How loaded beams bend under weight
Experiments on beam curvature versus weight, position and length
Five horizontal beams are drawn on the right, each hung with weights to reveal how the beam curves under load. Leonardo poses a series of experiments: how to shift a weight along a beam yet keep its curvature constant, how doubled weights affect the bending, and how a bundle of four beams behaves compared with a single one. Number series such as 1/2 1/4 1/8 1/16 1/32 and 1 8 64 track the doubling of weights and the proportional loading of beams of different lengths. Further untranscribed notes accompany the diagrams.
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Keeping a beam's curvature constant as the weight moves
Leonardo asks how to hang a weight at various points along a horizontally set beam so that the beam always keeps the same curvature. The question is how large the weight must be and by how much it should be increased or lightened depending on where it is placed. It is a study of the interplay between load, position and bending.
Doubled weights and doubled curvatures
Various weights, each double the heaviness of the last, are set at the middle of a horizontal beam to see how much more or less the beam bends under each. Leonardo believes that if the test is carried out with weights doubled one against the other, the curvatures too will follow the same proportion among themselves.
A bundle of four beams bends like one
Four beams bound together in a single bundle four braccia long will bend upon themselves just as much as one beam a single braccio long, when both are loaded at the middle by an equal weight. The note compares stiffness across length and cross-section.
Three beams of doubling length to an equal sagitta
Leonardo takes three beams of equal thickness, each double the other in length, and wants to load each at its centre with such a weight that the arc of their curvature has the same sagitta (rise) among them. The figures 1 8 64 record the proportional loading this requires.