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Shaping Structural Systems
Irmgard Lochner
1 Background and Motivation: Approaching Structures
During the past decades, the professional profiles of architects and engineers have been increasingly diverging - even though they initially have a common origin. Considering the complexity of building procedures and of specialist knowledge this process is comprehensible; considering the demand for holistic understanding and planning on the other hand, it is destructive.
The profession of the structural engineer is relatively young - compared to other professional profiles, and above all, compared to the history of building. Since the beginning of the ninetennth century, methods of structural analysis have been established and continuously developed. The characteristics of structural analysis can be described as a reduction or de-construction of a building into parts that can be calculated and dimensioned: In order of increasing scale the typical elements are nodes, columns and beams, plates and slabs, cables and arches, frames, shells and membranes. These structural elements can be isolated from a system and are part of a common language of architects and engineers, which can be very useful under diverse aspects: they facilitate communication, and serving as prototypes for the design and construction of a building, they can also support architectural concepts.
On the other hand, thinking in structural prototypes can also mean a loss of intuition and creativity.
From the author‘s point of view, there are two approaches to the analysis and design of structures: classical structural analysis on one side, and the principles of lightweight structures on the other side. The images shown in Fig. 1 represent these two views. Galileo Galilei in the later periods of his life was dealing with mathematics and mechanics and drew this in his „Discorsi e dimonstrationi matematiche“ (Galilei 1638). His cantilever illustrates the thinking in analysable elements. Michell‘s approach in „The Limits of Economy of Material in Frame-Structures“, published in the Philosophical Magazine in 1904, represents another approach: Taking off with a force to be transmitted from A to B, the material layout within a given design space is developed. The construction of structural element is the following step. Michell‘s approach today finds applications in numerical analysis and optimisation.
New design approaches and tools in architecture and engineering contribute to blur the borders of what is architecture and what is engineering. Designers of diverse backgrounds discuss and develop new methods from design development to construction. The method of structural optimisation is a very valuable design tool for designers of all backgrounds. This can contribute extensively to establish a common language for architects and engineers: It is the common, broadly adaptable tool that brings designers of various specialisations back together. Liberating the designer from pre-defined structural elements opens up a conceptual view that can strongly influence and support the design process.
Irmgard Lochner
Biberach University of Applied Sciences, Biberach, Germany
Fig. 1 Galileo Galilei‘s illustration of a cantilever (1st); Michell structure (2nd)
2 Method: Topology Optimisation Using the Homogenization Method
2.1 Structural Optimisation
Topology optimisation deals with the optimum layout of material within a given design space. When performing structural optimisation, the general formulation is the minimisation of the objective function f(x)
minimise f(x) objective function
such that the constraints are fulfilled:
gj (x)≤ 0; j=1,mg Inequality constraints
hk=0; k=1,mh Equality constraints
xil ≤ xi ≤ xiu; i=1,n Side constraints, upper and lower bounds
More descriptively, the optimisation goal can be described for example as
Develop a structure with minimum weight with given loadings and support conditions, with the constraint of the deflection not exceeding a given value (otherwise the optimisation algorithm would develop a structure with zero weight), or
Develop a structure with minimum compliance (maximum stiffness) with given loadings and support conditions, with the constraint of only a given ratio of the design space, the so-called volume fraction, to be filled with material - otherwise the optimisation algorithm would fill up all of the design space with material.
The Homogenization Method is a gradient based optimisation method as an addendum to the Finite Element Method, with the basic idea of subdividing the design space into small domains (pixels or voxels). The initial process of dividing the design space into areas with material and areas without material (0-1 problem) is re-formulated into a continuum problem. The optimisation problem is then to find an optimal structure described by pixels with density 1 (full material) and pixels density 0(no material) plus intermediate values between 0 and 1(„porous“ material). The material layout generated by the optimisation algorithm is a design proposal, which can then be interpretated and futher developed by the designer.
Fig. 2 shows a high beam and an optimised structure under a point load (Ramm 1996). The basic idea is, the optimised structure shows a very homogeneous stress distribution and carries the same load with less material.
A MATLAB code for the application of the homogenization method to structural optimisation is presented in (Lochner, Schumacher 2014). The studies carried out in is contribution use the commercial software Altair OptiStruct, which also uses the homogenization method.
Fig. 2 Typical structure (left) and optimised structure (right) with homogenuous stress conditions
3 Shaping Structural Systems