scholarly journals CRITERION OF MINIMUMMATERIAL CONSUMPTIONOF FLANGE OF I-SHAPED BAR WITH A VARIATION IN ITS THICKNESS ANDOUTLINE OF THE WIDTH, WITH RESTRICTION TO THE VALUE OF THE CRITICALFORCE OR RESTRICTIONTO THE VALUE OF THEFIRSTNATURAL FREQUENCY INTWO PRINCIPAL PLANESO

2018 ◽  
Vol 14 (2) ◽  
pp. 96-108
Author(s):  
Leonid S. Lyakhovich ◽  
Pavel A. Akimov ◽  
Boris A. Tukhfatullin
Keyword(s):  
Natural Frequency ◽  
Cross Section ◽  
Critical Force ◽  
Principal Plane ◽  
Material Consumption ◽  
The Cross ◽  
The Stability

We have already presented original criterion of minimum material consumption within the design of the outline of the width of the I-shaped bar and the stability constraints or restriction to the value of the first natu­ral frequency in one principal plane of inertia of the cross-section. This paper is devoted in its turn to a criterion for the minimum material capacity of the I-shaped bar with a variation in its thickness and outline of the width, with restrictions to the value of the critical force or restriction to the value of the first natural frequency in two principal planes of inertia of the section

scholarly journals ASSESSMENT CRITERIA OF OPTIMAL SOLUTIONS FOR CREATION OF RODS WITH PIECEWISE CONSTANT CROSS-SECTIONS WITH STABILITY CONSTRAINTS OR CONSTRAINTS FOR VALUE OF THE FIRST NATURAL FREQUENCY. PART 1: THEORETICAL FOUNDATIONS

2019 ◽  
Vol 15 (4) ◽  
pp. 88-100 ◽  
Author(s):  
Leonid Lyakhovich ◽  
Pavel Akimov ◽  
Boris Tukhfatullin
Keyword(s):  
Natural Frequency ◽  
Cross Section ◽  
Cross Sections ◽  
Optimization Methods ◽  
Optimal Solutions ◽  
Continuous Change ◽  
Material Consumption ◽  
Piecewise Constant ◽  
Ideal Object ◽  
The Cross

The special properties of optimal systems have been already identified. Besides, criteria has been for­mulated to assess the proximity of optimal solutions to the minimal material consumption. In particular, the cri­teria were created for rods with rectangular and I-beam cross-section with stability constraints or constraints for the value of the first natural frequency. These criteria can be used for optimization when the cross sections of a bar change continuously along its length. The resulting optimal solutions can be considered as an idealized ob­ject in the sense of the limit. This function of optimal design allows researcher to assess the actual design solu­tion by the criterion of its proximity to the corresponding limit (for example, regarding material consumption). Such optimal project can also be used as a reference point in real design, for example, implementing a step-by­step process of moving away from the ideal object to the real one. At each stage, it is possible to assess the changes in the optimality index of the object in comparison with both the initial and the idealized solution. One of the variants of such a process is replacing the continuous change in the size of the cross sections of the rod along its length with piecewise constant sections. Boundaries of corresponding intervals can be selected based on an ideal feature, and cross-section dimensions can be determined by one of the optimization methods. The dis­tinctive paper is devoted to criteria that allow researcher providing reliable assessment of the endpoint of the op­timization process.

scholarly journals CRITERIA OF MINIMUM MATERIALS CONSUMPTION FOR BARS WITH RECTANGULAR CROSS-SECTION AND RESTRICTIONS ON STABILITY OR LIMITATIONS ON THE VALUE OF THE FIRST NATURAL FREQUENCY

2017 ◽  
Vol 13 (1) ◽  
pp. 9-22 ◽  
Author(s):  
Leonid S. Lyakhovich ◽  
Pavel A. Akimov ◽  
Boris A. Tukhfatullin
Keyword(s):  
Mathematical Programming ◽  
Natural Frequency ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Optimal Solutions ◽  
Material Consumption ◽  
Variable Parameters ◽  
Special Cases ◽  
The Stability ◽  
Minimal Consumption

