The application of the thin-walled box beam element to multibox bridge analysis

An application of the recently developed thin-walled box beam element to the analysis of multibox bridges which arises in practical design, is presented. The thin-walled box beam element, which also takes account of warping distortional effects, when combined with traditional beam elements into a grillage model may adequately represent the three-dimensional behaviour of multibox superstructure. Equivalent sectional properties for the transverse grillage members across individual boxes are computed from a frame analysis. A numerical iterative procedure is introduced to take account of the interaction due to distortion.Comparisons with other numerical methods and model experiments demonstrate the accuracy and economy of the proposed method.     

大展弦比复合材料机翼动力学分析

针对结构非线性对大展弦比机翼动力学特性影响很大的问题,使用MSC Patran和MSC Nastran软件进行有限元建模及分析,将大展弦比机翼建成薄壁盒型梁模型．研究大变形对机翼动力学特性的影响,比较复合材料盒型梁模型和金属盒型梁模型的计算结果,并讨论复合材料铺层顺序的改变对机翼动力学特性的影响．研究表明：复合材料机翼的各阶固有频率明显高于铝合金机翼;铺层顺序会影响复合材料机翼的固有频率．     

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.     

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.     

空间薄壁梁单元面向对象程序的实现

针对传统有限元分析软件主要面向过程设计,其可维护性和可扩展性等较差的问题,基于面向对象程序设计方法,建立具有内部节点的空间薄壁截面梁单元模型,给出线弹性空间薄壁梁单元的UML类图,介绍矩阵类、截面类、材料类、节点类、单元类和结构类等6种类成员的主要属性和方法.用C#编制相应的有限元程序,通过T形框架算例比较和验证其位移和弯曲转角计算值、理论解和ANSYS的BEAM 189梁单元的数值解,结果表明该程序精度良好,可用于空间薄壁结构的有限元分析.     

On the hybrid-mixed formulation of C0 curved beam elements

Two C⁰ curved beam elements based on the hybrid-mixed formulation are studied in the form of membrane-shear locking, mesh convergence, and stress predictions. At the element level, both the displacement and stress fields are approximated separately. The stress parameters are then eliminated from the stationary condition of the Hellinger-Reissner variational principle so that the standard stiffness equations are obtained. The stress functions are chosen from two important considerations: (i) kinematic deformation modes must be avoided, and (ii) the constraint index counting of the element, when applied to limiting cases, must be equal to or greater than one. Based on these considerations, two curved beam elements are derived by including the effect of shear deformation and with linear and quadratic displacement fields. The elements are found to be lock-free for thin-walled beams. Several numerical examples are given to demonstrate the performance of the two curved elements.     

Finite strip analysis of curved composite girders with incomplete interaction

This paper presents a finite strip analysis of curved composite girders with incomplete interaction. In the present analysis curved composite girder bridges are modelled by curved strip elements for the concrete slab and steel girder and spring elements for the shear connectors. The shear connectors are assumed as a two-dimensional spring element along the nodal line. The proposed method is applied to the analysis of curved box girders, curved plates and curved composite girder bridges with complete and incomplete interaction. Some analytical results obtained by this method are compared with the test results and theoretical values obtained by other methods, and they are in good agreement. Slip behavior of curved composite box girders is also discussed based on the results by the proposed method.     

Post-buckling strength of thin-walled members

The finite element displacement method considering both geometrical nonlinearity and material non-linearity has been used to investigate the post-buckling behaviour and the ultimate strength of thin-walled nonplanar (three-dimensional) structural members. The two types of nonlinearities are based on Lagrangian coordinates and the flow theory of plasticity, and the formulations are developed using the variational principle and the incremental variational principle. The tangent stiffness matrix which is derived explicitly up to a point prior to volume integration, has been found to be quite efficient. The cases of a hat-section beam under a concentrated load for a web crippling study and a channel section subjected to combined bending and torsion are used to show the capabilities of the computer program. Results indicate that the conventional linear, elastic analysis over-estimates the strength of thin-walled members and may not even be a useful approximation and that the structure may be excessively deformed when approaching the ultimate load. The study also demonstrates the merit of using the finite element method for detailed investigations of particular problems.     

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.     

