A non-linear model for the dynamics of open cross-section thin-walled beams—Part I: formulation

A non-linear one-dimensional model of inextensional, shear undeformable, thin-walled beam with an open cross-section is developed. Non-linear in-plane and out-of-plane warping and torsional elongation effects are included in the model. By using the Vlasov kinematical hypotheses, together with the assumption that the cross-section is undeformable in its own plane, the non-linear warping is described in terms of the flexural and torsional curvatures. Due to the internal constraints, the displacement field depends on three components only, two transversal translations of the shear center and the torsional rotation. Three non-linear differential equations of motion up to the third order are derived using the Hamilton principle. Taking into account the order of magnitude of the various terms, the equations are simplified and the importance of each contribution is discussed. The effect of symmetry properties is also outlined. Finally, a discrete form of the equations is given, which is used in Part II to study dynamic coupling phenomena in conditions of internal resonance.     

Static behavior and bifurcation of a monosymmetric open cross-section thin-walled beam: Numerical and experimental analysis

The aim of the paper is the numerical and experimental validation of a previously developed nonlinear one-dimensional model of inextensional, shear undeformable, thin-walled beam with an open cross-section. Nonlinear in-plane and out-of-plane warping and torsional elongation effects are included in the model. To better understand the role of these new contributions a beam with a section with one symmetry axis, undergoing moderately large flexural curvatures and large torsional curvature is taken into account. To obtain a section of a cantilever beam for which the torsional curvature is expected to prevail with respect to the flexural ones, a preliminary study is performed. The attention is focused on the response to static forces and on the stability of the equilibrium branches. Analytical results are compared with results of two different nonlinear finite element models and mainly with experimental results to confirm the validity of the analytical model. Interesting results are obtained for the critical values of the flexural–torsional instability loads.     

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.     

Non-linear flexural–torsional dynamic analysis of beams of arbitrary cross section by BEM

In this paper, a boundary element method is developed for the non-linear flexural–torsional dynamic analysis of beams of arbitrary, simply or multiply connected, constant cross section, undergoing moderately large deflections and twisting rotations under general boundary conditions, taking into account the effects of rotary and warping inertia. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading in both directions as well as to twisting and/or axial loading. Four boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to the angle of twist and solved using the Analog Equation Method, a BEM based method. Application of the boundary element technique leads to a system of non-linear coupled Differential–Algebraic Equations (DAE) of motion, which is solved iteratively using the Petzold–Gear Backward Differentiation Formula (BDF), a linear multistep method for differential equations coupled to algebraic equations. The geometric, inertia, torsion and warping constants are evaluated employing the Boundary Element Method. The proposed model takes into account, both the Wagner's coefficients and the shortening effect. Numerical examples are worked out to illustrate the efficiency, wherever possible the accuracy, the range of applications of the developed method as well as the influence of the non-linear effects to the response of the beam.     

Nonlinear analysis of members curved in space with warping and Wagner effects

The centroidal axis of a member that is curved in space is generally a space curve. The curvature of the space curve is not necessarily in the direction of either of the principal axes of the cross-section, but can be resolved into components in the directions of both of these principal axes. Hence, a member curved in space is primarily subjected to combined compressive, biaxial bending and torsional actions under vertical (or gravity) loading. In addition, warping actions in particular may occur in curved members with an open thin-walled cross-section, and as the deformations increase, significant interactions of the compressive, biaxial bending and torsional actions occur and profoundly nonlinear deformations are developed in the nonlinear range of structural response. This makes the nonlinear behaviour of a member curved in space very complicated, making it difficult to obtain a consistent differential equation of equilibrium for the nonlinear analysis of members curved in space. In addition, because torsion is one of the primary actions in these members, when the torsional deformations become large, the Wagner effects including both Wagner moment and the conjugate Wagner strain terms are increasingly significant and need to be included in the nonlinear analysis. This paper takes advantage of the merits of so-called “geometrically exact beam theory” and the weak form formulation of the differential equations of equilibrium in beam theory, and it develops consistent differential equations of equilibrium for the nonlinear elastic analysis of members curved in space with warping and Wagner effects. The application of the nonlinear differential equations of equilibrium to various problems is illustrated.     

