First, an analytical asymptotic method to construct quasi-periodic solutions in autonomous dynamical systems governed by a nonlinear second order set of ordinary differential equations with delay is presented. The approach is based on the double asymptotic expansion of two independent perturbation parameters and is supported by symbolic computation using `Mathematica' package. Both resonance and non-resonance cases are successfully analyzed and the catastrophes of the torus solutions are classified and discussed. Second, a new method for numerical calculations of the quasi-periodic orbits, which is based on a concept of the general Poincare map, is addressed. In both cases considered examples support the introduced theory.

The paper presents the optimization of thin-walled structures such as vertical cylindrical reservoirs subject to pitting corrosion. The function of the structure utility is taken as an optimization criterion. The choice of an optimal thickness distribution of the reservoir shell along its height is determined from the conditions of its uniform reliability.

A computer method for elastic-plastic continuous beams with rotation constraints is proposed. Such structures belong to a particular class of slackened systems, i.e. systems with gaps at structural joints. The mathematical model of slackened structures represents a discrete form of the Kuhn-Tucker's conditions, and is equivalent to dual Quadratic Programming Problems (QPPs). The uniqueness of a solution to the problem of the beams under considerations is assured, excluding cases where the structure converts into a mechanism, and the solution corresponds to dual Linear Programming Problems (LPPs). In order to calculate both the structure and mechanism a concept of the finite element with elastic rotation supports is used in the computer program. The uniqueness of solutions makes it possible to use a pure elastic analysis with some additional constraints superimposed on state variables. It allows us to avoid the time-consuming mathematical programming methods. Several examples illustrate the behaviour of beams under multiparameter loads. Results relate the cyclic loading (shakedown) as well as elastic, sublimit and limit surfaces. The work presents the characteristic features of slackened structural systems.

An original idea of the Stochastic Finite Element Method (SFEM) application in numerical modelling of random fluctuations of elastic properties of fibre composites components has been presented in this paper. The displacement and the stress random fields have been analysed for various contents of a fibre periodicity cell of such a composite, and for different coefficients of variation of the Young modulus of both phases

Today the solution of mechanical problems in engineering practice is often routinely carried out by means of finite element packages. These packages are powerful and efficient and are able to solve many complicated problems of technical practice on a routine basis. The packages are more and more automated. In some cases, the user is even 'deprived' of solving meshing problems - the so called meshless finite element approach is being advocated. In other cases the packages take care of the correct determination of time step in transient problems. These packages offer a lot of options to choose from; the options themselves are described in particular manuals to a variable extent of details. As the Murphy law states, however, the manuals are as a rule read only if nothing else helps. It is thus worthwhile to recall some of the essentials from the finite element theory, show pitfalls which should be avoided and to present modern programming tools which help a lot in the derivation of necessary relations and in subsequent understanding of the matter. The behaviour of a rectangular membrane element and that of some finite element packages when solving simple problems will be shown in this paper with the intention to answer the question whether a modern engineer is supposed to know the theoretical details of the finite element theory and the essentials of programming.