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Design optimization and additive manufacturing of nodes in gridshell structures
Abstract
In this paper, two new design approaches are proposed to design structural nodes of complex shapes for gridshell structures, seeking improved structural performance and design efficiency by using the transitional section method and the bi-directional evolutionary structural optimization (BESO) method, respectively. Detailed design methodologies are introduced and compared with two types of conventionally designed structural nodes in real use, i.e., Seele node and Sun Valley node. Finite element analyses are conducted to evaluate the stress distribution, maximum stress and mean compliance of different structural nodes. The results show that, although the BESO node has the highest stiffness and the lowest structural volume, it seems to have a higher maximum stress level. Multiple load cases are considered in the analysis and design optimization processes. Moreover, the application of Laplacian smoothing in structural node design effectively reduces the stress concentration. Prototypes of the newly designed structural nodes are fabricated by additive manufacturing, which enables the rapid and precise manufacturing of customized structural nodes optimized for specific loading and boundary conditions.
... Gridshells which are also called lattice shells or reticulated shells are generally defined as structures with the shape and rigidity of a double curvature shell consisting of a grid not a continuous surface [1]. Although gridshells come to several forms, they are usually designed with triangular, quadrilateral, or hexagonal faces (or grid cells) [1][2][3][4][5][6][7][8][9][10]. Forming and optimizing gridshell structures have been very attractive problems in the past decades. ...
... Forming and optimizing gridshell structures have been very attractive problems in the past decades. Several approaches, such as inversion method [1], dynamic relaxation [2,4,11], force density method [3,12], and so forth [10,13], have been studied so far in the literature to address the problem of forming a grid shell structure. Moreover, various techniques from gradient-based to evolutionary methods have been employed for optimization of gridshells taking into account various aspects of a gridshell such as economic, structural, or aesthetic [11][12][13][14][15][16][17][18]. ...
... This is why we employ evolutionary techniques in this work. Among the evolutionary techniques, genetic algorithms (GAs) have been used the most in optimization of gridshells [1,3,5,8,10,11,[13][14][15][16][17][18]. Another well-known evolutionary method is particle swarm optimization (PSO) to which less attention has been paid for improving the gridshell structures so far. ...
Designing and optimizing gridshell structures have been very attractive problems in the last decades. In this work, two indexes are introduced as “length ratio” and “shape ratio” to measure the regularity of a gridshell and are compared to the existing indexes in the literature. Two evolutionary techniques, genetic algorithm (GA) and particle swarm optimization (PSO) method, are utilized to improve the gridshells’ regularity by using the indexes. An approach is presented to generate the initial gridshells for a given surface in MATLAB. The two methods are implemented in MATLAB and compared on three benchmarks with different Gaussian curvatures. For each grid, both triangular and quadrangular meshes are generated. Experimental results show that the regularity of some gridshell is improved more than 50%, the regularity of quadrangular gridshells can be improved more than the regularity of triangular gridshells on the same surfaces, and there may be some relationship between Gaussian curvature of a surface and the improvement percentage of generated gridshells on it. Moreover, it is seen that PSO technique outperforms GA technique slightly in almost all the considered test problems. Finally, the Dolan–Moré performance profile is produced to compare the two methods according to running times.
... The production of gridshells using additive manufacturing is an area in expansion and demands additional studies. A crescent number of projects use additive manufacturing to provide solutions for connection systems of complex gridshells with entirely 3D printed nodes (or printed molds for casting) [6][7][8][9]. However, the use of standardized hubs in combination with 3D printed interconnecting parts have not been found in the English literature to provide a solution for gridshell connection systems. ...
... It is only natural that, with the ever more available manufacturing techniques, there are developments of new types of systems. The 3D metal printed nodes [6][7][8] frequently make use of topology optimization, which allows the use of less material. This type of optimization withdraws material from the least solicited parts of the domain, according to the boundary conditions, restrictions, and objectives. ...
... Connections can also be fabricated with molds and later cast with any appropriate material [6,18]. A disadvantage may be that a complex surface may require many different molds. ...
Lightweight structures in grid form made of elements joined together with nodes are called reticulated shells, or gridshells. We propose a new connection system and method for gridshell structures. The connection system applies to grids that have a high variability of curvature and connection parameters. The connection uses standardized hub geometries and interconnecting parts. The elements of the structure connect to the hubs through these interconnecting parts, which absorb varying parameters of the connection, such as incidence, twist, and adjacency angles, and the fabrication tolerance of element lengths and bolted connections. The aim is to provide a system that can be assembled quickly, disassembled and reused. We use parametric modeling environment to model global and local geometry of meshes and nodes. Inputs for the programmed parametric definition include a discretized surface, geometry of the hub notch, and element cross-sections; outputs include element lengths, interconnecting parts geometries, and assembly information. We present a physical model that serves as a proof of the concept developed within the computational modeling environment. Finally, we discuss the scalability of the model and future work.
... Among these methods, the ESO method and its advanced version, the bi-directional evolutionary structural optimization (BESO) method, are popular among many researchers and designers because of their simplicity and the availability of well-developed commercial software such as Ameba [11]. The ESO method and the BESO method have been widely used in various fields such as structure, architecture, aerospace, medicine, and biomechanics [12][13][14]. Besides, some new structural optimization algorithms have been developed, such as YUKI, Jaya, and Cuckoo [15][16][17], which provide new strategies and perspectives for structural optimization and are promising to be applied to future topology optimization to enhance the efficiency of the computation. However, there are still some modifications to be done to combine these novel algorithms with topology optimization algorithms. ...
... One thing that needs to be pointed out is the volume evolution scheme shown in Eqs. (12) and (13) is specially designed for this study, and other slowly changing volume evolution schemes are also acceptable. ...
A novel method is proposed to optimize the topology of composite structures made of more than two materials with different mechanical properties in tension and compression. In this method, the design domain of the structure is divided into tensile and compressive regions according to the first invariant of the stress tensor. Then two groups of materials suitable for tension and compression are arranged in the tensile and compressive regions of the structure, respectively. Using a bridge-type beam with a concentrated force as an example, the study of the four-material topologically optimized structures reveals the effects of the volume fractions and material mechanical properties on the optimization results. Further, the three-material topology optimization method, which is derived from the four-material topology optimization method, is used to design a series of novel sandwich structures. Application examples demonstrate that the proposed method can achieve a balance between enhancing structural stiffness and saving material costs, providing solutions competitive in various aspects and exploiting the performance and potential of different materials better than previous single- or dual-material topology optimization methods.
Keywords: Bi-directional evolutionary structural optimization (BESO), multi-material topology optimization, multiple materials, sandwich structure.
... Based on maximum and minimum feature size control techniques, a method that can impose uniform feature size to the optimal topology has been proposed [154], which is particularly applicable to WAAM structures. The continuum topology optimisation methods may be suitable for design connections [155] and structural members [156] (Fig. 9). ...
... Fig. 8. Local buckling resistance of unstiffened stainless steel plates (tested by equal angle section stub columns produced by WAAM [128]), with the Eurocode 3 design equation [93]. Fig. 9. Examples of structural parts designed by topology optimisation (reprinting permission for these images has been obtained from the publisher [154,155]). ...
This paper provides a review of the capabilities of WAAM for manufacturing steel components for use in the construction industry, with a focus on the structural stability and design of WAAM builds. Manufacturing techniques that can be used for WAAM construction are first discussed. This is followed by a detailed review of the material and geometric properties, and the resulting structural stability performance of WAAM steel structures to date. To exploit the advantage of WAAM in building free-form shapes, structural optimisation techniques suitable for WAAM construction are discussed. Lastly, conclusions and future research directions are provided.
... Therefore, joint rigidity is an important factor when evaluating gridshells as using pinned joints in the evaluation often leads to unstable structures, and rigid joints could result in too unconservative estimates. Several studies report this, including [5][6][7]. ...
... For timber gridshells, the nodes are typically developed for each project. Analyses of gridshell nodes are typically done using solid elements as seen in the recent studies by Castriotto et al. [12] and Seifi et al. [7]. Like in the Printshell project presented in Section 2.1, the analyses require exporting to separate tools for performing the meshing and analyses. ...
Algorithms-Aided Design, a computational design methodology, has seen a seen
an increased interest in the past decades. Especially for complex free-form geometries such as
gridshells. The growing potential of controlling the design through algorithms allows for merging
structural and architectural design methods by creating customised design tools according to the
project requirements. In this paper, a plugin for Grasshopper based on the Finite Element Method
is presented with a study case in which it was used and developed. We hope this plugin will
help close the gap between the disciplines by quickly verifying boundary representation objects
in real-time during conceptual design. A general discussion about gridshell connectors regarding
their structural design and digital fabrication is provided.
... Therefore, joint rigidity is an important factor when evaluating gridshells as using pinned joints in the evaluation often leads to unstable structures, and rigid joints could result in too unconservative estimates. Several studies report this, including [5][6][7]. ...
... For timber gridshells, the nodes are typically developed for each project. Analyses of gridshell nodes are typically done using solid elements as seen in the recent studies by Castriotto et al. [12] and Seifi et al. [7]. Like in the Printshell project presented in Section 2.1, the analyses require exporting to separate tools for performing the meshing and analyses. ...
Linking architectural models to structural analyses can be demanding and time-consuming, especially when the architectural models cannot be accurately analysed using readily available one- or two-dimensional finite elements. This paper presents a tool for finite element analysis using solid elements developed as a plugin for Grasshopper 3D® that enables designers to include analyses of complex objects within the same software as the design exploration. A benchmark using the tool on a cantilever beam is compared with both ANSYS® and the theoretical solution, before the versatility of the tool is demonstrated by analyzing the metal part in timber gridshell nodes. The results were satisfying and the tool can prove especially useful for early phase design and collaboration between diciplines.
... As seen in Figure 11, the most common research type found was theoretical studies with 204 articles. This was followed by experimental study (65), project presentation (43), and review article (15). ...
... Furthermore, the studies on discrete gridshells are mostly steel regarding material. Examples from the discrete facet are [64][65][66]. The smooth gridshells, marked black in Figure 12, are relatively few and placed with no particular trends regarding part or material. ...
Gridshells are shells where the structural system is some kind of grid of linear members rather than a surface. With today’s focus on environmentally friendly solutions, gridshells have gained increased relevance as inherently material-efficient structures. This paper investigates the recent research on gridshells, who performs it and what their contributions are, and will thus provide an overview of the research field of gridshells. This study is performed as a systematic mapping. The articles were categorised by research type, motivation, contribution, gridshell type, material, and scientific field. The study shows that most articles are within structural engineering, whereas contributions from architecture were hard to find. The typical study was theoretical studies performing analyses on a specific load or structural behaviour. Some possible knowledge gaps were also identified, including review articles on loads and behaviour, research on bending active metal gridshells and development of gridshell nodes.
