The term 'space structure' refers to a structural system that involves three dimensions. This is in contrast with a 'plane structure', such as a plane truss, that involves no more than two dimensions. To elaborate, in the case of a plane structure, the external loads as well as the internal forces are in a single plane. This is the plane that also contains the (idealised) structure itself, both in its initial unloaded state and in its deformed loaded state. In the case of a space structure, the combination of the configuration, external loads, internal forces and displacements of the structure extends beyond a single plane.

The above definition is the 'formal' definition of a space structure. However, in practice, the term 'space structure' is simply used to refer to a number of families of structures that include grids, barrel vaults, domes, towers, cable nets, membrane systems, foldable assemblies and tensegrity forms. Space structures cover an enormous range of shapes and are constructed using different materials such as steel, aluminium, timber, concrete, fibre reinforced composites, glass, or a combination of these.

• 'lattice space structures' that consist of discrete, normally elongated, elements,
• 'continuous space structures' that consist of components such as slabs, shells, membranes, and
• 'biform space structures' that consist of a combination of discrete and continuous parts.

There are numerous examples of space structures that are built for sports stadiums, gymnasiums, cultural centres, auditoriums, shopping malls, railway stations, aircraft hangars, leisure centres, transmission towers, radio telescopes, supernal structures (that is, structures for outer space) and many other purposes.

The term 'spatial structure' is sometimes used instead of 'space structure'. The two terms are considered to be synonymous.

Space structure forms are at the centre of attention in the present review with emphasis on the geometric characteristics of lattice space structures and, in particular, the families of grids, barrel vaults and domes.

A 'grid' is a structural system involving one or more planar layers of elements [1]. A 'single layer grid', or 'flat grid', consists of a planar arrangement of rigidly connected beam elements. The external loading system for a flat grid consists of forces perpendicular to the plane of the grid and/or moments whose axes lie in the plane of the grid. The reason for classification of a flat grid as a space structure is that its external loads and displacements do not lie in the plane that contains its (idealised) configuration.

A number of basic grid patterns are illustrated in Fig. 1. The 'two-way' pattern, shown in Fig. 1a, is the simplest pattern for a flat grid. It consists of two sets of interconnected beams that run parallel to the boundary lines. The diagonal pattern, shown in Fig. 1b, consists of two parallel sets of interconnected beams that are disposed obliquely with respect to the boundary lines. Figs 1c to 1f show some basic three-way and four-way grid patterns. The basic grid patterns of Fig. 1 are frequently used in practice. However, there are also many other grid patterns that are commonly used. These patterns are normally derived by removal of some elements from the basic patterns of Fig. 1. Two examples of this type of operation are shown in Fig. 2. The grid pattern in Fig. 2a is obtained from a three-way pattern by omitting every other beam line. This is illustrated in Fig. 2c, showing a part of the grid of Fig. 2a with the omitted beam lines shown by dotted lines. The grid of Fig. 2b is obtained from a four-way pattern by removal of a number of beam lines as indicated in Fig. 2d.

Figure 1: Some basic patterns

In designing a grid configuration, one would like to find the most suitable pattern for the particular application. A question that arises naturally in this relation is: Are there some general principles or guidelines through which the structural behaviour of different grid patterns can be classified and used for selecting the 'right' pattern for every design case? The answer is that different grid patterns do indeed have their own characteristics. However, there are no inherent 'good' or 'bad' grid patterns and the suitability of a pattern for each particular case should be considered with regard to the shape and size of the boundary, support positions, loading characteristics, material(s) to be used and the manner in which the structure is to be constructed. These comments also apply in relation to all other space structure forms.

A 'double layer grid' consists of two (nominally) parallel layers of elements that are interconnected together with 'web' elements [1]. Views of some commonly used patterns of double layer grids are shown in Fig. 3. In this figure, the 'top' layer elements are shown by thick lines and the 'bottom' layer elements as well as the 'web' elements are shown by thin lines. The double layer grid of Fig. 3a consists of a two-way top layer and a two-way bottom layer. In the case of the grid of Fig. 3b, both the top and bottom layers have a diagonal pattern. There are also many double layer grids built with a two-way pattern for one of the layers and a diagonal pattern for the other layer.

A double layer grid of a different kind is shown in Fig. 3c. Here, the top and bottom layers are of an identical shape and are positioned such that their plan views are coincident. Also, in this case all the web elements lie in vertical planes. The result is a double layer grid that effectively consists of a number of intersecting plane trusses. A grid of this type is referred to as a 'truss grid'. A truss grid may be regarded as a flat grid whose elements are trusses.