Spatial stability of shear deformable curved beams with non-symmetric thin-walled sections. II: F. E. solutions and parametric study

In the companion paper, an improved formulation for spatial stability analysis of shear deformable thin-walled curved beams with non-symmetric cross-sections is presented based on the displacement field considering both constant curvature effects and the second-order terms of semi-tangential rotations. Thus the elastic strain energy and the potential energy due to initial stress resultants are consistently derived. Also closed-form solutions for in-plane and lateral-torsional buckling of curved beams subjected to uniform compression and pure bending are newly derived for mono-symmetric thin-walled curved beams under simply supported and clamped end conditions. In this paper, F. E. procedures are developed by using curved and straight beam elements with non-symmetric cross-sections. Analytical and numerical solutions for spatial buckling of shear deformable thin-walled circular beams are presented and compared in order to illustrate the accuracy and the practical usefulness of this study. In addition, the extensive parametric studies are performed on spatial stability behavior of curved beams. Particularly transition and crossover phenomena of buckling mode shapes with change in curvature and length of beam on buckling for curved beams are investigated for the first time.     

Curved box beam bridge analysis

A finite difference procedure is used to determine the response of a single and multi span curved single box beam bridge with any number of interior diaphragms. The bending and torsional distortions as well as cross-sectional distortions are determined throughout the box girder. The forces that are determined include bending moment and shear, pure torsion, warping torsion, and bimoment. These forces, in addition to distortional functions, yield resulting normal bending, normal warping, and normal distortional stresses.The entire analysis scheme has been programmed for use on an UNIVAC 1108 computer, FORTRAN IV language, as given herein.     

Radiation pattern of modified box‐horn antenna

The expression for radiation field of modified box‐horn antenna is derived using plane wave spectra for three‐dimensional fields. Modified box‐horn is a modified version of conventional box‐horn, in which the horn exciting the box is flared in both E‐ and H‐planes to increase its aperture size. It supports TE₁₀‐ and TE₃₀‐modes at its aperture. The radiation patterns of modified box‐horn antenna in E‐ and H‐planes are computed at 10 GHz and corresponding radiation parameters are extracted from the plots and compared for different flare angles in E‐plane of pyramidal horn exciting the box. The results for half power beam‐widths in E‐plane demonstrate that the radiation pattern of modified box‐horn in E‐plane can be made narrower by increasing the flare angle in E‐plane of pyramidal horn exciting the box. The radiation patterns of modified box‐horn are also compared with those for TE₁₀‐mode open ended waveguide of same aperture size and it is found that the former is narrower beam in H‐plane than the latter. The analysis has been validated against the experimental results available in the literature. The antenna presented here may find potential application as an element in a phased array for microwave wireless communication/radar beam steering, a feed for reflectors in point‐to‐point microwave communication/satellite communication. © 2008 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2008.     

Curved beam element stiffness matrix formulation

Stiffness matrix of order 12 × 12, for a curved beam element has been formulated involving all the forces together, using Castigliano's theorem. Effects of transverse shear forces and tangential thrust are also taken into account. In earlier works, stiffness submatrices for the two uncoupled systems of forces are formulated independently and then they are combined to give the overall 12 × 12 matrix. A program subroutine, NEWCBM for the stiffness matrix formulation of curved beams has been written in FORTRAN which can be added to the element library of general purpose computer programs like SAP-IV and its improved versions. Example problems have been worked to check the accuracy of this formulation.     

Pre-bent shape design of full free-form curved beams using isogeometric method and semi-analytical sensitivity analysis

In this paper, isogeometric analysis (IGA) is employed to solve the problem of a curved beam with free-form geometry, arbitrary loading, and variable flexural/axial rigidity. The main objective of the study is to develop a unified approach for full free-from curved beam problems that can be integrated with a newly developed semi-analytical sensitivity analysis to solve pre-bent shape design problems. The required set of B-spline control points are calculated using an interpolation technique based on chord-length parameterization. The one-to-one correspondence is considered for parameters of the geometry, loading, and rigidity which is proven to have extreme importance. An IGA curved beam element is suggested based on the Euler-Bernoulli beam theory for the general curvilinear coordinate. The validity and effectiveness of the proposed formulation is confirmed by application to a variety of examples. Moreover, three shape optimization examples are taken into consideration. In the first two examples, the pre-bent shapes of spiral and Tschinhausen curved beams with free-form geometry under distributed loading are obtained. In the third example, the pre-bending problem of wind turbine blades is addressed as an industrial example.     

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.     

