Sensibility of shells to buckling disturbances in connection with parameters of critical loading spectrum

This paper considers the critical loading density in the vicinity of the first eigenvalue for the stability problems of both a structurally orthotropic longitudinally-compressed cylindrical shell and a spherical panel. For structurally orthotropic shells it is shown that the critical loading spectrum in the above-mentioned problems can begin with the condensation point only when definite relations between structurally orthotropic parameters take place. In connection with the presence of eigenvalue condensation points in the stability problems it becomes expedient to obtain approximate solutions for nonhomogeneous problems where the right-hand part of the equation describes disturbances. The linear systems of such a type show a strong selectivity as regards to initial deflections and to small external disturbances. The experimental results were treated. The process of the additional shell deflection development was analyzed. It was established by experiment that the shells with more rigid longitudinal stiffeners are less sensitive to initial imperfections and disturbances that explains a good agreement of critical loadings of these imperfect shells with Euler loadings. The number of equivalent forms for the shells with longitudinally stiffened, transversely stiffened and non-stiffened shells has the following order, respectively: 0(1), 0(10), 0(100).     

Postbuckling analysis of axially-loaded functionally graded cylindrical shells in thermal environments

A postbuckling analysis is presented for a functionally graded cylindrical thin shell of finite length subjected to compressive axial loads and in thermal environments. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The governing equations are based on the classical shell theory with von Kármán–Donnell-type of kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of functionally graded cylindrical shells. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of axially-loaded, perfect and imperfect, cylindrical thin shells with two constituent materials and under different sets of thermal environments. The effects played by temperature rise, volume fraction distribution, shell geometric parameter, and initial geometric imperfections are studied.     

Optimization of composite plates with piezoelectric stiffener-actuators under in-plane compressive loads

Stiffeners which are used to strengthen a plate can be constructed of piezoceramic materials and subsequently used as piezo actuators to improve the load carrying capacity of the plate. In the present study, a fibre composite plate with initial imperfections and under in-plane compressive loads is studied with a view towards minimizing its deflection using the piezo actuators and the fibre orientations. Piezoceramic stiffeners are bonded symmetrically on the top and bottom of the plate and deployed as actuators. Two cases of electric fields, namely, the in-phase and out-of-phase voltages are applied to the actuators. The presence of initial deflections leads to deformation under the in-plane compressive loads which should be less than the critical buckling load. Two cases of initial imperfections are considered, and the first one is the deterministic initial deflections which are known a priori and as such they are given as input parameters for the problem. In the second case the initial deflections are uncertain and they have to be obtained according to a given criterion. In the present study they are determined to produce the least favourable initial deflection (largest deflection) at a given point and the solution is obtained by convex modelling. The effect of the actuators, the ply angles and the voltage are studied and their effects on the transverse deflection are investigated. A performance index involving the L₂ norm of the deflections is minimized using the piezo effect and as well as the ply angles the optimal values of which are determined for various problem parameters.     

A comparative study on the elastic-plastic collapse strength of initially imperfect deep spherical shells

A finite difference method for the large deformation elastic-plastic analysis of spherical caps is applied to predict the collapse strength of initially imperfect deep spherical shells. Twelve uniformly loaded hemispherical shell models with flat spots at their apex are analysed. For each model, a number of shallow spherical regions containing the flat spot are selected from its domain. One of these selected shallow regions yields a minimum buckling pressure; this minimum value is taken as the theoretical buckling load for the shell model under consideration. Present solutions are in good agreement with existing experimental and empirical results. The good comparison suggests that initially imperfect deep spherical shells may be analysed by using a much simpler mathematical model—the spherical cap—and thus the analytical cost may be greatly reduced. This also demonstrates that the collapse of imperfect spherical shells is primarily a local phenomenon and therefore dependent on local geometry. Consequently, the presence of initial imperfections must be fully taken into consideration in any large deformation inelastic buckling analysis before such analysis can be expected to quantitatively predict the collapse strength of practical shell structures.     

On the imperfection sensitivity of complete spherical shells

The imperfection sensitivity of elastic complete spherical shells under external pressure is studied for axisymmetric deformations and qualitatively different types of imperfections by means of a numerical analysis of the Reissner shell equations. It is shown that strong reductions of the critical load are obtained for small deviations of the middle surface of the shell from the perfect spherical configuration whereas imperfections of the shell thickness do not have a substantial influence on the critical load.     

Imperfection sensitive variation of critical loads at hilltop bifurcation point

Variation of critical loads due to initial imperfections at the hilltop bifurcation point is described by elastic stability theory. We derive a system of bifurcation equations for a potential system expressing local behavior at this bifurcation point, which is a double critical point occurring as a coincidence of a simple pitchfork bifurcation point and a limit point. The piecewise linear law of imperfection sensitivity of critical loads in Thompson and Schorrock [J. Mech. Phys. Solids 23 (1975) 21] is revised by extending initial imperfections to be considered in the bifurcation equations. Based on this sensitivity law, a procedure to determine the most influential (worst or optimum) initial imperfection is formulated. As the most essential development of this paper, under the assumption that initial imperfections are subject to a multi-variate normal distribution, we derive the probability density function of critical loads that follows a Weibull-like distribution. The validity of theoretical developments is assessed through its application to elastic truss structures.     

