The 200 m timber tower A study of building geometry and construction feasibility Master’s thesis in Structural Engineering and Building Technology LIEN TRINH HONG ZHANG DEPARTMENT OF ARCHITECTURE AND CIVIL ENGINEERING CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2021 www.chalmers.se MASTER’S THESIS ACEX30 The 200 m timber tower A study of building geometry and construction feasibility Master’s Thesis in Structural Engineering and Building Technology LIEN TRINH HONG ZHANG Department of Architecture and Civil Engineering Division of Structural Engineering Research Group for Lightweight Structures CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 2021 I The 200 m timber tower A study of building geometry and construction feasibility Master’s Thesis in Structural Engineering LIEN TRINH HONG ZHANG © LIEN TRINH & HONG ZHANG, 2021 Examensarbete ACEX30 Institutionen för arkitektur och samhällsbyggnadsteknik Chalmers tekniska högskola, 2021 Department of Architecture and Civil Engineering Division of Structural Engineering Research Group for Lightweight Structures Chalmers University of Technology SE-412 96 Göteborg Sweden Telephone: + 46 (0)31-772 1000 Cover: The design of the 200 m timber tower Department of Architecture and Civil Engineering Göteborg, Sweden, 2021 I The 200 m timber tower A study of building geometry and construction feasibility Master’s thesis in Structural Engineering and Building Technology LIEN TRINH HONG ZHANG Department of Architecture and Civil Engineering Division of Structural Engineering Research Group for Lightweight Structures Chalmers University of Technology ABSTRACT There are a lot of building materials used in the building sector such as concrete, steel and timber, each with its own advantages and disadvantages. For a long time, timber has not been seen as a common material for advanced structures in civil engineering field and architecture field. With the continuous development of engineered wood products, timber has nowadays become a more common building material used for multi-floor buildings. The 20th century witnessed a construction boom in tall buildings worldwide. So far, the knowledge of how to build a high-rise building with a timber structure is still limited. The current world’s tallest timber building is Mjöstornet in Norway, with a height of 85m, which is only about one-tenth of the world’s highest building Burj Khalifa in Dubai. The main objective of this report is to study and evaluate different building geometries with main concerns about economy, environment aspect and structural performance of a 200m timber building. The building site is assumed to be 30m x 30m and is in Gothenburg with the ground condition of solid rock. The study divided into two main steps and was mostly performed in Grasshopper and Karamba 3D. In the first step two concepts were selected through an evaluation based on the results of deflection, rental area and building mass of various concepts. In the second evaluation several criterions were added such as dynamic performance, base support force and building mass per rental area ratio to find a promising concept. The final proposal is a diagrid structure in hyperboloid form and was chosen through comprehensive evaluation. This geometry has significantly better performance regarding dynamic performance compared to all other diagrid structures in convex form and traditional braced frame structure that has been studied, though providing much less rentable areas. Key words: Acceleration, Aerodynamics, Burj Khalifa, Dynamics, Grasshopper, Karamba3D, Lateral deflection, Mjöstornet, Tall timber buildings, Timber, Timber connections, Rental area, Stiffness, Wind load. II CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 III Contents ABSTRACT I CONTENTS III PREFACE VII 1 INTRODUCTION 1 1.1 Background 1 1.2 Aim 2 1.3 Scope and limitation 2 1.4 Method 2 2 THEORETICAL FRAMEWORK 4 2.1 Definition of a high-rise, timber building 4 2.1.1 Definition of highness of timber building 4 2.1.2 Definition of timber building 6 2.1.3 Summary 8 2.2 Considerations regarding structural design for high-rise buildings 8 2.2.1 Lateral resisting stiffness and strength 8 2.2.2 Dynamic behavior 9 2.2.3 Gravity resisting system 11 2.3 Geometry and modification 11 2.3.1 Plan shape and corner modification 12 2.3.2 Modification in elevation 12 2.3.3 Consideration regarding structural stiffness 13 2.4 Structural systems for tall buildings 13 2.4.1 Rigid frame 14 2.4.2 Shear wall system 14 2.4.3 Shear wall and frame interaction system 15 2.4.4 Outrigger system 15 2.4.5 Tube system 16 2.4.6 Spaces truss structure 18 2.4.7 Super frame structure 18 2.4.8 Exoskeleton 18 2.4.9 Classification and considerations regarding timber buildings 19 2.5 Timber as a building