"Вестник МГСУ"

The theory of the strength of materials has produced a substantial influence on the development
and practical implementation of computer methods of the strength analysis of beams and
beam systems. Beams are modeled through the employment of one-dimensional elements within
the overwhelming majority of the finite element method software programmes; the stiffness matrix
is derived on the basis of the hypothesis of flat sections, and end forces concentrate in the centres
of the gravity of cross sections. This approach makes it possible to develop effective algorithms,
although it has several drawbacks. They include an incorrect transmission of forces from beams
to plates and massive elements of structures, difficulties in taking account of the warping effect of
the beam, and the complexity of taking account of physical and geometrical nonlinearities. Some
authors suggest using the three-dimensional theory with account for the flat sections hypothesis. It
encompasses the patterns of rotations of sections in the analysis of structures, although the problems
of warping and shear deformations remain.
The authors propose a new approach to rectangular column modeling by means of the finite
element analysis of building structures. Each column is presented as a set of three-dimensional
8-node elements with arbitrary discretization alongside the cross section and the height of the column.
The inner nodes of the finite element mesh are excluded sequentially layer by layer, thus,
reducing the stiffness matrix and other characteristics of the column with reference to its top and
bottom cross sections. The finite element method has been adapted to PRINS software programme.
The comparative analysis of the two structures has been completed with the help of this software.
The structures exposed to the structural analysis included slabs and columns. In one case,
columns were modeled with the help of one-dimensional elements, and in the another case, the
proposed elements were used. The comparison of the results demonstrates that the employment
of the proposed elements makes it possible to avoid problems associated with the transmission of
the force in a particular point.

In the article, the authors provide their summarized findings concerning the difference between
the mean velocity determined on the basis of the discharge rate and through the integration
of the velocity profi le alongside the pipe radius. The authors have identified the ratio between these
velocities, which is confirmed by the experimental data obtained for pipes and channels. The equation
characterizing the ratio of these velocities has also been derived.
The analysis of compatibility of dynamic and kinematic characteristics of in-pipe and wide
flows has been performed. This analysis demonstrates that the coefficient of hydraulic resistance
of in-pipe and wide flows can vary up to 10-20 % despite the identical hydraulic radius and the
tension of friction. This difference is caused by the difference in the mean velocities of in-pipe flows.
The authors demonstrate that the coincidence between the equations of hydraulic resistance
of in-pipe and wide flows is attainable when the numerical exponent of the velocity profile inside
pipes and channels is different.
The quantitative correlation between the hydraulic resistance coefficient and the numerical
exponent of the velocity profile for channel flows is identified. This correlation is substantiated by
the experimental data.

The problem concerning the relation between the intensity of an inclined uniformly distributed
load and the displacement of a section of the boundary half-plane is considered in the article. The
value of the load intensity, physical and mechanical properties of the media may be used to determine
the displacement of the section of the soil mass boundary at any point of the lower half-plane.
The authors have also identified a relation between the settlement value and the intensity of the
inclined load. The value of the settlement, physical and mechanical properties of the media may
be used to identify the intensity of the inclined load, applied to the pre-set section of the soil mass
boundary.