... Additive manufacturing is an automatic manufacturing method which could eliminate or minimize human mistakes during manufacturing process. The challenges of combining BESO as a design tool and additive manufacturing as manufacturing method have been formerly investigated by many other researches [14][15][16][17][18][19]. In these studies, BESO was employed to optimise the design of a node so as to achieve satisfactory structural performance with minimum amount of material, which also enabled the additive manufacturing of structural nodes. ...
... To introduce the design procedures of the new test setup, the arbitrary geometry condition used for the nodes in the authors' previous study [14] is used as an example herein. The geometrical parameters which are used in the node designs are listed in Table 1. ...
The design and manufacture of structural nodes for gridshell structures are complicated due to complex geometries and loading conditions. The validation of the design concepts of these nodes is even more challenging, because complex design loads are difficult to be applied in a laboratory test. In this paper, a testing rig is proposed and manufactured to test two different additively manufactured nodes. The two nodes are symmetrical and each connecting three members. These nodes are designed to sustain pure axial loads and pure out-of-plane bending moments, respectively. The results of the experiments show the importance of the bolt tolerance in the design of such a testing setup. Subsequently an innovative and inexpensive experimental setup is developed for testing nodes under dominant design loading conditions in gridshell structures. The proposed testing method is generalized, which can be applied to both individual and combined loading conditions, and the new testing rig can be easily fabricated at a low cost.
... Şekil 7. C tipi tonoz ızgara kabuk sistemde asimetrik yükleme durumunda oluşan yer değiştirme [21] Yapılan literatür araştırmasında ızgara kabuk sistemlerin birçok farklı formda tasarlanabildiği görülmüştür [21,[23][24][25][26][27]. Izgara kabuk sistemlerde yüzey geometrisinin, modül geometrisinin yanında yapının yükseklik/açıklık (Basıklık Oranı) oranının büyük önem arz ettiği görülmüştür [17,18-,23-27,]. ...
Günümüzde kent nüfusları gitgide artış göstermektedir. Kent nüfuslarının artmasıyla beraber yüksek insan kullanım kapasitesine sahip geniş ve ferah kapalı mekânlar oluşturulma ihtiyacı doğmuştur. Bu ihtiyaç doğrultusunda geniş açıklıkları kolaylıkla geçebilecek yeni taşıyıcı sistemler arayışına gidilmiştir. Yakın zamanda keşfedilen modern taşıyıcı sistemlerin başında ızgara kabuk (Gridshell) taşıyıcı sistemler gelmektedir. Izgara kabuk taşıyıcı sistemler ile geniş açıklıklar ekonomik olarak geçilebilmektedir. Bu çalışmada tonoz tipi çelik ızgara kabuk sistemler sayısal olarak incelenmiştir. Oluşturulan sayısal modellerde tonoz tipi çelik ızgara kabuk sistemler sabit basıklık oranı için dörtgen, üçgen ve altıgen modül geometrileri dikkate alınmıştır. Çalışma kapsamında tonoz tipi çelik ızgara kabuk sistemleri, 3 farklı modül geometrisi için analiz edilmiştir. Analizlerde değişken açıklıklar için en uygun modül geometrisi, en uygun modül geometrisine ait en uygun modül boyu ve taşıyıcı sistemin m2 düşen taşıyıcı sistem ağırlığı belirlenmiştir. Elde edilen sonuçlara göre tüm açıklıklarda dörtgen modül geometrisine sahip modellerde en az m2’ye düşen taşıyıcı sistem ağırlığına ulaşılmıştır. Açıklık artıkça altıgen modül geometrisinin dörtgen modül geometrisine alternatif olabileceği görülmüştür. Çalışma da ayrıca analiz edilen sistemler simetrik kar, asimetrik kar ve rüzgâr yükleme durumları altında analizleri gerçekleştirilmiş ve elde edilen düşey ve yatay yer değiştirmeler karşılaştırılmıştır. Analizler sonucunda altıgen modül geometrisi diğer 2 modül geometrisine göre yer değiştirmelere en dirençli modül geometrisi olduğu tespit edilmiştir. Yer değiştirme karşılaştırmaları incelendiğinde; tonoz tipi çelik ızgara kabuk sistemlerde asimetrik yükleme durumunda sistemlerin yer değiştirmelerinde olumsuz ölçüde artış meydana geldiği belirlenmiştir. Bu kapsamda elde edilen sonuçlar tonoz tipi çelik ızgara kabuklar sistemlerde farklı modül geometrileri için tasarımcılara bir perspektif sunacaktır.
... In this framework, Zhu et al. [56] performed a multi-objective topology optimization for joints connecting six circular hollow cross-sections to employ in spatial-structures: starting from a spherical node, the optimal topology has been found by removing material, in order to minimize both the compliance and the first natural frequency. Seifi et al. [57] proposed a topology optimization of six-way structural nodes for triangulated gridshells composed of hollow rectangular cross-sections, by considering different load conditions; then the optimized joints have been realized by means of additive manufacturing. In a similar way, Wang et al. [58] proposed a topology optimization for spatial structures characterized by a quadrangular mesh and members with rectangular hollow cross-sections, which joints have to be realized by means of additive manufacturing. ...
Structural optimization techniques are becoming popular and effective approaches for the design of constructions, able to support architects and structural designers in the complex process of searching competitive solutions, usually in terms of structural weight, cost and accounting for specific structural/functional requirements. In case of gridshells, the structural weight is strictly related to the susceptibility of the structure to global buckling, which is often the governing design criterion. The susceptibility to global buckling is mainly due to the global stiffness of the structures, primarily related to the stiffness of the joints, to the boundary conditions, and to the presence of imperfections. In this context, the paper presents design strategies based on optimization techniques that specifically take into accounts the presence of semi-rigid, rigid and hinged joints in order to guarantee light solutions safe from global buckling phenomena. In particular, two approaches are proposed: the joint stiffness approach, which considers the gridshell composed by semi-rigid joints, all characterized by the same rotational stiffness, and the rigid/hinged approach, which considers the gridshell composed by most hinged joints, and by a low number of rigid joints arranged in optimal positions. The approaches have been applied to a case study characterized by different boundary conditions, different rise-to-span ratios and also considering both perfect and imperfect shapes. The results of the proposed optimization processes highlight the beneficial effect of a finite value of the rotational stiffness of the joints in the susceptibility of the gridshell to global buckling phenomena, leading to light structural solutions.
... A further example is the so-called Takenaka node [100]. Topology optimised nodes, suitable for additive manufacture and use in a range of structural applications, including hybrid construction, have been described in [103][104][105][106], while a series of innovative additively manufactured steel connection designs, including a beam hook (see Figure 20(a)), a brace clamping element, a node for connecting four members and a beam-to-beam connection system, have been described by Lange et al. [15]. Optimised joints between tubular members, which are widely used in structural engineering applications, were simulated, manufactured and tested in Load WAAM steel T-stub ...
Although still in its infancy, metal additive manufacturing (AM) or 3D printing has now arrived at a scale suitable for use in construction. The new technology offers the potential for improved economy, sustainability, safety and productivity through greater automation, enhanced customisation, reduced material usage and reduced wastage. In this paper, a review of recent developments in metal AM in structural engineering is presented, including the latest research advances, current trends and applications in practice. Emphasis is placed on Directed Energy Deposition-arc (DED-arc) AM or wire arc AM (WAAM) since this is deemed to be the most promising technique for the requirements of the construction sector. A description of the observed material response of both steel and stainless steel WAAM elements, as well as the structural behaviour of cross-sections, members, connections and systems, is provided. The challenges surrounding the inherent geometric variability of as-built WAAM material, as well as the implications of possible anisotropy, are discussed.
Optimization and additive manufacturing go hand in hand, with the latter now enabling the former to be more readily realised in practice. Every kilogramme that can be saved by optimization reduces production times and makes additive manufacturing more economical, more sustainable and ultimately more viable. Recent examples of optimized, additively manufactured structural components are presented. Additive manufacturing is likely to complement, rather than replace existing construction methods – with this in mind, hybrid applications, as well as opportunities for strengthening and repair, are explored. The economics and sustainability of WAAM in construction, together with the key influencing parameters, are then discussed. Finally, with a view to the future, the opportunities, outlook and challenges for the wider use of metal additive manufacturing in structural engineering, are presented.
... Liu [20] took the spherical node as the research object and compared the mechanical properties of multi-objective optimization and single-objective optimization. Seifi [21] used the over-section method and the BESO method to study the topology optimization design of a six-bar box node in Sunshine Valley. These studies have greatly broadened the application scope of additive manufacturing and topology optimization, as well as accelerated the continuous innovation of manufacturing technology and industrial applications. ...
At present, a large number of scholars have conducted related research on topology optimization for additive manufacturing (AM). However, there are few relevant research reports on the impact of different directions of additive manufacturing on the optimal design and manufacturing results. In this paper, using the bidirectional evolutionary optimization (BESO) method, anisotropic optimization analysis was carried out on space nodes that are currently popular in the field of additive manufacturing and topology optimization. The elastic constants in different directions were used as anisotropic material properties for optimization research in this paper through tensile testing, which was carried out on 316L stainless-steel specimens fabricated using Selective Laser Melting (SLM) technology. In addition, SEM analyses were performed to explore the microscopic appearance of the material. The study found that additive manufacturing is affected by the printing direction in terms of both macroscopic mechanical properties and microscopic material structure; the deformation obtained by anisotropic optimization was about 1.1–2.3% smaller than that obtained by isotropic optimization.
... Therefore, cross-cast-steel joints have been widely applied in engineering during the last few years. However, the conventional fabrication process of cast-steel joints is mandrel, sandbox, casting, and polishing [23], and therefore the manufacturing cycle is long. Meanwhile, a large number of manual operations lead to huge labor consumption, especially when the types and numbers of joints in an entire structure are substantial, as shown in Figure 1b. ...
The integrated process of design and fabrication is invariably of particular interest and important to improve the quality and reduce the production cycle for structural joints, which are key components for connecting members and transferring loads in structural systems. In this work, using the generative design method, a pioneering idea was successfully realized to attain a reasonable configuration of the cross joints, which was then consecutively manufactured using 3D printing technology. Firstly, the initial model and generation conditions of a cross joint were constructed by the machine learning-based generative design algorithm, and hundreds of models were automatically generated. Then, based on the design objective and cost index of the cross joint, three representative joints were selected for further numerical analysis to verify the advantages of generative design. Finally, 3D printing was utilized to produce generative joints; the influences of printing parameters on the quality of 3D printing are further discussed in this paper. The results show that the cross joints from the generative design method have varied and innovative configurations and the best static behaviors. 3D printing technology can enhance the accuracy of cross joint fabrication. It is viable to utilize the integrated process of generative design and 3D printing to design and manufacture cross joints.
