Stress analysis of axisymmetric shear deformable cross-ply laminated circular cylindrical shells

A generalized mixed theory for bending analysis of axisymmetric shear deformable laminated circular cylindrical shells is presented. The classical, first-order and higher-order shell theories have been used in the analysis. The Maupertuis–Lagrange (M–L) mixed variational formula is utilized to formulate the governing equations of circular cylindrical shells laminated by orthotropic layers. Analytical solutions are presented for symmetric and antisymmetric laminated circular cylindrical shells under sinusoidal loads and subjected to arbitrary boundary conditions. Numerical results of the higher-order theory for deflections and stresses of cross-ply laminated circular cylindrical shells are compared with those obtained by means of the classical and first-order shell theories. The effects, due to shear deformation, lamination schemes, loadings ratio, boundary conditions and orthotropy ratio on the deflections and stresses are investigated.     

Thermal Postbuckling Analysis of Imperfect Reissner-Mindlin Plates on Softening Nonlinear Elastic Foundations

A thermal postbuckling analysis is presented for a simply supported, moderately thick rectangular plate subjected to uniform or nonuniform tent-like temperature loading and resting on a softening nonlinear elastic foundation. The initial geometrical 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 plate-foundation interaction and thermal effects. The analysis uses a deflection-type perturbation technique to determine the thermal buckling loads and postbuckling equilibrium paths. Numerical examples are presented that relate to the performances of perfect and imperfect, moderately thick plates resting on softening nonlinear elastic foundations. The effects played by foundation stiffness, transverse shear deformation, plate aspect ratio, thermal load ratio and initial geometrical imperfections are studied. Typical results are presented in dimensionless graphical form and exhibit interesting imperfection sensitivity.     

Benchmark three-dimensional elasticity solutions for the buckling of thick orthotropic cylindrical shells

Benchmark solutions to the problem of buckling of orthotropic cylindrical shells, which are based on the three-dimensional theory of elasticity, are presented in this review article. It is assumed that the shell is under external pressure or axial compression or a combination of these loadings. These solutions provide a means of accurately assessing the limitations of the various shell theories in predicting critical loads. A comparison with some classical shell theories shows that the classical shell theories may produce, in general, highly non-conservative results on the critical load of composite shells with thick construction. One noteworthy exception: the Timoshenko shell buckling equations produce conservative results under pure axial compression.     

复合材料层合厚圆柱壳高阶理论的改进及其应用

建立了一个改进的LCW型的精化高阶理论,以分析厚圆柱壳的振动。提出u,v为三次多项式、w为二次多项式的位移模式,并利用上、下自由表面横向剪应力为零的边界条件,对所假定的位移场作了化简,将三阶剪切变形理论的未知数缩减为7个,在此基础上建立了相应的有限元列式。通过一个典型算例,与Soldatos和Lam的高阶剪切变形理论的解析解作了比较,说明笔者的精化高阶理论是可行的,而且具有较高的精确性,比LCW高阶理论更具有实用性。还通过频率参数随长度半径比L/R的变化,说明由于考虑了法向应力和法向应变,本文方法更适用于长度半径比较小的结构。     

Stretching and bending deformations due to normal and shear tractions of doubly curved shells using third-order shear and normal deformable theory

ABSTRACT

We analyze static infinitesimal deformations of doubly curved shells using a third-order shear and normal deformable theory (TSNDT) and delineate effects of the curvilinear length/thickness ratio, a/h, radius of curvature/curvilinear length, R/a, and the ratio of the two principal radii on through-the-thickness stresses, strain energies of the in-plane and the transverse shear and normal deformations, and strain energies of stretching and bending deformations for loads that include uniform normal tractions on a major surface and equal and opposite tangential tractions on the two major surfaces. In the TSNDT the three displacement components at a point are represented as complete polynomials of degree three in the thickness coordinate. Advantages of the TSNDT include not needing a shear correction factor, allowing stresses for monolithic shells to be computed from the constitutive relation and the shell theory displacements, and considering general tractions on bounding surfaces. For laminated shells we use an equivalent single layer TSNDT and find the in-plane stresses from the constitutive relations and the transverse stresses with a one-step stress recovery scheme. The in-house developed finite element software is first verified by comparing displacements and stresses in the shell computed from it with those from either analytical or numerical solutions of the corresponding 3D problems. The strain energy of a spherical shell is found to approach that of a plate when R/a exceeds 10. For a thick clamped shell of aspect ratio 5 subjected to uniform normal traction on the outer surface, the in-plane and the transverse deformations contribute equally to the total strain energy for R/a greater than 5. However, for a cantilever shell of aspect ratio 5 subjected to equal and opposite uniform tangential tractions on the two major surfaces, the strain energy of in-plane deformations equals 95–98% of the total strain energy. Numerical results presented herein for several problems provide insights into different deformation modes, help designers decide when to consider effects of transverse deformations, and use the TSNDT for optimizing doubly curved shells.     

