Geometrically non-linear formulation for three dimensional curved beam elements with large rotations

This paper presents a geometrically non-linear formulation (GNL) for the three dimensional curved beam elements using the total Lagrangian approach. The element geometry is constructed using co-ordinates of the nodes on the centroidal or reference axis and the orthogonal nodal vectors representing the principal bending directions. The element displacement field is described using three translations at the element nodes and three rotations about the local axes

1 The element displacement field has also been described in the literature using Euler parameters, Milenkovic parameters, or Rodriges parameters representing the effects of large rotations.

. The GNL three dimensional beam element formulations based on these element approximations are restricted to small nodal rotations between two successive load increments. The element formulation presented here removes such restrictions. This is accomplished by retaining non-linear nodal terms in the definition of the element displacement field, and the consistent derivation of the element properties. The formulation presented here is very general and yet can be made specific by selecting proper non-linear functions representing the effects of nodal rotations. The details of the element properties are presented and discussed. Numerical examples are also presented to demonstrate the behaviour and the accuracy of the elements. A comparison of the results obtained from the present formulation with those available in the literature using a linearized element approximation clearly demonstrate the superiority of the formulation in terms of large load steps, large rotations between two load steps and extremely good convergence characteristics during equilibrium iterations. The displacement approximation of these elements is fully compatible with the isoparametric curved shell elements (with large rotations), and since the elements possess offset capability, these elements can also serve as stiffeners for the curved shells.     

A new approach for displacement functions of a curved Timoshenko beam element in motions normal to its initial plane

In existing literature, either analytical methods or numerical methods, the formulations for free vibration analysis of circularly curved beams normal to its initial plane are somewhat complicated, particularly if the effects of both shear deformation (SD) and rotary inertia (RI) are considered. It is hoped that the simple approach presented in this paper may improve the above‐mentioned drawback of the existing techniques. First, the three functions for axial (or normal to plane) displacement and rotational angles about radial and circumferential (or tangential) axes of a curved beam element were assumed. Since each function consists of six integration constants, one has 18 unknown constants for the three assumed displacement functions. Next, from the last three displacement functions, the three force–displacement differential equations and the three static equilibrium equations for the arc element, one obtained three polynomial expressions. Equating to zero the coefficients of the terms in each of the last three expressions, respectively, one obtained 17 simultaneous equations as functions of the 18 unknown constants. Excluding the five dependent ones among the last 17 equations, one obtained 12 independent simultaneous equations. Solving the last 12 independent equations, one obtained a unique solution in terms of six unknown constants. Finally, imposing the six boundary conditions at the two ends of an arc element, one determined the last six unknown constants and completely defined the three displacement functions. By means of the last displacement functions, one may calculate the shape functions, stiffness matrix, mass matrix and external loading vector for each arc element and then perform the free and forced vibration analyses of the entire curved beam. Good agreement between the results of this paper and those of the existing literature confirms the reliability of the presented theory. Copyright © 2005 John Wiley & Sons, Ltd.     

On the dynamic behaviour of the Timoshenko beam finite elements

First, various finite element models of the Timoshenko beam theory for static analysis are reviewed, and a novel derivation of the 4 × 4 stiffness matrix (for the pure bending case) of the superconvergent finite element model for static problems is presented using two alternative approaches: (1) assumed-strain finite element model of the conventional Timoshenko beam theory, and (2) assumed-displacement finite element model of a modified Timoshenko beam theory. Next, dynamic versions of various finite element models are discussed. Numerical results for natural frequencies of simply supported beams are presented to evaluate various Timoshenko beam finite elements. It is found that the reduced integration element predicts the natural frequencies accurately, provided a sufficient number of elements is used. The research reported herein is supported by theOscar S. Wyatt Endowed Chair.     

Geometrically nonlinear formulation for the curved shell elements

This paper presents a geometrically nonlinear formulation using total lagrangian approach for the three-dimensional curved shell elements. The basic element geometry is constructed using the coordinates of the middle surface nodes and the mid-surface nodal point normals. The element displacement field is described using three translations of the mid-surface nodes and the two rotations about the local axes. The existing shell element formulations are restricted to small nodal rotations between two successive load increments. The element formulation presented here removes such restrictions. This is accomplished by retaining nonlinear nodal rotation terms in the definition of the displacement field and the consistent derivation of the element properties. The formulation presented here is very general and yet can be made specific by selecting proper nonlinear functions representing the effects of nodal rotations. The element properties are derived and presented in detail. Numerical examples are also presented to demonstrate the behaviour and the accuracy of the elements.     

