Optimization of plastic conical shells of piece-wise constant thickness

Thin-walled conical shells subjected to uniformly distributed external pressure loading are considered assuming that the thickness of the shells is piece-wise constant. Minimum weight designs are established under the condition that the load carrying capacity of the optimized shell coincides with that of the reference shell of constant thickness. The material of the shell is assumed to obey Tresca's yield condition and the associated flow law. The exact yield surface in the space of generalized stresses corresponding to the Tresca condition is approximated with the squares on the planes of membrane forces and moments, respectively.     

Frequency response of shell-plate combinations

Natural frequencies of cylindrical shells with a circular plate attached at arbitrary locations are determined for various boundary conditions and L/D ratios. The semi-analytical finite element method is used for the analysis. A conical shell element with four degrees of freedom per node and two nodes per element is used. For clamped-clamped and simply-supported boundary conditions the plate is attached at the center of the shell. For a clamped-free boundary condition the plate is at the free end of the shell. The effects of plate thickness and L/D ratio of the shell on the frequencies of the shell-plate combination are investigated.     

功能梯度薄壁圆柱壳的自由振动

研究了由功能梯度材料制成的薄壁圆柱壳的自由振动.采用幂律分布规律描述功能梯度材料沿厚度的梯度性质,根据Donnell壳体理论,导出了功能梯度材料薄壁圆柱壳线性振动的简化控制方程.基于此理论分析了功能梯度圆柱壳的自由振动特性,给出了两端简支功能梯度材料薄壁圆柱壳小挠度固有振动的频率公式.以简支圆柱壳作为算例,与前人结果及有限元法对比验证了该简化功能梯度薄壁圆柱壳理论的正确性,同时讨论了周向波数及梯度指数对其频率的影响.     

Large deflection analysis of plates and shells by discrete energy method

The discrete energy method—a special form of finite difference energy approach—is presented as a suitable alternative to the finite element method for the large deflection elastic analysis of plates and shallow shells of constant thickness. Strain displacement relations are derived for the calculation of various linear and nonlinear element stiffness matrices for two types of elements into which the structure is discretized for considering separately energy due to extension and bending and energy due to shear and twisting. Large deflection analyses of plates with various edge and loading conditions and of a shallow cylindrical shell are carried out using the proposed method and the results compared with finite element solutions. The computational efforts required are also indicated.     

Postbuckling behaviour of plates and shells using a mindlin shallow shell formulation

The paper presents a finite element Mindlin shallow shell formulation. Compared to a previous flat plate formulation it is shown that the addition of a shallow shell capability adds very little extra computational effort. Results are given for the postbuckling behaviour of square and circular plates subject to direct inplane loading and a square plate subject to inplane shear loading. Examples are also presented of the analyses of a shallow truss and cylindrical and spherical shells, all exhibiting snap through behaviour. Agreement with existing solutions is generally good and where possible the results are presented numerically.     

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.     

Finite element analysis of bimodulus composite stiffened thin shells of revolution

This paper is a sequel to the work published by the first and third authors[l] on stiffened laminated shells of revolution made of unimodular materials (materials having identical properties in tension and compression). A finite element analysis of laminated bimodulus composite thin shells of revolution, reinforced by laminated bimodulus composite stiffeners is reported herein. A 48 dot doubly curved quadrilateral laminated anisotropic shell of revolution finite element and it's two compatible 16 dof stiffener finite elements namely: (i) a laminated anisotropic parallel circle stiffener element (PCSE) and (ii) a laminated anisotropic meridional stiffener element (MSE) have been used iteratively.The constitutive relationship of each layer is assumed to depend on whether the fiberdirection strain is tensile or compressive. The true state of strain or stress is realized when the locations of the neutral surfaces in the shell and the stiffeners remain unaltered (to a specified accuracy) between two successive iterations. The solutions for static loading of a stiffened plate, a stiffened cylindrical shell. and a stiffened spherical shell, all made of bimodulus composite materials, have been presented.     

A three-dimensional state space finite element solution for laminated composite cylindrical shells

On the basis of the theory of three-dimensional elasticity, this paper presents a state space finite element solution for stress analysis of cross-ply laminated composite shells. This is a continuation of the authors’ previously published work on laminated plates [Compos. Struct. 57 (1–4) (2002) 117; Comput. Methods Appl. Mech. Engrg. 191 (37–38) (2002) 4259]. Once again a state space formulation is introduced to solve for through-thickness stress distributions, while the traditional finite elements are used to approximate the in-surface variations of state variables. A three-dimensional laminated shell element is established in an arbitrary orthogonal curvilinear coordinate system, while the application of the element is shown by calculating stresses in laminated cylindrical shells. Compared with the traditional finite element method, the new solution provides accurate continuous through-thickness distributions of both displacements and transverse stresses.     

