Considerations on the numerical analysis of initial post-buckling behaviour in plates and beams

In this work, an exceedingly simple examination of the von Kármán theory for the buckling analysis of thin plates is carried out. It is shown that the von Kármán assumption, based on the interaction between axial and flexural deformation, is intrinsically less “robust” in capturing the target phenomenology than the usual treatment of Euler inextensible strut. Finally, the same observation is carried out in the case of a hinged cantilever, which constitutes the simplest structural example of a similar kind.     

Auto-parametric resonance in thin cylindrical shells using the slow fluctuation method

Internal auto-parametric instabilities in the non-linear vibrations of an undamped and unloaded cylindrical shell are studied. The focus is on the coupling between two modes that can combine to break the in–out symmetry and give an energetically favourable pattern of deformation. When the ratio of the natural frequencies is close to 2, internal resonance triggers significant energy transfer between the modes. This energy exchange has both regular and chaotic features, which were previously investigated by other methods of analysis in [J. Sound Vibrat. 227 (1999) 65; J. Sound Vibrat. 248 (2001) 395]. A Rayleigh-Ritz discretization of the von Kármán–Donnell equations is employed to derive the Hamiltonian equations in R⁴. A perturbation analysis using action-angle coordinates gives slow fluctuation equations that can be solved by quadrature in Jacobian elliptic functions: and, a simple analytical criterion for the parametric instability is presented. This analytical solution gives a good approximation to the regular motions, and provides a framework in which to examine the chaotic motions that are studied in [J. Sound Vibrat. 248 (2001) 395].     

Buckling of elastic cylindrical shells considering the effect of localized axisymmetric imperfections

The effect of localized axisymmetric initial imperfections on the critical load of elastic cylindrical shells subjected to axial compression is studied through analytical modeling. Some classical results regarding sensitivity of shell buckling strength with respect to distributed defects having axisymmetric or asymmetric forms are recalled. Special emphasis is placed after that on the more severe case of localized defects satisfying axial symmetry by displaying an analytical solution to the Von Kármán–Donnell shell equations under specific boundary conditions. The obtained results show that the critical load varies very much with the geometrical parameters of the localized defect. These variations are not monotonic in general. They indicate, however, a clear reduction of the shell critical load for some defects recognized as the most hazardous isolated ones. Reduction of the critical load is found to reach a level which is up to two times lower than that predicted by general distributed defects.     

Spline FSM postbuckling analysis of shear-deformable rectangular laminates

A spline finite strip method is developed for the prediction of the geometrically non-linear response of rectangular, composite laminated plates to progressive in-plane loading. The development takes place within the context of the use of the first-order shear deformation plate theory and the non-linearity is introduced in the strain-displacement equations in the manner of the von Karman assumption. A number of applications of the new capability is described, involving laminates subjected to progressive uniform end shortening and to progressive in-plane shearing. In all the applications a close comparison of the finite strip results with independent finite element results is demonstrated.     

Nonlinear buckling of microfabricated thin annular plates

Annular plate structures are commonly used in MEMS devices, particularly in pumps and valves. In MEMS applications, large nonlinear deflections are routinely achieved. In this paper, the nonlinear buckling of a thin elastic annular plate under a compressive radial force acting in the plane of the plate is considered. Although the critical loads at which buckling starts can be determined by solving a linear eigenvalue problem, the large deflection behavior of a buckled plate beyond the critical loads can be described by the nonlinear theory proposed by von Kármán. The buckling loads of the annular plate for various boundary conditions are calculated and the post-buckling behaviors are examined. Interestingly, the buckling of an annular plate exhibits a supercritical pitchfork bifurcation. Thus, the post-buckling behavior is stable and will not collapse catastrophically. Several design implications for microfabricated structures, in particular microvalves, are also given.     

