make these choices we rely on two main tools:
Our intuition – Much like meeting a new person, much of our decision-making process is carried out in the first few moments that we are presented with a new problem. We ‘see’ solutions to the problem at hand and instantly set about planning them. Our intuition is honed through the combination of our academic teaching and our professional experience and we like to think that as both our training and experience increase, so too does our intuitive ability.
Our analytical nature – As designers we naturally like to analyse solutions to find optimal ways of tackling problems. Our approach to ‘analysis’ is often quite rigid, constrained by the fact that the basis of our analysis (geometry, intended load paths, structural arrangement) is pre-determined by our intuition. The analysis tools available to us are typically able to give very accurate answers to the very specific questions that we ask them allowing us to create optimal solutions within the original constraints. However, our analysis tools are typically not capable of telling us that we asked the wrong question in the first place or that our engineering logic was flawed.
This article, and subsequent presentation at Stålbyggnadsdagen (7th November 2019), will look at how Topology Optimisation can enable structural designers to deliver optimal solutions, offering material savings over our intuitive ‘text book’ responses to engineering problems. Before you read on, I have a question for you. Please look at the image in Fig.2. and, with a pen, draw the most efficient (minimum weight/ maximum stiffness) simply supported truss geometry for the load/ supports shown, staying within the bounding box. This could represent a column transfer truss in a high-rise building.
This question has been chosen as one of the most basic structural systems that we all come across – a simply supported truss – with a trivial loading condition. We would expect our intuition to lead us directly to the optimal solution for something as basic as this. I have asked this question to over 100 practicing structural engineers from fresh graduate to company director and, surprisingly, have received many different responses. Why do we have such a range of answers when solving problems like this should be hard coded into our engineering DNA? Having studied the responses, the likelihood is that your sketch will look like one of the following:
- Shortest distance (Fig.7a – Benchmark 0%) – Depending on the scale of the system, this may be a cost-effective solution. The shape of the truss also reflects the bending moment diagram and ‘feels’ like it should be efficient. However, the geometry results in higher forces and longer unbraced buckling lengths which, for larger scale systems, prove less efficient.
- Pratt Truss (Fig.7b +22%) – It is generally a sound approach to set the geometry to induce tension in the longest diagonal members but, in the example chosen, it results in a longer force path with a greater number of high stressed members.
- Howe Truss (Fig.7c 0%) – Despite long diagonals in compression, this geometry results in a shorter force path than the Pratt truss and is typically both lighter and stiffer for the load shown in the example.
- Warren Truss (Fig.7d -7%) – The diagonal internal elements result in a more direct force path than the Pratt and Howe trusses, with fewer internal members resulting in a more efficient truss.
These responses are based on text-book solutions that we have seen and learned, but we have no real quantitative way of comparing them in our intuitive thought process without doing more in-depth analysis. A small number of you may have drawn an arched system, which offers further material savings (Fig.7e/f -8%, Fig.7g -12%) and, finally, Fig.7h shows the theoretical optimal solution with a 16% saving over the benchmark geometry. This theoretical solution has little practical use but it is clear that savings in excess of 10% (over 30% if you drew the Pratt truss!) are easily achievable with small modifications to the truss geometry. As this example shows, our gut feel for structural efficiency is often flawed due to a lack of insight into the effect of geometry on the precise force distribution in a system. We need to challenge our intuition and make use of tools that are currently available to help optimise structural systems.
Topology optimisation (TO) is a computational method that has been described as “an intellectual sparring partner” [Bendsøe, M.P., and Sigmund, O. (2003). ”Topology Optimization: Theory, Methods and Applications”] for designers as it often offers solutions that we wouldn’t have proposed ourselves. The technique uses iterative finite element (FE) analysis to assign material in the most effective manner to reduce the quantity of material required for a set loading condition. It is one of a set of tools known as Evolutionary Algorithms due to the way that the structure grows from stage to stage and we like to use the term “reject lazy material”. Counterintuitively, while reducing material usage, TO also increases structural stiffness for a fixed volume of material by minimising the overall force path length. Therefore, it often provides a dual benefit of a system that is both lighter AND stiffer.
