LECTURE 5 ARC

1.

LECTURE 5
Spatial Coverings of
Single-Storey Buildings
Precast shells, multi-wave vaults, and steel space-frames that span the column-free industrial floor
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2.

OVERV IEW
Seven sheets, three structural ideas
Precast RC shells · 4.01 – 4.02
Multi-wave shells · 4.03 – 4.04
Flat or curved reinforced-concrete plates assembled into a spherical or toroidal shell,
stiffened by a perimeter contour truss.
The same 3×6 m curved plate repeated as parallel waves across the building, closed by RC
or steel edge members.
Ferrocement structural plate · 4.05
Steel space-frames · 4.06 – 4.07
A shallow space-plate of pyramidal and ribbed ferrocement elements, spanning a full
column bay as a regular grid.
Bar systems of hot-rolled profiles or tubular ball-joints, forming a lightweight reticular plate
for large or irregular grids.
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3.

WHY GO SPAT IAL
Shape carries the load
• A shell or space-frame resists load mainly through in-plane (membrane) or axial forces
rather than bending, so far less material is needed per square metre than a flat beamand-truss roof.
24–36 m
typical shell / vault span
• The weight saved lets these roofs cover very large column grids — up to 18×36 m and
beyond — while leaving the floor below completely open for process equipment,
cranes, and future changes of use.
• Every system in this chapter is precast or shop-fabricated for site assembly, continuing
the standardization of frames and trusses studied in Chapters 2 and 3.
≤ 850 kgf/m²
design roof-load range
Prefab
shop-cast or shop-welded elements
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4.

4.01
SHEET 4.01
Spherical shells built from flat 3×3 m plates
• A shell with a spherical surface can be assembled entirely from flat square plates — the shell becomes a faceted
polyhedron with rhombic faces inscribed in the sphere.
• Square 3×3 m plates are completed into rhombi by small adjustments to joint width, so a single plate mould serves almost
the whole surface.
• Precast RC shells built this way span 24, 30, or 36 m, carried on a perimeter contour truss.
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5.

SHEET 4.01 — GEOMETRY
Plate types and the contour truss
• Plates in plan are 3×3 m with diagonal ribs 0.2 m high, produced in three types:
ordinary (mid-field), corner, and edge plates.
• Corner and edge plates along the shell's boundary carry reinforced contour ribs,
since they sit on the supporting edge.
• The perimeter contour truss is a segmental, curved-chord truss built from linear RC
elements; its corner elements are prestressed with high-strength tendon bundles.
• Available spans: 24, 30, and 36 m.
Fig. 24. Spherical shell assembled from 3×3 m plates, with corner column and contour truss (Sheet
4.01).
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6.

SHEET 4.01 — CONNECTIONS
Site joints and the corner column
• Ordinary plates meet at concrete shear keys cast into
grooved joints; rebar outlets are welded across the joint
before it is grouted with grade-300 concrete.
• The contour ribs of the four corner and edge plates
converge on a single reinforced corner column, which
carries the shell's perimeter thrust down to its footing.
Fig. 25. Assembly joints of the shell: plate-to-plate connection (view from below), corner column, and eaves detail.
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7.

SHEET 4.01 — ERECTION
Built span-by-span on movable scaffolding
1
2
3
4
Scaffold
Set plates
Grout & tie
Close & strike
Steel scaffolding (kruzhala) is erected
inside the bay; it can be moved from
span to span as work proceeds.
Plates are lifted onto the scaffold in
their design position and joined by
welding the projecting rebar outlets.
Joints are grouted; contour ribs are
linked by hinge joints into one
continuous perimeter truss.
The truss lower chord, assembled on
the ground, is raised and welded to the
upper chord; scaffolding is then struck
and reused.
Elements stay simple to form and transport, but are relatively small and depend on metal-heavy scaffolding — a limitation later designs (enlarged, crane-assisted dome assembly)
partly resolve.
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8.

