Jaan Lellep

Optimization of clamped plastic shallow shells subjected to initial impulsive loading

An optimization technique is suggested for shallow spherical shells made of a ductile material and subjected to initial impact loading. The shell under consideration is pierced with a central hole and clamped at the outer edge. The optimal design of the shell of piece-wise constant thickness is established under the condition that the maximal residual deflection attains its minimal value for given total weight. The material of the shell is assumed to obey the Tresca yield condition and associated flow law. By the use of the method of mode form motions the problem is transformed into a particular problem of non-linear programming and solved numerically.

Buckling of Beams and Columns with Defects

Beams and columns subjected to the axial pressure are studied. Critical buckling loads are established for stepped beams clamped at one end and elastically fixed at the other end. The beams under consideration are of piecewise constant thickness and are weakened by cracks emanating from re-entrant corners of steps. The cracks are assumed to be stable part-through surface cracks. The influence of a crack on the stability of the beam is modeled by the method of distributed line spring known in the elastic fracture mechanics. Numerical results are presented for beams with a single step making use of different stress correction functions.

OPTIMIZATION OF INELASTIC CYLINDRICAL SHELLS

An optimization procedure is developed for circular cylindrical shells accounting for geometrical non-linearity. The material of the shells is assumed to be rigid-plastic obeying a piece-wise linear or a smooth yield condition and the associated deformation law. The shells under consideration are subjected to a quasi-static or dynamic loading. Necessary conditions for optimality are derived with the aid of variational methods of the theory of optimal control. Minimum weight designs are established for geometrically linear Tresca shells and non-linear Mises shells, respectively, in the case of static loading. Cylindrical tubes of piece wise constant wall thickness are studied in the case of impulsive loading.

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

Abstract The minimum weight problem is studied under the condition that the considered shell has a piecewise constant thickness. The shell with free internal edge and clamped outer edge is subjected to uniformly distributed internal pressure. Moderately large deflections are taken into account and a deformation-type theory of plasticity is employed. The optimization problem includes the additional restriction, which demands that the maximal deflections of the shell of piecewise constant thickness and of the reference shell, of constant thickness, coincide. Employing the variational methods of the optimal control theory, necessary optimality conditions are established. The results obtained are used to establish the optimal parameters for the shell of piecewise constant thickness.

Axisymmetric vibrations of orthotropic circular cylindrical shells with cracks. Part I

Vibrations of circular cylindrical shells made of unidirectionally reinforced composite materials are considered. The shells have stepped cross sections and are weakened with cracks emanating from the reentrant corners of steps. The influence of circular cracks with a constant depth on the vibration of the shell is described with the aid of a matrix of local flexibility coupled with the stress intensity factor known in the linear elastic fracture mechanics. Numerical results are presented for shells with one and two steps.

Natural Vibrations of Stepped Arches

Free vibrations of elastic arches simply supported at both ends are studied. The arches under consideration have piece wise constant thickness and constant width. Keywords—vibration, arch, crack, elasticity

Optimization of Stepped Elastic Plastic Plates

A method of analysis and optimization of stepped plates made of elastic plastic materials is developed. The stress-strain of the plate is defined for the initial elastic and subsequent elastic plastic stages of deformation. Necessary optimality conditions are derived with the aid of variational methods of the theory of optimal control. This results in a differential-algebraic system of equations. The latter is solved numerically. The effectivity of the design established is assessed in the cases of one-and multi-stepped plates assuming the material obeys the Tsai-Wu or von Mises yield condition.

Optimization of Inelastic Cylindrical Shells with Stiffeners

A method of optimal design of inelastic cylindrical shells with absolutely rigid hoop stiffeners is suggested. The shells under consideration are subjected to internal pressure loading and axial dead load. Taking geometrical non-linearity of the structure into account optimal locations of hoop stiffeners are determined so that the cost function attains its minimum value. A particular problem of minimization of the mean deflection of the shell with cracks at the cross sections where stiffeners are located is treated in a greater detail.

Elastic Plastic Analysis of Elliptical Plates

The elastic response of elliptical plates to distributed transverse loading is considered. Resorting to the Haar’s wavelets a numerical algorithm for determination of stress and strain components is developed. The material of the plates is assumed to be an ideal elastic plastic material satisfying corresponding yield condition and associated flow law. Particular problems are solved in the cases of stepped plates and Hill’s and Tsai-Wu materials. Numerical results are presented for one- and two-stepped plates.

Optimization of Stepped Plates in the Elastic Plastic Range

An optimization technique is developed for circular plates of piecewise constant thickness. The plates under consideration have been manufactured of an ideal elastic plastic material obeying Tresca’s yield criterion. Necessary optimality conditions are derived with the aid of the theory of optimal control. Obtained system of differential-algebraic equations is solved numerically in the case of the plate with single step of the thickness. Effectiveness of the design is assessed numerically.