A primary double layer grid pattern, such as the one shown in Fig. 3a, is often used as a basis for the creation of various 'reduced forms' by removing a number of elements. An example of this is shown in Fig. 3d. This grid is obtained from the grid of Fig. 3a by removing the bottom layer and web elements that are connected to a number of bottom layer nodes. A similar process is used for obtaining the reduced grid of Fig. 3e from the grid of Fig. 3b. Also, the diagonal truss grid of Fig. 3f is obtained by removing the non-boundary third-direction trusses of the grid of Fig. 3c.

Figure 3: Examples of double layer grids

Grids may also involve more than two layers of elements, allowing for greater structural depth to cater for longer spans.

(Click on it to enlarge 180k)

There is a fundamental difference between the structural behaviour of flat grids and that of double layer (or multilayer) grids. Namely, flat grids are 'bending dominated' with the elements being under bending moments, shear forces and torques. In contrast, the main internal forces in the elements of double layer (or multilayer) grids are axial forces. Bending moments, shear forces and torques are also present in the elements of double layer (or multilayer) grids in various proportions depending on the cross-sectional properties of the elements and the jointing system. However, the non-axial force effects in these cases are normally secondary.

• Flat grids are often built in reinforced concrete with an integrated slab at the top. Also, a steel flat grid may be combined with a reinforced concrete slab.
• There are many instances of steel double layer (or multilayer) grids with the top layer replaced by (or embedded in) reinforced concrete slabs.
• There are also many examples of double layer (and multilayer) grids with incorporated membrane parts.

A biform space structure that consists of a combination of discrete steel elements and continuous reinforced concrete parts is traditionally referred to as a 'composite space structure'. Also, a biform space structure that consists of a combination of discrete elements (in any material) and continuous membrane parts is referred to as a 'hybrid space structure'. However, it should be noted that, normally, a space structure is considered to be biform provided that both the discrete and continuous parts play significant structural roles. Thus, a double-layer grid, which has a reinforced concrete slab at the top, will not be considered as a composite double layer grid unless the slab and the grid are designed to interact structurally. Also, in the case of a hybrid space structure, the lattice part of the structure is expected to be self-contained (that is, to be load-bearing by itself). Thus, a membrane structure with a number of individual support poles will not be regarded as a hybrid structure.

A 'barrel vault' is obtained by 'arching' a grid along one direction [2]. 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 Fig. 4. Fig. 4a 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 Fig. 4b is similar to the one in Fig. 4a but has a three-way pattern. A double layer barrel vault is shown in Fig. 4c with both the top and bottom layers having a two-way pattern. Also, the barrel vault of Fig. 4d 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 Figs 4e and 4f. The surface of the lamella barrel vault of Fig. 4e is a part of a hyperboloid of revolution. Also, the surface of the barrel vault of Fig. 4f is a part of an ellipsoid of revolution.

An example of a 'compound barrel vault' is shown in Fig. 4g. A compound barrel vault consists of two or more barrel vaults that are connected together along their sides. The compound barrel vault of Fig. 4g is obtained by combining three barrel vaults identical to the one in Fig. 4b.



Figure 4 : Examples of barrel vaults

The cross-sections of the barrel vaults in Fig. 4 are circular. However, a barrel vault may have a cross-section which has an elliptic, a parabolic or many other shapes.

A 'dome' is a structural system that consists of one or more layers of elements that are 'arched' in all directions [3]. The surface of a dome may be a part of a single surface such as a sphere or a paraboloid, or it may consist of a patchwork of different surfaces. Some commonly used basic single layer dome configurations are shown in Fig. 5. The dome shown in Fig. 5a 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 Fig. 5b 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' (after the nineteenth century German engineer J. W. Schwedler who built many domes of this kind). A simple example of a Schwedler dome is shown in Fig. 5c. Another example is shown in Fig. 5d. This dome configuration also involves trimming to avoid overcrowding of the elements at the upper part of the dome.

An example of a 'lamella dome' is shown in Fig. 5e. A lamella dome has a diagonal pattern and may involve one or more rings. An example of a trimmed lamella dome with rings is shown in Fig. 5f.

