Geometrically nonlinear analysis of thin-walled composite box beams

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

A beam element for coupled torsional-flexural vibration of doubly unsymmetrical thin walled beams axially loaded

A coupled torsional-bending finite element with shear deformations and rotatory inertia for vibration of nonsymmetric thin walled beams axially loaded is developed. The equations of motion are based on Vlasov’s theory of thin-walled beams, which are modified to include an axial load. The formulation is also applicable to solid beams. The Hermite cubic polynomials are adopted as shape functions. Mass, elastic stiffness and geometrical stiffness matrices of unsymmetrical cross-section beams are presented. In order to verify the accuracy of this theory and the corresponding beam element developed, a numerical study is presented and compared with the literature and experimental tests.     

A geometrical and physical nonlinear finite element model for spatial thin-walled beams with arbitrary section

Based on the theories of Bernoulli-Euler beams and Vlasov's thin-walled members,a new geometrical and physical nonlinear beam element model is developed by applying an interior node in the element and independent interpolations on bending angles and warp,in which factors such as traverse shear deformation,torsional shear deformation and their coupling,coupling of flexure and torsion,and second shear stress are all considered.Thereafter,geometrical nonlinear strain in total Lagarange(TL) and the correspondin...     

A corotational procedure that handles large rotations of spatial beam structures

A practical motion process of the three dimensional beam element is presented to remove the restriction of small rotations between two successive increments for large displacement and large rotation analysis of space frames using incremental-iterative methods. In order to improve convergence properties of the equilibrium iteration, an n-cycle iteration scheme is introduced.

The nonlinear formulation is based on the corotational formulation. The transformation of the element coordinate system is assumed to be accomplished by a translation and two successive rigid body rotations: a transverse rotation followed by an axial rotation. The element formulation is derived based on the small deflection beam theory with the inclusion of the effect of axial force in the element coordinate system. The membrane strain along the deformed beam axis obtained from the elongation of the arc length of the beam element is assumed to be constant. The element internal nodal forces are calculated using the total deformational nodal rotations. Two methods, referred to as direct method and incremental method, are proposed in this paper to calculate the total deformational rotations.

An incremental-iterative method based on the Newton-Raphson method combined with arc length control is adopted. Numerical studies are presented to demonstrate the accuracy and efficiency of the present method.     

A nonlinear two-node superelement for use in flexible multibody systems

A nonlinear two-node superelement is proposed for the modeling of flexible complex-shaped links for use in multibody simulations. Assuming that the elastic deformations with respect to a corotational reference frame remain small, substructuring methods may be used to obtain reduced mass and stiffness matrices from a linear finite element model. These matrices are used in the derivation of potential and kinetic energy expressions of the nonlinear two-node superelement. By evaluating Lagrange’s equations, expressions for the internal and external forces acting on the superelement can be obtained. The inertia forces of the superelement are derived in terms of absolute nodal velocities and accelerations, which greatly simplifies the dynamic formulation. Three examples are included. The first two examples are used to validate the method by comparing the results with those obtained from nonlinear beam element solutions. We consider a benchmark simulation of the spin-up motion of a flexible beam with uniform cross-section and a similar simulation in which the beam is simultaneously excited in the out-of-plane direction. Results from both examples show good agreement with simulation results obtained using nonlinear finite beam elements. In a third example, the method is applied to an unbalanced rotating shaft, illustrating the potential of the proposed methodology for a more complex geometry.     

Rigid-flexible coupling dynamics of three-dimensional hub-beams system

In the previous research of the coupling dynamics of a hub-beam system, coupling between the rotational motion of hub and the torsion deformation of beam is not taken into account since the system undergoes planar motion. Due to the small longitudinal deformation, coupling between the rotational motion of hub and the longitudinal deformation of beam is also neglected. In this paper, rigid-flexible coupling dynamics is extended to a hub-beams system with three-dimensional large overall motion. Not only coupling between the large overall motion and the bending deformation, but also coupling between the large overall motion and the torsional deformation are taken into account. In case of temperature increase, the longitudinal deformation caused by the thermal expansion is significant, such that coupling between the large overall motion and the longitudinal deformation is also investigated. Combining the characteristics of the hybrid coordinate formulation and the absolute nodal coordinate formulation, the system generalized coordinates include the relative nodal displacement and the slope of each beam element with respect to the body-fixed frame of the hub, and the variables related to the spatial large overall motion of the hub and beams. Based on precise strain-displacement relation, the geometric stiffening effect is taken into account, and the rigid-flexible coupling dynamic equations are derived using velocity variational principle. Finite element method is employed for discretization. Simulation of a hub-beams system is used to show the coupling effect between the large overall motion and the torsional deformation as well as the longitudinal deformation. Furthermore, conservation of energy in case of free motion is shown to verify the formulation.     

