Collapse of long, noncircular, cylindrical shells under pure bending

This paper describes an extension of a method developed in a previous paper to determine the moment carrying capacity of elastoplastic noncircular cylindrical shells with infinite length by the finite element method. As a result of the shape change in the cross section of a shell during deformation, the bending moment reaches a global maximum value and then decreases as the bending curvature further increases. The shell would consequently collapse at the maximum moment. However, a bifurcation buckling may occur before the maximum moment can be developed. This bifurcation buckling could induce collapse of the shell under a moment less than the maximum. Determination of the likelihood that the bifurcation buckling would generate shell collapse may be made from the initial post-buckling behavior. An initial post-buckling analysis based on the J₂ deformation theory of plasticity has been developed in this paper. The finite element method with one spatial variable is used to locate the bifurcation point as well as to analyze the initial post-buckling behavior. Numerical examples of cylindrical shells with various cross-sectional shapes are shown. In particular, for a shell of square cross section, the moment at the bifurcation is much lower than the maximum value; however, the initial post-buckling analysis reveals that the state of equilibrium is still stable. Deep post-buckling analysis is required to determine the moment carrying capacity of a shell with such cross section.     

Buckling and postbuckling of imperfect cylindrical shells under axial compression

A solution methodology is described for the complete analysis of a geometrically imperfect, thin, circular, cylindrical shell loaded by a uniform axial compression. The analysis includes pre-limit point behavior, the establishment of critical conditions (limit point) and post-limit point behavior. The solution scheme is then utilized to study the effects of various geometrical parameters (radius to thickness and length to radius ratios) on the response characteristics of an imperfect, unstiffened, thin, cylindrical shell. These effects are assessed for a virtually axisymmetric-type of geometric imperfection.     

Parametric instability of a circular cylindrical shell with geometric imperfections

The static and dynamic behavior of a compressed circular cylindrical shell having geometric imperfections is analyzed. The analysis is mainly performed by means of the Donnell’s nonlinear shallow-shell theory. However, the refined Sanders shell theory is also used for comparison. A suitable expansion of the radial displacement, able to describe both buckling and dynamic behaviors is developed; the effect of geometric imperfections is accounted for by means of a modal representation. The response of the shell subjected to a sinusoidal axial excitation at its ends, giving rise to a parametric excitation, is considered. The effect of imperfections on the critical value of the dynamic load, that causes the loss of stability of the system, is analyzed. Interesting nonlinear dynamic phenomena are observed: direct resonance with softening behavior and parametric instability with period doubling response.     

含初应力复合材料柱壳结构的双稳态特性

为分析初应力对复合材料圆柱壳结构双稳态特性的影响,采用经典板壳理论建立复合材料圆柱壳力学模型,基于层合结构本构关系推导用双参数表达的系统应变能公式;根据最小势能原理得到双稳态产生的条件和稳态时的曲率表达式。利用Abaqus软件构建圆柱壳的有限元模型,通过附加边界弯矩对柱壳稳态跃迁过程进行模拟。理论计算结果与有限元结果的对比验证理论模型的正确性。分析结果表明：当初应力满足一定条件时,复合材料柱壳结构在其变形过程中有2个稳定平衡位置,并且在稳定平衡位置结构都不产生扭转变形;2个稳定平衡位置的曲率方向可以相同或相反,这与无初应力时反对称复合材料柱壳双稳态曲率方向只能相同的情况有区别。     

Finite element analysis of buckling and post-buckling behaviors of arches with geometric imperfections

The buckling and post-buckling behavior of arches is very sensitive to their geometric imperfections. The purpose of this paper is to develop a refined curved finite element that might accurately represent the actual geometry of arches so that the imperfection effects on their buckling behavior could be properly investigated. For an arch with known geometric imperfections, the element stiffness matrix is precisely formulated in terms of Lagrangian variables for a perfect arch from a general incremental variational principle. In general, the element stiffness matrix contains Lagrangian strain, first and second order incremental strain and imperfection terms. For any general planar imperfect arch with a variable curvature, the element stiffness matrix is evaluated by numerical integration; however, for a nominally circular arch, it can be represented in closed form. Numerical results in terms of load-deformation curves are presented for a number of circular arches with and without imperfections and compared with existing solutions.     

Developable polynomial surface approximation to smooth surfaces for fabrication parameters of a large curved shell plate by Differential Evolution

This paper presents a simple and efficient method to approximate a developable surface to a compound design surface by a polynomial. It is required to predict a final shape of roll bending in the fabrication of a curved shell plate. The roll bending process usually makes the cylindrical or conical curvature from an initial flat plate. It means that the final shape is developable or the surface representation has zero Gaussian curvature. The fabrication shape is important in order to estimate process parameters of roller bending.An optimization problem is formulated to determine the polynomial surface which is in the closest proximity to the design surface or the given shell plate, which is subjected to developability. The results and the efficiency of this algorithm are verified and evaluated by applying it to some shell plates which are obtained from a real ship model. The predicted bending shape becomes fundamental information in determining more process parameters for the fabrication of a compound curved shell plate.     

