North-Holland Publishing Cy. A sysematic simplification of the general equations in the linear theory of thin shells. Introduction to the post-buckling behaviour of flat plates. Elastic stability and post-buckling behaviour. The concept of stability of equilibrium for continuous bodies. The effect of axisymmetric imperfections on the buckling of cylindrical shells under axial compression.
Koiter's Elastic Stability of Solids and Structures. Dealing with the elastic stability of solids and structures, this book is concerned not only with buckling, or linear instability, but most importantly with nonlinear post-buckling behavior and imperfection-sensitivity. Koiter s Elastic Stability of Solids and Structures. Chapter 1: Stability.
Chapter 2: Continuous Elastic Systems. Chapter 3: Applications. Simple Two-Bar Frame. Selected Publications of W. According to Hall , "As a student of statics and hydrostatics, even in their most practical aspects, he is heavily dependent upon earlier workers and borrows especially from al-Biruni and al-Asfizari; but his competence is not to be denied, and Kitab mizan al-hikma is of outstanding importance to the historian of mechanics, whatever its claims of originality or comprehensiveness may prove to be".
The Science of Weights was no longer seen as a respectable field for serious scholars in Iran after al-Khazini's times, and it became more a concern of those fabricating or using balances. We may conclude that the Islamic science of weight concerning stability was predominantly based on Greek thought and principles, but tackled new problems in a more systematic way than what had been done by the Greek philosophers.
This allowed the development of new instruments to achieve greater precision based on traditional stability concepts. Finally, the Arab science was also enriched by knowledge brought from India thanks to expeditions made around the year , in which an Arab scientist would spend several years in India to learn and teach discoveries in the two parts of the world.
The discipline known as " Scientia de Ponderibus ", or the Medieval Science of Weights [5, 35], developed first in the Islamic countries and then was learned in Europe from translations. The propositions of diferent medieval authors attempted to correctly predict the relations between weights and distances from the center of suspension of a balance arm or fulcrum. One of the main concerns was to predict the consequences of slight modifications in a balance. The descriptions of those problems may be easily correlated with our modern concepts of stability: For example, Moody and Clagett  discuss the original medieval texts in terms of stable and unstable equilibrium.
Similar discussions are used nowadays to introduce topics of structural stability for engineers, see for example Croll and Walker . Mechanics research in the European Middle Ages was carried out by university professors who were concerned with establishing equilibrium conditions and movements of bodies. The professors were not limited to study the most obvious features of the Science of Weights, but were also interested in importing concepts from Physics to this field.
Other non-academic scholars not linked with universities were also interested in the topic; they were geometers who used simpler methodologies in their work. The main scholar in Mechanics during the Middle Ages in Europe was perhaps Jordanus Nemorarius or Jordanus de Nemore , who was a well respected scholar in his own times, was forgotten for centuries, and was re-discovered towards the end of the XIX Century .
Nothing is known about the personal life of Jordanus, except that he lived and worked between the end of the XII Century and the beginning of the XIII Century; dating his birth and death has not been possible. According to Sarton , he was a German mathematician and physicist who joined the Dominican Order in Paris by ; however, this is based on the assumption that Jordanus Nemorarius was the same person as Jordanus Saxo, a claim that has been denied by many historians of science.
Other historians [11, 19] state that he was born in France, and may have taught at the University of Tolouse an institution that was created in Based on his writings, it seems that Jordanus was a gifted mathematician and physicist, who used knowledge from the Greek masters and went beyond them . It has been argued that it is discult to distinguish the writings of Jordanus from those of his disciples and commentators. The problem of authorship in the Middle Ages is complex because manuscripts were copied by hand and often the copyists altered the texts and introduced modifications according to their own believes.
Again, much controversy exists among historians who attempt to emphasize a continuity between medieval and renaissance mechanics  and those who understand that there was a rupture in knowledge during the Renaissance .
Theory of Elastic Stability
It is now acknowledged that the book " Elementa Jordanis super demonstrationem ponderum ", containing seven postulates and nine theorems with demonstrations, was written by Jordanus Nemorarius. Following the Archimedean tradition, Jordanus presented the laws of the lever, considering both the weights and the distances to the fulcrum as variables of the problem.