material 20 2.5.1 Property of timber 20 2.5.2 Wood product for tall buildings 23 2.5.3 Connection in timber buildings 24 2.6 Other considerations 25 2.6.1 Fire safety 25 2.6.2 Sound and vibration issues of the floor 26 2.6.3 Daylight 27 2.6.4 Vertical transportation 27 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 IV 2.7 Structural design 28 2.7.1 Design of the structure and the structural components in ULS 28 2.7.2 Design of the structure and the structural components in SLS 28 2.7.3 Dynamic design in SLS 30 3 GEOMETRY STUDY 32 3.1 Parametric volume geometry 33 3.2 Structural model 35 3.2.1 Simplified structural model 35 3.2.2 Properties of elements 36 3.2.3 Load 38 3.3 Finite element models for the general analysis 41 3.3.1 Structural elements 41 3.3.2 Boundary condition 42 3.3.3 Loads 43 3.3.4 Mesh 44 3.3.5 Verification of FE models 44 3.4 Evolutionary design 45 3.4.1 Optimizing algorithm 45 3.4.2 Description of the models used in geometry optimization 46 3.4.3 Optimized geometry 49 3.5 Typical geometries 51 3.6 Hyperboloid 52 3.7 Result 54 3.7.1 Rental area 54 3.7.2 Mass and mass per area 55 3.7.3 Lateral displacement with consideration to structural stiffness, under equal loads 56 3.7.4 Lateral displacement with consideration to both structural stiffness and aerodynamics, under loads varying between geometries 57 3.8 Evaluation 58 3.8.1 Evaluation criteria 58 3.8.2 Primary evaluation 59 3.8.3 Secondary selection 61 4 STABILIZING STRUCTURE STUDY 62 4.1 Geometries to investigate 62 4.2 Structural models 63 4.2.1 Structural models 63 4.2.2 Structural elements 64 4.2.3 Boundary conditions 66 4.2.4 Loads 66 4.3 Cross section optimization 66 4.4 The benchmark building 67 CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 V 4.5 Convex geometry 69 4.5.1 Parameters 70 4.5.2 Results 73 4.6 Hyperboloid 79 4.6.1 Parameters 80 4.6.2 ULS design and optimization 80 4.6.3 Result 81 4.7 Evaluation 86 4.7.1 Criteria 86 4.7.2 Summarized results for evaluation 87 4.7.3 Evaluation 89 5 PRELIMINARY STRUCTURAL DESIGN 91 5.1 Design of structural members 91 5.2 Joints 91 5.2.1 Deviations/risk due to joints 92 5.2.2 Design of joints 93 5.3 Overview of structural performance 95 6 DISCUSSION 96 6.1 Slenderness and comparability 96 6.2 Simplifications and design codes 97 6.3 The three geometries 98 6.4 The hyperboloid 99 6.5 Long term effect on structure 100 6.6 Joints 101 6.7 Potential error source 101 6.8 Improvement and future work 102 6.9 Software 103 7 CONCLUSION 104 8 FURTHER RESEARCH 105 9 REFERENCES 106 APPENDIX A ESTIMATION OF FORCE COEFFICIENT A1 APPENDIX B CALCULATION OF CHARACTERISTIC WIND LOADS A4 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 VI APPENDIX C1 GEOMETRY STUDY: PERSPECTIVE VIEW OF THE 25 TYPICAL GEOMETRIES A9 APPENDIX C2 GEOMETRY STUDY: CONVERGENCE STUDY FOR FEM ANALYSIS A10 APPENDIX C3 GEOMETRY STUDY: VERIFICATION OF FE MODELS A13 APPENDIX C4 GEOMETRY STUDY: CONTROL OF THE VALIDITY OF THE SIMPLIFIED MODELS IN EVOLUTIONARY DESIGN A14 APPENDIX C5 GEOMETRY STUDY: RESULTS FOR EVOLUTIONARY OPTIMIZATION A16 APPENDIX C6 GEOMETRY STUDY: NORMALIZATION AND SCALING OF THE RESULTS FOR PRIMARY EVALUATION A17 APPENDIX D COMBINATION OF LOADS A21 APPENDIX E ULS DESIGN OF STRUCTURAL ELEMENTS OF TIMBER A22 APPENDIX F CALCULATION OF PEAK ACCELERATION A27 APPENDIX G STRESSES FOR TRUSSES/COLUMNS FOR STRUCTURAL PROPOSALS A28 APPENDIX H CALCULATION OF TIMBER CONNECTION A31 CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 VII Preface The topic of tall buildings and skyscrapers is very popular in the field of architecture and engineering. There is always a demand to research new methods and innovative processes to build taller and more environmentally friendly. As future structural engineers, we would like to contribute what we have investigated to the constant development of the construction industry. This master thesis is our work in the master’s program Structural Engineering and Building Technology during spring term 2021. The project put a focus on the possibility of building a 200m tower in timber and the feasibility of constructing it. The project is being carried out at the research group for Lightweight Structures at the Department of Architecture and Civil Engineering, Chalmers University of Technology, Sweden. Structural engineers from the firm VBK are the initiators of this project, who came up with this topic and have been involved in the entire project. During this period, we have enriched ourselves with new knowledge that we can use in the future, and we were very happy to work in this project. Due to the outbreak of Covid-19 most of the work has been done from home. We are grateful to VBK who provided us with workstations and computer software for us to work from home. First and foremost, we