Seville, June 1996

Expandable arches.

R. Sastre

School of Architecture of the Vallès, Department of Architectural Technology 1, Polytechnic University of Catalunya,

08190 Sant Cugat del Vallès, Spain


Taking profit of the knowledge of lamps with expandable arms, fixed on the walls, or on the table, we have decentered the pinned join in order to achieve an arch, when the arm is expanded. The analysis of this structure has to deal with two situations: when the arch is wholly expanded, working in compression and when the arch is expanding, working in flexion. Delimiting cables can be fixed on the external joints so that certain improvement of flexion resistance can be gained.

If a textile membrane is attached to the inner joints, not only an arch but a real vault is obtained when unfolding the structure. Posterior pretension of this membrane can solve the stability of the whole construction.

Half arches can produce a dome, by solving the top joint where all arches converge. Automatic expansion is also studied in this research.

1 Geometry

In fact, expandable arches are a clear instance of how a geometrical analysis can develop into a particular structure. In this case we'll begin this study by defining the smallest object, the nucleus of a complex whole whose properties or applications may be of great interest.

Let's call X-Element to the simplest structure formed by two bars pin joined through a hole equally placed in both bars. This element can be geometrically defined by L (the length of the bars), a and b (the distance of the hole to the ends of the bar) and (the angle formed by the a-a and b-b lengths).

Being the angle variable, any X-Element can show different positions, but what is really more important is the form that we achieve by connecting a set of X-Elements side by side. We should difference two main possibilities:

EXPANDABLE BEAM. The pin joint is positioned at the middle of the bar, i.e. a = b = L/2. In this case the axis of the achieved form is a line and, by changing , we obtain an expandable beam. This structure has been widely used in small devices as lamps, hangers, etc. or big devices as raisable platforms used to repair street lights or to reach high objects anywhere.

EXPANDABLE ARCH. The pin joint is positioned out of the middle of the bar, i.e. a b; a + b = L. In this case the shape obtained is an arch and, by changing a, we obtain a different arch, it is: different radii (Ri, Re) and different f, being f the whole angle covered by this arch.

It exists a direct relation between L (a + b) and a, on one hand, and the radii (Ri, Re), f and k (number of X-Elements used), on the other hand. Here we show a list of different, interesting equations: (being b = f / k)

a / b = Ri / Re = m / n

b = Re · L / (Ri + Re) ; a = Ri · L / (Ri + Re)

m = a · sin ( a / 2) = Ri · sin ( b / 2)

a = 2 · arcsin (Ri · sin ( b / 2) / a) (1)

b = 2 · arcsin (a · sin ( a / 2) / Ri)

L² = Ri² + Re² - 2 · Ri · Re · cos b

It is not necessary that all the X-Elements forming a set are equal. However, in order to obtain a structure that could be expanded and folded properly we must follow some elementary rules. All bars must have the same length L, which is the same to say that two different X-Elements having the joint fixed at a1 and b1 (a1 + b1 = L) and at a2 and b2 ( a2 + b2 = L) are perfectly compatible in the same expandable structure.

What do we obtain by mixing different but compatible X-Elements? Something that can be very useful: a pluricentered arch. Many applications in building and architecture require large, broad spaces with a not so important height. In these situations we can choose between a single centred arch (which implies that the sides of the building will be useless due to their lack of height) or a three centred arch (what means two small radius arches at each side connected tangentially to a single, big radius arch in the central area.

Not easily, but following strict rules of design, we can obtain expandable structures which can be assimilated to those arches and, still, they'll be able of folding and expanding as a simple, single X-Element arch.

2 Expansion process of an arch

As most elementary expandable arms, the process of expanding an arch is succeeded by lessening or increasing the thickness h of the arch. Evidently h, a, b form a triangle. When the structure is completely folded h equals to L, (h = L = a + b) and the angle is 180º. By decreasing the value of h, the angle a^b decreases as well and, as alfa = 180º - a^b, the angle b increases, what means opening or expanding the arch. According to the equations (1) we observe that there is not direct relation between the variation of the value of h and b. This means that, although we get a uniform variation in h value, the arch will expand or fold in an apparently accelerated motion.

