# Vol 15, No 6 (2019)

**Year:**2019**Articles:**9**URL:**https://journals.rudn.ru/structural-mechanics/issue/view/1273**DOI:**https://doi.org/10.22363/1815-5235-2019-15-6

## Full Issue

## Analysis and design of building structures

### Effective modules of two-phase construction composites with grain filler

#### Abstract

In the book of R.M. Christensen, “Introduction to the Mechanics of Composites” (1982), a calculation formula is given for the bulk module of polydisperse composites with spherical inclusions. This formula has been known to the Russianspeaking reader for almost 40 years, but unfortunately, it is not used in the practice of building materials science. To identify applied possibilities, R.M. Christensen's formula is modified and reduced to a dimensionless function k = k ( w , η, θ), which depends on three dimensionless parameters, i.e., it depends on three quantities: w is the volume fraction of the inclusion, η - the ratio of the shear modulus of the matrix material to the volume modulus of the same matrix, θ is the ratio of the volume moduli of the matrix materials and inclusion. Numerical studies of this function reveal that in two-phase granular composites, the range of effective moduli is significantly narrowed compared to the region limited by Voigt and Reuss estimates (in the sense of the upper and lower bounds of real values). At the same time, the lower Christensen score is the same as the Reuss score. Numerical and graphically presented results are given on the examples of the study of two characteristic groups of composite materials. In addition, the dimensionless form of the effective module allows to construct a system of visual graphic dependencies of the functions k ( w ) in a flat space k - w . For different values of θ, the function k = k ( w , η) displays a bunch of curved segments, which sets the position of the plane figure in flat space. Examples of constructing figures for characteristic regions of the values of the function k (η, θ, w ) are given.

**Structural Mechanics of Engineering Constructions and Buildings**. 2019;15(6):407-414

### Comparative calculation of optimal parameters of channel bent and bent closed profiles

#### Abstract

Relevance. A new technical solution for channel bent closed profiles (BCP), distinguished by a composite section and related to light steel thin-walled structures (LSTWS), which are distinguished by high technical and economic indicators and massive demand in industrial and civil construction, is presented. The main results of the comparative calculation of the optimal parameters of bent channels and channel horizontal bending sections are also given. Aim of the research. The purpose of the study is to show that the characteristics of LSTWS can be further improved by shaping modification profiles, combining in its composite section straight and round outlines of closed and open loops. Methods. Through experimental design and optimization and design calculations of channel profiles, their new technical solution has been developed, the originality of which is confirmed by patent examination. Results. Channel BCP consists of two tubular shelves and one wall of double thickness. For its manufacture without welded, bolted or riveted joints, the outer and inner blanks are made along the entire length with serrated longitudinal edges, the teeth of which are staggered relative to each other and mutually bent in grooves after closing a bent profile along its shelves. The bends of the gear mounts increase the collapse thickness, provide an increase in local stability and shear strength of the thin-walled elements, and also allow not to reduce the design sections. A comparative calculation of the optimal parameters of bent channels and channel bending sections for bending showed that in the first of them the strength is maximum when the ratio of the width and height of the cross section is 1/6, and in the second - 1/5.68.

**Structural Mechanics of Engineering Constructions and Buildings**. 2019;15(6):415-432

### Numerical studies of strength of concrete cylinders for compression

#### Abstract

Relevance. The choice of adequate models of materials and deformation diagrams is of great importance when performing structural calculations in a nonlinear setting. Since there are no instructions on how to use the deformation diagrams of concrete and reinforcement when working together, given in SP 63.13330.2018, it is necessary to introduce assumptions for modeling reinforced concrete structures with finite elements of the same type. The aims of the work are to conduct numerical experiments on testing concrete cylinders for uniaxial compression and to verify the results with normative data. Methods. Numerical experiments were performed in the LS-DYNA software package. This program complex allows to simulate the joint work of concrete and reinforcement with the help of volume (for concrete) and rod (for reinforcement) finite elements. A cylinder with a diameter of 150 mm and a height of 300 mm was taken as model. Samples were modeled by volumetric finite elements. The CSCM - Continuous Surface Cap Model is a nonlinear material used to model concrete. Tests were carried out with samples of the following classes of concrete for cylindrical compressive strength: C12, C16, C20, C25, C30, C35, C40, C45, C50, C55. This corresponds to the following classes of cubic compressive strength: B15, B20, B25, B30, B37, B45, B50, B55, B60, B67. Results. The conducted researches have shown that the character of destruction of samples at numerical experiment corresponds to the character of destruction at tests. The investigated concrete model CSCM can be used in the calculation of concrete and reinforced concrete structures for the main classes of concrete, when taking into account the transition from cubic to prismatic strength and additional correction factors to cylindrical strength.