考虑剪切效应的环肋圆柱壳的稳定性

基于Reissner-Naghdi壳体理论和Timoshenko环梁理论,采用Ritz法对环肋圆柱壳的稳定性进行了研究。在简支边界条件下考虑了壳体和环肋的剪切变形效应对临界载荷的影响。由于环肋会约束壳体的变形,壳体的前屈曲状态应该是有弯曲变形的,环向压缩力也是非均匀分布的,分析了这一因素的影响。算例的计算结果与文献中给出的实验值吻合较好,并表明壳体的剪切效应在一些情形下会对环肋圆柱壳的稳定性产生一定的影响。     

点支撑预应力中厚矩形板的横向振动

基于Reissner-Mindlin一阶剪切变形板理论,讨论在预加面内机械荷载或温度场作用下,点支撑中厚矩形板的横向振动.温度场假定沿板表面为均布,沿板厚方向为线性分布的.利用考虑剪切变形影响的Timoshenko梁函数,采用Rayleigh-Ritz法给出不同边界条件下点支撑中厚板的自振频率.结果表明,温度升高与预加面内压力将使板的自振频率下降,支撑点位置的变化、边界约束条件和横向剪切变形效应都对板的自振频率有显著影响.     

Analysis of laminated conical shell structures using higher order models

In this paper is presented a numerical method for the structural analysis of laminated conical shell panels using a quadrilateral isoparametric finite element based on the higher order shear deformation theory. The displacement expressions used for the longitudinal and circumferential components of the displacement field are given by power series of the transversal coordinate and the condition of zero stresses in the top and bottom surfaces of the shell is imposed. The shape functions used for the transversal displacement are C¹ conforming and the finite element is a conical/cylindrical panel with 8 nodes and 40 degrees of freedom. The model presented performs static analysis with arbitrary boundary conditions and loads, as well eigenvalue problems (free vibration and buckling). Illustrative examples are presented and discussed.     

Static and dynamic analysis of laminated composite and sandwich plates and shells by using a new higher-order shear deformation theory

A new higher order shear deformation theory for elastic composite/sandwich plates and shells is developed. The new displacement field depends on a parameter “m”, whose value is determined so as to give results closest to the 3D elasticity bending solutions. The present theory accounts for an approximately parabolic distribution of the transverse shear strains through the shell thickness and tangential stress-free boundary conditions on the shell boundary surface. The governing equations and boundary conditions are derived by employing the principle of virtual work. These equations are solved using Navier-type, closed form solutions. Static and dynamic results are presented for cylindrical and spherical shells and plates for simply supported boundary conditions. Shells and plates are subjected to bi-sinusoidal, distributed and point loads. Results are provided for thick to thin as well as shallow and deep shells. The accuracy of the present code is verified by comparing it with various available results in the literature.     

Analysis for buckling and vibrations of composite stiffened shells and plates

This paper deals with development of triangular finite element for buckling and vibration analysis of laminated composite stiffened shells. For the laminated shell, an equivalent layer shell theory is employed. The first-order shear deformation theory including extension of the normal line is used. In order to take into account a non-homogeneous distribution of the transverse shear stresses a correction of transverse shear stiffness is employed. Based on the equivalent layer theory with six degrees of freedom (three displacements and three rotations), a finite element that ensures C⁰ continuity of the displacement and rotation fields across inter-element boundaries has been developed. Numerical examples are presented to show the accuracy and convergence characteristics of the element. Results of vibration and buckling analysis of stiffened plates and shells are discussed.     