Lateral buckling of beams with shear deformations – A hyperelastic formulation

The equilibrium and buckling equations are derived for the lateral buckling of a prismatic straight beam. A consistent finite strain constitutive law is used, which is based on a hyperelastic model for an isotropic material. The kinematics of the cross-sectional deformations are based on a Timoshenko type beam displacement of the cross-sectional plane using Euler angles and two shear finite rotations coupled with warping taken normal to the displaced plane. Also derived are the second order approximations to the displacements, curvatures, twist and internal actions. The constitutive relationships for the internal actions reveal new coupling terms between the bending moments, torsion and bimoment, which are functions of the cross-sectional warping and shear deformations. New Wagner type nonlinear torsion terms are derived which are functions of the warping of the cross-sectional plane, and are coupled to the twisting and shear deformations of the cross-section. Solutions are determined for the lateral buckling of a prismatic monosymmetric beam under pure bending and the flexural–torsional buckling under axial compression. For the flexural–torsional buckling problem it is found that the Euler type column buckling formula is consistent with Haringx’s column buckling formula while the torsional buckling formula is different to conventional equations. The second variation of the total potential is also derived. The effects of shear deformations are explored by examining the non-dimensional lateral buckling equation for a simply supported beam.     

Coupled bending and torsional vibration of nonsymmetrical axially loaded thin-walled Bernoulli–Euler beams

The dynamic transfer matrix is formulated for a straight uniform and axially loaded thin-walled Bernoulli–Euler beam element whose elastic and inertia axes are not coincident by directly solving the governing differential equations of motion of the beam element. Bernoulli–Euler beam theory is used, and the cross section of the beam does not have any symmetrical axes. The bending vibrations in two perpendicular directions are coupled with torsional vibration and the effect of warping stiffness is included. The dynamic transfer matrix method is used for calculation of exact natural frequencies and mode shapes of the nonsymmetrical thin-walled beams. Numerical results are given for a specific example of thin-walled beam under a variety of end conditions, and exact numerical solutions are tabulated for natural frequencies and solutions calculated by the other method are also tabulated for comparison. The effects of axial force and warping stiffness are also discussed.     

Degenerate weakly non-linear elastic plane waves

Weakly non-linear plane waves are considered in hyperelastic crystals. Evolution equations are derived at a quadratically non-linear level for the amplitudes of quasi-longitudinal and quasi-transverse waves propagating in arbitrary anisotropic media. The form of the equations obtained depends upon the direction of propagation relative to the crystal axes. A single equation is found for all propagation directions for quasi-longitudinal waves, but a pair of coupled equations occurs for quasi-transverse waves propagating along directions of degeneracy, or acoustic axes. The coupled equations involve four material parameters but they simplify if the wave propagates along an axis of material symmetry. Thus, only two parameters arise for propagation along an axis of twofold symmetry, and one for a threefold axis. The transverse wave equations decouple if the axis is fourfold or higher. In the absence of a symmetry axis it is possible that the evolution equations of the quasi-transverse waves decouple if the third-order elastic moduli satisfy a certain identity. The theoretical results are illustrated with explicit examples.     

A NEW SPATIAL THIN-WALLED BEAM ELEMENT INCLUDING TRANSVERSE AND TORSIONAL SHEAR DEFORMATION

基于Timoshenko梁及Benscoter薄壁杆件理论,建立了考虑剪切变形、弯扭耦合以及翘曲剪应力影响的空间任意开闭口薄壁截面梁单元. 通过引入单元内部结点,对弯曲转角和翘曲角采用三节点Lagrange独立插值的方法,考虑了剪切变形和翘曲剪应力的影响并避免了横向剪切锁死问题;借助载荷作用下薄壁梁的截面运动分析,在位移和应变方程中考虑了弯扭耦合的影响. 通过数值算例将该单元的计算结果与理论解以及商用有限元软件和其他文献中的数值解进行对比和验证,结果对比表明该薄壁梁单元具有良好的精度和收敛性.     