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He employed geometric demonstrations, in which the main concern was the localization of the center of gravity. According to Moody and Clagett , "Most medieval definitions are of a topological nature, i. Thus they are of the form of proportions. The book was translated into English by Moody and Clagett , and includes 45 theorems and demonstrations. Some of those theorems had already been included in Elementa , whereas others were corrections of previous versions that had some mistakes. The third part of this text contains six theorems about Statics and is of relevance for our historical reconstruction because Jordanus included a correct version of the conditions to reach stable equilibrium.
To understand the ideas of stability in Jordanus it is necessary to consider the concepts of decomposition of forces and of virtual displacements. Jordanus employs the concept " gravitas secundum situm ", known as the principle of positional gravity, which means "the action of gravity depending on the position".
Positional gravity was included in Postulates 4 and 5 in Elementa : A weight has larger positional gravity whenever the path that it follows is less oblique. This concept is important to work with force components and is an original contribution that was not present in Greek or Islamic authors. Next, consider the concept that we now call virtual displacement.
Jordanus states that a body in equilibrium cannot have any real movement, so that to evaluate positional gravity it is necessary to open the possibility to introduce "virtual displacements. For example, "What susces to lift a weight w through a vertical distance h will susce to lift a weight kw through a vertical distance h-k and it will susce to lift a weight w-k through the vertical distance kh. The two ideas were put together to work and decide about the stability of given positions of a balance.
Jordanus investigated the stability at a state by evaluating the theoretical response i. On the whole, the general concepts of stability are correctly applied. Jordanus correctly identifies an equilibrium position and imposes virtual displacements to understand how the balance behaves. To illustrate the use of stability concepts, we may refer to Theorem R1. But if unequal weights are suspended, the balance will fall on the side of the heavier weight until it reaches the vertical position.
Jordanus introduced a mistake in this case when he assumes that equal weights acting with equal arms are in stable equilibrium. In fact, this would be a case of indiferent rather than stable equilibrium. The introduction of potential gravity in the stability problem may be observed in this statement:. According to Jordanus, to achieve stability, a small displacement from the equilibrium position should result in an increase in potential gravity of the weight that was raised by this displacement, in such way that the beam would return to its original equilibrium position.
For the demonstration of this theorem, refer to Figure 3. Theorem R1. Then, "if the ends are at the same distance from the vertical line that passes through the support, equal weights suspended from the ends will have the same heaviness". Our Figure 3. From the point of view of positional gravity, what matters is the obliquity of the displacement along the arc. To understand its stability, the balance is displaced a small angle but because the weights have diferent radius of rotation, then weight A has larger vertical and horizontal displacements than B.
An out-of-balance moment occurs which tends to restore the balance to its original position. The knowledge reported by Jordanus was the basis of the statics taught at medieval universities and received in the Renaissance through Descartes and Torricelli. The great influence of Jordanus is evident from the inclusion of his theorems into Renaissance texts and the spread of his ideas often without giving any acknowledgement to his work.
Some stability topics reported by Jordanus were heavily criticized by Guido Ubaldo, Marquis del Monte , so that it is clear that Jordanus was still important in the XVI Century. Specifically, Ubaldo notices that Jordanus is not clear about an indiferent equilibrium in the balance of equal arms .
An assessment of the contributions of Jordanus attempts to explain why he did not produce a tradition of work in the Middle Ages: "During the Middle Ages the scientific phase of the Greek heritage was maintained, but it was maintained at a low level of social interest.
- Buckling of Structures – Readings.
- W. T. Koiter's Elastic Stability of Solids and Structures: Stability.
- Structural stability concepts in medieval and renaissance mechanics.
- W T Koiter's Elastic Stability Of Solids and Structures.
A writer like Jordanus, who possessed a great deal of originality, managed to incorporate, along with erroneous ideas inevitable in the atmosphere of his time, several original demonstrations which were in advance of the achievements of the Greeks in that field. This work failed to germinate and bear fruit in the Middle Ages because not enough persons were interested in that line of investigation Ginzburg, , pp. Stability was an established concept in the texts by Jordanus and was part of the basic knowledge inherited by the Renaissance scholars.
The introduction of " gravitas secundum positio allowed having a criterion regarding what a hypothetical displacement would do to the system, which is what we now call stability criterion. Finally, the discussions only addressed problems relating rigid links and masses, whereas the idea of a strange behavior of columns was not present during medieval times.
Bernal writes that "The professions of artist, architect, and engineer were not separated in the Renaissance. The artist might be called on by his town or prince, or might ofer himself, to cast a statue, build a cathedral, drain a swamp, or besiege a town. The master craftsman always had to know the properties of materials and the means of handling them.