Dynamic stiffness matrix of non-symmetric thin-walled curved beam on Winkler and Pasternak type foundations

Using the technical computing program Mathematica, the dynamic stiffness matrix for the spatially coupled free vibration analysis of thin-walled curved beam with non-symmetric cross-section on two-types of elastic foundation is newly presented based on the power series method. For this, the elastic strain energy considering the axial/flexural/torsional coupled terms, the kinetic energy including the rotary inertia effect, and the energy due to the elastic foundation are introduced. Then, equations of motion are derived from the energy principle and explicit expressions for displacement parameters are derived based on power series expansions of displacement components. Finally, the exact dynamic stiffness matrix is determined using force–displacement relations. In order to demonstrate the validity and the accuracy of this study, the natural frequencies of thin-walled curved beams with mono-symmetric and non-symmetric cross-sections are evaluated and compared with the analytical solutions and finite element solutions using Hermitian curved beam elements and ABAQUS’s shell elements. In addition, some results by a parametric study are reported.     

Non-linear finite element analysis of composite beams with interlayer slips

This article presents a new non-linear finite element formulation for the analysis of two-layer composite plane beams with interlayer slips. The element is based on the corotational method. The main interest of this approach is that different linear elements can be automatically transformed to non-linear ones. To avoid curvature locking that may occur for low order element(s), a local linear formulation based on the exact stiffness matrix is used. Five numerical applications are presented in order to assess the performance of the formulation.     

Torsional vibrations of composite bars of variable cross-section by BEM

In this paper a boundary element method is developed for the nonuniform torsional vibration problem of doubly symmetric composite bars of arbitrary variable cross-section. The composite bar consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli and are firmly bonded together. The beam is subjected to an arbitrarily distributed dynamic twisting moment, while its edges are restrained by the most general linear torsional boundary conditions. A distributed mass model system is employed which leads to the formulation of three boundary value problems with respect to the variable along the beam angle of twist and to the primary and secondary warping functions. These problems are solved employing a pure BEM approach that is only boundary discretization is used. Both free and forced torsional vibrations are considered and numerical examples are presented to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. The discrepancy in the analysis of a thin-walled cross-section composite beam employing the BEM after calculating the torsion and warping constants adopting the thin tube theory demonstrates the importance of the proposed procedure even in thin-walled beams, since it approximates better the torsion and warping constants and takes also into account the warping of the walls of the cross-section.     

A torsion bending element for thin-walled beams with open and closed cross sections

A finite element is formulated for the torsion problems of thin-walled beams. The element is based on Benscoter's beam theory, which is valid for open and also closed cross-sections. The non-polynomial interpolation presented in this paper allows the exact static solution to be obtained with only one element. Numerical results are presented for three thin-walled cantilever beams, one with a channel cross-section and the two others with rectangular cross-sections. The influence of the transverse shear strain is investigated and the different models of torsion are compared. For one example, the results obtained with one-dimensional torsion elements are compared with those obtained using shell elements.     

Stiffness analysis of locally-buckled thin-walled continuous beams

A direct iterative numerical method is presented for predicting the post-local-buckling response of thin-walled continuous structures. Nonlinearities due to local buckling and non-linear material properties are accounted for by the nonlinear moment-curvature relations of the section derived with the aid of effective width concept. Since the effective width of the compression element decreases as the stress borne by the element edge increases, the effective flexural rigidity of the cross-section varies along the member length depending upon the magnitude of the moment at the section. In the post-buckling range, the member is treated as a nonprismatic section. For continuous thin-walled structures, it is further complicated by the fact that the bending moment distribution throughout the structure and the member stiffnesses are interdependent. The proposed direct iterative solution scheme includes a stiffness matrix method of analysis in conjunction with a numerical integration procedure for evaluating the member stiffnesses. The method is employed to analyze continuous beams in the post-buckling range. Using the moment distribution of an elastic prismatic continuous beam based on the nonbuckling analysis as a first approximation, it has been found that the iterative solution scheme converges rapidly.An excellent agreement has been obtained between the results based on the method presented and from an earlier study for continuous beams. The stiffness formulation is direct and is well suited for the analysis of continuous thin-walled structures.     