A finite strip analysis of locally buckled plate structures subject to nonuniform compression

The post-local-buckling behaviour of thin plate structures is governed by the mode of loading which can be either one of prescribed load eccentricity or displacement pattern of end sections. A finite strip approach to the analysis of post-local-buckling behaviour under either mode of loading is presented. A particularly simple initial postbuckling analysis based on the perturbation technique is discussed in detail. Examples are presented to illustrate the convergence of the solution and contrast the behaviour of the structure under the two modes of loading. In particular it is demonstrated that a channel-section strut is stiffer and attains a higher collapse load under uniform end shortening that when loaded through the centroid of the cross-section.     

Nonlinear thermomechanical post-buckling of circular FGM plate with geometric imperfection

Nonlinear thermomechanical post-buckling of an imperfect functionally graded material (FGM) circular plate, subjected to both mechanical load and transversely non-uniform temperature rise, is presented. The material properties of FGM plates are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Based on von Kármán's plate theory, equilibrium equations governing a large axi-symmetric deformation of the FGM circular plate under thermomechanical loads are derived. In the analysis, the geometric imperfections of the plate are taken into account. By using a shooting method the nonlinear ordinary differential equations with immovably clamped boundary conditions are solved numerically. Responses for the nonlinear thermomechanical post-buckling responses of the FGM plate are obtained. Numerical examples are presented that relate to the performances of perfect and imperfect, homogenous and graded plates. Characteristic curves of the post-buckling deformation of the imperfect FGM circular plate varying with thermal loads, imperfection parameters and volume fraction index are plotted. And then effects of the load parameters, materials constitution, and the geometric imperfection of the plate on the deformation are discussed in detail.     

Post-local-buckling of fiber-reinforced plastic composite structural shapes using discrete plate analysis

In this paper, post-buckling of rectangular composite plates rotationally restrained at the longitudinal unloaded edges and subjected to end shortening strain at the simply-supported loaded edges is analyzed using the first-order shear deformation plate theory-based spline finite strip method, and its application to post-local-buckling of fiber-reinforced plastic (FRP) composite structural shapes is illustrated with discrete plate analysis. Two cases of elastically- and rotationally-restrained plates are analyzed using the spline finite strip method: rotationally-restrained along both the unloaded boundary edges (RR) and one rotationally-restrained and the other free along the unloaded edges (RF). The two cases of rotationally-restrained plates (i.e., the RR and RF plates) are further treated as the discrete plates of closed and open section FRP shapes, and by considering the effect of elastic restraints at the joint connections of flanges and webs, post-local-buckling of various FRP shapes under end shortening is studied. The numerical comparisons with the finite element modeling demonstrate that the proposed discrete plate analysis technique and spline finite strip method can be used as an efficient and valid tool for post-local-buckling analysis of FRP shapes.     

Vibration and buckling of thin-Walled structures by a new finite strip

This paper presents the application of a new finite strip to the analysis of folded-plate structures. The displacement function of a flat shell finite strip is made up of two parts, namely, the two in-plane displacement interpolations and the out-of-plane displacement interpolation. Each of the three displacement components is interpolated by a set of computed shape functions in the longitudinal direction and, as usual, one-dimensional shape functions in the transverse direction. Only standard beam shape functions are involved in the longitudinal computed shape functions. When compared with other finite strips, the present finite strip is relatively simple in dealing with boundary and internal support conditions. In addition, the method can be easily implemented by incorporating a standard finite strip program with a continuous-beam program. The computation of the stiffness matrix involves no numerical integration. To verify the accuracy and efficiency of the new finite strip, a few numerical experiments are conducted in which the present finite strip results are compared with those using other finite strips and/or finite elements for the vibration and buckling of folded-plate structures with varying complexity.     

The effects of bend–twist coupling on the post-buckling characteristics of composite laminated plates using semi-energy finite strip approach