An effective finite rotation formulation for geometrical non-linear shell structures

This paper presents a simple stress resultant 4-node shell element for geometrical non-linear analysis. In order to model smooth surfaces and/or stiffened structures, a simple and efficient technique for finite rotation is adopted. By means of suppressing the component of singular rotation effectivley, convenient use of six degrees of freedom is possible without deteriorating the robustness and the convergence rate of the classical 5-dof formulation. In the formulation of shell element, section eccentricity is also considered to model stiffened structures. Through numerical experiments the effectiveness of the proposed method is demonstrated. Received 16 November 2000     

Numerical transfer-method with boundary conditions for arbitrary curved beam elements

A novel numerical transfer-method is presented to solve a system of linear ordinary differential equations with boundary conditions. It is applied to determine the structural behaviour of the classical problem of an arbitrary curved beam element. The approach of this boundary value problem yields a unique system of differential equations. A Runge–Kutta scheme is chosen to obtain the incremental transfer expression. The use of a recurrence strategy in this equation permits to relate both ends in the domain where boundary conditions are defined. Semicircular arch, semicircular balcony and elliptic–helical beam examples are provided for validation.     

Finite elements with embedded displacement discontinuity: a generalized variable formulation

This paper is intended to bring a contribution towards a satisfactory simulation of those fracture phenomena which result in the appearance and development of discrete cracks. To this purpose, a general mixed finite element formulation is proposed, based on the concept of generalized variables in Prager's sense. The displacement field inside an element is modelled by the sum of two contributions: a regular (continuous) part which is governed by standard shape functions, and a possibly discontinuous one which is introduced soon after a suitable criterion is satisfied. The formulation is first specialized to a one‐dimensional case, then a triangular element for two‐dimensional problems is described in detail. Analytical and numerical examples are presented in order to clarify the formulation and to point out the essential role of inter‐element conformity. Copyright © 2000 John Wiley & Sons, Ltd.     

A consistent finite element formulation for non-linear dynamic analysis of planar beam

A co-rotational finite element formulation for the dynamic analysis of planar Euler beam is presented. Both the internal nodal forces due to deformation and the inertia nodal forces are systematically derived by consistent linearization of the fully geometrically non-linear beam theory using the d'Alembert principle and the virtual work principle. Due to the consideration of the exact kinematics of Euler beam, some velocity coupling terms are obtained in the inertia nodal forces. An incremental-iterative method based on the Newmark direct integration method and the Newton–Raphson method is employed here for the solution of the non-linear dynamic equilibrium equations. Numerical examples are presented to investigate the effect of the velocity coupling terms on the dynamic response of the beam structures.     

Total Lagrangian formulation for the large displacement analysis of rectangular plates

In this paper, a method for the non-linear dynamic analysis of rectangular plates that undergo large rigid body motions and small elastic deformations is presented. The large rigid body displacement of the plate is defined by the translation and rotation of a selected plate reference. The small elastic deformation of the midplane is defined in the plate co-ordinate system using the assumptions of the classical theories of plates. Non-linear terms that represent the dynamic coupling between the rigid body displacement and the elastic deformation are presented in a closed form in terms of a set of time-invariant scalars and matrices that depend on the assumed displacement field of the plate. In this paper, the case of simple two-parameter screw displacement, where the rigid body translation and rotation of the plate reference are, respectively, along and about an axis fixed in space, is first considered. The non-linear dynamic equations that govern the most general and arbitrary motion of the plate are also presented and both lumped and consistent mass formulations are discussed. The non-linear dynamic formulation presented in this paper can be used to develop a total Lagrangian finite element formulation for plates in multibody systems consisting of interconnected structural elements.     

On a geometrically exact curved/twisted beam theory under rigid cross-section assumption

 A geometrically exact curved/ twisted beam theory, that assumes that the beam cross-section remains rigid, is re-examined and extended using orthonormal frames of reference starting from a 3-D beam theory. The relevant engineering strain measures with an initial curvature correction term at any material point on the current beam cross-section, that are conjugate to the first Piola-Kirchhoff stresses, are obtained through the deformation gradient tensor of the current beam configuration relative to the initially curved beam configuration. The stress resultant and couple are defined in the classical sense and the reduced strains are obtained from the three-dimensional beam model, which are the same as obtained from the reduced differential equations of motion. The reduced differential equations of motion are also re-examined for the initially curved/twisted beams. The corresponding equations of motion include additional inertia terms as compared to previous studies. The linear and linearized nonlinear constitutive relations with couplings are considered for the engineering strain and stress conjugate pair at the three-dimensional beam level. The cross-section elasticity constants corresponding to the reduced constitutive relations are obtained with the initial curvature correction term. Along with the beam theory, some basic concepts associated with finite rotations are also summarized in a manner that is easy to understand. Received: 17 June 2002 / Accepted: 21 January 2003 The work was partly sponsored by a grant (CDAAH04-95-1-0175) from the Army Research Office with Dr. Gary Anderson as the grant monitor. We would also like to thank Prof. Raymond Plaut of Dept. of Civil and Environmental Engineering at Virginia Polytechnic Institute and State University for his technical help.     

