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.

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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.

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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).

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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.

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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