The authors consider the problem of conveyance of non-spherical solid particles in an open
rectangular channel. The process of glass container manufacturing is accompanied by formation of
waste glass at 1150...1350 °С. As a result, hot glass mass flows into cold water and transforms into
glass granules. Granules are used in the production of glass, and they can be loaded back into the
industrial furnace.
At this stage, there arises a problem of conveyance of waste glass granules into the gallery,
in the direction of the furnace. The pipeline-based method requires an engine, which will increase
the cost of glass containers. Hydraulic transportation of waste glass is a cheaper method. In this
connection, there is a practical problem of identifying the slope angle sufficient for the transportation
of waste glass in an open rectangular channel. Thus, we must determine the hydraulic characteristics
of the two-phase flow to solve the problem.
A laboratory research of the particle size distribution pattern was conducted in 2011 at the
glass factory in operation in the Tula region. The shape of particles and the condition of the glass surface affect the parameters of their hydraulic transportation. These characteristics are taken into
account when calculating the formula and introducing the correction coefficient.
The problem of determining the slope of the open channel needed to transport waste glass into
the glass melting furnace can be formulated as follows. What should be the angle of the bottom of
the channel for hydraulic transport of waste glass, when the particle speed reaches its critical value?
The input data are as follows: channel length - 70 meters, cross-section area - 1.4 m2. Hydraulic
transport of waste glass is produced under the influence of gravity, due to the difference in the height
of the upper and lower points of transportation.
Chezy coefficient helps determine the appropriate slope of the bottom of the channel. As
a result of the calculation of the angle of inclination of the bottom of the channel, the difference
between the upper and lower points was 2.17 m, the particle size of glass 4.76...17.97 mm, the
channel length - 70 m, height - 1 m, width - 1.4 m.
The benefits of free flow hydraulic transport include small operating costs. The main
disadvantage of hydraulic transport is the need for a substantial difference in the heights of upper
and lower points.
As a result, the authors have worked out their recommendations concerning the
transportation of y solid particles of waste glass.

The article represents an overview of the field studies of the intensity and distribution of probability
of longitudinal turbulent velocity fluctuations in river flows with different sizes of beds and
hydrological characteristics. The authors demonstrate that the normalizing transformation of velocity
fluctuations performed by the local friction velocity makes it possible to get the changes of velocity
fluctuations deep inside the flow close to universal.
The authors have also identified that the intensity of turbulent velocity fluctuations exceeds
the friction velocity 2.5-3-fold in the area close to the river bottom, while their intensities demonstrate
their gradual decline closer to the surface of the flow. The authors have derived an approximation
formula, describing the change of the intensity of longitudinal velocity fluctuations
deep inside river flows.
Probability distributions of longitudinal velocity fluctuations were compared to those based on
the law of Gauss. It is proven that they have a kurtosis of a frequency curve as well as an asymmetry
in comparison with the distribution of Gauss, which are most vivid in the area close to the bottom of the flow. Due to the fact that the coefficient of asymmetry includes a third degree of velocity fluctuations,
and a kurtosis of the frequency curve, experimental identification of these characteristics
is problematic for the reason of their instability. The new information concerning the intensity and
probability properties of the river flow turbulence can be used in projecting the mixture formation and
mass exchange processes ongoing inside river flows.

The objective of this research is to study the fungicidal properties of the cement composition
modified by chitosan (deacetylation rate - 95%, molecular weight - 200 kDm (MM 2,7x105)). The
optimal concentration of chitosan is identified so that its infusion into the cement paste did not deteriorate
the physical and mechanical characteristics, structure and composition of the latter. The
authors have identified that the infusion of 1% chitosan (in relation to the cement mass) into the
cement composition is optimal. It has resulted in (1) the slight improvement of the strength properties
of modified samples, (2) the reduction of dimensions of the porous space alongside with the
increase in the number of gel pores (20%), (3) the reduction of the number of capillary pores (5%).
The other subject of this research represents interaction of 1% (and lower concentrations of)
chitosan with calcium hydroxide. No interaction between 2% chitosan and calcium hydroxide is
identified. The conclusion is that the infusion of 1% chitosan into the cement composition provides
it with fungicidal and fungistatic properties, while the strength characteristic of the cement paste is
slightly improved.

Two points of an elastic and perfectly plastic material exposed to the plane stress are examined by the author. One point is located on the stress concentrator surface. The other one is located at a certain distance from the first one (it is considered as a secondary point within the framework of the kinetic theory of a plastic flow).
As a result of the finite element analysis of the stress-strain state it has been discovered that the material in the point located in the front area of the kinetic plastic flow remains linearly elastic in terms of its physical condition, and the load is applied to it in accordance with a curved trajectory. This trajectory is represented by $$





U
0




-


U











coordinates, where Uф and U0 are the density-related components of dilatation and distortion strain. For the purposes of modeling, the trajectory is represented as a two-component broken line.
As a result, the kinetic plastic flow prolongation is limited. This effect intensifies while the value of the elastic Poisson ratio (µ) goes down. For example, for ? < 0.5, dimensions of the plastic zone outstretched along the crack curve are smaller than those identified using the Irwin plastic zone solution. Furthermore, in case of ? = 0.25, the effective crack length is $$



l

eff


=l-
1

18 π




(


K


σ
Y






)


2







, and the modified stress distribution is below the singular stress distribution according to the laws of linear elastic fracture mechanics.