4.02
SHEET 4.02
Multi-wave spherical shells, 18×24 and 18×30 m
• Precast RC shells sized 18×24 or 18×30 m, built from 3×6 m plates on steel contour trusses (Series 1.466-1).
• In plan the roof reads as a set of convex, cylindrical vaults — a "multi-wave" covering — all inscribed in one toroidal
surface.
• Each shell behaves as an "individually standing shell": it needs no structural tie to its neighbours over most of the span.
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SHEET 4.02 — GEOMETRY
A toroidal surface at a 1:3 maximum slope
• Rise and curvature follow a maximum roofing slope of 1:3 at the shell edges.
• Adjacent shells are grouted together only along the support contour; over the middle
1/7–1/8 of the span the plates are free to move, so no connection is needed there.
• Because neighbouring shells aren't tied along their length, each carries its own load
independently — the "individually standing shell" static scheme.
Fig. 26. Multi-wave spherical shell, 18×24/18×30 m, on steel contour trusses (Sheet 4.02).
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10.

SHEET 4.02 — ELEMENTS
Main, edge, and additional plates
250 / 150 mm
longitudinal / transverse rib height
30 / 40 / 60 mm
flange: mid-field / contour / openings
18 × 3 m
enlarged arch-block assembly unit
• Main plates are nominally 3×6 m; edge and infill plates measure 0.7×6 m and 0.4×6 m — all curved in the direction of greatest curvature.
• Shear between plates is carried by concrete keys cast into triangular grooves, without a separate mechanical connector.
• For small roof areas, precast plates are replaced by monolithic infill, cutting down the number of mould types needed on site.
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11.

SHEET 4.02 — ERECTION
Arch blocks raised onto steel edge trusses
• Main plates are pre-assembled on the ground into 18×3 m
arch blocks with temporary strut ties, then lifted into place as
one unit.
• Erection starts with the steel edge trusses, braced by
removable ties; arch blocks are then set onto them and
welded closed at the ridge.
• A suspended crane track (up to 5 t) can hang from the shell on
roughly 6 m centres, anchored between the contour-plate
ribs.
Fig. 27. Plate-layout detail, contour-plate joint, and roofing build-up at the shell edge.
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12.

4.03 /
4.04
SHEETS 4.03 – 4.04
Multi-wave shells — two edge-structure variants
• Both sheets repeat the same 3×6 m curved plate across the building as parallel "waves" — they differ in how each
wave meets the building's edge.
• Sheet 4.03: cylindrical multi-wave shells, closed by reinforced-concrete edge beams.
• Sheet 4.04: shells of double positive curvature (curved in two directions), closed by steel edge arches for a lighter,
longer-span perimeter.
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13.

SHEET 4.03 — CYLINDRICAL MULTI -WAVE SHELL
3×6 m plates on reinforced-concrete edge beams
• Cylindrical shell plates rest on longitudinal RC edge beams, which also form the building's gutter line.
• A raised trapezoidal light-and-vent lantern runs the length of the ridge on every wave.
• The contour-ribbed soffit uses the same rib-height system (250 mm / 150 mm) as the other multi-wave sheets.
• Columns bear on monolithic RC footings (Series 1.412 / 1.415), with expansion joints where the frame steps
down.
Fig. 28. Cylindrical multi-wave shell on RC edge beams and
columns, with monolithic footings (Sheet 4.03).
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14.

SHEET 4.04 — DOUBLE -CURVATURE MULTI -WAVE SHELL
Doubly-curved plates on steel edge arches
• The steel edge arch — spanning up to 30 m — replaces the
concrete edge beam, letting the shell close at a slimmer,
longer-span perimeter.
• Ordinary and end-wave plates are curved in both directions,
so their formwork is more complex than the single-curvature
cylindrical wave.
• A steel tie rod closes the arch's horizontal thrust at every
support.
Fig. 29. Multi-wave shell of double positive curvature, 3×6 m plates on steel edge arches (Sheet 4.04).
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15.