In composite laminated plates various mechanical properties may be attained by changing the lay-up arrangement. The bent–twist coupling is one of the significant properties which dominate the mechanical performance of a laminated plate structure in large out-of-plane deformations. The effects of bend–twist coupling on the post-buckling behavior of composite laminated plates have been studied in this paper by implementing a finite strip approach based on the concept of a rigorous post-buckling solution for composite plates and plate structures, namely the semi-energy approach. All the plates are assumed to be symmetrically balanced laminates having uniform in-plane stiffness properties. As far as the out-of-plane stiffness properties are concerned, all the properties, except for the bend–twist coupling terms which are assumed to change from one laminate to another, are the same among different laminates. The orthotropic solutions have also been determined by precluding the bend–twist coupling terms. A comprehensive set of parametric studies have been carried out to investigate the effects of bend–twist coupling terms on the post-buckling load–displacement path, post-buckling deformations and stress distribution. The comparison between the results revealed markedly different behaviors among different laminates due to the bend–twist coupling effects. The outcome of the paper can help structural designers to have a better understanding of the effects of bend–twist coupling terms on the post-buckling behavior of laminated composite plates at the design stage.     

Solution to the problem of axisymmetric and asymmetric dynamic instability of three-layered annular plates

This paper presents the solution to the dynamic stability problem of three-layered, annular plate loaded by compressive stress increasing in time. The solution enables the evaluation of the critical, dynamic loads corresponding to the various modes of plate buckling. The symmetrical cross-section structure of plate is described by the classical theory of sandwich plate with the broken line hypothesis, the nonlinear Kármán’s plate equations and linear physical relations. The solution is based on Bubnov–Galerkin method and finite difference method. The values of critical dynamic loads have been calculated by means of the stability criterion presented by (Volmir (1972) [1]). The comparison of values of critical dynamic and static loads is presented, using the dynamic ratio. The obtained results have been compared with those obtained for plate model built by means of finite element method. The calculations were carried out using the ABAQS system. The dynamic response of plate models examined by two methods is consistent.     

有限元、组合条元与映射无穷元联合应用研究

提出了将有限元、组合条元与映射无穷元联合应用求解无限域问题的思想 .首先介绍 6种映射无穷元的位移模式和坐标映射关系 ;然后基于联合应用组合条元的要求 ,提出了一种新型单元—组合映射无穷元 ,并建立了相应的公式 ;编制了计算程序 ,经过算例分析证明了有限元、组合条元与映射无穷元联合应用方法的正确性和有效性     

复合荷载作用下钢梁弯扭屈曲的临界弯矩： (Ⅰ)理论研究

边界约束条件是影响钢梁弯扭屈曲临界弯矩的重要因素之一,但当前复合荷载作用下钢梁临界弯矩的研究仅为平面内、外均简支的情况。本文基于完备的总势能方程,采用Euler微分方程推导了完备的钢构件弯扭屈曲平衡微分方程组,进而简化得到钢梁弯扭屈曲侧移与扭转角独立及耦合时的平衡微分方程。分别采用Galerkin法、Rayleigh-Ritz法求解平衡微分方程(组)、总势能方程,得到了复合荷载作用下钢梁临界弯矩Mcr的计算式、双重求和形式的复合弯矩系数计算式以及单一荷载作用下的Mcr三系数(C1,i、C2,i、C3,i)通式,揭示了单一荷载两两共同作用时相关系数C1,ij的互等性和统一性特征,得到了平面外不同边界约束条件下比值C2,i/C1,i和C3,i/C1,i的关系,对影响Mcr三系数数值的因素进行了分析。研究表明：钢梁临界弯矩Mcr的计算式、复合弯矩系数Cb计算式以及Mcr的三系数通式适用于平面外4种不同边界约束条件的简支钢梁和固支钢梁;三系数形式的临界弯矩Mcr的计算式是基于扭转角试函数的基函数取1项得到的,系数C1,i的精度仅受侧移与扭转角是否耦合的影响,比值C2,i/C1,i和C3,i/C1,i不受侧移与扭转角是否耦合的影响,与采用Galerkin法还是Rayleigh-Ritz法无关。     

Postbuckling analysis of cross-ply laminated cylindrical shell panels under parabolic mechanical edge loading

