown kinematic parameters. In the first model, these are considered as equal to the generalized cross-section torsional curvature and shear strains and a classical displacement-based formulation is adopted to derive the element governing equations. In the second model, warping parameters are assumed as independent kinematic quantities and a mixed approach is considered to derive the FE formulation. Third model, also relying on a mixed formulation, independently interpolates warping by introducing additional degrees of freedom on the cross-section plane, thus, resulting in a richer description of the out-of-plane deformations. This latter is also adopted to propose a numerical procedure for the warping profile evaluation of thin-walled beams subjected to torsional and shear forces, for general cross-section geometry.The efficiency and accuracy of the proposed FE formulations are validated by simulating the response of thin-walled structures under torsion and coupled torsion/shear actions and the influence of the kinematic assumptions characterizing each formulation is discussed.