Adoption of TO as a concept design tool has previously been used by the aerospace and automotive industries but there are limited examples of the use of TO for structural engineering applications. This is largely because it has historically been computationally demanding; however, advancements in the efficiency of current optimisation routines has significantly reduced processing times to practical level for many structural applications. Current progress in the field has resulted in numerous tools being readily accessible to practicing structural engineers, many of these are also integrated into a parametric modelling environment to benefit from the flexibility that this allows at concept stage. Grasshopper, the parametric modelling platform within Rhino3D, has five such TO tools available in one environment (BuildOpt, tOpos, TopOpt, Millipede, Karamba3D). There are two distinct TO methods available, described below, each of which has benefits and disadvantages.
The Continuum Method begins with a constant thickness FE mesh across the design domain. The iterative TO analysis gradually reduces individual mesh element thicknesses and stiffnesses to deactivate areas of low stress until a defined volume fraction remains (e.g. 10% of the original material). The remaining material distribution typically follows lines of principal stress on the original constant thickness mesh and results in organic shapes, often copying naturally occurring forms in a process often referred to as biomimicry.
The freeform nature of this output (Fig.3 and Fig.4) is often difficult to process into a rationalised and buildable structural form and it is necessary to either apply rationalisation manually or use a second stage analysis approach. Following the initial topology optimisation, we have found it useful to create a parametric model with real constraints such as column grids, floor levels, symmetry etc. This model is set up to allow a geometry similar to the TO results with the right combination of input parameters. Structural analysis carried out in the parametric environment (Karamba3D) then allows the optimisation of this model using a Genetic algorithm solver. This offers an optimised solution within the specific constraints of the project and can be easily coordinated with the design team. Modifications of the idealised geometry can also be made easily via the parametric model. This approach was used in the example shown in Fig.5.
The second TO method uses discrete elements and is often referred to as the Ground Structure Method. This overcomes the freeform nature of the continuum method as the resulting structure consists of straight elements. The method begins with a large number of interconnected members spanning between a grid of nodes within the design domain. The optimisation routine eliminates ineffective members whilst retaining effective members resulting in a structure that can be optimised of minimum weight or maximum stiffness to weight ratio. Models can often be set up to fit project constraints and there are powerful post processing tools available to rationalise and simplify results to deliver practical solutions. As such, the discrete method is often more appropriate for structural applications although the solutions are limited by the density of the grid which can be computationally demanding for dense grids.
With increasing demand on engineers to make good use of natural resources on projects we are turning to computational processes to enhance the conceptual development of our structures as our natural intuition is often misguided. This promotes the computer from being a slave to our analysis processes to an active participant in the design of our structures. These evolutionary algorithms are the true definition of Digital Design and offer solutions (and savings) that may not otherwise be available to us. Whilst being open minded to the solutions offered by these tools, we must remain pragmatic and practical in our application of them to ensure that our designs are cost effective and buildable which will usually require a degree of rationalisation and modification for our projects. Understanding the basis of the tools and guiding them towards truly optimal solutions that consider project and construction constraints remains the preserve of our engineering intuition. We should embrace these new practices as a helpful addition to the creative process of design.
Notes on solutions for
– Buckling is not explicitly included, but the optimal solutions have been developed with a penalty on compression over tension (reduced allowable stress).
– The percentages show material usage efficiency and allow high level comparison of different systems but are not exact final tonnage comparisons.
– The theoretical optimum is nearly always highly complicated. However, there is usually a rationalised version that is a similar level of complexity to our intuitive solution that is close to the optimum efficiency.
– These solutions are specific to the design problem in Fig.2, it is likely we would see different optimal solutions for a different load condition (e.g load applied to the bottom chord). TO analysis should always be set up to consider a range of load cases to ensure that solutions are balanced for the full operation of the structure. Additionally, actual solution for a particular application may depend on the scale of the system and restraint conditions.
Nick Cole, Robert Bird Group