Dome
Shapes and internal forces

A masonry dome produces thrusts downward and outward. They are thought of in terms of two kinds of forces at right angles from one another: meridional forces (like the meridians, or lines of longitude, on a globe) are 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 at the base, with the transition in a hemispherical dome occurring at an angle of 51.8 degrees from the top. The thrusts generated by a dome are directly proportional to the weight of its materials. Grounded hemispherical domes generate significant horizontal thrusts at their haunches.

The 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, although cracking along the meridians is natural. 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.

Unlike voussoir arches, which require support for each element until the keystone is in place, domes are stable during construction as each level is made a complete and self-supporting ring. 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. 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.

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. For a particular material, the optimal dome geometry is called the funicular surface, the comparable shape in three dimensions to a catenary curve for a two-dimensional arch. 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.
keywords:existence