# Vol 16, No 5 (2020)

**Year:**2020**Articles:**8**URL:**https://journals.rudn.ru/structural-mechanics/issue/view/1375**DOI:**https://doi.org/10.22363/1815-5235-2020-16-5

## Full Issue

## Analysis and design of building structures

### Multitudes of Voigt - Reuss forks and Voigt - Christensen - Reuss tridents

#### Abstract

In the literature, there are many studies of the representative volume of a composite material, in particular, those calculated using the formulas of Christensen, Voigt and Reiss. The aim of this work is to study the features of evaluating the set of forks of effective modules. Methods. On the basis of solving the Lame problem (for a thick-walled sphere), a spherical model of a representative volume (cell) of a composite material with a granular (spherical) filler is compiled and the value of the effective modulus of elasticity of a two-phase composite is determined. The study of the obtained formula for the effective modulus, expressed in dimensionless quantities, for the cell material revealed its identity with the R.M. Christensen’s formula, expressed in dimensional values, for the bulk modulus of composites with a spherical filler. In this case, Christensen’s solution was previously obtained by a different method when he considered the polydisperse model of the composite. The dimensionless form of the function (effective module) of three dimensionless parameters made it possible in flat spaces (two coordinate planes) to construct graphical images of the function of the named modules according to Christensen, which are compared and combined in one figure with similar images of the functions of estimating the values of the modules (real composites) according to Voigt and Reiss. Graphical studies in relation to the spherical representative volume model show that in the flat space of the set of Voigt - Reuss forks, these forks are not “narrowed”, but they are partially filled by the flat space of the set of Christensen - Reiss forks. The graphs of the functions of the modules, at the same time, form, simultaneously with the sets of two-toothed forks, a set of Voigt - Christensen - Reiss trident forks (tridents), which, depending on the size of the intervals of the numbers of the studied parameters, have “forks” of different sizes. Results. Graphic illustrations of numerical examples have been obtained showing that for given values of the module of the matrix and filler and the volume fraction of the latter, it is possible to determine the effective volumetric module and shear module of two-phase composites, and to perform a comparison with the conclusions of the applied plan. The dimensionless form of the obtained expressions makes it possible to solve the inverse problems of the mechanics of polydisperse composites, for example, to determine the volume module of the composite components by the effective modulus obtained by mechanical testing of standard samples.

**Structural Mechanics of Engineering Constructions and Buildings**. 2020;16(5):323-333

### I-shaped bent closed profiles with tubular shelves and calculation of the optimal layout of their composite sections

#### Abstract

I-shaped bent closed profiles with tubular shelves are distinguished by a composite section and related to light steel thin-walled structures (LSTWS), which are characterized by high technical and economic indicators and mass demand in industrial and civil construction, which determines the relevance of the development of their new technical solutions. The aim of the work - to show that the characteristics of LSTWS can be further improved by forming profiles, combining straight and round outlines of closed and open circuits in a composite section. Methods. New technical solution, the originality of which is confirmed by patent examination, has been developed through experimental design and optimization and design calculations of I-shaped profiles. The calculation of the optimal bending layout of the composite sections of I-shaped profiles of horizontal billets from sheet blanks, identical and unequal in thickness, including bisteel modifications, is made. Results. The I-shaped bent closed profiles consists of two tubular shelves and one wall of double thickness. For its manufacture without welded, bolted or riveted joints, the outer pair and inner pair 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 the local stability and shear strength of the thin-walled elements, and also allow not to reduce the design sections. The calculation of the optimal layout of I-shaped profiles horizontal bend for bending showed that its strength is maximum when the ratio of the width and height of 1/5.2 and equal thicknesses of shelves and walls.