An aeroelastic model for composite rotor blades with straight and swept tips. Part I: Aeroelastic stability in hover

An analytical model for predicting the aeroelastic behavior of composite rotor blades with straight and swept tips is presented. The blade is modeled by beam type finite elements along the elastic axis. A single finite element is used to model the swept tip. The non-linear equations of motion for the finite element model are derived using Hamilton's principle and based on a moderate deflection theory and accounts for: arbitrary cross-sectional shape, pretwist, generally anisotropic material behavior, transverse shears and out-of-plane warping. Numerical results illustrating the effects of tip sweep, anhedral and composite ply orientation on blade aeroelastic behavior are presented. It is shown that composite ply orientation has a substantial effect on blade stability. At low thrust conditions, certain ply orientations can cause instability in the lag mode. The flap-torsion coupling associated with tip sweep can also induce aeroelastic instability in the blade. This instability can be removed by appropriate ply orientation in the composite construction.     

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.     

Dynamic analysis of 3-D beam elements including warping and shear deformation effects

In this paper the dynamic analysis of 3-D beam elements restrained at their edges by the most general linear torsional, transverse or longitudinal boundary conditions and subjected in arbitrarily distributed dynamic twisting, bending, transverse or longitudinal loading is presented. For the solution of the problem at hand, a boundary element method is developed for the construction of the 14 × 14 stiffness matrix and the corresponding nodal load vector of a member of an arbitrarily shaped simply or multiply connected cross section, taking into account both warping and shear deformation effects, which together with the respective mass and damping matrices lead to the formulation of the equation of motion. To account for shear deformations, the concept of shear deformation coefficients is used, defining these factors using a strain energy approach. Eight boundary value problems with respect to the variable along the bar angle of twist, to the primary warping function, to a fictitious function, to the beam transverse and longitudinal displacements and to two stress functions are formulated and solved employing a pure BEM approach that is only boundary discretization is used. Both free and forced transverse, longitudinal or torsional vibrations are considered, taking also into account effects of transverse, longitudinal, rotatory, torsional and warping inertia and damping resistance. Numerical examples are presented to illustrate the method and demonstrate its efficiency and accuracy. The influence of the warping effect especially in members of open form cross section is analyzed through examples demonstrating the importance of the inclusion of the warping degrees of freedom in the dynamic analysis of a space frame. Moreover, the discrepancy in the dynamic analysis of a member of a spatial structure arising from the ignorance of the shear deformation effect necessitates the inclusion of this additional effect, especially in thick walled cross section members.     

Non-linear buckling and postbuckling behavior of thin-walled beams considering shear deformation

The static stability of thin-walled composite beams, considering shear deformation and geometrical non-linear coupling, subjected to transverse external force has been investigated in this paper. The theory is formulated in the context of large displacements and rotations, through the adoption of a shear deformable displacement field (accounting for bending and warping shear) considering moderate bending rotations and large twist. This non-linear formulation is used for analyzing the prebuckling and postbuckling behavior of simply supported, cantilever and fixed-end beams subjected to different load condition. Ritz's method is applied in order to discretize the non-linear differential system and the resultant algebraic equations are solved by means of an incremental Newton-Rapshon method. The numerical results show that the beam loses its stability through a stable symmetric bifurcation point and the postbuckling strength is in relation with the buckling load value. Classical predictions of lateral buckling are conservative when the prebuckling displacements are not negligible and the non-linear buckling analysis is required for reliable solutions. The analysis is supplemented by investigating the effects of the variation of load height parameter. In addition, the critical load values and postbuckling response obtained with the present beam model are compared with the results obtained with a shell finite element model (Abaqus).     