The artist of the Renaissance had to know all that and much more: he had to instill into his work geometry and mechanics in the conscious imitation of antiquity. It was in this field that Leonardo da Vinci. Leonardo was valued as an artist both in his own times and during the following centuries; however, it seems that his own interests were more closely related with engineering, mechanics, physics, anatomy and mathematics than with art. He was a self-educated person in those topics, and recorded his readings and thoughts in unpublished notebooks.
What Leonardo read about mechanics has been a topic of research and discussion among science historians [11, 12, 21, 22]. It is now acknowledged that Leonardo was familiar with the works of Archimedes on equilibrium of balances and levers, and that he also read Jordanus Nemorarius on the Medieval Science of Weights. His readings were oriented at understanding the topics in order to do something with them, such as applying the ideas to solve practical problems. More than one hundred years after his death, Leonardo's writings were re-organized according to topics of interest, such as painting, architecture, elements of machines, and anatomy.
As said before, Leonardo was mainly concerned with what he could do with knowledge to solve problems, rather than with the prestige that authoring books could give him. After all, it was his ability to understand and solve problems that gave him work opportunities in the territories that are now Italy and France.
Further, Leonardo lived during a transition in written communication with the newly invented printing press playing an important role. This was a changing environment with evolving forms of communication and information technologies. As said before, Leonardo did not have a formal education nor did he learn Latin at school which was the language for technical writing in his own times , so that part of the knowledge that he acquired was from conversations with travelers and colleagues, and partly from his own speculations and sometimes from observations and experiments that he thought or conducted.
In his writings, he remarks the need to repeat experiments to be certain about the outcome. Whenever he conducted experiments, he made a series of them in which some parameter was changed in a gradual form to understand its influence. His aim was to derive quantitative rules from experiments; and if a new experiment contradicted a rule, then perhaps it was time to abandon the old rule. This is important in the present context, because he would attempt to derive rules for the lateral displacements of columns. It is possible that he wrote in an encrypted form to protect his writings; but because his notebooks were not prepared to be published it is more discult for us to understand his ideas on mechanics and statics.
They are presented in the form of short statements derived from observations, and they sometimes contradict other statements on the same subject written a few pages before. The notebooks that survived were mutilated by their temporary owners who attempted to give them a new organization, with the consequence that writings about the same topics are scattered in various notebooks. His annotations on mechanics deal with movement, weight, force and percussion.
There are several references about the strength of columns under compression, observing the lateral deflection of such columns. Those are his private thoughts and speculations on what we now call buckling problems. Specifically, there are annotations on these topics in Codex Atlantis with texts written between and and in Codex of the Institute of France , which includes texts written between and There are also studies on mechanics in Codex Madrid I , possibly written between and , which were re-discovered in .
The discovery of Codex Madrid I was made after Truesdell  and Hart  wrote on the topic and much later than the writings of Duhem . A dominant form of expression in the notebooks is the drawing, whereas the text is inserted to explain information that was shown in the graphics. Those are carefully drawn figures with text filling the spaces between them.
Some drawings show schematics of deflections of columns. Projects are developed to measure quantities from experiments, but there is no evidence that he actually performed such experiments. According to Truesdell , whenever Leonardo writes about experiments he does that in future tense, showing that those were plans of things to be done to find something.
On the other hand, it seems clear that he was constantly observing from nature as a source of evidence provided by the physical world, and it was from observations that he could draw conclusions and write tentative laws. This was the case for columns under compression. Codex Madrid I is devoted to mechanics and statics, partly on fundamentals and in part on applications. At various places there are entries on pillars supporting weight in the axial direction; his annotations in this Codex are of a qualitative nature rules on pillars are given in Codex Institute of France .
In page of Codex Madrid I  we read:. The influence of column length is considered in page , showing that Leonardo was aware that length has a negative influence on the column capacity to bear loads:. This is understood in cases of frame columns made of timber, columns, pillars and things like that and the rule may be checked as we did in the case of a cord, because one could imagine a pillar so high that its own weight would reduce its base to thin powder. The same thing would happen if it were loaded by superimposed excessive weight".
Here Leonardo argues using an analogy between the pillar and a cord, probably because at that time he had no opportunity to develop a solid argument for the pillar. The most important part is found at the reverse of page , with reference to Figure 4 :. And whatever the position in which the center of weight is placed on the support, this will always bend in such a way that at mid-height it will deflect towards the inside. Any part of the support placed below the center of weight will be subjected to this torsion, except for the center of support. Whenever this [center of support] is placed below the center of weight, it will not be possible that the support bends to anywhere.