A boundary element formulation with equilibrium satisfaction for thermo-mechanical problems considering transient and non-linear aspects

Boundary element formulations incorporating consistent transient potential theory, satisfying exact energy balance, and dynamic equilibrium satisfaction with respect to the co-ordinate axis directions and moments, including inertial forces, elastoplastic deformations and thermal loadings, are presented. The procedures are quite general and can be implemented into existing boundary element codes. The required expressions for the transient potential analysis and the dynamic formulation are discussed, and we include two examples that take into account linear and non-linear material behaviour to illustrate the potential of the proposed methodology.     

On the strain-displacement equations of a geometrically nonlinear curved beam

The paper is concerned with the strain-displacement equations of a curved beam undergoing arbitrarily large deflections/rotations. Two alternative procedures are used for this derivation: a mathematically consistent and an engineering-type approach. It is demonstrated that upon making certain simplifying assumptions regarding higher-order terms both approaches lead to identical results. VDI     

On curved finite elements for the analysis of circular arches

This paper proposes a straightforward criterion to warrant the displacement functions being used in the finite element approximation of circular arches. The criterion was established by studying the natural shape function, i.e. the exact solution of the deformed shape, of the circular arch element. The exact stiffness matrix [ K ]_exact is derived from the natural shape and is confirmed to be the inverse of the well-known flexibility matrix [ F ]_exact in the curved beam theory. The present paper compares the inverse [ K ]^?1 of the stiffness matrix derived from the assumed displacement function with the [ F ]_exact. It is shown that the procedure also guarantees the implicit inclusion of rigid-body modes in the pertinent stiffness matrix [ K ]. Case studies on typical approximate displacement functions assure the appropriateness as well as the ease of application of the proposed method.     

Winkler地基上修正Timoshenko梁振动分析

利用Bernoulli-Euler梁理论建立的弹性地基梁模型应用广泛,但其在高阶频率及深梁计算中误差较大,利用修正的Timoshenko梁理论建立新的弹性地基梁振动微分方程,由于其在Timoshenko梁的基础上考虑了剪切变形所引起的转动惯量,因而具有更好的精确度。利用ANAYS beam54梁单元进行振动模态的有限元计算,所求结果与理论基本无误差,从而验证了该理论的正确性。基于修正Timoshenko梁振动理论推导出了弹性地基梁双端自由-自由、简支-简支、简支-自由、固支-固支等多种边界条件下的频率超越方程及模态函数。分析了弹性地基梁在不同理论下不同约束条件及不同高跨比情况下的计算结果,从而论证了该理论计算弹性地基梁的适用性。分析了不同弹性地基梁理论下波速、群速度与波数的关系。得到了约束条件和梁长对振动模态及地基刚度对振动频率有重要影响等结论。     

On the contact behavior of a buckled Timoshenko beam constrained laterally by a plane wall

This paper discusses the effect of shear deformation of the cross-section on the contact behavior of a buckled Timoshenko beam in frictionless contact with a rigid plane wall. The deformation is assumed to be small, and only planar deformation is considered. The results are compared with those from the Euler beam model. In the case when a buckled Euler beam is in line contact with a plane wall, the plane wall exerts concentrated forces at the boundary points of the line-contact segment but no distributed force. In the case of a Timoshenko beam, on the other hand, the concentrated forces are replaced by a distributed contact force. The distribution of the contact force approaches the two concentrated forces at the boundary points when the shear effect is negligible. The line-contact deformation may become unstable when the edge thrust reaches a secondary buckling load. It is found that the normalized critical length of the contact segment is always smaller when the shear effect is more significant. On the other hand, the normalized secondary buckling load reaches a maximum when the slenderness ratio (beam length over thickness) is in the order of 45.     

Free vibration behavior of a thermally post-buckled FG Timoshenko beam under large deflection using a tangent stiffness-based method

For thermally postbuckled configurations, the free vibration behavior of functionally graded (FG) Timoshenko beams are investigated. The postbuckling configurations are obtained through a geometrically nonlinear static problem. The free vibration problem around the postbuckled configuration is formulated using its tangent stiffness. The energy based governing equations are solved following the Ritz method. The elements of the tangent stiffness matrix are obtained using the Ritz coefficients. The results are shown to exhibit the effects of FG material, material profile parameter, and length-thickness ratio. The comparative results are presented for both the cases of the physical neutral surface and the geometrical neutral surface.     

On the stability of a Timoshenko beam on an elastic foundation under a moving spring-mass system

Summary The stability problem of densely distributed oscillators moving along a Timoshenko beam on an elastic foundation is considered. The forward speed of the moving subsystem is assumed to be constant. The friction at the contact line between the beam and the oscillator set is neglected. A qualitatively new instability region is found. It is pointed out that the critical velocity for some system parameters takes smaller values than the velocity of shear waves or the velocity of longitudinal waves.With 8 Figures