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Innovative design solutions for contemporary Tou-Kung based on topological optimisation

Innovative design solutions for contemporary Tou-Kung based on topological optimisation Tou-Kung, which is pronounced in Chinese and known as Bracket Set (Liang & Fairbank, A pictorial history of Chinese architecture, 1984), is a vital support component in the Chinese traditional wooden tectonic systems. It is located between the column and the beam and connects the eave and pillar, making the heavy roof extend out of the eaves longer. The development of Tou-Kung is entirely a microcosm of the development of ancient Chinese architecture; the aesthetic structure and Asian artistic temperament behind Tou-Kung make it gradually become the cultural and spiritual symbol of traditional Chinese architecture. In the contemporary era, inheriting and developing Tou-Kung has become an essential issue. Several architects have attempted to employ new materials and techniques to integrate the traditional Tou-Kung into modern architectural systems, such as the China Pavilion at the 2010 World Expo and Yusuhara Wooden Bridge Museum. This paper introduces the topological optimisation method bi-directional evolu- tionary structural optimisation (BESO) for form-finding. BESO method is one of the most popular topology optimisa- tion methods widely employed in civil engineering and architecture. Through analyzing the development trend of Tou-Kung and mechanical structure, the authors integrate 2D and 3D optimisation methods and apply the hybrid methods to form-finding. Meanwhile, mortise and tenon joint used to create stable connections with components of Tou-Kung are retained. This research aims to design a new Tou-Kung corresponding to “structural performance-based aesthetics”. The workflow proposed in this paper is valuable for Architrave and other traditional building components. Keywords Bi-directional Evolutionary Structural Optimisation (BESO), TOU-KUNG (Bracket Set), Mortise and tenon joint, Structural form-finding, Heritage building, Architectural component dynasties. The development of the timber frame has 1 Introduction also become the central vein of China’s architectural Chinese ancient architecture has a long history and development. Tou-Kung plays an essential role in the a self-contained system. Regarding building materi- Chinese wood tectonic system. It is an indispensable als, China’s vast territory, various timber, and large supporting element in Chinese timber-frame buildings bases have led to the gradual formation of the tectonic that supports  the  weight  of  the  overhanging  eaves. system based on a wooden framework through the Meanwhile, it symbolises the feudal hierarchy and an indispensable cultural character in traditional Chinese *Correspondence: architecture. It is a complex member located at the Ding Wen Bao nic.bao@rmit.edu.au junction of the column and the beam, made up of arms Xin Yan (Kung) and blocks (Tou). At the base of Tou-Kung, it yanxin2022@tsinghua.edu.cn is a large block called Lu-Tou. Crossed arms spread- China University of Mining and Technology, Jiangsu 221116, China RMIT University, Melbourne 3000, Australia ing in four directions are set into Lu-Tou. These in Tsinghua University, Beijing 10084, China © The Author(s) 2023. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Duan et al. Architectural Intelligence (2023) 2:10 Page 2 of 16 turn bear smaller blocks (Shan-Tou) that carry longer 2.1 The dendritic column with structural bionics design arms. Hua-Kung extends outward, while the others The application of dendritic columns in the spatial struc - parallel to the wall are called Transverse Kung as a ture solves the problem that a single vertical column group, including Ni-Tao-Kung, Man-Kung etc. Long with an equal cross-section is challenging to set off the cantilever arms called Ang extend out to support the far-reaching eave, in accordance with the function of the outermost purlins (Fig. 1) (Liang & Fairbank, 1984). In tiered projections of Tou-Kung. The dendritic column is an era where digital design is booming, the research upright and looks like a tree trunk (Qian et  al. 2020). It present here tries to combine the conventional Tou- has various branches at the top of the column, geomet- Kung with advanced structural form-finding methods rically dispersed to form multiple points of support and to create a new derivative of Tou-Kung that incorpo- radiating away from the centre of the column (Fig.  2A). rates ancient and modern elements. The inheritance Although the dendritic column satisfies the integration of and innovation of Tou-Kung have become a significant “force” and “form”, achieved straightforward transferred concern. This paper focuses on using the evolution force and material efficiency, and it differs from Tou- of traditional building components to document the Kung in appearance. development of architectural techniques in a new era. 2.2 Archaise Tou‑Kung using contemporary tectonic methods Archaise Tou-Kung is constructed from high-strength 2 Translations and derivation practices materials such as cement and aluminum alloy, imitat- of Tou‑Kung ing or copying Tou-Kung from various historical periods From a contemporary standpoint, several architects (Fig.  2B). The same are the beams, columns, Tou-Kung have attempted to translate the traditional Tou-Kung, and pillars in archaise buildings. Still, they are not opti- employing new materials and techniques to integrate mised for the mechanical properties of the individ- it into modern architectural systems. These can be ual materials. As a result, the overall stiffness and the broadly divided into three categories according to structural frequency of archaize buildings are excessive structural role, form, and cultural symbolism. Fig. 1 Principal parts of a Chinese timber-frame building D uan et al. Architectural Intelligence (2023) 2:10 Page 3 of 16 Fig. 2 A Dendritic column: B Archaize Tou-Kung; C The cultural imagery of Tou-Kung (Wang, 2011), leading to enormous material waste and The development of Tou-Kung can be briefly divided the loss of characteristics of the traditional Tou-Kung as into five stages. Firstly, from the Western Chou Dynasty simple, efficient, and exquisite. to the Han Dynasty, the image of Tou-Kung was depicted in various historical relics (Fig.  3A). After the Han 2.3 C ultural symbols of Tou‑Kung Dynasty, it appeared between the columns. Secondly, in This type of building has a superficial structure that the period of Wei, Jin, Southern and Northern Dynasties, exhibits layer-by-layer stacking, achieving a cultural the form of Tou-Kung gradually became standardised, homage to Tou-Kung (Fig.  2C). In the case of the China and the classic forms of inverted-V brackets appeared Pavilion at the 2010 World Expo, the traditional Tou- (Fig. 3B). Thirdly, from the Sui and Tang Dynasties to the Kung is simplified through dislocation, orthogonality, Five Dynasties, Tou-Kung was vigorously gorgeous and and classification in three-dimensional construction. magnificent, characterised by the huge body (Fig.  3C). While the mechanical characteristics of force transferred Seen from the elevation, the height of Tou-Kung can from one layer to another are retained (Xu & Liu, 2018). reach the half-height of the column. Transverse Kung However, this type of abstract translation detracts from and column-top joists overlap to form a lattice-shaped the essential structural role of Tou-Kung and the cor- framework. As a strengthened node, Tou-Kung also responding scale of its components. It serves only as an becomes an indispensable part of the structure. Fourthly, abstract derivative and a metaphor for Tou-Kung culture. during Song and Yuan Dynasties, the image of Tou- Kung became tenderly beautiful. Since Song Dynasty, 3 Introduction and analysis of Tou‑Kung the number of Tou-Kung has increased, and the volume 3.1 The evolution of Tou‑Kung has decreased. A certain decorative effect was intro - The evolution of Tou-Kung is a nearly comprehensive duced. The size of components started to be unified, and presentation of the development of architectural skills the modular system appeared (Fig.  3D). The last was the and the aesthetic trends of building components in each Ming and Qing Dynasties. By the Qing Dynasty, Tou- period of Chinese history. Tou-Kung is not only a single Kung adopted the modular of the mortise of Lu-Tou, structural component, but also has multiple-layered sig- which was used to standardise the dimensions of each nificance in cultural inheritance, decoration and beauti - component of it (Fig. 3E). Decorative and coloured paint- fication, class identity, measurement, etc. The variations ings appeared on the surface of Tou-Kung, which further of Tou-Kung have also provided important evidence for strengthened the decoration and weakened the structure identifying the age of ancient architecture and observing roles. At this stage, Tou-Kung ceased playing the role of the style of different times. keeping the overall framework stable. Duan et al. Architectural Intelligence (2023) 2:10 Page 4 of 16 Fig. 3 The evolution of Tou-Kung Throughout its history, Tou-Kung has evolved from and stability. So, Tou-Kung rises in tiers or “jumps” and large to small, from simple to complex, with its struc- extends outward in steps, and through the mortise-tenon tural role diminishing and its decorative role strengthen- joint, several beams become whole, which increases the ing. As technology and materials developed, Tou-Kung bending and shearing resistance of Tou-Kung. As sup- has switched its role from load-bearing to decorative porting the eaves, the beam sometimes appears extremely and from function to form in the process of meeting long, and the distance between the bearing beam end the demands of the times. The ideal state of the devel - and the bearing support is excessively large, leaving a opment and derivation of the contemporary Tou-Kung large force arm and a weak beam bearing capacity, which is to reflect the technological development of the new makes the beam easy to break. Therefore, from the end era while preserving the historical heritage. The focus to the starting point of the overhanging part, Tou-Kung of this paper is not to simply restore Tou-Kung of a cer- adopts the way of layer-by-layer superposition and lay- tain period or use modern materials to make a tradi- ered force transmission to ensure overall stability. tional form of Tou-Kung, but to combine new technology Tou-Kung is a combined structural member, which and new materials with traditional architectural tech- needs to be disassembled into several components when niques, exploring the profound traditional architectural studying its force transmission characteristics to present spirit with a high degree of unity of structure and form its layer-by-layer force transmission properties clearly (Yuan et  al., 2017), and seeking resemblance rather than (Fig.  5). The key mechanical components of Tou-Kung mimesis. can be dissembled into Tou (blocks), Kung (arms) and Ang (long cantilever arms) (Liang & Fairbank, 1984). 