Optimization of the crushing characteristics of triangulated aluminum beverage cans

In this study, a cylindrical shell body of the aluminum cans is triangulated and optimized for being folded down easily and safely for recycling. The triangulated cylindrical shell is constructed by a family of triangular surfaces based on one set of helical strips and circles lying on a cylindrical side surface. The intersections of helical strips and circles are used as the vertexes of the triangular surfaces. By changing the helical angle of the strips, the number of the strips, and the number of the circles, various triangulated cylindrical shells with different crushing characteristics can be developed. On the basis of the numerical simulation, the minimization problem of the crushing force of the triangulated cylindrical shells is solved using the crashworthiness maximization technique for tubular structures that combines the techniques of design of experiment, response surface approximation as well as usual mathematical programming.     

A finite element model for the analysis of 3D axisymmetric laminated shells with piezoelectric sensors and actuators: Bending and free vibrations

This paper addresses the bending and free vibrations of multilayered cylindrical shells with piezoelectric properties using a semi-analytical axisymmetric shell finite element model with piezoelectric layers using the 3D linear elasticity theory. In the present 3D axisymmetric model, the equations of motion are expressed by expanding the displacement field using Fourier series in the circumferential direction. Thus, the 3D elasticity equations of motion are reduced to 2D equations involving circumferential harmonics. In the finite element formulation the dependent variables, electric potential and loading are expanded in truncated Fourier series. Special emphasis is given to the coupling between symmetric and anti-symmetric terms for laminated materials with piezoelectric rings. Numerical results obtained with the present model are found to be in good agreement with other finite element solutions.     

Optimal design of underwater stiffened shells

The optimal design parameters of stiffened shells are determined using a rational multicriteria optimization approach. The adopted approach aims at simultaneously minimizing the shell vibration, associated sound radiation, weight of the stiffening rings as well as the cost of the stiffened shell. A finite element model is developed to determine the vibration and noise radiation from cylindrical shells into the surrounding fluid domain. The production cost as well as the life cycle and maintenance costs of the stiffened shells are computed using the Parametric Review of Information for Costing and Evaluation (PRICE) model. A Pareto/min-max multicriteria optimization approach is then utilized to select the optimal dimensions and spacing of the stiffeners. Numerical examples are presented to compare the vibration and noise radiation characteristics of optimally designed stiffened shells with the corresponding characteristics of plain un-stiffened shells. The obtained results emphasis the importance of the adopted multicriteria optimization approach in the design of quiet, low weight and low cost underwater shells which are suitable for various critical applications. Received September 14, 2000 Communicated by J. Sobieski     

Analysis of stresses in internally loaded cylindrical shells

Simply supported cylindrical shells under internal fluid and granular loading are investigated. A Fourier series solution is presented and results are shown for various shell geometries. The accuracy of the results is discussed and a comparison is made with other approximate methods available in the materials handling industry.     

Comparison of shell theories in the analysis of cylindrical shells with discontinuity in the thickness

Cylindrical shells with discontinuity in the thickness and that are subjected to axisymmetric loading have been analysed. Two types of finite elements are used: the first is based on thin shell theory and the second on thick shell theory. The loadings considered are a uniform internal pressure and a circular ring load at the mid-section. The effect of these loads for various end conditions and various step-ratios in the thickness have been analysed. Numerical results are presented and compared for both the theories. It has been shown that the transverse normal stress acting along the thickness direction is not negligible compared to other stresses at places of discontinuity either in the thickness or in the loading. The weight of the shell is kept constant for various step-ratios.     

Instability of short stiffened composite cylindrical shells under bending with prebuckling displacements

This paper presents results for buckling of a stiffened cylindrical shell with cutouts and both isotropic and composite shells without cutouts acting under end bending moments. The STAGS-C program has been used in the analysis.     

A finite element analysis of an axisymmetrically loaded orthotropic shell of revolution

The objective of this study was to develop a finite element matrix method of analysis for symmetrically loaded orthotropic shells of revolution using closed form elasticity solutions for the element. A computer program for structural analysis was developed based on this method.