**Structural Mechanics of Engineering Constructions and Buildings**. 2020;16(5):334-350

### Analytical assessment of the frequency of natural vibrations of a truss with an arbitrary number of panels

#### Abstract

The aim of the work is to derive a formula for the dependence of the first frequency of the natural oscillations of a planar statically determinate beam truss with parallel belts on the number of panels, sizes and masses concentrated in the nodes of the lower truss belt. Truss has a triangular lattice with vertical racks. The solution uses Maple computer math system operators. Methods. The basis for the upper estimate of the desired oscillation frequency of a regular truss is the energy method. As a form of deflection of the truss taken deflection from the action of a uniformly distributed load. Only vertical mass movements are assumed. The amplitude values of the deflection of the truss is calculated by the Maxwell - Mohr’s formula. The forces in the rods are determined in symbolic form by the method of cutting nodes. The dependence of the solution on the number of panels is obtained by an inductive generalization of a series of solutions for trusses with a successively increasing number of panels. For sequences of coefficients of the desired formula, fourth-order homogeneous linear recurrence equations are compiled and solved. Results. The solution is compared with the numerical one, obtained from the analysis of the entire spectrum of natural frequencies of oscillations of the mass system located at the nodes of the truss. The frequency equation is compiled and solved using Eigenvalue search operators in the Maple system. It is shown that the obtained analytical estimate differs from the numerical solution by a fraction of a percent. Moreover, with an increase in the number of panels, the error of the energy method decreases monotonically. A simpler lower bound for the oscillation frequency according to the Dunkerley method is presented. The accuracy of the lower estimate is much lower than the upper estimate, depending on the size and number of panels.

**Structural Mechanics of Engineering Constructions and Buildings**. 2020;16(5):351-360

## Numerical methods of structures’ analysis

### Stress-strain state of shell of revolution analysis by using various formulations of three-dimensional finite elements

#### Abstract

The aim of the work is to perform a comparative analysis of the results of analyzing arbitrarily loaded shells of revolution using finite element method in various formulations, namely, in the formulation of the displacement method and in the mixed formulation. Methods. To obtain the stiffness matrix of a finite element a functional based on the equality of the actual work of external and internal forces was applied. To obtain the deformation matrix in the mixed formulation the functional obtained from the previous one by replacing the actual work of internal forces in it with the difference of the total and additional work was used. Results. In the formulation of the displacement method for an eight-node hexahedral solid finite element, displacements and their first derivatives are taken as the nodal unknowns. Approximation of the displacements of the inner point of the finite element was carried out through the nodal unknowns on the basis of the Hermite polynomials of the third degree. For a finite element in the mixed formulation, displacements and stresses were taken as nodal unknowns. Approximation of the target finite element values through their nodal values in the mixed formulation was carried out on the basis of trilinear functions. It is shown on a test example that a finite element in the mixed formulation improves the accuracy of the strength parameters of the shell of revolution stress-strain state.

**Structural Mechanics of Engineering Constructions and Buildings**. 2020;16(5):361-379

### Dynamic method for determining critical loads in the PRINS computer program

#### Abstract

Relevance. Buckling analysis is important in the design of buildings and structures. It is used in various fields of engineering - mechanical engineering, aircraft and shipbuilding, civil engineering, etc. Until the second half of the twentieth century, mainly analytical methods of buckling were applied in practice. With the appearance of computers, numerical methods, in particular, the finite element analysis, began to prevail. Buckling analysis was implemented in programs of finite element analysis, such as NASTRAN, ANSYS, ABAQUS, ADAMS, DIANA, and others. In view of great responsibility, buckling analysis of structure should be carried out using at least two different programs. However, due to the high cost of software products, not all project organizations are able to have a number of programs. An alternative is to develop programs that can complete buckling analysis using several methods. This would increase the reliability and quality of calculation results. The PRINS computer program has opportunity for buckling analysis using two methods - static and dynamic. The aims of the work - to show the theoretical aspects and practical implementation of the dynamic principle of buckling analysis in buildings and structures using finite element method, as well as to give the algorithm implemented in the PRINS program and the results of verification calculations confirming its reliability. Results. The algorithm presented in this article and implemented in the PRINS computer program allows to determine critical loads using a dynamic buckling criterion. On the basis of numerous verification calculations, it was established that the implemented algorithm was effective for determining critical loads in frame, thin-walled and ribbed plate structures. The use of the PRINS computer program enables to use an alternative method for determining critical loads for a wide class of engineering problems in addition to the classical (static) method.

