Vol 18, No 1 (2022)

The study of the stability of plates under shear under the action of dynamic loads is one of the important problems of structural mechanics. The plates are widely used in construction, mechanical engineering, shipbuilding and aircraft building. The paper presents a method for calculating plates for shear buckling, taking into account the physical nonlinearity of the material. A plate is considered under the action of a shearing dynamic load along the edges. The calculation is based on the Kirchhoff - Love hypotheses and the hypothesis of a non-linear elastic body. The plate material is assumed to be physically nonlinear. The deformation diagram is approximated as a cubic polynomial. The deflection of the plate points is determined in the form of Vlasov - Kantorovich expansions. Basic non-linear differential equations are derived using the energy method. Lagrange’s equations are used to obtain the resolving equations for plate buckling. On the basis of the developed technique, a calculation was made for the stability of a physically nonlinear square plate under the action of a shear dynamic load. The edges of the plate are hinged. The finite system of nonlinear differential equations is integrated numerically by the Runge - Kutta method. Based on the results of calculations, plots of the dependence of the relative value of the deflection of the central point of the plate on the dynamic coefficient Kd (with and without taking into account the physical nonlinearity of the material) are plotted. The influence of the degree of physical nonlinearity of the material, the parameter of the rate of change of the shear load on the criteria for the dynamic stability of a square plate is studied.

Some approaches to the derivation of rheological equations of the mechanical state of concrete are considered and the principle of superposition of fraction deformations is justified in a nonlinear statement. In linear creep theory, this principle is known as L. Boltzmann’s superposition principle of fraction creep deformations. The concept of the strength structure of the constructive material is the basis for substantiating the statements given in this work. The statistical distribution of the strength of the fractions forming a structural element in the union allows the derivation of nonlinear equations of state. At the same time, the so-called structural stresses of fractions that capable to force resistance are considered. The overlay principle of fraction deformations in non-linear statement is justified. This means the modification of L. Boltzmann’s principle of superposition allowing its applicability also under the nonlinear dependence of deformations on stresses. It is established that the integral equation of state, which is nonlinear with respect to calculated stresses, is linear with respect to structural stresses. It is this circumstance that permits its reduction to a simple linear differential equation, which, in particular, simplifies the solution of relaxation problems. These problems are closely related to the calculation of structures for long-term safety.

The work is devoted to improving the methods of experimental determination of internal friction parameters in materials. The aim of the laboratory experiments is to obtain physical parameters of the material that allow to take into account the damping forces in a uniaxial stress state. The research is focused on the internal friction model, which is based on the use of the generalized Prandtl model, that gives frequency-independent internal friction and allowing for the dependence of internal friction on the level of time-varying stresses. Damped oscillations during pure bending are recorded on a specially made laboratory installation. The description of the installation, the reference points of which coincide with the fixed points of the realized form of natural oscillations, is provided. The algorithm of cameral processing of experimental data is obtained. It is proposed to use a virtual system equivalent in damping. This is a system with one dynamic degree of freedom. The involvement of an imaginary system permits, after performing tests of the sample for pure bending, to acquire data corresponding to stretching - compression. The technique grants the use of long samples, which reduces the negative effect of stress concentration in the anchorages. The damping equivalent scheme makes it possible to use samples with an arbitrary cross-section. The found damping parameters for low-carbon steel are given.

In 1972, I.I. Kotov proposed to separate the surfaces of congruent sections into a separate class and to include the surfaces of plane-and-parallel translation, surfaces of revolution, carved surfaces of Monge, cyclic surfaces with a generatrix circle of constant radius, rotative, spiroidal, and helical surfaces in it. The aim of the research is to obtain generalized parametric equations of surfaces of congruent sections of the pendulum type on right cylinders with plane-and-parallel translation of movable rigid superellipses. Analytical geometry methods are used. Computer systems MathCad and AutoCAD are applied to visualize surfaces. The results consist in the derivation of parametric equations of the studied surfaces in a general form convenient for the use of computer modeling methods. The technique is demonstrated on five examples with congruent mobile superellipses. The possibility of using obtained surface shapes in parametric architecture, free-form architecture, and in shaping of the surfaces of some technical products is noted.

The article presents the design of the architectural and artistic composition “Memorial stele on a local spherical support” (KB-1 Stele), as well as the analysis of design solutions and the calculated justification of the strength and stability of the KB-1 Stele to external influences in conditions of the most intense climatic influences with parameters set according to building codes and regulations. The design of the KB-1 Stele is a vertically installed overall mass model of a peaceful nuclear explosive device weighing 2.8 tons and 2.8 meters high with hemispherical covers. In accordance with the terms of reference, it was required to preserve the appearance of the device as accurately as possible by placing it on a small local support. The peculiarity of the design lies in its angular mobility relative to the support. To ensure a stable equilibrium state of the stele, a hinge is organized at the place of support, the stele is installed freely by the lower sphere on an intermediate stand with a specially selected spherical surface. Additionally, the center of mass of the stele is significantly reduced by lightening (thinning the walls) of the upper part of the body and installing additional cargo in its lower part. As a result, under the action of external lateral loads, the stele acquires a stable position with possible fluctuations in the tumbler type and with a return to its original state. Analytical and numerical computational studies were performed to substantiate the strength and stability of the KB-1 Stele under external loads. It is shown that the developed design meets the regulatory requirements and is safe in operation.