... Complex (Prayudhi, 2016), Right: Reduced-weight floor slab by (Jipa et al., 2016), derived from: https:// dbt.arch.ethz.ch/project/ topology-optimisationconcrete-slab/ grid shell nodes designed using TO have been previously explored in steel structures (van der Linden 2015; Prayudhi 2016;Seifi et al. 2018) finding that significant weight savings can be achieved while retaining structural integrity. To the knowledge of the authors, structural grid shell nodes made of glass are a novelty. ...
Up to now, fabricating cast glass components of substantial mass and/or thickness involves a lengthy and perplex annealing process. This has limited the use of this glass manufacturing method in the built environment to simple objects up to the size of regular building bricks, which can be annealed within a few hours. For the first time, structural topological optimization (TO) is investigated as an approach to design monolithic loadbearing cast-glass elements of substantial mass and dimensions, with significantly reduced annealing times. The research is two-fold. First, a numerical exploration is performed. The potential of reducing mass while maintaining satisfactory stiffness of a structural component is done through a case-study, in which a cast-glass grid shell node is designed and optimised. To achieve this, several design criteria in respect to glass as a material, casting as the manufacturing process and TO as a design method, are formulated and applied in the optimisation. It is concluded that a TO approach fully suited for three-dimensional glass design is as of yet not available. For this research, strain- or compliance based TO is selected for the optimization of the three-dimensional, cast glass grid shell node; in our case, we consider that a strain based TO allows for a better exploration of the thickness reduction, which, in turn, has a major influence on the annealing time of cast glass. In comparison, in a stress-based optimization, the considerably lower tensile strength of glass would become the main restrain, leaving underutilized the higher compressive strength. Furthermore, it is determined that a single, unchanging and dominant load-case is most suited for TO optimisation. Using ANSYS Workbench, mass reductions of up to 69% compared to an initial, unoptimized geometry are achieved, reducing annealing times by an estimated 90%. Following this, the feasibility of manufacturing the resulting complex-shaped glass components is investigated though physical prototypes. Two manufacturing techniques are explored: lost-wax casting using 3D-printed wax geometries, and kiln-casting using 3D-printed disposable sand moulds. Several glass prototypes were successfully cast and annealed. From this, several conclusions are drawn regarding the applicability and limitations of TO for cast glass components and the potential of alternative manufacturing methods for making such complex-shaped glass components.
... All different approaches resulted in the same pattern classes that were used to define the type of defect with high accuracy 35,41 . Fringe patterns are described as visual mathematical entities exhibiting symmetry, order, and limitation, the qualities of which are found in symmetry property exploration [88][89][90][91][92][93][94][95][96][97][98][99][100][101] . ...
Previous collaborative studies have shown the main fringe patterns and their typical classification with regard to defects. Nevertheless, the complexity of the results prevents defect detection automation based on a fringe pattern classification table. The use of fringe patterns for the structural diagnosis of artwork is important for conveying crucial detailed information and dense data sources that are unmatched compared to those obtained using other conventional or modern techniques. Hologram interferometry fringe patterns uniquely reveal existing and potential structural conditions independent of object shape, surface complexity, material inhomogeneity, multilayered and mixed media structures, without requiring contact and interaction with the precious surface. Thus, introducing a concept that from one hand allows fringe patterns to be considered as a powerful standalone physical tool for direct structural condition evaluation with a focus on artwork conservators' need for structural diagnosis while sets a conceptual basis for defect detection automation is crucial. The aim intensifies when the particularities of ethics and safety in the field of art conservation are considered.There are ways to obtain the advantages of fringe patterns even when specialized software and advanced analysis algorithms fail to convey usable information. Interactively treating the features of fringe patterns through step-wise reasoning provides direct diagnosis while formulates the knowledge basis to automate defect isolation and identification procedures for machine learning and artificial intelligence (AI) development. The transfer of understanding of the significance of fringe patterns through logical steps to an AI system is this work's ultimate technical aim. Research on topic is ongoing.
... The combination of topology optimization and AM has made breakthroughs in the fields of machinery, medicine, and aerospace engineering (Phillips et al., 2020;Wang & Qian, 2020). The application of AM technology to spatial structures provides a new path for the design and manufacture of new spatial joints (Hertlein et al., 2021;Seifi et al., 2018;Wang et al., 2020b;Zhu et al., 2019). ...
Topology optimization and additive manufacturing are of particular importance in solving the problems of stress concentration, excessive steel consumption, and excessive displacement in spatial structure joints. In this work, we investigate the best topological form for such joints under multi-objective topology optimization (MTO). We conducted a single-objective topology optimization (STO) analysis on the joint first, in the optimization process, the influences of penalty parameters, minimum member size, symmetry constraints, and the checkerboard phenomenon were comprehensively considered. Through comparative analysis, the optimal values for various load conditions were obtained, according to the optimal value obtained by STO, the MTO calculation of joint is carried out by using the compromise programming method. Then, the Nurbs modeling method is used to redesign the preliminarily optimized result, and the smooth joint is obtained. The mechanical performance of single-condition and multi-condition, MTO and STO, MTO joint and hollow sphere joint are compared with the same steel consumption. The results show that the mechanical performance of the joint obtained by MTO is more uniform than those obtained by STO, and the mechanical performance of the joint obtained by MTO is better than those of hollow sphere joint under the same steel consumption. Finally, the optimized joints were fabricated via additive manufacturing, which produced joints with novel shapes, smooth surfaces, and light weights.
... Further, their cost is related to different parameters, such their rigidity (which greatly affects the structural behavior), the number of elements converging in the joints, and the level of stress to which they are subject, which also depends on the presence and intensity of pretension in the elements [77]. In this context, recent studies propose topology optimization techniques combined with additive manufacturing processes in order to obtain joint configurations optimal from the point of view of structural weight and stiffness [76,78,79]. This suggests that future development of the present work may be to integrate the optimization of the structural elements with the optimization of the joints. ...
Gridshell are fascinating examples of structures able to combine aesthetic qualities and optimal structural performances thanks to a complete merging between “shape” and “structure”. To make gridshell structurally efficient is their capacity to cover large spans with light systems, exploiting the inherent strength of a double curvature shell. The design of gridshell is often the result of strategies based on both form-finding and structural-optimization techniques, sometimes combined together to obtain efficient structural solutions and architectural conception of space. Pre-tensioned rods, in some cases incorporated in the structure of gridshell, could particularly influence the structural performance of gridshell in terms of both its global deformability and stress distribution within its members. Consequently, pre-tensioned rods, when present, represent additional structural components to necessarily consider in the design optimization process of gridshell. Aim of the present paper is, first, to analyze the influence of pre-tensioned rods in the design optimization of gridshell and, then, to propose an optimization procedure for gridshell equipped with pre-tensioned rods. To this end, some simple gridshell examples and the case study of the Smithsonian Museum canopy in Washington are analyzed in the paper. The obtained results underline the importance of considering the presence of pre-tensioned rods both in the phase finalized to find the free-form shape and, moreover, in the structural optimization process.
... Meanwhile, the bypassing strategy turns fiber into an opposite direction to avoid crossing, resulting in a discontinuity of fiber reinforcement. Studies [22,23] about the joints of AM fabricated lattice structure are rare, and no existing work has been reported for CFRPC-AM. This necessitates a study on the fabrication strategy of joints to improve the structural integrity of continuous fiber reinforced composite lattice structures fabricated by AM. ...
Three-dimensional (3D) printing has revolutionized the fabrication process of continuous fiber reinforced composite lattice structures (CFRCLSs) with complex designs. However, optimal strategies for fabricating specific features of lattice structures, particularly joints, are not established, limiting the fabricated parts’ geometry accuracy and mechanical property. To address the above issues, we propose an interlayer-based adaptive process planning strategy for printing joints of CFRCLSs. More specifically, six strategies with different printing paths and parameter settings were used to fabricate an orthogrid structure. Their mechanical properties were compared and the part fabricated with the proposed strategy outstands the others. In addition, the failure modes and internal microstructure at the joints were analyzed using the digital image correlation (DIC) and micro-CT techniques for providing deep insights. Results indicate that the proposed strategy can significantly improve the geometry accuracy and mechanical properties of fabricated parts. Moreover, the effectiveness of the proposed joint strategy has been validated with two representative metamaterial designs, i.e., horseshoe and auxetic structures.
... It converts the problem of finding the optimal topology of a structure into the problem of finding the optimal material distribution in a given designable area. Representative topology optimization algorithms, such as bidirectional evolutionary structural optimization (BESO), smooth-edged material distribution for optimizing topology (SEMDOT), and solid isotropic material with penalization model (SIMP) are widely used in engineering fields, including machinery, aerospace, auto, and construction [8][9][10]. However, the current topology optimization algorithms require pre-specification of the designable area, setting of the optimization objectives, and determination of the loads and constraints, so the human workload is still large [11]. ...
Computer-aided design has been widely used in structural calculation and analysis, but there are still challenges in generating innovative structures intelligently. Aiming at this issue, a new method was proposed to realize the intelligent generation of innovative structures based on topology optimization and deep learning. Firstly, a large number of structural models obtained from topology optimization under different optimization parameters were extracted to produce the training set images, and the training set labels were defined as the corresponding load cases. Then, the boundary equilibrium generative adversarial networks (BEGAN) deep learning algorithm was applied to generate numerous innovative structures. Finally, the generated structures were evaluated by a series of evaluation indexes, including innovation, aesthetics, machinability, and mechanical performance. Combined with two engineering cases, the application process of the above method is described here in detail. Furthermore, the 3D reconstruction and additive manufacturing techniques were applied to manufacture the structural models. The research results showed that the proposed approach of structural generation based on topology optimization and deep learning is feasible, and can not only generate innovative structures but also optimize the material consumption and mechanical performance further.
... e loads acting at joint are the forces transferred in from the surrounding components. During topology optimization, one of its edges or its surfaces usually is assumed totally fixed [33]. ...
