The article provides information about method of sustainability calculation of compressed polymer rods taking into account nonlinear creep . As the law of the relationship between stress and strain is used nonlinear equation of Maxwell-Gurevich. Derived from the analysis of the resolving equations with time tends to infinity, we obtain an expression for a long critical force in the case of constant rigidity of the rod.

Keywords: nonlinear creep, rod, stability, Maxwell-Gurevich, finite difference method, long critical force, relaxation viscosity, viscoelasticity, high elasticity modulus.

The technique of determining the stress-strain state of the polymer thick cylindrical shells in flat tension conditions with effects of temperature and creep deformation. As the law of the relationship between stress and strain is used nonlinear equation of Maxwell-Gurevich. Solution is performed numerically by finite element method.

Keywords: nonlinear creep, cylinder, the Maxwell-Gurevich equation, finite element method, relaxation viscosity, viscoelasticity, high elasticity modulus, plane stress and temperature.

We obtained the solution of the optimization problem for the step-parallel beam with constant width of the cross-section. The problem was solved by minimizing the strain energy while the volume of beam is constant. The solution was made for the case of simple support at the ends and evenly distributed along the length load. We introduced the parameter α, which is the ratio of height of the average part to a height of side parts. The optimal value of α was found. At this value the stiffness of the beam is maximum at constant volume. The obtained beam of maximum stiffness is not equal strength.

Keywords: optimization, speed-prismatic beam, the strain energy, variable stiffness, minimum

We solved the problem of optimizing the gable beam by minimizing the strain energy at constant volume. The problem reduced to an integral equation for determining the optimum angle of the beam. This equation was solved numerically by the method of bisection. Integral was calculated using the method of trapezoids. Solution was made in software package Matlab. The optimum angle depends on the width of the cross section b, and the volume of the beam. It was found that with increasing of width of the cross section the optimum tilt angle decreases. Also the volume increases by increasing the width of the beam.

Keywords: optimization, gable beam, strain energy, variable rigidity, minimum volume, method of bisection

The phenomenon of buckling under the creep of concrete arches was investigated. Solution of the problem carried out by means of the finite element method. To analyze the stability we used Newton-Raphson method. It has been established that there is a long-term critical load, beyond which the growth of the deflection has not fading character. As the equation of the relationship between the creep deformation and strains we used viscoelastoplastic hereditary model of aging concrete. To determine the creep strain we used a linear approximation with respect to time. It was found that the long critical load for considered arch was in 1.44 times lower than the instant critical load.

Keywords: reinforced concrete arch, stability, creep, geometric nonlinearity, finite element method, Newton-Raphson method

The resolving equations for determining stress-strain state of thick-walled polymeric cylindrical shell under plane strain state with changes in temperature and highly elastic strains. Into law that describes the relation between stress and creep strain, the nonlinear equation of Maxwell-Gurevich. The solution is performed numerically using the finite element method.

Keywords: nonlinear creep, plastic cylinder, highly elastic deformation, the equation of Maxwell-Gurevich, finite element method, viscoelasticity, modulus of viscoelasticity, plane strain state, temperature.

Solved the inverse problem for a thick-walled cylinder, experiencing temperature and force action, under the plane of the axisymmetric problem of elasticity theory. By the variation of the modulus of elasticity, in which the cylinder is equally stressed by the Mohr's theory of strength. The problem is reduced to a differential equation of the first order. This equation was solved numerically, using the Runge-Kutta method of fourth order.

Keywords: thick-walled cylinder, optimization, heterogeneity, method, Runge-Kutta, temperature, flat axisymmetric problem

The problem of optimization of thick-walled sphere loaded by internal and external pressures was solved. The essence of the method consists in the variation of the modulus of elasticity. The problem of finding the distribution of the characteristics of material in which the stress state is given, is called the inverse problem. The idea of the inverse method was proposed by academician of RAACS prof.  V.I. Andreev. The analytical dependence of modulus of elasticity from the radius at which the calculated stress on the strength theory of Mor is constant throughout the thickness of the shell was found. This shell will be equally stressed. If the strength characteristics of the material do not depend on the modulus of elasticity it will also be equiresistant. By creation of indirect heterogeneity we reduced the maximum tensions in 1.6 times. It was also shown that in the case of a centrally symmetric problem the second theory of strength is a particular case of the strength theory of Mor.

Keywords: thick-walled spherical shell, optimization, strength theory of Mor, uniform strength, equal stress

We solved the problem of optimization of thick-walled prestressed concrete cylinder loaded by internal pressure. Preliminary stresses in such a cylinder are created by wounding with tension cables on the outer surface. The idea of the method is to find the law of variation of modulus of elasticity , in which the state of stress is given . The analytical dependence of modulus of elasticity of the radius at which the tensile stress in the entire thickness of the shell does not occure was found. By creation of indirect heterogeneity we reduced the consumption of rebar by 10%.  The direct problem - determining the stress- strain state of a cylinder with a constant modulus of elasticity was also resolved.

Keywords: thick-walled prestressed concrete cylinder, optimization, inverse problem, theory of elasticity, heterogeneity

Method of successive approximations was used to solve the problem of optimization of multispan continuous beam. Cross section of a beam was a welded I-beam.  By the variation of the width of the shelf we ensured that the maximum equivalent stress for energy theory of strength in each section is the same throughout. The solution is made numerically using complex MatLab. Bearing capacity of beams with variable stiffness along the length increased in 2 times in comparison with the beam with constant stiffness and the same mass. The proposed method can be applied to frames, taking into account the contribution of longitudinal forces in the equal stresses. Another effective way to change stiffness is to change the height of the wall of I-beam. In practice curvilinear shape of beams or shelves are not used because of complexity of manufacturing. They are replaced by the discrete form of cross section.

Keywords: multispan beams, work method, optimization, uniform strength, welded I-beam, variable stiffness

The resolving equations for the calculation of the stability of polymer compressed rods with the creep using Galerkin method are obtained. The problem is solved in the case of a simply supported pivotally-rod having an initial bending in the plane of the lower stiffness, and in the case of off-center application of force. Galerkin method used in combination with the finite element method, that is the shape functions are taken as the basis functions . The solution is made numerically, using the software package Matlab. There are two variants considered: a variant of the rod with constant and variable in length stiffness. It is shown that for rods with variable cross-sections and the same mass, a critical time has increased by almost 4.5 times. The solution for the rods of constant cross section agrees well with the solutions obtained by the method of finite differences. This may indicate the reliability of the proposed method  

Keywords: polymer rod, creep, Galerkin method, stability, variable stiffness