Postbuckling of cross-ply laminated cylindrical shells with piezoelectric actuators under complex loading conditions

A postbuckling analysis is presented for a cross-ply laminated cylindrical shell with piezoelectric actuators subjected to the combined action of mechanical, electric and thermal loads. The temperature field considered is assumed to be a uniform distribution over the shell surface and through the shell thickness and the electric field is assumed to be the transverse component E_z only. The material properties are assumed to be independent of the temperature and the electric field. The governing equations are based on the classical shell theory with a 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 hybrid laminated cylindrical shells. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, cross-ply laminated cylindrical thin shells with fully covered or embedded piezoelectric actuators subjected to combined mechanical loading of external pressure and axial compression, and under different sets of thermal and electric loading conditions. The effects played by temperature rise, applied voltage, shell geometric parameter, stacking sequence, as well as initial geometric imperfections are studied.     

Vibration and buckling of cross-ply laminated composite circular cylindrical shells according to a global higher-order theory

Natural frequencies and buckling stresses of cross-ply laminated composite circular cylindrical shells are analyzed by taking into account the effects of higher-order deformations such as transverse shear and normal deformations, and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for laminated composite circular cylindrical shells made of elastic and orthotropic materials is derived through Hamilton's principle. Several sets of truncated approximate higher-order theories are applied to solve the vibration and buckling problems of laminated composite circular cylindrical shells subjected to axial stresses. The total number of unknowns does not depend on the number of layers in any multilayered shells. In order to assure the accuracy of the present theory, convergence properties of the first natural frequency and corresponding buckling stress for the fundamental mode r=s=1 are examined in detail. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. Modal transverse shear and normal stresses can be calculated by integrating the three-dimensional equations of equilibrium in the thickness direction, and satisfying the continuity conditions at the interface between layers and stress boundary conditions at the external surfaces. It is noticed that the present global higher-order approximate theories can predict accurately the natural frequencies and buckling stresses of simply supported laminated composite circular cylindrical shells within small number of unknowns.     

Postbuckling analysis of imperfect stiffened laminated cylindrical shells under combined external pressure and thermal loading

A postbuckling analysis is presented for a stiffened laminated cylindrical shell of finite length subjected to combined loading of external pressure and a uniform temperature rise. The formulation is based on a boundary layer theory of shell buckling which includes the effects of nonlinear prebuckling deformations, nonlinear large deflections in the postbuckling range and initial geometrical imperfections of the shell. The “smeared stiffener” approach is adopted for the stiffeners. The analysis uses a singular perturbation technique to determine the interactive buckling loads and the postbuckling equilibrium paths. Numerical examples are presented that relate to the performance of perfect and imperfect, stiffened and unstiffened cross-ply laminated cylindrical shells. Typical results are presented in dimensionless graphical form for different parameters and loading conditions.     

Deformation and failure mechanisms of lattice cylindrical shells under axial loading

The lattice cylindrical shells wound from the planar lattice plates, which have significant applications in aerospace engineering, exhibit different deformation modes with their planar counterparts because of the curvature of the cell wall. In this paper, deformation mechanisms are systematically investigated and failure analyses are conducted for the lattice cylindrical shells with various core topologies. Analytical models are proposed to predict the axial stiffness, critical elastic buckling load or effective yield strength of these shells. Finite element simulations are carried out to identify the validity of the models. The models can be employed for the optimal design. As an example, we construct the failure map for the Kagome lattice cylindrical shell made from an elastic ideally-plastic material. Various failure mechanisms, including yielding, global elastic buckling and local elastic buckling are taken into account. Moreover, optimizations are performed to minimize the weight for a given stiffness or load-carrying capacity for three types of lattice cylindrical shells. It is found that the Kagome and triangular lattice cylindrical shells have almost equivalent load-bearing capacity and both significantly outperform the hexagonal one under axial compression.     

Postbuckling of shear deformable cross-ply laminated cylindrical shells under combined external pressure and axial compression

A postbuckling analysis is presented for a shear deformable cross-ply laminated cylindrical shell of finite length subjected to combined loading of external pressure and axial compression. The governing equations are based on Reddy's higher order shear deformation 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 shear deformable laminated cylindrical shells under combined loading cases. A singular perturbation technique is employed to determine interactive buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, unstiffened or stiffened, moderately thick, antisymmetric and symmetric cross-ply laminated cylindrical shells for different values of load-proportional parameters.     