How do we get this motion? There are two distinct ways widely confirmed by use: an hydraulic piston or an endless threaded rod. In the first case the stroke of the piston will cover the variation of h, while in the second case

we'll only have to keep rotating the threaded rod and fix the end of one of the bars forming the X-Element to the one end of the rod, and the end of the other bar to a movable nut progressing along this spinning rod.

No matter which method we use, we'll need power enough to do some work. Mainly, the work to be done consists in moving up and down the centre of gravity of the structure. By studying step by step the motion of the structure as it folds or expands we may assure that the maximum power is needed at the very beginning of the expansion (when the centre of gravity rises quickly), while from the middle on the centre of gravity height decreases and the power is needed to restrain the motion.

As it can be observed in the figures the process of expansion shows a curious behaviour. During more than the half of the time the structure tends to expand vertically, as it was an expandable beam, but quite suddenly in begins to fold and gets its final position in a rather brusque movement.

This peculiarity involves two questions. The first one is the difficulty in setting the movable end of the arch in its final position, since a small movement in the expanding device turns into a big displacement of that end. The other question, which cannot be avoided at all, is the fact that the structure, during its expansion, gets higher an higher to a dimension that almost equals the final span of the arch. This fact supposes many problems in the stability of the whole structure during expansions and foldings.

But not everything is negative in the expansion process. Apart from pluricentered arches, which can be as complicated as the designer may like, the superposition of the different situations of the arch during expansion shows how the final covered space is never invaded by any part of the structure. This means that we can cover an object without moving it previously.

3 Analysis

An expandable arch can be analysed as a typical bar structure. Each X-Element is composed of 4 members, since we suppose that each bar is divided into two members: one from one end to the pin joint and another from the pin joint to the other end. However there is a main trouble when we try to define the nodes of this structure. We know that in matricial analysis of 2D structures nodes mean two displacements and one rotation, but in an X-Element the pin joint means two displacements (X & Y) and two rotations (one for each bar). In fact this is something that cannot be applied straight to most of computer programs, but there exist several methods to simulate this physical device to a group of members and joints. We have used the program WinEVA which shows one of these methods.

Analysis should consider different loading. First of all we must care for self weight, mainly during expansion. If we think about a technique of expansion as the one illustrated in this paper, there will be an important flexion, as a cantilever, when the free end of the arch is about to get the opposite side. This bending moment has to be carried by the two bars of the X-Element connected to the expansion device.

We should consider compression as the normal behaviour of the arch under loads applied to the joints. It's not easy to determine which actions are to be supported, but it seems quite reasonable to associate expanding structures to membrane (or any type of folding) structures. In this case we could use the joints of the arch to attach this membrane. Forces transmitted by this membrane would be diverse according to the load: wind, snow, etc., but would produce an almost pure compression on the structure.

As it happens in any type of arch, we cannot forget the cross stability. Not only during normal "life", because of unsymmetrical loads, but during expansion, if the structure attain a significant height. Under these circumstances we must think of a 3D X-Element, which gives a cross stiffness to the whole, without changing the analysed behaviour of the whole.

4 Conclusion

When this paper was written the author is studying lots of possibilities to us these expandable arches. Some of them have been developed as models of certain significance, but most of them are still waiting for their materialisation. The following pages will show some of this possible applications and we hope to be able to show more pictures at the Conference time.

First, we may look at the results we can obtain by using half arches. Not only we'll reduce height and lack of balance during expansion but, by using a suitable connecting part, we can get closed spaces as domes.

A simple tent can be obtained by fixing a membrane to and 3D-arch, so that by expanding the arch we stretch, at the same time, the membrane. A real case, 6 m in diameter, has been build and carried in a small trailer.