不同参数影响下球面网壳表面风压系数分布规律研究

大跨度球面网壳在其风工程分析中包含了风的分离、再附着等最复杂部分,曲面表面风压系数的分布变得异常复杂。针对风压系数分布问题利用FLUENT 和计算流体力学(CFD)技术,在选用SST k-w 湍流物理模型的基础上,改变球面网壳矢跨比、球面网壳高度、风速、风向攻角及球面网壳半径等不同参数,计算分析了球面网壳表面风压系数的变化规律。为快速准确的得出网壳表面风压系数,提出了风压系数分布的二维几何平面拟合方法并得出拟合公式。最后将公式应用到已有的风洞实验结果,得到其风压分布趋势和实验结果基本相符。     

单层网壳结构稳定性分析方法研究

为了准确地分析具有初始几何缺陷的单层网壳结构整体稳定性,运用随机缺陷模态法、一致缺陷模态法和N阶特征缺陷模态法,对4个不同矢跨比的K8型单层网壳进行了近1300例弹塑性荷载-位移全过程分析,探讨不同分析方法的合理性和可行性。研究表明:随机缺陷模态法能较为科学地评估初始几何缺陷对结构稳定性的影响,但计算量较大;对稳定承载力系数样本进行统计特性分析时,空间样本数量n不应小于100;运用随机缺陷模态法确定网壳结构最终的稳定承载力系数时,建议采用“3σ”原则;采用最低阶屈曲模态模拟初始几何缺陷分布,求得的稳定承载力并非最不利,其保证率得不到有效保证;N阶特征缺陷模态法能够通过较少的计算量,得到满足“3σ”原则要求的稳定承载力,并能较为合理、安全地评估网壳结构的稳定性能;在运用N阶特征缺陷模态法时,建议N=20。     

Rigidity of spherical bands

A method of numerical calculation of an imperfect spherical band (truncated spherical shell) subjected to external pressure is proposed and tested. It is shown that the calculated and experimental values of the critical loads agree with introduction into the calculation of local axially symmetric initial imperfections.Translated from Problemy Prochnosti, No. 10, pp. 118–120, October, 1990.     

单双层球面扁网壳连续化方法非线性稳定理论临界荷载的确定

连续化方法是研究网壳结构稳定问题的一种重要途径,目前用连续化理论分析球面扁网壳的稳定问题还存在欠缺和不足。运用经典的壳体理论,将单层和双层球面扁网壳等代为实体薄壳并建立非线性稳定理论混合法基本方程,再用李兹法求出球面扁网壳上下临界荷载计算公式。通过参数分析,首次从1000多个算例中得出了正三角形网格单层和双层常用球面扁网壳临界荷载系数的精确解。与国内外现有文献的计算公式相比,结果更为完善和正确。即便在有限元技术日益成熟的今天,用连续化方法计算的网壳结构临界荷载仍然对工程设计有重要指导作用,也是有限元方法分析网壳稳定性的对比和补充。     

A note on impulsive sphere motion beneath a free-surface

A general framework is presented for solving the impulsive oblique motion of a spherical body in close proximity and below a free-surface. The fluid is considered to be impulsive and the flow as incompressible. The irrotational flow field is deduced from a velocity potential. The full nonlinear problem is reduced to a sequence of boundary-value problems by employing a small-time expansion technique. The mixed boundary conditions are of a Dirichlet type on the undisturbed free-surface and of a Neumann type on the equilibrium spherical shape. The solution is obtained by employing a Green's function and the method of multipoles expansions. General expressions, correct to each order in the small-time, are given for the free-surface deflections and the pressure force experienced by the moving sphere.     

初始几何缺陷对仓壁柱承钢筒仓稳定性能的影响

仓筒壁直接支承于支柱是小型钢筒仓的一种简单、经济的支承方式。薄壳结构通常对初始几何缺陷十分敏感。引入周向轴对称焊缝凹陷和特征值屈曲模态两种缺陷形式,通过几何非线性分析,研究初始缺陷对轴压下仓壁柱承筒仓稳定性能的影响。研究表明,周向焊缝凹陷会显著降低结构的屈曲荷载,且与焊缝位置密切相关,而特征值模态缺陷只有在支柱接近仓顶时才会显著影响其屈曲强度。     

Random field of initial deflections and strength of thin rectangular plates

The use of an integral measure of initial deflections of thin plates, based on the strain energy, has proved to be of importance in studying the influence of the imperfections on the strength of plates subjected to in-plane compression. The energy measure allows for a straightforward definition of compound modes and for a computational determination of the lower bound strength. The last quantity may be essential for assessment of plate design strength considering the imperfections as a random field.     