A co-rotational formulation for nonlinear dynamic analysis of curved euler beam

A co-rotational finite element formulation for the dynamic analysis of a planar curved Euler beam is presented. The Euler-Bernoulli hypothesis and the initial curvature are properly considered for the kinematics of a curved beam. Both the deformational nodal forces and the inertial nodal forces of the beam element are systematically derived by consistent linearization of the fully geometrically nonlinear beam theory in element coordinates which are constructed at the current configuration of the corresponding beam element. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for the solution of the nonlinear dynamic equilibrium equations. Numerical examples are presented to demonstrate the effectiveness of the proposed element and to investigate the effect of the initial curvature on the dynamic response of the curved beam structures.     

Analysis of Large Flexible Body Deformation in Multibody Systems Using Absolute Coordinates

To consider large deformation problems in multibody system simulations afinite element approach, called absolute nodal coordinate.formulation,has been proposed. In this formulation absolute nodal coordinates andtheir material derivatives are applied to represent both deformation andrigid body motion. The choice of nodal variables allows a fullynonlinear representation of rigid body motion and can provide the exactrigid body inertia in the case of large rotations. The methodology isespecially suited for but not limited to modeling of beams, cables andshells in multibody dynamics.This paper summarizes the absolute nodal coordinate formulation for a 3D Euler–Bernoulli beam model, in particular the definition of nodal variables, corresponding generalized elastic and inertia forces and equations of motion. The element stiffness matrix is a nonlinear function of the nodal variables even in the case of linearized strain/displacement relations. Nonlinear strain/displacement relations can be calculated from the global displacements using quadrature formulae.Computational examples are given which demonstrate the capabilities of the applied methodology. Consequences of the choice of shape.functions on the representation of internal forces are discussed. Linearized strain/displacement modeling is compared to the nonlinear approach and significant advantages of the latter, when using the absolute nodal coordinate formulation, are outlined.     

Finite element formulation for dynamics of planar flexible multi-beam system

In some previous geometric nonlinear finite element formulations, due to the use of axial displacement, the contribution of all the elements lying between the reference node of zero axial displacement and the element to the foreshortening effect should be taken into account. In this paper, a finite element formulation is proposed based on geometric nonlinear elastic theory and finite element technique. The coupling deformation terms of an arbitrary point only relate to the nodal coordinates of the element at which the point is located. Based on Hamilton principle, dynamic equations of elastic beams undergoing large overall motions are derived. To investigate the effect of coupling deformation terms on system dynamic characters and reduce the dynamic equations, a complete dynamic model and three reduced models of hub-beam are prospected. When the Cartesian deformation coordinates are adopted, the results indicate that the terms related to the coupling deformation in the inertia forces of dynamic equations have small effect on system dynamic behavior and may be neglected, whereas the terms related to coupling deformation in the elastic forces are important for system dynamic behavior and should be considered in dynamic equation. Numerical examples of the rotating beam and flexible beam system are carried out to demonstrate the accuracy and validity of this dynamic model. Furthermore, it is shown that a small number of finite elements are needed to obtain a stable solution using the present coupling finite element formulation.     

Stability of composite thin-walled beams with shear deformability

In this paper, a theoretical model is developed for the stability analysis of composite thin-walled beams with open or closed cross-sections. The present model incorporates, in a full form, the shear flexibility (bending and non-uniform warping), featured in a consistent way by means of a linearized formulation based on the Reissner’s Variational Principle. The model is developed using a non-linear displacement field, whose rotations are based on the rule of semi-tangential transformation. This model allows to study the buckling and lateral stability of composite thin-walled beam with general cross-section. A finite element with two-nodes and fourteen-degrees-of-freedom is developed to solve the governing equations. Numerical examples are given to show the importance of the shear flexibility on the stability behavior of this type of structures.     

Absolute nodal coordinate plane beam formulation for multibody systems dynamics

A new plane beam dynamic formulation for constrained multibody system dynamics is developed. Flexible multibody system dynamics includes rigid body dynamics and superimposed vibratory motions. The complexity of mechanical system dynamics originates from rotational kinematics, but the natural coordinate formulation does not use rotational coordinates, so that simple dynamic formulation is possible. These methods use only translational coordinates and simple algebraic constraints. A new formulation for plane flexible multibody systems are developed utilizing the curvature of a beam and point masses. Using absolute nodal coordinates, a constant mass matrix is obtained and the elastic force becomes a nonlinear function of the nodal coordinates. In this formulation, no infinitesimal or finite rotation assumptions are used and no assumption on the magnitude of the element rotations is made. The distributed body mass and applied forces are lumped to the point masses. Closed loop mechanical systems consisting of elastic beams can be modeled without constraints since the loop closure constraints can be substituted as beam longitudinal elasticity. A curved beam is modeled automatically. Several numerical examples are presented to show the effectiveness of this method.