For any other place in which the centers are located, they will make that the support will twist or bend to the opposite side, as shown here below". We may infer from these annotations that Leonardo qualitatively understood the phenomenon of lateral bending due to a vertical load acting on a column. Finally, the reverse of page contains writings on the strength of pillars:. There are many entries in the notebooks in which Leonardo drew columns supported at the base under an axial load. We find especially interesting those annotations in which he goes beyond observations and advances rules that he had found.
Figure 5 , from Codex Atlanticus , shows two compressed columns of the same cross section and diferent lengths, indicating the relative loads that each can take. For the short column with unit length a load of is indicated, whereas a load of is written in connection with the long column with length equal to According to this, one sees an inverse relation between column length and strength. In Codex Institute of France , Leonardo writes that if several pillars are tied so that they resist together, the strength of the group will be larger than one single equivalent pillar:.
Of 1, such rushes of the same thickness and length which are separated from one another, each one will bend if you stick it upright and load it with a common weight. And if you bind them together with cords so that they touch each other, they will be able to carry a weight such that each single rush is in the position of supporting twelve times more weight than formerly" . In the same page, he considers the relative load capacity of two bars with equal length but diferent cross section:.
Hart  considers that the proof is not convincing. According to Leonardo, the load capacity is directly proportional to the area of the cross section and inversely proportional to the ratio between height and thickness. Finally, a comparison is made between the loads in pillars of the same cross section and diferent lengths, to conclude that they are inversely proportional to their lengths.
Truesdell translates Leonardo's estimates on vertical pillars from Codex Institute of France A as follows:.
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Truesdell has identified two rules on the strength of pillars loaded by P :. Truesdell acknowledges the disculties in interpreting those rules, on account that Leonardo did not provide definitions and did not specify what he kept constant in each case and what he changed. In summary, we observe that the annotations in Codex Madrid I are of a qualitative nature and help to assure that Leonardo had clear ideas about the behavior of compressed pillars. He made significant contributions to understanding the conditions under which columns develop bending.
From the writings of Codex Institute of France , Truesdell  has ventured that Leonardo understood that the strength of pillars was a function of the cube of the thickness and the inverse of the length. There are at least six areas of disculties to fully understand the contributions of the notebooks to our field. The impact of his ideas is also discult to assess.
On any given topic we have no direct means of assessing what was his original idea and what was a recollection of the work of others with whom he was in contact or whom he read. In the first case, due to the private nature of the notebooks, the impact would have been negligible on Renaissance scholars and after, but in the second case we would be referring to public domain knowledge that should show in the writings of the XVI and XVII Centuries. Thus, an indirect means of assessing the private-public nature of his knowledge is with reference of the work of others in the years to follow.
This is the topic of the next section. Leonardo left extremely interesting short notes in his voluminous notebooks on the bending of clamped columns under compression. He also produced quantitative information on the maximum load that the column can resist. Had this been part of an open chain of public literature, perhaps others would have taken his observations and refine them; however, the manuscripts were never in the hands of scientifically oriented persons and were only available when they were of historical rather than scientific interest.
One hundred years after the death of Leonardo, the load that could be taken by a column was considered by Frere Marini Mersenne , a priest of the order of Minimis in France . Mersenne shown in Figure 7 was better known for his role in establishing what became known as the Republic of Letters for savants, a system of communication between scientists of diferent parts of Europe. Thus, Mersenne may be considered as an extremely well informed scholar in his own time, thanks to his contacts with other French and European researchers.
If any new contribution was made to the scholarly field of mechanics, he would be one of the first to know. There are two places to read Mersenne about the strength of pillars: his translation of the work of Galileo  and his own treatise of mechanics . In the first case, it is not easy to recognize the work of Galileo in the translation by Mersenne: Galileo presented his Dialogues on Two New Sciences in the form of a theatrical dialogue between three characters, but Mersenne eliminated the dialogues and converted everything into impersonal statements.
Lenoble  compared the contents of both versions original and translation to find that the translation was much shorter. Although Galileo's work was exceptional in developing strength of materials, there was no discussion on compressed columns, in part because apparently he did not distinguish between tension and compression. The treatise of mechanics is the third part of the book Cogitata by Mersenne , containing a summary of what was known in his times. There are no equations employed to represent knowledge, but there are basically propositions and arguments on various topics.
