Thin-Shell Concrete Domes: Membrane Action and Buckling
A dome only a few inches thick can span a hundred feet or more, which looks like it violates common sense until you look at how the shell actually carries load. A flat concrete slab of the same thickness spanning the same distance would fail under its own weight through bending long before it reached that span. A dome mostly avoids bending altogether by carrying load through membrane action, in-plane compression and tension distributed across the curved surface, the same reason an eggshell resists a surprising amount of point load despite being almost paper-thin.
Meridional Compression and Hoop Tension
Picture the dome's surface divided into meridians, curves running from the crown down to the base like lines of longitude, and hoops, horizontal rings like lines of latitude. Under uniform gravity load, the meridional direction is everywhere in compression, carrying load down toward the base the way an arch carries load along its own curve. The hoop direction tells a more interesting story: near the crown, the hoops are also in compression, but below a specific latitude, roughly 51.8 degrees from the crown for a uniformly loaded spherical dome under its own weight, the hoop stress flips to tension.
That transition matters because concrete is poor in tension, so the lower portion of a dome shell, the zone where hoop tension develops, either needs hoop reinforcement to carry that tension or needs the shell thickened or stiffened in some other way to keep the tension stress within what unreinforced concrete can sustain. This is directly analogous to how reinforced concrete fundamentals put steel wherever the concrete would otherwise see tension it can't carry alone, except here the tension zone is defined by shell geometry rather than by a beam's bending diagram.
At the base of the dome, meridional compression has to resolve into whatever supports the shell, and if that support is a simple ring beam rather than a continuous wall, the ring beam picks up substantial hoop tension from the accumulated meridional thrust pushing outward at the perimeter, the same ring-and-thrust logic used to close the load path in tension-roof structures anchored to a compression ring.
Buckling, Not Strength, Usually Governs Shell Thickness
A thin shell rarely fails because the concrete crushes under the calculated membrane compressive stress; it fails because the shell buckles, snapping into a different, lower-capacity equilibrium shape under load well below its material crushing strength. Shell buckling capacity is extremely sensitive to geometric imperfections, small deviations from the ideal curved shape introduced during formwork and construction, which is why classical theoretical buckling loads for thin shells are almost never used directly in design; actual design capacity is knocked down substantially from the theoretical value using empirical reduction factors calibrated against tested shells that buckled well below the textbook prediction.
Asymmetric loading, snow drifted to one side, wind suction over part of the surface, or a construction load parked on one quadrant, is usually more dangerous to a thin shell than uniform load, because the membrane stress state that keeps a dome working so efficiently assumes a load pattern reasonably close to axisymmetric; a shell that comfortably carries uniform snow load can develop local bending and buckling risk under the same total load applied unevenly, a concern parallel to how silo and bunker design has to account for unevenly distributed granular pressure rather than assuming a tidy symmetric fill.
Ribbed and Waffle Domes Trade Membrane Purity for Buckling Resistance
Many historic and modern large-span domes aren't smooth continuous shells at all but ribbed or waffle-patterned surfaces, a grid of stiffening ribs cast into or projecting from the shell. The ribs add local bending stiffness that raises the buckling capacity substantially above what a smooth shell of the same average thickness could achieve, at the cost of introducing some bending action into what would otherwise be a pure membrane stress state, a deliberate compromise rather than a design failure.
The American Concrete Institute's guidance on thin shell and folded plate concrete structures, along with classical shell theory texts, remain the primary references engineers use for both the membrane stress calculations and the empirical buckling knockdown factors, maintained by the American Concrete Institute.