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1 Emeritus Reader, Civil and Environmental Engineering Department, University College London, etc. SOME SIMPLE THOUGHTS ON BUCKLING T. Barta 1 This paper is dedicated to: - my beloved wife, with the same feelings as from old, with grateful thanks for her patience, support and loving care. - to the memory of my mentor and friend C. F. Kollbrunner, from whom I learned so much (in all ways of life) and who died too soon. ABSTRACT The author’s previous paper [1] is completely rewritten. Great emphasis is put on the historical development of structural engineering, and the life of some key-figures. We follow Kollbrunner who wrote (quoting from memory) “-learn from the past -be creative in the present -think of the future” The paper shows the important achievements at UCL under Prof. H. Chilver’s inspired guidance. The paper has also a strong Romanian flavor and connections are also made to Timisoara. E.g. it is shown that D. Cantemir was the very first Romanian involved in column problems, the importance of the construction of the Vauban–type fortress in Timisoara and its military engineers, etc. Key Words: Columns, Buckling, Historical developments and dominant personalities in structural engineering 1. INTRODUCTION As it is obvious from this paper, this is a generalized and extended version of an earlier paper [1]. We will repeat in this paper the original derivation of the proposed formula, as well as the necessary historical references which are updated by using new sources (ref. [2] to [7]) The title of [1] and of this paper is a deliberate paraphrase of a paper by his friend John Roorda, “Some simple thoughts on the Southwell-plot”. * The paper contains biographical data on the important personalities connected with our subject, and also some (perhaps extended) historical references. We mention the now well

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by El Naschie's late thesis supervisor, Thomas Barta

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Page 1: Some simple thoughts on buckling

1 Emeritus Reader, Civil and Environmental Engineering Department, University College London, etc.

SOME SIMPLE THOUGHTS ON BUCKLING

T. Barta1

This paper is dedicated to: - my beloved wife, with the same feelings as from old, with grateful thanks for her patience, support and loving care. - to the memory of my mentor and friend C. F. Kollbrunner, from whom I learned so much (in all ways of life) and who died too soon.

ABSTRACT

The author’s previous paper [1] is completely rewritten. Great emphasis is put on the

historical development of structural engineering, and the life of some key-figures. We follow Kollbrunner who wrote (quoting from memory)

“-learn from the past -be creative in the present

-think of the future” The paper shows the important achievements at UCL under Prof. H. Chilver’s

inspired guidance. The paper has also a strong Romanian flavor and connections are also made to Timisoara. E.g. it is shown that D. Cantemir was the very first Romanian involved in column problems, the importance of the construction of the Vauban–type fortress in Timisoara and its military engineers, etc. Key Words: Columns, Buckling, Historical developments and dominant personalities in structural engineering

1. INTRODUCTION

As it is obvious from this paper, this is a generalized and extended version of an earlier paper [1]. We will repeat in this paper the original derivation of the proposed formula, as well as the necessary historical references which are updated by using new sources (ref. [2] to [7]) The title of [1] and of this paper is a deliberate paraphrase of a paper by his friend John Roorda, “Some simple thoughts on the Southwell-plot”.

*The paper contains biographical data on the important personalities connected with

our subject, and also some (perhaps extended) historical references. We mention the now well

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known fact that the study of Young’s work [13] would have saved a lot of time and energy to his and following generations.

*As we deal with standards, as well as with mathematical methods, e.g. the

Pythagorean triangle, stereotomy, etc. They have been around since time immemorial. See the comprehensive work of Parvopassu [2] the rather unique work of Benvenuto [3] and the works of Matschoss [4], Straub [5], Salvadori [6] and Mainstone [7].

*At this Colloquium takes place in Timisoara (Romania) we have emphasized the

Romanian connection, (Cantemir, Bolyai). Referring again to the historical development, we will mention two interesting cases,

not mentioned in the historical references: - the gauge of most railways worldwide (1435 mm) is in fact the gauge of the

imperial roman chariot; - the guild of the master-mason was highly secretive, well-respected and well-paid;

they are actually the predecessors of today’s freemasons. The pulpit of the gothic St. Stevens cathedral in Vienna is adorned by the portrait of

its hidden builder, master Pilgram, holding in his hand the coat of arms of his guild, with the freemasons’ symbol. According to legend (the version remembered by my mother), he asked for the devil’s help to finish the second tower, on condition that he will build his wife into the tower. When he tried to cheat the devil, he went to hell and the second tower was never finished. Similar legends are associated with gothic structures all over Europe; e.g. in Romania the legend of master Manole and Curtea de Arges monastery.