**Structural Mechanics of Engineering Constructions and Buildings**. 2020;16(5):380-389

## Theory of elasticity

### Modern interpretation of Saint-Venant’s principle and semi-inverse method

#### Abstract

Relevance. The progressive development of views on the Saint-Venant formulated principles and methods underlying the deformable body mechanics, the growth of the mathematical analysis branch, which is used for calculation and accumulation of rules of thumb obtained by the mathematical results interpretation, lead to the fact that the existing principles are being replaced with new, more general ones, their number is decreasing, and this field is brought into an increasingly closer relationship with other branches of science and technology. Most differential equations of mechanics have solutions where there are gaps, quick transitions, inhomogeneities or other irregularities arising out of an approximate description. On the other hand, it is necessary to construct equation solutions with preservation of the order of the differential equation in conjunction with satisfying all the boundary conditions. Thus, the following aims of the work were determined: 1) to complete the familiar Saint-Venant’s principle for the case of displacements specified on a small area; 2) to generalize the semi-inverse Saint-Venant’s method by finding the complement to the classical local rapidly decaying solutions; 3) to construct on the basis of the semi-inverse method a modernized method, which completes the solutions obtained by the classical semi-inverse method by rapidly varying decaying solutions, and to rationalize asymptotic convergence of the solutions and clarify the classical theory for a better understanding of the classic theory itself. To achieve these goals, we used such methods , as: 1) strict mathematical separation of decaying and non-decaying components of the solution out of the plane elasticity equations by the methods of complex variable theory function; 2) construction of the asymptotic solution without any hypotheses and satisfaction of all boundary conditions; 3) evaluation of convergence. Results. A generalized formulation of the Saint-Venant’s principle is proposed for the displacements specified on a small area of a body. A method of constructing asymptotic analytical solutions of the elasticity theory equations is found, which allows to satisfy all boundary conditions.

**Structural Mechanics of Engineering Constructions and Buildings**. 2020;16(5):390-413

## Experimental researches

### Investigation of the probability of existence of defects with a size exceeding the allowed value

#### Abstract

Relevance. Non-destructive testing of metal determines the actual state of the metal, the presence of discontinuities and their sizes, and also allows to determine what mechanisms of metal degradation were subjected to. One of the main characteristics of the quality of non-destructive testing is the detectability of discontinuities and defects. If no defects were missed, then it’s possible to guarantee the reliable operation of the facility until the next scheduled inspection. The article is devoted to the study of the probability function of detecting defects and determining the probability of the existence of a residual defect with a size exceeding the permissible value. The aim of the work - to develop a method to determine the probability of the existence of a residual defect with a size exceeding the permissible value after non-destructive testing and repairs of equipment and pipelines of a nuclear power plant. Methods. During the work formulas for the probability of detecting a defect and initial defectiveness, regulatory requirements in the field of certification of flaw detectors, and the results of research on non-destructive testing were used. Results. A method for determining the probability of defects with a size exceeding the allowed value, using the example of a reactor vessel, is presented. The method is based on residual defects, which takes into account the detectability of defects. The value of the coefficient that takes into account the influence of the human factor, instrument and methodological shortcomings or complexity of access to the control point is determined, which reduces the degree of uncertainty in determining the residual defect. The results of this work permit to evaluate the probability of the existence of a defect with a size exceeding the allowed value. The development of a residual defect to critical values characterizes the initial event for the destruction of the integrity of the structure. Thus, the probability of a residual defect can be used when performing a safety analysis of the water-water energetic reactor vessel.

**Structural Mechanics of Engineering Constructions and Buildings**. 2020;16(5):414-423

### Improving the compressive strength of lightweight cylindrical concrete column with basalt fiber reinforced polymer acting under imposed load

#### Abstract

**Structural Mechanics of Engineering Constructions and Buildings**. 2020;16(5):424-434