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.     

Optimization of plastic spherical shells of von Mises material

An optimization procedure is developed for spherical shells pierced with a central hole. The outer edge of the shell is simply supported whereas the inner edge is absolutely free. The material of the shell is assumed to be an ideal plastic material obeying the von Mises yield condition. Resorting to the lower bound theorem of limit analysis, shells with constant and piece-wise constant thickness are considered. The designs of spherical shells corresponding to maximal load carrying capacity are established for a given weight. Necessary optimality conditions are derived with the aid of variational methods from the theory of optimal control. The obtained set of equations is solved numerically.     

Optimization of plastic cylindrical shells with stepwise varying thickness in the case of von Mises material

The problems of the optimization of rigid-plastic cylindrical shells are studied under the condition that the shell wall thickness is piece-wise constant. It is assumed that the deflections of the shell are moderately large and the material obeys the von Mises yield condition and the associated deformation law. The optimization problem is posed as an optimal control problem and necessary optimality conditions are derived with the aid of the variational methods of optimality theory. The set of equations obtained is solved numerically. An example regarding the minimization of the central deflection of the shell with two steps in the thickness is presented.     

Optimization of inelastic conical shells with cracks

Inelastic conical shells loaded by the central rigid boss with vertical load are studied. The thickness of the conical part of the structure is piecewise constant. The connection between the shell and the boss is weakened with a stable crack. The designs with the maximal load-carrying capacity are established under a given material consumption of the shell. Material of the shell wall obeys the von Mises yield condition.     

Optimal design of plastic cylindrical shells of von Mises material

Optimal design of plastic circular cylindrical shells of von Mises material is studied. The optimization problem is stated as the maximization problem of the load carrying capacity for given weight of the shell. Shells with constant and piecewise-constant thickness are considered. The maximization problem is performed under the requirement that the material volume of the stepped shell is equal to the case of the reference shell of constant thickness. The material of the shell is assumed to be an ideal rigid plastic obeying von Mises yield criterion. The considered nonlinear problems are solved by using the CASes method.     

Optimization of shallow shells with different yield stresses in tension and compression

The minimum weight problem of thin rigid-plastic shallow spherical shells is studied. The thickness of the shell is piece-wise constant and the material has different yield stresses in tension and compression. The flow theory of plasticity is employed. Both solid and sandwich shells are considered. Necessary optimality conditions are derived with the aid of optimal control theory.     

Optimization of conical shells of Mises material

Conical shells made of a von Mises material are considered. The shells are subjected to unifromly distributed lateral loading and are simply supported at outer edges whereas inner edges are absolutely free. The shell wall is assumed to be of piece-wise constant thickness. Resorting to the lower bound theorem of limit analysis, optimal designs of shells are established under given weight (material volume of a shell) which corresponds to the maximum load carrying capacity. Received May 15, 2000     

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.     

Optimal design of shells against buckling subjected to combined loadings

In this paper, the problem of optimal design of shells against instability is considered. A thin-walled shell is loaded, in general, by overall bending moment, constant or varying along an axis of a shell, by the appropriate shearing force and by an axial force and a constant torsional moment. We look for the shape of middle surface as well as the thickness of a shell, which ensures the maximum critical value of the loading parameter. The volume of material and the capacity of a shell are considered as equality constraints. The concept of a shell of uniform stability is applied.     

Optimization of plastic spherical shells with a central hole

An optimization method for plastic spherical shells is presented. The shells under consideration are clamped at the outer edge and contain a central hole. The material of the shells obeys the generalized square yield condition and the associated flow rule. The problem of maximization of the load carrying capacity under the condition that the weight (material volume) of the shell is fixed is transformed into a problem of nonlinear programming. The latter is solved with the aid of Lagrangian multipliers. The solution obtained is compared with the optimal solution of the minimum weight problem for given load carrying capacity. Received September 9, 1999     

Optimization of inelastic cylindrical shells with internal supports

A non-linear programming method is developed for optimization of inelastic cylindrical shells with internal ring supports. The shells under consideration are subjected to internal pressure loading and axial tension. The material of shells is a composite which is considered as an anisotropic inelastic material obeying the yield condition suggested by Lance and Robinson. Taking geometrical non/linearity of the structure into account optimal locations of internal ring supports are determined so that the cost function attains its minimum value. A particular problem of minimization of the mean deflection of the shell with weakened singular cross sections is treated in a greater detail.     

Optimal design of axially symmetrical shells under hydrostatic pressure with respect to their stability

In this paper, the problem of optimal design of axially symmetrical shells against instability is considered. We look for the shape of middle surface as well as the thickness (constant or variable) of a shell, which ensure maximal value of the critical hydrostatic pressure. As the equality constraints the volume of material and the capacity of a shell are considered. The concept of a shell of uniform stability is applied. Received November 23, 1998     

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.     