want to thank our supervisors Andreas Lindelöf, Felicia Flink and Johan Örnborg, structural engineers from VBK, for their engagement in guiding us and giving us positive support despite their busy schedule. We also want to give our thanks to our supervisor and examinator from Chalmers, Robert Jockwer, for his informative advice during the entire project. Gothenburg, August 2021 Lien Trinh and Hong Zhang CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 VIII CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 1 1 Introduction The world we live in is developing, with society constantly changing and new demands emerging. The population in the world is increasing but the available land area that can be built on is decreasing. In order to maximize the use of land in densely urbanized areas, high-rise buildings were constructed. The 20th century witnessed a construction boom in tall buildings worldwide. Traditionally concrete and steel are the dominating building materials in the construction of high-rise buildings. However, in the recent years sustainable development is an important aspect in the construction industry and therefore using timber as the main structural material has been a tendency in modern buildings. 1.1 Background The number of tall buildings with a height of over 200 meters completed between 2000-2020 is 1384, 5 times more than what completed under the period 1885-2000 (Al-Kodmany, 2020). The average height of the world’s top 100 highest buildings has been increasing in the recent years, approaching 400m in 2020 (CTBUH, 2021). With the continuing urbanization, building skyscrapers and high-rise buildings is still the upcoming trend in the construction industry. The prevailing building materials in high-rise building construction, i.e. concrete, steel and composites or combinations thereof, are not environmentally friendly. Despite the production with new technologies making them less harmful to the environment than before, emissions of carbon dioxide are still at a high level. Especially for tall buildings, with the rise in the height, the dimensions of structural elements increase dramatically because of the large self-weight of the building, meaning that the emission of carbon dioxide per rental floor can be even higher (SOM, 2013). In comparison with other materials, timber is both renewable and is formed through an ecological cycle that absorbs carbon dioxide from the air, making it a more sustainable substitution for concrete and steel. In addition, the high strength- mass ratio in timber also decreases the loads to the foundation and in this way saves the foundation materials (Svenskt trä, n.d.). A lot of research and practice has been done regarding timber buildings. The appearance of engineered timber products, such as cross-laminated timber (CLT) and glued laminated timber (glulam), which to a large degree enhance the strength of timber and remedy the weakness arising from defects and anisotropy, increased the usage of timber in building construction. However, the usage of timber is still mostly limited to low-rise buildings nowadays, as a secondary structural element such as floor or combined or as a supplement to other main materials. Due to its low stiffness and strength, the resistance to horizontal load is relatively low compared to steel and concrete. Timber also has a low density and mass that makes it more challenging to build higher due to the problem of poor dynamic performance. So far, the knowledge of how a tall timber tower performs is still limited. At this time the tallest timber building in the world is Mjöstornet, located in Norway. It has a height of 85m and is far from comparable to the world’s highest building, Burj Khalifa in Dubai, with a height of 828 m, almost 10 times higher than the highest timber tower. (Abrahamsen, 2017) CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 2 1.2 Aim The aim of this project is to find the appropriate geometrical form with an applicable structural system for a 200 m tall timber tower with regards to structural stiffness and ease of construction. The design should not only fulfill criteria for lateral deformation and dynamic criteria but also provide as much rental area as possible. 