*The early work of Heron of Alexandria (about 150 BC to AD250?) is mentioned in

[1]. The genius of the Renaissance, who could be called the first “engineer” in the modern sense, is Leonardo da Vinci (1452-1519). His “Codex Atlanticus” contains his research on column buckling. The pages have ended up in different parts of the world; the one on buckling in Paris. The drawings show real tests on central symmetrical cross-sections, and by some kind of dimensional analysis [6] arrives at formula equivalent to Euler’s erroneous formula in [9].

Leonardo made notes when he tested his theories, investigated if these results were repeatable. He also extended his thoughts to consequences (good or evil) of his research. 2. LEONHARD EULER

Euler (1707-1783) was born in a little village near Basel. In 1720 he entered Basel-university when he got his masters degree at the age of sixteen. Johann Bernoulli (1667-1748) recognizing his talents, gave him additional private tuition. His first non-linear column theory appeared in 1744 (see [1] ref. [15]).

*The Euler Committee of the Swiss Academy was founded in 1907 with the task to

publish Euler’s complete works, his correspondence, his unpublished manuscripts, notebooks and diaries. This huge endeavor could only succeed with the collaboration of internationally famous scientists who acted as editors and commentators. This monumental work [13] has been published for the last 90 years, 75 volumes have already been published, and 6 more volumes are in preparation and should appear in the early years of this century. We shall refer to it briefly as “Op. omn”.

*

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C. Truesdell, editing Euler’s work on mechanics, claims that Euler was not interested in the linearised theory. This statement is obviously infirmed by Euler’s papers [9] and [10]. Euler was interested in the experimental verification of his linear theory [10]. In [11] Series Quarta D (Commercium epistolarum, vol.1), appears a letter from Euler – then residing in St. Petersburg, where he was a member of the academy to a follow member, the exiled prince of Moldavia Dmitrij Kantemir (in Romanian Dimitrie Cantemir), asking him to intercede with the Russian mechanic of the academy, in order to do some tests to check the applicability of the linearised theory. Unfortunately, this is the last volume of op. own available in UCL-library.

We will consider now Euler’s hyperbola (which is so well known that a figure is not necessary). The hyperbolic approaches two asymptotes, which are open to rather odd interpretations.

a) The “short column” degenerates into an infinitely large infinitely thin sheet. This obviously does not make practical sense. We can imagine that there is a certain load Ns where the hyperbola will be cut by a horizontal line. Ayrton and Perry [12] define this, in 1886, for ductile steel through the onset of yield in the extreme fiber (i.e. a set of incorrectly defined ultimate load). The behavior of this column is obviously stable, and experimental data are available.

b) The “long column” degenerates into an infinitely thin, infinitely long line. Here Euler’s generally highly non-linear theory (ref [15] in [1]) should be applied: the post buckling behavior is unstable and experimental data are available up to a realistic limit.

3. THOMAS YOUNG

Young (1773-1829) [13] had a much clearer understanding of column buckling, stating: “… a permanent alteration of form … limits the strength of materials with regard to practical purposes, almost as much as fracture, since in general the force which is capable of producing this effect, is sufficient, with a small addition, to increase it till fracture takes place”. He notices the different behavior of stocky and slender columns and gives limits for various materials for the two types of behavior. His understanding of what we call today inhomogeneity of material properties and imperfections is amazing, and has its origin in analyzing experimental results; “… considerable irregularities may be observed in all the experiments … and there is no doubt but some of them were occasioned by the difficulty of applying the force precisely at the extremities of the axis, and others by the accidental inequalities of the substances, of which the fibers must often have been in such directions as to constitute originally rather bent than straight columns”. P.H. Smith [1] recognized that: “… the whole question of the strength of struts is one of probability”.

It can be seen that the physical model of flexural buckling was reasonably well established almost 100 years ago (or even longer) but unfortunately not well known or understood. Rayleigh’s remarks about Young as quoted by TIMOSHENKO [20] (1953)[1] that he “… did not succeed in gaining due attention from his contemporaries. Positions which he had already occupied were in more than one instance reconquered by his successors at great expense of intellectual energy” this applies equally well to other 18-th and 19-th century scientists ([1], [28]).

*Given Young’s importance, we consider adequate to give some details about his life.

