# Vol 17, No 6 (2021): Prospects for the application of shell structures and thin shells in the first half of the 21st century

**Year:**2021**Articles:**11**URL:**https://journals.rudn.ru/structural-mechanics/issue/view/1541**DOI:**https://doi.org/10.22363/1815-5235-2021-17-6

## Full Issue

## Articles

### A message of greetings

**Structural Mechanics of Engineering Constructions and Buildings**. 2021;17(6):551-552

## From the editor-in-chief of the thematic number of the journal

### Shell structures and shells at the beginning of the 21st century

#### Abstract

Researchers know that “golden century of shells” falls on 1920-1960 when the finishing of building of a thin-walled shell became an important event in life of country where this shell was erected. Every built shell was analyzed in tens of scientific works with a point of view of used method of analysis, applied constructive materials, cost of erection. Later on, an interest to thin-walled shells fell down. On the base of the fulfilled research in a paper, it is shown that application of shell structures is increasing in the 21st century because it was closely connected with needs of different branches of human activity. It is proved, that practically in all countries of the world, design and building of shell structures and shells was carried out. Only priority in application constructive materials changed. In the main, reinforced concrete was used earlier but now bar curvilinear structures, composite shells, and bar structures with the glass filling are in priority. It is shown that young and prominent architects and engineers tale part in construction of considered structures and thin-walled shells. All conclusions are confirmed by references containing 38 used original sources.

**Structural Mechanics of Engineering Constructions and Buildings**. 2021;17(6):553-561

## Shell forming

### Geometry of the normal ruled surfaces

#### Abstract

The wide circle of the surfaces formed by the motion of the right line in the normal plain of some base directrix curve is regarded. The generate right line may rotate at some low at the normal plane of the base curve. The vector equation of the surface with any plane or space base curve is received. There are given the formulas of the geometry characteristics of the surfaces, on the base of them there is shown that the coordinate system of the normal ruled surfaces is orthogonal but there is not conjugated in common, that is that the normal ruled surfaces there are not developable surfaces in common way. The condition of the rotation of directrix plane line when the coordinate system of the normal ruled surfaces will be conjugated and the normal ruled surface will be developable is received. The condition that the normal ruled surface with space base curve will be the developable surface there is connected with its curvature of base curve. The developable normal ruled surface with plane base curve is formed by motion of right line at the normal plane of the base curve with the constant angle to the plane of the base curve; the received surface is a surface of constant slope. On the base of the vector equation of the surfaces there are made the figures of the normal ruled surfaces with the help of program complex MathCAD.

**Structural Mechanics of Engineering Constructions and Buildings**. 2021;17(6):562-575

## Theory of thin elastic shells

### Diagnostics of thin-walled structures of complex geometry and structure

#### Abstract

The main stages of the birth of thin-walled structures, changes in their relative thickness and mass of a unit area are given; ways of creating perfect thin-walled structures are indicated. The problems arising during the operation of thin-walled structures of complex geometry, as well as approaches and methods of their calculation are noted. To ensure trouble-free operation of a thin-walled structure with a thin-layer coating, under load and exposed to physical fields and environments, it is necessary to correctly diagnose the condition of structural elements. The spline variant of the finite element method in two-dimensional (SV FEM-2) and three-dimensional (SV FEM-3) productions is noted, as well as the synthesis of these variants - SV FEM-2 + SV FEM-3. The combination of the idea of parametrization of the entire domain and approximation of the desired variables within the element by Hermitian cubic splines makes it possible to obtain high-precision consistent finite elements. The developed variants of the finite element method make it possible to evaluate the stress-strain state of structures of complex geometry, including the calculation of multilayer, thin-walled structures with coating and local defects, as well as to take into account specific surface properties other than those of the main array. Studies of stress concentration near local depressions are considered. Two-dimensional experimental and theoretical methods are noted for evaluating the stiffness properties and adhesion of thin-walled, thin-layer and composite structural elements of complex structure, which, along with a distributed complex structure, may have distributed defects. The developments were used in solving specific tasks of a number of enterprises.

