## The history of the theory of beam bending – Part 1

The theory of the flexural strength and stiffness of beams is now attributed to Bernoulli and Euler, but developed over almost 400 years, with several twists, turns and dead ends on the way.

Galileo Galilei is often credited with the first published theory of the strength of beams in bending, but with the discovery of “The Codex Madrid” in the National Library of Spain in 1967 it was found that Leonardo da Vinci’s work (published in 1493) had not only preceded Galileo’s work by over 100 years, but had also, unlike Galileo, correctly identified the stress and strain distribution across a section in bending.

The complete scanned text of the Codex Madrid (20 pdf files, each of about 40 MB) may be downloaded from: http://historical.library.cornell.edu/kmoddl/toc_leonardo1.html

The relevant page is Folio 84, reproduced below.

The page is translated with further commentary here:

“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 spite of Leonardo’s accurate appreciation of the stresses and strains in a beam subject to bending, he did not provide any way of assessing the strength of a beam, knowing its dimensions, and the tensile strength of the material it was made of.  This problem was addressed by Galileo in 1638, in his well known “Dialogues concerning two new sciences.  Illustrated with an alarmingly unstable looking cantilever beam supported by a wall, Galileo assumed that the beam rotated about the base at its point of support, and that there was a uniform tensile stress across the beam section equal to the tensile strength of the material.

From this he derived a bending strength equal to the tensile strength of the beam multiplied by half its depth.

Due to the incorrect assumptions of uniform stress across the section and rotation about the base of the section Galileo’s result was three times higher than the correct value for a brittle material which has approximately linear behaviour up to its failure load.  The disparity between Galileo’s calculations and actual breaking loads did not go unnoticed, and in 1686 Edme Mariotte’s alternative approach was published.  Mariotte initially retained Galileo’s position of the axis or rotation, but proposed a triangular stress distribution, varying  from the failure stress at the top to zero at the base.  He then proposed (without proof) that the neutral axis should be at the centre of the section, but also introduced an error in his working resulting in the section modulus being double its correct value, as it would be if the neutral axis were at the base of the section as he had initially assumed.

The correct formula was eventually derived by Antoine Parent in 1713 who correctly assumed a central neutral axis and linear stress distribution from tensile at the top face to equal and opposite compression at the bottom, thus deriving a correct elastic section modulus of the cross sectional area times the section depth divided by six.  Unfortunately Parent’s work had little impact, and it was many more years before scientific principals were regularly applied to the analysis of the strength of beams in bending.

To be continued.

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### 7 Responses to The history of the theory of beam bending – Part 1

1. James says:

I found this fascinating – thanks for posting it. It seems so obvious now, but it’s amazing to think that someone thought “the center of its height has become much like a balance for the sides” hundreds of years ago.

2. dougaj4 says:

Thanks for the comment James.

Reminds me that I have to finish the story (from Euler to the present day)!

3. Raghu says:

This as well as the parts 2 and 3 are really fascinating to read.

I quote from the paper by Professor Bellarini. The first translated sentence reads:

“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.”
Reading between the lines, I presume that “thinner” means contracting and “thicker” implies stretching. If that is so isn’t there a contradiction? When a beam bends the convex part stretches and the concave part contracts.

I will be happy to get clarification. Thanks.

4. dougaj4 says:

Raghu – I agree, it doesn’t seem right. I don’t know what the explanation is. It is possible that the translation is misleading, but my Latin is not good enough for me to know (especially when the original is in mirror writing script!).

5. Raghu says:

Dougaj4,

Your reply was as fast as lightening. Thanks. I feel this “contradiction” has to be investigated.

Curiously, I have reasons to believe that I am not the first one to point out this contradiction.

6. Ghostdog says:

Raghu and dougaj4
“concave is hollowed or rounded inward like the inside of a bowl and convex is the curving out or bulging outward, as opposed to concave.
If your mind can visualise a beam bending, the concave part is thicker (compression) due to the contraction of the fibers whilst the convex part is thinner (tension) due to the streching of the fibers”

• dougaj4 says:

Ghostdog – but why do you say that “thicker” is compression? When you compress a spring you make it thinner.