Nonlinear flexural oscillations of orthotropic shallow spherical shells

The effect of curvature and polar orthotropy on nonlinear dynamic behaviour of a shallow spherical shell is investigated in the present paper. Numerical solutions based on an assumed two-term modeshape for the axisymmetric, forced (uniform pressure) and free vibrations are obtained for different shell geometries and orthotropic material constants. The results, when specialised for the case of isotropic material, are in good agreement with those available in the literature. Based on a one-term modeshape solution, the values of the geometric parameter at which the transition from hardening to softening type of nonlinearity takes place and where the reversal of the softening trend occurs are obtained for different values of the orthotropic constants.     

Axisymmetric static and dynamic buckling of orthotropic shallow spherical cap with circular hole

Nonlinear axisymmetric static and dynamic buckling of clamped isotropic and cylindrically orthotropic elastic cap with central circular hole have been investigated. The governing equations are expressed in terms of normal displacement ω and stress function ψ. The orthogonal point collocation method has been used in the space domain and Newmark-β scheme in the time domain. The cases of shallow cap with free hole and with a hole plugged by rigid central mass are considered. Analysis has been carried out for uniformly distributed load and a ring load at the hole. Dynamic load is taken as a step function load. Detailed new results for static and dynamic buckling loads have been presented for the isotropic and orthotropic cases.     

Axisymmetric vibration of layered orthotropic spherical shells of variable thickness

Axisymmetric free vibration of thick orthotropic spherical shells with linearly varying thickness along the meridian is analysed. Both deep and shallow shells are considered for the analysis. The effect of thickness variation and lay-up are considered. The results are presented for clamped and hinged boundary conditions. A thick shell finite element is used for the analysis. It is observed that the thickness variation and lay-up have a pronounced effect on the natural frequencies and a considerable increase of the natural frequencies can be achieved by selecting a proper combination of lay-up and thickness variation.     

Axisymmetric static and dynamic buckling of orthotropic truncated shallow conical caps

This investigation deals with the axisymmetric static and dynamic buckling of a cylindricaliy orthotropic truncated shallow conical cap with clamped edge. The cases of conical caps with a free central circular hole and with a hole plugged by a rigid central mass have been considered. The governing equations are formulated in terms of normal displacement w and stress function Ψ. The orthogonal point collection method is used for spatial discretisation and the Newmark-β scheme is used for time-marching. Analysis has been carried out for a uniformly distributed conservative load normal to the undeformed surface and a central axial ring load at the hole. Dynamic load is taken as a step function load. The influence of orthotropic parameter β and annular ratio on the buckling loads has been investigated. New results for static and dynamic buckling loads have been presented for the isotropic and orthotropic truncated conical caps. Dynamic buckling loads obtained from static analysis have been found to agree well with the dynamic buckling loads based on transient response.     

Optimization of rigid-plastic shallow spherical shells of piecewise constant thickness

An approximate method developed earlier for the investigation of large plastic deflections of circular and annular plates is accommodated for shallow spherical shells. The material of the shells is assumed to obey Tresca's yield condition and the associated deformation law. The minimum weight problem concerning shells operating in the post-yield range is posed under the conditions that (i) the thickness of the structure is piece-wise constant and (ii) the maximal deflections of the optimized shell and a reference shell of constant thickness, respectively, coincide. Necessary optimality conditions are derived with the aid of the variational methods of the optimal control theory. The set of equations obtained is solved numerically.     

具有损伤扁球面网壳非线性稳定性

基于Lematire等效应变损伤原理,计及扁球面网壳各个杆件的损伤影响,根据薄壳非线性动力学理论推导出含有损伤扁球面网壳非线性动力学方程和协凋方程,在固定夹紧边界条件下,用Galerkin方法得到一个含二次和三次非线性振动微分方程,并对具有损伤扁球面网壳的非线性自由振动方程求解．用Floquet指数法研究系统分叉问题给出了平衡点的状态．并通过数字仿真绘出了不同损伤状态下系统的分叉图和平衡点的相对位置图,发现损伤对系统的平衡点的状态影响较大．     

Dynamic buckling of axisymmetric spherical caps with initial imperfections

Dynamic buckling loads are obtained for axisymmetric spherical caps with initial imperfections. Two types of loading are considered, namely, step loading with infinite duration and right triangular pulse. Solutions of perfect spherical caps under step loading are in excellent agreement with previous findings. Results show that initial imperfections do indeed have the effect of reducing the buckling capacity for both dynamic and static responses, although they are affected in a different manner. From the solutions obtained for triangular pulse situations, it is revealed that pulse duration has a very significant impact on the magnitude of the dynamic buckling load. When comparing these solutions with those of step loading, it is concluded that the step loading with infinite duration is the limiting case of a triangular pulse, and that the step loading provides the most severe loading situation for dynamic analysis.     