The dome configurations shown in Figs 5g and 5h are two examples of a family of domes that are referred to as 'diamatic domes' [4]. The dome shown in Fig. 5g is an example of a basic diamatic form consisting of triangulated sectors. The pattern of the diamatic dome of Fig. 5h is obtained from a denser version of the dome of Fig. 5g by removing every other line of elements in a manner similar to that shown in Fig. 2c.

The domes shown in Figs 5i and 5j represent two examples of the family of 'grid domes' [3, 5]. A grid dome is obtained by projecting a plane grid pattern onto a curved surface. The grid dome of Fig. 5i is obtained by projection of a denser version of the pattern of Fig. 2a onto a spherical surface. The grid dome of Fig. 5j is obtained in a similar manner using a denser version of the pattern of Fig. 2b. 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 Fig. 5k. 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 [3, 6]. The dome of Fig. 5k 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 Fig. 5l is obtained in a similar manner with the initial pattern chosen such that the resulting dome has a honeycomb appearance.


The configurations shown in Fig. 5 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 snapthrough buckling. This comment also applies to the case of single layer barrel vaults.

There are many examples of biform domes and barrel vaults both in composite and hybrid forms. A composite dome or barrel vault consists of a steel lattice framework with an incorporated reinforced concrete shell which is designed to interact with the lattice part structurally. A hybrid dome or barrel vault consists of a load bearing lattice framework and structurally active membrane parts.

Continuous domes and barrel vaults have been constructed in various masonry materials since the ancient times and there exist thousands of such structures, including some very impressive ones, throughout the globe. Also, there is the ingenious igloo which is a continuous dome structure made from snow blocks.

• [1] Makowski, Z S (Editor). Analysis, Design and Construction of Double Layer Grids, Applied Science Publishers Ltd, 1981 (Obtainable from Chapman & Hall Publishers)
• [2] Makowski, Z S (Editor). Analysis, Design and Construction of Braced Barrel Vaults, Elsevier Applied Science Publishers Ltd, 1985 (Obtainable from Chapman & Hall Publishers)
• [3] Makowski, Z S (Editor). Analysis, Design and Construction of Braced Domes, Granada Publishing Ltd, 1984
• [4] Nooshin, H and Tomatsuri, H. Diamatic Transformations, Proceedings of the Symposium on Spatial Structures: Heritage, Present and Future, Edited by G C Giuliani, Milan, Italy, June 1995, pp 71-82
• [5] Nooshin, H. A Technique for Surface Generation, Proceedings of the International Symposium on Conceptual Design of Structures, Edited by K U Bletzinger et al, Published by Institute fur Konstruktion und Entwurf II, Stuttgart, Germany, October 1996, pp 331-338
• [6] Nooshin, H, Disney, P and Champion, O. Computer Aided Processing of Polyhedric Configurations, chapter 12 in the book: Beyond the Cube: The Architecture of Space Frames and Polyhedra, Edited by: J Francois Gabriel, John Wiley & Sons, 1997, pp 343-384

Sources of general information about space structures

• [1] International Journal of Space Structures (currently in its twelfth volume), Editors: H Nooshin and Z S Makowski, Published by Multi-Science Publishing Co Ltd, 107 High Street, Brentwood, Essex CM14 4RX, UK
• [2] Makowski, Z S. Steel Space Structures, Michael Joseph Ltd, London, 1965
• [3] Davies, R M (Editor). Proceedings of the First International Conference on Space Structures (held in London, UK, September 1966), Blackwell Scientific Publications, 1967
• [4] Supple, W J (Editor). Proceedings of the Second International Conference on Space Structures, held at the University of Surrey, UK, September 1975
• [5] Nooshin, H (Editor). Proceedings of the Third International Conference on Space Structures, Elsevier Applied Science Publishers Ltd, 1984 (Obtainable from Chapman & Hall Publishers)
• [6] Nooshin, H (Editor). Studies in Space Structures, Multi-Science Publishing Co Ltd, 1991
• [7] Parke, G A R and Howard, C M (Editors). Proceedings of the Fourth International Conference on Space Structures, Thomas Telford Services Ltd, 1993
• [8] Journal of the International Association for Shell and Spatial Structures (currently in its 38th volume), Editor: J Abel, Published by the International Association for Shell and Spatial Structures, Alfonso XII, 3-28014 Madrid, Spain
• [9] Proceedings of Past Conferences Organised by the International Association for Shell and Spatial Structures, Alfonso XII, 3-28014 Madrid, Spain