Owing to the capacities of generating structural configuration with both reasonable mechanical properties and high material utilization, topology optimization has been widely adopted in engineering design. Although numerous architects have tried to apply topology optimization tools to assist architectural morphology design in practical projects, topology optimization, like other quantitative analysis techniques, has not been systematically incorporated into the architectural morphology design. In this study, by integrating topology optimization toolsets and parametric design theory, combined with multiattribute decision-making analysis, a design method is proposed that could efficiently obtain several architectural structural architectural morphologies with both structural rationality and aesthetic rules and complete the evaluation and performance ranking of alternatives. In this study, the essential architectural application scenarios are divided into surface application scenarios and volumetric application scenarios, and the possible variation range of topology optimization parameters of architectural application scenarios is defined. By iteratively adjusting the influence parameters, diverse results of structural morphology are obtained. It is found that small changes in optimization parameters will bring great differences in topological results. Such a sensitive relationship can be utilized to generate a set of rational topological structures, and these topological results can be regarded as alternatives for architectural morphology design. For the performance evaluation and ranking analysis of alternatives, the application of FANP-TOPSIS multiattribute decision-making model is put forward in this study. The case study shows that this decision-making analysis model is efficient, convenient, and applicable in the architectural morphology design. The results of this study can provide new ideas and key references for scholars and architects in the field of architecture to explore the process and method of architectural morphology design and other related issues.
1. Introduction
Developments in construction industry design software and the maturity of related manufacturing techniques over the past two decades have led to the construction of buildings with complex and eye-catching appearance [1]. Whilst many have received praise and are considered to be iconic landmarks for their region, others are criticized for the lack of harmony between their architectural design and structural considerations. The challenge therefore remains to obtain satisfying designs that can simultaneously embrace architectural operational functions and aesthetic appealing effects, as well as maintaining rational structural performance [2]. Inspired by structural morphology (Rene Motro, an anthology of structural morphology), which involves form, forces, material, and structures and aiming at developing a structural system with harmony synthesis of these four aspects, architectural morphology is defined by extending the connotation of structural morphology, which simultaneously deals with structural performance, architectural functions, and aesthetical requirement, aiming at developing an architectural system with a balance between these factors.
Topology optimization, a mathematical method to optimize material distribution in a given area according to given conditions and objective index, has attained its popularity in civil engineering and architectural design owing to its potential to generate rational and aesthetic-artistic morphology [3]. Topology optimization was initially developed for applications in aeronautic and mechanical engineering [4], where the design space represents a continuum of material, and even small savings in weight are significant, for example, by saving fuel on thousands of journeys and/or saving material on thousands of mass-produced products. Amongst the many topology optimization methods that have been developed, common approaches include the solid isotropic material with penalization (SIMP) method [5–7], the (bidirectionally) evolutionary structural optimization (ESO or BESO) method [8, 9], level set methods [10–12], the moving morphable components (MMC) method [13, 14], and the independent continuous mapping (ICM) [15] method. Many of these approaches have been adopted for the application to the architectural morphology problem domain. For the design of bracing systems for high-rise buildings, Beghini et al. [16] proposed a topology optimization framework to integrate architecture and engineering. The generation of optimized shell- and large-scale spatial structures was investigated by Ohmori [17], who developed an extended ESO method, whereas Peng [18] applied the ICM method to designs of dendriform structures with hierarchical topologies similar to tree branches.
Whilst a wide range of literature can be found relating the application of topology optimization methods to architectural design, there still exist a number of gaps that necessitate further investigation, which this paper address. Firstly, researchers usually focus on a particular type of application scenario, such as beams, walls, or large-scale spatial structures, whereas a comprehensive study of how to use topology optimization to generate architectural morphology across many different application scenarios is still missing. Additionally, the relationship between the inputs to a topology optimization and the resulting morphology has not been investigated in detail. This lack of understanding of the sensitivity of the outputs to the inputs is one of the main obstacles preventing the architects from using topology optimization tools in practice. Thirdly, little research has been carried out to discuss and compare the topology and morphology of optimized architectural design from topology optimization in the perspective of aesthetic.
This paper first extracts and classifies the most common architectural scenarios based on their geometrical features and structural properties. It then derives the key parameters that affect the topological results and discusses their relative impact on these results. A methodology combining parametric modelling and topology optimization is then adopted for architectural morphology generation. By making use of the sensitive relationship between the resulting topology and the input parameters for optimization, a single solution, or a cluster of solutions, can be obtained. They are viewed as potential candidates for building designs, thus solving the problem of architectural morphology generation. Finally, a numerical case is adopted to compare morphology of different optimized shell results and provides some basic aesthetic evaluation from architect’s perspective.
The outline of this article is as follows. In Section 1, the context of the study and the required background knowledge is presented. In Section 2, the morphology generation procedure is proposed, and the influential parameters are identified. The essential architectural application scenarios are classified in Section 3, along with a discussion on how the influential parameters relate to each architectural application. Section 4 assesses the relationship between optimization parameters and the topological results for each classification, and in Section 5, the specific example of the morphology generation of a shell structure is investigated. Finally, Section 6 highlights the conclusions of the work and discusses the implications for morphology generation in practice.
2. Morphology Generation Methodology
Topology optimization of structure generally involves the addition, subtraction, or elimination of material from within a design domain. Through iterative adjustment of material, the optimal topology, representing the force flow within the domain, will gradually emerge. In addition to having the best mechanical performance, it is often the case that the obtained topology is also highly aesthetic. This successful combination of engineering and art is therefore viewed as a desirable candidate for architectural morphology design. However, there is no guarantee that the configuration produced though topology optimization would always be suitable for direct employment in the next design stage, and usually some modification is necessary, which can be achieved by adjusting the influential parameters.
Before considering how to adjust the influential parameters accordingly, a method for solving the problem of architectural morphology generation via topology optimization is introduced below, and the parameters that play key roles in determining the resulting morphology are considered.
2.1. Influential Parameters
In this paper, topology optimization is used to generate architectural morphology; therefore, the optimization parameters for topology optimization of different structures and structural members are also used as the parameters for morphology generation of them. Some additional parameters are required for the topology optimization, such as load scenarios, boundary conditions, and material properties, which are not directly related to the morphology.
The first parameter to be considered, the design domain, is represented by a geometry with planar or spatial features. This is usually defined based on consideration of architectural functions, such as space division, people-flow, light, and ventilation requirements. For example, it can be a wall with openings representing doors and windows, a hemispherical shell with holes on the top representing skylights, or a trimmed solid box representing an entire building. It should be noted that, during the optimization process, only materials in the design domain can be removed, retained, or reintroduced. This means that the optimal topology can only be made up of material within the design domain. In this way, the design domain, on the one hand, provides space for the morphology to change but, on the other hand, constrains the scope of that variation. Therefore, this essential relationship between the design domain and the resulting optimal topology makes the design domain one of the most dominant parameters that influences the optimization results of original structures.
The second consideration is the different loading scenarios on original structures. The purpose of topology optimization is to generate structural configuration with best mechanical performance under the external loads. The loads acting on buildings include gravity, live-, wind-, and snow-load, as well as concentrated (point) forces applied at certain positions to represent specific objects. With a small change in external loads, major variation of optimal topology can occur, since it is the mechanical response of the structure under these loads that determines the evolutionary direction of the optimization process.
Boundary conditions are the third parameter to consider. For buildings, boundary conditions usually include pin-supports, roller-supports, or fixed-supports. These supports can be present at specific discrete points, applied continuously along lines or curves, or even distributed across an entire surface. The boundary condition specifies the positions where the structure transfers its external loads to the foundations. Therefore, slight variations in boundary conditions also introduce significant changes in the optimization results.
Material properties also need to be carefully defined, and it is often the case that there will be more than one type of material being used within one architectural design of buildings or any specific structural members. For example, many high-rise buildings are constructed from steel beams, columns and decks, with a reinforced concrete slab poured on the deck in-situ to make a composite floor system. Specifying different material properties in different areas of a building can have a significant effect on its structural response, and hence its optimal topology. However, architectural morphology generation is usually carried out at an early stage of architectural design, at which point it is usually considered acceptable for only one material to be used for topology optimization. It is also a common assumption during early stage design that only linear elastic deformation would occur within the structure. In this case, the optimal topology for one material is also the optimal topology for another material. Therefore, for the purpose of this paper, the difference in the topology optimization results caused by the variation of materials can be assumed to be negligible.
Besides the optimization initialization parameters outlined above, the formulation of the topology optimization itself also involves the defining of parameters that have an impact on the results. Generally, the formulation of a topology optimization problem requires the definition of objective functions and constraint functions. The objective functions use objective index or performance index as dependent variable and input parameters as independent variable, and objective index or performance index is what researches want to maximize or minimize, for example, maximizing overall structure stiffness. Besides, researchers can use the constraint functions to apply specific geometric or mechanical constraint to optimized structures, such as minimum/maximum feature size [19, 20] and symmetry and pattern repetition [21]. These two kinds of functions are usually determined based on consideration of mechanical properties or geometric features of the design of original structure and can involve measures of deformation, stress, stability, material volume, etc. The influential parameters introduced above are classified into two categories as summarized in Table 1.
Influential parameters
Parameters for initialization of optimization problem
Design domain
Load scenario
Boundary condition
Material property
Parameters for formulation of optimization problem
Objective function
Constraint function
... For the reticulated shell structures in civil engineering [29,30], the slender beam members are generally used to realize the arbitrary shape. The configuration of these beam members strongly influences the mechanical performance and aesthetics of the spatial structures. ...
Evolutionary node shifting is an effective approach to the shape optimization of spatial structures for excellent mechanical performance. In this paper, a novel shape optimization algorithm is proposed, which is applicable to complex gridshell structures of irregular geometries and non-uniform grids. With the objective of maximizing the structural stiffness, the nodal coordinates are iteratively updated according to the sensitivity information. The perturbation displacements referring to the inverse hanging method are applied to the initial flat model. By analysis, we found that the irregular grids give rise to non-smooth gradient fields, which results in jagged surfaces. To normalize the non-smooth gradient fields and to prevent the optimization process from falling into local optima, a double filter scheme is introduced in the process of optimization. A variety of examples are presented to demonstrate that the proposed algorithm can effectively solve shape optimization problems of general gridshell structures.
... There has been extensive research on the topology optimization of the single-material continuum based on the BESO method [8,9]. Apart from the applications to architectural design [10] and mechanical design [11], the BESO method has also been introduced to the fields of advanced materials [12], aircraft design [13] and biomechanics [14,15]. ...
Topology optimization techniques based on finite element analysis have been widely used in many fields, but most of the research and applications are based on single-material structures. Extended from the bi-directional evolutionary structural optimization (BESO) method, a new topology optimization technique for 3D structures made of multiple materials is presented in this paper. According to the sum of each element's principal stresses in the design domain, a material more suitable for this element would be assigned. Numerical examples of a steel-concrete cantilever, two different bridges and four floor systems are provided to demonstrate the effectiveness and practical value of the proposed method for the conceptual design of composite structures made of steel and concrete.
... 3D printing technology applied in the turbine fabrication sector faces many obstacles, such as inadequate space for construction, cost, engineering effort, and time-consuming, in addition to the lack of 3D models. Additive manufacturing is widely used in many areas, such as buildings [5][6][7][8], bridges [9], automobiles [10][11][12], sports and medical equipment [13][14][15], wind turbines [16,17], marine vehicles [18], engines [19][20][21], rotating wings [22], and water tunnel testing [23]. The environmental impact of additive manufacturing using 3D printing technology is the reduction of waste; it can reduce waste by 40% [24]. ...
