Barrel vaults also known as tunnel vaults (shells with single curvature) are the earliest kind of vaulting used. They are simple structures for roofing rectangular ground plans. How to enclose the arched openings of the barrel vault has always been a challenging structural and aesthetic problem.
The standard solutions are plane lattice structures which resist wind loads via bending. A much more efficient solution is a double curved space frame, which resist wind loads principally in an axial manner (via compression or tension). The figure show a barrel vault capped by half domes. Its geometry is a geodesic envelope!
It is not generally recognised that a series of circles (arches) parallel to one another, and a series of straight lines parallel to one another, intersecting the circles at right angles, form the simplest case of a geodesic pattern on the surface of a cylinder. Indeed, structures with geodesic patterns on single curved surfaces (Euclidean geometry) are nothing new! The geometry of the half dome is derived from the spherical cube.
Returning to lightweight roofs
The transept roof of the Crystal Palace, for the Great Exhibition of 1851, with its barrel vault construction was given lateral stability via diagonal bracing made from wrought iron rods. It was a highly efficient system. It is very instructive to note how Paxton used hierarchical systems for the transfer of loads — nature’s way. This seminal roof structure inspired engineers to develop ligtweight grid shells with rectangular meshes and diagonally pre-stressed cables. If the cladding is transparent the cables become almost invisible!