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.@en

Defining a suitable truss model is one of the most important steps of applying the strut-and-tie modeling (STM) method to design D-regions in reinforced concrete (RC) structures. The truss model is a discrete representation of the stress field developed within a region of a concrete element. Topology optimization (TO) methods have been investigated by researchers for about two decades to generate suitable models for the STM method. Several truss models and numerous continuum TO results that could serve as an inspiration for suitable truss models have been proposed. However, limited attention has been paid to the evaluation of various TO results in the perspective of the STM method. As a result, it is at present unclear to what extent TO results offer a benefit for STM modeling, and which method should be preferred. In order to address this gap, an automatic and objective evaluation procedure is proposed in this paper. First, a TO result extraction method is proposed to systematically convert optimized topologies to truss-like structures. Next, based on the extracted structures, three evaluation measures are formulated to evaluate TO results. These measures indicate whether an analyzable truss model could be extracted, to which extent tensile stress regions are covered by tensile ties and how economical the design will be. The effectiveness of the proposed evaluation procedure is validated using known STM solutions. Subsequently, the evaluation procedure is applied to 23 TO results from the literature, covering three different design problems. Most TO results show a good performance in covering tensile regions and would result in economical designs, and some undesired topologies are also identified by the evaluation method. Nevertheless, the use of continuum TO is most hampered by difficulties in identifying a suitable truss from the TO results.

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Strut-and-Tie modelling (STM) has proven to be a powerful method to design D-regions of reinforced concrete structures. STM uses truss analogy models to represent the stress field within the D-regions of concrete elements. Schematizing a suitable truss model is one of the most important aspects in the STM. Over the past decades, the use of Topology Optimization (TO) methods has been explored to inspire efficient truss models for the STM. However, it is at present still unclear whether these TO-based truss models benefit the STM. In order to investigate this question, this paper presents a case study based on a deep beam with an opening. Firstly, a TO-based STM model is generated which satisfies requirements of the axial force equilibrium and stress constraints of the STM method. Secondly, different STM designs are analysed by Nonlinear Finite Element Analysis (NFLEA), based both on classical STM models and TO-based models. The resulting steel usage, ultimate capacity and failure mode are used as measures to evaluate the designs. Finally, based on the generated STM model, two practical steel designs considering construction complexity are proposed and evaluated by NLFEA. The effectiveness of the TO-based STM model for this deep beam case has been demonstrated based on the evaluation results.

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Strut-and-Tie modelling (STM) is a well-known approach to design D-regions in reinforced concrete structures. Because the STM method is based on lower-bound analysis, finding a suitable truss-analogy model is the most important aspect to guarantee good structural and economic performance of a resulting design. Continuum topology optimization (TO) methods have been studied for two decades to solve this problem. However, while these studies provide inspiration to designers, they lack the capability to automatically generate valid truss-analogy models as needed in the STM method. In order to prevent manual interpretation and automatically generate suitable STM models for various D-regions, a method is proposed for the generation of optimization-based STM (OPT-STM) models. The proposed method includes three phases: the TO phase, the topology extraction phase and the shape optimization phase. Next, in order to evaluate the effectiveness of the generated OPT-STM models, an evaluation using Nonlinear Finite Element Analysis (NLFEA) is performed to analyze the performance of STM models. For two D-region problems, two OPT-STM models and 11 manually constructed STM models from literature are evaluated, and their performance is compared and discussed, demonstrating the validity and effectiveness of the proposed automated generation method.

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Strut-and-tie modelling (STM) is an effective and widely used method to design disturbed regions (D-regions) of reinforced concrete structures. Among the various steps of STM, finding a suitable truss-analogy model is the most challenging part. Even for experienced engineers it is difficult to find representative models for complex D-regions, and this task is even harder for three-dimensional (3D) D-regions. To date, only a few 3D STM models have been proposed by researchers for several complex D-regions, which leaves practitioners with little guidance. In order to solve this problem, a method is proposed to automatically generate 3D optimization-based STM (3D-OPT-STM) models. The generation method comprises a compliance-based topology optimization process that generates optimized structural forms by maximizing stiffness, a topology extraction method, and a shape optimization method. In this study, three 3D-OPT-STM models are generated for typical 3D D-regions, and their performances are compared to manually created STM models. The generated 3D-OPT-STM models result in more economical designs. Moreover, geometrical and loading parameter studies demonstrate the applicability and robustness of the proposed method.

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In current optimization methods for free-form shells, the shape and topology are usually optimized separately. These methods are based on the assumption that the shape and topology of a shell influence each other only slightly, but this is not always correct. Moreover, different parameterization models are used in the shape optimization and topology optimization of free-form shells, which brings difficulties to carry out the integrated optimization. To solve this problem, an integrated method is proposed for simultaneously optimizing shape and topology for free-form shells. A uniform parameterization model based on NURBS solids is established to parameterize the free-form shells. In this model, only a small number of variables are used to describe both the shape and topology of the shell. In this way, the integrated optimization problem can be simplified, thus decrease the computational complexity. The integrated optimization of shape and topology is a complicated and large-scale optimization problem. Solving this problem requires a suitable optimization method. In this paper, the Method of Moving Asymptotes (MMA) is adopted. Finally, numerical examples are presented to address the importance of the optimization sequences and show the effectiveness and application of the proposed method.@en