Earthquake-Proof Arches and Reinforced Ceiling Beams
The chord rule for an arch, inverted arches against earthquakes, and composite beams jointed with an iron clasp
Headed 'On protection against earthquakes,' the folio combines a rule for arch strength with a long series of beam-reinforcement studies. Leonardo states that an arch will not break so long as the chord of its outer arc does not touch the inner arc (a o n against x b y), growing weaker the more the inner arc cuts that chord, and draws inverted round arches between columns and a tripod of half-arches to brace against earthquakes—an arch that directs its weight perpendicularly onto its roots works whatever its orientation, inverted, prone, or upright. Most of the page details composite ceiling beams whose members interlock along diagonal or stepped joints, with a grooved principal beam m n whose tongued upper member i K is fixed by a square metal clasp l. A closing note observes that an arch loaded on one side alone drives the load to the top of the other half, down to the spring, and breaks farthest from its ends and its chord.
On this page
The chord rule for an arch's strength (a o n; x b y)
An arch will not break so long as the chord of the outer arch does not touch the inner arch. Experience shows that whenever chord a o n of the outer arch n r a touches the inner arch x b y the arch is weak, and weaker the more the inner arch cuts across that chord.
Inverted arches and a tripod against earthquakes
Three inverted round arches set between four columns reinforce their standing strength as protection against earthquakes, beside a tripod of three half-arches meeting at one peak. An arch that directs its weight perpendicularly onto its roots performs its office whatever its orientation—inverted, prone, or upright.
Composite ceiling beams with interlocking joints (a b, c d, e f, g h)
A series of experimental ceiling beams that rise to an angle at the middle, each reinforced by narrower members meeting along long diagonal inclines or stepped rectangular patterns, their ends labelled a b, c d, e f, and g h.
The grooved principal beam and its tongued member (m n, i K)
The principal beam m n is whole and angled at the middle, grooved with a channel on its upper face to receive the tongue of the upper member i K, which fits in from above.
The square metal clasp l
A square metal clasp, labelled l and drawn as an iron bent into a handle-shape, fixes the upper member to the lower beam through fastenings at the two upper corners.
One-sided loading breaks the arch farthest from its ends (a b)
When an arch is loaded on only one of its sides, the weight burdens the top of the other half, and this weight passes down to the spring of the arch; the arch then breaks at the point farthest from its ends and from its chord.