Postbuckling of functionally graded fiber reinforced composite laminated cylindrical shells, Part I: Theory and solutions

Buckling and postbuckling behavior are presented for fiber reinforced composite (FRC) laminated cylindrical shells subjected to axial compression or a uniform external pressure in thermal environments. Two kinds of fiber reinforced composite laminated shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The governing equations are based on a higher order shear deformation shell theory with von Kármán-type of kinematic non-linearity and including the extension-twist, extension-flexural and flexural-twist couplings. The thermal effects are also included, and the material properties of FRC laminated cylindrical shells are estimated through a micromechanical model and are assumed to be temperature dependent. The non-linear prebuckling deformations and the initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths of FRC laminated cylindrical shells.     

Thermal effects on nonlinear vibrations of functionally graded doubly curved shells using higher order shear deformation theory

Geometrically nonlinear vibrations of functionally graded (FG) doubly curved shells subjected to thermal variations and harmonic excitation are investigated via multi-modal energy approach. Two different nonlinear higher-order shear deformation theories are considered and it is assumed that the shell is simply supported with movable edges. Using Lagrange equations of motion, the energy functional is reduced to a system of infinite nonlinear ordinary differential equations with quadratic and cubic nonlinearities which is truncated based on solution convergence. A pseudo-arclength continuation and collocation scheme is employed to obtain numerical solutions for shells subjected to static and harmonic loads. The effects of FGM power law index, thickness ratio and temperature variations on the frequency–amplitude nonlinear response are fully discussed and it is revealed that, for relatively thick and deep shells, the Amabili–Reddy theory which retains all the nonlinear terms in the in-plane displacements gives different and more accurate results.     

Computational approach of piezoelectric shells by the GDQ method

A piezoelectric laminated cylindrical shell with shear rotations effect under the electromechanical loads and four sides simply supported boundary condition was studied by using the two-dimensional generalized differential quadrature (GDQ) computational method. The typical hybrid composite shells with 3-layered cross-ply [90°/0°/90°] graphite–epoxy laminate and bounded PVDF layers are considered under the sinusoidal pressure loads and electric potentials on the shell. The governing partial differential equation with first-order shear deformation theory in terms of mid-surface displacements and shear rotations can be expressed in series equations by the GDQ formulation. Thus we obtain the GDQ numerical solutions of non-dimensional displacement and stresses at center position of laminated piezoelectric shells. Displacement is generally affected by the thickness of laminated piezoelectric shells under the action of mechanical load. Stresses are generally affected by the thickness and the length of laminated piezoelectric shells under the actions of mechanical load and electric potential.     

Thermo-Electro-Mechanical Buckling of Shear Deformable Hybrid Circular Functionally Graded Material Plates

Buckling analysis of perfect circular functionally graded plates with surface-bounded piezoelectric layers based on the first-order shear deformation theory is presented in this article. The material properties of the functionally graded (FG) layer are assumed to vary continuously through the plate thickness by distribution of power law of the volume fraction of the constituents. The plate is assumed to be under constant electrical field and two types of thermal loadings, namely, the uniform temperature rise and nonlinear temperature gradient through the thickness. Also, the stability of a plate under radial mechanical compressive force is examined. The equilibrium and stability equations are derived based on the first-order shear deformation plate theory using a variational approach. The boundary condition of the plate as an immovable type of the clamped edge is considered. Resulting equations are employed to obtain the closed-form solution for the critical buckling temperature for each loading case. The effects of electric field, piezo-to-host thickness ratio, and power law index of functionally graded plates subjected to thermo-mechanical-electrical loads are investigated. The results are compared with the classical plate theory and verified with the available data in the open literature.     

