Bending of framed beams and the loading of pointed arches
Opposite curvatures in a timber frame, and why loading an arch at mid-height is dangerous
The top figures show a horizontal beam carried at its ends by two upright beams, forming a portal whose members curve, while below are pointed arches loaded with weights. Leonardo argues that when the horizontal beam bends toward the load, the two uprights curve oppositely, and that a frame of three equal timbers behaves the same way. He warns that loading a pointed arch at the middle of its height makes it break inward, and that arches so loaded risk buckling toward the centre of their circle. He defers a full account of the proofs to his projected treatise on mathematics.
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Opposite curvatures in a beam on two uprights
A beam supported at its ends by two upright beams of equal figure, weight and power will, when curved at its middle, bow toward the load, while the two supporting uprights curve the opposite way. Because the upper beam a b has greater motion at n than at a, the support n r is pressed more than the side a c.
A three-timber square frame under load
If three equal timbers are set up as the three sides of a standing square and the middle of the upper timber is curved by excess weight, the supporting timbers will curve too. Their curvature, Leonardo notes, is created oppositely to that of the loaded beam.
A pointed arch loaded at mid-height breaks inward
If the pointed arch is loaded at the middle of its height, Leonardo states, it will break inward. The drawn pointed arches carry weights at their crowns to illustrate the failure.
The danger of loading arches at mid-height
Loading arches at the middle of their height is often done but should be avoided as far as possible, for too much load makes them unsafe and liable to buckle toward the centre of their circle. Leonardo says he will not extend himself further here, promising a distinct and precise account in his treatise on mathematics.