The problem of forced vibrations of plates exposed to the thermal impact is interesting both
as a theoretical implication and an issue of practical importance. A thermal impact causes formation
of a non-steady temperature field. Thereafter, some materials turn fragile and cannot withstand the
exposure to the impact of a thermal field.
The authors propose a solution to the problem of influence of a thermal impact onto an isotropic
plate that demonstrates special boundary conditions, if its two opposite edges are simply
supported, and the surface temperature is equal to zero, while the two other edges might have an
arbitrary type of fixation and an arbitrary thermal mode.
In the first part of the paper, the authors provide their derivation of the elastic plate vibration
equation, if the plate is exposed to the thermal impact under the pre-set boundary conditions.
In the second part of the paper, the authors provide their solution to the aforementioned problem
based on a strictly mathematical approach. Their solution is presented as an integral function
of the plate deflection. The solution in question may be reduced to algebraic frequency equations
by using the method of expansion of trigonometric functions. Thus, it is possible to identify natural
frequencies of the plate vibration caused by the thermal impact.

The article represents a summarized research of the influence of thickness of elements connected
by multiple-bolt friction joints produced onto the bearing ability of the connection. Different
thicknesses of structural elements connected by multiple-bolt friction joints produce both qualitative
and quantitative influence on the stress-strain state of the connection.
The relevance of the research is driven by the technology of Light Steel Framing Construction
(LSFC) that is intensively developed and implemented in the construction industry.
The objectives of the research encompass:
the experimental study of the influence of the thickness or thickness difference of connected
elements produced onto the bearing capacity of friction joints;
identification of the bearing capacity of high-strength bolts with controlled tension in the
bolted connection, depending on the thickness of connected elements.
The results of experimental tests of samples of connected plates, the thicknesses of which
vary from 1.5 mm to 5 mm, are presented in the paper.
The analysis of connections that have, at least, one plate the thickness of which is less than
5 mm, should incorporate the coefficients that take account of the bearing capacity fluctuations.
The authors demonstrate that the bearing capacity of friction joints of steel plates is highly
dependent on the thickness and thickness difference of connected plates. Different combinations of
thicknesses of plates within the range of 1…5 mm cause the bearing capacity of the joints to vary
within the range of 130 %.

The operator symbol is usually its image at homomorphy into some function algebra or even into other operator algebras. In this case it is usually supposed, that the kernel of homomorphism is an ideal of rather continuous operators. In this case for Fredholm property of an operator the inversibility of its symbol is needed and enough. The authors consider the algebra generated by a complex matrix. The authors proved the existence and uniqueness of random matrix representation in the form of the sum of nilpotent matrix and linear combination of minimal idempotent matrixes combination.
The obtained results allow generalization for infinite-dimensional operators and can be used in systems of linear differential equations and in mathematical statistics.

