Lateral-torsional buckling of beams and the Southwell plot

The Southwell plot is a well-known technique for determining experimentally the elastic critical load of a structure, without having to subject the structure to loading in the vicinity of critical. But several authors have suggested that when the structure is a beam which undergoes lateral-torsional buckling, a modified version of the Southwell plot is called for. In this paper we demonstrate that the modified form of the Southwell plot is not needed, and that the standard version is indeed satisfactory. We do this by plotting and re-plotting some experimental data; by drawing attention to some very clear work by Meck; and by explaining the practical coupling between the variables describing the lateral deflection and the rotation when lateral-torsional buckling occurs. Finally, we examine an argument based on symmetry which appears to support the idea that a modification of the standard Southwell plot is needed in the case of lateral-torsional buckling: but we show that a correct deployment of the argument from symmetry leads to the conclusion that the modified form of the Southwell plot is valid only for special, unrealistic cases.     

FRP 层压复合梁结构屈曲变形在线监测方法

为了实现梁结构屈曲变形在线监控,提出了一种FRP层合梁屈曲变形重构方法。首先,依据高阶剪切变形理论,提出了一种复合梁结构变形场描述方法,并基于冯卡门应变梯度理论,推导出了中性轴应变表述方式。然后,利用最小二乘变分法建立了位移重构模型。其中,利用四次B-样条基函数构造了屈曲变形位移插值函数,推导了理论中性轴应变计算公式。并基于少量应变测量,提出了非线性项应变解耦方法,建立了测量应变与实测中性轴应变转换关系。最后,为了验证所提方法的精确性,以25层碳纤维复合梁为样件,搭建固定-简支梁试验平台,对其进行数值计算和试验论证。结果表明,建立的屈曲变形重构模型在不同轴向载荷作用时,位移场重构误差均小于8%。     

Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams

An improved third order shear deformation theory is employed to investigate thermal buckling and vibration of the functionally graded beams. A power law distribution is used to describe the variation of volume fraction of material compositions. The functionally graded material properties are assumed to vary smoothly and continuously across the thickness of the beams. The Ritz method is adopted to solve the eigenvalue problems that are associated with thermal buckling and vibration in various types of immovable boundary conditions. The parametric study covered in this paper includes the effects of material composition, temperature-dependent material properties, and slenderness ratio.     

Free vibration of laminated cross-ply plates including shear deformation by spline method

Spline function approximation technique is used to analyze the free vibration of symmetric and anti-symmetric cross-ply plates under shear deformation theory. The equations of motion of the plate are derived using YNS theory. A system of coupled differential equations in terms of displacement and rotational functions are obtained by assuming the solution in a separable form. These functions are approximated using Bickley-type splines of suitable orders. A generalized eigenvalue problem is obtained on applying the process of point collocation with suitable boundary conditions. Parametric studies have been made to investigate the frequency response of the plates with reference to the material properties, number of layers, fiber orientation, side-to-thickness ratio, aspect ratio and relative layer thickness. Some results are compared with existing solution obtained by FEM.     

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.     

Buckling of shallow spherical shells including the effects of transverse shear deformation

The large deflection equation of a shallow spherical shell under uniformly distributed transverse loads is established in this paper with consideration of effects of transverse shear deformation on flexural deformation. Using an updated iteration method, an analytical solution for nonlinear stability of a shallow spherical shell is obtained. Formulae for estimating the critical buckling loads are presented for two types of boundary conditions. Discussions on the influences of the geometric and physical parameters on the critical buckling loads are given.     

An exact solution for buckling of functionally graded circular plates based on higher order shear deformation plate theory under uniform radial compression

In this research, mechanical buckling of circular plates composed of functionally graded materials (FGMs) is considered. Equilibrium and stability equations of a FGM circular plate under uniform radial compression are derived, based on the higher order shear deformation plate theory (HSDT). Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations are established. A buckling analysis of a functionally graded circular plate (FGCP) under uniform radial compression is carried out and the results are given in closed-form solutions. The results are compared with the buckling loads of plates obtained for FGCP based on the first order shear deformation plate theory (FSDT) and classical plate theory (CPT) given in the literature. The study concludes that HSDT accurately predicts the behavior of FGCP, whereas the FSDT and CPT overestimates buckling loads.     

A closed-form approximate solution of lateral buckling of doubly symmetric thin-walled members considering shear lag

To our knowledge, apart from a closed-form solution for a simply supported I-beam subjected to uniform moment, most buckling loads are obtained by numerical methods. By using Galerkin's method of weighted residuals, an accurate approximate solution for lateral buckling of simply supported I-beam is proposed in this paper. Due to fast convergence of the method, the solution can be written in closed form. The results compare well with the previous work including experimental results and optimal and finite element analytical results.     

