Research by:

Dipl.-Ing. Frank Jensen

and

Prof. Sergio Pellegrino

Sponsored by:

Søren Jensen
Consulting Engineers as

Planar Retractable Roofs

The work presented below is based on an existing type of circular bar structure which forms a series of interconnected rhombuses and that is capable of retracting towards the outer boundary of the structure. The text below is only considering circular structures though any plan shape can be constructed.

An animation of the scheme can be found by clicking here.

Figure 1: Artist impression of retractable roof structure by Lake Associates

By analysing the motion of this type of bar structure a method has been derived for finding simple rigid shapes that can be attached to the bar structure and which do not interfere with each other when the structure moves. An analytical solution for the shape of these rigid cover elements has also been derived.

Figure 2: Existing type of retractable structure

The cover elements forms a gap free surface in both the open and closed position. The shape of these elements is basically triangular, but a number of features - which need to fulfill certain conditions of periodicity - can be incorporated. For example it is possible to chose a shape consisting of circular arcs such that a perfectly circular opening is obtained when the structure is retracted, see Figure 1.

If certain geometric conditions are fulfilled, the joints of the bar structure fall within the covering elements, in which case the bar structure is no longer needed and the whole structure can be made from rigid plate elements alone.

Two physical models of these plate structures have been made to demonstrate some of these new solutions. For larger animations please click here.

Figure 3: Physical model of plates structure

A unified design method for both types of structures is proposed in the report: Cover Elements for retractable roof structures (in pdf).

Scissor Hinges

Retractable structures using scissor hinges were largely pioneered by the Spanish engineer Pinero and have been further developed by Escrig and Zeigler.

A large number of structures that can be opened and closed are based on the well know concept of scissor hinge system. The scissor hinge or cylindrical joint allows one relative rotation, about is own axis, between connected members while other relative rotations and translations are inhibited. These hinges allow structures to be made so that they can be compacted using the principle of the lazy-tong. This type of structure has been used by designers to construct many demountable structures, some of which could be classified as retractable roofs though they require manual intervention for opening and closing. A considerable problem for structures that make use of scissor hinges is how to connect them to a permanent foundation.

One major advantage of these structures is the relative simplicity of the joints, compared to other deployable structures.

The Iris Dome

The American engineer Hoberman made a considerable advance in the design of retractable roof structures based on scissor hinges when he discovered the simple angulated element. This element consists of two identical angulated bars connected together by a scissor hinge at node E, see Figure 1, and forms the basis of a new generation of retractable structures.


Figure 4: Angulated element

This element is able to open and close while maintaining the end nodes A, B, C, D on radial lines that subtend a constant angle. Using these elements Hoberman created the retractable roof of the Iris Dome, shown in Figure 2, and other foldable structures.

Figure 5: Iris Dome at EXPO 2000 in Hanover, Germany

An enclosure can be created by covering angulated elements with rigid plates which are allowed to overlap in the retracted position. Several different designs have been proposed by Hoberman. The Iris Dome consists of a large number of plates and bars, and requires many hinges. This can causes potential problems with reliability and at present no large scale structure has been built using this system.

Multi-angulated elements

Further progress on this type of structures was made possible by the discovery by Z. You & S. Pellegrino of using so-called multi-angulated elements. Each multi-angulated element is composed of a number of bars, which are rigidly connected to each other, instead of separate angulated elements as used by Hoberman. Retractable structures made from two layers of such elements connected only by cylindrical joints are shown below. The two interconnected layers create a series of rings, each consisting of a ring of rhombuses.

An entire family of these structures has been identified; they all have the ability to retract radially towards the perimeter and can be generated for any plan shape. This makes them particularly interesting for sporting venues where retractable roofs must be able to retract towards the perimeter of the structure.

Figure 6: Deployable structure using angulated elements

However, so far the system has only been made to work for motion in a plane. To design practical structures any solution that would work in two dimensions, can be projected vertically onto a 3-dimensional surface. Such a structure is shown below. As the perimeter of the structure varies during retraction the support conditions need to allow this motion. A possible solution, where the structure is mounted on pinned columns, is demonstrated by the model shown in Figure 4. Further work on support conditions by Kassabian showed that, if a rigid body rotation of the structure is allowed, then the motion of each angulated element is a pure rotation about a fixed point and hence can be described by a circle. Therefore it is possible to support the structure on a number of fixed points each corresponding to the centre of one of these circles. This has been demonstrated by Kassabian & Pellegrino (P.E. Kassabian and S. Pellegrino, Retractable roof structures, Proceedings Institution of Civil Engineers Structures & Buildings 134, Feb 1999, 45-56).

Figure 7: Deployable "spherical" structure using angulated elements

Cover Plates

Kassabian suggested that to provide a retractable roof using this type of structure, a series of rigid cover elements used should be attached to the multi-angulated elements, instead of using membranes which repeatably fold and unfold. The shape of these cover elements should be such that they neither interfere nor overlap during motion. They should also provide a continuous, i.e. gap free, covering surface in both the open and closed positions of the structure. Using a kinematical approach, two possible designs were developed. Each cover is attached to a single angulated element so the motion of the structure is not inhibited.