**Structural Mechanics of Engineering Constructions and Buildings**. 2019;15(6):433-437

## Theory of thin elastic shells

### Simplified selection of optimal shell of revolution

#### Abstract

Relevance. Architects and engineers, designing shells of revolution, use in their projects, as a rule, spherical shells, paraboloids, hyperboloids, and ellipsoids of revolution well proved themselves. But near hundreds of other surfaces of revolution, which can be applied with success in building and in machine-building, are known. Methods. Optimization problem of design of axisymmetric shell subjected to given external load is under consideration. As usual, the solution of this problem consists in the finding of shape of the meridian and in the distribution of the shell thickness along the meridian. In the paper, the narrower problem is considered. That is a selection of the shell shape from several known types, the middle surfaces of which can be given by parametrical equations. The results of static strength analyses of the domes of different Gaussian curvature with the same overall dimensions subjected to the uniformly distributed surface load are presented. Variational-difference energy method of analysis is used. Results. Comparison of results of strength analyses of six selected domes showed that a paraboloid of revolution and a dome with a middle surface in the form of the surface of rotation of the z = - a cosh( x/b ) curve around the Oz axis have the better indices of stress-strain state. These domes work almost in the momentless state and it is very well for thin-walled shell structures. New criterion of optimality can be called “minimum normal stresses in shells of revolution with the same overall dimensions, boundary conditions, and external load”.

**Structural Mechanics of Engineering Constructions and Buildings**. 2019;15(6):438-448

### Vizualizing of semi-regular polyhedrons in AutoCAD environment

#### Abstract

The paper examines the automated formation by the kinematic method of the surfaces of Archimedes' semi-regular polyhedra of three forms: truncated tetrahedron, truncated octahedron and truncated icosahedron. To solve this problem, AutoCAD and the built-in programming language AutoLISP were used. Each of these five semi-regular polyhedra of Archimedes has faces of two kinds. In this regard, the surface of a separate polyhedron is considered to consist of two structural forms. Each structural shape is formed in the AutoCAD environment from the compartments of the surfaces of the faces of the polyhedron of the same type, and each compartment is assigned to a specific layer of the drawing. The formation of constructive forms is provided by user-defined functions developed in the functional programming language AutoLISP. User-defined functions not only form images of surfaces, but also perform all the necessary calculations. The electronic model of each polyhedron is formed by the union of its structural forms. A block is formed from it. The surface formation of each polyhedron performs user-defined functions that provide “freezing” of drawing layers intended for surface compartments, insertion of a block with an electronic model of the polyhedron, and sequential “defrosting” of drawing layers. When there is a “thawing" of the layers of the drawing, the process of forming a polyhedron is shown on the monitor screen. As a result of research software that includes userdefined functions for the formation of an electronic model of selected polyhedrons and visualization of the process of formation of their surfaces in a dynamic mode was created.

**Structural Mechanics of Engineering Constructions and Buildings**. 2019;15(6):449-457

## Numerical methods of structures’ analysis

### Reliability and applicability of modern numerical analyses of dams

#### Abstract

Relevance. At present the application of numerical analyses to real problems of dam engineering has suffered at times from the gaps between the specialists of mathematical modeling and dam engineers and managers. The first group usually includes information system specialists because they are able to develop the computer models to their full potential. The professionals belonging to the second group often prefer to revert to traditional methods of calculation and empirical methods based on their proven experience. The aim of the work - based on recommendations of International workshops seminars, organized by the ICOLD Committee on Computational Aspects of Dam Analysis and Design, help dam engineers to interact with mathematical modeling specialists and to work with them without language barriers or gaps in knowledge. In this relation the assessment of reliability and applicability of numerical analyses of dams allows engineers to develop the optimal dam design. Methods. Assessment of the reliability of numerical methods of analyses of dam behavior was based on data of 10 International benchmark-workshop seminars, organized by the Committee in Italy (1991 and 1992), France (1994 and 2009), Spain (1996), USA (1999), Austria (2001), Romania (2003), China (2005), Russia (2007), in which specialists of these countries also took part.