Insight into finite element shell discretizations by use of the “basic shell mathematical model”

The objective of this paper is to gain insight into finite element discretizations of shells using the basic shell mathematical model and, in particular, regarding the sources of “locking”. We briefly review the “basic shell mathematical model” and present a formulation of shell finite elements based on this model. These shell finite elements are equivalent to the widely-used continuum mechanics based shell finite elements. We consider a free hyperboloid shell problem, which is known to be difficult to solve accurately. Using a fine mesh of MITC9 elements based on the basic shell mathematical model, a detailed analysis is performed giving the distributions of all strain terms. A similar analysis using the MITC6 shell element shows why this element locks when the shell thickness is very small.     

A posteriori initial imperfection identification in shell buckling problems

The current research seeks to demonstrate that an inverse solution approach, leveraging nonlinear finite element analysis with a divide and conquer type stochastic search algorithm, can identify the presence of localized denting imperfections in cylindrical shell structures. This imperfection field identification is achieved using rather sparse displacement measurements taken at safe, service loading conditions. Both the existence and nature of the imperfection field present in a given shell structure instance are determined. These inferred imperfections are subsequently used to make reasonably accurate predictions regarding the actual shell structure strength at ultimate loading.     

Nonlinear thermo-mechanical response of temperature-dependent FG sandwich nanobeams with geometric imperfection

In this paper, the nonlinear dynamic response of functionally graded (FG) sandwich nanobeam associated with temperature-dependent material properties by considering the initial geometric imperfection is investigated. The size-dependent behavior of the FG sandwich nanobeam is simulated based on the nonlocal strain gradient theory, and Von Karman nonlinear hypothesis is used to model the geometrical nonlinearity. Moreover, the geometric imperfection is considered as a slight curvature satisfying the first mode shape, and four different FG sandwich patterns including two asymmetric configurations and two symmetric configurations are taken into account. The governing equation of the FG sandwich nanobeam subjected to thermal and harmonic external excitation loadings is derived on the basis of Hamilton’s principle. The numerical results are obtained by employing the multiple-scale method, which are also validated by comparison with two previous studies. Furthermore, comprehensive investigations into the influences of size-dependent parameters, external temperature variation, geometric imperfection amplitude, gradient index and sandwich configuration on the nonlinear characteristics of imperfect FG sandwich nanobeams are conducted through numerical results.

     

Axisymmetric shell optimization under fracture mechanics and geometric constraint

This paper describes a problem of axisymmetric shell optimization under fracture mechanics and geometric constraints. The shell is made from quasi-brittle materials, and through crack arising is admitted. It is supposed that the shell is loaded by cyclic forces. A crack propagation process related to the stress intensity factor is described by Paris fatigue law. The problem of finding the meridian shape and the thickness distribution (geometric design variables) of the shell having the smallest mass subject to constraints on the cyclic number for fatigue cracks and geometrical constraint on the shell volume is investigated. Special attention is devoted to different possibilities of problem transformation and analytical methods of their solution. Using minimax approach, optimal shapes of the shells and their thickness distributions have been found analytically.     

Biaxial buckling and post-buckling of composite laminated plates on two-parameter elastic foundations

A post-buckling analysis is presented for a simply supported, composite laminated rectangular plate under biaxial compressive loading and resting on a two-parameter (Pasternak-type) elastic foundation. The analysis uses a perturbation technique to determine the interactive buckling loads and post-buckling equilibrium paths. The initial geometrical imperfection of the plates is taken into account. Numerical examples are presented that relate to the performances of perfect and imperfect, antisymmetrically angle-ply and symmetrically cross-ply laminated rectangular plates. Typical results are presented in dimensionless graphical form.     

Design sensitivity analysis with isoparametric shell elements

This paper deals with design sensitivity calculation by the direct differentiation method for isoparametric curved shell elements. Sensitivity parameters include geometric variables which influence the size and the shape of a structure, as well as the shell thickness. The influence of design variables, therefore, may be separated into two distinct contributions. The parametric mapping within an element, as well as the influence of geometric variables on the orientation of an element in space, is accounted for by the sensitivity calculation of geometric variables, and efficient formulations of sensitivity calculation are derived for the element stiffness, the geometric stiffness and the mass matrices. The methods presented here are applied to the sensitivity calculations of displacement, stress, buckling stress and natural frequency of typical basic examples such as a square plate and a cylindrical shell. The numerical results are compared with the theoretical solutions and finite difference values.     