Free vibration of multi-layered circular cylindrical shell with cross-ply walls,including shear deformation by using spline function method

The spline function technique is used to analyze the vibration of multi-layered circular cylindrical shells with cross-ply walls including first-order shear deformation theory. Both antisymmetric and symmetric cross-ply laminations are considered in this analysis. The governing equilibrium equations are obtained in terms of displacement and rotational functions. A system of coupled ordinary differential equations in terms of displacement and rotational functions are obtained by assuming the solution in a separable form. These functions are approximated by using Bickley-type splines of suitable order to obtain the generalized eigenvalue problem by applying point collocation techniques with appropriate boundary conditions. Parametric studies are performed to analyze the frequency response of the shell with reference to the material properties, number of layers, fiber orientation, thickness to radius ratio, length to radius ratio and circumferential node number. Reasonable agreement is found with existing results obtained by FEM and other methods. Valuable results are presented as graphs and discussed. This paper was recommended for publication in revised form by Associate Editor Maenghyo Cho Dr. K. K. Viswanathan was born in 1962 in Vellore District, India. He received his B.Sc. in Mathematics from University of Madras and M.Sc. in 1992 and Ph.D. in 1999 from Anna University, India. Later he was a Project Associate in Indian Institute of Science, Bangalore. He served as lecturer in Crescent Engg. College and as Asst. Professor in SRM University, India. He did his post doctoral research in Korea for three years. At present he serves as Professor in the Dept. of Naval Architecture, Inha University, Incheon, Korea. His research areas of interest includes vibration of plates, shells and the application of numerical techniques in Engineering problems. Dr. Kyung Su Kim was born in Korea in 1954. He is a professor in Naval Architecture and Ocean Engineering at Inha University, Korea. He obtained his B.Sc. degree in Naval Architecture and Ocean Engineering from Seoul National University, Korea, in 1981. He worked for KR (Korean Register of Shipping) from 1981 to 1983. He obtained M.Sc. degree in Naval Architecture and Ocean Engineering in 1986, and Ph.D. degree in Structural Mechanics in 1991 from Rheinisch — Westfaelische Technische Hoch-schule Aachen, Germany. From 1986 to 1992, he was a Post Doctoral Research Engineer of Engineering Research Institute at Rheinisch — Westfaelische Technische Hochschule Aachen. He was appointed as a professor of Inha University, Korea, in 1994. His major area of study is Impact and Fatigue Fracture. Dr. Jang Hyun Lee was born in Korea in 1969. Currently, he is an Assistant professor of the Department of Naval Architecture and Ocean Engineering at Inha University, Korea. He obtained his B.Sc., M.Sc. and Ph.D. degrees in Naval Architecture and Ocean Engineering from Seoul National University, Korea, in 1993, 1995 and 1999 respectively. From 1999 to 2002, he was a Post Doctoral Research Engineer of Engineering Research Institute at Seoul National University. He joined the Inha University in 2005 after holding the Chief Technology Officer at Xinnos for four years. His research interests include press forming of thick plates and shells, computational welding mechanics and Product Lifecycle Management.     

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.     

薄壁球壳受刚性平面压缩的连续变形测量

建立了非接触光学测量实验系统,对薄壁球壳受刚性平面压缩的连续变形及失稳行为进行了实验研究。实验以理想弹塑性材料制成的薄且壁厚均匀的乒乓球为压缩对象,采用透明石英玻璃板作为刚性平面,用相机透过玻璃板采集清晰的散斑图像。然后利用具有非接触、全场、高精度等优点的三维数字图像相关(3D-DIC)技术对与刚性平面接触的乒乓球表面的连续变形进行直接测量,同时利用二维数字图像相关技术结合压力传感器得到对应的力-位移关系。实验研究表明,基于3D-DIC方法测量三维表面变形场,实现了对受刚性平面压缩的乒乓球表面从局部压平阶段到轴对称屈曲阶段,进而演化到非轴对称屈曲阶段等力学行为的可视化。实验结果表明,对乒乓球半径的测量精度可达0.5%,利用Abaqus有限元方法得到的结果与实验结果非常吻合,验证了所提方法的有效性。     

Thermoelastic lateral-torsional buckling of fixed slender beams under linear temperature gradient