**Structural Mechanics of Engineering Constructions and Buildings**. 2021;17(6):576-587

### Iterative methods for constructing an equations of non-closed shells solution

#### Abstract

The elasticity relations are transformed to a form that allows, in accordance with the previously proposed Saint-Venant - Picard - Banach method, to iteratively calculate all the required unknowns of the problem. The procedure for constructing a solution is reduced to replacing eight first-order differential equations of the original system of shell theory with eight corresponding integral equations with a small parameter that has the meaning of the ratio of the shell width to its length or the variability of the stress-strain state in the transverse direction. The fifteen unknowns of the original problem calculated by direct integration are expressed in terms of five main unknowns. The fulfillment of the boundary conditions on the long sides of the strip leads to the solution of eight ordinary differential equations for slowly varying and rapidly varying components of the main unknowns. Slowly varying components describe the classical stress-strain state. The rapidly changing ones determine the edge effects at the points of discontinuity of the slowly changing classical solution and the fulfillment of the boundary conditions unsatisfied by them due to the lowering of the order of the differential equations of the classical theory based on the Kirchhoff hypothesis. In the general case, the solution is represented as asymptotic series in a small variability parameter with coefficients in the form of power series in the transverse coordinate. The presentation is illustrated by an example of constructing an iterative process for a long circular cylindrical panel. By virtue of the fixed-point theorem, the iterative process is convergent.

**Structural Mechanics of Engineering Constructions and Buildings**. 2021;17(6):588-607

## Numerical methods of shell analysis

### Numerical analysis of cylindrical shell stability interacting with inhomogeneous soil

#### Abstract

The research is aimed at determining the critical buckling load of the spatial model “shell - soil” system in the case of inhomogeneous physical and mechanical soil properties along the longitudinal axis of the cylindrical shell in a nonlinear formulations of the task. Methods. The task is solved by a numerical method using a finite element complex ANSYS. Two calculated cases of the spatial model “shell - soil” system are compiled. The soil is divided into two equal parts with different physical and mechanical properties. The problem was solved in geometrically, physically and constructively nonlinear statement. Nonlinearity is due to the need to find the contact zone through an iterative process and determine the time-varying position of the shell. The soil is modeled by volumetric elements, each consisting of twenty nodes. The shell is modeled by flat elements, each consisting of four nodes. Contact elements of one-side action are used. Critical buckling load are determined relative to the actual load of its own weight. Results. Critical loads are obtained from two calculated cases of the spatial model “shell - soil” system. There is a comparative analysis of the results. An assessment of the stability margin of the shell relative to the actual load is given.

**Structural Mechanics of Engineering Constructions and Buildings**. 2021;17(6):608-616

### Investigation of the accuracy and convergence of the results of thin shells analysis using the PRINS program

#### Abstract

The theoretical foundations of compatible finite elements construction for static and dynamic analysis of single-layer and multilayer shells are discussed. These finite elements are implemented in the PRINS computer program. The paper presents verification tests to investigate the accuracy and convergence of the results of calculating various shells using these finite elements. Shell structures are widely used in various fields of technology - construction, mechanical engineering, aircraft construction, shipbuilding, etc. Specialists on the design and calculation of such structures need a reliable and accessible tool for the practical problems solving. Computer program PRINS can be one of such tools. It can be effectively used by engineers of design and scientific organizations to solve a wide class of engineering problems related to the calculations of shell structures. The paper describes the finite elements of the shells, implemented in the PRINS program. The results of verification calculations are presented, which confirm the high accuracy of this program.

**Structural Mechanics of Engineering Constructions and Buildings**. 2021;17(6):617-627

## Shell dynamics

### Determination of natural vibration frequencies of reinforced cylindrical shell

#### Abstract

Free vibrations of a reinforced cylindrical shell filled with liquid are investigated. The case of an orthotropic shell is considered when the cord filament is placed symmetrically with respect to the meridian of the shell. The motion of a fluid is potential and is described by a wave equation. The fluid moves without separation from the walls of the cylinders. The fluid pressure is taken into account in the equations of motion of the shells, and the velocities of the fluid and the shell are equalized at the boundaries. Representing a solution in a harmonic form reduces to a system of transcendental equations. Comparison of the solutions of the problems without a liquid and with a liquid shows the dependence of the frequency of the system without a liquid at the frequency of the system with the liquid. An inverse method is proposed for solving the equation. The inverse method for solving the problem has made it possible to construct a more accurate frequency spectrum of free oscillations of the system. For some values of the system parameters, the natural frequencies of the cylinder are determined.