The 3D printing technology used for small tidal and wind turbines has great potential to change and overcome certain weaknesses in traditional manufacturing techniques. In rural areas and isolated communities, small turbine systems could be locally fabricated and assembled by using additive manufacturing machines and also can be employed to decrease residential energy consumption. The objective of the paper is to study the thermomechanical performance of 3D printing of a small-scale tidal turbine blade and their process using Digimat-AM because more research efforts are needed in this area. In this work, the tidal turbine blade is printed by using the Selective Laser Sintering SLS method with Polyamide 12 (PA12) and Polyether ether ketone (PEEK) polymers reinforced by carbon beads (CB) and glass beads (GB). This research examines conceptual considerations of small tidal turbines including material properties and aerodynamic parameters. Once the finite element evaluation has been completed, the deflection, residual stresses, temperature distribution, and the deformed blade or warpage can be obtained. It is concluded that PA12-CB has warpage higher than PA12-GB by 3.78%, and PEEK-CB has warpage lower than PEEK-GB by 8.4%. Also, the warpage of PA12-CB is lower than PEEK-CB by 10.31%, and the warpage of PA12-GB is lower than PEEK-GB by 20.95%. Therefore, the lowest warpage is observed for PA12-GB. Finally, the results showed that 3D printing presents an excellent opportunity in the design and development of tidal energy systems in the future.
... Wang et al. [13] studied the topology optimization of treelike joints with the goal of minimizing compliance and manufactured them through additive manufacturing. Seifi et al. [14] studied the topology optimization of structural joints of Sunshine Valley of World Expo under axial loading. Bai et al. [15] presents an explicit three-dimensional topology optimization method to obtain the hollow structures using moving morphable components (MMCs), which is achieved by combining two topology description functions, namely, internal and external topology description functions. ...
To realize the static and dynamic multiobjective topology optimization of joints in spatial structures, structural topology optimization is carried out to maximize the stiffness under static multiload conditions and maximize the first third-order dynamic natural frequencies. According to the single-objective optimization results, the objective function of the multiobjective topology optimization of joints is established by using the compromise programming method, and the weight coefficient of each static load condition is determined by using the analytic hierarchy process. Subsequently, under the constraint of the volume fraction, the multiobjective topology optimization of joints is realized by minimizing the multiobjective function. Finally, the optimized structure is smoothed to obtain a smoother joint, and its mechanical properties are compared with those of the hollow ball joint. The results indicate that the multiobjective topology optimization that considers the static stiffness and dynamic frequency can effectively improve the mechanical properties of the structure. Through the research on multiobjective topology optimization, a new type of spatial joint with reasonable stress, a novel form, and aesthetic shape can be obtained, which mitigates the shortcomings of single-objective topology optimization. In comparison to hollow spherical joints with the same weight, topology-optimized joints have a superior ability to resist deformation and improve low-order frequency, which verifies the feasibility of applying multiobjective topology optimization to the lightweight design of joints.
... For computing the complex nonlinear response of structures, the arclength method has become the de-facto incremental-iterative numerical technique in computational structural mechanics using the finite element method (FEM). Numerical methods for computing buckling instabilities are becoming even more important in the design and computer modelling of metamaterials [1][2][3][4][5], soft structures [6][7][8][9] and additively manufactured components [10,11]. ...
This paper presents a simplified implementation of the arc-length method for computing the equilibrium paths of nonlinear structural mechanics problems using the finite element method. In the proposed technique, the predictor is computed by extrapolating the solutions from two previously converged load steps. The extrapolation is a linear combination of the previous solutions; therefore, it is simple and inexpensive. Additionally, the proposed extrapolated predictor also serves as a means for identifying the forward movement along the equilibrium path without the need for any sophisticated techniques commonly employed for explicit tracking. The ability of the proposed technique to successfully compute complex equilibrium paths in static structural mechanics problems is demonstrated using seven numerical examples involving truss, beam-column and shell models. The computed numerical results are in excellent agreement with the reference solutions. The present approach does not require prohibitively small increments for its success.
... The Smart Nodes Project revealed the application potential of organic geometry generated using BESO algorithm. Another application case of customoptimisation of structural nodes was documented in research on "design optimization and additive manufacturing of nodes in gridshell structures" (Seifi 2018). We argue that reducing the structural complexity to the level of structural nodes allowed to rethink the design process on a smaller scale. ...
Virtual prototypes enable performance simulation for building components. The presented research extended the application of generative design using virtual prototypes for interactive optimisation of structural nodes. User-interactivity contributed to the geometric definition of design spaces rather than the final geometric outcome, enabling another stage of generative design for the micro-structure of the structural node. In this stage, the micro-structure inside the design space was generated using fixed topology. In contrast to common optimisation strategies, which converge towards a single optimal outcome, the presented design exploration process allowed the regular review of design solutions.
User-based selection guided the evolutionary process of design space exploration applying Online Classification. Another guidance mechanism called Shape Comparison introduced an intelligent control system using an inital image input as design reference. In this way, aesthetic guidance enabled the combined evaluation of quantitative and qualitative criteria in the custom-optimisation of structural nodes. Interactive node design extended the potential for shape variation of custom-optimized structural nodes by addressing the geometric definition of design spaces for multi-scalar structural optimisation.
... The optimisation process used in this research, as shown in figure 6, contains the core elements from the processes established by previous relevant research [12,13]. Steps 1 and 2 in the design process started by creating the conventional connection model in Fusion 360 and visually optimise its shape [14]. ...
Structural Topology Optimisation (STO) is a prevalent optimisation technique used nowadays to reach desired weight-to-stiffness ratios via highly complex and efficient designs unable to achieve otherwise. Additive manufacturing (AM) is widely known in the manufacturing industry and provides designers with a higher degree of freedom in realising highly optimised designs through a layer-based fabrication process. This paper focuses on reticulated structures and proposes using STO and AM to design and fabricate alternative connection designs with outstanding bespoke performance and drastically reduced weight. It studies the optimisation of a conventional node-connection found in reticulated timber structures under four loading cases, to producing state-of-the-art opti-mised connection designs, each capable of withstanding one of the four selected loading cases. The results are compared with the conventional node-connection, and the optimised configurations achieved up to 46.9% weight reduction. A selection of the highly bespoke scaled-down designs was additively manufactured in two different materials (metallic and polymer) as a proof of concept for the capacity of the technologies available for future testing.
... The optimisation process used in this research, as shown in figure 6, contains the core elements from the processes established by previous relevant research [12,13]. Steps 1 and 2 in the design process started by creating the conventional connection model in Fusion 360 and visually optimise its shape [14]. ...
Structural Topology Optimisation (STO) is a prevalent optimisation technique used nowadays to reach desired weight-to-stiffness ratios via highly complex and efficient designs unable to achieve otherwise. Additive manufacturing (AM) is widely known in the manufacturing industry and provides designers with a higher degree of freedom in realising highly optimised designs through a layer-based fabrication process. This paper focuses on reticulated structures and proposes using STO and AM to design and fabricate alternative connection designs with outstanding bespoke performance and drastically reduced weight. It studies the optimisation of a conventional node-connection found in reticulated timber structures under four loading cases, to producing state-of-the-art optimised connection designs, each capable of withstanding one of the four selected loading cases. The results are compared with the conventional node-connection, and the optimised configurations achieved up to 46.9% weight reduction. A selection of the highly bespoke scaled-down designs was additively manufactured in two different materials (metallic and polymer) as a proof of concept for the capacity of the technologies available for future testing.
... Different types of joints were proposed and their semi-rigid characteristics were investigated. For example, some of the existing joints are the TEMCOR joint [7], the socket joints [8], the welded hollow spherical joints [3], the bolted-arm joints [9,10], the gear joints [11], and the welded grid shell joints [12]. Besides, aluminum dome joints were also investigated [13][14][15], trying to take advantage of the lightness of the material. ...
This paper proposes a new type of bolted glulam joint for small-span and medium-span reticulated timber dome structures. The joint fastens the timber elements and the angled slotted-in steel plates together with steel bolts. Reasonably simplified experiments were designed and conducted to understand the mechanical properties of the proposed joint. Finite element models were also developed and calibrated with the tested results. A four-line model was provided to explain the mechanical properties of the joints, which were observed from the tests and simulations. To facilitate the future use of the proposed joint, theoretical derivations were provided to estimate its mechanical features. According to the estimation equations, bilinear moment–rotation curves could be easily obtained for the joints with different wood species, member sizes, joint designs, and/or bolt diameters. Finally, full-size structural models were created to investigate the static stability of K6 single-layered reticulated timber domes with the proposed joints. The influences on the ultimate structural stability capacity from the span, the rise-to-span ratio, the joint model (i.e., stiffness), the initial geometric imperfection introduced from the construction, and the load distribution were systematically investigated.
... The corresponding results show that increasing joint stiffness within a certain range can remarkably improve the dome stability. To improve the stiffness of joints, Seifi et al. [27,39] performed a topology optimization of the joints used in gridshell structures. Lu and Ye [40] proposed a method that combined optimization and design to develop various forms of joints. ...
... Also, in [57] a novel assembled hub joint with good mechanical performance and economic benefit is introduced in order to satisfy the demand for industrial assembly of single-layer reticulated domes. Furthermore, two design approaches have been proposed to design structural nodes of complex shapes for grid shell structures, seeking improved structural performance and design efficiency by using the transitional section method and the bi-directional evolutionary structural optimization method, respectively [58]. Similarly, other methods that involve topology optimization in conjunction with robotic construction procedures, try to simplify or standardize the structural elements, yet allow a custom logic to come to the fore towards the feasibility of the construction process. ...
This paper demonstrates an integrated computational design and fabrication workflow, implemented in actual construction scale, in order to test its feasibility towards a productive and customizable manufacturing process. The project suggests the computational development and the semi-automated fabrication of a free-form shell structure that consists of customized concrete members. The development process is based on two pillars of investigation that are bi-directionally connected; the computational design optimization and the fabrication. The first part of investigation refers to the form-finding of the overall shell structure, as well as its structural members by using topology optimization principles. The second part deals with the development of a reconfigurable modular formwork, which adapts to all shape alternations of the structural components, enabling their effective fabrication by using a single mechanism. Within this framework, decisions taken in the optimization stage are evaluated based on limitations and potentials of the suggested formwork, whose reconfigurability influences fabrication outcomes and vice versa.
