A higher order model for thin-walled structures with deformable cross-sections

A higher order model for the analysis of linear, prismatic thin-walled structures that considers the cross-section warping together with the cross-section in-plane flexural deformation is presented in this paper. The use of a one-dimentional model for the analysis of thin-walled structures, which have an inherent complex three-dimensional (3D) behaviour, can only be successful and competitive when compared with shell finite element models if it fulfills a twofold objective: (i) an enrichment of the model in order to as accurately as possible reproduce its 3D elasticity equations and (ii) the definition of a consistent criterion for uncoupling the beam equations, allowing to identify structural deformation modes.The displacement field is approximated through a linear combination of products between a set of linear independent functions defined over the cross-section and the associated weights only dependent on the beam axis; this approximation is not constrained by any ab initio kinematic assumptions. Towards an efficient application of the approximation procedure, the cross-section is discretized into thin-walled elements, being the displacement field approximated for each element independently of the displacement direction. The approximation is thus hp refined enhancing the “capture” of the 3D structural mechanics of thin-walled structures. The beam model governing equations are obtained through the integration over the cross-section of the corresponding elasticity equations weighted by the cross-section global approximation functions.A criterion for uncoupling the beam governing equations is established, allowing to (i) retrieve the classic equations of the thin-walled beam theory both for open and closed sections and (ii) derive a set of uncoupled deformation modes representing higher order effects. The criterion is based on the solution of the polynomial eigenvalue problem associated with the beam differential equations, allowing to quantify the Saint-Venant principle for thin-walled structures. In fact, the solution of the non linear eigenvalue problem yields a twelve fold null eigenvalue (representing polynomial solutions) that are verified to represent beam classic solutions and sets of pairs and quadruplets of non-null eigenvalues corresponding to higher order modes of deformation.     

Free vibration and spatial stability of non-symmetric thin-walled curved beams with variable curvatures

An improved formulation for free vibration and spatial stability of non-symmetric thin-walled curved beams is presented based on the displacement field considering variable curvature effects and the second-order terms of finite-semitangential rotations. By introducing Vlasov’s assumptions and integrating over the non-symmetric cross-section, the total potential energy is consistently derived from the principle of virtual work for a continuum. In this formulation, all displacement parameters and the warping function are defined at the centroid axis and also thickness-curvature effects and Wagner effect are accurately taken into account. For F.E. analysis, a thin-walled curved beam element is developed using the third-order Hermitian polynomials. In order to illustrate the accuracy and the practical usefulness of the present method, numerical solutions by this study are presented with the results analyzed by ABAQUS’ shell elements. Particularly, the effect of arch rise to span length ratio is investigated on vibrational and buckling behaviour of non-symmetric curved beams.     

Prandtl’s formulation for the Saint–Venant’s torsion of homogeneous piezoelectric beams

The Saint–Venant torsional problem for homogeneous, monoclinic piezoelectric beams is formulated in terms of Prandtl’s stress function and electric displacement potential function. The analytical approach presented in this paper generalizes the known formulation of Prandtl’s solution which refers to homogeneous elastic beams. The Prandtl’s stress function and electric displacement potential function satisfy the so called coupled Dirichlet problem (CDP) in the cross-sectional domain. A direct and a variational formulation are developed. Exact analytical solutions for solid elliptical cross-section and hollow circular cross-section and an approximate solution based on a variational formulation for thin-walled closed cross-section are presented.     

A finite element beam model including cross-section distortion in the absolute nodal coordinate formulation

A new class of beam finite elements is proposed in a three-dimensional fully parameterized absolute nodal coordinate formulation, in which the distortion of the beam cross section can be characterized. The linear, second-order, third-order, and fourth-order models of beam cross section are proposed based on the Pascal triangle polynomials. It is shown that Poisson locking can be eliminated with the proposed higher-order beam models, and the warping displacement of a square beam is well described in the fourth-order beam model. The accuracy of the proposed beam elements and the influence of cross-section distortion on structure deformation and dynamics are examined through several numerical examples. We find that the proposed higher-order models can capture more accurately the structure deformation such as cross-section distortion including warping, compared to the existing beam models in the absolute nodal coordinate formulation.     