Apparatus of mathematical programming is normally used in the most part of research works, dealing with structural optimization. However, the special properties of optimal systems have been identified in several studies. Besides, corresponding criteria, which have been formulated as well, can be used for assessments of proximity of optimal solutions to minimal material consumption. Particularly relevant criteria for bars with rectangular cross-section and restrictions on the stability or limitations on the value of the first natural frequency have been formulated. However, not all the features of some of the criteria have been observed. In addition it seems appropriate to identify relevant criteria for special cases set variable parameters. The distinctive paper contains additional property proximity criterion of optimal solutions to minimal consumption of materials for the bars with a rectangular cross-section and limitations on the value of the first natural frequency, modification of one of the previously proposed criteria and formulation of appropriate criterion for the case where one of the parameters of variable rectangular cross-section is constant along the length of the bar.

scholarly journals ASSESSMENT CRITERIA OF OPTIMAL SOLUTIONS FOR CREATION OF RODS WITH PIECEWISE CONSTANT CROSS-SECTIONS WITH STABILITY CONSTRAINTS OR CONSTRAINTS FOR VALUE OF THE FIRST NATURAL FREQUENCY. PART 2: NUMERICAL EXAMPLES

2019 ◽  
Vol 15 (4) ◽  
pp. 101-110 ◽  
Author(s):  
Leonid Lyakhovich ◽  
Pavel Akimov ◽  
Boris Tukhfatullin
Keyword(s):  
Natural Frequency ◽  
Cross Section ◽  
Cross Sections ◽  
Optimal Solutions ◽  
Continuous Change ◽  
Material Consumption ◽  
Piecewise Constant ◽  
Numerical Examples ◽  
Ideal Object ◽  
The Cross

The special properties of optimal systems have been already identified. Besides, criteria has been for­mulated to assess the proximity of optimal solutions to the minimal material consumption. In particular, the cri­teria were created for rods with rectangular and I-beam cross-section with stability constraints or constraints for the value of the first natural frequency. These criteria can be used for optimization when the cross sections of a bar change continuously along its length. The resulting optimal solutions can be considered as an idealized ob­ject in the sense of the limit. This function of optimal design allows researcher to assess the actual design solu­tion by the criterion of its proximity to the corresponding limit (for example, regarding material consumption). Such optimal project can also be used as a reference point in real design, for example, implementing a step-by­step process of moving away from the ideal object to the real one. At each stage, it is possible to assess the changes in the optimality index of the object in comparison with both the initial and the idealized solution. One of the variants of such a process is replacing the continuous change in the size of the cross sections of the rod along its length with piecewise constant sections. Boundaries of corresponding intervals can be selected based on an ideal feature, and cross-section dimensions can be determined by one of the optimization methods. The dis­tinctive paper is devoted to criteria that allow researcher providing reliable assessment of the endpoint of the op­timization process, and the second part of the material presented contains corresponding numerical examples, prepared in accordance with the theoretical foundations given in the first part.

scholarly journals Numerical simulation of cutting layer in internal corners milling

Mechanik ◽  
2019 ◽  
Vol 92 (7) ◽  
pp. 412-414
Author(s):  
Jan Burek ◽  
Rafał Flejszar ◽  
Barbara Jamuła
Keyword(s):  
Numerical Simulation ◽  
Numerical Model ◽  
Cross Section ◽  
Cross Sectional Area ◽  
Sectional Area ◽  
Cross Sectional ◽  
Cross Section Area ◽  
Section Area ◽  
The Cross ◽  
The Stability

The analytical and numerical model of the cross-section of the machined layer in the process of milling of concave rounding is presented. Simulation tests were carried out to determine the cross-sectional area of the cutting layer. A strategy has been developed that allows to increase the stability of the cross-section area of the cutting layer when the mill enters the inner corner area.