He was a physician, physicist (natural philosopher), professor of natural philosophy at the Royal Institution and foreign secretary of the Royal Society. He discovered the principle of

Page 4: Some simple thoughts on buckling

interference of light. He introduced Young’s modulus and was among the first who deciphered the Egyptian hieroglyphs on the Rosetta stone in the British museum.

A quaker from a poor family, was supposed to study medicine at the request of a wealthy benefactor. As Cambridge and Oxford admitted only Anglicans, he attended a more tolerant university in Scotland. His real interest lay in mathematics and philosophy and he moved to Göttingen, where Gauss (1777-1855) lectured on mathematics and G.F. Lichtenberg on natural philosophy.

Lichtenberg was also a writer of the enlightenment, and attacked with brilliant satire some pseudo-scientific theories, and the “cult of the genius” as represented by Goethe, Schiller, et al. He was attracted by the English spirit (Swift, Sterne, Fielding) and made several trips to England writing about these trips, and about the painter and cartoonist W. Hogarth. His “aphorisms” were re-discovered and re-published in the 20-th century. Young became his lodger and life-long friend. He returned to England and practiced as a physician.

The situation in English universities changed rapidly after the foundation in the first quarter of the 19-th century of University College London (UCL), after the model of the University of Bonn. UCL did not discriminate (for students and staff) against nationality, religion or sex, and aimed to teach everything, except theology (hence the nickname “the godless people of Gowerstreet”). This signaled the erosion of the old system, and all universities adapted to the industrial revolution and mercantilism. Young gave lectures on physics at UCL; his lecture-notes and description of experiments are kept in the college library. He was a man of high moral principles. (e.g. –he did not eat sugar, a product of slave labor from the Caribbeans). Documents about his life are kept by the Royal Institution. 4. THE PERIOD BETWEEN EULER AND YOUNG

This period sees the advent of the “engineer”. The famous Vauban [5] created under Louis XIV the “Corps des ingénieurs du génie militaire“. He invented also the polygonal and the star-shaped fortifications which were built all over Europe. Other engineering schools followed, “École des Ponts et Chaussées” (1767), “École Polytechnique” (1794). Similar schools in other countries followed.

The engineers needed handbooks Belidor (1697-1761), published his “Science des Ingénieurs” in 1729 and it was re-issued in the 1830. In 1730 appears Lukas Voch`s ``Bruckenbaunkunst`` which dealt also with the engineering problems. The list of these books is too long to follow up; they all were a mixture of empirical data and proper mechanical theories and equations. When Prince Eugene of Savoy (the ``noble knight`` in folklore) general of the empress Maria Theresia liberated Belgrade and Temeswar (today Timisoara) after centuries of turkish rule (1717-18..) the wooden turkish fortress was replaced by a stone fortress after Vauban’s system. Its construction provided great difficulties due to the marshy soil. In 1823 a young engineer-lieutenant Janos Bolyai (1802-1860) [14], graduated of the military academy in Vienna was sent to Temeswar to consolidate the bastions. (We can only speculate on the handbooks he used). In 1832 he wrote an ``Appendix`` to the work of his father Farkas Bolyai (also a famous mathematician) containing a hyperbolic non-Euclidian geometry, which brought him world fame. Identical results have been obtained in 1829, by Lobachewsky. 5. THE PERIOD BETWEEN YOUNG AND KOLLBRUNNER

As the subtitle of Kollbrunner`s comprehensive work [19] suggests, this is a period when theory and practical calculations head to design codes (norms). To get an idea about the dynamic, colorful personality we should consult [20]. We supplement this with our own

Page 5: Some simple thoughts on buckling

experiences; Kollbrunner graduated at the ETH Zurich with a thesis on stability problems of structural steel construction. He researched also hydrotechnic structures, soil- and snow-mechanics etc. He encouraged his employees and colleagues to do independent research; if necessary, he facilited their publication considering this to be a more agreeable and efficient way than state sponsoring. Kollbrunner`s industrial activity was mainly the management of the ``Conrad Zschokke, Stahlbau AG`` in Dottingen (the company is still managed by his sons).