SHEET S 4.03 – 4.04 — WHERE THEY APPLY
Column grids from 12×24 to 18×36 m
≤5t
suspended-crane capacity
≤ 50 t
supported-crane capacity
450 / 650 / 850
kgf/m² design roof-load classes
• Suited to column-free halls, buildings with a technical floor above the suspended ceiling, or roofs that must carry large ducts run in any direction.
• Windowless variants are available alongside versions with longitudinal light-and-air or zenith lanterns.
• Raising the design load only changes the reinforcement — the formwork dimensions of every element stay the same.
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SHEET S 4.03 – 4.04 — PLATES AND JOINTS
Ordinary and end waves, sized by column spacing
• Ordinary waves are assembled from two or three edge-plate widths depending
on column spacing; end waves use trimmed (doborny) trapezoidal plates sized
for the lantern.
• Flange thickness grows from 30 mm at mid-span to 40 mm at the contour, and
to 60 mm around lantern or deflector openings.
• Joint width between waves varies 40–195 mm with the shell geometry; joint
depth is 120 mm without a suspended crane, or 250 mm where one is carried.
Fig. 30. Shell-to-edge-arch connection and wave-to-wave joint detail (Sheet 4.04).
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17.

SHEET S 4.03 – 4.04 — ERECTION
Waves assembled on the ground, closed "onto itself"
10 t
self-propelled erection crane
• Plates are first enlarged on the ground into end and ordinary waves, in a staging zone within the erection
bay.
• A self-propelled crane of 10 t lifts each wave "onto itself," closing it against the previously erected wave,
with diaphragms already in place between them.
• Diaphragms are reinforced concrete for 24 m spans; for 30–36 m spans they are steel or RC, assembled
in two halves where needed for transport.
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18.

SHEET S 4.03 – 4.04 — LANTERNS
Two lantern outlines for two shell geometries
Cylindrical shells · 4.03
Double-curvature shells · 4.04
A rectangular trapezoidal lantern runs the full length of the shell, giving continuous light
and ventilation along every wave.
The lantern outline is broken, opening only over the middle section of each wave, following
the shell's changing curvature.
6 or 12 m less
lantern length vs. shell span
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3m
steel lantern frame spacing
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19.

4.05
SHEET 4.05
A structural plate roof of ferrocement elements
• A roof of regular structure (Series 1.260-1), assembled from two basic ferrocement element types — a pyramidal element
and a ribbed plate.
• The pyramidal elements sit apex-down in a grid, forming a shallow space-truss only about 1 m deep overall.
• Developed for spanning a full column bay as a flat, lightweight ceiling-and-roof plate — no separate truss or purlin layer is
needed.
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SHEET 4.05 — ELEMENTS
Pyramidal elements and ribbed plates
• Each pyramidal element is 3×3 m in plan and 0.9 m high, formed as
four equilateral pyramids with thickened, reinforced edges and a
reinforced apex.
• Ordinary ribbed plates, 1.5×1.5 m, rest on cut corners at the
pyramid apexes; reinforced overhang plates rest directly on the
apexes and lock around them.
• Embedded steel plates at every pyramid corner and apex connect
the pieces to each other and to the ribbed plates above.
Fig. 31. Plan of the structural plate and isometric of pyramidal / ribbed elements (Sheet 4.05).
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21.

SHEET 4.05 — DIMENSIONS
A one-metre-deep plate spans an 18×18 m bay
3×3 / 0.9 m
pyramidal element: plan / height
1.5×1.5 m
ordinary ribbed plate
0.1 / 0.12 m
edging-rib height, ordinary / reinforced
18×18 m
column grid at 1 m plate depth
• Ordinary ribbed-plate flanges are 15 mm thick; reinforced overhang plates thicken to 25 mm, with ribs extending 20 mm into the pyramid recess.
• At this 1 m assembled depth, the plate can also carry a 24 m span where only two supporting sides are used.
• Faces and ribs near a column are reinforced further, since that is where support reactions concentrate.
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22.