Component-wise analysis of laminated anisotropic composites

This paper proposes a one-dimensional (1D) refined formulation for the analysis of laminated composites which can model single fibers and related matrices, layers and multilayers. Models built by means of an arbitrary combination of these four components lead to a component-wise analysis. Different scales can be used in different portions of the structure and this leads to a global–local approach. In this work, computational models were developed in the framework of finite element approximations. The 1D FE formulation used has hierarchical features, that is, 3D stress/strain fields can be detected by increasing the order of the 1D model used. The Carrera Unified Formulation (CUF) was exploited to obtain advanced displacement-based theories where the order of the unknown variables over the cross-section is a free parameter of the formulation. Taylor- and Lagrange-type polynomials were used to interpolate the displacement field over the element cross-section. Lagrange polynomials permitted the use of only pure displacements as unknown variables. The related finite element led straightforwardly to the assembly of the stiffness matrices at the structural element interfaces (matrix-to-fiber, matrix-to-layer, layer-to-layer etc). Preliminary assessments with solid model results are proposed in this paper; various numerical examples were carried out on cross-ply symmetrical fiber-reinforced laminates [0/90/0] and a more complex composite C-shaped model. The examples show that the proposed models can analyze laminated structures by combining fibers, matrices, layers and multilayers and by referring to a unique structural finite element formulation.     

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.     

On the cross-section distortion of thin-walled beams with multi-cell cross-sections subjected to bending

This paper deals with distortion of the cross-section contour of thin-walled beams with simple multi-cell closed rectangular cross-sections. The cross-section distortion is considered in the limit case. It is assumed that beam plates are hinged together along their longitudinal edges. Double symmetric three and two-cell closed cross-sections are considered. The stresses and displacements are obtained in the closed analytical form. The additional stresses and displacements due to distortion with respect to the stresses and displacements of the ordinary theory of bending are obtained. The boundary conditions are given in the general form. Some illustrative examples are given.     

Improved torsional analysis of laminated box beams

The improved torsional analysis of the laminated box beams with single- and double-celled sections subjected to a torsional moment is performed by introducing 14 displacement parameters. For this, a thin-walled laminated box beam theory considering the effects of shear and elastic couplings is presented. The governing equations and the force-displacement relations are derived from the variation of the strain energy. The system of linear algebraic equations with non-symmetric matrix is constructed by introducing the displacement parameters and by transforming the higher order simultaneous differential equations into first order ones. This numerical technique determines eigenmodes corresponding to 12 zero and 2 non-zero eigenvalues and derives displacement functions for displacement parameters based on the undetermined parameter method. Finally, the element stiffness matrix is determined using the member force-displacement 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 using the Lagrangian interpolation polynomials and three-dimensional analysis results using the shell elements of ABAQUS for coupled laminated beams with single- and double-celled sections.     

Non-linear elastic non-uniform torsion of bars of arbitrary cross-section by BEM

In this paper an elastic non-uniform torsion analysis of simply or multiply connected cylindrical bars of arbitrary cross-section taking into account the effect of geometric non-linearity is presented employing the boundary-element(BE) method. The torque-rotation relationship is computed based on the finite-displacement (finite-rotation) theory, that is the transverse displacement components are expressed so as to be valid for large rotations and the longitudinal normal strain includes the second-order geometric non-linear term often described as the “Wagner strain”. The proposed formulation does not stand on the assumption of a thin-walled structure and therefore the cross-section's torsional rigidity is evaluated exactly without using the so-called Saint-Venant's torsional constant. The torsional rigidity of the cross-section is evaluated directly employing the primary warping function of the cross-section depending on its shape. Three boundary-value problems with respect to the variable along the beam axis angle of twist, to the primary and to the secondary warping functions are formulated. The first one, employing the Analog Equation Method (a BEM-based method), yields a system of non-linear equations from which the angle of twist is computed by an iterative process. The remaining two problems are solved employing a pure BE method. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The developed procedure retains most of the advantages of a BEM solution over a pure domain discretization method, although it requires domain discretization.     