To its innovative character it won many contracts in competition with giants like Krupp. The company published more and better research papers (the familiar red and blue booklets) than its bigger competitors. During world-war II he was the colonel in command of all pioneer troops. He was also a military historian. (We can’t go into too many details). According to [20] he was the “yeast” in various technical societies; e.g. he was a driving force in the founding of the “European Convention of Structural Steel Associations” which evolved into the “European Convention for Structural Steelwork”. There appeared also the “Comité Européen du béton”. The ultimate aim of these organizations is the creation of “Euro-norms”. His belletristic work appears under the pseudonym Tse-Ef-Kah. The author is proceed to have an almost complete collection of numbered little booklets of aphorisms, published as “Privatdruck” for his friends. Last, but not least – some words about Kollbrunner the humanitarian. When, after the end of World War II, his friend and collaborator E. Chwalla, was expelled from Brunn (Brno) where he was a professor – he had to leave his library behind; Kollbrunner presented him with a number of essential books to compensate as far as possible for his loss.

The author had a similar experience when he passed in 1963 through Zurich. He was already in correspondence with Kollbrunner, while still in Romania, ([19] p. 235). When in 1963 in a situation not unlikely to Chwalla, he visited Kollbrunner in his beautiful home near Zurich he was presented with a comprehensive collection of books including [10], etc. He was offered a job in the Zschokke research department; but in a year time; as I could not wait, I moved with my aged mother to Britain (a decision I never regretted) and joined the Civil Engineering Department at UCL; where Henry Chilver, another dynamic and inspiring personality just became head. The Chilver –years were never surpassed. The author stayed in UCL from 1963 for the next 24 years, till his retirement.

To return now to Kollbrunner. On the numerous occasions when the author visited Zurich, he met him. Actually Kollbrunner became his mentor and friend; and many questions professional (e.g. [1] in its original, simpler form) and otherwise, were discussed in a congenial atmosphere.

*In this period there were already various design codes in use. The engineer had only to

choose an appropriate effective length and replace into a buckling equation. The collapse in 1891 of a railway bridge near Basel, due to the buckling of a diagonal of truss, lead to systematic tests by Tetmajer. ([19] p. 13); (also an earlier publication with the same title published in 1896 in Zurich). Tetmajer proposed a straight-line, based on his tests, intersecting Euler`s hyperbola ([19] p. 14).

The advent of the German code ``DIN 4114`` and supplementary approximate formulae and references in the ``Ri.zu DIN 4114`` opened the eyes of engineers and researchers to a new world. DIN4114 and commentary were mainly the work of E. Chwalla, (just as the equivalent Austrian norm). National norms were compared [19] (chapter1 C and D, p. 279-287).

The attempt for a European standard [19] p 292 lead to the formation of commission 8 (buckling) under the leadership of the late Prof. H. Beer. He reported on his investigations in 1960 ([19] p. 295).

Page 6: Some simple thoughts on buckling

Reports on the experimental (Sfintesco) and theoretical (Beer and Schultz) foundations of the buckling curves were published in “Construction Métallique” N03, 1970.

Committee 8, with its large international resources in funds and collaborators, soon realized that it faces numerous difficulties. Thus, the only real “material constant” E was difficult to measure. The column tests performed in different laboratories, in different countries gave erratic results. As it turned out later, one had to consider the stiffness of the system, the support of the specimen, the loading device etc, etc. The Gauss distribution was inadequate for the statistical interpretation of the situation tests. Even worse was the situation for other materials. For concrete, on an international scale the test specimens to determine the materials strength varied in shape, size, methods of comparison and storing. For timber we found the so-called bending strength to the fibers, is actually pure-shear strength to the fibers.

The only reason why, despite this rather chaotic situation, structures did not fail was corroboration with practice. So in a given country the columns designed were OK as long as no catastrophe occurred. Taking Switzerland as an example, we recall the bridge disaster which prompted Tetmajers systematic tests and subsequent changes of the norms (See again [19] p.17).

*At the same time (about 1962) the new head of UCL Civil Engineering Department,

Henry Chilver (now Lord Chilver, FRS) was an inspirational dynamic leader for his staff, and research students. To mention a few, J.M.T. Thompson (later professor of non-linear dynamics; FRS and editor of its publications). The monograph [18] reports successful solutions to some of the problems: (Ref: 3.1; 4.3; 5.2).

The Southwell-plot ([16] p. 23) can be used to determine E. (Oddly, this plot well known in the UK and North-America was not known on the continent). Later ultra-sound and vibration tests were used at UCL.

For the statistical evaluation of a small number of tests the ``Student`` or t distribution was proposed. (Later more refined methods were used).