This paper represents a summary of the iterative solution to the problem of linearized coupled
filtration. The formulation of the coupled filtration problem can be applied for the purposes of simulation
of the land surface subsidence caused by the pumping of the fluid out of a well located near the
land surface. The pumping process causes pressure redistribution and, consequently, undesirable
subsidence of the land surface. The filtration problem considered by the authors is a direct problem,
therefore, domain dimensions, ground properties and pumping characteristics are supposed to be
available. With this assumption in hand, coupled differential equations are derived on the basis of
the Biot's filtration model and the Darcy's law.
First, spatial discretization is based on the finite element method, while the finite-difference
scheme is used to assure discretization within the course of time. Discretization of the linear coupled
problem leads to the generation of a linear saddle system of algebraic equations. It is well-known
that the stability condition of such a system is usually formulated as the LBB condition (inf-sup
condition). The condition is satisfied for a differential problem (to say more accurately, for a variational
problem). The validity of the stability condition for an algebraic system depends on the finite
elements used for the purpose of the problem discretization. For example, the LBB condition is not
always satisfied for most simple Q1-Q1 elements. Therefore, first of all, stability of the finite element
system is studied in the paper. The filtration problem has a number of parameters; therefore, it is
not easy to identify analytically the domain in which the stability condition is satisfied. Therefore, the
stability condition is under research that includes some numerical tests and examination of physical
dimensionality. The analysis completed by the authors has ended in the derivation of the formula
that determines the stability condition formulated on the basis of the problem parameters.
Second, solution methods are explored numerically in respect of sample 3D problems. Dimensions
of domains under consideration are typically as far as 20 km in length and width and up to
5 km in depth. Thus, the resulting linear system is rather large, as it is composed of hundreds of
thousands to millions of equations. Direct methods of resolving these saddle systems can hardly be
successful and they are definitely inefficient. Therefore, the only choice is the iterative method. The
simplest and the most robust method is the Uzawa method applied in combination with the conjugate
gradients iteration method used for the Schur complement system solution. The computer code
that implements iterative solution methods is written in FORTRAN language of programming. The
conjugate gradients method is compared to its alternatives, such as the Richardson iteration and the
minimal residue methods. All three methods were tested as methods of solving the model problems.
The authors provide their numerical results and conclusions based on the comparative analysis of
the aforementioned iteration methods.

The author considers the variational formulations of the problem of stability of non-uniformly
compressed rectilinear elastic bars that demonstrate their variable longitudinal bending rigidity in the
event of different classical conditions of fixation of bar ends.
Identification of the critical bar loading value is presented as a minimax problem with respect
to the loading parameter and to the transversal displacement of the bar axis accompanied by the
loss of stability. The author demonstrates that the critical value of the loading parameter may be formulated
as a solution to the dual minimax problem. Further, the minimax formulation is transformed
into the problem of identification of eigenvalues in the bilinear symmetric and continuous form, which
is equivalent to the identification of eigenvalues of a strictly positive, linear and completely continuous
operator. The operator kernel is presented in the form of symmetrization of the non-symmetric
kernel derived in an explicit form.
Within the framework of the problem considered by the author, the bar ends are fixed as follows:
(1) both ends are rigidly fixed, (2) one end is rigidly fixed, while the other one is pinned, (3) one
end is rigidly fixed, while the other one is attached to the support displaceable in the transverse direction,
(4) one end is rigidly fixed, while the other one is free, (5) one end is pinned, while the other
one is attached to the support displaceable in the transverse direction, (6) both ends are pinned.

The author describes the application of certain conditions that deprive the boundaries of certain
areas from reflecting properties. A numerical simulation of the elastic wave propagation pattern
in the infinite media is to be incorporated into the study of the impact of seismic loads produced on
buildings and structures.
The problem of elimination of reflected waves from the set of boundaries in the course of
calculation of dynamic problems of the theory of elasticity is quite important at this time. The study
of interaction between elastic waves and various engineering facilities has been unfeasible for quite
a long time.
A well-known method of generating counter-propagating waves at the boundary is applied
to compensate for the accumulation of longitudinal and transverse waves. The boundary ratio is
derived for longitudinal, transverse and other types of waves, including conical surface Rayleigh
waves, to check the performance of the proposed methodology.
Longitudinal, transverse, and conical surface Rayleigh waves as the main carriers of the elastic
energy fail to represent the relation. The problem is solved numerically through the application
of the dynamic finite element method. The numerical solution is capable of taking account of the
internal points of the area.