Influence of the Steel Channel Shaped Shear Connectors on the Stability of the Simply Supported Steel Bridge without the Intermediate Bracing System in the Construction Stage

2021 ◽  
Vol 879 ◽  
pp. 151-168
Author(s):  
Van Khai Nguyen ◽  
Nghia Dai Van ◽  
Van Tuong Khanh Vo ◽  
Rin Anh Nguyen ◽  
Phuc Gia Nguyen ◽  
...  
Keyword(s):  
Cross Section ◽  
Steel Plate ◽  
Moment Of Inertia ◽  
Steel Bridge ◽  
Bridge Structure ◽  
Shear Connectors ◽  
The Cross ◽  
The Stability ◽  
The Moment ◽  
Main Girder

Steel bridge structure without intermediate bracing system (IBS) has been widely used in several countries and one of them is Japan. In this type of structure, the main steel girder is not reinforced by the stiffeners. The stiffness of the main girder is enhanced with steel plate directly welded to the top flange of the main girder, forming the “beams–system”. The reinforced concrete deck slab with the set of main girder and steel plate works compositely through steel shear connectors whose shape is C (channel) or I character. As for steel bridge structures, the main role of shear connectors is shear resistance between the concrete deck slab and steel girder plate in the exploitation stage. However, previous research has shown that the density of shear connectors influences on the stability as well as the stiffness of the bridge structure. Therefore, it has approved that this appurtenance is able to not only have the ability of shear resistance but also enhance the stiffness of the steel bridge structure which is particularly surveyed with the type of especial bridge structure – the steel bridge structure without IBS. Hence, the shear connectors in this kind of bridge structure are deliberately researched as an extra role in the construction stage. The following factors of the channel shape shear connectors would be researched for evaluating their impacting level on the stability of the special steel bridge structure: the properties (the length and the moment of inertia of the cross-section) and the density on the steel plate. Through the analysis of impacting level to the stability of three mentioned factors (the length and density of the shear connectors; the moment of inertia of the cross-section), the expected result is as following: 1) The minimum density of shear connectors is proposed. 2) The influence of the moment of inertia of the cross-section, the density, and the length on the stability is quite clear. 3) As for the economy, the optimal designed range among three factors is also suggested.

scholarly journals Calculation of wooden beams on the stability of a flat bending shape enhancement

2018 ◽  
Vol 196 ◽  
pp. 01003 ◽  
Author(s):  
Anton Chepurnenko ◽  
Vera Ulianskaya ◽  
Serdar Yazyev ◽  
Ivan Zotov
Keyword(s):  
Differential Equation ◽  
Cross Section ◽  
Stability Problem ◽  
Secular Equation ◽  
Rectangular Cross Section ◽  
Center Of Gravity ◽  
Critical Force ◽  
Distributed Load ◽  
Matlab Package ◽  
The Stability

Flat bending stability problem of constant rectangular cross section wooden beam, loaded by a distributed load is considered. Differential equation is provided for the cases when load is located not in the center of gravity. The solution of the equation is performed numerically by the method of finite differences. For the case of applying a load at the center of gravity, the problem reduces to a generalized secular equation. In other cases, the iterative algorithm developed by the authors is implemented, in the Matlab package. A relationship between the value of the critical force and the position of the load application point is obtained. A linear approximating function is selected for this dependence.

scholarly journals ASSESSMENT OF THE PROXIMITY OF DESIGN TO MINIMUM MATERIAL CAPACITY SOLUTION OF PROBLEM OF OPTIMIZATION OF THE FLANGE WIDTH OF I-SHAPED CROSS-SECTION RODS WITH ALLOWANCE FOR STABILITY CONSTRAINTS OR CONSTRAINTS FOR THE VALUE OF THE FIRST NATIONAL FREQUE

2020 ◽  
Vol 16 (2) ◽  
pp. 71-82 ◽  
Author(s):  
Leonid Lyakhovich ◽  
Pavel Akimov ◽  
Boris Tukhfatullin
Keyword(s):  
Objective Function ◽  
Natural Frequency ◽  
Cross Section ◽  
Wall Thickness ◽  
Optimal Solution ◽  
Computational Process ◽  
Flange Thickness ◽  
The Cross ◽  
Special Approach ◽  
Shaped Cross Section

There are known methods for optimizing the flange width of I-shaped cross-section rods with stability constraints or the constraints for the value of the first natural frequency. Corresponding objective function has the form of the volume of the flange material for the case when only the flange width varies and the cross-section height, wall thickness and flange thickness are specified. Special criterion for assessment of proximity of corresponding an optimal solution to the design of minimal material capacity was formulated for the considering problem. In this case, the resulting solution may not meet some other unaccounted constraints, for example, strength requirements. Modification of solution in order to meet previously unaccounted constraints does not allow researcher to consider such design as optimal. In the distinctive paper allowance for strength requirements, stability constraints or constraints for the value of the first natural frequency are proposed within considering problem of optimization. Special approach is formulated, which proposes to assess proximity to the design of minimum of material capacity obtained as a result of optimization. Increment of the objective function and criteria corresponding to constrains and restrictions are under consideration within computational process.