**Structural Mechanics of Engineering Constructions and Buildings**. 2021;17(6):628-638

## Calculation of seismic impacts

### Mathematical modeling of bending stress waves in an aboveground oil pipeline under unsteady seismic action

#### Abstract

The problem of numerical modeling of bending waves in an aboveground oil pipeline under nonstationary seismic action is studied. To solve the unsteady dynamic problem of elasticity theory with initial and boundary conditions the finite element method was applied. Using the finite element method in displacements, a linear problem with initial and boundary conditions was led to a linear Cauchy problem. A quasi-regular approach to solving a system of linear ordinary differential equations of the second order in displacements with initial conditions and to approximation of the studied domain is proposed. The technique is based on the schemes: point, line and plane. The area under study is divided by spatial variables into triangular and rectangular finite elements of the first order. According to the time variable, the area under study is divided into linear finite elements with two nodal points. The algorithmic language Fortran-90 was used in the development of the software package. The problem of the effect of a plane longitudinal wave in the form of six triangles on an elastic half-plane to assess physical reliability and mathematical accuracy is considered. A system of equations consisting of 8 016 008 unknowns is solved. The calculation results are obtained at characteristic points. A quantitative comparison with the results of the analytical solution is taken. Furthermore, the problem of the impact of a plane longitudinal seismic wave at an angle of 90° degrees to the horizon on an aboveground oil pipeline is considered. The seismic impact is modeled as a Heaviside function, which is applied at a distance of three average diameters from the edge of the pipe. The calculation results were obtained at the characteristic points of the object under study. A system of equations consisting of 32 032 288 unknowns is solved. Bending waves prevail in the problem under consideration.

**Structural Mechanics of Engineering Constructions and Buildings**. 2021;17(6):639-650

## Optimization of reinforced concrete shells

### Optimization of design solutions of protective structures of erections of nuclear power stations

#### Abstract

The ideas of optimization of constructive solutions of protective erections of nuclear power stations are presented. A problem of such optimization for the securing of nuclear and radiation safety for different regime of nuclear power stations exploitation, including extreme action, is very topical at present time. Modern home and international achievements on the considered subjects are demonstrated and modern demands, analysis methods, and problems of design of erections of nuclear power stations that give an opportunity to assure their safety exploitation under collision of flying objects are given as well. The results confirm wide opportunities of improvement of the constructive solutions of protective shells of reactor sections of nuclear power stations with the application of innovative materials that give the prospects to economize the material resources considerably and to raise the reliability and safety of exploitation of erections of nuclear power stations simultaneously.

**Structural Mechanics of Engineering Constructions and Buildings**. 2021;17(6):651-663

## Wooden shell structures

### The economic feasibility of taking into account the joint work of load-bearing and enclosing elements in large-span wooden spatial structures

#### Abstract

Large-span wooden spatial structures in the form of domes, developed in JSC Research Center of Construction under the leadership of A.A. Pogoreltsev, are built in large numbers for roofing various buildings. Such designs have high performance indicators and continue to improve. In 2020 the “Manual on accounting for the joint work of the frame and fencing in wooden spatial structures of buildings and structures” was developed as an addition to SP 64.13330.2017, containing examples of strength calculation and optimization of rib cross-section dimensions, as well as an example of determining the long-term strength of the shell cladding. These structures, in fact unique, are designed so far in the margin of safety without taking into account the participation of the enclosing part (panels, decking, etc.) in the bearing loads. Taking into account the work of the decking, especially when exposed to asymmetric loads, can lead to material savings. In addition to the above theory of nonlinear shell calculation, the authors have also developed a calculation of composite anisotropic panels operating under complex stress conditions, i.e. under biaxial compression (tension) and shear. The calculation of such structures under both short-term and long-term loads presents certain difficulties and requires the introduction of strength theories and criteria for their description that are unusual for specialists of design organizations.

**Structural Mechanics of Engineering Constructions and Buildings**. 2021;17(6):664-678