**Structural Mechanics of Engineering Constructions and Buildings**. 2019;15(6):458-469

## Dynamics of structures and buildings

### Free vibrations of anisotropic rectangular plate laying on a heterogeneous viscouselastic basis

#### Abstract

The aim of the work. Free, transverse vibrations are considered heterogeneous along the three spatial coordinates of rectangular plates lying on an inhomogeneous viscoelastic base. It is assumed that the boundary conditions are homogeneous. A closed solution for the problem of free vibration of an inhomogeneous rectangular orthotropic plate based on an inhomogeneous viscoelastic foundation is developed in the article. Young's moduli and the density of the orthotropic plate continuously change with respect to three spatial coordinates, while the characteristics of a viscoelastic base change depending on the coordinates in the plane. Methods. The corresponding equation of motion is obtained using the classical theory of plates. The solution to the problem was constructed using the method of separation of variables and the Bubnov - Galerkin method. Results. Explicit formulas of the fundamental tone of the frequency of the transverse vibration of an anisotropic plate lying on an inhomogeneous viscoelastic base are determined. The influence of heterogeneity of orthotropic materials, viscosity inhomogeneities, inelastic and elastic substrates at dimensionless plate frequencies have been studied in detail.

**Structural Mechanics of Engineering Constructions and Buildings**. 2019;15(6):470-476

### Mathematical modeling of unsteady elastic stress waves in a console with a base (half-plane) under fundamental seismic action

#### Abstract

The aim of the work is to consider the problems of numerical modeling of seismic safety of the console with the base in the form of an elastic half-plane under unsteady wave influences. Stress waves of different nature, propagating in the deformed body interact with each other. After three or four times the passage and reflection of stress waves in the body, the process of propagation of disturbances becomes steady, the body is in oscillatory motion. The problem of modeling problems of the transition period is an actual fundamental and applied scientific problem. Methods. The finite element method in displacements is used to solve the two-dimensional plane dynamic problem of elasticity theory with initial and boundary conditions. On the basis of the finite element method in displacements, an algorithm and a set of programs for solving linear plane two-dimensional problems have been developed, which allow solving problems with non-stationary wave effects on complex systems. The algorithmic language “Fortran-90” was used in the development of the complex of programs. The study area is divided by spatial variables into finite elements of the first order. According to the time variable, the study area is also divided into finite elements of the first order. Results. The problem of the influence of a plane longitudinal elastic wave in the form of a Heaviside function on a console with a base (the ratio of width to height is one to ten) is considered. The initial conditions are taken as zero. The system of equations from 16 016 084 unknowns is solved. Contour stresses and stress tensor components are obtained in characteristic areas of the problem. On the basis of the conducted researches it is possible to draw the following conclusions: the console (the ratio of width to height one to ten) is modeled with the elastic basis in the form of an elastic half-plane; the elastic contour stresses on the faces of the console are almost a mirror image of one another, that is, antisymmetric; the console under seismic action works as a rod of variable cross-section, that is, if there are tensile stresses on one face, then compressive stresses on the other; on the contours of the console under seismic action, bending waves mainly prevail.

**Structural Mechanics of Engineering Constructions and Buildings**. 2019;15(6):477-482

## Theory of Plasticity

### Principle of the overlay deformations in the theory of creep

#### Abstract

The aim of the research is to justify in the non-linear statement the overlay principle of fraction creep deformation, known in the linear creep theory as Bolzmann’s principle of superposition. Methods. In contrast to the traditional approach the material of constructive elements is considered as an union of its links with statistical disturbed strength. The model of structural strength allows the deduction of rheological equations. In loading process so called structural stresses of capable to resist links are considered. Results. The modification Bolzmann’s principle of superposition for fraction creep deformations is proposed. This permits its applicability also under non-linearly dependence of deformations on stresses. In according to concept of the statistical distribution of the strengths of links and linear dependence of determinations on structural stresses the rheological of mechanical statement is reduced. This equation implies the suitable on relation problems the linear integral equation. The relation of structural strength of material with its energy of entirety and with the experimentally known independency of specific to strength deformation on age of concrete is showed. The correct interpretations of certain known mechanical state equations for concrete are represented.

**Structural Mechanics of Engineering Constructions and Buildings**. 2019;15(6):483-496