3.2 The structural function and mechanical prototype Specifically, Tou is a square wooden block used to sup - of Tou‑Kung port Kung and Ang. The Tou mainly works as cushions Tou-Kung, placed between beams and columns, plays to transfer the pressure from the upper structure, trans- the key role of a connecting link. And the load of the roof porting the small buckets from top to bottom to the and the upper structure is transmitted to the spreading Kung connected below (Wei, 2007). Among these Kung, layer of Tou-Kung, and then to the column and founda- the Lu-Tou at the bottom of Tou-Kung, which is called tion. As a structural “transit hub”, Kung is paved up layer Lu-Tou, is the most important compression member of by layer, and the truss and purlin on the outermost layer the Tou-Kung system. Kung is a bow-like part. Hua-Kung are propped further for a longer distance, which supports is vertical to the building facade, responsible for over- the far-reaching eaves of the structure and makes it more hanging, while Transverse Kung is parallel to the build- imposing and magnificent (Fig. 4A). ing facade, which plays a balancing role. Hua-Kung and The mechanical prototype of Tou-Kung is essentially Transverse Kung bear the bending moment and longi- a short cantilever beam (Fig.  4B), whose main structural tudinal load, respectively. The pressure from the upper function is to support the load of eaves (Yu et al., 2006). structure is concentrated at the middle and the two ends For the cantilever beam, the increased height of the of Hua-Kung, resulting in a large bending and shear stress beam from outside to inside can ensure overall strength at the head of Hua-Kung (Lv, 2010). However, the use of D uan et al. Architectural Intelligence (2023) 2:10 Page 5 of 16 Fig. 4 The structural function and mechanical prototype of the Tou-Kung Fig. 5 Layer-by-layer force transmission system of Tou-Kung Duan et al. Architectural Intelligence (2023) 2:10 Page 6 of 16 Transverse Kung can support the upper structure out- is divided into several components according to the force side the Hua-Kung plane, strengthening the connection characteristics of the individual elements. Therefore, the between Tou-Kung and the upper structure, improving structural prototype of layer-by-layer force transmission the overall bending and torsional strength of Tou-Kung, is reserved. Through the digital form-finding and finite and making Tou-Kung stable spatial support in all direc- element analysis, a new Tou-Kung is designed on the tions. Ang is the longest member in Tou-Kung, which premise of material efficiency and ensuring definite force sags diagonally and is structured as a cantilever beam in transmission routes and a reasonable structure (Fig. 6). the oblique direction. Ang is a compression member due to its large cross-section area, which can bear the lift-4.1 Design methodology ing load of the eaves and transmit the pressure from the The Ameba plug-in is implemented to carry out topology upper parts. In addition to compression, Ang also car- optimisation and BESO algorithm based on the Rhino ries a certain level of bending moment. The components and Grasshopper platforms (Zhou et  al., 2018; Zhou, of Tou-Kung are interlaced and stacked, full of rhythm 2021). In the application of topology optimisation, the changes. rational distribution of Tou-Kung materials is realised by deciding the removal, reservation, or supplement of 4 Topology optimisation of Tou‑Kung materials through finite element analysis (Fig. 7). In this experiment, the bi-directional evolutionary struc- The design transforms Tou-Kung in the Ming and tural optimisation (BESO) method has been applied to Qing Dynasties. In that period, the Ang was fake, and the design. The concept of BESO is to find a preliminary didn’t have the structural character of lower Ang in the structural configuration which meets a predefined cri - Song Dynasty. The force transmission of Tou-Kung in terion based on finite element analysis. As an advanced the Ming and Qing Dynasties is neater and better aligned form-finding method in digital design (Lin, 2020; Zhao with the property of layer-by-layer force transmission and Chen, 2016), topology optimisation can optimise the from top to bottom. In topology optimisation, it is also layout of materials and explore new forms of a contempo- the most effective structural morphology to transmit rary bucket set on the premise of a reasonable structure the upper load with specific boundary support, material (Xie et  al., 2014). BESO method has been implemented consumption, and material type. To keep to the layer-by- into architectural design in many pieces of research due layer force bearing and transmission mode of each part of to its capacity to generate diverse organic forms with Tou-Kung, this study adopts the concept of “zoning opti- high structural performance (Bao et  al., 2022; Yan et  al., misation” in the Multi-volume BESO method (Yan et al., 2022). During the design process, the scale and posi- 2022) as the core logic of Tou-Kung optimisation. Spe- tion of the traditional Tou-Kung are inherited. Also, the cifically, Tou-Kung is divided into Tou, Kung, Ang, and hybrid method for optimization is introduced. Tou-Kung other parts, and then they are respectively optimised by Fig. 6 Topologically optimised Tou-Kung D uan et al. Architectural Intelligence (2023) 2:10 Page 7 of 16 Fig. 7 Workflow of Tou-Kung optimisation & exploded diagram of optimised Tou-Kung 4.2 Topology optimisation form‑finding of 2D Tou‑Kung BESO/Ameba; later, all optimised parts are reconstructed The 2D topology optimisation and planar force analy - and assembled. sis are conducive to the subsequent 3D structure form- This calculation framework (Fig.  7) is derived from the finding of Tou-Kung. In addition, the 2D optimised optimization process of Tou-Kung and is based on the results can be compared with the 3D optimised ones combination of the topology optimisation method and to decide the rationality of the results and provide a the zoning optimisation method. Zoning optimisation reference for adding supplements, resetting boundary can be applied to the optimization of the different scales conditions, and optimising parameters. In the zon- of components. More specifically, the components can be ing optimisation of 2D Tou-Kung, the primary parts, divided according to the number of storeys of the parti- including Lu-Tou, Shan-Tou, Ni-Tao-Kung, Ling-Kung, tions, the volume of the partitions and the different posi - Hua-Kung, and Kua-Tzu-Kung, are extracted. Kua- tions of the partitions in the building, in order to satisfy Tzu-Kung on the central axis of the column is selected the optimisation calculation of components in different as representative of component composition simula- scales. In the case of this chapter, the zoning optimisation tion analysis. Firstly, the planar angle of each compo- of Tou-Kung, a small-scale, multistorey building element, nent is selected, and the ratio scale of each element is will be calculated in Sections 4.2 and 4.3. Then the zoning determined according to the Tou-Kou module system of the architrave-column combination, a large-scale, low- in the Qing Dynasty. Taking “Tou-Kou” as the modu- storey building element, will be calculated in Section 4.4, lus unit, an accurate 2D model of each component is which can further illustrate the compatibility and ration- constructed. Then according to the descriptions of ality of the framework for optimising building elements the force bearing of each part of the structure and the at different scales. Duan et al. Architectural Intelligence (2023) 2:10 Page 8 of 16 Fig. 8 2D topological optimised components of Tou-Kung connection mode between Tou and Kung in Ying Zao 1. Low strength of the wall of Tou-Kung Fa Shi (Li & Liang, 1983) and Engineering Practice Rules and Examples from Qing Dynasty (Liang, 2003), The wall of Tou-Kung is the smallest individual com - the force diagram of components is drawn for 2D force ponent, and it has a large volume difference com - analysis. Secondly, the load and support are set as pared to other components. Considering the differ - boundary conditions after generating the mesh element ent shapes and low strength of the wall of Tou-Kung in Rhino/Ameba, and parameters