Bifurcation and snap-through phenomena in asymmetric dynamic analysis of shallow spherical shells

The asymmetric dynamic behavior of clamped shallow spherical shells under a uniform step pressure of infinite duration is investigated. The solution of a linear eigenvalue problem yields the bifurcation paths and also the lower bound for the asymmetric dynamic snap-through buckling pressure. The asymmetric dynamic response of shells with a shape imperfection is studied. The asymmetric dynamic snap-through buckling load is defined to be the threshold value of the step pressure at which the asymmetric response shows significant growth rate. The snap-through buckling loads are obtained for a few shell parameters. The numerical results are compared with the available experimental results and they are in good agreement. Finally, a preliminary study of the phase planes is presented.     

Analysis of shallow spherical and paraboloidal cantilever shells under symmetrically distributed loads

A mathematical stress and deformation analysis of thin shallow spherical and paraboloidal cantilever shells is presented. This paper describes a method of solution of Reissner's differential equations for the generally distributed loads, and then shows the determination of the particular solutions corresponding to rotationally symmetrical distributed loads. Computer results generated in the form of nondimensional curves for cantilever shells supported continuously around the apex by a rigid ring provide stresses and displacements in the most general form, allowing the engineer to perform an automatic stress analysis and to obtain an optimized design in a minimum length of time.     

A novel snap-through buckling behaviour of axisymmetric shallow shells with possible application in transducer design

In this study, the snap-through buckling behaviour of axisymmetric shells, subjected to axisymmetric horizontal peripheral load or displacement for various shell parameters and various boundary conditions, is investigated. Results obtained seem not to have been reported previously. An application of peripheral displacement type of loading is seen in metal-ceramic composite transducers developed by sandwiching a piezoelectric (PZT) ceramic between two metal end caps which serve as mechanical transformers for converting and amplifying the lateral displacement of the ceramic into an axial motion normal to the metal cap. In our numerical search, we have observed that snap-through and snap-back buckling is possible for shallow spherical caps for a very narrow range of the shell parameter used. When a hole is opened around the apex of the cap, buckling is possible for a larger range of the shell parameter. Obtaining the displacement amplification and the blocking or generative force for various material and geometric properties is necessary for the possible application of the findings in transducer design. The numerical results are presented in graphical forms.     

Multi-fidelity optimization of laminated conical shells for buckling

Optimum laminate configuration for minimum weight of filament-wound laminated conical shells is investigated subject to a buckling load constraint. In the case of a composite laminated conical shell, due to the manufacturing process, the thickness and the ply orientation are functions of the shell coordinates, which ultimately results in coordinate dependence of the stiffness matrices (A,B,D). These effects influence both the buckling load and the weight of the structure and complicate the optimization problem considerably. High computational cost is involved in calculating the buckling load by means of a high-fidelity analysis, e.g. using the computer code STAGS-A. In order to simplify the optimization procedure, a low-fidelity model based on the assumption of constant material properties throughout the shell is adopted, and buckling loads are calculated by means of a low-fidelity analysis, e.g. using the computer code BOCS. This work proposes combining the high-fidelity analysis model (based on exact material properties) with the low-fidelity model (based on nominal material properties) by using correction response surfaces, which approximate the discrepancy between buckling loads determined from different fidelity analyses. The results indicate that the proposed multi-fidelity approaches using correction response surfaces can be used to improve the computational efficiency of structural optimization problems.     

Exact internal controllability for shallow shells

The internal control problem is considered, based on the linear displacement equations of shallow shell. It is shown, with some checkable geometric conditions on control region, that the undergoing shallow shell is exactly controllable by using Hilbert uniqueness method (HUM), piecewise multiplier method and Riemannian geometry method. Then some examples are given to show the assumed geometric conditions.     

Optimal orientation of orthotropic materials for plates designed against buckling

The paper deals with the optimization of anisotropic plates loaded by in-plane forces and designed against buckling. The internal structure of the plate is seen as a twodimensional locally orthotropic solid and the orientations of the axes of orthotropy are taken as the design variables. The problem of optimization consists in determining the best orientation of the axes of orthotropy from the condition of the maximum behaviour of the critical buckling loads. General properties of the problem, such as multiplicity of the critical parameters and splitting of multiple eigenvalues, are studied. Optimization algorithms are developed to improve the anisotropic properties of the plate. The results of numerically finding the optimal orientation of orthotropic properties are compared with conventional layouts for square and rectangular plates loaded by normal in-plane forces.     

