Index
1. Introduction................................................................................................................1
2. Barrel vaults...............................................................................................................2
3. Domes........................................................................................................................4
4. Domes Construction...................................................................................................7
4.1. Framing........................................................................................................................7
4.2. Cladding........................................................................................................................7
4.3. Stability.........................................................................................................................7
4.4. Analysis.........................................................................................................................9
5. Modern tendencies...................................................................................................11
6. References................................................................................................................12
Lattice shells 1/13
1. Introduction
The main difference between double layer grids and latticed shells is the form.
For a double layer grid, it is simply a flat surface. For latticed shell, the variety of forms
is almost unlimited. A common approach to the design of latticed shells is to start with
the consideration of the form a surface curved in space. The geometry of basic surfaces
can be identified, according to the method of generation, as the surface of translation
and the surface of rotation. A number of variations of form can be obtained by taking
segments of the basic surfaces or by combining or adding them. In general, the
geometry of surface has a decisive influence on essentially all characteristics of the
structure: the manner in which it transfers loads, its strength and stiffness, the economy
of construction, and finally the aesthetic quality of the completed project.
Latticed shells can be divided into three distinct groups forming singly curved,
synclastic, and anticlastic surfaces. A barrel vault (cylindrical shell) represents a typical
developable surface, having a zero curvature in the direction of generatrices. A spherical
or elliptical dome (spheroid or elliptic paraboloid) is a typical example of a synclastic
shell. A hyperbolic paraboloid is a typical example of an anticlastic shell.
The inherent curvature in a latticed shell will give the structure greater stiffness.
Hence, latticed shells can be built in single layer grids, which is a major difference from
double layer grid. Of course, latticed shells may also be built in double layer grids.
Although single layer and double layer latticed shells are similar in shape, the structural
analysis and connecting detail are quite different. The single layer latticed shell is a
structural system with rigid joints, while the double layer latticed shell has hinged
joints. In practice, single layer latticed shells of short span with lightweight roofing may
also be built with hinged joints. The members and connecting joints in a single layer
shell of large span will resist not only axial forces as in a double layer shell, but also the
internal moments and torsions.
Since the single layer latticed shells are easily liable to buckling, the span should
not be too large. There is no distinct limit between single and double layer, which will
depend on the type of shell, the geometry and size of the framework, and the section of
members.
Spatial Structures
Lattice shells 2/13
2. Barrel vaults
A barrel vault is obtained by arching a grid along one direction. The result is a
cylindrical form that may involve one, two or more layers of elements. Some examples
of barrel vault configurations are shown in Figure 1. Figure 1.a shows a single layer
barrel vault that is obtained by arching a diagonal flat grid. A barrel vault with a
diagonal pattern is often referred to as a lamella barrel vault. The barrel vault in Figure
1.b is similar to the previous but has a three-way pattern. A double layer barrel vault is
shown in Figure 1.c with both the top and bottom layers having a two-way pattern.
Also, the barrel vault of Figure 1.d has a top layer and a bottom layer with
interconnecting web elements. However, in this case the disposition of the elements
results in a truss barrel vault, that is, a barrel vault that consists of intersecting curved
trusses.
The shape of the cross-section of a barrel vault may vary along its longitudinal
axis. Examples of this are shown in Figure 1.e and f. The surface of the lamella barrel
vault of Figure 1.e is a part of a hyperboloid of revolution. Also, the surface of the
barrel vault of Figure 1.f is a part of an ellipsoid of revolution.
An example of a compound barrel vault is shown in Figure 1.g. A compound
barrel vault consists of two or more barrel vaults that are connected together along their
sides. The compound barrel vault of Figure 1.g is obtained by combining three barrel
vaults identical to the one in Figure 1.b.
The cross-sections of the barrel vaults in Figure 1 are circular. However, a barrel
vault may have a cross-section which has an elliptic, a parabolic or many other shapes.
Spatial Structures
Lattice shells 3/13
Figure 1
Spatial Structures
Lattice shells 4/13
3. Domes
A dome is a structural system that consists of one or more layers of elements
that are arched in all directions. The stresses in a dome are generally membrane and
compressive in the most part of the shell except circumferential tensile stresses near the
edge and small bending moments at the junction of the shell and the ring beam. Most
domes are surfaces of revolution. The curves used to form the synclastic shell are
spherical, parabolic, or elliptical covering circular or polygonal areas. Some commonly
used basic single layer dome configurations are shown below.
The dome shown in Figure 2 is a ribbed dome.
