Editing Dome (section)

== Shapes and internal forces ==
A masonry dome produces [[thrust]]s downward and outward. They are described as two kinds of forces at right angles to one another: meridional forces (like the [[Meridian (geography)|meridians]], or lines of longitude, on a globe) are [[Compression (physics)|compressive]] only, and increase towards the base, while hoop forces (like the lines of [[latitude]] on a globe) are in compression at the top and [[Tension (physics)|tension]] at the base, with the transition in a hemispherical dome occurring at an angle of 51.8 degrees from the top.{{sfn|Hourihane|2012|p=242}} The thrusts generated by a dome are directly proportional to the weight of its materials.{{sfn|Hourihane|2012|p=301}} 

When hoop forces at the base of a masonry dome exceed the tensile strength of the dome, vertical cracks develop that make the dome act as a series of concentric wedge-shaped arches that do not necessarily compromise the overall structure.{{sfn|Pavlovic|Reccia|Cecchi|2014|pp=1-2}} Although some cracking along the meridians is natural, excessive outward thrusts in the lower portion of a hemispherical masonry dome can be counteracted with the use of chains incorporated around the circumference or with external buttressing.{{sfn|Hourihane|2012|p=242}} Grounded hemispherical domes can still generate significant horizontal thrusts at their haunches.{{sfn|Robison|1991|p=395}} For small or tall domes with less horizontal thrust, the thickness of the supporting arches or walls can be enough to resist deformation, which is why drums tend to be much thicker than the domes they support.{{sfn|Gye|1988|p=142}}

Meridian forces can cause dangerous horizontal cracking when not enclosed in the structure. When such compression is focused on an inside surface, for example, the corresponding outside surface will be in tension and crack, with the inside surface acting as a [[hinge]] in a potential collapse.{{sfn|Pavlovic|Reccia|Cecchi|2014|p=2}}

Unlike voussoir arches, which require support for each element until the [[Keystone (architecture)|keystone]] is in place, domes are stable during construction as each level is made a complete and self-supporting ring.{{sfn|Hourihane|2012|p=302}} The upper portion of a masonry dome is always in compression and is supported laterally, so it does not collapse except as a whole unit and a range of deviations from the ideal in this shallow upper cap are equally stable.{{sfn|Gye|1988|pp=141–142}} Because voussoir domes have lateral support, they can be made much thinner than corresponding arches of the same span. For example, a hemispherical dome can be 2.5 times thinner than a semicircular arch, and a dome with the profile of an [[equilateral arch]] can be thinner still.{{sfn|Fernández|Hernández-Ros|1989}}

The optimal shape for a masonry dome of equal thickness provides for perfect compression, with none of the tension or bending forces against which masonry is weak.{{sfn|Robison|1991|p=395}} For a particular material, the optimal dome geometry is called the [[funicular curve|funicular surface]], the comparable shape in three dimensions to a [[catenary]] curve for a two-dimensional arch.{{sfn|Rovero|Tonietti|2012|p=183}}{{sfn|Blockley|2014|p=22}} Adding a weight to the top of a pointed dome, such as the heavy cupola at the top of [[Florence Cathedral]], changes the optimal shape to more closely match the actual pointed shape of the dome. The pointed profiles of many Gothic domes more closely approximate the optimal dome shape than do hemispheres, which were favored by Roman and Byzantine architects due to the circle being considered the most perfect of forms.{{sfn|Larson|Tyas|2003|pp=32, 38}}