1.3 Scope and limitation The study is limited to the structural behavior in service limit state (SLS), although the dimensions of the main structural elements will be estimated in ultimate limit state (ULS). Furthermore, the following assumptions and simplifications are made: ● The location is chosen to be in central Gothenburg and the environmental data in Gothenburg is applied throughout the project when needed. Terrain category IV according to SS-EN 1991-1-4 is assumed. ● The entire building will be designed for commercial use as offices. ● Floor slabs and their properties are selected from the existing products in the market that may fulfill the requirement for serviceability. No deeper investigation will be done about this. ● The footprint area is limited to be 30m by 30m and the core is 10m by 10m. The estimation of sizes is chosen based on experience without check of architectural requirements. ● Only superstructure above the ground will be studied, i.e. the connection between upper structure and foundation will be given as boundary conditions. The foundation will not be incorporated. ● Long-term effects will be discussed only. 1.4 Method The analysis was conducted in Rhino 3D and Grasshopper 3D and its plugin program Karamba 3D. Following steps were carried out for this project: 1. Literature study. Theories and previous study about slender buildings, geometries, timber as building material, structural systems and relevant standards and guidelines were studied. 2. General geometry study. In this early stage a series of volume geometries were generated, based on literature study and design workflow. These geometries were analyzed as shell elements by finite element analysis (FEA). The geometries were then assessed, and three geometrical forms were selected for further structural study. 3. Structural study. Based on earlier literature study of stabilizing systems, the selected concepts from the geometry study were further modelled in structural systems. The optimization of controlling parameters such as angles and sizes CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 3 were done with regards to static and dynamic performance and compared to the traditional braced frame structure. 4. Suggestions for feasible connections at critical places were given and roughly designed based on the loads on nodes from the former step. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 4 2 Theoretical framework This section consists of general knowledge and information of high-rise buildings, stabilizing systems and their structural behavior, design of geometries and their modifications, the effect of wind load on a tall building. The theory study of this project also includes information on timber as a building material and its connections. The design process and standards that were applied are also presented in this section. Theories and discussions presented in this section are the basis of further study. 2.1 Definition of a high-rise, timber building The initial step is to define what a high-rise timber building is. The aim of this project is to investigate the structure of a 200 m timber tower. Although the height of the tower has been determined, it is still supposed to be high in the view of structural design in order to make the study meaningful. On the other hand, a clear definition of timber building in the view of the application of material is important since timber is usually used in combination with other materials, especially in high-rise buildings. 2.1.1 Definition of highness of timber building There is no exact definition of what a “high-rise” building is because the definition varies from one culture and context to another. According to Council on Tall Buildings and Urban Habitat (CTBUH), there are three requirements for a building to be classified as “high”: Height relative to context, proportions and building technology (Al-Kodmany, 2020). Buildings that have a height in the range of 50 to 300 m are considered as tall buildings. Meanwhile, supertall buildings should have a height of 300 to 600 m and buildings with a height surpassing 600m are called mega- tall buildings (CTBUH). Height relative to context means how the building is in comparison to the surrounding buildings in a certain place. For example, a 14-story building is not considered as a tall building in a highly urbanized city such as Hong Kong or Chicago but in a provincial city it may be higher than the urban norm and still be referred to as a tall building (Al-Kodmany, 2020). However, from the structural point of view, it is the absolute height rather than the relative height that makes sense for the design such as an overall height or a height to minimum plan dimension. The taller a building is, the larger the wind load impact on the building. This means that with the rise in height, the wind load increases dramatically. CTBUH mentioned that that even if a building is very tall but has a large footprint, it is still not considered as a high building due to its proportions. Some buildings are not that high but are very slender and still give a vibe of tall building. In structural design, the proportion is usually measured as “slenderness”. The slenderness ratio (SR) of a building is defined as in Figure 2.1. