Link zur deutschen Version: Grammatik für Fachwerkmodelle erklärt
Strut-and-tie models are a simplified, safe and typically manual design approach for reinforced concrete structures such as beams or walls. The iterative procedure to develop strut-and-tie models requires time and expertise from engineers which researchers have been attempting to reduce for decades. In my previous blogpost, I introduced two main approaches to automate the generation of strut-and-tie models for reinforced concrete structures and their advantages and drawbacks. Continuous topology optimisation aims to remove as much material as possible to minimise the weight or volume of a structure while meeting some structural constraints (e.g. yield strength of the reinforcing steel), where subsequently a skeleton truss needs to be derived from. However, this method has difficulty in incorporating fabrication aspects such as preferences for a regular layout or orthogonal reinforcement. Discrete layout optimisation aims to find a connectivity structure based on pre-defined grid-like nodes under certain constraints, but struggles to accommodate for user-defined adaptations. Due to these, both methods have not yet been able to gain widespread application in practice.
To tackle this research gap, we developed an approach based on rules and grammars. Inspired by theoretical computer science and formal languages, the concept of grammars has been already transferred to design, architecture and engineering applications. Domain knowledge or preferences are incorporated into rules that can generate diverse designs. We illustrate this concept with a simple example from formal languages. Typically, so-called context-free grammars consist of the symbols S (starting symbol), P (production rules), T (terminals, generation stops; here denoted with a rectangular shape) and V (non-terminals, generation continues; here denoted with an elliptical shape), all shown in Fig. 1. Suppose we want to generate a sentence, then each right-hand side of a rule can replace a non-terminal (left-hand side of the rule) in the generation process. We start with the rule S → subject-verb-object. As the next rule, we replace subject with “the student” and so on until we have no non-terminals left anymore and have generated the sentence “The student reads a book”.
This concept has also been applied in generative design. Our research interests lies in whether we can generate strut-and-tie models using such an approach, and whether these generated strut-and-tie models can be more practical than optimisation-based suggestions.
As a first step, a graph is chosen to represent the structural problem (Fig. 2). We are typically familiar with Euclidean data, which is represented in matrix form. However, when we want to capture more complex relationships within data (e.g. hierarchical structures or dependencies), we can use non-Euclidean representations such as graphs, groups or cells. Among these, graphs are advantageous for us because they efficiently represent structured data using nodes and edges. A strut-and-tie model can be easily transformed into a graph representation, where the nodes of the truss are the nodes of the graph and the struts and ties are the edges. Subsequently, we obtain a graph with the same connectivity structure as the truss. However, the usual matrix-based structural analysis scheme to obtain the forces of the edges is due to the graph representation no longer applicable. Therefore, we use the equilibrium condition at the nodes to find a static equilibrium. For the computational implementation, additional information such as the length or angle of an edge is added as node or edge attributes.
The main difficulty with the grammar-based approach is the derivation of rules. The set of rules should be sufficiently general to generate a variety of strut-and-tie models, but also specific enough to incorporate certain engineering judgement, such as adhering to minimum strut angles. To create these rules, some structural and geometric understanding is therefore required. Details of such rules can be found in our recently published journal paper [1] and some examples are shown in Fig. 3.
Once the rules have been defined, the procedure for generating strut-and-tie models is similar to the example for generating a sentence, see Fig. 3. An initial truss (the starting symbol) is derived from the structural problem, e.g. by connecting the boundary conditions. Unlike the example from formal languages, this generation does not directly have terminals and non-terminals. Edges can be fixed as terminals, to reduce the number of possible generations. Otherwise, based on the set of production rules, a subgraph matching algorithm1A subgraph matching algorithm tries to efficiently match a given subgraph to subgraphs of a graph. In this case, we want to find the left-hand side of a rule in the truss to know, where we can apply the rule. detects which left-hand sides of the rules can be found in the current truss. These possible combinations are suggested to the user, who can then choose which rule to apply. After transforming the truss with the chosen rule, performance metrics help to assess the quality of the truss, both in terms of validity (e.g. geometry, crossing of edges and strut angles) and suitability (e.g. reinforcement complexity and steel consumption) of strut-and-tie models.
The generation with grammars is illustrated for the deep beam with opening in Fig. 4, a common example2This example was also discussed in my last blogpost. from the literature. As a first step, a rule is applied to suspend the left node above the opening. The parameters of this rule can be adjusted to influence the position of the newly introduced node (compare the first step of each solution). By varying rules and parameters different strut-and-tie models can be generated. Both a simple and efficient design, such as Solution A with inclined reinforcement, and a complex design, such as Solution C, with orthogonal reinforcement can be generated using the grammar-based approach.
This and more examples [1] demonstrate that the grammar-based generation of strut-and-tie models allows for generating more practical strut-and-tie models as well as human-computer interation compared with the previously mentioned optimization methods3e.g. in Mozaffari et al. 2020. However, this generation might become cumbersome with increasing complexity of the strut-and-tie model and number of rules. We are currently exploring alternative computational or AI-based methods to navigate the design space with more automation and hope to update you later this year. If you have questions or suggestions, feel free to reach out to Karin Yu.
Karin Yu
Literature
| [1] Yu, K. L., Kraus, M. A., Chatzi, E., Kaufmann W. (2024). Grammar-Based Generation of Strut-and-Tie Models for Designing Reinforced Concrete Structures. Computers & Structures. Volume 305. 107549. doi.org/10.1016/j.compstruc.2024.107549 |