SHEET 4.05 — ERECTION
Assembled in 3×12 m blocks on temporary supports
• Elements are cast by vibro-rolling in double steel forms, then assembled into spatial
frames in dedicated jigs before erection.
• The roof is erected in enlarged blocks of 3×12 m; over large covered areas, blocks are held
on temporary supports until joints are grouted and the concrete reaches 70% of design
strength.
• Gaps between pyramids double as space for air ducts and other services — useful for
administrative halls and single free-standing pavilions.
• Developed by the Leningrad Institute of Typology and Experimental Design (LenZNIIEP).
Fig. 32. Pyramidal-element joint at mid-span and at the column, with reinforcement detail (Sheet
4.05).
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23.

4.06
SHEET 4.06
A space-frame of hot-rolled steel profiles
• A "structure"-type spatial bar system, assembled from standard blocks, for covering buildings on a 12×18 or 12×24 m
column grid.
• Each covering block is built from linear chords and diagonals plus planar end trusses — all lattice bars between top and
bottom chords lie in bisector planes.
• Blocks are classed as ordinary (mid-field) or wall-adjacent — the latter carry consoles that support the profiled roof
decking.
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SHEET 4.06 — GEOMETRY
Standard blocks on a 12×18 / 12×24 m grid
• Upper chords are rolled I-sections and directly support the profiled steel decking; every other member —
diagonals and the lower chord — is built from single angles.
• Block support centres are shifted 0.18 m off the building axis; longitudinal system axes are spaced 2.91 m
apart.
• Roofing is roll roofing over insulation on the profiled deck, with skylight lanterns optional along the ridge.
Fig. 33. Space-frame roof block, cross-sections at ridge and end
wall, and the corner support (Sheet 4.06).
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SHEET 4.06 — DIMENSIONS
A shallow, repetitive steel lattice
2.91 m
spacing of longitudinal system axes
1.45 m
block height at the chord axes
1.62–1.67 m
overall structural depth
• The 0.18 m support offset and 2.91 m axis spacing together set the block height of 1.45 m measured at the chord axes.
• Overall depth grows to 1.62–1.67 m once the actual rolled-section profiles are added, depending on which sizes are used.
• Elements are shop-fabricated, delivered to site in containers, and joined by high-strength bolts at typical nodes and welding at the end-truss joints.
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SHEET 4.06 — CONNECTIONS & MOVEMENT JOINTS
Bolted field joints, a fluoroplast-film sliding support
• Corner joints within a block use normal-precision high-strength bolts with welded-on gusset plates; end-truss joints are
welded.
• Temperature (expansion) sections are limited to 72×72 m. Transverse joints sit on single columns; longitudinal joints use
paired columns with a 1 m insert.
• Sliding movement at single-column joints is achieved with a 0.4 mm fluoroplast film bonded to the support plate and bent 20
mm around the anchor-bolt washer edges.
• At roof level, the transverse temperature joint is closed with a flexible cylindrical apron (fartuk); longitudinal joints and edgecolumn ties are covered by consoles.
Fig. 34. Block truss fragment, upper-chord block
joint, and insulated joint fill (Sheet 4.06).
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SHEET 4.06 — SERVICES & CRANES
Suspended monorail cranes to 3.2 t
• Blocks can carry suspended monorail electric cranes of up to 3.2 t in either direction of the grid.
≤ 3.2 t
suspended monorail electric crane
• Where a heavier suspended-transport load is needed, forces are spread to neighbouring lower-chord
nodes through distributing beams.
• In multi-span buildings with internal drainage, water-intake funnels sit at the middle of the end truss;
roof slope is carried by the decking, not the frame.
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4.07
SHEET 4.07
A space-frame of electric-welded steel tubes
• A tubular "structure"-type spatial bar system, blocked for conveyor-line (assembly-line) fabrication — the MArkhI system,
from the Moscow Institute of Architecture.
• Inclined tubular rods and spherical nodal elements form a reticular plate of repeating pentahedral (square-base pyramid)
modules.
• Applicable to roofs on grids up to 24×24 m, and to multi-storey floors on a 9×9 m grid carrying up to 0.5 t/m².
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SHEET 4.07 — GEOMETRY
A reticular plate of pentahedral modules
The reticular plate is built from single-type tubular rods and universal spherical nodal elements — combined,
they form an equilateral pyramid on a square base (a pentahedron).
Support height runs from 1/10 to 1/25 of the span for contour (edge) support, and from 1/12 to 1/40 for
internal-contour or multi-span support.
Flat plate surfaces suit low-slope roofing and light suspended ceilings; the inter-truss space carries air ducts
and other services.
This drawing's own roof build-up: profiled steel decking, vapour barrier, 50 mm polystyrene insulation, and a
four-layer roll roofing membrane.
Fig. 35. Roof build-up and support column of the tubular space-frame,
spanning 27 m (Sheet 4.07).
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30.