The program was used to analyze orthotropic cylindrical shells with edge loads, orthotropic spherical shells with edge loads, and pressurized ellipsoidal shells.

For the ellipsoidal shells, the ratio of the major to minor axis (a/b) varied from 0.2 to 1.8. The orthotropic materials used had ratios of Young's modulus in the meridional direction to Young's modulus in a direction tangent to a parallel circle (E₁/E₂) that ranged from 0.2 to 1.8.

For the structures and orthotropic materials studied, it was found that the edge effect, as signified by the meridional moment, was affected by the Young's moduli ratio E₁/E₂, the radius of curvature R₂ in the plane containing a normal to the shell surface and a tangent to a parallel circle, and Poisson's ratio v₂, the latter being more prominent for large E₁/E₂ values. The range of the E₁/E₂ ratio caused the meridional edge moment to double, increasing as the E₁/E₂ ratios increased from 0.2 to 1.8, for pressurized ellipsoidal shells. The meridional edge moment more than doubled as the ellipsoidal axes ratio, a/b, ranged from 0.2 to 1.8.     

Buckling of cylindrical shells under the action of nonuniform external pressure

Over the last few decades, storage tanks have become bigger and thinner. Because of this, the buckling capacity of these cylindrical shells may well be the determining factor of shell thickness. In this paper, the critical buckling load of isotropic and orthotropic cylinders subjected to different types of wind load distributions is investigated. The prebuckling displacements are obtained by using the membrane theory of shell analysis. The principle of minimum potential energy in conjunction with Ritz's approach is used to obtain the stability matrix. The size of the stability matrix in this analysis is (81 × 81). By solving the stability matrix as an eigenvalue problem, the critical pressures are obtained as eigenvalues and the deflection shapes as eigenvectors. In the present study cylindrical shells of various dimensions, which are fixed at the base and free at the top, are investigated. The buckling load curves for isotropic and orthotropic cylinders of various dimensions are given for practical use.     

3D characterization of hot metallic shells during industrial forging

During industrial forging of hot metallic shells, it is necessary to regularly measure the dimensions of the parts, especially the inner and outer diameters and the thickness of the walls. A forging sequence lasts 2 h or more during which the diameter of the shell is regularly measured in order to decide when to stop the forging process. For better working conditions, for the safety of the blacksmiths, and for a faster and more accurate measurement, we have developed a novel system based on two commercially available time of flight laser scanners for the measurement of the diameters of hot cylindrical metallic shells during the forging process. The advantages of using laser scanners are that they can be placed very far from the hot shell, more than 15 m, while at the same time giving an accurate point cloud from which three-dimensional views of the shell can be reconstructed and diameter measurements done. Moreover, more accurate measurement is achieved in less time with the laser system than with the conventional method using a large ruler. The system has been successfully used to measure the diameters of hot cylindrical metallic shells.     

Optimal design of a cylindrical shell loaded by internal pressure

The problem of the optimal design of a noncircular cylindrical shell loaded by uniform internal pressure is presented. As an optimization criterion the minimal ratio of structure mass to medium mass is taken into account. Solution constraints are geometrical and strength conditions. Numerical analysis is realized using the finite element method. Optimal shapes of steel and aluminium shells are determined.     

Application of penalty functions to a curved isoparametric axisymmetric thick shell element

A curved axisymmetric shell element with three nodes is developed. Quadratic interpolation is used and as the transverse shearing strain is included only first derivatives are required in the calculation of the strains. The element is found to yield accurate solutions for thick circular plates but a penalty factor must be used when the ratio of plate radius to thickness is of the order of 100. With appropriate values of the penalty factor, though, thin plate behaviour is reproduced with reasonable accuracy. Further, it is shown that for all practical purposes the penalty factor need only be based on the plate thickness. This is a useful conclusion in relation to shell analysis where different penalty factor values would otherwise need to be evaluated on the basis of the radius to thickness ratio. Finally the element is shown to give good results for cylindrical and spherical shells.     

Stability of cylindrical shell panels subjected to follower forces based on a mixed finite element formulation

A mixed finite element formulation is developed from a weak variational priniciple. This formulation is applied to stability analysis of cylindrical shell structures subjected to follower loading. Bilinear trial functions are used for all field variables. The rectangular curved elements presented here satisfy the continuity requirements for the field variables at the element interface. Two examples of a cantilevered cylindrical shell panel under different kinds of loading are solved.