A ribbed dome consists of a number of intersecting
ribs and rings. A rib is a group of elements that lie
along a meridional line and a ring is a group of
elements that constitute a horizontal polygon. A
ribbed dome will not be structurally stable unless it is
designed as a rigidly-jointed system.
When the number of ribs is large then there
could be a problem regarding the overcrowding of
the elements near the crown. One way of avoiding
this problem is to cut back the upper parts of some
of the ribs. Such an operation is referred to as
trimming. An example of a trimmed ribbed dome is
shown in Figure 3 when every other rib is trimmed
to the level of the fourth ring from the top.
A modified form of a ribbed dome is obtained by bracing the quadrilateral
panels of the dome. The result is a dome configuration that is referred to as a Schwedler
dome. A simple example of a Schwedler dome is shown in Figure 4. Another example
is shown in Figure 5. This dome configuration also involves trimming to avoid
overcrowding of the elements at the upper part of the dome.
Spatial Structures
Figure 2
Figure 3
Lattice shells 5/13
Figure 4 Figure 5 Figure 6
An example of a lamella dome is shown in Figure 6. A lamella dome has a
diagonal pattern and may involve one or more rings.
The dome configurations shown in Figure 7 and Figure 8 are two examples of a
family of domes that are referred to as diamatic domes. The dome shown in Figure 7 is
an example of a basic diamatic form consisting of triangulated sectors. The pattern of
the diamatic dome of Figure 8 is obtained from a denser version of the dome of Figure 7
by removing every other line of elements.
Figure 7 Figure 8
The dome shown in
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found represents an example of
the family of grid domes. A
grid dome is obtained by
projecting a plane grid pattern onto a curved surface.
Spatial Structures
Figure 9
Lattice shells 6/13
Grid domes are normally rather shallow with
their rise to span ratios being smaller than the other
types of domes. A geodesic dome configuration is
shown in Figure 10 and Figure 11. A dome of this kind
is obtained by mapping patterns on the faces of a
polyhedron and projecting the resulting configuration
onto a curved surface.
The dome of Figure 10 is obtained by mapping
a triangulated pattern on five neighbouring faces of an
icosahedron (20-faced regular polyhedron) and
projecting the result onto a sphere which is concentric
with the icosahedron. The geodesic dome of Figure 11
is obtained in a similar manner with the initial pattern
chosen such that the resulting dome has a honeycomb
appearance. The main advantage of this type of dome
is that all members are of approximately equal length and the dome surface is
subdivided into approximately equal areas.
The configurations shown above represent the basic dome patterns but there are
many other dome patterns that are obtained as variations of the basic forms. Also, there
are a large number of double layer (and multilayer) dome patterns that may be obtained
from the combinations of the basic patterns. Included in these are truss domes that
consist of intersecting curved trusses. An important point that should be borne in mind
is that one should be careful in using single layer domes unless the jointing system
provides sufficient rigidity for the connections and that the elements are designed for
resisting bending and shear in addition to the axial forces. Otherwise, the structures will
be prone to snap through buckling. This comment also applies to the case of single layer
barrel vaults.
Spatial Structures
Figure 10
Figure 11
Lattice shells 7/13
4. Domes Construction
4.1.Framing
Dome framing may be single or double layer. Large domes must be double layer
to prevent buckling. All types of members have been used. Hollow sections with welded
joints are attractive where the steelwork is exposed. Members are usually straight
between nodes. The dome must be broken down into suitable sections for shop
fabrication. Lattice double-layer domes, can be assembled on site using bolted joints.
4.2.Cladding
Cladding causes problems because panel dimensions vary in most domes and
twisted surface units are often needed. The systems used are:
• Roof units, triangular or trapezoidal in shape, supported on the dome frame.
These may be in transparent or translucent plastic or a double-skinned metal
sandwich construction.
• Timber decking on joists with metal sheet or roofing felt covering.
• Steel decking on purlins and dome members with insulation board and
roofing felt. This can only be used on flat surfaces.
4.3.Stability
Flexible domes, that is shallow or single-layer large-span domes present a
stability problem. There are three distinct types of buckling:
1) Member buckling: occurs when an
individual member becomes unstable,
while the rest of the space frame
(members and nodes) remain unaffected
(Figure 12). The buckling load Pcr of a
straight prismatic bar under axial
compression is given by:
Pcr=∝π2 Ee I
l2
Spatial Structures
Figure 12
Lattice shells 8/13
where ‘Ee’ is de effective modulus of elasticity that coincides with the
Young’s modulus in the elastic range, ‘I’ is the moment of inertia of the
member, ‘l’ is the length of the member and ‘α’ is a coefficient that takes
different values depending on some parameters.