Nonlinear stability analysis of infinitely long laminated cylindrical shallow shells including shear deformation under lateral pressure

This paper presents a theoretical analysis for the various kinds of buckling behaviour of infinitely long laminated cylindrical shallow shells subjected to lateral uniform pressure. The exact solutions of the nonlinear equilibrium equations, in which first-order shear deformation is included, are obtained and the buckling criteria corresponding to different kinds of buckling are constructed taking into account the effects of the transverse shear deformation.     

弹性圆柱壳的稳定性优化设计

研究任意轴对称边界条件下和受均布法向载荷作用圆柱壳的稳定性优化设计问题，即极大化屈曲临界载荷。利用能量原理分析轴对称变厚度圆柱壳的分支点屈曲，将求解屈曲临界载荷变成求解广义特征值方程，使圆柱壳稳定性优化设计成为极大化最小特征值问题。实际算例验证了本方法的有效性。研究结果可用于圆柱壳的加肋优化设计。     

Postbuckling of 3D braided composite cylindrical shells under combined external pressure and axial compression in thermal environments

A postbuckling analysis is presented for a three-dimensional (3D) braided composite cylindrical shell of finite length subjected to combined loading of external pressure and axial compression in thermal environments. Based on a micro–macro-mechanical model, a 3D braided composite may be a cell system and the geometry of each cell is highly dependent on its position in the cross-section of the cylindrical shell. The material properties of epoxy are expressed as a linear function of temperature. The governing equations are based on a higher order shear deformation shell theory with a von Kármán–Donnell-type kinematic nonlinearity and includes thermal effects. A singular perturbation technique is employed to determine interactive buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, braided composite cylindrical shells with different values of shell geometric parameter and of fiber volume fraction under combined loading conditions. The results show that the shell has lower buckling loads and postbuckling paths when the temperature-dependent properties are taken into account. The effects of temperature rise, fiber volume fraction, shell geometric parameter, load-proportional parameter, as well as initial geometric imperfections are studied.     

Thermal buckling behavior of laminated composite plates: a finite-element study

In this paper, the thermal buckling behavior of composite laminated plates under a uniform temperature distribution is studied. A finite element of four nodes and 32 degrees of freedom (DOF), previously developed for the bending and mechanical buckling of laminated composite plates, is extended to investigate the thermal buckling behavior of laminated composite plates. Based upon the classical plate theory, the present finite element is a combination of a linear isoparametric membrane element and a high precision rectangular Hermitian element. The numerical implementation of the present finite element allowed the comparison of the numerical obtained results with results obtained from the literature: 1) with element of the same order, 2) the first order shear deformation theory, 3) the high order shear deformation theory and 4) the three-dimensional solution. It was found that the obtained results were very close to the reference results and the proposed element offers a good convergence speed. Furthermore, a parametrical study was also conducted to investigate the effect of the anisotropy of composite materials on the critical buckling temperature of laminated plates. The study showed that: 1) the critical buckling temperature generally decreases with the increasing of the modulus ratio E _L/E _T and thermal expansion ratio α _T/α _L, and 2) the boundary conditions and the orientation angles significantly affect the critical buckling temperature of laminated plates.     

Post-buckling and delamination propagation behavior of composite laminates with embedded delamination

Delamination occurred due to poor manufacturing process or in-service actions significantly affects the mechanical and failure behavior of laminated composite structures. In this study, the buckling and post-buckling delamination behavior of laminated composite with an embedded initial delamination under in-plane compression was studied experimentally and numerically. First, compression tests for laminated composite specimens with embeded initial delamination were performed and the buckling and delamination responses were obtained. Then the experimental test was numerically simulated using finite element methods with the progressive failure accounted for by using cohesive zone modeling. The load-displacement curve, strain behavior and delamination shapes of experimental specimens obtained from load cells, strain gages installed at different locations, and C scan images, respectively, were compared with the FEM results, and good agreements were attained. The effect of the buckling modes, laminate stacking sequence and shape of initial delamination on the buckling load and propagation behavior was studied by considering different ply stacking and shapes of initial delaminations. It was found that the buckling mode determined the growth direction of the delamination propagation, and the stacking sequence influenced the extent of the propagation area, while the orientation of the delamination affected the buckling loads.     