... Topology optimization can also be used to optimize the layout for the global shape for better load distribution [16,17] which in turn could be interpreted as the network topology for a reticulated shell. With additive manufacturing, topology optimized nodes have been made in steel for several cases [18][19][20]. Additive manufacturing has also recently used for various custom-made splice connection for timber [21] and aluminium [18] members. Another project shows a free form façade node in aluminium [22] adapted to window profiles. ...
This paper discusses and describes the geometry and manufacturing of nodes in free form reticulated shell structures. Reticulated shells are commonly produced with steel nodes, providing new design solutions aluminium has different manufacturing constraints. The paper presents manufacturing methods for mass customized aluminium nodes and discusses the future potential of aluminium as a structural material for free form shells and spatial structures. The node's relation to the global shape, the grid division and the design choices is presented. Based on this, the future potential of aluminium nodes in free form reticulated shells is discussed.
... Likewise, much of the research efforts in topology optimization are focusing on additive manufacturing (AM) (Chen et al. 2015;Tang et al. 2018;Seifi et al. 2018). The recent advances in AM allow the exploitation of the full capacities of ESO/BESO techniques. ...
The challenges that a shape or design stands are central in its evolution. In the particular domain of stress/strain challenges, existing approaches eliminate under-demanded neighborhoods from the shape, thus producing the evolution. This strategy alone incorrectly (a) conserves disconnected parts of the shape and (b) eliminates neighborhoods which are essential to maintain the boundary conditions (supports, loads). The existing analyses preventing (a) and (b) are conducted in an ad-hoc manner, by using graph connectivity. This manuscript presents the implementation of a meta-graph methodology, which systematically lumps together finite element subsets of the current shape. By considering this meta-graph connectivity, the method impedes situations (a) and (b), while maintaining the pruning of under-demanded neighborhoods. Research opportunities are open in the application of this methodology with other types of demand on the shape (e.g., friction, temperature, drag, and abrasion).
... The optimisation process used in this research as shown in figure 5 contains the core elements from the processes established by previous relevant research [11,12]. Steps 1 and 2 in the design process started by creating the traditional connection model in Fusion 360 and visually optimise its shape [13]. ...
Abstract. Structural Topology Optimisation (STO) is a prevalent optimisation technique used nowadays to reach highly complex and efficient designs (weight-to-stiffness ratio) unable to achieve otherwise. Additive Manufacturing (AM) is a developing manufacturing process which overcomes many of the manufacturing limitations and realises highly optimised products through a layer-based fabrication process. Recent research on reticulated structures proposed using STO and 3D printing to design and fabricate alternative bespoke complex connection designs which have shown its significance through obtaining substantial weight reductions for the same structural capacity. This paper builds on previous research through optimising a single-layer S355 traditional node-connection under four loading cases, producing a state-of-the-art optimised connection design capable of withstanding the four loading cases considered and comparing the results to the traditional ones. In all loading cases, optimised shapes with 46.90% weight reduction were obtained with varying stress levels. A selection of the highly bespoke designs were 3D printed as a proof of concept for the applicability of AM.
... The optimisation process used in this research as shown in figure 4 contains the core elements from the processes established by previous relevant research [11,12]. Steps 1 and 2 in the design process started by creating the typical connection model in Fusion 360 and visually optimise its shape [13]. ...
The wire arc additive manufacturing (WAAM) technology in combination with computational design shows a big potential for realising novel force‐flow optimised and material‐efficient connections. This contribution deals with point‐by‐point WAAM, a material deposition strategy that allows to place material precisely where structurally needed or aesthetically desired. This could be applied, among others, for realising a novel optimised type of steel nodes between custom‐oriented profiles, as they occur in freeform steel‐glass grid‐shells. In this paper, the structural behaviour of robotically fabricated straight WAAM steel bars under uniaxial tensile and compressive loading is discussed. The focus is set on the ductility exhibited by such components as well as on the buckling behaviour observed under compressive loading. Experimental tests were conducted, both under tensile and under compressive loading to assess the influence of the irregular geometry on the structural performance. Furthermore, it was studied to what extent a prediction of the ductile structural behaviour, of the compressive load‐bearing capacity and of the post‐buckling behaviour is possible by finite element simulations. This contribution presents and discusses highlights of the obtained results.
Casting nodes are used to connect branch members at joints of freeform grid shell structures. In this study, the structural performance of a rectangular hollow section (RHS) branch-to-casting node connection was investigated. A RHS branch member with an access hole (i.e., opening) for temporary bolting was weld-connected to the T-shaped node fabricated by a conventional casting method. Six RHS branch-to-node connection specimens were tested under tensile and compressive loading. The eccentricity of axial loads was considered as the main test variable. Tests showed that, despite the opening near the weld joint, the RHS branch-to-casting node connections exhibited good performances. For tensile specimens, the ultimate strengths were greater than the yield strengths of the casting nodes, and weld rupture eventually occurred at the stiffener-to-casting node wall joints. On the other hand, for compressive specimens, the ultimate strengths were limited by flexural and torsional buckling modes accompanying local buckling and crippling of the casting node walls. The ultimate strengths degraded by the eccentric loading and openings were evaluated in accordance with current design codes. The failure modes and consequent strengths of compressive specimens were further investigated through finite-element analysis. Based on the investigations, design recommendations of RHS branch-to-casting node connections with openings were given.
To explore the connection reliability of double-ring joints, three groups of full-scale specimens with different structural parameters were designed. Loading tests for these specimens were conducted under the action of cyclic moments, and the failure modes and hysteretic performance of the double-ring joints were studied. In addition, the influences of various structural parameters on the hysteretic performance of the joints were explored using a finite element analysis model. The following conclusions were obtained: (1) Two failure modes, central ring fracture and bolt fracture, were observed in the experiment. For the joints in which central ring fracture occurred, the deformation and energy dissipation capacity were the strongest, and bearing capacity degradation and rigidity degradation both occurred. For the joints in which bolt fracture occurred, only rigidity degradation occurred during the loading process, and slippage existed in the moment-rotation hysteresis loops due to deformation of the bolts. (2) Energy dissipation mainly occurred in the connection region, and the energy dissipation value of the connection region was more than 70% of the total energy dissipation value. (3) By increasing the section height and central ring thickness, the rotational stiffness and ultimate bearing capacity of the joint could be improved. However, the increase in section height had little effect on the energy dissipation capacity of the joint.
Purpose
There are many investigations in design methodologies, but there are also divergences and convergences as there are so many points of view. This study aims to evaluate to corroborate and deepen other researchers’ findings, dissipate divergences and provide directing to future work on the subject from a methodological and convergent perspective.
Design/methodology/approach
This study analyzes the previous reviews (about 15 reviews) and based on the consensus and the classifications provided by these authors, a significant sample of research is analyzed in the design for additive manufacturing (DFAM) theme (approximately 80 articles until June of 2017 and approximately 280–300 articles until February of 2019) through descriptive statistics, to corroborate and deepen the findings of other researchers.
Findings
Throughout this work, this paper found statistics indicating that the main areas studied are: multiple objective optimizations, execution of the design, general DFAM and DFAM for functional performance. Among the main conclusions: there is a lack of innovation in the products developed with the methodologies, there is a lack of exhaustivity in the methodologies, there are few efforts to include environmental aspects in the methodologies, many of the methods include economic and cost evaluation, but are not very explicit and broad (sustainability evaluation), it is necessary to consider a greater variety of functions, among other conclusions
Originality/value
The novelty in this study is the methodology. It is very objective, comprehensive and quantitative. The starting point is not the case studies nor the qualitative criteria, but the figures and quantities of methodologies. The main contribution of this review article is to guide future work on the subject from a methodological and convergent perspective and this article provides a broad database with articles containing information on many issues to make decisions: design methodology; optimization; processes, selection of parts and materials; cost and product management; mechanical, electrical and thermal properties; health and environmental impact, etc.
Taking double-ring joint as research object, the mechanical performances under pure bending, bending-shear and eccentric loading conditions of the joints were studied in this paper. Double-ring joints with different structural parameters were analyzed using finite element analysis and theoretical derivation methods and combined with experimental results. The following conclusions were obtained: 1) under the pure bending condition, the spring model of the double-ring joint was established, and the theoretical calculation formulas of the ultimate bearing capacity and initial rotational stiffness were derived; 2) under the bending-shear condition, the accuracy of finite element analysis models was verified by comparing the numerical with experimental results. The mechanical properties of double-ring joints under the bending-shear condition were similar to those under the pure bending condition; 3) based on the full-section plasticity criterion, the relationship between ultimate bending moment ratio η0N and axial force eccentricity δ was derived. With the exponential function for nonlinear fitting, the relationship between initial rotational stiffness ratio ξ0N and axial force eccentricity δ was derived.
Gridshells are light structures that allow the obtention of large spans, as well as the use of a wide variety of architectural shapes. However, in Brazil, there are no records of the construction of this type of structure, neither literature available on this type of structural composition. This paper presents the main theoretical aspects, as well as design and construction details of a gridshell built in wood in real scale. The construction costs are also reported. The structure was built by using a quadrangular flat mesh configuration, formed by double slatted Lyptus wood (Class D40), joined orthogonally, using 6.35 mm diameter screws. The plane mesh (with total dimensions of 9 m x 9 m) was bent until the desired curve was achieved. The system was subsequently locked by diagonal bracing elements. The knowledge obtained through this practical application led to a significant improvement in the original design idea related to controlling the shape of the structure during the building phase and the the optimisation of the executing strategies. The building recommendations presented in this paper will be a contribution to the scarce literature on the subject currently available in Brazil.
The majority of glass used in load-bearing structures is as planar elements. Some projects exist that use linear glass elements. This paper discusses in broad terms the design, engineering, and fabrication of a unique vector active glass structure consisting of glass bundles and partly printed steel connections. A structure was conceived that utilizes the glass bundles in a way that can be directly experienced by the users: a swing. To create a non-standard form for the swing, a structural optimization procedure was used. To realize the structure, a novel steel node was developed and produced using an additive manufacturing technique in steel. These novel applications have made the project innovation heavy, particularly considering the limited timeframe for its development and construction. Description is given of the several optimization techniques incorporated in the digital process, the assembly and testing of the glass bundles, and the manufacturing of the steel nodes by Wire and Arc Additive Manufacturing.
