Towards a systematic design approach of D-regions in reinforced concrete: Optimization-based generation of Strut-and-Tie models

Research output: ThesisDissertation (TU Delft)

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Reinforced Concrete (RC) structures are widely used in our society for more than a century. In order to design safe and economical RC structures, various methods have been proposed by engineers and researchers. Remarkably, it still is a challenging task for engineers to design D-regions of RC structures, regions with nonlinear strain distributions. Strut-and-Tie Modelling (STM) is a well-known method for designing such regions. The STM method uses a truss-analogy model to represent the force flow within the D-region, thereby providing insight to engineers for reinforcement design. The relative simplicity of the method and the fact that STM leads to a safe design are beneficial to engineering practice. However, in investigations of the STM method, the creation of suitable truss-analogy models has been identified as the key problem for a systematic application of STM. During the past three decades researchers have conducted intensive efforts to find systematic approaches for obtaining truss-analogy models for the STM method. Adopting topology optimization (TO) methods to assist the making of Strut-and-Tie (ST) models appears the most promising direction. For this reason, various TO methods have been proposed, however which method leads to the most suitable ST models is still unknown. Very few investigations have been carried out regarding the systematic evaluation of TO results from the perspective of the STM method. In this thesis, a procedure to evaluate the TO result for STM is presented. Using this procedure, an evaluation of TO methods revealed an urgent and challenging problem of generating a suitable ST model in the TO process. Currently, TO methods only provide optimized material layouts as inspiration for creating ST models. Manual and subjective adjustments are required to convert TO results into adequate ST models. These additional processes not only affect the performance of the desired design, but also hinder the application of TO methods for STM. In this thesis, first a 2D generation method that integrates TO, topology extraction and shape optimization is proposed to solve this problem. The proposed method successfully generates valid ST models for D-regions automatically and without manual adjustments. In addition, an evaluation procedure adopting nonlinear finite element analysis (NLFEA) is proposed to evaluate the performance of the generated ST models. Based on the evaluation results, the generated ST models show a high stiffness and sufficient, yet not overly conservative load capacity. By comparing the generated ST models with various previously manually-created ST models, the generated ST models lead to the most economical steel usage relative to load capacity. Through two case studies and three parameter investigations, the effectiveness of the proposed generation method is validated. For 3D D-regions, generating suitable ST models is an even more challenging task. Therefore, subsequently, the proposed generation method was extended to 3D conditions. In the 3D generation method, three additional measures are adopted to improve the computational efficiency of the TO process, and a new robust procedure is proposed to extract 3D truss-like structures from the TO results. Three 3D D-regions are investigated, and the corresponding ST models are generated based on the proposed method. Again, the generated ST models lead to economically superior designs compared to the manually created ST models. In addition, the proposed generation method is used to investigate three other aspects of the STM method: 1) a parametric study of four-pile caps; 2) STM generation considering complex load conditions; 3) the influence of load discretization. The robustness and effectiveness of the 3D generation method are validated through these investigations. In spite of these improvements, challenges remain for engineers in application of the STM method. The standard STM method involves human choices, which depend on the engineer’s experience and preferences. These subjective factors hinder the application of the STM method and bring uncertainties and variations to the STM design. Developing a systematic STM method that reduces the subjective choices and uncertainties is identified as an important future research direction. In order to explore this problem, in this thesis, the main choices and uncertainties are identified and discussed. In addition, the proposed generation method can be used to investigate these subjective aspects. Therefore, next to being of value for engineers already in the design of D-regions, the proposed generation method is expected to also form a fruitful basis for future refinements in this research direction.
Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • Delft University of Technology
  • Hendriks, M.A.N., Supervisor
  • Langelaar, M., Supervisor
Award date22 Feb 2021
Publication statusPublished - 2021


  • reinforced concrete
  • Strut-and-Tie
  • topology optimization
  • Integrated optimization
  • Nonlinear Finite Element Analysis


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