The sectorial co-ordinate of elastic-ideally-plastic beams of three thin-walled open cross-sections

A beam lamina of thin-walled open cross-section is considered. The sectional forces are a bimoment B and a Vlasov torsional moment M_w. The material is assumed to be elastic-ideally-plastic. The change in sectorial co-ordinate with increasing bicurvature is studied for three frequently used cross-sections.     

Vlasov torsion of non-linearly elastic beams of thin-walled open cross-section

A beam lamina of thin-walled open cross-section is considered. The sectional forces are a bimoment (warping moment) B and a Vlasov torsional moment M_w. A formula for the sectorial co-ordinate containing three unknowns and being suitable for numerical calculations is given. The material is assumed to be non-linearly elastic with the constitutive equation = K|σ|ⁿ sgn σ in uniaxial tension. The relation M_w = B′ is shown to hold also for n ≠ 1. The location of the centre of twist as a function of n is determined for a monosymmetric I cross-section. The torsion-bending analogue for beams is found to be valid also for n ≠ 1.     

Vlasov torsion of elastic-ideally-plastic beams of thin-walled open cross-section

Prismatic beams of thin-walled open cross-section are studied. The sectional forces are a bimoment B and a Vlasov torsional moment M_w. The material is assumed to be elastic-ideally-plastic. Special consideration is given to the monosymmetric I-section. Exact and approximate solutions are derived for a cantilevered I-beam. These solutions are compared in a numerical example.     

Torsion of thin-walled beams of open cross-section with influence of shear

An approximate analytical approach to the torsion of thin-walled beams of open cross-section with influence of shear is presented. It is assumed that the normal stresses in the cross-section contour direction are small compared to the normal stresses in the beam longitudinal direction that can be ignored in the stress–strain relations. The stresses and displacements are obtained in the closed analytical form. It is assumed that transverse load can be reduced to moments of torsion only, with respect to the cross-section shear centre. The beam will be subjected to torsion, with respect to the shear centre, i.e. the principal cross-section sectorial coordinate, in the case of cross-sections with two axes of symmetry; to torsion and bending in the case of cross-sections with one axes of symmetry and to torsion, bending and tension in the case of general cross-sections. An illustrative example of beam with one axis of symmetry is given.     

A consistent approach to linear stability of thin-walled beams of open section

This paper presents a stability criterion for thin-walled beams of curvilinear section, based on the positive definiteness of the second variation of the potential energy. The internal stresses are derived according to a shell model which incorporates the kinematic restraints of thin-walled beams. In addition to the classical terms of the potential energy, the geometric effects of an applied torque, as well as flexural-extensional and torsional-extensional coupling terms, are accounted for.     

Non-linear non-uniform torsion of thin-walled beams

A theoretical and experimental investigation is made to study the non-linear behaviour of thin-walled elastic beams of open sections subjected to non-uniform torsion. The differential equations describing the behaviour and the associated boundary conditions are presented. A perturbation solution is carried out to obtain the torque-rotation characteristics and also the axial strain distribution. The accuracy of the perturbation solution is checked against a numerical solution of the governing equation.

A set of experimental tests on two I-sections is carried out to verify the theoretical calculation. It is shown that the torque-rotation behaviour exhibits a hardening non-linear characteristic. The agreement between theory and test results is reasonable.     

Bending crashworthiness of thin-walled square tubes with multi-cell and double-tube cross-sections

Journal of Mechanical Science and Technology - For structural bending crashworthiness, previous studies mostly focused on thin-walled tubes with multi-cell cross-sections, while the double-tube...     

Vibration analysis of thin-walled curved beams using DQM

The differential quadrature method (DQM) is applied to computation of the eigenvalues of small amplitude free vibration for thin-walled curved beams including a warping contribution. Natural frequencies are calculated for singlespan, curved, wide-flange uniform beams having a range of nondimensional parameters representing variations in warping stiffness, torsional stiffness, radius of curvature, included angle of the curve, polar mass moment of inertia, and various end conditions. Results are compared with existing exact and numerical solutions by other methods for cases in which they are available. It is found that the DQM gives good accuracy even when only a limited number of grid points is used. In addition, results are given for a cantilever beam not previously considered for this problem. Finally, parametric results are presented in dimensionless form.     

Analysis of structures with thin-walled open sections

The problems of incorporating the effects of cross-sectional warping, offset shear centre and arbitrary orientation of an element in an assembled structure are examined. The kinematics of a variety of joint intersections and stiffening schemes are incorporated and transformation matrices are developed which enable solutions to be obtained comparing with a number of experimental and analytical studies.     

Coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams

A dynamic transfer matrix method of determining the natural frequencies and mode shapes of axially loaded thin-walled Timoshenko beams has been presented. In the analysis the effects of axial force, warping stiffness, shear deformation and rotary inertia are taken into account and a continuous model is used. The bending vibration is restricted to one direction. The dynamic transfer matrix is derived by directly solving the governing differential equations of motion for coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams. Two illustrative examples are worked out to show the effects of axial force, warping stiffness, shear deformation and rotary inertia on the natural frequencies and mode shapes of the thin-walled beams. Numerical results demonstrate the satisfactory accuracy and effectiveness of the presented method.     