for preprocessing are after topology optimisation, Shan-Tou is combined set to run the BESO algorithm for seeking optimal 2D with Kung in the 3D prototype, and a typical combi- topology structural forms (Fig. 8). nation of Tou and Kung, which can be described as The 2D topological optimized results and the force one block on the bottom, one arm in the middle and analysis shows a significant reduction in materials at three small blocks on the top, is formed (Fig. 9). both ends and wall of Tou-Kung. Ancient carpenters 2. The dangerous critical cross-section used the entasis method to make the ends of Tou-Kung During the 2D optimisation, the rabbets of Lu-Tou, into gentle curves or fold with a full and soft appearance. the first layer Kua-Tzu-Kung and the first layer Hua- The material layout of Tou-Kung after topology optimi - Kung complement each other with overlapped dan- sation is similar to the form of the traditional Tou-Kung gerous sections with hidden safety hazards. There - after entasis treatment. From force analysis, the corner in fore, we combined the three components with the the traditional Tou-Kung is a non-force bearing part or a central part as the integrated component to supple- part with less bearing load. Therefore, the entasis part’s ment the strength of the critical section further. material deletions outnumber that of other parts after 3. Recurrence of entasis topological optimisation. The ancients artificially omitted To further verify the significance of entasis in terms of the corner material based on their construction experi- rational force transfer and material saving, the geomet- ence, thereby developing the practice of entasis. This is ric prototype of Tou-Kung is simplified during the 3D an experimental result of artificially saving materials and optimisation, and the entasis chamfer was cancelled efficiently expressing structural force transmission. to form a simplified, integrated cube. The purpose is to determine whether material deletion and chamfer would reappear at the original entasis position in the 4.3 Topology optimisation form‑finding of 3D Tou‑Kung optimised structure. At last, four combinations were After previous 2D component decomposition optimisa- formed: one Tou with three Shan-Tou, three Tou with tion, we found three issues that should be considered in three Shan-Tou, the combination of the Ang and Hua- the 3D optimisation: Kung, and the combination of Lu-Tou and Hua-Kung. Then 3D optimisation is performed and comparatively D uan et al. Architectural Intelligence (2023) 2:10 Page 9 of 16 Fig. 9 3D topological optimised components of Tou-Kung analysed with the 2D optimisation results. Parameters structural prototype of the tree-like column and the frac- are adjusted to achieve the optimal solution for topo- tal structure of tree branches in nature, proving the high logical optimised Tou-Kung. correlation and similarity between Tou-Kung and tree According to the results of the 3D optimisation, sev- branches in the mechanical prototype. The upper and eral combinations of Tou-Kung have presented structural lower surfaces have complete materials for the structure shapes of oblique fractal bifurcation from the support to of Shan-Tou, while the middle part is a porous struc- the load ends. This mechanical structure is similar to the ture. Massive materials were cut for the original entasis Fig. 10 Topology optimisation results of the entasis part of Tou-Kung Duan et al. Architectural Intelligence (2023) 2:10 Page 10 of 16 position or even an “entasis style” was formed. The cham - optimisation was applied to form-finding of topologi - fering amplitude of these designs was more obvious than cally optimised Tou-Kung. This approach can be applied in traditional Tou-Kung (Fig.  10). The topology optimisa - not only to the form-finding of the scale of Tou-Kung, tion results also reflected the wisdom of ancient crafts - but also to the scale of the building. In the following, we men and the excellent craftsmanship of Tou-Kung, which will verify the feasibility of our workflow through the is absolutely an architectural gem that integrated form optimisation of the architrave and column. During the and force (Fig.11). architrave topology optimisation of the Hall of Prayer for Good Harvests, the combined member of a ring of archi- 4.4 A case study and t opological optimisation experiment trave and eave column can be disassembled into six same of the Hall of Prayer for Good Harvests units according to the symmetry. Then the 1/12 mini - Architrave is also an important part of the Chinese tim- mum optimisation unit can be obtained based on the ber frame. Usually, an architrave is installed on the top symmetric properties of the units (Fig. 12). of a column to link Tou-Kung between columns with the For the load arrangement, the downward uniform load load-bearing horizontal member. In some architectures, can be set on the upper critical plane connecting Archi- large and small architraves were stacked and juxtaposed trave and Tou-Kung, and a side thrust can be im- posed with the middle connected by a clamp pad (Cao & Wang, on the sparrow brace, which ensures that all the units 2010). These architraves were simplified as a complete are closely connected in the partition optimisation. The geometric graph and were used as the optimised pro- base was installed at the position of the symmetrically totype together with the eave column. In the former optimised vertical cross-section and column bottom. The research, in spite of the layer-by-layer force transmis- overall effect after optimisation was shown in the image sion and mortise-tenon joint of Tou-Kung, the zoning (Fig. 13). Fig. 11 3D topological optimised Tou-Kung models Fig. 12 3D topological optimised architrave-column combination D uan et al. Architectural Intelligence (2023) 2:10 Page 11 of 16 Fig. 13 3D topological optimised components in ancient buildings After the optimization of Architrave, the results with a structural layer that transmits the layer-by-layer force the previous optimization results of Tou-Kung were between the roof and columns. As a structural connec- combined. Then, the corresponding original structure of tion method, the mortise-tenon joint connects indi- the Hall of Prayer for Good Harvests was replaced while vidual members without using a single nail or bolt retaining the traditional paintings, doors, windows, tiles (Zhang, 2022), which only uses the geometric connection and other decorative components. The overlap of new between the members with concave-convex compensa- and old elements formed a stark contrast in appear- tion for the shape to achieve an efficient and secure con - ance (Fig.  14) while maintaining the same mechanical nection (Wang, 2019). In the mortise-tenon joint, “tenon” essence. This kind of hidden link ensures a more rigor - is the convex part, and “mortise” is the concave part ([Liu ous connection between culture and structure based on and Song, 2019), with a variety of derived forms, which the unity of aesthetic form and force in contemporary realise the horizontal and vertical connection between all structure. the members of Tou-Kung (Fig. 15). The use of the mortise-tenon technique in Tou-Kung 5 Structural optimisation strategy can be summarized into two types: the mortise-tenon for the members of Tou‑Kung based joint and the mortise-mortise joint. In the first type, the on the mortise‑tenon joint concave and convex parts interlock with each other and 5.1 O verview and classification of the mortise‑tenon joint form a strong, stable and flexible framework (Fig.  15B). in Tou‑Kung Mortise-tenon joint is commonly seen in the connec- In the structure of the traditional Tou-Kung, multiple tion of the top of the column with the bottom of Lu-Tou, members stepped layer outwards layer by layer, forming as well as Shan-Tou with cross mortise with their lower Fig. 14 Topological optimised Hall of Prayer for Good Harvests Duan et al. Architectural Intelligence (2023) 2:10 Page 12 of 16 Fig. 15 The use and classification of mortise-tenon joint in Tou-Kung members. This type of mortise-tenon technique allows the mortise-mortise joint is suitable for the connection of the connection structure to be hidden in the members’ members at the same horizontal layer (Liang et al., 2021). own volume without protruding from the body block, The joint use of the two methods forms the basis for the thus preserving the integrity of the appearance of Tou- layer-by-layer force transmission in Tou-Kung, which is Kung. Meanwhile, the height of the two layers of mem- also an indispensable technical path to achieve the façade bers is also retained with this combination. modelling of “corbelling”. The second type is mortise-mortise joint (Fig.  15A). “Mortise” is the concave part. Two “concave” members 5.2 Problems and solutions to the structural optimisation are lapped in the mutual perpendicular direction in Tou- in mortise‑tenon joint Kung between columns and are buckled in other angular Topology optimisation often presents structural reten- directions in the corner Tou-Kung. This type of connec - tion of unit volume, and mortise-tenon joint usually tion is widely used between Transverse Kung and Hua- raises a high-volume correspondence requirement to Kung, or Transverse Kung and Ang. As two “concave” ensure sufficient interlocking between the concave and members are lapped together in a position that cuts 1/2 convex parts. Therefore, it became the focus of this of the height of the single-layer member, the complemen- study to make topology optimisation of the members tary joint becomes a finished type and still retains the in Tou-Kung to eliminate low-efficiency units to ensure height of the single-layer