The imperfect linearly viscoelastic column

Columns made of a linear three-element model material and with initial lateral deflections, referred to as ‘imperfections’, are analyzed by the dynamic and static approaches. Both analyses provide approximately the same time-deflection response for the structure. It is shown that imperfections may be regarded as a special class of disturbances which do not affect the value of the viscoelastic critical load. However, the influence of such ‘permanent’ disturbances on the rate of deformation is much more severe than that due to disturbances acting on a perfect column. Consequently, any possible safe operational period, related to the so-called ‘critical time,’ for the imperfect structure is much shorter than a corresponding period obtained from an analysis of the perfect column.     

On the complete extinction of selected imperfection wavelengths in dynamically expanded ductile rings

In this work the inception and development of multiple necks in dynamically expanded ductile rings with ab initio geometric imperfections has been addressed. Finite element simulations and linear perturbation analysis have been applied for that task. In the numerical calculations a selected wavelength is included into the model defining along the circumference of the ring an array of periodic geometric imperfections of predefined amplitude. In the stability analysis a perturbation of a given mode is added to the background solution and the growth rate of the perturbation is evaluated. The attention has been focused on the extinction of both long and short wavelength imperfections and the appearance of a dominant necking pattern which emerges when the geometric imperfections are vanished. The role played by the loading rate on the extinction of imperfections is also addressed. Moreover, the necking strain is found to be dependent on the imperfection pattern and the loading rate. Its maximum value is registered for the loading cases in which the initial imperfections distribution is completely extinguished.     

Random field models of geometrically imperfect structures with “clamped” boundary conditions

The subject of the paper is the stochastic modeling of geometrical imperfections of one- and two-dimensional structures. The imperfections are considered as initial deflections from a perfect axis of a bar or a mid-surface of a plate. The main goal is to define random field models whose sample paths satisfy clamped boundary conditions. The approach based on conditional random fields is adopted and formulas allowing conditioning on derivatives of the random fields are presented. Conditions for random field sample differentiability are discussed as well. The proposed approach is illustrated with a numerical example.     

Oblique impact of inflated balls at large deflections

A planar theory for oblique impact of thin-walled spherical balls against a rough rigid surface has been developed on the basis of an assumed deformation field—an initially spherical ball is assumed to flatten against the constraint surface while the remainder of the ball remains undeformed. For inflated thin-walled balls, which are represented by these assumptions (basketballs, soccer balls, volleyballs, etc) the normal reaction force acting on the flattened contact patch is predominately due to the internal gas pressure—the reaction due to shell bending is insignificant in comparison with this gas force. During impact of a thin-walled ball there also is a non-conservative momentum flux reaction that is caused by the flow of momentum into and out-of the flattened contact patch. If the ball is translating as well as rotating about an axis perpendicular to the plane of motion, the distribution of the normal component of velocity for material entering and exiting the flattened contact patch results in a distribution of momentum flux force intensity around the periphery of the contact patch and consequently, a momentum flux torque acting on the flattened sphere. The effect of these reaction forces and torque on oblique impact of thin-walled spherical balls is calculated as a function of the ball deflection (or normal component of impact velocity). In comparison with rigid body calculations for oblique impact of a spinning ball against a rough surface at angles<45° from normal, the effect of maximum deflections as large as half the initial radius is to slightly accentuate the effect of friction on angle of rebound and moderately decrease the angular velocity of the ball. However, for angles >45° from normal, the final angular velocity can be as small as 40% of that predicted by rigid body theory. The most significant changes in rebound angle are for cases with initial backspin—a technique commonly used in many ball sports.     

Sensitivity analysis and optimal design of non-linear structures

Sensitivity analysis for non-linear elastic structures in regular and critical states is first discussed including design parameters and initial imperfections. Next, the optimal design problem is formulated by considering imperfect structures and setting constraints on deflections and stresses. For structures with unstable post-critical response the limit load constraint is introduced in the optimization procedure. Several examples of truss optimization are provided. The level of initial imperfections can be regarded as design parameter and specified from the optimal solution. © 1998 John Wiley & Sons, Ltd.     

局部缺陷对旋转壳内力分布的影响

本文应用有初始几何缺陷的壳体非线性应变分量公式及内时本构理论，对冷却塔在重力作用下的内力进行了弹塑性大变形计算，分析了常见的几种缺陷对冷却塔内力分布的影响。对于工程实际中有缺陷的冷却塔的整体性能分析具有重要意义，可以为工程设计提供理论基础。     

超高模板支架的极限承载能力研究

近年来超高模板支架坍塌事故经常发生,承载力不足是主要原因,该文对模板支架的初始缺陷进行实测,统计分析测量数据,考虑较大的初始弯曲率和不足的钢管壁厚,提出一个新的极限承载力公式;通过非线性有限元理论计算,给出了初始弯曲率和钢管壁厚对极限承载力的影响系数;通过弹性稳定性分析,找出了不同立杆超出顶层水平杆长度下超高模板支架的临界荷载及失稳模态的变化规律,为今后的深入研究打下基础。