## The history of the theory of beam bending – Part 2

Before moving on from Galileo, Mariotte and Parent, let’s examine a peculiarity arising from the testing of their formulas.

Jacques Heyman in “The Science of Structural Engineering” states:

“… then the new calculation of bending strength [by Parent] gave a coefficient or 1/6 instead of the 1/2 of Galileo or 1/3 of Mariotte. For the first time, a logical and correct mathematical description had been given of the way a beam might fracture, but in fact none of the three values of coefficient accorded with tests – 1/2 might be better for stone and 1/3 for wood, while the mathematically correct value of 1/6 seemed to be useless as a predictor of fracture.”

This statement seems to be based on the work of Coulomb, who in 1773 (60 years after Parent’s work) reinvented the theory that had already been discovered by others. Coulomb found that stone and wood behaved in different ways, and presented two theories, based on a common approach, that satisfied the requirements of mechanics. He concluded that Galileo’s coefficient of 1/2 seemed best for stone, but he could not verify his (and Parent’s) coefficient of 1/6 for timber. By the turn of the century the standard text of the Ecole Polytechnique still stuck to Galileo’s formula for stone, and Mariotte’s coefficient of 1/3 for wood (Heyman).

This is curious, because although timber might approximate a plastic material, and thus Mariotte’s coefficient of 1/3 would give a good approximation of the bending strength, stone is a brittle material that will fail in tension while the stress distribution is close to triangular, and thus the coefficient of 1/6 should be appropriate, rather than 1/2.

Unfortunately Heyman gives no details of Coulomb’s experimental work, but the only way I can explain this anomaly is to assume that either Coulomb calculated the tensile strength of stone by back-calculation from bending tests, using Galileo’s formula, or that he carried out tensile tests that gave results much lower than the correct value, perhaps due to stress concentrations in the testing apparatus.

Either way, the end result was that by the early 19th century, over 300 years after Leonardo Da Vinci published the basis of the correct theory of bending, engineering texts still recommended a formula for the bending strength of stone that was incorrect by a factor of 3.

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### 3 Responses to The history of the theory of beam bending – Part 2

1. Raghu says:

Reference: History of S of M by Timoshenko, page 23. Mariotte also has given a value of 1/6. In the book, however, there is no subsequent reference to this “corrected” value by Mariotte.

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2. dougaj4 says:

Raghu – I have been meaning to continue this series for quite some time, so thanks for the reminder!

I think the passage in Timoshenko’s book indicates that Mariotte calculated the 1/6 factor as the contribution of the fibres in tension. The compressive zone will contribute the same resistance, giving a total load of Sh/3l. This is wrong because when he moved the position of the neutral axis from the base of the section to the centre line, he should have used S/2 instead of S, as well as reducing the lever arm.

The end result was that although he correctly identified the location of the neutral axis, his equation for the load required to cause bending failure in a brittle material was still too high by a factor of 2.

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