member. the lightweight structure of Tou-Kung while retaining In summary, the mortise-tenon joint is suitable for the the unit at the mortise-tenon joint part and ensuring connection between different layers of members, while the complementary concave and convex volumes at the D uan et al. Architectural Intelligence (2023) 2:10 Page 13 of 16 Fig. 16 The process of optimising the mortise-tenon joint in Tou-Kung mortise-tenon joint with high accuracy. This issue can two worked as a whole and are connected to the cross- be further summarized as the “structural optimisation shaped recess of the Lu-Tou below, and the pressure is strategy for the members of Tou-Kung based on mortise- transmitted from the above to the Lu-Tou and then to tenon joint constraints” (Fig. 16). the columns below (Fig.  17A). As the symmetry axis In the selection of the original domain of Tou-Kung, parts of Kua-Tzu-Kung and Hua-Kung have removed we chose the representative Tou-Kung with two “Jumps” 1/2 of the height of the single-layer member respec- on the architrave between two columns and selected the tively, the middle part becomes relatively weak. If the five-member combination of “Lu-Tou - Kua-Tzu-Kung weak domain of Kua-Tzu-Kung and Hua-Kung is not on the central axis of the column - Shan-Tou on the top protected, the materials in the weak domain would be of Kua-Tzu-Kung - Hua-Kung - Shan-Tou” as the final further removed after structural optimisation. On the matrix for structural optimisation. This combination one hand, it will cause a hidden danger in structural encompasses the two above-mentioned mortise-tenon strength; on the other hand, the unrestrained volume techniques (the mortise-tenon joint and the mortise- cut would destroy the tightness of the connection of the mortise joint) in the mortise-tenon joint of a traditional mortise-mortise joint. Tou-Kung. For the selection of Shan-Tou, two represent- To deal with the above problems, the following meas- ative types, namely the two sidebar constraints and the ures are taken during the structural optimisation of Kua- four corner point constraints, are also presented. Tzu-Kung and Hua-Kung to retain the mortise-tenon Kua-Tzu-Kung and Hua-Kung are used as an exam- joint based on the consideration of the force transmis- ple to study the optimisation treatment of mortise- sion between Kua-Tzu-Kung and Hua-Kung and the mortise joint members. Due to the layer-by-layer force ensurance of volume integrity of critical sections of transmission nature of Tou-Kung, Hua-Kung receives tenon-tenon joint: (1) setting the action and reaction the downward pressure from the above Shan-Tou force respectively on the force contact surface by setting and transmits the force to Kua-Tzu-Kung. As Kua- loads at the connection part; (2) setting the rectangular Tzu-Kung and Hua-Kung are oriented perpendicular mortise-tenon joint members at the connection point to each other, the height of the single-layer member between the two ends of Hua-Kung and Kua-Tzu-Kung would have remained after the two arms were locked. with Shan-Tou as non-design domains, making them no After being connected by a mortise-mortise joint, the change in shape and materials distribution during the Duan et al. Architectural Intelligence (2023) 2:10 Page 14 of 16 Fig. 17 Methodology and solutions for the optimisation of mortise-tenon construction process of topology optimization, (Xie, 2022) (Fig.  17B); structural optimisation calculation is carried out with (3) setting sub-design domains: dividing the original the same multiplicity of the constrained volume frac- design domain so that the parameters of each divided tion (VF) and the filter radius (Rmin) value calculated design domain can be set individually. Then, reducing the on 2 times of the elemental size, the optimised mini- material cut factor in the structural optimization of the mum rod size of Shan-Tou will be too small, which is not dangerous section ensures strength rationality at the dan- in line with the rationality of the structure and would gerous section. affect the harmony and unity of the overall appearance In the optimisation of mortise-tenon joint members, (Xie, 2022). Therefore, after multiples times of attempts, Shan-Tou played as the small joint members to main- we increased the constrained volume fraction of Shan- tain the layer-by-layer force transmission of Tou-Kung. Tou from 0.3 to 0.45, raised the filtering radius from 2 Due to the differences in the angle and function of the times to 3-4 times of the original (Fig. 17D), and set the receiving Kung, the forms of the Shan-Tou wall are also non-design domain and pre-treatment parameters cor- different, which can be divided into two types: two side - responding to the mortise-tenon joint. The final mem - bar constraints and four corner point constraints. In ber bar dimensions of Shan-Tou are similar to those the structural optimisation, loads were set on the sur- of other members, the appearance of Tou-Kung tends face joints of the upper walls of both types of Shan-Tou, to be harmonious in general, and the upper and lower and supports were set in the lower mortise-tenon joint mortise-tenon joint functions are also retained. Besides, recesses to ensure the materials at the structural joints to address the small size and the thin wall of Shan-Tou, of the upper and lower recesses of the Shan-Tou were we used a combined optimisation method to design an not optimised and removed (Fig.  17C). Meanwhile, the independent Shan-Tou in addition to the above-men- volume of Shan-Tou is significantly smaller than that of tioned method of changing the pre-treatment param- other members such as Lu-Tou and Hua-Kung. If the eters. This combined optimisation method of Shan-Tou D uan et al. Architectural Intelligence (2023) 2:10 Page 15 of 16 and Kung can be describe as “one Tou with three Shan- only an appreciative value but also a practical signifi - Tou” or “three Tou with three Shan-Tou”, which can bal- cance. In the case of damaged traditional architectural ance the large volume difference between independent components, we do not necessarily have to follow the members and obtain a balanced and combined member rule of “repairing the old as the new”. In the topology with uniform bar size (Fig. 17E). optimisation of Tou-Kung, the traditional connection In the member structure optimisation strategy of Tou- method - mortise and tenon joint is reserved. Therefore, Kung based on mortise-tenon joint constraints, a free damaged components of Tou-Kung can be replaced and strong connection between structural optimised with new topological optimised ones, achieving a par- members was realized through the traditional mortise- tial restoration and creative nostalgia. Meanwhile, we tenon technique, while retaining zoning optimisation of also applied the complete workflow to the architrave- Tou-Kung and layer-by-layer force transmission. During eave column combination. The results also turned out the exploration of the mortise-tenon joint stage, mul- to be enlightening. Through these attempts, we aim to tiple independent Tou-Kung structure members were encourage more scholars to explore the possibilities of obtained through the following methods: (1) setting inheritance and derivation of traditional architectural non-design domain; (2) setting a subdomain for division components both technically and culturally. of original design domain; (3) applying proper loads and Acknowledgements supports at the Tou-Kung domain; (4) changing the input The authors would like to express their deepest gratitude to the DigitalFutures parameters VF and Rmin in the topology algorithm rea- organization and Tongji University for providing the platform; Nanjing Ameba Engineering Structure Optimisation Research Institute for providing the sonably according to the member volume and bar size educational version of Ameba software (For Ameba, see Ameba.xieym.com); constraints. These independent members can be con - Professor Yi Min ‘Mike’ Xie (RMIT University Centre for Innovative Structures & nected through traditional mortise-tenon techniques to Materials), for mentoring authors, to support this project. form a contemporary Tou-Kung that is highly material Authors’ contributions saving and efficient in force transmission with the unity The conceptualization, methodology, and writing of this manuscript were of force and form. under- taken by Cheng Bi Duan, Su Yi Shen, Ding Wen Bao, and Xin Yan. The authors read and approved the final manuscript. Funding 6 Conclusion and future work No funding was received to assist with the preparation of this manuscript. Bi-directional evolutionary structural optimisation (BESO) is performed on Tou-Kung in this study. 2D and Availability of data and materials Data sharing is not applicable to this article as no datasets were generated or 3D comparison and partitioned optimisation are applied analysed during the current study. to the design. The new Tou-Kung is designed based on high and reasonable structural performance, material Declarations efficiency, and traditional tectonic rule, with the features of aesthetic form and force-form united. In terms of the Competing interests The author has no competing interests to declare that are relevant to the material problem, the traditional Tou-Kung are made up content of this article. of wood, which is an anisotropic material. Although, the anisotropy of materials has been studied for considera- Received: 6 February 2023 Accepted: 16 April 2023 tion in topology optimization at the present time. 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Innovative design solutions for contemporary Tou-Kung based on topological optimisation

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Abstract

