Metaheuristics for Structural Design and Analysis. Yusuf Cengiz Toklu
to safety, functionality, economy, sustainability, aesthetics and comfort. The first question, that of safety, is directly related to structural analysis. It involves characterizing the materials to be used in the structure, making a mathematical model of the structure after making some meaningful assumptions, giving meaningful dimensions to the elements of the structure, determining design forces with a distinction between dead and live forces and calculating internal effects in the structures under different loadings, so as to support reactions, stresses, strains, deflections and rotations. Then comes the stage for determining the safety of the structure, i.e. deciding whether the structure, whose behavior has been calculated, is sufficiently strong to resist the acting forces with a meaningful factor of safety, whether the stresses are less than the materials can safely carry and whether the deflections and rotations are not severe enough to create a sense of discomfort. If any of the answers to these questions are negative, then either the dimensions of the structure will be changed or, in some cases, the entire model will be changed as a more radical measure. A change in the dimensions may be in two directions: increase the dimensions if the element is overstressed, or decrease them if the element is highly under-stressed to avoid having a structure that is far from economical.
The flowchart of this process is shown in Figure 1.1. It should be noted that the algorithm represented in this chart is not fully applicable, even in our age, though engineers are close to using it only for some simple structures. We can estimate that full use of the algorithm shown will be possible with advances yet to be made in the field known as artificial intelligence.
In the early ages, there was of course no knowledge about concepts such as stress, strain, force, deformations, stability, buckling, bending and torsion. Formulating a mathematical model of the structure was unthinkable, and there was no knowledge of calculus to do any kind of computation. Despite all of this, there was the intuition, experience, intelligence and observation power of some extraordinary people.
Figure 1.1. General flowchart for structural design
In the beginning, the above algorithm comprised only the first step shown; the engineers/architects started building structures based only on conceptual design: a temple, a ziggurat, a pyramid, an obelisk, etc. and made improvements to their designs, according to their feelings, as construction went on. The most important characteristic of the flowchart shown above is the existence of iterations at various steps. It is easy to assume that iterations during these early times were being performed in the field, i.e. during the construction process. Perhaps tens of years or more were necessary for field iterations for the cutting of stones; carrying them to their places, carving figures wherever necessary, placing small stones and erecting T-shaped columns at Göbeklitepe – the first known edifices of humankind, recently uncovered at Urfa, Turkey – built some 12,000 years ago (see Figure 1.2).
Figure 1.2. Representative drawing showing construction operations at Göbeklitepe, Urfa, Turkey (https://www.cnnturk.com/kultur-sanat/dunyanin-en-eski-tapinagi-gobekli-tepe?page=1). For a color version of this figure, see www.iste.co.uk/toklu/metaheuristics.zip
We also see field iterations in structures that have collapsed and been rebuilt. The best example that can be given from those years is the Hagia Sophia (see Figure 1.3) in Istanbul, Turkey, which was constructed between February 532 and December 537, following the design created by Anthemius of Tralles and Isidorus of Miletus (Mainstone 1988; Ozkul and Kuribayashi 2007; Çakmak et al. 2009). This design was carried out using the experience and intuition of the two architects, perhaps with some computations with respect to the sizes of the columns. It was not possible to perform the computational iterations depicted in Figure 1.1, but they forcedly took place to end up with a safe structure. In May 558, i.e. almost 20 years after construction had finished, “parts of the central dome and its supporting structural system collapsed” (Çakmak et al. 2009). Those failures were probably due to earthquakes that took place in August 553 and December 557. Subsequently, a new dome – approximately 6.24 m higher than the original – was designed by Isidorus the Younger, nephew and successor to Isidorus of Miletus. This can be considered as an iterational process on the conceptual design of the structure, performed in the field as a real construction.
Figure 1.3. Current photo of Hagia Sophia in Istanbul, Turkey. For a color version of this figure, see www.iste.co.uk/toklu/metaheuristics.zip
The real iterations started with advances in structural analysis; its history written by Timoshenko (1983), Benvenuto (1990), Mainstone (1997), Felippa (2001), Addis (2003), Kurrer (2018) and many others.
Strength of materials is a primordial field in structural analysis. Following an introduction that charts its early beginnings, Timoshenko (1983) – in his book originally published in 1953 – explores the history of this subject across 14 chapters, beginning with the 17th century and ending with the period 1900–1950.
Mainstone (1997) states that structural analysis, as we now know it, began in the 18th century to assess the safety of buildings to be constructed. By the mid-19th century, this area is extended for analyzing earlier structures like Gothic cathedrals. The first idealization was Hooke’s law of direct proportionality; this idealization went together with increasing knowledge about statically determinate structures, ignoring those that were statically indeterminate or hyperstatic. The second half of the 19th century was marked by graphical methods.
Matrices hold a particularly important place in the history of analyses of structures. Felippa (2001) addresses this aspect under three titles: creation, unification and FEMinization, the last term meaning the launching of “[...] the direct stiffness method [...] as an efficient and general computer implementation, as yet unnamed, finite element method (FEM)”. Addis (2003) states that the main causes of failure in historical structures were wind loads, foundation problems and fires. We must add earthquakes to these effects for some parts of the world. Addis mentions the importance of full-scale tests on small physical models in historical times.
Two books, authored by Benvetuno (1990) and Kurrer (2008), are especially remarkable in following the history of structures. In both, the subject is traced from the very first works up until the time each book was written, dividing the advances into eras with historical drawings and anecdotes. Benvetuno’s book is in two parts: “Statics and resistance of solids” and “Vaulted structures and elastic systems”. Kurres’s book, titled The History of the Theory of Structures – From Arch Analysis to Computational Mechanics was published originally in German, in 2002.
Although the design, analysis, and construction of structures have always shown continuous advances in history, there was a period when this was the opposite. Indeed, between the time of the Romans and the Renaissance, let aside making of constructions more important than in previous times, the knowledge accumulated until then was lost (Timoshenko 1983).
1.2. From empirical rules and intuition to FEM
In the beginning, the