Loaded Struts and the Sharing of Weight on Supports
How a strut's lean governs its strength, and how the perpendicular through a triangle divides a beam's weight between two supports.
This page of Codex Madrid I studies the statics of loaded struts and beams. In the left-hand diagrams an inclined strut, described by its 'axis' (assis) and 'hypotenuse' (ipotenissa), carries a hanging weight; Leonardo argues that the more the strut leans from the vertical the longer and weaker it grows, so that more weight is needed to keep it as strong, the ratio of axis to hypotenuse governing the loss of strength. The right-hand triangular figures show how a beam's weight divides between two supports according to where the perpendicular falls within the angle: equally when it bisects the angle, and in a one-to-two (subduple) ratio when it does not. A final figure notes that a continuous support follows a different rule, the load varying along its length rather than being shared alike.
On this page
A leaning strut grows longer and weaker under its load
The more the line f g departs from its perpendicular, the longer it becomes while having to keep the same height, and the longer it becomes the less it resists. To make it as strong as before you must increase its weight in proportion to its added length, as seen in b a and a d. Leonardo marks the note 'tested' (sperimentate).
Axis-to-hypotenuse ratio fixes the loss of strength
The same proportion that the axis b a has to the hypotenuse a d, the weakness of the hypotenuse has to that of the axis. The note is headed 'tested' (sperimentata).
Construction rule: cut the angle with the perpendicular
Always cut the angles with the perpendicular and set that cut, the axis, in proportion with the side, that is the hypotenuse. The two sides of such triangles are to be taken equal, a straight line drawn from one extreme to the other, and the length of the axis reckoned against the length of the side.
Weight shared equally when the perpendicular bisects the angle
In this figure the perpendicular b r falls in the middle of the angle a r c, between equal parts a b and b c. Therefore the weight divides equally between its two supports.
Unequal (subduple) sharing when it does not bisect
In the second figure the perpendicular e t does not fall in the middle of the angle d t f; the parts flanking the central line stand in a one-to-two ratio, e f being subduple to d e. Hence the weight at d is half the weight at f.
A continuous support obeys a different rule
Because this support is continuous, the rule above does not hold; instead one follows that of the poles, a support of continuous quantity. The weights are distributed inversely at the ends in the ratio of length m x to x a, and, unlike the cords where the load is alike everywhere, here the weight goes on varying in every part.