Tou-Kung, which is pronounced in Chinese and known as Bracket Set (Liang & Fairbank, A pictorial history of Chinese architecture, 1984), is a vital support component in the Chinese traditional wooden tectonic systems. It is located between the column and the beam and connects the eave and pillar, making the heavy roof extend out of the eaves longer. The development of Tou-Kung is entirely a microcosm of the development of ancient Chinese architecture; the aesthetic structure and Asian artistic temperament behind Tou-Kung make it gradually become the cultural and spiritual symbol of traditional Chinese architecture. In the contemporary era, inheriting and developing Tou-Kung has become an essential issue. Several architects have attempted to employ new materials and techniques to integrate the traditional Tou-Kung into modern architectural systems, such as the China Pavilion at the 2010 World Expo and Yusuhara Wooden Bridge Museum. This paper introduces the topological optimisation method bi-directional evolu- tionary structural optimisation (BESO) for form-finding. BESO method is one of the most popular topology optimisa- tion methods widely employed in civil engineering and architecture. Through analyzing the development trend of Tou-Kung and mechanical structure, the authors integrate 2D and 3D optimisation methods and apply the hybrid methods to form-finding. Meanwhile, mortise and tenon joint used to create stable connections with components of Tou-Kung are retained. This research aims to design a new Tou-Kung corresponding to “structural performance-based aesthetics”. The workflow proposed in this paper is valuable for Architrave and other traditional building components. Keywords Bi-directional Evolutionary Structural Optimisation (BESO), TOU-KUNG (Bracket Set), Mortise and tenon joint, Structural form-finding, Heritage building, Architectural component dynasties. The development of the timber frame has 1 Introduction also become the central vein of China’s architectural Chinese ancient architecture has a long history and development. Tou-Kung plays an essential role in the a self-contained system. Regarding building materi- Chinese wood tectonic system. It is an indispensable als, China’s vast territory, various timber, and large supporting element in Chinese timber-frame buildings bases have led to the gradual formation of the tectonic that supports  the  weight  of  the  overhanging  eaves. system based on a wooden framework through the Meanwhile, it symbolises the feudal hierarchy and an indispensable cultural character in traditional Chinese *Correspondence: architecture. It is a complex member located at the Ding Wen Bao nic.bao@rmit.edu.au junction of the column and the beam, made up of arms Xin Yan (Kung) and blocks (Tou). At the base of Tou-Kung, it yanxin2022@tsinghua.edu.cn is a large block called Lu-Tou. Crossed arms spread- China University of Mining and Technology, Jiangsu 221116, China RMIT University, Melbourne 3000, Australia ing in four directions are set into Lu-Tou. These in Tsinghua University, Beijing 10084, China © The Author(s) 2023. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Duan et al. Architectural Intelligence (2023) 2:10 Page 2 of 16 turn bear smaller blocks (Shan-Tou) that carry longer 2.1 The dendritic column with structural bionics design arms. Hua-Kung extends outward, while the others The application of dendritic columns in the spatial struc - parallel to the wall are called Transverse Kung as a ture solves the problem that a single vertical column group, including Ni-Tao-Kung, Man-Kung etc. Long with an equal cross-section is challenging to set off the cantilever arms called Ang extend out to support the far-reaching eave, in accordance with the function of the outermost purlins (Fig. 1) (Liang & Fairbank, 1984). In tiered projections of Tou-Kung. The dendritic column is an era where digital design is booming, the research upright and looks like a tree trunk (Qian et  al. 2020). It present here tries to combine the conventional Tou- has various branches at the top of the column, geomet- Kung with advanced structural form-finding methods rically dispersed to form multiple points of support and to create a new derivative of Tou-Kung that incorpo- radiating away from the centre of the column (Fig.  2A). rates ancient and modern elements. The inheritance Although the dendritic column satisfies the integration of and innovation of Tou-Kung have become a significant “force” and “form”, achieved straightforward transferred concern. This paper focuses on using the evolution force and material efficiency, and it differs from Tou- of traditional building components to document the Kung in appearance. development of architectural techniques in a new era. 2.2 Archaise Tou‑Kung using contemporary tectonic methods Archaise Tou-Kung is constructed from high-strength 2 Translations and derivation practices materials such as cement and aluminum alloy, imitat- of Tou‑Kung ing or copying Tou-Kung from various historical periods From a contemporary standpoint, several architects (Fig.  2B). The same are the beams, columns, Tou-Kung have attempted to translate the traditional Tou-Kung, and pillars in archaise buildings. Still, they are not opti- employing new materials and techniques to integrate mised for the mechanical properties of the individ- it into modern architectural systems. These can be ual materials. As a result, the overall stiffness and the broadly divided into three categories according to structural frequency of archaize buildings are excessive structural role, form, and cultural symbolism. Fig. 1 Principal parts of a Chinese timber-frame building D uan et al. Architectural Intelligence (2023) 2:10 Page 3 of 16 Fig. 2 A Dendritic column: B Archaize Tou-Kung; C The cultural imagery of Tou-Kung (Wang, 2011), leading to enormous material waste and The development of Tou-Kung can be briefly divided the loss of characteristics of the traditional Tou-Kung as into five stages. Firstly, from the Western Chou Dynasty simple, efficient, and exquisite. to the Han Dynasty, the image of Tou-Kung was depicted in various historical relics (Fig.  3A). After the Han 2.3 C ultural symbols of Tou‑Kung Dynasty, it appeared between the columns. Secondly, in This type of building has a superficial structure that the period of Wei, Jin, Southern and Northern Dynasties, exhibits layer-by-layer stacking, achieving a cultural the form of Tou-Kung gradually became standardised, homage to Tou-Kung (Fig.  2C). In the case of the China and the classic forms of inverted-V brackets appeared Pavilion at the 2010 World Expo, the traditional Tou- (Fig. 3B). Thirdly, from the Sui and Tang Dynasties to the Kung is simplified through dislocation, orthogonality, Five Dynasties, Tou-Kung was vigorously gorgeous and and classification in three-dimensional construction. magnificent, characterised by the huge body (Fig.  3C). While the mechanical characteristics of force transferred Seen from the elevation, the height of Tou-Kung can from one layer to another are retained (Xu & Liu, 2018). reach the half-height of the column. Transverse Kung However, this type of abstract translation detracts from and column-top joists overlap to form a lattice-shaped the essential structural role of Tou-Kung and the cor- framework. As a strengthened node, Tou-Kung also responding scale of its components. It serves only as an becomes an indispensable part of the structure. Fourthly, abstract derivative and a metaphor for Tou-Kung culture. during Song and Yuan Dynasties, the image of Tou- Kung became tenderly beautiful. Since Song Dynasty, 3 Introduction and analysis of Tou‑Kung the number of Tou-Kung has increased, and the volume 3.1 The evolution of Tou‑Kung has decreased. A certain decorative effect was intro - The evolution of Tou-Kung is a nearly comprehensive duced. The size of components started to be unified, and presentation of the development of architectural skills the modular system appeared (Fig.  3D). The last was the and the aesthetic trends of building components in each Ming and Qing Dynasties. By the Qing Dynasty, Tou- period of Chinese history. Tou-Kung is not only a single Kung adopted the modular of the mortise of Lu-Tou, structural component, but also has multiple-layered sig- which was used to standardise the dimensions of each nificance in cultural inheritance, decoration and beauti - component of it (Fig. 3E). Decorative and coloured paint- fication, class identity, measurement, etc. The variations ings appeared on the surface of Tou-Kung, which further of Tou-Kung have also provided important evidence for strengthened the decoration and weakened the structure identifying the age of ancient architecture and observing roles. At this stage, Tou-Kung ceased playing the role of the style of different times. keeping the overall framework stable. Duan et al. Architectural Intelligence (2023) 2:10 Page 4 of 16 Fig. 3 The evolution of Tou-Kung Throughout its history, Tou-Kung has evolved from and stability. So, Tou-Kung rises in tiers or “jumps” and large to small, from simple to complex, with its struc- extends outward in steps, and through the mortise-tenon tural role diminishing and its decorative role strengthen- joint, several beams become whole, which increases the ing. As technology and materials developed, Tou-Kung bending and shearing resistance of Tou-Kung. As sup- has switched its role from load-bearing to decorative porting the eaves, the beam sometimes appears extremely and from function to form in the process of meeting long, and the distance between the bearing beam end the demands of the times. The ideal state of the devel - and the bearing support is excessively large, leaving a opment and derivation of the contemporary Tou-Kung large force arm and a weak beam bearing capacity, which is to reflect the technological development of the new makes the beam easy to break. Therefore, from the end era while preserving the historical heritage. The focus to the starting point of the overhanging part, Tou-Kung of this paper is not to simply restore Tou-Kung of a cer- adopts the way of layer-by-layer superposition and lay- tain period or use modern materials to make a tradi- ered force transmission to ensure overall stability. tional form of Tou-Kung, but to combine new technology Tou-Kung is a combined structural member, which and new materials with traditional architectural tech- needs to be disassembled into several components when niques, exploring the profound traditional architectural studying its force transmission characteristics to present spirit with a high degree of unity of structure and form its layer-by-layer force transmission properties clearly (Yuan et  al., 2017), and seeking resemblance rather than (Fig.  5). The key mechanical components of Tou-Kung mimesis. can be dissembled into Tou (blocks), Kung (arms) and Ang (long cantilever arms) (Liang & Fairbank, 1984). 