Non-uniform warping including the effects of torsion and shear forces. Part II: Analytical and numerical applications

This two-part contribution presents a beam theory (BT) with a non-uniform warping (NUW) including the effects of torsion, and shear forces and valid for any homogeneous cross-section made of isotropic elastic material. In part I, the governing equations of the NUW-BT has been established and simplified-NUW-BT versions has been deduced, wherein the number of degrees of freedom is reduced. In this part II, these theories are used to analyze, for a representative set of cross-sections (CS) (solid-CS and thin-walled open/closed-CS, bi-symmetric or not), the elastic behavior of cantilever beams subjected to torsion or shear-bending. For bi-symmetrical-CS, torsion and shear-bending are analyzed separately: analytical and numerical results are given for the distributions along the beam axis of the cross-sectional displacements and stresses, for the NUW-BT and its simplified versions. Numerical results are also given for the three-dimensional stress distributions close to the embedded section: the stress predictions of the NUW-BT are compared to those obtained by three-dimensional finite elements computations. It can be drawn from all these results indications that can help to decide when the simplified theories may be applied, and hence when the warping parameters may be reduced. As specified in NUW-BT, torsion and bending are coupled for non-symmetrical-CS, even if the bending moments refer to the centroid while the torsional moment refers to the shear center. To illustrate this coupling effect, the particular example of the channel-CS presented in Kim and Kim [Kim, N.-I., Kim, M.-Y., 2005. Exact dynamic/static stiffness matrices of non-symmetric thin-walled beams considering coupled shear deformation effects. Thin-Walled Structures 43, 701–734.] is analyzed and the results are compared.     

Shear deformable bars of doubly symmetrical cross section under nonlinear nonuniform torsional vibrations—application to torsional postbuckling configurations and primary resonance excitations

In this paper a boundary element method is developed for the nonuniform torsional vibration problem of bars of arbitrary doubly symmetric constant cross section, taking into account the effects of geometrical nonlinearity (finite displacement—small strain theory) and secondary twisting moment deformation. The bar is subjected to arbitrarily distributed or concentrated conservative dynamic twisting and warping moments along its length, while its edges are subjected to the most general axial and torsional (twisting and warping) boundary conditions. The resulting coupling effect between twisting and axial displacement components is also considered and a constant along the bar compressive axial load is induced so as to investigate the dynamic response at the (torsional) postbuckled state. The bar is assumed to be adequately laterally supported so that it does not exhibit any flexural or flexural–torsional behavior. A coupled nonlinear initial boundary value problem with respect to the variable along the bar angle of twist and to an independent warping parameter is formulated. The resulting equations are further combined to yield a single partial differential equation with respect to the angle of twist. The problem is numerically solved employing the Analog Equation Method (AEM), a BEM based method, leading to a system of nonlinear Differential–Algebraic Equations (DAE). The main purpose of the present contribution is twofold: (i) comparison of both the governing differential equations and the numerical results of linear or nonlinear free or forced vibrations of bars ignoring or taking into account the secondary twisting moment deformation effect (STMDE) and (ii) numerical investigation of linear or nonlinear free vibrations of bars at torsional postbuckling configurations. Numerical results are worked out to illustrate the method, demonstrate its efficiency and wherever possible its accuracy.

     

开口厚壁截面短构件的约束扭转1）

为了分析开口厚壁截面短构件的约束扭转问题,采用统一分析梁模型与有限节线法,对T形和L形厚壁截面短构件约束扭转时横截面的翘曲和应力分布情况等问题进行了分析研究.算例计算结果表明：开口厚壁截面短构件存在与其横截面形心位置不一致的扭转（弯曲）中心,构件在不过扭转中心的外力作用下会产生弯扭耦合变形,其横截面将产生不均匀翘曲,横截面上的翘曲正应力和扭转剪应力均呈非线性分布.     