scholarly journals XVII. The complete system of the periods of a hollow vortex ring

1895 ◽  
Vol 186 ◽  
pp. 603-619 ◽  
Keyword(s):  
Kinetic Theory ◽  
Vortex Ring ◽  
Cross Section ◽  
Steady Motion ◽  
Dynamical Theory ◽  
Vortex Theory ◽  
Wide Range ◽  
The Cross ◽  
Hollow Vortex ◽  
The Stability

According to the vortex theory of matter, atoms consist of vortex rings in an infinite perfect liquid, the æther. These rings may be either hollow or filled with otating liquid. The cross section of the hollow or rotating core is in the simplest ase small and the ring is circular. Such vortices have been investigated. It has been hown that they can exist, and that they are stable for certain types of deformation, in this paper the stability of the hollow vortex ring is investigated further, with a view to proving that it is stable for all small deformations of its surface. An attempt also made to make the vortex theory of matter agree with the kinetic theory of ases as regards the relation between the velocity and the energy of an atom. On he latter theory the energy of an atom varies as the square of its velocity, while on he former theory the energy decreases as the velocity increases. As the two theories liffer on a fundamental point, while the consequences of the kinetic theory agree over wide range with experiment, those of the vortex theory are likely to be in discrepancy therewith. However, no account has been taken of the electric change which an atom must hold if electrolysis is to be explained. This electrification will evidently alter the relation between the energy and the velocity. The nature of the change thus produced is here discussed for the case of a hollow vortex, the surface of which behaves as a conductor of electricity, a representation which is dynamically realised by the theory of a rotationally-elastic fluid æther developed in Mr. Larmor’s paper, “A Dynamical Theory of the Electric and Luminiferous Medium.” The small oscillations also are worked out with a view to the discussion of the stability of an electrified vortex. 2. The velocity of translation of the vortex in its steady motion is constant and perpendicular to its plane. By impressing on the whole liquid a velocity equal and opposite to this, the hollow is reduced to rest. Since the cross section of the hollow is small, any small length of it may be regarded as cylindrical. A cylindrical vortex must, by reason of symmetry, have its cross section a circle, so that the cross section of the hollow of the annular vortex is approximately circular, and the hollow itself approximately a tore.

scholarly journals Analytical-numerical method for calculating the stability of rod elements of lightweight steel structures

2020 ◽  
Vol 17 (4) ◽  
pp. 39-46
Author(s):  
G. I. Belyy ◽  
Keyword(s):  
Numerical Method ◽  
Cross Section ◽  
Steel Structures ◽  
Longitudinal Force ◽  
Initial Section ◽  
Limiting State ◽  
Lightweight Steel ◽  
Combined Action ◽  
The Cross ◽  
The Stability

The article proposes an analytical-numerical method for calculating the rod elements of lightweight steel structures for general stability, in which the reduced section is replaced with the unreduced one with compensating loading with a fictitious longitudinal force. The dependence of this force with uniaxial or biaxial eccentricities on the acting longitudinal force is established when they are combined acting on fictitious stresses in the center of the weakening gravity of the rod most stressed section. Therewith, the effect of displacements corresponding to the stability tasks (bending, bending-torsional or spatial ones) is taken into consideration. The analytical part of the stability problems` solution is constructed for the known parameters of the real abatement of the cross section, which is determined numerically under the combined action of all efforts using the EN 1993-1-3-2004 in combination with the "Cross-section" algorithm. To reduce the volume of calculations, there is proposed a reverse course of solving the stability problem in dimensionless parameters. The section reduction and the actual forces are numerically determined according to the limiting state of the rod in the most loaded initial section at given force parameters (relative eccentricities of the longitudinal force, taking into account displacements). Then, depending on the flexibility of the rod, the corresponding loading of the rod on its supports is specified by the reverse analytical solution.

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