Wind tunnel testing is a reliable means for aircraft design. The wind tunnel models are the objects used in the tests. The accuracy and economy of the model design and fabrication have an important impact on the quality and cycle of aircraft development. Additive Manufacturing (AM, or Rapid Prototyping, 3D printing) can directly fabricate 3D parts through accumulating raw materials, and is widely regarded as a revolutionary advancement in manufacturing technology. In the very early development period, AM was soon introduced and was studied by many groups worldwide. Firstly, the introduction of AM is an advancement for the fabrication of models, which can greatly improve the fabrication economy of current models, such as reducing the number of parts, and shortening the processing cycle etc. Secondly, the introduction of AM can also improve the design of models, which is helpful to develop new types of models and even new test methods. Thirdly, AM has blurred the boundaries between real aircraft and experimental models, and promoted the development of new concept aircraft. The review first introduces the design requirements of the wind tunnel test models and recalls the AM application history for models. Next, detailed technologies concerning the design procedure and fabrication processing of the AM-based models are presented. Finally, the application of AM-based model in the development of current air vehicles and new-concept vehicles is introduced. The review provides an overview of techniques of AM in wind tunnel test models, and provide typical examples for reference to designers and researchers in the aerospace industry.
In the safety assessment of existing spatial structures, sampling measurements are widely used to investigate structural geometries. The information from sampling measurements is limited and leads to the positional uncertainty of unmeasured nodes. Uncertain nodal positions have a significant influence on the stability assessment of the spatial structures with high geometric-imperfection sensitivity. To deal with the uncertainty of nodal positions, a probabilistic framework based on the randomness of nodal positional deviations was proposed. In the framework, uncertain nodal positions are modeled by random field, whose relevant parameters were inferred with the data from sampling measurements; based on the random field model, a full-probability analysis is adopted to determine the load-bearing capacity and further perform the stability assessment. According to the analysis of the correlation between nodal positions, correlated and uncorrelated random fields were respectively introduced to model the correlated and uncorrelated nodal deviations. And the detailed modeling methods were discussed, including the inference of distribution types, parameters and correlations. Finally, an existing single-layer reticulated shell was used to validate the reliability of the proposed probabilistic framework.
The innovative architect, Frei Otto, developed the concept of Gridshells which could bedesigned by a funicular modelling method and constructed from an equal net mesh of timber laths bent into the planned shape. In 1970 this technique was used to construct a 9,000 m sq curved roof structure from 5 cm square timber laths. This paper summarises the design and engineering work that went into the construction of this remarkable building.
An initial research has been carried out, exploring the opportunities of using the design freedom created by Additive Manufacturing in metal on structural elements. The application of Additive Manufacturing (AM) bears the potential of increasing efficiency and shortening lead time by reducing processing steps, material use and labor intensity.
The aim of this research is the exploration of AM technology potentials as a feasible and robust design and manufacturing solution for structural building elements. This research is based on the redesign of an existing node, applying the new production opportunities of AM. This resulted in insights into the different design steps involved and knowledge on process, costs and structural characteristics. The start was a topology optimization on an existing design for a structural node, followed by a further rationalized design for production. A comparison in aesthetics, structural behavior and costs has been carried out between the new design and the conventional design. The first results look very promising and the work will be continued to enable the use of these new opportunities in structural designs in the near future.
Cast steel nodes are being increasingly popular in steel structure joint application as their advanced mechanical performances and flexible forms. This kind of joints improves the structural antifatigue capability observably and is expected to be widely used in the structures with fatigue loadings. Cast steel node joint consists of two parts: casting itself and the welds between the node and the steel member. The fatigue resistances of these two parts are very different; the experiment results showed very clearly that the fatigue behavior was governed by the welds in all tested configurations. This paper focuses on the balance fatigue design of these two parts in a cast steel node joint using fracture mechanics and FEM. The defects in castings are simulated by cracks conservatively. The final crack size is decided by the minimum of 90% of the wall thickness and the value deduced by fracture toughness. The allowable initial crack size could be obtained through the integral of Paris equation when the crack propagation life is considered equal to the weld fatigue life; therefore, the two parts in a cast steel node joint will have a balance fatigue life.
Free-form geometries are very popular in today's architecture. Computer controlled fabrication methods allow for structures, which would not have been possible a few years ago. But how does one transfer 3D geometries into a load bearing structure without loosing the architectural vision of smoothly shaped building envelopes? An intense discussion is necessary to navigate architectural visions of elegant 3D shapes through the technical and economic constraints of realization. The role of the engineer is conceived as that of a mediator between the aesthetic ambitions of the architect, the budget of the client and the technical capabilities of the contractor. From the very early stages of form finding to the assembly on site, a consistent design process is absolutely necessary to achieve high quality free formed structures.
In this paper different examples are presented to demonstrate this. The projects vary from a single layer grid shell for Westfield's new shopping centre at White City, London and a large space structure in Frankfurt to a timber shell for a department store in Cologne
Promoted by buildings like the DZ-Bank in Berlin (Frank Gehry), the Arts Center in Singapore (Vikas Gore), the British Museum in London (Norman Foster) and recently the New Fair in Milan (Massimiliano Fuksas) with its roofs above the Central Axis and the Service Center, free form shaped structures became more and more popular in recent years.
This paper is a brief review of the geometrical and structural issues related to the design of reticulated structures on free form surfaces.
The need to simplify the construction issues of complex structures leads to definition of SmartNodes project as a research which aims to confine the complexity of structure to a limited area (nodes) in order to decrease processing steps and labor intensity by application of additive manufacturing (AM) techniques. Bi-Directional Evolutionary Structural Optimization (BESO) is used to design efficient and elegant nodal connections of large scale spatial structures and minimise the volume of nodes to be printed and to ultimately replace welded, forged and cast connections by 3D printed connections. The prototypes discussed in this paper demonstrate BESO design process through two generic cases.
Steel/Aluminum grid shells—a key structural concept selected by notable Architects such as HOK, Fosters, AECOM, and others for institutional & commercial projects resulted in the design/construction of innovative skylights, walkways & bridges. Built facilities offered clear view of the sky that makes it feel like outdoor spaces with controlled temperatures.
Grid-shells are lightweight structures used to cover long spans with few load-bearing material, as they excel for lightness, elegance and transparency. In this paper we analyze the stability of hex-dominant free-form grid-shells, generated with the Statics Aware Voronoi Remeshing scheme introduced in Pietroni et al. (2015). This is a novel hex-dominant, organic-like and non uniform remeshing pattern that manages to take into account the statics of the underlying surface. We show how this pattern is particularly suitable for free-form grid-shells, providing good performance in terms of both aesthetics and structural behavior. To reach this goal, we select a set of four contemporary architectural surfaces and we establish a systematic comparative analysis between Statics Aware Voronoi Grid-Shells and equivalent state of the art triangular and quadrilateral grid-shells. For each dataset and for each grid-shell topology, imperfection sensitivity analyses are carried out and the worst response diagrams compared. It turns out that, in spite of the intrinsic weakness of the hexagonal topology, free-form Statics Aware Voronoi Grid-Shells are much more effective than their state-of-the-art quadrilateral counterparts.
Single-layer reticulated domes are typical structures with high degree of static indeterminacy. The stability of the reticulated dome structures is complicated and closely related to both the member buckling and the overall buckling of the structure. However, little theoretical research work deals with this very complex phenomenon. Meanwhile, for the statically indeterminate structures, it is a common belief that the local failure would occur first and result in the change of internal force paths, while the structure could support load continually before the overall collapse of the structure. In this paper, the interaction between the member buckling and the overall buckling of the structure was investigated, and this common belief was explored. The geometrically and materially nonlinear analyses (GMNA) were carried out for the reticulated domes with eight types of different grid forms, and the examination of member buckling for each dome was conducted based on the examination method of the P–δend curve or the P–δmid curve of the member. The relationship between the member buckling and the global instability of the structure was obtained by evaluation results of member buckling. Consequently, for the common reticulated domes, two instability patterns were identified, i.e. progressive instability and synchronous instability. In the case of progressive instability, the member buckling occurs in advance of the overall buckling of the structure. Then the number of the instable member increases, and finally the overall buckling of the structure occurs. In the case of synchronous instability, the member buckling and the overall buckling of the structure happen simultaneously. The interaction mechanism between the member buckling and the global instability of the structure was analyzed for both instability patterns.
The present paper discusses the sensitivity of the global and local stability of a hybrid single layer grid shell to a set of Equivalent Geometric Nodal Imperfections representative of the actual structural and construction imperfections. Since imperfections are hard to be measured and controlled in experimental facilities, the stability of the structure is extensively investigated in numerical experiments. The imperfections are set by means of the so-called Eigenmode Imperfection Method. The method parameter space is densely sampled, and different structural models are adopted. The results are given in terms of two bulk parameters: the well established Load Factor and the proposed Buckling Shape Length, the latter being introduced to provide a continuous measure of the degree of “globalness” of the instability. Significant and non monotonic changes in both the Load Factor and Buckling Shape Length are observed versus the growth of the imperfection amplitude. Further, a local metrics of the grid shell geometry, named nodal apex, is introduced for each structural node. Special emphasis is given to the analysis of the correlation between the apex of the initial imperfect geometry and the apex of the deformed shape at collapse. The observed high correlation suggests that the mechanical behavior of the imperfect grid shell is significantly influenced by this local geometrical feature.
A new type of joint was adopted in the single-layer lattice structure of the Sun Valleys for Expo Axis project. The design idea of the new joints was that only upper and lower flanges of rectangular steel tube members and webs of two main members with maximum internal forces were welded to the core zone of the joint. According to processing technology, the core zone has two types of configuration: two end plates connected with stiffener and solid cylinder. Ten typical full-scale joints selected from three Sun Valleys were tested, in order to investigate the structural behavior, failure mechanism and load bearing capacity of the joints so that the safety and reliability of joints could be assured, and to verify the joints fabricated with different processing technologies could meet the design requirement. Test results indicate that the joints adopted in the project have sufficient safety reserves and can meet the requirement of 'strong-joints/weak-member' concept. For joints with a solid cylinder as the core zone, the stress level is lower and stress concentration is relatively mild. Results from finite element modeling of the test joints indicate that, the elasto-plastic nonlinear analysis considering large deformations can simulate the bearing behavior of the joint satisfactorily.
For free-form structures with a single layer envelope, the node is subjected to axial load, as well as moment. The purpose of this paper is to develop and evaluate a new hollow spherical connector, termed the FREE (Flexible, Resilient, Efficient, Economic) node, for single layer free-form spatial envelopes. Reducing material and efficient shape are important points in the space structure node. Therefore, hollow spherical node sections with different holes were proposed and finite element analysis was performed, to determine the prototype among these parameters. The parameters are the size and shape of the hollow sphere. Bending experiments were also performed, to evaluate the structural capacities of the proposed node; and the corresponding test results were compared with finite element analysis. As a result, it may be said the proposed FREE node satisfied the required capacities.