Modelling and optimization of laminated adaptive shells of revolution

In this paper two shell finite element models are presented for the structural analysis of composite laminated piezoelectric shells. One is an axisymmetric conical frustum with two nodal rings and the other is a conic shell panel with eight nodes. Both models are based in a mixed laminated theory that combines a higher order shear deformation theory for the mechanical displacement field with a layerwise representation with linear functions for the electric potential through each piezoelectric layer. In order to obtain the optimal design sensitivities analysis and optimization techniques based in the nonlinear mathematical programming are used. The design objectives can be the minimization of the deformed structure or the maximization of the natural fundamental frequency and the design variables are the electric potential difference applied to the actuators or the ply thicknesses among others.     

Thermal Postbuckling of Imperfect Shear-Deformable Laminated Plates on Two-Parameter Elastic Foundations

Thermal postbuckling analysis is presented for a simply supported, shear-deformable composite laminated plate subjected to uniform or nonuniform parabolic temperature loading and resting on a two-parameter (Pasternak-type) elastic foundation. The initial geometric imperfection of the plate is taken into account. Reddy's third-order shear-deformation plate theory with von Karman nonlinearity is used. The governing equations also include the plate-foundation interaction and thermal effects. The analysis uses a mixed Galerkin-perturbation technique to determine thermal buckling loads and postbuckling equilibrium paths. Numerical examples are presented that relate to the performances of perfect and imperfect, symmetric cross-ply laminated plates resting on Pasternak-type elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The influence played by a number of effects, among them foundation stiffness, transverse shear deformation, plate aspect ratio, fiber orientation, thermal load ratio, and initial geometric imperfections, is studied. Typical results are presented in dimensionless graphical form.     

Thermal buckling of various types of FGM sandwich plates

The sinusoidal shear deformation plate theory is used to study the thermal buckling of functionally graded material (FGM) sandwich plates. This theory includes the shear deformation and contains the higher- and first-order shear deformation theories and classical plate theory as special cases. Material properties and thermal expansion coefficient of the sandwich plate faces 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. The core layer is still homogeneous and made of an isotropic material. Several kinds of symmetric sandwich plates are presented. Stability equations of FGM sandwich plates include the thermal effects. The thermal loads are assumed to be uniform, linear and non-linear distribution through-the-thickness. Numerical examples cover the effects of the gradient index, plate aspect ratio, side-to-thickness ratio, loading type and sandwich plate type on the critical buckling for sandwich plates.     

Nonlinear vibrations of laminated circular cylindrical shells: Comparison of different shell theories

The geometrically nonlinear forced vibrations of laminated circular cylindrical shells are studied by using the Amabili–Reddy higher-order shear deformation theory. An energy approach based on Lagrange equations, retaining modal damping, is used in order to obtain the equations of motion. An harmonic point excitation is applied in radial direction and simply supported boundary conditions are assumed. The equations of motion are studied by using the pseudo-arclength continuation method and bifurcation analysis. A one-to-one internal resonance is always present for a complete circular cylindrical shell, giving rise to pitchfork bifurcations of the nonlinear response with appearance of a second branch with travelling wave response and quasi-periodic vibrations. The numerical results obtained by using the Amabili–Reddy shell theory are compared to those obtained by using an higher-order shear deformation theory retaining only nonlinear term of von Kármán type and the Novozhilov classical shell theory.     

Super convergent shear deformable finite elements for stability analysis of composite beams

The super convergent finite beam elements are newly presented for the spatially coupled stability analysis of composite beams. For this, the theoretical model applicable to the thin-walled laminated composite I-beams subjected to the axial force is developed. The present element includes the transverse shear and the warping induced shear deformation by using the first-order shear deformation beam theory. The stability equations and force–displacement relationships are derived from the principle of minimum total potential energy. The explicit expressions for the seven displacement parameters are then presented by applying the power series expansions of displacement components to simultaneous ordinary differential equations. Finally, the element stiffness matrix is determined using the force–displacement relationships. In order to demonstrate the accuracy and the superiority of the beam element developed by this study, the numerical solutions are presented and compared with the results obtained from other researchers, the isoparametric beam elements based on the Lagrangian interpolation polynomial, and the detailed three-dimensional analysis results using the shell elements of ABAQUS. The effects of shear deformation, boundary condition, fiber angle change, and modulus ratios on buckling loads are investigated in the analysis.