Concept

Curved surface pressure — where it appears

Fluid or soil pressure acting on a surface that is not flat, such as a pipe, a tank or an arch dam. Its resultant on any curved surface equals the pressure times the surface's projection on a plane square to the push, which is how a ring's hoop force pR is found without integrating.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as hoop stress — the same set of essays touches all of them, so they are one junction rather than several.

The water pushes on the chord, and the abutments push back along the arch. One horizontal ring of an arch dam, one metre high, 50 m below the water, spanning a 100 m gap with a central angle of 133.6°, drawn in plan with the water on the convex side. The water presses 491 kPa normal to the face everywhere (arrows); by the projection rule its resultant is that pressure times the chord, 49,050 kN per metre of height, pointing straight downstream. Each abutment pushes back along the ring's tangent, at 66.8° to the chord, so the two thrusts are N = pw/2sin(θ/2) = 26,686 kN — the pressure times the ring's radius, 54.4 m, which is the hoop formula of a pipe, arrived at without a pipe.

The angle that saves the most concrete

Cut an arch dam into horizontal rings and each ring is a curved surface with water on one side. The projection rule gives the water's whole push on it without an integral — the pressure times the chord — and the two abutments, pushing back along the ring, turn that into a thrust equal to the pressure times the radius, the hoop formula of a pipe arrived at without a pipe. From there one line of calculus says that the ring using least concrete has a central angle of 133.6°, whatever the canyon, the depth or the concrete. That number explains why arch dams in narrowing canyons are drawn at a constant angle rather than a constant radius, and why the angle that suits the concrete is not the one that suits the rock.

equilibrium · Curved surface pressure

In a V-shaped canyon the rings carry the water, and the top rings more than theirs. The water pressure down the crown of a dam 100 m high in a V-shaped canyon 200 m wide at the crest, 5 m thick at the crest and 25 m at the base, with its rings at 133.6° (dashed), divided between the horizontal ring at each depth (solid) and the vertical cantilever at the crown (thin) so that the two deflect alike. At 4 m below the crest the ring carries 195% of the water there — the cantilever, loaded from below and free at the top, leans on it; at 49 m the cantilever carries 11%; at 89 m, 6%. Over the whole reservoir the cantilever carries 10% of the water and the rings the rest.

The water the valley floor takes

An arch dam is not a stack of rings. At its crown it is also a vertical cantilever fixed into the valley floor, and the water at each depth is shared between the ring there and the cantilever in whatever proportion makes them bend alike. In a V-shaped canyon the short lower rings are stiff and carry nearly everything, and the top rings carry up to twice their own water because the cantilever leans on them. In a U-shaped canyon the cantilever takes half the reservoir and bends the heel into tension that the ring formula never sees.

equilibrium · Curved surface pressure

Named alongside it

The objects these essays reach for when they reach for this one.

Hoop stress Abutment Arch Arch dam Cantilever Compatibility Hydrostatic pressure Load sharing Optimisation Stiffness Thrust

All concepts