3.2 The structural function and mechanical prototype Specifically, Tou is a square wooden block used to sup - of Tou‑Kung port Kung and Ang. The Tou mainly works as cushions Tou-Kung, placed between beams and columns, plays to transfer the pressure from the upper structure, trans- the key role of a connecting link. And the load of the roof porting the small buckets from top to bottom to the and the upper structure is transmitted to the spreading Kung connected below (Wei, 2007). Among these Kung, layer of Tou-Kung, and then to the column and founda- the Lu-Tou at the bottom of Tou-Kung, which is called tion. As a structural “transit hub”, Kung is paved up layer Lu-Tou, is the most important compression member of by layer, and the truss and purlin on the outermost layer the Tou-Kung system. Kung is a bow-like part. Hua-Kung are propped further for a longer distance, which supports is vertical to the building facade, responsible for over- the far-reaching eaves of the structure and makes it more hanging, while Transverse Kung is parallel to the build- imposing and magnificent (Fig. 4A). ing facade, which plays a balancing role. Hua-Kung and The mechanical prototype of Tou-Kung is essentially Transverse Kung bear the bending moment and longi- a short cantilever beam (Fig.  4B), whose main structural tudinal load, respectively. The pressure from the upper function is to support the load of eaves (Yu et al., 2006). structure is concentrated at the middle and the two ends For the cantilever beam, the increased height of the of Hua-Kung, resulting in a large bending and shear stress beam from outside to inside can ensure overall strength at the head of Hua-Kung (Lv, 2010). However, the use of D uan et al. Architectural Intelligence (2023) 2:10 Page 5 of 16 Fig. 4 The structural function and mechanical prototype of the Tou-Kung Fig. 5 Layer-by-layer force transmission system of Tou-Kung Duan et al. Architectural Intelligence (2023) 2:10 Page 6 of 16 Transverse Kung can support the upper structure out- is divided into several components according to the force side the Hua-Kung plane, strengthening the connection characteristics of the individual elements. Therefore, the between Tou-Kung and the upper structure, improving structural prototype of layer-by-layer force transmission the overall bending and torsional strength of Tou-Kung, is reserved. Through the digital form-finding and finite and making Tou-Kung stable spatial support in all direc- element analysis, a new Tou-Kung is designed on the tions. Ang is the longest member in Tou-Kung, which premise of material efficiency and ensuring definite force sags diagonally and is structured as a cantilever beam in transmission routes and a reasonable structure (Fig. 6). the oblique direction. Ang is a compression member due to its large cross-section area, which can bear the lift-4.1 Design methodology ing load of the eaves and transmit the pressure from the The Ameba plug-in is implemented to carry out topology upper parts. In addition to compression, Ang also car- optimisation and BESO algorithm based on the Rhino ries a certain level of bending moment. The components and Grasshopper platforms (Zhou et  al., 2018; Zhou, of Tou-Kung are interlaced and stacked, full of rhythm 2021). In the application of topology optimisation, the changes. rational distribution of Tou-Kung materials is realised by deciding the removal, reservation, or supplement of 4 Topology optimisation of Tou‑Kung materials through finite element analysis (Fig. 7). In this experiment, the bi-directional evolutionary struc- The design transforms Tou-Kung in the Ming and tural optimisation (BESO) method has been applied to Qing Dynasties. In that period, the Ang was fake, and the design. The concept of BESO is to find a preliminary didn’t have the structural character of lower Ang in the structural configuration which meets a predefined cri - Song Dynasty. The force transmission of Tou-Kung in terion based on finite element analysis. As an advanced the Ming and Qing Dynasties is neater and better aligned form-finding method in digital design (Lin, 2020; Zhao with the property of layer-by-layer force transmission and Chen, 2016), topology optimisation can optimise the from top to bottom. In topology optimisation, it is also layout of materials and explore new forms of a contempo- the most effective structural morphology to transmit rary bucket set on the premise of a reasonable structure the upper load with specific boundary support, material (Xie et  al., 2014). BESO method has been implemented consumption, and material type. To keep to the layer-by- into architectural design in many pieces of research due layer force bearing and transmission mode of each part of to its capacity to generate diverse organic forms with Tou-Kung, this study adopts the concept of “zoning opti- high structural performance (Bao et  al., 2022; Yan et  al., misation” in the Multi-volume BESO method (Yan et al., 2022). During the design process, the scale and posi- 2022) as the core logic of Tou-Kung optimisation. Spe- tion of the traditional Tou-Kung are inherited. Also, the cifically, Tou-Kung is divided into Tou, Kung, Ang, and hybrid method for optimization is introduced. Tou-Kung other parts, and then they are respectively optimised by Fig. 6 Topologically optimised Tou-Kung D uan et al. Architectural Intelligence (2023) 2:10 Page 7 of 16 Fig. 7 Workflow of Tou-Kung optimisation & exploded diagram of optimised Tou-Kung 4.2 Topology optimisation form‑finding of 2D Tou‑Kung BESO/Ameba; later, all optimised parts are reconstructed The 2D topology optimisation and planar force analy - and assembled. sis are conducive to the subsequent 3D structure form- This calculation framework (Fig.  7) is derived from the finding of Tou-Kung. In addition, the 2D optimised optimization process of Tou-Kung and is based on the results can be compared with the 3D optimised ones combination of the topology optimisation method and to decide the rationality of the results and provide a the zoning optimisation method. Zoning optimisation reference for adding supplements, resetting boundary can be applied to the optimization of the different scales conditions, and optimising parameters. In the zon- of components. More specifically, the components can be ing optimisation of 2D Tou-Kung, the primary parts, divided according to the number of storeys of the parti- including Lu-Tou, Shan-Tou, Ni-Tao-Kung, Ling-Kung, tions, the volume of the partitions and the different posi - Hua-Kung, and Kua-Tzu-Kung, are extracted. Kua- tions of the partitions in the building, in order to satisfy Tzu-Kung on the central axis of the column is selected the optimisation calculation of components in different as representative of component composition simula- scales. In the case of this chapter, the zoning optimisation tion analysis. Firstly, the planar angle of each compo- of Tou-Kung, a small-scale, multistorey building element, nent is selected, and the ratio scale of each element is will be calculated in Sections 4.2 and 4.3. Then the zoning determined according to the Tou-Kou module system of the architrave-column combination, a large-scale, low- in the Qing Dynasty. Taking “Tou-Kou” as the modu- storey building element, will be calculated in Section 4.4, lus unit, an accurate 2D model of each component is which can further illustrate the compatibility and ration- constructed. Then according to the descriptions of ality of the framework for optimising building elements the force bearing of each part of the structure and the at different scales. Duan et al. Architectural Intelligence (2023) 2:10 Page 8 of 16 Fig. 8 2D topological optimised components of Tou-Kung connection mode between Tou and Kung in Ying Zao 1. Low strength of the wall of Tou-Kung Fa Shi (Li & Liang, 1983) and Engineering Practice Rules and Examples from Qing Dynasty (Liang, 2003), The wall of Tou-Kung is the smallest individual com - the force diagram of components is drawn for 2D force ponent, and it has a large volume difference com - analysis. Secondly, the load and support are set as pared to other components. Considering the differ - boundary conditions after generating the mesh element ent shapes and low strength of the wall of Tou-Kung in Rhino/Ameba, and parameters for preprocessing are after topology optimisation, Shan-Tou is combined set to run the BESO algorithm for seeking optimal 2D with Kung in the 3D prototype, and a typical combi- topology structural forms (Fig. 8). nation of Tou and Kung, which can be described as The 2D topological optimized results and the force one block on the bottom, one arm in the middle and analysis shows a significant reduction in materials at three small blocks on the top, is formed (Fig. 9). both ends and wall of Tou-Kung. Ancient carpenters 2. The dangerous critical cross-section used the entasis method to make the ends of Tou-Kung During the 2D optimisation, the rabbets of Lu-Tou, into gentle curves or fold with a full and soft appearance. the first layer Kua-Tzu-Kung and the first layer Hua- The material layout of Tou-Kung after topology optimi - Kung complement each other with overlapped dan- sation is similar to the form of the traditional Tou-Kung gerous sections with hidden safety hazards. There - after entasis treatment. From force analysis, the corner in fore, we combined the three components with the the traditional Tou-Kung is a non-force bearing part or a central part as the integrated component to supple- part with less bearing load. Therefore, the entasis part’s ment the strength of the critical section further. material deletions outnumber that of other parts after 3. Recurrence of entasis topological optimisation. The ancients artificially omitted To further verify the significance of entasis in terms of the corner material based on their construction experi- rational force transfer and material saving, the geomet- ence, thereby developing the practice of entasis. This is ric prototype of Tou-Kung is simplified during the 3D an experimental result of artificially saving materials and optimisation, and the entasis chamfer was cancelled efficiently expressing structural force transmission. to form a simplified, integrated cube. The purpose is to determine whether material deletion and chamfer would reappear at the original entasis position in the 4.3 Topology optimisation form‑finding of 3D Tou‑Kung optimised structure. At last, four combinations were After previous 2D component decomposition optimisa- formed: one Tou with three Shan-Tou, three Tou with tion, we found three issues that should be considered in three Shan-Tou, the combination of the Ang and Hua- the 3D optimisation: Kung, and the combination of Lu-Tou and Hua-Kung. Then 3D optimisation is performed and comparatively D uan et al. Architectural Intelligence (2023) 2:10 Page 9 of 16 Fig. 9 3D topological optimised components of Tou-Kung analysed with the 2D optimisation results. Parameters structural prototype of the tree-like column and the frac- are adjusted to achieve the optimal solution for topo- tal structure of tree branches in nature, proving the high logical optimised Tou-Kung. correlation and similarity between Tou-Kung and tree According to the results of the 3D optimisation, sev- branches in the mechanical prototype. The upper and eral combinations of Tou-Kung have presented structural lower surfaces have complete materials for the structure shapes of oblique fractal bifurcation from the support to of Shan-Tou, while the middle part is a porous struc- the load ends. This mechanical structure is similar to the ture. Massive materials were cut for the original entasis Fig. 10 Topology optimisation results of the entasis part of Tou-Kung Duan et al. Architectural Intelligence (2023) 2:10 Page 10 of 16 position or even an “entasis style” was formed. The cham - optimisation was applied to form-finding of topologi - fering amplitude of these designs was more obvious than cally optimised Tou-Kung. This approach can be applied in traditional Tou-Kung (Fig.  10). The topology optimisa - not only to the form-finding of the scale of Tou-Kung, tion results also reflected the wisdom of ancient crafts - but also to the scale of the building. In the following, we men and the excellent craftsmanship of Tou-Kung, which will verify the feasibility of our workflow through the is absolutely an architectural gem that integrated form optimisation of the architrave and column. During the and force (Fig.11). architrave topology optimisation of the Hall of Prayer for Good Harvests, the combined member of a ring of archi- 4.4 A case study and t opological optimisation experiment trave and eave column can be disassembled into six same of the Hall of Prayer for Good Harvests units according to the symmetry. Then the 1/12 mini - Architrave is also an important part of the Chinese tim- mum optimisation unit can be obtained based on the ber frame. Usually, an architrave is installed on the top symmetric properties of the units (Fig. 12). of a column to link Tou-Kung between columns with the For the load arrangement, the downward uniform load load-bearing horizontal member. In some architectures, can be set on the upper critical plane connecting Archi- large and small architraves were stacked and juxtaposed trave and Tou-Kung, and a side thrust can be im- posed with the middle connected by a clamp pad (Cao & Wang, on the sparrow brace, which ensures that all the units 2010). These architraves were simplified as a complete are closely connected in the partition optimisation. The geometric graph and were used as the optimised pro- base was installed at the position of the symmetrically totype together with the eave column. In the former optimised vertical cross-section and column bottom. The research, in spite of the layer-by-layer force transmis- overall effect after optimisation was shown in the image sion and mortise-tenon joint of Tou-Kung, the zoning (Fig. 13). Fig. 11 3D topological optimised Tou-Kung models Fig. 12 3D topological optimised architrave-column combination D uan et al. Architectural Intelligence (2023) 2:10 Page 11 of 16 Fig. 13 3D topological optimised components in ancient buildings After the optimization of Architrave, the results with a structural layer that transmits the layer-by-layer force the previous optimization results of Tou-Kung were between the roof and columns. As a structural connec- combined. Then, the corresponding original structure of tion method, the mortise-tenon joint connects indi- the Hall of Prayer for Good Harvests was replaced while vidual members without using a single nail or bolt retaining the traditional paintings, doors, windows, tiles (Zhang, 2022), which only uses the geometric connection and other decorative components. The overlap of new between the members with concave-convex compensa- and old elements formed a stark contrast in appear- tion for the shape to achieve an efficient and secure con - ance (Fig.  14) while maintaining the same mechanical nection (Wang, 2019). In the mortise-tenon joint, “tenon” essence. This kind of hidden link ensures a more rigor - is the convex part, and “mortise” is the concave part ([Liu ous connection between culture and structure based on and Song, 2019), with a variety of derived forms, which the unity of aesthetic form and force in contemporary realise the horizontal and vertical connection between all structure. the members of Tou-Kung (Fig. 15). The use of the mortise-tenon technique in Tou-Kung 5 Structural optimisation strategy can be summarized into two types: the mortise-tenon for the members of Tou‑Kung based joint and the mortise-mortise joint. In the first type, the on the mortise‑tenon joint concave and convex parts interlock with each other and 5.1 O verview and classification of the mortise‑tenon joint form a strong, stable and flexible framework (Fig.  15B). in Tou‑Kung Mortise-tenon joint is commonly seen in the connec- In the structure of the traditional Tou-Kung, multiple tion of the top of the column with the bottom of Lu-Tou, members stepped layer outwards layer by layer, forming as well as Shan-Tou with cross mortise with their lower Fig. 14 Topological optimised Hall of Prayer for Good Harvests Duan et al. Architectural Intelligence (2023) 2:10 Page 12 of 16 Fig. 15 The use and classification of mortise-tenon joint in Tou-Kung members. This type of mortise-tenon technique allows the mortise-mortise joint is suitable for the connection of the connection structure to be hidden in the members’ members at the same horizontal layer (Liang et al., 2021). own volume without protruding from the body block, The joint use of the two methods forms the basis for the thus preserving the integrity of the appearance of Tou- layer-by-layer force transmission in Tou-Kung, which is Kung. Meanwhile, the height of the two layers of mem- also an indispensable technical path to achieve the façade bers is also retained with this combination. modelling of “corbelling”. The second type is mortise-mortise joint (Fig.  15A). “Mortise” is the concave part. Two “concave” members 5.2 Problems and solutions to the structural optimisation are lapped in the mutual perpendicular direction in Tou- in mortise‑tenon joint Kung between columns and are buckled in other angular Topology optimisation often presents structural reten- directions in the corner Tou-Kung. This type of connec - tion of unit volume, and mortise-tenon joint usually tion is widely used between Transverse Kung and Hua- raises a high-volume correspondence requirement to Kung, or Transverse Kung and Ang. As two “concave” ensure sufficient interlocking between the concave and members are lapped together in a position that cuts 1/2 convex parts. Therefore, it became the focus of this of the height of the single-layer member, the complemen- study to make topology optimisation of the members tary joint becomes a finished type and still retains the in Tou-Kung to eliminate low-efficiency units to ensure height of the single-layer member. the lightweight structure of Tou-Kung while retaining In summary, the mortise-tenon joint is suitable for the the unit at the mortise-tenon joint part and ensuring connection between different