Leschenko A.P. The fundamental structural mechanics of elastic systems:
Stability

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Introduction and a brief historical review
Conventional symbols used in the book
Chapter I.	Description of the author's invention
	1.1.	Calculation method to control catastrophic destruction
	1.2.	Description of invention
		1.2.1.	Object – phenomenon
		1.2.2.	Introduction
		1.2.3.	Justification of invention
		1.2.4.	Formula of invention
	1.3.	Description of invention
		1.3.1.	Object – law
		1.3.2.	Introduction
		1.3.3.	Justification of invention
		1.3.4.	Formula of invention
	1.4.	Description of invention
		1.4.1.	Object – law
		1.4.2.	Introduction
		1.4.3.	Justification of invention
		1.4.4.	Formula of invention
Chapter II.	The method of analogy in stability of thin walled constructions (a general linear theory of stability)
	2.1.	Setting of the problem on stability of eccentrically compressed bar and ways of its solution
	2.2.	Loss of stability "in small" and "in big"
	2.3.	Possible forms of stability loss "in big". Solution of differential equation of the form
	2.4.	Classification of load combination in stability. A concept of analogy
	2.5.	The theorem on analogy in stability
	2.6.	The analogy method in calculations on stability of centrally and eccentrically compressed thin walled bars. Calculation of the examples was performed using a computer
	2.7.	Experimental grounding of the analogy method and correction provided by the experiment for the calculations on normative technique
	2.8.	The analogy method in calculations on stability of the beam loaded with a concentrated load in the middle of the span
	2.9.	The analogy method in calculations on stability of the beam loaded with the load evenly distributed by the length
	2.10.	The analogy method in calculations on stability of the beam loaded with concentrated moments at bearings
	2.11.	Method of analogy in calculations on stability of thin plates and gentle cylindrical shells
Chapter III.	Application of the method of analogy in calculation on stability of construction elements of bridges and flying apparatus
	3.1.	Introduction
	3.2.	Solution of the sets of differential equations of stability of the analogy method for thin walled bars with rigidity variable by the length
	3.3.	The qualitative method of solution of some equations on stability of bars
	3.4.	General informations about frame/beam bridges and description of their constructions
	3.5.	Calculation on stability of the span structure of the frame/beam bridge
	3.6.	The computer programs at calculation of the bridge's girder on stability
	3.7.	The forces effecting for carrier rocket in flight
	3.8.	The forces effecting an aircraft in flight
	3.9.	On selection of the design model of construction elements of flying apparatus at calculation on stability
	3.10.	The calculation on stability of a carrier rocket
	3.11.	Calculation on stability of construction elements of the aircraft
Chapter IV.	Corrections provided by new concepts on state of the problems on strength, stability and dynamics of thin-walled constructions
	4.1.	LaGrange-Castiliano's principle in the theory of elasticity
	4.2.	A variational method in construction mechanics
	4.3.	The corrections provided by new concepts to a state of the problem on strength of thin walled constructions
	4.4.	The corrections provided by new concepts to the state of a problem on stability and oscillations of thin walled constructions
Conclusion
References

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The second part of the XX century was characterized by the rapid process of scientific/technical revolution which embraced all the spheres of human activity.

The actual task for theory as well as for practice is the scientific understanding and maximum application of scientific/technical achievements born by scientific/technical revolution.

Development of the new branches of engineering as cosmonautics, rocket construction, the study of hydrosphere depth sets the more complicated tasks on calculation of constructions on stability and oscillations before engineers and designers. Accidents and catastrophes of different constructions that still take place in practice testify that the existing methods of calculation of constructions do not take into account all the variety of factors effecting their durability and stability.

In many researches of late the cases of discrepancy between the existing methods of calculation of constructions on stability and practice of designing and experiments were indicated.

The investigators were faced with a need of immediate search of new ways of solving this problem caused by these discrepancies.

