4 Spatial rings and domes
4.1 Introduction
The conventional scissor- like element is one of the most popular mechanisms utilised to form large foldable assemblies like pop- up display stands,
folding chairs, portable shelters and even swimming pool covers. However,
readers should be aware that some of the assemblies are in fact deformable
structures, as the expansion induces strain within structural components.
They further differ from normal deformable structures in that the strain in
these structures falls to zero in the initial folded and final expanded configurations, resulting in self- locking once being completely folded or
expanded because any shape change away from these zero- strain configurations requires energy. It is difficult to find a generic design solution for
this type of assemblies since detailed analysis at every expansion configuration is required to ensure that the stress does not exceed appropriate limits.
Examples of this group of assemblies include Zeigler’s dome (1981), structures proposed by Gantes (1991) and a swimming pool cover by Escrig et
al. (1996).
The number of strain-free motion assemblies using the conventional
scissor- like element is rather limited. Typically they are open chains, with the
notable exception of the domes proposed by Meurant (1993) and SánchezCuenca (1996). Since we have shown in Section 3.1 that the conventional
scissor- like element alone cannot be used to build a mobile planar closed
chain which has mobility, the mobile assembly always takes a spatial form.
This only becomes possible with the utilisation of the Sarrus linkage for connection of planar chains of conventional scissor- like elements.
The Sarrus linkage (Sarrus, 1853) was the first reported spatial overconstrained linkage. It consists of six links, Figure 2.11 and reproduced in
Figure 4.1, connected together by two sets of three parallel hinges. A
simple spatial closed chain of four conventional scissor- like elements can
be constructed as shown in Figure 4.2(a). These elements are identical and
have pivots at the middle of the straight rods. At each of the corners of the
assembly, the upper and lower joint pieces and four beams that are hinged
together by them form a Sarrus linkage, which are given by their axes 1,
4.1 Introduction
The conventional scissor- like element is one of the most popular mechanisms utilised to form large foldable assemblies like pop- up display stands,
folding chairs, portable shelters and even swimming pool covers. However,
readers should be aware that some of the assemblies are in fact deformable
structures, as the expansion induces strain within structural components.
They further differ from normal deformable structures in that the strain in
these structures falls to zero in the initial folded and final expanded configurations, resulting in self- locking once being completely folded or
expanded because any shape change away from these zero- strain configurations requires energy. It is difficult to find a generic design solution for
this type of assemblies since detailed analysis at every expansion configuration is required to ensure that the stress does not exceed appropriate limits.
Examples of this group of assemblies include Zeigler’s dome (1981), structures proposed by Gantes (1991) and a swimming pool cover by Escrig et
al. (1996).
The number of strain-free motion assemblies using the conventional
scissor- like element is rather limited. Typically they are open chains, with the
notable exception of the domes proposed by Meurant (1993) and SánchezCuenca (1996). Since we have shown in Section 3.1 that the conventional
scissor- like element alone cannot be used to build a mobile planar closed
chain which has mobility, the mobile assembly always takes a spatial form.
This only becomes possible with the utilisation of the Sarrus linkage for connection of planar chains of conventional scissor- like elements.
The Sarrus linkage (Sarrus, 1853) was the first reported spatial overconstrained linkage. It consists of six links, Figure 2.11 and reproduced in
Figure 4.1, connected together by two sets of three parallel hinges. A
simple spatial closed chain of four conventional scissor- like elements can
be constructed as shown in Figure 4.2(a). These elements are identical and
have pivots at the middle of the straight rods. At each of the corners of the
assembly, the upper and lower joint pieces and four beams that are hinged
together by them form a Sarrus linkage, which are given by their axes 1,