Mobile and Rapidly Assembled Structures III, C.A. Brebbia & F.P. Escrig (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-817-1
Variable rigidity as a tool of shaping of selferecting structures
R. Tarczewski
Department of Building Structures, Faculty ofArchitecture,
Wroclaw University of Technology, Poland
Abstract
The paper presents a concept of shaping of the space structures formed by means of
post tensioning. Basic structure is completed as a plane form with a bottom chord
made of the cable. Shortening of the bottom chord allows "self-erection" of the
structure. Final shape i.e. curvature of the upper chord is a result of interaction
between bending rigidity of the structure and the tensile force in the cable. Structures more rigid in the central part than at the sides gains curvature with smaller
radius while structures more rigid at the sides than at the central part gains curvature with bigger radius. The simple way of changing global rigidity of the structure
is to change its height. This concept allows to easily create a wide range of structures suitable for any purpose especially of large span.
1 Introduction
The paper presents a concept of shaping of self-erecting structures by means of
predetermination its initial topology.
Application of large deformation for shaping geometrical form of structures is known
and presented in publications (e.g. Schmidt [1] [2]). This method can be applied for
plane metal structures as well as for space structures. Significant for this method is
that the shaping and erecting of the structure are combined into one technological
process. Construction of this type is shaped in a sequence of successive stages.
After assembling of the structure at the ground level, the cable is mounted in place
of the bottom chord. Than the structure is 'deformed' by means of shortening of the
cable i.e. its longitudinal axis becomes a curve. Rigid elements of bottom chord are
mounted in position and the cable can be removed as the structure presents its final
configuration.
Mobile and Rapidly Assembled Structures III, C.A. Brebbia & F.P. Escrig (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-817-1
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In his former papers the author proposed to modify this method in the way that the
cable used for erection of the structure serves also as its bottom chord with no rigid
elements added (Tarczewski [3] [4]).
2 General description of proposed self-erecting structures
Unlike names of traditional structures made of bars, term 'self-erecting' concerns
both structures' forms and methods of assembling. It is significant for these structures that the tensioned element used for shaping is than used as a permanent component part of the completed structure. Material solution for this element is a cable.
These structures can be realised as a plane or as space systems.
2.1 Main components of the system
The system proposed consists of the same main group of element that any other
plane or space truss. These are: upper and bottom chord (or layer) and cross braces.
Some differences appears in requirements to be fulfilled by them.
Upper chord must be an element with capacity to concentrate energy of elastic
deformation (i.e. non-zero bending rigidity). This allows the structure keeping both
the initial shape and the balance after deformation. Bottom chord presents a flexible
cable which takes tensile forces only. Cross braces are connected with upper chord
in articulated joints and with bottom chord in passing joints. Passing joints allows
rotation of cross braces and sliding of the bottom chord.
2.2 Order of realisation
The structure is completed in initial configuration with non-deformed upper chord
and cross braces mounted. Than the bottom chord (cable) is placed (Figure 1). It
goes through the all passing joints. The cable is fixed in the one of the terminal
joints ('fixed support point' - articulated joint connected to the supporting structure:
foundations, columns etc.) while it goes through and turns back in the opposite one
('mobile support point') (Figure 1).
Tension of the bottom chord causes displacement of the free support point and
deformation of the upper chord with cross braces (Figure 2).
AA^
bottom chord (cable)
mobile support point
fixed support point
Figure 1: Assembling of self-erecting structure in initial configuration.
Mobile and Rapidly Assembled Structures III, C.A. Brebbia & F.P. Escrig (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-817-1
Mobile and Rapidly Assembled Structures III
shortening of the bottom chord
mobile support point
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fixed support point
Figure 2: Erection of the structure by the means of post-tensioning.
The shortening of bottom chord is performed till the structure achieves required
erection (Figure 3).
Figure 3: Final configuration of the structure, after erection. The cable is fixed in
the mobile support point.
In the final configuration the cable is fixed in the mobile support point and this
point is connected to the supporting structure below.
3 Variable rigidity method for self-erecting structures
The curvature of self-erecting structure with a cable as a bottom chord is a result of
interaction between bending rigidity of structure and tensile force in the cable.
In order to pre-determinate the final curvature of the structure it is useful to
differentiate its bending rigidity along the longitudinal axis. Easiest way to increase
rigidity in chosen part of the structure composed of bars is to make structure higher
in that part by means of placing longer cross braces (Figure 4).
\/vV\/\^
Figure 4: An example of variable rigidity in the plane truss: longer cross braces
allows to increase it in the central part of the structure
Mobile and Rapidly Assembled Structures III, C.A. Brebbia & F.P. Escrig (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-817-1
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Mobile and Rapidly Assembled Structures III
It is noted that the final curvature is smaller when bending rigidity increases at the
central part of the structure and higher when rigidity decreases at the central part
(Figure 5).
Figure 5: Variable bending rigidity results in different curvature achieved.
The shortening of distance between supporting points was the same
for all structures
Figure 6 presents variable rigidity method applied to the sample plane trusses. Different height in the central part after erection and different curvatures were achieved.
More complicated curvatures can be formed when the height of structure's crosssection is changing according to a parametric function.
Figure 6: Differrent shapes of structures achived due to variable rigidity.
The method presented can be applied also to multiple structures, where one selferected structure is supported by another one (Figure 7).
Figure 7: Variable rigidity method applied to multiple structures.
Mobile and Rapidly Assembled Structures III, C.A. Brebbia & F.P. Escrig (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-817-1
Mobile and Rapidly Assembled Structures III
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4 Conclusions
The presented 'variable rigidity' method extents scope of application of self-erecting structures. The general advantages of these structures: easy and fast assembling, fast erection and low cost guides to creation a suitable and impressive architectural form. Structures like these can be used for covering any type of object,
however they are most efficient for large span domes.
The present develop of this method is focused on defining in mathematical language the final geometry of the structure in relation to the rigidity function and
tensile force in the cable. Application of energetic method has given some interesting results and seems to be the most effective way.
5 References
[1] Schmidt, L.C., Dehdashti, G. Shape creation and erection of metal structure
by means of post-tensioning, Space Structures 4, Thomas Telford: London,
Vol. 1, pp. 69-77, 1993.
[2] Schmidt, L.C., Dehdashti, G. Hypar space trusses formed by means of posttensioning, Proc. of the Int. Conf. LSCE'95, ed. J.B. Obrebski: Warsaw,
pp. 629-636, 1995.
[3] Tarczewski, R. Shaping of space structures by means of shortening of cabletype bottom chord, Proc. of the Local Seminar oflASS Polish Chapter '96, ed.
J.B. Obrebski: Warsaw, pp. 156-163, 1996.
[4] Tarczewski, R. Bar and pneumatic self-erecting space structures, Proc. of the
Int. Conf. on Challenges to Civil and Mechanical Engineering in 2000 and
beyond, ed. R. Ciesielski, B. Ciszewski, J. Gronostajski, H. Hawrylak, J. Kmita,
S. Kobielak: Wroclaw, pp. 505-513, 1997.