The application of Additive Manufacturing (AM) technologies promises much innovation across the manufacturing sector, and has generated great interest in the Architecture, Engineering and Construction (AEC) industry. This paper presents and reflects upon early prototypes for integrating AM in a construction application, through the design of a prototype frame structure with linear members connected by nodes of unique shapes. As an enabler for design, a system is developed and implemented to integrate expertise across architecture, structural engineering and advanced manufacturing in order to design and detail components for single layer canopies. This includes the topology optimisation and additive manufacture of nodes, both of which require the control of material behaviour at small scales. A scaled pavilion structure and full-scale node prototypes were successfully realised. However, this first stage of research presented a number of challenges to modelling material behaviour across scales— both the physical size and production volumes.
The quality of finite element meshes is one of the key factors that affects the accuracy and reliability of finite element analysis results. In order to improve the quality of hexahedral meshes, we present a novel hexahedral mesh smoothing algorithm which combines a local regularization for each hexahedral mesh, using dual element based geometric transformation, with a global optimization operator for all hexahedral meshes. The global optimization operator is composed of three main terms, including the volumetric Laplacian operator of hexahedral meshes and the geometric constraints of surface meshes which keep the volumetric details and the surface details, and another is the transformed node displacements condition which maintains the regularity of all elements. The global optimization operator is formulated as a quadratic optimization problem, which is easily solved by solving a sparse linear system. Several experimental results are presented to demonstrate that our method obtains higher quality results than other state-of-the-art approaches.
A new classification system for the joints in lattice shells is proposed. The stiffness and moment capacity of the joints together with the overall structural behavior of the lattice shells are considered in order to establish the classification system. According to this new system, joints in lattice shells can be classified into unique categories: rigid, semi-rigid or pinned. The rigid joints have both high bending stiffness and moment capacity; the semi-rigid joints have both moderate bending stiffness and moment capacity, and the pinned joints have either low bending stiffness or low moment capacity. Determination coefficients α and β are defined, based on the stiffness and moment capacity of the joints, and these are used to establish clear boundaries between the different categories. Some numerical examples are included to demonstrate the validity of the classification system. With the help of the classification system, an efficient process for practical engineering design is proposed, which can help designers choose the appropriate analysis method for lattice shells with different joints.
Geometric non-linearities in single-layer domes can cause the loss of global stability, which is strongly influenced by both geometric parameters and joint rigidity. The rigidity of the joint requires deeper study, since it significantly affects the behaviour of these structures. To this end, a model is proposed for the ORTZ joint. The model is established from the dimensions and properties of the different elements of which the joint is composed and considers the possibility of the material reaching its yield point. After experimental verification, the model is implemented in a computer application. Finally, experimental tests have been conducted on two structures possessing very different features related to geometry and rigidity of joints. In both cases the proposed model has given a good estimation of the experimentally observed behaviour of the structures.
Although space trusses have seen continuous development during the last few decades, their share in the markets of large-span structures is still quite small. Primarily, because of their relative high cost, space trusses have almost been limited in use to applications where a pleasing appearance is of high priority, and cost is not the determining factor [Codd, 1984, low technology space frames. Third Int. Conf. on Space Structures, Surrey, UK, 955–960]. It is argued that the special nodes required for space truss assembly are most responsible for their high cost. Attempts to create space truss systems that have continuous chord members and do not need node components have been successful in reducing the cost substantially with the chord continuity and eccentricity creating unexpected structural advantages. This paper presents an assessment of the current technology of space truss construction. It also presents a new space truss system that attempts to reduce the cost further without compromising the system's structural reliability. The paper also introduces some added features of the new system that are expected to increase its appeal.
The analysis of tubular single layer structures involves geometric nonlinear methods that can consider large displacements of nodes. In addition, the analysis of single-layer structures must take into account not just member buckling, but also local and overall buckling situations. Collapse is strongly influenced by both geometric parameters and joint rigidity. Even though current computing power does allow designers to run nonlinear analyses as many times as needed, efficiency of the design process is based on the designer’s ability to define an initial design close enough to the final geometry of the structure. This paper presents results obtained from analysis of the nonlinear behaviour of single-layer domes with different geometric parameters and joint rigidity. The study is performed for two load cases and concludes with the proposal of a new formula which allows designers a rapid estimation of buckling loads for semi-rigid jointed single-layer latticed domes under symmetric loading conditions.
In the design of single-layer structures, the hypothesis of pinned joints leads to structures with low capacity in terms of stability and resistance. Therefore, one of the main concerns of structural designers in recent years has been to find an appropriate joint design which would endow the joint with sufficient stiffness. In this paper, the results of experimental tests conducted with the aim of establishing geometrical parameters for a semi-rigid joint that may be used in single-layer structures are presented. They showed how the combination of different parameters can improve the stiffness of the joint and its rotational capacity. At the same time, the experimental tests provided the initial rotational stiffness of the tested joints which is to be introduced into the analysis of the structure. The paper presents an analytical method for the determination of the initial rotational stiffness of the joint. The method follows a technique similar to the component method of Eurocode 3 part 1.8, although it has been adapted to the geometry of this particular joint.
Investigation of elasto-plastic stability of reticulated shells has gradually become more and more attractive to researchers. However, the analysis of the elasto-plastic stability is much more complicated than the elastic analysis, because of the involvement of both the geometrical nonlinearity and the material nonlinearity and the interactions of the two. Taking into consideration of material nonlinearity, the elasto-plastic stability behaviour of reticulated shells will be significantly different, which can only be revealed by large amount of geometrically and material nonlinear analysis of the structures. The elasto-plastic stability of seven types of commonly used single-layer reticulated shells (Kiewitt-8, Kiewitt-6, Geodesic, Schwedler Bidirectional, Schwedler Monoclonal, Sunflower, and Radial Rib) was investigated systematically with ANSYS, in which more than 2000 cases of geometrically and material nonlinear analysis were conducted with different initial geometrical imperfections, geometrical, structural and load parameters. Based on the analytical results and via comparison between elastic and elasto-plastic buckling loads, the plasticity influence coefficients were summarized for each type of the reticulated domes. These coefficients can be used to predict accurately the elasto-plastic buckling load in design practice of the reticulated domes, which is an easy way for designers to evaluate stability of the reticulated shells.
The effects of physical and geometrical non-linearities on the buckling and post-buckling behaviour of reticulated shells are analysed, paying particular attention to the influence of imperfections.Two types of random imperfections are considered: joint imperfections, which are often caused by the technical assemblying of the elements, and geometry (or scheme's) imperfections, which occur when the real configuration of the shell does not respect the designed one. The static analysis consists firstly of modelling the material non-linearity of a tubular element under eccentrical normal load and secondly of developing the iterative incremental procedure to solve the non-linear equilibrium conditions. Finally, the problem of tracing secondary equilibrium paths is faced, by means of a non-linear stability analysis. The results obtained by a test structure hexagonal in plan are compared in the final section for the perfect and imperfect cases.
In contrast to most triangulation algorithms which implicitly assume that triangulation point locations are fixed, ‘Laplacian’ smoothing focuses on moving point locations to improve triangulation. Laplacian smoothing is attractive for its simplicity but it does require an existing triangulation. In this paper the effect of Laplacian smoothing on Delaunay triangulations is explored. It will become clear that constraining Laplacian smoothing to maintain a Delaunay triangulation measurably improves Laplacian smoothing.
Due to their aesthetic and structural advantages, tubular space truss structures are enjoying increasing popularity in modern bridge construction. The use of cast steel nodes for the joints between the circular hollow section members is also becoming increasingly popular. The fatigue design of such joints, however, requires additional knowledge with respect to their fatigue resistance. Previous experimental investigations showed very clearly that the fatigue behaviour is governed by the welds between the casting stubs and the hollow section members. This paper presents a methodology for the determination of allowable initial sizes of casting defects as a function of the required fatigue resistance of the welds. The relative influence of the main parameters is quantitatively discussed and recommendations for design are given.
Buckling collapse and its analytical method of steel reticulated domes with semi-rigid ball joints are discussed, on the basis of a nonlinear elastic–plastic hinge analysis formulated for three-dimensional beam-columns with elastic perfectly plastic hinges located at both ends and the mid-span for each member. The second-order elastic analysis for reticulated domes, composed of members with semi-rigid connections modelled with elastic springs at the member ends, in conjunction with an update Lagrangian formulation, is carried out. Reductions in collapse loads due to semi-rigidity of the connections, as well as to the geometric imperfections of nodal coordinates and member crookedness, are investigated. The reductions of member strength are also plotted and summarized as the function of generalized slenderness, on the basis of which, the axial strength in the member is expressed ready for use to estimate the collapse loads of reticulated domes.
A procedure is presented to improve the quality of hexahedral element meshes, especially for the meshes adaptively generated using a modified grid-based method. Insertion technique and collapsing technique are proposed to modify the mesh topology and improve the quality of the degenerated elements which are on the mesh boundaries and cannot be improved by any nodes position smoothing methods. A curvature-based Laplacian smoothing method is employed to improve the shape-quality of boundary elements and ensure that the boundary characters of mesh are well preserved. A method for improving the shape-quality of the mesh surface and the initial elements which is based on the objective function of the mesh quality metric is proposed. For the mesh surface, the Condition Number of the Jacobian metric associated with the quadrangle elements is taken as the optimization objective function. For the initial elements, after smoothing with the Laplacian method, the Scaled Jacobian metric associated with the hexahedral elements is employed as the optimization objective function. The proposed optimization methods have been applied to the adaptive generation of the hexahedral element mesh using grid-based method. The effectiveness and robustness of the proposed optimization methods are tested through several complex three-dimensional models.
Buckling analyses of several reticulated shell structures are carried out using both an approximate equivalent shell analysis and a discrete analysis which is essentially exact. The structures considered are: an infinite reticulated beam under axial compression and resting at equally spaced intervals on elastic springs; a shallow section of a reticulated sphere with an equilateral triangle grid subject to normal loading; and an infinite reticulated cylindrical shell with an equilateral triangle grid subject to axial compression. The discrete analysis is used to evaluate the accuracy of the predictions of the equivalent shell analysis. The buckling load computed on the basis of the equivalent shell analysis is nonconservative when a characteristic wavelength of the buckling deformation is on the order of the member length or the axial load in a member at buckling is on the order of the Euler buckling load of a simply supported column. The effect on the buckling load of reducing the rigidity of the joints is investigated for both the beam-spring model and the reticulated spherical shell. Finally, the importance of the discrete analysis is illustrated by the determination of the optimum properties of a shallow section of a reticulated sphere subject to a prescribed normal loading and designed against buckling.
Cast steel nodes in tubular construction-Canadian experience
- De Oliveira
- J C Willibald
- S Packer
- J A Christopoulos
- C Verhey
De Oliveira JC, Willibald S, Packer JA, Christopoulos C, Verhey T. Cast steel nodes
in tubular construction-Canadian experience. In: 11th International symposium and
IIW international conference on tubular structures, Quebec 2006, p. 523-529.