The thermal expansions and rotations that result from a linear in-plane temperature gradient field are fully restrained at the ends of a fixed beam. These restrained expansions and rotations will produce internal bending and compressive actions in the beam, and these actions increase with an increase of the temperature differential and average temperature of the linear temperature gradient field. When these actions reach critical values, the fixed beam may bifurcate from its primary equilibrium state to a buckled equilibrium configuration. This paper presents a systematic treatment of classical buckling analysis for thermoelastic lateral-torsional buckling and for in-plane thermoelastic flexural buckling of a fixed beam of doubly symmetric open thin-walled cross-section that is subjected to a linear temperature gradient field over its cross-section. It is shown that the effective centroid and shear centre, rather than the geometric centroid and shear centre, should be used in formulating the thermoelastic prebuckling and buckling analysis and that the effects of temperature on the buckling resistance need to be considered. The thermoelastic lateral-torsional buckling of a fixed beam under a linear temperature gradient field is more complicated than its mechanical counterpart for uniform bending or for uniform compression, and iterative methods are needed to obtain accurate solutions.     

The shear buckling behaviour of thin composite plates with particular reference to the effects of bend–twist coupling

The effect of bend–twist coupling on the shear buckling behaviour of laminated composite plates is examined in this paper using a finite strip procedure. The complex buckled shapes which are associated with shear loading are duly accounted for in the analysis approach through the multi-term facility of the strip formulation employed and, of course, through the appropriate level of structural modelling. The degree of bend–twist coupling in the laminated composite plates is varied by changing the level of anisotropy in the plies and by altering the lay-up configuration of the plies in the laminated stack. Symmetric laminates of a balanced and unbalanced nature are given consideration. It is shown that, for a given degree of anisotropy in the plies of a laminate and for a given laminate thickness, the stacking sequence of the plies significantly alters the degree of bend–twist coupling. The shear buckling performance of composite plates having the same dimensions and being made from the same material are therefore shown in the paper to be quite different. The preclusion of the bend–twist coupling coefficients in the solution procedure of the finite strip method allows the shear buckling orthotropic solution to be determined. Comparisons between the coupled and orthotropic solutions are shown in the paper to be markedly different with respect to critical shear performance level and also buckled mode shape. For square plates or plates with a moderate aspect ratio the influence of bend–twist coupling on buckled mode shape is shown in the paper to be noticeable through increased distortion. For the larger aspect ratio plates it is shown that the presence of bend–twist coupling can cause a complete change in the mode shape from a symmetric to an antisymmetric nature or vice versa. Amplitude modulation is shown in the paper to be clearly evident in the shear buckling mode shapes of long plates.     

A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate

A new hyperbolic shear deformation theory taking into account transverse shear deformation effects is presented for the buckling and free vibration analysis of thick functionally graded sandwich plates. Unlike any other theory, the theory presented gives rise to only four governing equations. Number of unknown functions involved is only four, as against five in case of simple shear deformation theories of Mindlin and Reissner (first shear deformation theory). The plate properties are assumed to be varied through the thickness following a simple power law distribution in terms of volume fraction of material constituents. The theory presented is variationally consistent, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. Equations of motion are derived from Hamilton's principle. The closed-form solutions of functionally graded sandwich plates are obtained using the Navier solution. The results obtained for plate with various thickness ratios using the theory are not only substantially more accurate than those obtained using the classical plate theory, but are almost comparable to those obtained using higher order theories with more number of unknown functions.     

Local and distortional buckling of cold-formed zed-section beams under uniformly distributed transverse loads

This paper presents a numerical investigation on the local and distortional buckling behaviour of cold-formed steel zed-section beams subjected to uniformly distributed transverse loads. The analysis is performed using a semi-analytical finite strip method. The beams investigated include both detached sections and restrained sections. The results obtained from the present study highlight the differences in the local and distortional buckling behaviours of the thin-walled sections between pure bending and the uniformly distributed loading.     

Static, dynamic, and buckling analysis of partial interaction composite members using Timoshenko's beam theory

The static, dynamic, and buckling behavior of partial interaction composite members is investigated in this paper by taking into account for the influences of rotary inertia and shear deformations. The governing differential equations obtained are very comprehensive, covering and extending the current models for the problems that are based on Euler–Bernoulli beam theory. The analytical solutions of the deflection are then found for the beam with uniformly distributing load under common boundary conditions. The free vibration and buckling behavior are also studied and the analytical expressions of the frequencies of the simply supported beam are obtained explicitly, as are the buckling loads. For other boundary conditions, the eigen-equations are transcendental and thus some numerical examples are presented to demonstrate the effects of the shear deformation and rotary inertia on the resonant frequencies and buckling loads.     