SHEET 4.07 — THE BALL -JOINT CONNECTION
One universal spherical node, bolted rods
• Tubular rods are ⌀57–102 mm, wall thickness 3.5–7 mm, cut to a module length of 1.5 or 3 m
between node centres.
• The nodal element is a universal hemispherical piece, ⌀100–120 mm, with radial threaded holes
matching each rod's angle in the lattice.
• Each rod ends in a fitting — a liner, pin, and bolt — that screws into the node through a hex
sleeve; tightening the bolt locks the connection for both tension and compression.
Fig. 36. Lower node (spherical element with tube rods) and the support-column / eaves
expansion detail (Sheet 4.07).
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31.

SHEET 4.07 — ERECTION & USE
Light enough to lift and rotate into place
20–30 kg/m²
assembled plate self-weight
≤ 10 t
erection-crane lifting capacity
• Because the assembled plate is so light (20–30 kg/m²), roofs over small point buildings are fully assembled at ground level, then lifted whole by a self-propelled crane of
up to 10 t.
• Roofs over elongated spans are erected in blocks at least two modules wide, then screwed together with intermediate connecting rods once in position.
• The system also enables a "free" flexible plan below — supports can sit off the structural grid, at whichever node suits the layout, without a subdivided ceiling.
• Developed by the Department of Building Structures, Moscow Institute of Architecture.
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32.

SUMMARY
Seven sheets at a glance
4.01
Spherical shell, flat 3×3 m plates, RC contour truss
4.02
Multi-wave spherical shell, 3×6 m plates, steel contour truss
4.03
Cylindrical multi-wave shell, RC edge beams
12×24 – 18×36 m grid
4.04
Double-curvature multi-wave shell, steel edge arches
12×24 – 18×36 m grid
4.05
Ferrocement pyramidal + ribbed structural plate
4.06
Space-frame, hot-rolled profiles, bolted + welded nodes
4.07
Space-frame, welded tubes, spherical ball-joint nodes
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24 / 30 / 36 m
18×24 / 18×30 m
≤ 18×18 m, or 24 m span
12×18 / 12×24 m grid
≤ 24×24 m (9×9 m floors)
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CHAPTER 4 · CLOSING
From flat plates to spherical shells,
from rolled bars to a welded space-frame
• All seven systems trade a beam-and-truss roof for a shell or reticular plate that carries load mainly in membrane or axial action — less material,
longer spans, a clearer column-free floor.
• Precast RC shells (4.01–4.04) and the ferrocement plate (4.05) suit standard industrial spans up to about 36 m; the steel space-frames (4.06–
4.07) reach the largest or most irregular grids, up to 24×24 m and beyond.
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