Vibration and buckling analysis of axisymmetric polar orthotropic circular plates

The vibration and stability analysis of polar orthotropic circular plates using the finite element method is discussed. In order to formulate the eigenvalue problems associated with the vibration and stability analyses, the clement stiffness, mass, and stability coefficient matrices are presented. By assuming the static displacement function, which is an exact solution of the polar orthotropic circular plate equation, approximates the vibration and buckling modes, the mass and stability coefficient matrices are readily derived from the given displacement function. Results showing the effects of orthotropy on natural frequencies and buckling loads are compared with their isotropic counterpart.     

Optimization of plastic shallow shells exhibiting nonstable behaviour

An optimal design technique is suggested for axisymmetric shallow shells exhibiting nonstable behaviour in the post-yield point range. The material of the shells is ideally rigidplastic obeying the von Mises yield condition, which is satisfied in the average and the associated deformation law. Spherical shells pierced with a hole and subjected to uniformly distributed transverse pressure are studied. Different cases of the support conditions are considered. Making use of the methods of the optimal control theory the problem is transformed into a boundary value problem which is solved numerically.     

Observability inequalities for the transmission of shallow shells

In this paper, we establish some observability inequalities from both boundary and distributed controls for the transmission of shallow shell with a middle surface of any shape. In particular, if only the boundary controls are allowed, we obtain the condition for the observability inequality on Young's modulus and Poisson's coefficient of the material which is similar as those in the case of wave and plate equations.     

Fiber composite thin shells subjected to thermal buckling loads

The results of parametric studies to assess the effects of various parameters on the buckling behavior of angle-ply, laminated thin shells in a hot environment are presented in this paper. These results were obtained by using a three-dimensional finite element analysis. An angle-ply, laminated thin shell with fiber orientation of [θ/ −θ]₂ was subjected to compressive mechanical loads. The laminated thin shell has a cylindrical geometry. The laminate contained T300 graphite fibers embedded in an intermediate-modulus, high-strength (IMHS) matrix. The fiber volume fraction was 55% and the moisture content was 2%. The residual stresses induced into the laminated structure during the curing were taken into account. Parametric studies were performed to examine the effect on the critical buckling load of the following parameters: cylinder length and thickness, internal hydrostatic pressure, different ply thicknesses, different temperature profiles through the thickness of the structure, and different layup configurations and fiber volume fractions. In conjunction with these parameters the ply orientation varied from 0° to 90°. Seven ply angles were examined: 0°, 15°, 30°, 45°, 60°, 75°, and 90°. The results show that the ply angle θ and the laminate thickness had significant effects on the critical buckling load. The fiber volume fraction and the internal hydrostatic pressure had important effects on the critical buckling load. The cylinder length had a moderate influence on the buckling load. The thin shell with [θ/−θ]₂ or [θ/−θ]^s angle-ply laminate had better buckling-load performance than the thin shell with [θ]₄ off-axis laminate. The temperature profiles through the laminate thickness and various laminates with the same thickness but with the different ply thickness had insignificant effects on the buckling behavior of the thin shells.     

Nonlinear dynamic buckling of spherical caps with initial imperfections

A finite difference method is developed for the large deformation elastic-plastic dynamic buckling analysis of axisymmetric spherical caps with initial imperfections. The problem formulation is based on governing differential equations of motion, treating the plastic deformation as an effective plastic load. Both perfectly plastic and strain hardening behavior are implemented in the program. Strain hardening is incorporated through use of the Prager-Ziegler kinematic hardening rule, so that the Bauschinger effect is accounted for. The solution for the large deformation elastic-plastic dynamic response of a spherical cap is compared very favorably with other findings. Two spherical cap models are selected to study the title problem. Results obtained indicate that both plastic yielding and initial imperfection play significant roles in reducing the load carrying capacity of these shell structures. Both increase their influence as the thickness to radius ratio and the imperfection magnitude increase, respectively. It is also found that dynamic effect has the influence of lowering load carrying capacity of perfect spherical caps; however, its influence on imperfect spherical caps depends on the magnitude of initial imperfections.     

Nonlinear vibration and stability of a shallow unsymmetrical orthotropic sandwich shell of double curvature with orthotropic core

In this paper, nonlinear behaviours for a shallow unsymmetrical, orthotropic sandwich shell of double curvature with orthotropic core having different elastic characteristics have been studied by a new set of uncoupled differential equations. The face sheet may be of unequal thickness of different materials. However, a restriction that the elements radii of curvature be large compared to the overall thickness of the sandwich has been imposed.

A simple approach used in the present analysis can be applied for stability as well as vibration. For the symmetrical case, where the face sheets are of equal thickness and of same materials, these equations can be shown to reduce to those given by Grigolyuk in 1957. Numerical results of a square rectangular simply supported curved plate, and of a rectangular sandwich cylindrical shell under mechanical and dynamic loading, have been computed and compared with other known results.