Consistent curved beam element

Curved beam finite elements with shear deformation have required the use of reduced integration to provide improved results for thin beams and arches due to the presence of a spurious shear strain mode. It has been found that the spurious shear strain mode results from an inconsistency in the displacement fields used in the formulation of these elements. A new curved beam element has been formulated. By providing a cubic polynomial for approximation of displacements, and a quadratic polynomial for approximation of rotations a consistent formulation is ensured thereby eliminating the spurious mode. A rotational degree of freedom which varies quadratically through the thickness of the element is included. This allows for a parabolic variation of the shear strain and hence eliminates the need for use of the shear correction factor k as required by the Timoshenko beam theory. This rotational degree of freedom also provides a cubic variation of displacements through the depth of the element. Thus, the normal to the centroidal axis is neither straight nor normal after shearing and bending allowing for warping of the cross section. Material nonlinearities are also incorporated, along with the modified Newton-Raphson method for nonlinear analysis. Comparisons are made with the available elasticity solutions and those predicted by the quadratic isoparametric beam element. The results indicate that the consistent beam element provides excellent predictions of the displacements, stresses and plastic zones for both thin and thick beams and arches.     

Finite element mesh deformation with the skeleton-section template

To develop fast finite element (FE) adaptation methods for simulation-driven design optimization, we propose a radial basis functions (RBF) method with a skeleton-section template to globally and locally deform FE meshes of thin-walled beam structures.The skeleton-section template is automatically formulated from the input mesh and serves as a hierarchical parameterization for the FE meshes. With this hierarchical parameterization, both the global and the local geometries of a thin-walled beam can be processed in the same framework, which is of importance for designing engineering components. The curve skeleton of the mesh is constructed with Voronoi decomposition, while the cross-sections are extracted from the mesh based on the curve skeleton.The RBF method is employed to locally and globally deform the mesh model with the cross-sections and the skeleton, respectively. The RBF method solves the spatial deformation field given prescribed deformations at the cross-sections. At the local scale, the user modifies the cross-sections to deform a region of the surface mesh. At the global level, the skeleton is manipulated and its deformation is transferred to all cross-sections to induce the mesh deformation.In order to handle curved mesh models and attain flexible local deformations, the input mesh is embedded into its skeleton frame field using an anisotropic distance metric. In this way, even strip-like features along arbitrary directions can be created on the mesh model using only a few cross-sections as the deformation handles. In addition, form features can be rigidly preserved at both deformation levels.Numerical examples demonstrate that intuitive and qualified FE mesh deformations can be obtained with manipulation of the skeleton-section template.     

Exact dynamic and static element stiffness matrices of nonsymmetric thin-walled beam-columns

An improved numerical method to exactly evaluate 14 × 14 dynamic and static element stiffness matrices is proposed for the spatial free vibration and stability analysis of nonsymmetric thin-walled straight beams subjected to eccentrically axial loads. Firstly equations of motion and force-deformation relations are rigorously derived from the total potential energy for a uniform beam element with nonsymmetric thin-walled cross-section. Next a system of linear algebraic equations with nonsymmetric matrices is constructed by introducing 14 displacement parameters and transforming the higher order simultaneous differential equation into the first order simultaneous equation. And then explicit expressions for displacement parameters are exactly evaluated by solving a generalized eigenproblem with complex eigenvalues. Finally exact element stiffness matrices are determined using force-deformation relations. Particularly straightforward application of the present method may not give the exact static stiffness because of existence of multiple zero eigenvalues in case of static buckling problems. Accordingly, a modified numerical method to resolve this difficulty is developed for two cases depending on the initial state of stress resultants. In order to demonstrate the validity and the accuracy of this method, the natural frequencies and buckling loads of nonsymmetric thin-walled beam-columns having bending-torsional deformation modes are evaluated and compared with analytical and F.E. solutions or results analyzed by ABAQUS’s shell element.     

A curved-beam bistable mechanism

This paper presents a monolithic mechanically-bistable mechanism that does not rely on residual stress for its bistability. The typical implementation of this mechanism is two curved centrally-clamped parallel beams, hereafter referred to as "double curved beams". Modal analysis and finite element analysis (FEA) simulation of the curved beam are used to predict, explain, and design its bistable behavior. Microscale double curved beams are fabricated by deep-reactive ion etching (DRIE) and their test results agree well with the analytic predictions. Approaches to tailor the bistable behavior of the curved beams are also presented.