In conventional prismatic beam design, the
sectional area of concrete and steel is sized to resist the maximum bending
moment, and this section is applied uniformly across the entire length
of the beam – despite the fact that the bending moment of any beam
varies across its length from zero to maximum.
Because a conventional rectangular beam maintains a
constant depth and structural section along its span, tension and compression
stress levels will necessarily vary along its length in direct proportion
to changes in the magnitude of the bending moment. This strategy causes
the great majority of material in a uniform section beam to be used in
a less than efficient manner.
A beam that follows the shape of its bending moment diagram, on the other
hand, has a depth (moment arm) that varies in proportion to its bending
moments. In an efficiently proportioned beam, the tension and compression
materials both work at optimum stress levels along the entire length of
the beam. The net effect of this strategy is a beam that places material
only where it is needed and uses that material at optimum stress levels
at every point along its span. This strategy is well known, being used
most often for long span structures where dead weight reduction is a controlling
factor of the design.
It is simply too expensive, however, to use conventional formwork materials
(rigid wood or steel panels) to construct a more efficient, curved, variable
section mould. Because rigid moulds have been used since the invention
of concrete, we have grown entirely accustomed to the construction economies
dictated by rigid formworks, as well as the structural inefficiencies
imposed by conventional formwork technology.
More efficient beams
Reinforced concrete beams with shapes defined by efficient
structural curves can be easily formed in a prestressed fabric membrane
using a simple formwork rig. The first prototype beam using this production
method was cast in the spring of 2003 at the Con-Force Structures precast
concrete factory in Winnipeg, Manitoba, Canada. It was developed by Mark
West and his students, Christopher Wiebe, Phillip Christensen, and Fariborz
Hashemian in the Centre for Architectural Structures and Technology (C.A.S.T.)
at the University of Manitoba.
12-meter prototype reinforced concrete beam cast from a prestressed fabric
form
 Schematic drawing of formwork method used to produce a mould for casting variable section concrete members |
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A single flat, rectangular sheet of fabric, with two hems sewn along opposite edges, is allowed to drape down into the space formed between two flat tables. Flexible splines are inserted into the hems on both sides. When the splines are brought closer together, the downward deflection of the fabric increases. When the splines are moved apart the deflection is reduced. This simple method provides a mould for a wide range of variable section members.
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The longitudinal shape of the member is determined by the geometry given to the splines; if the splines are curved, the longitudinal profile of the member will be correspondingly curved. The complex three-dimensional curves of this prototype beam were achieved by stretching a flat rectangular piece of woven polyolefin geotextile fabric in a basic wooden framework made of 2x4s and plywood.
 Formwork fabric weighs less than 10 kg |
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The formwork fabric for this 12-meter beam weighs less than 10
kg, and can be stored in a space of less than .028 m3.
Wrinkles and other unwanted undulations in the fabric sheet can
be eliminated or reduced by prestressing the formwork membrane in
one or more directions. These same prestressing forces are used
to reduce lateral deflections in the formwork membrane caused by
the fluid pressure of the wet concrete. The tension forces required
to do this are small, accomplished in this case by one worker with,
at most, a simple lever or block and tackle. |
The beam's transverse sections vary along its length to reduce concrete in the tension zones, while distributing concrete to the compression zones. We estimate that this 12-meter prototype beam requires half the concrete and 40 percent of the reinforcing steel compared to an equivalent conventional rectangular beam.1
Reinforcement
In addition to finding an efficient formwork system,
a simple method of reinforcing a variable section beam needed to be found
as well. The standard use of vertical stirrups for shear reinforcement
was rejected as being too complex; a variable section beam would require
each stirrup to be a different shape and size. A simpler method, suitable
to the demands of construction, was used. Two pairs of identical vertical
stirrups were installed, one pair over each support, forming a small "tower"
to support horizontally (rather than vertically) placed shear reinforcement
bars and to provide containment of the concrete at these crucial points.
Primary tension reinforcement is provided by a combination of 15-mm rebars
and steel cables. The steel cables are draped along the mid-span curve
and continue across the tension zones of the two cantilever sections.
The 15-mm "top steel" rebars are hung from temporary wooden
supports lying transversely across the void of the form.
Other observations
It is important to note that variable section beam
shapes are designed for specific loading envelopes. So, for example, the
prototype beam illustrated in this article should be less capable of sustaining
a concentrated load at one of its inflection points than an equivalent
prismatic rectangular beam would. The appropriate use of such lightweight
beams in different applications needs to be carefully studied. Where the
loading envelope is well known, however, the advantages of such an approach
are evident.
 12-meter prototype beam |
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This first prototype beam, and the
ease and economy of its production, serve as a proof of concept,
and indicate the larger implications of this new method. The fabric
membrane used to form this beam cost less than $75 (U.S.) including
sewing. By simply altering the tension of the fabric, an identical
flat sheet can be used to form a wide range of efficient beam geometries;
a single formwork rig can produce a multitude of individual beam
shapes, each configured to efficiently resist the bending forces
produced by any specific combination of support and loading conditions.
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The beauty of these beams also represents an important advantage over
conventionally formed concrete structures. In this respect our work follows
closely that of structural engineers such as Robert Maillart, Pier Luigi
Nervi, Eladio Dieste, Heinz Isler and others, who can rightfully be called
"structural artists". In this tradition, sculptural and architectural
beauty is discovered by taking dictation from natural law.
In each case, engineering and structural elegance is matched by an innate
sculptural beauty, and accompanied by constructional simplicity and an
economy of means appropriate to a specific building culture. The development
of fabric-cast beams aspires to the inseparability of science and poetics.
Acknowledgements
Support for this work was given by the Canadian Precast Concrete Institute
(Manitoba Chapter).
For further information, please contact:
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Professor Mark West Director, Centre for Architectural Structures and Technology University of Manitoba, Faculty of Architecture 204-474-7427 westm@cc.umanitoba.ca
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