Consider his Proposition XIX, in which he lectures on the strength of cylinders by discussing the influence of length and thickness on strength. With reference to Figure 8 , Mersenne presents four lines of cylinders. First, he takes those identified as AC and DF, with equal length but diferent diameters to be more precise, Mersenne refers not to diameters but to thickness- crassitudine. He states that the strength of DF depends on the triple relation cube of the strength of AB, and to illustrate his point he provides numbers: if the diameter of DE is three times that of AB, then the strength of DE is 27 times that of AB.
Next, he illustrates the incidence of length by comparing a long cylinder KP with a short one HI.
If the length of KP is twice that of HI and has twice the diameter, then there will be a cubic relation due to diameters plus a linear relation due to length, leading to a quadruple relation. Clearly, he believed that the strength increases with length, having a positive influence. Notice that Mersenne postulates relations between strengths but provides no indication of the way the cylinders are loaded.
His narrative with reference to Figure 8 does not provide experimental information, but at some point he discusses the response of metal columns made of diferent materials as they stand compression. He acknowledges that a perplexing behavior occurs under compression:. It may be added that iron, copper and other metals, even single bodies, subject to force of weight, curve and bend to the form of an arch before breaking.
This produces a new disculty which escaped even Galileos notice. If there is a physico-mathematician who is capable of finding a solution, he will deserve more credit than the inventor of quadrature. This illustrates the astonishment of Mersenne as he faced the phenomenon of lateral displacement of a compressed member before breaking. Mersenne also speculates that this efect will surprise many readers, but the sole observation did not advance to the comprehension about causes or parameters that would be involved.
In his times, this was one of several topics for which no answers were available. Notice that this is a significant step backwards with respect to the writings of Leonardo, who was not astonished and even provided quantitative estimates of the response under compression. Benvenuto states that: "Mersenne and his contemporaries had no idea what the root cause of resistance might be.
The physical reasons for resistance were still wrapped in the deepest mystery. But this did not exclude the possibility of gathering useful information about the measure of resistance a purely phenomenological description, but hardly a trivial one. Mersenne is explicit on that point, and promises to return to the subject if the Lord gives him strength. Benvenuto, , pp. This astonishment in Mersenne, rather than showing a search direction to his contemporaries, illustrates his vision of frontiers that cannot be reached by the use of reason as Galileo attempted in other topics and belonged to what was only accessible through observation and experimentation.
The limitations in the understanding of Mersenne show that the knowledge of Leonardo did not permeate to Mersenne and his contemporaries, thus serving as an indirect proof that Leonardos thoughts were only private in nature. Even years later, Petrus van Musschenbroek  carefully cited Mersenne with great respect but did not mention Leonardo. His own work was based on a similar philosophy, i.
Koiter, W. T. (Warner Tjardus)
Duhem, on the other hand, had less favorable comments on the work of Mersenne, stating that his Tractatus Mechanicus was only a compilation of work done by others on statics . As said before, it is not simple to make a fair assessment of the knowledge that researchers had before the XVI Century. Part of the disculties are the limited early original manuscripts that survived up to our times, language problems texts being written in Arabic or Latin , but there are also problems with the translations of concepts, especially with those that were employed during medieval times as mentioned by Kuhn .
The symbolic value of balances throughout the centuries is a fascinating topic. In Greece, balances were used not just to weight goods, but also for the weighing of the souls in the afterlife an activity known as psychostasia. In the medieval Islamic world, the balance was associated to moral and justice: a good precision was required to assure honesty in the transactions . However, to understand the importance of the study of a balance in the medieval world it is crucial to understand not just the cultural context or the practical applications of the studies but also their theoretical importance within the discipline: The methodology to treat Statics problems of any kind in the Science of Weights was to reduce each problem to the case of a lever or a balance.
Thus, the study of the balance played a role similar to the use of equilibrium equations in the mechanics of the XIX Century. One key point about the mind frame in which authors developed their ideas is that following the scientific revolution, a sense of knowledge accumulation existed and this led to a sense of progress. Each contribution was seen as an advance in the frontier of knowledge towards some form of "truth". Herbert Butterfield argued that such ideas of progress did not exist during medieval and renaissance times:.
Men assumed rather the existence of a closed culture, assumed that there were limits to human achievement, the horizon reaching only to the design of recapturing the wisdom of antiquity, as though one could do no more than hope to be as wise as the Greeks or as politic as the Romans.