To exclude these discrepancies from practice, to explain all the paradoxes in the field of elastic systems' stability, to create a complete theory of constructions' stability – is the main purpose of investigations.

The author has involved to some extent in solving these problems for thin walled constructions.

Over 200 years ago Leonard Eiler for the first time considered the problem on buckling of prism bar.

These results have not been used for about one hundred years and only in the second part of the XIX century Eiler's research attracted attention.

During this period which was characterized by a rapid growth of industry and construction engineering, the engineering constructions of primary importance which had been built without strict theoretical basis and calculations have appeared. It was the reason of many accidents and catastrophies. Desriptions of some of them already appeared at the end of the XIX century on the pages of technical journals.

V.L.Kirpichev in his book "The course on strength of materials" gives interesting informations that 251 bridges had collapsed in the period from 1875 to 1888 in the USA. Bridges are the most important engineering constructions. All this required development of science of construction mechanics and in particular the part of construction's stability. When accidents of the bridges had been investigated rich material was accumulated which have impulses to development of engineering ideas and construction mechanics.

Thus, after the catastrophy of the bridge across the Kevda river (1875) at Syzran – Morshansk railway F.S.Yassinsky developed the calculation technique of the trusses' compressed belts.

As the result of the analysis of the Tei Bridge wreckage (1879) an effort of wind load at calculation on stability against turning over was taken into account.

After the Quebeck bridge across St.Laurentis river (Canada, 1907) has collapsed, stability of the compressed elements of composed section was considered.

The absence of calculation technique for stability of compressed elements is also the main reason of the bridge wreck across the river Birs at the village of Menhenstein (Switzerland, 1891), the author of which was famous engineer Eifel.

The fields of developing technique as shipbuilding, machine building etc also required strict theoretical grounding. The fact that a field of stability of thin walled constructions and plates had not been studied caused a wreck of the ship "Machchi" at the Spain coast in the seventies of the last century. The elements of the body's covering lost stability at a comparatively small wave and the ship has broken in two. It all happened at the edge of XIX and XX centuries and it pushed forward new problems and profound studies in the field of stability.

Thus, S.P.Timoshenko in 1905 first set the problem on stability of a flat form of i-beam bar.

Russian scientists I.G.Bubnov, S.P.Timoshenko, B.G.Galerkin, P.F.Papkovich, A.N.Krylov, L.S.Leibenson, V.Z.Vlassov and others played the great part in development of theory and calculation technique of constructions.

I.G.Bubnov in his capital paper "Construction mechanics of a ship" solved many particular problems and for the first time completely solved one of the most difficult task – the problem of a plate buckling loaded evenly with distributed load.

In B.G.Galerkin's book "Elastic plates" (1933) the method of a differential equation integration of a plate buckling was proposed.

P.F.Papkovich, the Soviet engineer and the scientist brought invaluable contribution to research in the field of stability of constructions. His eleven theorems on stability of elastic systems are fundamental in this field. Brilliantly combining engineering intuition and analysis of a scientist he stepped far forward. The vast monograph by P.F.Papkovich on building mechanics of a ship is the text book for scientists and engineers.

A merit of the Soviet scientist V.Z.Vlassov should be noted in particular. In 1935 he formulated the law of section areas which he proposed as a basis of practical method of calculation of ribbed arches-shells, general theory of strength and stability of thin walled bars of an open profile.

The author of the present book had to encounter not once with practical aspects of the construction operation.

Thus, analysing the catastrophy at Savinsk concrete plant in Archangelsk region (1970) I established that the designers accepted wrong concepts at calculation of pressed bent elements of columns of raw materials storage. Designed according to generally accepted technique the columns did not endure overloads from the cover of the workshop and became unstable which resulted in the storage collapse. The State suffered great loss. This experiment set by the life made the author to critically observe some aspects of the theory of bars' stability. The analysis of the catastrophy may be seen in Fig. 2.1.