The objective of this article is to present the analysis of a double-span beam that has disabled
constraints, including its analysis in the state of static equilibrium and in the event of forced
vibrations. Hereinafter, the original system is entitled System 1, while the system that has disabled
constraints is System 2.
The analysis is performed in furtherance of the following pattern. First, System 1 static analysis
and System 2 static and dynamic properties analysis is executed. Later, we calculate the deflection
and the internal force of System 2 as the consequence of disabled constraints. By comparing
the process of static equilibrium of System 2 and the process of free vibrations of System 2, we
identify that the moment of flexion in the mid-span increases by 85 %, while the support moment
increases by 66 %.
The analysis of the system that has disabled constraints in the process of forced vibrations is
the same as the analysis demonstrated hereinbefore, except that the initial condition is calculated
differently. By disabling constraints, we can both reduce and increase the peak values of displacement
of the system in the process of forced vibrations.
This research proves that the proposed method can be used to calculate defl ection and the
internal force of static and dynamic systems having disabled constraints. That can be very important
in evaluation of the safety of structures after destruction of their individual elements.

The authors present a method of homogenization used to solve nonlinear equilibrium problems
of laminated plates exposed to transversal loads.
The homogenization technique is a general and mathematically rigorous solution to elasticity
problems. It describes the processes of deformation of composite structural elements. It was originally
developed for linear problems. This method encompasses the calculation of all characteristics
related to deflection by combining solutions to local and global homogenization problems. Thus, it
implements the general idea of the domain decomposition into subdomains.
The homogenization method has been most widely used in cases of periodical heterogeneity
because of significant simplification that happens due to periodicity. This simplification implies that
any cell of periodicity appears to be the material representative volume element (RVE). Therefore,
it is sufficient to solve local problems within a single periodicity cell. Hence, with reference to local
problems, conditions of periodicity are a mere consequence of the periodicity of the material
structure. Decomposition of the domain causes decomposition of the solution. The latter means that
displacements, stresses and strains are represented by functions that depend on both global and local
coordinates. Global coordinates are associated with the whole body scale and local coordinates
vary in the periodicity cell, i.e. in RVE only.
If the material structure is not periodic, but its properties do not depend on global coordinates,
material effective properties can be determined by solving local problems in any RVE. That is not the ase of nonlinear materials. Now local problems have to be solved in every RVE because of the homogenized
properties dependence on global coordinates. Another complication arises due to nonlinearity.
Indeed, the homogenization method employs the superposition principle to represent the solution to the
elasticity problem as summarized solutions to global and local problems. This principle doesn't work in
the case of nonlinearity. We suggest combining the standard homogenization technique with linearization
by using the loading history to solve the nonlinear problem. On the contrary, local linear problems
have to be solved in every RVE. Certainly, this method involves numerous calculations.
As for the problem considered in the paper, its nonlinearity is caused by material plastic properties.
Most plasticity-related principles are formulated as tensorial linear relationships between the
stress and strain rates. Hence, here we identify a perfect opportunity to employ the homogenization
method combined with linearization with regard to the load parameter. This combined technique is
implemented to resolve the heterogeneous plate bending problem. Heterogeneous materials are of
the two types: laminates and functionally graded materials (FGM).
The computer code is developed for the purpose of numerical plate bending simulation. It employs
the parallel programming MPI technique and the Euler type explicit and implicit methods. For
example, laminated plate bending due to the distributed transversal load was the subject of research.
Each layer of the plate was composed of FGM or a homogeneous material. The authors have discovered
that FGM plates have a higher yield stress then the plates composed of homogeneous layers.

The authors provide solutions to the relevant issue of development and construction of
foundations resting on a sliding layer as a method of external seismic protection of buildings and
structures. The authors have developed a system of automated operating control over the external
seismic protection of buildings (structures) that represents an automatic switch of the emergencylevel
seismic load. The authors have filed an application for the registration of their invention. The
proposed solution has everything in place to be widely applied to improve the seismic protection of
buildings and structures, especially those resting on problematic soils.
The authors provide their description of a paradox of foundations: foundations are analyzed
in terms of "upside-down" loads, whereas seismic loads have an opposite direction. The authors
provide their solutions to this problem.
The authors argue that methods of seismic protection incorporated into effective regulations
are limited, and they do not constitute any external seismic protection methods, whereas the
application of seismic isolation inside buildings is unreasonable, as it is limited by the requirement
to install it "above the foundation". Presently, the above methods are being reworked into foundation
platforms resting on a sliding layer. Their efficiency has been proven by a computerized model and
a theoretical analysis.
The authors also provide their argumentation in favour of the conclusion that relevant seismic
protection development trends are to incorporate advanced structural solutions, including methods
of external seismic protection.