Analytical test functions for free vibration analysis of rotating non-homogeneous Timoshenko beams

In this paper, the governing equations for free vibration of a non-homogeneous rotating Timoshenko beam, having uniform cross-section, is studied using an inverse problem approach, for both cantilever and pinned-free boundary conditions. The bending displacement and the rotation due to bending are assumed to be simple polynomials which satisfy all four boundary conditions. It is found that for certain polynomial variations of the material mass density, elastic modulus and shear modulus, along the length of the beam, the assumed polynomials serve as simple closed form solutions to the coupled second order governing differential equations with variable coefficients. It is found that there are an infinite number of analytical polynomial functions possible for material mass density, shear modulus and elastic modulus distributions, which share the same frequency and mode shape for a particular mode. The derived results are intended to serve as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of rotating non-homogeneous Timoshenko beams.     

Stability of Thin-Walled Open Beams Under Nonconservative Loads

ABSTRACT

A finite element formulation for free-vibration analysis of straight prismatic beams of general thin-walled open cross-section, under conservative and nonconservative loads, is presented. The formulation is used to calculate the flutter load for a number of beam problems and is verified by comparison with pre-existing numerical solutions.     

The dimensional reduction modelling approach for 3D beams: Differential equations and finite-element solutions based on Hellinger–Reissner principle

This paper illustrates an application of the so-called dimensional reduction modelling approach to obtain a mixed, 3D, linear, elastic beam-model.We start from the 3D linear elastic problem, formulated through the Hellinger–Reissner functional, then we introduce a cross-section piecewise-polynomial approximation, and finally we integrate within the cross section, obtaining a beam model that satisfies the cross-section equilibrium and could be applied to inhomogeneous bodies with also a non trivial geometries (such as L-shape cross section). Moreover the beam model can predict the local effects of both boundary displacement constraints and non homogeneous or concentrated boundary load distributions, usually not accurately captured by most of the popular beam models.We modify the beam-model formulation in order to satisfy the axial compatibility (and without violating equilibrium within the cross section), then we introduce axis piecewise-polynomial approximation, and finally we integrate along the beam axis, obtaining a beam finite element. Also the beam finite elements have the capability to describe local effects of constraints and loads. Moreover, the proposed beam finite element describes the stress distribution inside the cross section with high accuracy.In addition to the simplicity of the derivation procedure and the very satisfying numerical performances, both the beam model and the beam finite element can be refined arbitrarily, allowing to adapt the model accuracy to specific needs of practitioners.     

薄壁杆系结构的梁元分析模型

本文导出了用于薄壁杆系结构弹性分析的薄壁梁元分析模型，在空间梁元分析模型^[3]的基础上，采用了一种改进的位移模式，考察了薄壁杆件可能发生的拉压，剪切，弯曲，扭转和翘曲等各变形形式以及它们的耦合效应，得出了相应的单元形函数，同时从工程应变的定义出发，采用Taylor级数展开的方法，建立了单元的五阶近似正交变表达式，并建立了相应的薄壁单元刚度方程，从而得出了局部坐标系下单元刚度矩阵的显式，根据本文所导出的薄壁梁元分析模型，编制了相应的结构计算程序，通过算例验证了本文所推导的单元刚度矩阵，同时通过与传统空间梁元计算模型计算结果的比较，阐述了截面翘曲对薄壁杆系结构的影响。     

Stochastic response of an axially loaded composite Timoshenko beam exhibiting bending–torsion coupling

A spectral finite element method is proposed to investigate the stochastic response of an axially loaded composite Timoshenko beam with solid or thin-walled closed section exhibiting bending–torsion materially coupling under the stochastic excitations with stationary and ergodic properties. The effects of axial force, shear deformation (SD) and rotary inertia (RI) as well as bending–torsion coupling are considered in the present study. First, the damped general governing differential equations of motion of an axially loaded composite Timoshenko beam are derived. Then, the spectral finite element formulation is developed in the frequency domain using the dynamic shape functions based on the exact solutions of the governing equations in undamped free vibration, which is used to compute the mean square displacement response of axially loaded composite Timoshenko beams. Finally, the proposed method is illustrated by its application to a specific example to investigate the effects of bending–torsion coupling, axial force, SD and RI on the stochastic response of the composite beam.     

Free vibration of horizontally curved thin-walled beams with rectangular hollow sections considering two compatible displacement fields

A finite element is presented for vibration analyses of horizontally curved thin-walled rectangular hollow beams. Eight cross-section deformation modes are employed to describe the mid-surface contour displacement field with the modal superposition method. Focused on the in-plane moment equilibrium condition and the displacement continuity condition, two compatible displacement fields are constructed to calculate the strain energy and the kinetic energy of the beam, respectively. With the application of Hamilton’s principle the dynamic governing equations are formulated, and then approximated for the finite element implementation. Finally, numerical examples are illustrated to verify the validity of the present theory.     