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 5 Figure 2.1 Definition of slenderness ratio (MPA & fib,2014) The slenderness ratio is a basic proportion of the structure and influences the structural behavior of a high building significantly (Awida, 2011). For a slenderness ratio larger than h/8 the requirements on dynamic performance can be dominant during the design process. Slenderness ratios around h/6 or less are more appropriate and more common in tall buildings (MPA & fib, 2014). The last criterion, high building technology in terms of e.g., structural system and material, was stated as weakest by CTBUH (Al-Kodmany, 2020). By this, they meant a building with which height-related technologies might be suggested to be high. For example, a building with specific vertical transport technology can still be categorized as a tall building. The number of floors is a bad measurement for a tall building because the dimension of floor-to-floor height varies with the building’s function. Therefore, a building of 14 or more stories or higher than 50 m is a threshold for a tall building. There exists an obvious gap between the slenderness ratio of highest building in prevailing building materials such as concrete and that in timber. The existing highest timber building, Mjöstornet, cannot even be classified as a slender building with a slenderness ratio larger than 1:10 according to the general classification regardless of building materials. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 6 At the initial stage of the project, it is important to decide the slenderness ratio for the 200 m timber tower. A few examples of the slenderness ratio from the existing tall buildings were studied. Figure 2.2 shows the width height ratios of some popular tall buildings and the highest timber tower. Figure 2.2 Slenderness ratio of high- rise buildings in concrete and that of Mjöstornet 2.1.2 Definition of timber building Tall buildings can be divided into four categories according to the building materials that are utilized for the main vertical and horizontal load-bearing structural elements: concrete, steel, composite and mixed-structure buildings. The “main” structural elements refer to the primary structural members, where the secondary structural elements like the floor would not be considered in classification of a tall building with regards to structural materials. According to CTBUH a steel-framed tall building with concrete slabs resting on steel beams should be called a steel building rather than mixed. Furthermore, a building is defined as single-material when a single material makes up 85% or more of the height or the floor area of the building. (Foster et al., 2016). To extend this classification criteria to cover timber buildings, when more than 85% of the height or floor area of the primary structures are constructed of any kind of timber the building can be referred to as a single-material timber building. The light weight of timber is one of the properties that contributes to the poor dynamic performance. It is common to supplement the structure with extra mass by using concrete floor slabs for better dynamic behavior. For example, the prefabricated wooden decks were replaced by concrete decks for floor 12-18 on the top part of the building in Mjöstornet to fulfill the requirements on acceleration (Abrahamsen, 2017). As stated before, regardless of the concrete slabs the building can be still classified as a timber building, though the diaphragm action and the mass contributed from the floor are very important for the global structural behaviour. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 7 Traditional carpentry wooden connections are not commonly used in modern timber buildings, especially in high-rise buildings. For these weakest points, steel connection components such as plates and dowel are generally applied. Therefore, it is reasonable that the material used for timber joints are not included in the classification criteria. (Foster et al., 2017). Another general arrangement in timber buildings is that the lowest floors, often up to the first or second floor, are built in concrete. In this case the whole building should be regarded as mixed structure and the height of the upper section consisting of only timber can be regarded as the height of single-material timber structure. For better understanding several examples of schemes are presented in Figure 2.3. Figure 2.3 Examples of building typology by structural material (Foster et al. 2017) CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 8 2.1.3 Summary In this project, 200 m is aimed to be the height of the single-material timber building. The footprint is assumed to be 30m x 30m and the prescribed slenderness ratio 6.6 is the starting point for the design, which is not slender according to general classification criteria. However, considering the material properties for timber a height of 200 m and a slenderness ratio of 6.66 can be challenging and worth exploring. Steel connections and concrete floors slabs will not influence the fact that the building is a timber building as long as the primary structural elements, such as columns, beams and load-bearing walls are in timber. The application of concrete in flooring system will also not influence the classification of the building. However, from a sustainability perspective the concrete slabs will be introduced only as necessary when the mass of wooden decks is not enough to fulfill dynamic criteria. 