The authors provide data on the influence of a round cap installed in the upper compressed
zone of soil onto displacement of laterally loaded single-pile supports identified on the basis of the
analytical method that they have developed. The authors demonstrate that both the front surface of
the round cap and its bottom contribute to formation of the supplementary resistance to displacements
of a laterally loaded pile due to (i) the friction force that arises alongside its contact with soil
in the event of a lateral displacement, and (ii) a vertical pressure under the impact of a moment load
in the event of a tilted slab. A round cap in the upper displaced zone of soil may cause a substantial
reduction (4-8-fold) of displacements of a laterally loaded pile if compared to the pile that has no
round cap.

The authors analyze rheological equations of viscoplastic landslides under natural conditions
and in the presence of a sparse row of piles as an anti-slide barrier.
Rheology of viscoplastic slides in the presence of buttresses that significantly alter the pattern
and speed of their motion has enjoyed little attention of researchers, although it plays an important
role in the analysis of stabilization of landslide flows. Elements of tensor calculus are used to analyze
the motion of the viscoplastic matter. An exact solution to the problem of gravitational motion
alongside an inclined plane was used as the main one. It is proven that the motion pattern of the
viscoplastic matter contains rigid zones where the flow velocity is equal to zero.
In the event of motion of a layer alongside an inclined surface, the rigid zone moves together
with the viscous surface, and the overall velocity of the viscoplastic flow will be determined by the
rheology of the viscous surface.
This paper provides solutions designated for the identification of rigid zones of cohesive soils,
as well as soils that demonstrate internal friction and cohesion.
The authors have proven that whenever piles are used, the nucleus of the landslide mass is
stabilized.

Plates are widely used as flat structural elements in various spheres of construction and engineering.
Development of industrial and residential building techniques furthers development of
the construction science. Therefore, the design and the refined theory of plate vibrations is one
of signifi cant issues considered within the framework of the applied theory of elasticity, which is of
practical interest to the researchers. The authors provide a summary of frequency equations that
describe self-excited oscillations of a transversally isotropic plate resting on the strain foundation
bed, if one edge of the plate is flexibly fixed and the other three edges are hinged. Oscillations are
described by partial differential equations of the fourth order. The problem is resolved through the
employment of an approximate method of decompositions. As a result, the frequency equation is
derived to identify self-excited lateral oscillations of the plate. The equations derived by the authors
for the purpose of identifi cation of the frequencies of self-excited transverse vibrations of a plate are
fit for practical use, and they may be applied in calculations to identify the dependence of the selfexcited
frequency of the plate on its geometry.

The authors present their methods of experimental determination of parameters of statically
indeterminate pre-stressed reinforced concrete bar systems in the out-of-limit state. The authors
also present their method of the out-of-limit state modeling and experimental determination of dynamic
strain and stress values typical for such systems in the event of the abrupt failure of one
system element.
Methods of experimental determination of parameters and dynamic loadings typical for deformations
caused by the abrupt failure of one of connecting elements drive special attention to
the analysis of deformations and cracking of pre-stressed structures on the basis of the research
of the nature of destructions and redistribution of dynamic forces within the above elements. The
following parameters have been identified in the course of tests: longitudinal strain of compressed
and cracked concrete elements; strain-stretched connecting pieces (calibrated covers); defl ections
of beams, including the application of loads beyond design parameters; load intensity and nature
of cracks and crack openings before and after the application of loads beyond design parameters;
overall patterns of destructions and prototypes of destructions caused by a sudden fracture of a
welded connecting element; increments of displacements and oscillations of pre-stressed elements
of beams after a sudden destruction of a connection under pressure.
Experimental parameters are needed to draw the curves of diagrams of experimental static
and dynamic deformations necessary for the survivability analysis.