脱层复合材料圆柱壳屈曲的实验研究

脱层的存在将会大大降低层台结构的屈曲载荷。本文首先针对在轴压载荷作用下含脱层的复合材料层合圆柱壳，设计了测试其稳定性的实验方案。然后利用位移传感器、应变传感器、声发射仪器和红外热像仪等实验仪器，对含任意位置脱层与不含脱层的圆柱壳的屈曲载荷进行了实验测试，并对实验进行了精度分析。最后给出了实验结果并进行了讨论。     

Neural potentials and micro-signals of non-linear deep and shallow conical shells

Conventional sensors, such as proximeters and accelerometers, are add-on devices usually adding additional weights to structures and machines. Health monitoring of flexible structures by electroactive smart materials has been investigated over the years. Thin-film piezoelectric material, e.g. polyvinylidene fluoride (PVDF) polymeric material, is a lightweight and dynamic sensitive material appearing to be a perfect candidate in monitoring structure's dynamic state and health status of flexible shell structures with complex geometries. The complexity of shell structures has thwarted the progress in studying the distributed sensing of shell structures. Linear distributed sensing of various structures have been studied, e.g. beams, plates, cylindrical shells, conical shells, spherical shells, paraboloidal shells and toroidal shells. However, distributed microscopic neural signals of non-linear shell structures has not been carried out rigorously. This study is to evaluate microscopic signals, modal voltages and distributed micro-neural signal components of truncated non-linear conical shells laminated with distributed infinitesimal piezoelectric neurons. Signal generation of distributed neuron sensors laminated on conical shells is defined first. The dynamic neural signal of truncated non-linear conical shells consists of microscopic linear and non-linear membrane components and linear bending component based on the von Karman geometric non-linearity. Micro-signals, modal voltages and distributed neural signal components of two different truncated non-linear conical shells are investigated and their sensitivities discussed.     

Viscoelastic and geometrically nonlinear behavior of laminated cylindrical shells

A viscoelastic and geometrically nonlinear finite element analysis is performed to investigate the stress relaxation and deflection of a laminated cylindrical shell under thermal loading. Incremental viscoelastic constitutive equations are derived to predict the stress relaxation. The finite element program is developed using a 3-D degenerated shell element, the first order shear deformation theory and the updated Lagrangian formulation. The viscoelastic and geometrically nonlinear analysis is executed for laminated shells with cross-ply and angle-ply stacking sequences, and its results are compared with those obtained from geometrically linear and viscoelastic analyses. The numerical results show that viscoelasticity and geometrical nonlinearity affect on the deflections and stresses of laminated cylindrical shells.     

Nonlinear elastic buckling and postbuckling of axially compressed functionally graded cylindrical shells

Based on the nonlinear large deflection theory of cylindrical shells as well as the Donnell assumptions, this paper presents nonlinear buckling and postbuckling analyses for axially compressed functionally graded cylindrical shells by using the Ritz energy method and the nonlinear strain-displacement relations of large deformation. The material properties of the shells vary smoothly through the shell thickness according to a power law distribution of the volume fraction of constituent materials. Meanwhile, by taking into account the temperature-dependent material properties, various effects of external thermal environment are also investigated. Numerical results show various effects of the inhomogeneous parameter, dimensional parameters and external thermal environments on nonlinear buckling and postbuckling behaviors. There is a mode-jumping observed after buckling. The present theoretical results are verified by those in the literature.     