Optimal design of dynamic absorbers on vibration and noise control of the fuselage

Optimum design of dynamic absorbers for reducing the vibration and the interior noise of an aircraft’s fuselage is studied. Herein, a thin, elastic cylindrical shell is adopted as a simple model of the fuselage. The sound source of the noise in the acoustic field comes from the vibration of the shell. Several dynamic absorbers are then attached to the shell for vibration and noise control. The vibration of the shell and its interior sound pressure, caused by the propellers or the engines are formulated. Optimum design of the absorbers is studied for obtaining the minimum vibration of the fuselage or the minimum noise level in the cylindrical cavity. From the numerical results, the absorbers are found to be effective for vibration and noise control of the fuselage. Some general guidelines on optimum absorber design are also offered in conclusion.     

Homoclinic and heteroclinic orbits underlying the post-buckling of axially-compressed cylindrical shells

A structural system with an unstable post-buckling response that subsequently restabilizes has the potential to exhibit homoclinic connections from the fundamental equilibrium state to itself over a range of loads, and heteroclinic connections between fundamental and periodic equilibrium states over a different (smaller) range of loads. It is argued that such equilibrium configurations are important in the interpretation of observed behaviour, and govern the minimum possible post-buckling loads.

To illustrate this, the classical problem of a long thin axially-compressed cylindrical shell is revisited from three different perspectives: asymptotic conjecture, analogy with nonlinear dynamics, and numerical continuation analysis of a partial spectral decomposition of the underlying equilibrium equations. The nonlinear dynamics analogy demonstrates that the structure of the heteroclinic connections is more complicated than that indicated by the asymptotics: this is confirmed by the numerics. However, when the asymptotic portrayal is compared to the numerics, it turns out to be surprisingly accurate in its Maxwell-load prediction of the practically-significant first minimum to appear in the post-buckling regime.     

Adaptive enrichment meshfree simulation and experiment on buckling and post-buckling analysis in sheet metal forming

Sheet buckling, a form of instability, is one of the major considerations in the design of part shape, die geometry and processing parameters of sheet metal forming. In this study, an adaptive enrichment meshfree method is developed to capture wrinkling and post-buckling behavior in sheet metal forming. A three-dimensional meshfree continuum approach is applied to the large deformation of plate/shell structures. A stress-based wrinkling predictor is used to predict the onset of buckling within effective compressive regions. Enrichment particles with a proper enrichment function are inserted/deleted in those regions to capture the buckling mode and therefore post-buckling behavior. For verification of the simulation results, a high-resolution wedge strip test is designed to study the onset and post-buckling behavior of a sheet under different boundary conditions.     

Stochastic inverse identification of geometric imperfections in shell structures

It is now widely known that the presence of geometric imperfections in shell structures constitutes an important contribution to the discrepancy between theoretical and experimentally realizable ultimate loads governed by buckling. The present paper describes a method by which an actual initial imperfection field may be estimated using the service load response of a shell structure. The approach requires solving a stochastic inverse problem wherein uncertainty regarding initial imperfection predictions is expressed within the context of a Bayesian posterior distribution. The proposed approach could be applied to condition assessment and performance evaluation activities in practice.     

基于有限差分法的端部约束同轴圆柱壳的流致失稳分析

本文基于经典壳体和理想势流理论建立圆柱壳流固耦合系统的运动方程,并引入有限差分法(FDM)对运动方程进行离散.为将壳体表面的扰动压力离散到差分网格节点,基于差分离散的原理,提出了以分段函数作为基函数的展开方法.本文对结构与压力控制方程均采用FDM方法进行求解,发展了一种基于FDM的同轴圆柱壳流固耦合求解策略.首先,利用本文方法计算了同轴圆柱壳在静态流体环境中的振动频率,并与有限元软件(ANSYS)的计算结果相比较,验证了本文方法的正确性；然后,探究了在静流体环境中同轴圆柱壳的结构参数变化对其振动频率的影响规律；最后,研究了同轴圆柱壳系统在运动流体环境中的流弹失稳问题.     

Thermal post-buckling analysis of imperfect shear-deformable plates on two-parameter elastic foundations

A thermal post-buckling analysis is presented for a shear-deformable rectangular plate subjected to uniform or non-uniform tent-like temperature loading and resting on a two-parameter elastic foundation. The initial goemetrical imperfection of the plate is taken into account. The formulations are based on the Reissner-Mindlin plate theory considering the first order shear deformation effect and including thermal effects. The analysis uses a deflection-type perturbation technique to determine the thermal buckling loads and post-buckling equilibrium paths. Numerical examples cover the performances of perfect and imperfect, shear-deformable plates resting on Winkler or Pasternak-type elastic foundations.     

恒定磁场中简支圆柱壳的磁弹性振动分析

依据电磁场方程及相应的电磁本构关系,给出了作用于圆柱壳体上的电磁力及力矩表达式.在此基础上,分别推得了纵向和横向磁场中圆柱壳体的磁弹性轴对称振动方程.针对两端简支约束条件,通过位移函数的设定,得到了相应的有阻尼振动微分方程.通过算例,给出了系统衰减振动的响应曲线图和相图,分析了磁感应强度和壳体厚度对系统振幅衰减速度的影响.结果表明,通过改变磁感应强度可以达到控制系统振动的目的.