Conclusion: since as at e –> 0, N_pred == N_э, then introducing off-center e > 0 into calculation scheme the conditions of the construction operation get worse and, consequently it should be N_pred < N_э. There is an error admitted in the Inquiry Book [14, p.246, R. 17.7] as N_pred > N_э.

After the catastrophy the practical question arises: what expenses, time and force will be required to restore the workshop? Simultaneously, an idea occurred: what forces are needed to be applied and what operation they will perform in order to straighten the columns' elements that lost their stability and put them back into the previous position? It appeared that to "uncoil" the columns' twisted elements to the previous position it is necessary to apply a load of a diametrical compression (draft) which could cause a torsion form of stability loss. When the problem was set up the area of elastic and plastic deformations was not limited, but the problem on a load character capable to put the elements into the previous position was set up. In the considered case a load character was of interest but not a fact if this load was real which could take place in practice or unreal, fictitious load. According to loads' classification by P.F.Papkovich, the load of a diametrical compression is a specific one and it corresponds totally to a torsion form of stability loss. Thus, the problem occurred on stability of a diametrically compressed bar.

The further analysis demonstrated that this is a purely bifurcation problem and at certain value, the forces of cross compression may cause purely a torsion form of stability loss the same as according to Eiler, the force of central compression may cause a longitudinal curve of a bar.

Revealing of the new type of a load and the search of the ways of its realization in stability problems made the author to turn to the theorems by P.F.Papkovich, to his classification of loads.

In development of P.F.Papkovich's ideas and in relation the introduction of new types of load into consideration a more general classification of loads combinations in stability is given in the second chapter of the present book, the concept "analogy" is introduced and a new theorem and a corollary to it are formulated and proved.

For a consistent and argumented realization of new concepts it was necessary to carry out the profound analysis of the existing methods of calculation of stability in practice. In the previous book such analysis for thin walled bars is given, contradictions have been revealed in V.Z.Vlassov's theory, the new hypothesis on division of a torsion angle has been proposed and experimental basis of this hypothesis in a theory of stability of thin walled bars has been given.

In order to ground new concepts in the theory of stability a series of experiments on testing eccentrically compressed thin walled bars had been conducted in the laboratories of Leningrad Engineering/Construction Institute which produced positive results and allowed to develop calculation technique of thin walled constructions based on analogy in stability- an analogy method.

Taking part in construction of Kalinin Nuclear Power Plant, the author introduced the analogy method into the practice of construction. Thus, bearing elements of the columns of machinery hall of NPP and the Club of construction workers in t. Udomla, Kalinin region had been recalculated according to the new technique. It produced economical effect of 10 thousand roubles.

A further development of the analogy method allowed to the author to elaborate an oscillation theory and flutter of thin walled constructions.

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Prof. Alexander Petrovich Leschenko was born in 1939. He has got PhD degree in Civil Engineering.

Prof. Leschenko has 3 Certificates on his discoveries in the field of Structural Mechanics:

1) Certificate DO N 000008 on discovery Principle of pairing of force factors;

2) Certificate DO N 000006 on discovery Phenomena of separating torsional strains of elastic bars;

3) Certificate DO N 000007 on discovery Specific analogy law in stability and oscillation of an elastic system;

and 2 patents on his inventions:

1) Patent N 2150098 of 27.05.2000 on invention Testing method for buckling failure of metal constructions;

2) Patent N 542435 of 21.09.1978 on invention Breakdown controller of pile driver.

His current research concerns various aspects of Civil Engeneering and Structural Mechanics.

The publication list comprises the following books:

– Structural mechanics of thin-walled structures (in Russian), Moscow, Stroyizdat, 1989;

– New principles in structural mechanics of thin-walled structures (in Russian), Moscow, Stroyizdat, 1995;

– Fundamental structural mechanics of elastic systems (in Russian), Taganrog, Sphinx, 2003.