*Following Chilver’s invitation, Beer joined his department, for a period, as visiting

professor. A rapport was formed between two congenial personalities, and the result of UCL-research influenced the further work of Committee 8. 6. THE COLUMN CURVE

(This chapter should be read in conjunction with [1] Chapter 3.) We have discussed Euler’s hyperbola in Chapter 2. Young [13] considers pin-ended elastic columns with an initial sinusoidal curvature of

amplitude e, and a straight column with a load N applied with an eccentricity e0.We will transcribe his results for these :imperfect columns” into a more modern form so as to express the second-order moment MII by multiplying the first order moment MI=Ne0 (1)

With an amplification factor α, so that: MII=MIα (2)

We will use systematically the “normalisation” technique, i.e. variables varying

between 0 and 1. (This brings considerable advantages in calculations). Normalised variables will be designated with a bar on top. Thus the normalised Euler load

EE N

NN ≡(3)

Page 7: Some simple thoughts on buckling

0 1

1

NE

N0

N0=1

NE=1

N0+NE=0

N0+NE=1

0 1M0

1

N0

N0+M0=1

A numerical calculation by Ayrton and Perry (1886) [12] showed that the initially curved column can be used to define α, thus:

EN−=

11α

(4) Due to (doubtful) desire to curtail all diagrams in the domain of the stocky column, a

lot of (unnecessary) complications have occurred. The original, much simpler form of this paper (discussed with Kollbrunner in 1969/70) shall be presented here.

The source of all evil is the introduction of limit for the stocky column, defined by eq. (10) p.282 in [1]. The simple present theory is obtained by taking systematically: 00 ≡E

N (5)

in all subsequent calculations. A faster approach (which does not help understanding) as suggested in ([1] p.288) to

substitute (7) in the final formula ([1] formula (27-d)). *

With the elimination of some obvious misprints and errors in notations, we obtain the desired simple formulation.

We define

yN

NN =0 (6)

We introduce the use of “interaction diagram” (Fig. 1 and 2).

Fig. 1. Interaction diagrams All experimental results will lie within the upper triangle defined by

10 =+ ENN (7) which is usually attributed to Rankine [13]. This has been proved by Horne and Merchant [16] in the wider contact of frame stability. The (not yet) defined curve will be the desired column curves.

Chilver and his student Britvec [17] have explained the large scatter of experimental results in the region of “highest imperfection sensitivity” (the dotted diagonal in Fig. 1).

We define now a second interaction diagram (Fig. 2).

Fig. 2. N0-M0 interaction diagram

Page 8: Some simple thoughts on buckling

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 ?

0.2

0.4

0.6

0.8

1.0

Highest imperfection sensitivity (?=1)

N0

Euler

Lower bound

Upper bound

Young’s approach will be used to find an expression for the experimental curve in Fig. 1.

On substituting (2) into (9) we obtain with (8), after some algebraic manipulations

( )( ) 011 00 =−−− NNN E η (8) With the (nondimensional) generalized imperfection

00

0

/ NMe=η (9)

Equations (8) and (9) represent the modern version of Young’s formulae (in view of the historical account in Chapter 3)

But our aim is to obtain a nondimensional “buckling curve” through systematic substitution of (7) we arrive finally at a second-degree equation, with the solution (best for numerical calculations):

222 )1(4)1()1(

21λλλ c−−+++

=(10)

(where c appears as a tracer to define the upper and lower bounds) The desired column curve is shown in fig. 3.

Fig. 3. Upper and lower bounds of buckling curve The Rankine formula and the line of maximum imperfection sensitivity will appear as

curves in Fig. 3. The three curves purposed by Beer and Schulz, could be obtained from our column

curve, upon the proper specialisation.

a) b) Fig. 4. Buckling curves

Fig. 4-a shows a buckling curve similar to Fig. 4-b in [1], which creacomplications. Fig. 4-b represents another aberration.

0

N0

λ

N0

λ

(λ=1)

ted all the

Page 9: Some simple thoughts on buckling

When Prof. K.W. Johansen (of yield line theory fame) gave a lecture at UCL, we discussed, over lunch, the problems we investigated at the time. We showed figures 4-a and 4-b. Laughingly, he said “obviously the dromedary and the camel.” Explaining further, he said What is a dromedary respectively a camel? A horse designed by a committee. 7. OTHER APPLICATIONS