Optimal design of stepped circular plates with allowance for the effect of transverse shear deformation

Presented herein is a canonical exact deflection expression for stepped (or piecewise-constant thickness) circular plates under rotationally symmetric transverse loads. The circular plates may be either simply supported or clamped at the edges. As the plates may be very thick or certain portions of the optimal design may become rather thick, the significant effect of transverse shear deformation on the deflections cannot be ignored. This effect was taken into consideration in accordance to the Mindlin plate theory. Based on the analytical deflection expression, necessary conditions are derived for the optimal values of segmental lengths and thicknesses that minimize the maximum deflection of stepped circular plates of a given volume. These optimality conditions are solved using the Newton method for the optimal segmental lengths and thicknesses. Local minima are observed for this nonlinear problem at hand and they may pose some difficulties in getting the solutions. The shear deformation effect increases the plate deflections, but interestingly it affects the thickness variation marginally.     

Elastic lateral-torsional buckling of circular arches subjected to a central concentrated load

An arch under an in-plane central concentrated radial load is subjected to combined axial compressive and bending actions. When these combined axial compressive and bending actions reach a certain value, the arch may suddenly deflect laterally and twist out of its plane of loading and fail in a lateral-torsional buckling mode. This paper derives analytical solutions for the elastic lateral-torsional buckling load of pin-ended circular arches that are subjected to a central concentrated load, using the principle of stationary potential energy in conjunction with the Rayleigh-Ritz method. Analytical solutions of the buckling load for in-plane fixed and out-of-plane pin-ended arches and for the case of the load acting above or below the shear centre are also derived. The analytical solutions are compared with results of a commercial finite element package ANSYS and a finite element code developed by authors elsewhere for arches with different slendernesses, included angles, and cross-sections. The agreement between the analytical solutions and the finite element results is very good.     

An advanced analysis of axisymmetric plastic sheet bending including transverse shear deformation

Finite deformation theory for axisymmetric elastic-plastic sheet bending is developed. The theory incorporates transverse shear deformation by adopting the extended Kirchhoff-Love's hypothesis, i.e. during bending deformation, a line element normal to the undeformed mid-surface is allowed to change the angle between itself and the mid-surface while its straightness is retained. A new idea, called “equivalent curvature”, is proposed which plays a similar role to the curvature in conventional plate bending theory but incorporates the effects of transverse shear deformation. Numerical calculations based on this theory have been performed for two examples of sheet-forming processes, i.e. the hydrostatic bulging of a circular sheet and the U-type sheet-bending process. Results show that the proposed theory can predict more precise information concerning the forming processes for a moderately thick sheet than the conventional sheet bending theory which ignores transverse shear deformation.     

A series solution for spatially coupled deflection analysis of thin-walled Timoshenko curved beam with and without elastic foundation

The exact solutions for the spatially coupled deflection and the normal stress at an arbitrary location of a crosssection of the thin-walled Timoshenko curved beam with symmetric and non-symmetric cross-sections with and without two types of elastic foundations are newly presented using series solutions for the displacement parameters. The equilibrium equations and the force-deformation relations are derived from the elastic strain energy including the effects of shear deformation and the axial-flexural-torsional coupling, and the strain energy considering the foundation effects. The explicit expressions for displacement parameters are derived by applying the power series expansions of displacement components to the simultaneous ordinary differential equations. Next, the element stiffness matrix is determined by using the force-deformation relationships. The normal stress at any arbitrary location of the cross-section for a curved beam is evaluated from the stiffness matrix. To verify the validity and the accuracy of this study, the displacements and the normal stresses of curved beams are presented and compared with the analytical solutions, the finite element results using the isoparametric curved beam elements based on the Lagrangian interpolation polynomial, and the detailed three-dimensional analysis results using the shell elements of SAP2000. This paper was recommended for publication in revised form by Associate Editor Maenghyo Cho Nam-Il Kim received his B.S. degree in Civil and Environmental Engineering from Sungkyunkwan University, Korea, in 1996. He then received his M.S. and Ph.D. degrees from Sungkyunkwan University in 1998 and 2004, respectively. Dr. Kim is currently a research professor at Civil and Environmental Engineering at Myongji University in Korea. Dr. Kim’s research interests include stability and vibration of steel and composite structures. Dong Ku Shin received his B.S. and M.S. degrees in Civil Engineering from Seoul National University, Korea, in 1983 and 1985, respectively. He then received his Ph.D. degree from Virginia Tech. at Blacksburg, VA, USA, in 1990. Dr. Shin is currently a professor of Civil and Environmental Engineering Department at Myongji University in Korea. Prof. Shin’s research interests include LRFD design of steel bridges and stability of composite structures.