Hierarchical Beam Truss:
Fractal for Light-Weight Structure
The feature of self-similar repetition is not a new concept even in the field of construction. In the language of civil engineering, this type of configuration is called as hierarchical configuration, and the structures having this configuration are called hierarchical structures. Many of the lattice structures, such as trusses, are the examples of hierarchical structures. The fundamental reason for adopting hierarchical arrangement of members and their sub-members in a structure is to achieve high strength but light weight.
In this context, with reference to the design of the Eiffel tower, Benoit Mandelbrot claimed in his book The Fractal Geometry of Nature,
"My claim is that (well before Koch, Peano, and Sierpinski), the tower that Gustave Eiffel built in Paris deliberately incorporates the idea of a fractal curve full of branch points. . . . However, the A's and the tower are not made up of solid beams, but of colossal trusses. A truss is a rigid assemblage of interconnected submembers, which one cannot deform without deforming at least one submember. Trusses can be made enormously lighter than cylindrical beams of identical strength. And Eiel knew that trusses whose 'members' are themselves subtrusses are even lighter."
This cliam clarifies the strength lies not in mass, but in cleverly designed geometric shapes and in the case of Eiffel tower, it lies in branching points. Fractal geomtery can be a tool that aid to develop such branching point in many hierarchical scales.
Hierarchical trusses are the assemblage of the self-similar members that are further assembled by their own sub-members in a similar fashion. In the case hierarchical truss beam, each beam member is replaced by self-similar truss whose sub-members are further replaced by its corresponding truss beams. This geometric shape follows a rule that can be constructed by a generative process such as the Iterated Function System. In this research, the Barnsley's contraction mapping method has been applied to produce such beam. In the beginning, a single line has been taken as a beam B0, and initial shape. In the rst iteration, B0 is transformed into a truss beam B1 which is an assemblage of 25 self-similar copies (b1; b2; b3; ...; b25) of B0. There are 4 groups of dierent self-similar copies that are produced by the contractions of Y1; Y2; Y3 and Y4. In the next steps, this process of repetition is continued using the same transformation rules. Thus, it results a fractal gure which is an attractor. It is an intersection set of all the identical subsets of B0, as shown in Figure 5:5.
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The computational morphogenesis and optimization of the fractal-based truss beam shows that the high iterated design is structurally efficient and optimal. This experiment confirms Mandelbrot's claim about merit of fractal geometry for developing light-weight and high-strength lattice structures. It also ensures and gives a confidence to explore further applicability in designing efficient and innovative structural shapes that can oer novel and inventive design possibilities in architecture. In this optimization process, the cross-sectional sizes of all the members are uniform. However, the ideal process of optimization to find the optimal size of the cross-section of each member is based on its minimum capacity to carry internal distributing force within the allowable limit. In this case, the process will result different sizes of all the members and they will not be uniform any more.