热-机荷载作用的P-FGM扁球壳跳跃屈曲分析

基于经典壳体理论和Sanders非线性应变-位移关系,导出了幂律型功能梯度材料(P-FGM)扁球壳在热-机械荷载作用下的几何非线性常微分控制方程。推导过程考虑了沿厚度存在一维热传导温度场和法向均布荷载作用。采用打靶法求解了由控制方程和固定夹紧边界条件构成的两点边值问题。得到了FGM扁球壳的一些典型的屈曲平衡路径和双稳态构形。对热-机械荷载作用的FGM扁球壳的跳跃屈曲行为进行了参数影响分析。结果表明：温度上升时,球壳上临界荷载显著增加、下临界荷载变化不明显。梯度指数增加时,球壳上、下临界荷载均显著减小。组分材料模量增加时,球壳上、下临界荷载均显著增加。当底圆半径和厚度给定时,随壳体中面曲率半径增加,球壳上、下临界荷载均显著增加。当中面曲率半径和厚度给定时,随底圆半径增加,球壳下临界荷载显著减小,上临界荷载几乎不变。     

Thermomechanical buckling of laminated composite and sandwich plates using global–local higher order theory

In this paper, a global–local higher order theory has been used to study buckling response of the laminated composite and sandwich plates subjected to thermal/mechanical compressive loads. The present global–local theory satisfies the free surface conditions and the geometric and stress continuity conditions at interfaces, and the number of unknowns is independent of the layer numbers of the laminate. Based on this higher-order theory, a refined three-noded triangular element satisfying C¹ weak-continuity conditions has been also proposed. The present theory not only predicts accurately the buckling response of general laminated composite plates but also calculates the critical buckling loads of the soft-core sandwich plates. However, numerical results show that the global higher-order theories as well as first order theories encounter some difficulties and overestimate the critical buckling loads for the sandwich plates with a soft core.     

Influence of the initial imperfection on the non-linear buckling response of FGM truncated conical shells

Non-linear buckling analyses of imperfect functionally graded truncated conical shells with simply supported boundary conditions and subjected to an axial compressive load have been presented in this work. The material properties of functionally graded shells are assumed to vary continuously through the thickness of the shell. The non-linear prebuckling deformations and initial geometric imperfections of an FGM truncated conical shell are both taken into account. The fundamental relations, modified Donnell type non-linear stability and compatibility equations of an imperfect FGM truncated conical shell are obtained and are solved by superposition and Galerkin methods, and the upper and lower critical axial loads has been found analytically. The numerical illustrations concern the non-linear buckling response of FGM truncated conical shells with different values of truncated conical shell parameters, initial imperfections and compositional profiles. Comparing the results of this study with those in the literature validates the present analysis.     

复合材料薄壁圆柱壳热振动特性分析

采用有限元软件ANSYS参数设计语言（APDL）实现复合材料薄壁圆柱壳结构在线性分布温度场作用下的非线性热振动特性分析。计算不同约束条件、不同铺层方式和铺层层数的复合材料薄壁圆柱壳在线性温度场作用下的各阶固有频率,分析边界约束、铺层角度及铺层层数对结构热振动频率的影响。结果表明,温度效应、边界约束、铺层层数显著影响结构的固有频率,适当的铺层角度会改变结构的固有频率,不同的铺层方式对结构固有频率影响程度不同。这些结论将对复合材料结构设计、抗热设计有一定的指导作用。     

Theory and comparison of the effect of composite and shape memory alloy stiffeners on stability of composite shells and plates

The effects of composite and shape memory alloy stiffeners on stability of composite cylindrical shells and rectangular plates subjected to a compressive load are compared. The governing equations for reinforced cylindrical shells are developed based on the Love first approximation theory and smeared stiffeners technique. It is shown that composite stiffeners are more efficient in cylindrical shells, while shape memory alloy stiffeners may be preferable in plates or in long shallow shells. It is also proven that shape memory alloy stiffeners increase the upper and lower buckling loads, i.e. the linear buckling load and the minimum postbuckling load-carrying capacity of cylindrical shells modeled as single-degree-of-freedom systems by the same amount.     

Analytical solutions for thick closed laminated cylindrical shells

Discarding any assumptions about displacement models and stress distributions, the state equation for orthotropy is established in full in a cylindrical coordinate system. The analytical solutions are presented for the statics, dynamics and buckling of thick closed laminated cylindrical shells by means of dividing any layer into several thin plies. No matter how many layers are considered, the calculation always leads to the solution of a set of linear algebraic equations in three unknowns. Every equation of elasticity can be satisfied and all the elastic constants can be taken into account. Arbitrary precision of a desired order can be obtained.