The structural action of vaults depends on their final shape rather than on their method of construction. Intersecting semicircular barrel vaults evolved into pointed Gothic vaults which remain stable but need much less material. The vaults between the groins can be slightly domed, so they can be analysed much like fuller domes, by both membrane and slicing techniques. The ribs at the groins carry severe stresses; this is their structural purpose. The lines of thrust escape from the ribs into vaulting pockets filled with rubble, whence they pass through the walls into the buttresses. Ungewitter’s tables show how thrusts vary with vault materials and rise-to-span ratios. Vaults develop cracks of different types (as do arches); these can respond differently to unexpected loads, such as those due to fires and firefighting. Technical analyses of vaults can illuminate historical debates, about the original presence and purpose of flying buttresses, for example. Fan vaults are more demanding technically than other vaults but can still be analysed using membrane techniques to obtain profiles and lines of thrust. Henry VII Chapel at Westminster provides a case study of cracks.

Both solid spire tips and hollow spire bodies, regarded as circular cones, can be considered using simple statics (applied, as an example, to the spire at Hemingbrough). Solid spire tips are at risk from wind forces if they are too short (or too light), but they may be stabilised by hanging weights from them inside the spire. Hollow spire bodies are at risk if they are too thin-walled (or too light); they can also be analysed with membrane techniques, which show that tensile stresses start to develop in their bases at about half the wind force that would be needed to overturn them. Spires often have eight sides; however, circular cones are demonstrably good models for them that conform reasonably well with an empirical safety rule. This is so even for decorative spires like that of Freiburg, made from open stonework tracery. Because of their low centres of gravity, spires can lean at visually alarming angles before overturning; again this can be shown by simple statics or membrane techniques. This tilting (and indeed twisting) is more common in timber than in stone, because timber spires can suffer through differential shrinkage of their frameworks.

Accounts of building collapses at Venice and Beauvais help to demonstrate that structural failures can occur through changes in soil (perhaps in the level of the water table) or masonry (from mortar shrinkage or stone decay). Stabilisation works carried out on the tower at Ely by the author have involved removal of nineteenth century external straps, corner tie bars (possibly unnecessary) and grout forming a solid core encircling the inner wall surface and reinforced by rods inserted through the outer wall surface. The vibration and cracking of towers due to bell-ringing are potentially significant, as are the effects of wind; square solid towers intended as pinnacles can be overturned by the wind if they are too tall. The development of cracks in both solid walls and square hollow towers can be explored using simple equilibrium approaches to find the angles at which the walls and towers lean enough to first crack and later be overturned. Cracks appear in walls at smaller angles of leaning than in comparable thin-walled towers, but overturning occurs at rather greater angles for walls than for towers.

Various structural elements have construction methods and potential problems that deserve attention. Points of note include: the ‘ratchet effect’ on rubble-filled walls of repeated freezing and thawing; the possibility of mortar shrinkage or of stone decay through excessive stress; the crucial role of crossing piers which carry a tower, and how they can be strengthened (as at Milan and Worcester); the (maybe counterintuitive) structural contribution of pinnacles; the detailed actions of flying buttresses, and how they may fail (as at Amiens) if they are not ‘flat arches’; the importance of binding ribs to walls by single ‘through-stones’; how stone windows handle thrusts from the wall above and wind outside, starting with rectangular windows and moving on to rose windows; and the actions in response to live and dead loads on cantilevered stone stairs, whether piecewise straight with corner landings or geometrical (as in a round tower). Calculations about structural actions (of flying buttresses, stone windows and stone stairs) can be based on simple statics.

Unlike an arch, a dome can be thought of as a thin shell, with forces acting smoothly within its surface. It is then treated as if the minimum thickness is set mainly to avoid local buckling. The compressive stress required to support the dome is independent of the thickness, for the dome as for other thin shells, such as cones. However, the thickness is often combined with the stress in the ‘stress resultant’ of membrane techniques. The techniques demonstrate that tensile stresses can develop near the base of the dome. If its supports move, a hemispherical dome can crack into orange-like segments along lines from its base towards its crown. It can be assembled from such notional segments. Opposite segments paired at their crown as ‘arches’ can be analysed separately to find the minimum thickness. From the use of ‘arches’ for complete domes comes the use of slices for incomplete domes, which have lost some adjacent segments. The results show that complete domes can be thinner than incomplete ones. There remain difficulties, though: in a dome that has (say) eight sides, stresses focussed on the ribs between the sides need analysis.

Geometry and proportion have always been fundamental to expertise in building; they emerge even in the record of constructing a great temple in the biblical book of Ezekiel. The books on architecture of the Roman author Vitruvius were copied widely and fed directly into the secrets of the medieval lodges, which are now known in part from Villard de Honnecourt’s sketchbook. The disputes at Milan about how to proceed with the cathedral illustrate how the time-honoured rules of proportion persisted, even though their intuitive justifications appeared to be getting lost. Ultimately, Renaissance thinking and the invention of printing opened a new era. This is well represented by St Paul’s Cathedral but also gave rise to the distinction between engineers and architects and the belief that every gentleman with money and a copy of Vitruvius could design his own buildings.

Idealised assumptions are made about masonry: it has zero tensile strength, unlimited compressive strength and zero sliding. These assumptions allow calculations about masonry structures that equilibrate and accommodate small changes in boundary conditions. Such small changes produce cracks that in an arch form ‘hinges’ through which the line of thrust passes. With four or more hinges, large loads make the arch collapse as the line of thrust strays outside it; a ‘flat arch’, whose abutments can be joined by an internal straight line, does not collapse. To keep the line of thrust within it, an arch must have at least a minimum thickness. The ratio of the actual thickness to this minimum is the ‘geometrical factor of safety’. Often it exceeds 2, but by the so-called ‘safe’ theorem, if it exceeds 1 then the arch is safe. New cathedral buildings have collapsed within two decades, a typical period for soil settlement. Later collapses may be due to changes in the soil or the masonry. Without evidence of recent shifts, cracks are just responding to previous change, and should merely be filled with mortar to keep them dry.