2) Local buckling: consists of a snap-through
buckling which takes place at one joint. Snap-
through buckling is characterized by a strong
geometrical non-linearity (Figure 13). Local
buckling is apt to occur when the ratio ‘t/R’ is
small (where ‘t’ is the equivalent shell
thickness and ‘R’ is the radius of curvature).
Local buckling is greatly affected by the stiffness of and the loads on the
adjacent members. Buckling load ‘qcr’ in terms of uniform normal load per
unit area can be expressed as:
A E l
12 R3≤qcr ≤
A E l
6 R3
Where ‘A’ is de cross-sectional area of the member, ‘E’ is the modulus of
elasticity, ‘l’ is the length of the member and ‘R’ is the radius of an
equivalent spherical shell.
3) Overall buckling: occurs when a relatively
large area becomes unstable, and a relatively
large number of joints is involved in the
buckle (Figure 14). For most cases, in overall
buckling, the wave length is significantly
greater than the member length. Local
buckling often plays the role of trigger for overall buckling. The buckling
formula for a spherical shell subjected to a uniformly distributed load
normal to the middle surface can be expressed as:
qcr=k E( tR )
2
Where ‘E’ is the modulus of elasticity, ‘t’ is the thickness and ‘R’ is the
radius of an equivalent spherical shell.
Spatial Structures
Figure 13
Figure 14
Lattice shells 9/13
The buckling of reticulated domes in most cases has a form of local concave on
the surface as shown below, starting from snap-through of some joint and gradually
expanding its area to become a concave. The concave emerges at different place for
different type of reticulated domes: it starts from some joint of a main rib for lamellar
domes (Figure 15), from some joint of the third ring (from bottom) for Schwedler
domes (Figure 16), and from some joint on the triangular surface for geodesic
domes(Figure 17). The first buckling of a dome is characterized as a limit point of the
load-deflection curve, and the corresponding critical load is taken as the limit load of
the dome.
Figure 15 Figure 16 Figure 17
4.4.Analysis
The ribbed dome with ribs hinged at the base and crown and the pin-jointed
Schwedler dome subjected to uniform load are statically determinate. The Schwedler
dome under non-uniform load and other types of domes are highly redundant. Shell
membrane theories can be used in the analysis of Schwedler domes under uniform load.
Standard matrix stiffness space frame programs can be used with accuracy to analyse
stiff or double-layer domes. The behaviour of flexible domes may be markedly
nonlinear and the effect of deflection must be considered. Dome stability must be
investigated through nonlinear analysis. A linear analysis will be sufficiently accurate
for design purposes in many cases.
Using membrane theory for spherical shells, the forces in the members of a
Schwedler dome can be determined approximately. Membrane theory gives the
following expressions for forces at P (Figure 18).
Spatial Structures
Lattice shells 10/13
Figure 18
The meridional or rib force (kN/m) is
NΦ= w R1+cosΦ
+ q R2
The hoop on ring force is
Nθ=w Rcos Φ−11+cosΦ
+ 12
q R cos2 Φ
Where ‘w’ is dead load, ‘q’ is the imposed load, ‘K’ is the shell radius and ‘ɸ’ is
the angle at point of force P.
Spatial Structures
Lattice shells 11/13
5. Modern tendencies
Glass or acrylic panels are used to cover vertical, horizontal and slanted
surfaces. It notes that they also become more and more popular for curved structures
with larger windows to cover. The trend is explained by the desire of the architect to
achieve a good balance between natural lighting and thermal insulation.
Traditional material covers for industrial buildings were the metal panels, which
have a poor insulation but acceptable for these buildings. The type depends on the load,
the slopes, the architectural views...
In recent years, the flexible textile has gain importance and has been used in
exhibition pavilions, shopping centers etc. Many materials are vinyl and polyester with
fire-resistant coating. Coating has possibilities endless, they can be transparent, opaque
or multilayer (with increased thermal insulation). One easy example is the Eden Project.
Spatial Structures
Lattice shells 12/13
6. References
• Steel Structures. Practical design studies. T. J. MacGinley. E & FN SPON. 2005
Second Edition.
• Space Frame Structures. Structural Engineering Handbook. Tien.T. Lan. Ed
Chen Wai-Fah. Boca Raton: CRC Press LLC. 1999.
• University of Surrey (UK) notes.
• Design formulas for stability analysis of reticulated shells. S.Z. Shen. Advances
in steel structures. ICASS ’99. S.L. Chan and J.G. Teng
• The Dome Builder’s Handbook. Edited by John Prenis. Running Press.
Spatial Structures
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