"The Da Vinci-Euler-Bernoulli Beam Theory," ME Online Web Exclusive, April 18, 2003

for 4/18/03

The Da Vinci-Euler-Bernoulli Beam Theory?

by Roberto Ballarini

Many mechanical engineers and educators are not aware of some of Leonardo da Vinci's fundamental contributions to solid mechanics, fluid mechanics and mechanical design. These contributions appear in Codex Madrid I, one of two remarkable notebooks that were discovered in 1967 in the National Library of Spain (Madrid), after being misplaced for nearly 500 years. Selected scientific and artistic works that appear in Codex Madrid I are summarized, translated, and discussed by Vincian scholars in The Unknown Leonardo (edited by L. Reti, McGraw Hill Co., New York, 1974), a book that should be of significant historical interest to the mechanical engineering community.

Mechanics textbooks and instructors often credit Galileo's analytical attempt to determine the load carrying capacity of a transversely loaded beam as the beginning of beam theory (see for example Mechanics of Materials, J.M. Gere and S.P. Timoshenko, Fourth Edition, PWS Publishing Company, 1997). This is consistent with Timoshenko's influential History of Strength of Materials (Dover Publications, Inc., New York, 1983), which summarizes the individual contributions made to beam theory by Galileo, Mariotte, (Jacob) Bernoulli, Euler, Parent, and Saint-Venant.

However, while his name does not appear in any textbook discussion of beam theory, the mechanics education community should be aware that da Vinci made a fundamental contribution to what is commonly referred to as Euler-Bernoulli (engineering) beam theory 100 years before Galileo. Historians of mechanics did not cheat Leonardo; they simply were not aware that he made the fundamental hypothesis upon which Euler-Bernoulli beam theory rests in Codex Madrid I.

The normal stress and the moment-curvature formulas for slender linear elastic beams,

constitute what is universally referred to as Euler-Bernoulli beam theory. In these formulas

represent, respectively, the normal stress, bending moment, distance from the neutral axis, moment of inertia, deflection of the neutral axis, and Young's modulus. The fact that the theory is not called Galileo-Euler-Bernoulli beam theory is understandable, since Galileo incorrectly assumed that under transverse loading a beam's cross-section develops a uniform stress distribution. Parent, on the other hand, who was the first to obtain the correct stress distribution and to relate the stress to the bending moment, was arguably short-changed.

As everyone well versed in mechanics knows the key to the development of the above referenced formulas is the assumption made by Bernoulli in the 17th century that the strain is proportional to the distance from the neutral surface, the constant of proportionality being the curvature. It is remarkable that da Vinci hypothesized this form of strain distribution two centuries earlier. As explained by Carlo Zammattio in his article "Mechanics of Water and Stone" (in The Unknown Leonardo), da Vinci established all of the essential features of the strain distribution in a beam while pondering the deformation of springs. For the specific case he considered of a rectangular cross-section, da Vinci argues for equal tensile and compressive strains at the outer fibers, the existence of a neutral surface, and a linear strain distribution. Of course, da Vinci did not have available to him Hooke's law and the calculus. If he did, it is conceivable that he would have derived the formulas listed above, so that beam theory would be referred to as Da Vinci Beam Theory.

As discussed by Zammattio, the Codex includes other remarkable contributions, including the partitioning of energy into potential and kinetic components, within the context of what is commonly referred to as the Bernoulli Equation for fluids. In fact, da Vinci introduces (in words and not equations) this fundamental theorem of fluid mechanics, which states that for frictionless flow along a stream line the total head is constant. Contributions to conceptual mechanical design include the gear drive used in a bicycle.

The following image is a scanned version of the reproduction of folio 84 of Codex Madrid I that appears in Zammattio's article, together with Zammattio's translation of da Vinci's notes. Ignoring the fact that da Vinci wrote right-to-left in mirror-image script, this discussion may as well have been copied from a modern mechanics of materials text.

Da Vinci's discussion of the deformation of a beam/spring with rectangular cross-section.

(Image taken from the book "The Unknown Leonardo," McGraw Hill Co., New York, 1974.)

Translation: "Of bending of the springs: If a straight spring is bent, it is necessary that its convex part become thinner and its concave part, thicker. This modification is pyramidal, and consequently, there will never be a change in the middle of the spring. You shall discover, if you consider all of the aforementioned modifications, that by taking part 'ab' in the middle of its length and then bending the spring in a way that the two parallel lines, 'a' and 'b' touch a the bottom, the distance between the parallel lines has grown as much at the top as it has diminished at the bottom. Therefore, the center of its height has become much like a balance for the sides. And the ends of those lines draw as close at the bottom as much as they draw away at the top. From this you will understand why the center of the height of the parallels never increases in 'ab' nor diminishes in the bent spring at 'co.'

In conclusion, I suggest that the mechanics community remove the question mark in the title of this paper, that future discussions on the mechanics of beams acknowledge da Vinci's fundamental contribution, and that everyone look for a copy of The Unknown Leonardo.

ASME member Roberto Ballarini is a professor of civil engineering at Case Western Reserve University in Cleveland. In August of 2003, he will be F.W. Olin Professor of Mechanical Engineering at F.W. Olin College of Engineering.