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.     

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.     

Shape optimization of dynamically loaded viscoelastic cylindrical shells

Axisymmetric deflections of cylindrical shells of variable thickness are examined. The shell material is linear viscoelastic. The loading is of the impulsive type—it induces inside the shell a radial velocity field. The amount of kinetic energy is prescribed. The thickness function includes some design parameters, which must be calculated so that deflections of the beam are minimal. Only designs with a given volume are considered.For solving this optimization problem the space variable and the time will be separated. For evaluating the minimum of the objective function the Nelder-Mead technique has been used. Computations show that the viscosity effect is essential only for very short shells. Some numerical examples are presented.     

Reducing the shell thickness of double emulsions using microfluidics

Double emulsion drops are well-suited templates to produce capsules whose dimensions can be conveniently tuned by adjusting those of the drops. To closely control the release kinetics of encapsulants, the composition and thickness of the capsule shell must be precisely tuned; this is greatly facilitated if the shell is homogeneous in its composition and thickness. However, the densities of the two drops that form the double emulsion are often different, resulting in an offset of the two drop centers and therefore in an inhomogeneous shell thickness. This difficulty can be overcome if the shell is made very thin. Unfortunately, a controlled fabrication of double emulsions with thin shells is difficult. In this paper, we present a microfluidic squeezing device that removes up to 93 vol% of the oil from the shell of water–oil–water double emulsions. This is achieved by strongly deforming drops; this deformation increases their interfacial energy to sufficiently high values to make splitting of double emulsions into double emulsions with a much thinner shell and a single emulsion oil drop energetically favorable. Therefore, we can reduce the shell thickness of the double emulsion down to 330 nm. Because this method does not rely on solvent evaporation, any type of oil can be removed. Therefore, it constitutes a new way to produce double emulsions with very thin shells that can be converted into thin-shell capsules made of a broad range of materials.     

Influence of edge conditions on the modal characteristics of cross-ply laminated shells

The dynamic and static behavior of cross-ply laminated shells are investigated using the third-order shear deformation shell theory of Reddy. The theory is a modification of the Sanders shell theory and accounts for parabolic distribution of the transverse shear strains through the thickness of the shell and does not require shear correction coefficients. The Lévy-type exact solutions for bending, buckling and natural vibration are presented for doubly curved, cylindrical and spherical shells under various boundary conditions.     

Closed-form solutions for the free-vibration problem of multilayered piezoelectric shells

This paper addresses the free-vibration problem of multilayered shells with embedded piezoelectric materials. A series of hierarchic, two-dimensional axiomatic shell theories are presented within the “Unified Formulation” introduced by the last author. Shells of constant curvature are considered, and no simplifying assumptions on the curvature terms are made in the geometric relations. Closed-form solutions are given for the free-vibration problem of simply supported, orthotropic piezoelectric laminates. The formulations are applied to study the influence of the electro-mechanical coupling on the resonant frequencies. It is demonstrated that the slenderness of plates with through-thickness polarized piezoelectric layers increases the electro-mechanical coupling. For comparison purposes, the fundamental axisymmetric mode of hollow cylinders has been exemplarily considered: with respect to flat plates, the thickness and the curvature of the shells have a less important effect on the piezoelectric coupling.     

A scaled thickness conditioning for solid- and solid-shell discretizations of thin-walled structures

The numerical discretization of thin shell structures yields ill-conditioned stiffness matrices due to an inherent large eigenvalue spectrum. Finite element parametrization that depends on shell thickness, like relative displacement shells, solid shells and other solid finite elements even add to the ill-conditioning by introducing high eigenmodes.To overcome this numerical issue we present a scaled thickness conditioning (STC) approach, a mechanically motivated preconditioner for thin-walled structures discretized with continuum based element formulations. The proposed approach is motivated by the scaled director conditioning (SDC) method for relative displacement shell elements. In contrast to SDC, the novel STC approach yields a preconditioner for the effective linear system. It is applicable independently of element technology employed, coupling to other physical fields, boundary conditions applied and additional algebraic constraints and can be easily extended to multilayer shell formulations.The effect of the proposed preconditioner on the conditioning of the effective stiffness matrix and its eigenvalue spectrum is studied. It is shown that the condition number of the modified system becomes almost independent from the aspect ratio of the employed elements. The improved conditioning has a positive influence on the convergence behavior of iterative linear solvers. In particular, in combination with algebraic multigrid preconditioners the number of iterations could be decreased by more than 85% for some examples and the computation time could be reduced by about 60%.