Flexural-torsional bifurcations of a cantilever beam under potential and circulatory forces I: Non-linear model and stability analysis

The stability of a cantilever elastic beam with rectangular cross-section under the action of a follower tangential force and a bending conservative couple at the free end is analyzed. The beam is herein modeled as a non-linear Cosserat rod model. Non-linear, partial integro-differential equations of motion are derived expanded up to cubic terms in the transversal displacement and torsional angle of the beam. The linear stability of the trivial equilibrium is studied, revealing the existence of buckling, flutter and double-zero critical points. Interaction between conservative and non-conservative loads with respect to the stability problem is discussed. The critical spectral properties are derived and the corresponding critical eigenspace is evaluated.     

On the uniqueness of large deflections of a uniform cantilever beam under a tip-concentrated rotational load

The problem of a uniform cantilever beam under a tip-concentrated load, which rotates in relation with the tip-rotation of the beam, is studied in this paper. The formulation of the problem results in non-linear ordinary differential equations amenable to numerical integration. A relation is obtained for the applied tip-concentrated load in terms of the tip-angle of the beam. When the tip-concentrated load acts always normal to the undeformed axis of the beam (the rotation parameter, β=0) there is a possibility of obtaining non-unique solution for the applied load. This phenomenon is also observed for other rotation parameters less than unity. When the tip-concentrated load is acting normal to the deformed axis of the beam (β=1), many load parameters are obtained for a tip-angle with different deformed configurations of the beam. However, each load parameter corresponds to a tip-angle, which confirms the uniqueness on the solution of non-linear differential equations.     

Some problems of linear elasticity for cylinders in micropolar orthotropic material

Some special problems for axisymmetric solids made of linearly elastic orthotropic micropolar material with central symmetry are dealt with. The first one is a hollow circular cylinder of unlimited length, subjected to internal and external uniform pressure. The second one is a hollow or solid circular cylinder of finite length, subjected to a relative rotation of the bases about its axis. In both cases, one of the axes of elastic symmetry is parallel to the cylinder axis; the other two are arbitrarily oriented in the plane of any cross-section of the solid. The elastic properties are invariant along the cylinder axis. It is shown that the two problems are governed by formally similar sets of ordinary differential equations in the kinematic fields (in-plane displacements and microrotations). In the general case, numerical solutions are derived. The solution for the cylinder subjected to radial pressure does not significantly differ from that obtained in classical elasticity, at least in terms of radial and hoop force stresses. In the case of a cylinder subjected to torsion the difference between the micropolar and the classical solutions is more pronounced. The torque induces twisting couple stresses about the cylinder axis of variable sign. Finally, size effects in terms of torsional inertia are pointed out.     

An asymptotic analysis of composite beams with kinematically corrected end effects

A finite element-based beam analysis for anisotropic beams with arbitrary-shaped cross-sections is developed with the aid of a formal asymptotic expansion method. From the equilibrium equations of the linear three-dimensional (3D) elasticity, a set of the microscopic 2D and macroscopic 1D equations are systematically derived by introducing the virtual work concept. Displacements at each order are split into two parts, such as fundamental and warping solutions. First we seek the warping solutions via the microscopic 2D cross-sectional analyses that will be smeared into the macroscopic 1D beam equations. The variations of fundamental solutions enable us to formulate the macroscopic 1D beam problems. By introducing the orthogonality of asymptotic displacements to six beam fundamental solutions, the end effects of a clamped boundary are kinematically corrected without applying the sophisticated decay analysis method. The boundary conditions obtained herein are applied to composite beams with solid and thin-walled cross-sections in order to demonstrate the efficiency and accuracy of the formal asymptotic method-based beam analysis (FAMBA) presented in this paper. The numerical results are compared to those reported in literature as well as 3D FEM solutions.