The dynamic analysis of nonuniformly pretwisted Timoshenko beams with elastic boundary conditions

The coupled governing differential equations and the general elastic boundary conditions for the coupled bending–bending forced vibration of a nonuniform pretwisted Timoshenko beam are derived by Hamilton's principle. The closed-form static solution for the general system is obtained. The relation between the static solution and the field transfer matrix is derived. Further, a simple and accurate modified transfer matrix method for studying the dynamic behavior of a Timoshenko beam with arbitrary pretwist is presented. The relation between the steady solution and the frequency equation is revealed. The systems of Rayleigh and Bernoulli–Euler beams can be easily examined by taking the corresponding limiting procedures. The results are compared with those in the literature. Finally, the effects of the shear deformation, the rotary inertia, the ratio of bending rigidities, and the pretwist angle on the natural frequencies are investigated.     

Bending of axially constrained beams on elastic foundation

Approximate analysis of bending of a pin-pointed beam under distributed load resting on elastic foundation is considered. The supports are assumed to be immovable, thereby giving rise to the axial stretching of the neutral surface under lateral loads. The resulting problem is nonlinear since the axial force of constraint is dependent on the lateral displacements. Quantitative results of axial force, benidng moment and deflections are given in terms of two dimensionless parameters which depend on the load intensity, foundation modulus and properties of the beam.     

开口薄壁结构声固耦合问题的计算方法

对于开口薄壁结构的声辐射问题,很难用解析法直接计算,可采用多种数值方法进行仿真分析。文中研究了有限元法、边界元法与无限元法几种数值方法及其耦合的理论基础,研究了求解不封闭声腔声辐射问题的几种数值方法。在考虑声固耦合的前提下,分析了间接边界元法、直接边界元内外声场耦合法、有限元与边界元耦合法、有限元与无限元耦合法4种方法求解开口结构声场问题的系统方程及求解过程,以具体开口薄壁结构为实例采用4种方法进行了模拟计算,并对结果进行了分析比较,得出相关结论。     

An investigation of the collapse of thin-walled rectangular beams in biaxial bending

Two different thin-walled closed sections, one square and one rectangular were tested in biaxial bending using displacement controlled cantilever bending tests.Two different constraint conditions were used, the first kept the deflection axis fixed and left the direction of the load axis free. The second kept the load axis direction fixed and allowed the beam to deflect freely.Each configuration was repeated at 15° axis intervals and was performed twice to assure repeatability. This gave a total of 42 cantilever bending tests.The results are used to compare the beam behaviour in the transition and collapse stages between the two constraint conditions. The behaviour is also contrasted with thick-walled theory predictions, since this is the only currently available reference.     

Geometrically nonlinear theory of thin-walled composite box beams using shear-deformable beam theory

A general geometrically nonlinear model for thin-walled composite space beams with arbitrary lay-ups under various types of loadings is presented. This model is based on the first-order shear deformable beam theory, and accounts for all the structural coupling coming from both material anisotropy and geometric nonlinearity. 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 beams under vertical load to investigate the effects of shear deformation, geometric nonlinearity and fiber orientation on axial-flexural-torsional response.     

Lateral-torsional buckling analysis of thin-walled beams including shear and pre-buckling deformation effects

In this paper, lateral-torsional buckling behavior of open-section thin-walled beams is investigated based on a geometrically nonlinear formulation, which considers the effects of shear deformations. A finite element numerical solution along with an incremental-iterative solution procedure is adopted to trace the pre-buckling as well as the post-buckling equilibrium paths. Formulation is applicable to a general type of open-section and load position effects are also included. Numerical results are validated through comparisons with experimental results and those based on other formulations presented in the literature. Comparisons have also been made between the results based on fully nonlinear analysis and linearized buckling analysis in order to illustrate the effects of pre-buckling deformations as well as the shear deformations on the buckling load predictions. Examples illustrate the influence of beam slenderness and moment gradient on the effects of pre-buckling deformations in predicting bucking loads.     

Numerical and analytical study on deep biaxial bending collapse of thin-walled beams

The objective of the paper is to get a fundamental understanding of large rotation response of thin-walled prismatic beams subjected to biaxial bending. Square and rectangular section beams were subjected to a cantilever bending with hinge rotation up to 40°. A general purpose nonlinear FE code PAM-CRASH was used to generate results. Altogether 94 computer runs were made by changing aspect ratio, condition of the loading and bending plane with respect to principal inertia axes of the cross section. The main emphasis was placed on the determination of initial and subsequent yield loci. It was also shown that the direction of the vector of generalized strain rates obeys the normality rule. Calculated in the paper was also a critical angle defining a preferable direction of bending. The present finding can be used to better interpret results of FE calculation of automotive bodies and to develop simplified crash-oriented design tools.