7.1. Other materials. As our buckling curve (Fig. 3) is material-independent, it is obvious that it can and is used for other materials then steel. E.g. reinforced concrete, concrete filled steel tubes, etc. 7.2. Frames. This has first been shown by Horne and Merchant in [16]. Most other approximative methods are based on their work. In [21] numerous examples are given. Work by the author and his friend and colleague J. Appeltauer, (together and separately) seem to be the first studies on stability problems (since 1969) in Timisoara. See also [19] p. 235. Prof. V. Valcovici published in the 30-ties of the last century, a solution for the column in an elastic medium; (to be applied to drills in the oil industry). Unfortunately, the author quotes from memory as he cannot get hold of the reference. 7.3. Shells. In [22] the author a shell theory in a rather theoretical form. Due to lack of space the paper was radically shortened. The author has the intention for an amplified version of this paper, in engineering terms which will include also buckling. ACKNOWLEDGEMENTS:

- My grateful thanks to Dr. R. Mc-Keran and Dr. Pumphrey; without them, this paper would have never been written;

- Grateful thanks to my friend Dr. Max Herzog (Solothurn) for his invaluable help in providing references.

8. REFERENCES

[1] Barta, T. A., Some simple thoughts on column buckling, International Congress on Column Strength, Paris, 1972, published in IVBH, Berichte der Arbeitskommissionen, Bd. 23, Zürich, 1975. [2] Parvopassu, C., Visione storica della scienza e della tecnica delle construzioni, Politecnico di Milano, Rendiconti e publicazioni del corso di perfezionamento per le costruzione in cemento armato, Fondazione fratelli Pesenti, Vol. V, 1953. [3] Benvenuto, E., La scienza delle Costruzioni e il suo sviluppo storico, Sansone Editore, Firenze, 1981. [4] Matschoss, C., Great Engineers, (translation from German), G. Bell and Sons Ltd., pp. 48-60, London, 1952. [5] Straub, H., A History of Civil Engineering, English translation by E. Rockwell, Leonard Hill Ltd., London, 1939.

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[6] Salvadori, M., Why Buildings Stand Up, W. N. Norton and Co., New York – London, 1980. [7] Mainstone, R. J., Developments in Structural Forms, Allen Lane / Penguin Books, 1975. [8] Leonardo da Vinci (1452 – 1519), Codex Atlanticus, Inst. France Ms. A.f.45v [9] Euler, Leonhard, Sur la force des colonnes, Mem. Acad. Berlin, Vol. 13, pp. 252-282, Berlin, 1759, presented in 1757. [10] Euler, Leonhard, Determinatio onerum quae columnae gestare valent, Acta Acad. Sci. Petrop., Vol. 2 pp. 163-193, 1780. [11] Leonhardi Euleri Opera Omnia, Birkhauser, Basel (referred briefly as: Euleri Op. [12] Ayrton, R. E., and Perry, D., On struts. The Engineer, 1880, pp. 464-465, and pp. 573-615. [13] Young, Thomas, A course of lecturer on natural philosophy and the mechanical arts, (2 volumes) London,, J.Johnson,, 1807. [14] Jancso, A., Temeswar u seine Brucken. . Mirton Verlag, Timisoara, p.34ff [15] Rankine, W. J. M., Useful Rules and Tables, London, 1866, as quoted by Horne and Merchant in [16] [16] Horne, M. R., and Merchant, W., The Stability of Frames, Pergamon Press, Oxford, 1963. [17] Chilver, A. H., and Britvec, S. J., The Plastic Buckling of Aluminum Columns, Proc. Symp. on aluminum in structural engineering, Aluminum Federation, London, 1963. [18] Croll, J. G. A., and Walker, A. C., Elements of Structural Stability, Macmillan, London, 1972. [19] Kollbrunner, C. F., and Meister, M., Knicken, Biegedrillknicken, Kippen (Theorie u. Berechnung von Knickstaben, Knickvorschriften, 2-d augmented edition, Springer, Berlin, 1961. [20] Dubas, Pierre, Curt F. Kollbrunner (1907-1983) Nekrolog, Schweizer Ingenieur u. Architekt, 51/52, 1983. [21] Kollar, L, (editor), Structural Stability in Engineering Practice [earlier book published in Hungarian in 1991], Chapter 5. Buckling of frames; 5.3. General theory by J., Appeltauer; 5.2. Approximative methods by L., Koller. [22] Roorda, J., and Srivastava N. K. (editors), Trends in Structural Mechanics, Kluver Academic Publishers, Dordrecht 1997. Symposium in honour of Prof. A. N. Sherbourne. 1. Solid mechanics: T., Barta: Cosserat continuum and shell theory, pp. 1-14.