Figure 8: Previous covers for retractable structure

Note that neither of the above solutions takes account of the physical size of joints and members in a real structure.

Hoberman have used a similar solution for his innovative Flight Ring, see Figure 9 below.

Figure 9: Hoberman's Flight Ring

Parallelogram

Consider a rigid plate attached to two parallel bars in a parallelogram. This rigid body eliminates the mechanism of the pin-jointed parallelogram. If a straight cut at an inclination angle, Λ, is made in the plate then the structure will regain its mechanism. The line of the cut is called the inclination line. Because the two plates are not allowed to overlap, the angle φ can only increase, or decrease - depending on the inclination angle. In each case, the motion of the parallelogram has to stop when the gap between the two plates is closed again. This is shown in Figure 10.

Figure 10: Motion of parallelogram with two rigid plates

Multi-angulated parallelograms

For the bar structure this can be used to find the shape of cover elements which can be attached to the bar structure without restricting its motion.

Two neighbouring angulated elements in the same layer form, together with bars in the elements of the other layer, a series of rhombuses. The limits to the rotation of these rhombuses are determined by the overall configuration of the structure. It is therefore possible to define two plates, using a single inclination line, that can be attached to the angulated elements without restricting their motion.

Considering the complete bar structure, a series of inclination lines can be drawn between the angulated elements. This creates a number of wedge shaped cover elements equal to the number of angulated elements in each layer, see Figure 11.

The tips of these cover elements must coincide at the centre of the structure to avoid gaps.

Figure 11: Motion of parallelogram with two rigid plates

Reduction angles

The practical limitations for the motion of real structures can be taken into account by introducing so called reduction angles. These simply modify the limits for the parallelograms and can thus easily be included in the definition of the inclination angle.

Further properties of parallelograms

During the motion of the parallelogram in Figure 12 the upper bar and cover element shifts by L relative to the lower bar and cover element. The shift is parallel to the inclination line and therefore a point on the boundary of one cover element will come into contact with two different points y on the other element, in the two extreme positions, see Figure 12. The distance between the two points of contact is the shift L.

Figure 12: Point y shifts by a distance L during motion

Consider two cover elements a and b, which have a non-straight boundary as shown in Figure 13. The shared boundary between the two elements is shifted as the parallelogram is distorted. Therefore the edge of plate b must have the same features at two different positions to prevent overlap and gaps between the plates. If the common boundary between the two elements is longer than L, then the same features of b must be repeated also in a. Hence, cover elements with common boundaries longer than L must be such that all features have a periodic pattern as shown in Figure 13. As the motion is parallel to the straight inclination line, the periodicity of the boundary is expressed in terms of a distance parallel to the inclination line.

Figure 13: Plates with period L

Limits on Modified Shapes

Not all modified shapes that satisfy the rule of periodicity are possible. The displacement increment of any point on a cover element is instantaneously perpendicular to the linking bars, and equal to the velocity of the bar that it is attached to. Thus, by describing the displacement of a joint in the parallelogram, the motion of all other points is also found. This can be used to determine the limits for shape changes of the cover elements that do not inhibit the motion.

Thus by defining a single point on the inclination line the limits for all possible shape changes can be determined. These limits are shown diagrammatically in Figure 14.

Figure 14: Upper and lower boundaries established through the motion of bars a and b respectively

Various designs

Using the methods described above two different types of structures can be designed. The first is a bar structure which is covered by a layer of plate elements. It would be possible to use two layers of cover elements fixed to different layers of bars. As the motion of the structure is determined by the bar structure, the cover elements do not control or restrict the kinematics of the structure and thus each cover element can be fixed to an angulated bar element in many different ways.

The second type of structure is purely a plate structure and all the joints are directly between plate elements. This structure is made from two layers of identical plate elements that are designed so that the mechanism of this structure is equal to that of the previous bar structure.

Both types of structures can be designed with straight or non-straight boundaries.

Circular opening

To form a circular opening in the open position, the boundary of the cover elements must be formed by circular arcs. For the arcs to form a circle they must also join up at the ends in this position. Therefore the length of the arcs along the inclination line must be equal to the period L and the radius of the arcs must be equal to the radius of the opening. These conditions are satisfied by the design shown in Figure 6.

Figure 15: Design with circular opening

Physical models

Two physical models have been built to demonstrate the plate structures described above. Both of these have been designed using periodical boundaries with arcs of a predetermined radius. They were constructed from spark-eroded, 16 gauge Al-alloy plate. The joints were made from plastic snap rivets.

Figure 16: First physical model

Figure 17: Second physical model with near-circular opening

Animation

Quicktime movie of a planar scheme [2.2 MB]