Coupled refined layerwise theory for dynamic free and forced response of piezoelectric laminated composite and sandwich beams

This work extends a previously presented coupled refined layerwise theory to dynamic analysis of piezoelectric laminated composite and sandwich beams. Contrary to most of the available theories, all the kinematic and stress boundary conditions are satisfied at the interfaces of the piezoelectric layers with the non-zero longitudinal electric field. Moreover, both electrical transverse normal strains and transverse flexibility are taken into account for the first time in the present theory. In the presented formulation a high-order polynomial, an exponential expression and a layerwise term containing the electric field are included in the describing expression of the in-plane displacement of the beam. For the transverse displacement, the coupled refined model uses a combination of continuous piecewise fourth-order polynomials with a layerwise representation of electrical unknowns. The electric field is also approximated as linear across the thickness direction of piezoelectric layers. One of advantages of the present theory is that the mechanical number of the unknown parameters is very small and is independent of the number of the layers. For validation of the proposed model, various free and forced vibration tests for thin and thick laminated/sandwich piezoelectric beams are carried out. For various electrical and mechanical boundary conditions, excellent correlation has been found between the results obtained from the proposed formulation with those resulted from the three-dimensional theory of piezoelasticity.     

确定性载荷作用下Timoshenko薄壁梁的弯扭耦合动力响应

建立了一种普遍的解析理论用于求解确定性载荷作用下Timoshenko薄壁梁的弯扭耦合动力响应。首先通过直接求解单对称均匀Timoshenko薄壁梁单元弯扭耦合振动的运动偏微分方程，给出了计算其自由振动的精确方法，并导出了Timoshenko弯扭耦合薄壁梁自由振动主模态的正交条件。然后利用简正模态法研究了确定性载荷作用下单对称Timoshenko薄壁梁的弯扭耦合动力响应，该弯扭耦合梁所受到的荷载可以是集中载荷或沿着梁长度分布的分布载荷。最后假定确定性载荷是谐波变化的，得到了各种激励下封闭形式的解，并对动力弯曲位移和扭转位移的数值结果进行了讨论。     

Transverse shear and normal deformation effects on vibration behaviors of functionally graded micro-beams

A quasi-three dimensional model is proposed for the vibration analysis of functionally graded(FG) micro-beams with general boundary conditions based on the modified strain gradient theory. To consider the effects of transverse shear and normal deformations, a general displacement field is achieved by relaxing the assumption of the constant transverse displacement through the thickness. The conventional beam theories including the classical beam theory, the first-order beam theory, and the higher...     

Buckling of short beams with warping effect included

A beam theory for the stability analysis of short beam that includes shear deformation and warping of the cross-section is developed. The warping of the cross-section is taken to be an independent kinematics quantity and corresponding force resultants are defined. For the beam subjected to the external loading only at the ends of the beam, equilibrium equations have been obtained by the principle of virtual work. The variations of lateral displacement, rotational angle of the cross-section and the multiplier of the warping shape along the beam axis are solved in closed form and expressed in terms of deformation quantities at the ends of the beam. Based on this beam theory, the lateral stiffness of the beam sustained an axial compression force and a lateral shear force at one end is explicitly derived, from which the equation of the buckling load is established and the buckling load can be solved. When the effect of cross-section warping is neglected, the derived lateral stiffness and buckling load converge to the solutions of the Haringx theory.     

The approximate analytical solution for the buckling loads of a thin-walled box column with variable cross-section

1.IntroductionThin-walledboxcolumnswithvariablecross-sectionareextensivelyusedascompressionmembersforhighbridgepiers,waterandtelevisiontowersandothersimilarstructures.Atpresent,however,fewpapersdealingwiththecriticalloadsoftorsional-fie-curalbucklingforthiskindofstructurescanbeseen.Inthispaper,bymeansoftheenergyprincipleandtheGalerkin'smethod,theapproximateexpressionsforcalculatingthecriticalloadsoffie-curalandtorsionalbucklingaredevelopedrespectively.Anapplicablecomputerprogramisworkedout,an…