Postbuckling equilibrium paths of simply supported cross-ply laminated cylindrical shell panels subjected to non-uniform (parabolic) inplane loads are traced in this paper. Love's shell theory with higher order shear deformation theory and von Kármán nonlinear strain–displacement relations are used in the mathematical formulation of the problem. In the first step, the plate membrane problem is solved to evaluate the stress distribution within the prebuckling range as the applied inplane edge load is non-uniform. The governing shell panel postbuckling equations are derived from the principle of minimum total potential energy using the above stress distributions. Adopting multi-term Galerkin's approximation, the governing equations are reduced into a set of non-linear algebraic equations. Newton–Raphson method in conjunction with Riks approach is employed to plot the postbuckling paths through limit points. Numerical results are presented for symmetric (0/90/0) crossply laminated cylindrical shell panels under parabolic inplane load, lateral distributed load and initial imperfections. Limit loads and snap-through behavior of shell panels are studied.     

基于广义协调变形的直桩弹塑性分析

考虑桩土作用界面存在虚拟剪切带,基于Mindlin位移解,用广义协调变形方程建立桩与土之间的协调方程。假设桩身为完全弹性,桩土界面为双曲线模型,建立方程组,采用有限差分方法求解。较好的模拟了桩土界面的相对滑移,对传统的桩土相互作用偏大做出了合理的解释。分析了多个工程实例,计算结果与实测结果比较一致。     

多室箱形梁非线性有限元分析

本文将单箱多室箱梁离散为若干个空间梁段单元,梁段单元的弯曲、扭转及其横截面翘曲均采用三次插值函数描述,其拉压变形采用线性插值。为了计算箱梁顶、底板的面外弯曲,剪力滞后及横截面畸变,采取横向有限条的概念,从梁段单元截取单位厚度的空腹桁架;取桁架的每根杆件为一平面梁单元,用三次插值函数描述其位移,单元节点位移参数用三次多项式表示为箱梁梁段单元节点位移参数的函数。箱梁各横隔板取为有结点转角的四结点平面单元,用基于连续介质力学的Ｕ．Ｌ．列式虚功增量方程,考虑剪切变形影响建立各种单元的弹塑性平衡方程。按照上述思路编制了计算程序,计算结果与模型测试结果接近。     

宽薄腹工形截面压弯构件的平面外稳定设计

《钢结构设计规范》(GB50017-2003)、《门式刚架轻型房屋钢结构技术规程》(CECS102∶2002)关于宽薄腹工形等截面压弯构件的平面外稳定计算,虽然基本原理相同,但具体方法差别很大。通过对试验数据、数值法的分析结果与按这两种方法的结果比较,发现“规范”的方法在很大范围内偏于保守,不甚合理;而“门规”的方法更为经济合理,建议规范也采用“门规”的方法对宽薄腹工形截面压弯构件的平面外稳定进行计算,使设计更加统一,经济合理。     

Plate bending analysis by unequally spaced splines

A cubic B-spline finite strip method (BFSM) is developed to analyze thin plates in bending. The basic mathematical relationships are derived for a direct stiffness formulation using a series type strip displacement function. Longitudinal behavior is modeled by a spline series in which unequal spline spacing is permitted. This feature allows local refinement of the discretization near patch and concentrated loads. Accuracy and convergence vis-à-vis alternative methods are compared. These include various finite element models, the conventional finite strip method and the BFSM with equally spaced splines. Comparisons show comparable accuracy with improved convergence. Oscillatory convergence due to Gibb's phenomenon, evident in some of the models, is avoided in the BFSM.     

Elastic-plastic analysis of internally pressurized torispherical shells

The constitutive equation for an elastic-plastic material model was derived using the von Mises yield criterion and assuming isotropic strain hardening. A layered finite element permitting geometrically linear and geometrically nonlinear elastic-plastic analysis of thin shell structures is presented. The effect of linear strain hardening on the size of plastic regions and the distribution of internal forces in an internally pressurized torispherical shell was analyzed. At sufficiently high pressures a significant difference in the distribution of internal forces was observed between elastic, perfectly plastic and strain hardening material. The effect of the size of plastic regions on the difference in the magnitude of internal forces obtained by geometrically linear and geometrically nonlinear computations of the torispherical shell was studied. An increase in the size of the plastic region was found to produce greater differences in the computation of meridional bending moments than in the computation of hoop stress resultants.