layers of members, while the complementary concave and convex volumes at the D uan et al. Architectural Intelligence (2023) 2:10 Page 13 of 16 Fig. 16 The process of optimising the mortise-tenon joint in Tou-Kung mortise-tenon joint with high accuracy. This issue can two worked as a whole and are connected to the cross- be further summarized as the “structural optimisation shaped recess of the Lu-Tou below, and the pressure is strategy for the members of Tou-Kung based on mortise- transmitted from the above to the Lu-Tou and then to tenon joint constraints” (Fig. 16). the columns below (Fig.  17A). As the symmetry axis In the selection of the original domain of Tou-Kung, parts of Kua-Tzu-Kung and Hua-Kung have removed we chose the representative Tou-Kung with two “Jumps” 1/2 of the height of the single-layer member respec- on the architrave between two columns and selected the tively, the middle part becomes relatively weak. If the five-member combination of “Lu-Tou - Kua-Tzu-Kung weak domain of Kua-Tzu-Kung and Hua-Kung is not on the central axis of the column - Shan-Tou on the top protected, the materials in the weak domain would be of Kua-Tzu-Kung - Hua-Kung - Shan-Tou” as the final further removed after structural optimisation. On the matrix for structural optimisation. This combination one hand, it will cause a hidden danger in structural encompasses the two above-mentioned mortise-tenon strength; on the other hand, the unrestrained volume techniques (the mortise-tenon joint and the mortise- cut would destroy the tightness of the connection of the mortise joint) in the mortise-tenon joint of a traditional mortise-mortise joint. Tou-Kung. For the selection of Shan-Tou, two represent- To deal with the above problems, the following meas- ative types, namely the two sidebar constraints and the ures are taken during the structural optimisation of Kua- four corner point constraints, are also presented. Tzu-Kung and Hua-Kung to retain the mortise-tenon Kua-Tzu-Kung and Hua-Kung are used as an exam- joint based on the consideration of the force transmis- ple to study the optimisation treatment of mortise- sion between Kua-Tzu-Kung and Hua-Kung and the mortise joint members. Due to the layer-by-layer force ensurance of volume integrity of critical sections of transmission nature of Tou-Kung, Hua-Kung receives tenon-tenon joint: (1) setting the action and reaction the downward pressure from the above Shan-Tou force respectively on the force contact surface by setting and transmits the force to Kua-Tzu-Kung. As Kua- loads at the connection part; (2) setting the rectangular Tzu-Kung and Hua-Kung are oriented perpendicular mortise-tenon joint members at the connection point to each other, the height of the single-layer member between the two ends of Hua-Kung and Kua-Tzu-Kung would have remained after the two arms were locked. with Shan-Tou as non-design domains, making them no After being connected by a mortise-mortise joint, the change in shape and materials distribution during the Duan et al. Architectural Intelligence (2023) 2:10 Page 14 of 16 Fig. 17 Methodology and solutions for the optimisation of mortise-tenon construction process of topology optimization, (Xie, 2022) (Fig.  17B); structural optimisation calculation is carried out with (3) setting sub-design domains: dividing the original the same multiplicity of the constrained volume frac- design domain so that the parameters of each divided tion (VF) and the filter radius (Rmin) value calculated design domain can be set individually. Then, reducing the on 2 times of the elemental size, the optimised mini- material cut factor in the structural optimization of the mum rod size of Shan-Tou will be too small, which is not dangerous section ensures strength rationality at the dan- in line with the rationality of the structure and would gerous section. affect the harmony and unity of the overall appearance In the optimisation of mortise-tenon joint members, (Xie, 2022). Therefore, after multiples times of attempts, Shan-Tou played as the small joint members to main- we increased the constrained volume fraction of Shan- tain the layer-by-layer force transmission of Tou-Kung. Tou from 0.3 to 0.45, raised the filtering radius from 2 Due to the differences in the angle and function of the times to 3-4 times of the original (Fig. 17D), and set the receiving Kung, the forms of the Shan-Tou wall are also non-design domain and pre-treatment parameters cor- different, which can be divided into two types: two side - responding to the mortise-tenon joint. The final mem - bar constraints and four corner point constraints. In ber bar dimensions of Shan-Tou are similar to those the structural optimisation, loads were set on the sur- of other members, the appearance of Tou-Kung tends face joints of the upper walls of both types of Shan-Tou, to be harmonious in general, and the upper and lower and supports were set in the lower mortise-tenon joint mortise-tenon joint functions are also retained. Besides, recesses to ensure the materials at the structural joints to address the small size and the thin wall of Shan-Tou, of the upper and lower recesses of the Shan-Tou were we used a combined optimisation method to design an not optimised and removed (Fig.  17C). Meanwhile, the independent Shan-Tou in addition to the above-men- volume of Shan-Tou is significantly smaller than that of tioned method of changing the pre-treatment param- other members such as Lu-Tou and Hua-Kung. If the eters. This combined optimisation method of Shan-Tou D uan et al. Architectural Intelligence (2023) 2:10 Page 15 of 16 and Kung can be describe as “one Tou with three Shan- only an appreciative value but also a practical signifi - Tou” or “three Tou with three Shan-Tou”, which can bal- cance. In the case of damaged traditional architectural ance the large volume difference between independent components, we do not necessarily have to follow the members and obtain a balanced and combined member rule of “repairing the old as the new”. In the topology with uniform bar size (Fig. 17E). optimisation of Tou-Kung, the traditional connection In the member structure optimisation strategy of Tou- method - mortise and tenon joint is reserved. Therefore, Kung based on mortise-tenon joint constraints, a free damaged components of Tou-Kung can be replaced and strong connection between structural optimised with new topological optimised ones, achieving a par- members was realized through the traditional mortise- tial restoration and creative nostalgia. Meanwhile, we tenon technique, while retaining zoning optimisation of also applied the complete workflow to the architrave- Tou-Kung and layer-by-layer force transmission. During eave column combination. The results also turned out the exploration of the mortise-tenon joint stage, mul- to be enlightening. Through these attempts, we aim to tiple independent Tou-Kung structure members were encourage more scholars to explore the possibilities of obtained through the following methods: (1) setting inheritance and derivation of traditional architectural non-design domain; (2) setting a subdomain for division components both technically and culturally. of original design domain; (3) applying proper loads and Acknowledgements supports at the Tou-Kung domain; (4) changing the input The authors would like to express their deepest gratitude to the DigitalFutures parameters VF and Rmin in the topology algorithm rea- organization and Tongji University for providing the platform; Nanjing Ameba Engineering Structure Optimisation Research Institute for providing the sonably according to the member volume and bar size educational version of Ameba software (For Ameba, see Ameba.xieym.com); constraints. These independent members can be con - Professor Yi Min ‘Mike’ Xie (RMIT University Centre for Innovative Structures & nected through traditional mortise-tenon techniques to Materials), for mentoring authors, to support this project. form a contemporary Tou-Kung that is highly material Authors’ contributions saving and efficient in force transmission with the unity The conceptualization, methodology, and writing of this manuscript were of force and form. under- taken by Cheng Bi Duan, Su Yi Shen, Ding Wen Bao, and Xin Yan. The authors read and approved the final manuscript. Funding 6 Conclusion and future work No funding was received to assist with the preparation of this manuscript. Bi-directional evolutionary structural optimisation (BESO) is performed on Tou-Kung in this study. 2D and Availability of data and materials Data sharing is not applicable to this article as no datasets were generated or 3D comparison and partitioned optimisation are applied analysed during the current study. to the design. The new Tou-Kung is designed based on high and reasonable structural performance, material Declarations efficiency, and traditional tectonic rule, with the features of aesthetic form and force-form united. In terms of the Competing interests The author has no competing interests to declare that are relevant to the material problem, the traditional Tou-Kung are made up content of this article. of wood, which is an anisotropic material. Although, the anisotropy of materials has been studied for considera- Received: 6 February 2023 Accepted: 16 April 2023 tion in topology optimization at the present time. 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Journal

Architectural Intelligence – Springer Journals

Published: Apr 25, 2023

Keywords: Bi-directional Evolutionary Structural Optimisation (BESO); TOU-KUNG (Bracket Set); Mortise and tenon joint; Structural form-finding; Heritage building; Architectural component

References

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