2.2 Considerations regarding structural design for high-rise buildings The structural considerations for a high-rise building are in principle involved in the design process for a low-rise building as well. However, some of the factors might be more critical and require more attention when it comes to design for high-rise buildings. 2.2.1 Lateral resisting stiffness and strength With increasing height of a building, the load acting on the surface of the building increase exponentially. Apart from this, the cantilever arm from the tip to ground gets longer and high-rise buildings are usually very slender. The higher the building will be, the more dominant the lateral loads are in the design process, in relation to gravity loads. The overturning moment at the base increases by a power of 2 and the sway at the tip increases by a power of 4 when the height of the building increases. The limit of the maximum lateral displacement and the interstory displacement (relative displacement between adjacent stories) are of great importance regarding not only comfort demands but also requirements related relating to for example façade cladding and placement of partition walls. For a low-rise building in steel up to 10 stories, the design of structural elements designed for gravity loads only are adequate to resist lateral loads. For a building higher than 10 stories the sizes of the structural elements in the stabilizing system need to be enlarged to increase the global stiffness of the structure to satisfy the horizontal deflection control. (Balendtra, 1993). In the design of the 300 m tall timber building Oakwood Tower, the design was governed by the lateral stabilizing system. (Ramage et al., 2017). It was also pointed out that the strength and the stiffness of the connections in the timber structures plays a key role in the structural behavior and a minor movement in the connection can weaken the global stiffness significantly. (Ramage et al., 2017). CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 9 Some ways to improve the lateral stiffness are: • Increase the material stiffness parameters E and G • Increase the size of the structural elements • Increase geometrical stiffness and make use of the depth of the building, for example, place the stabilizing elements around the perimeter of the structure Despite this a careful designed form of the building and reduction of the loads can improve the lateral deflection of the structure. The net lifting forces in structural elements occurs when the tensile forces caused by overturning due to horizontal loads exceed gravity load. The design and construction for structural elements in tension are more complex and difficult. For concrete structures, the net uplifting force is not so common because of the high weight of concrete (SOM, 2013). Due to the light-weight nature of timber the occurrence of a net uplifting force is more probable. It is important to lead as much of the vertical load into the stabilizing system in the perimeter as possible to make use of the self- weight to resist overturning moments. (Ramage, 2017). In the design of the prototype building in the same research conducted by SOM, the concrete joints were adopted to provide enough ballast to avoid tensions (SOM, 2013). 2.2.2 Dynamic behavior When the building is exposed to time-dependent loading such as wind loads and seismic loads, dynamics need to be investigated in detail. For buildings located in seismic zones, seismic actions can be critical. These loads are dominant cross a certain spectrum of frequencies. How a structure behaves under these loads depends on its natural frequency and damping ratios. (Ramage et al., 2017). When the structural frequency coincides with the frequency of the loading, the deflection of the structure will be amplified to the fullest. For a building located in a seismic zone, the seismic actions are more critical. Due to the low stiffness the natural frequency of a tall and slender building is usually low, and the performance of the structure can be influenced more by wind turbulence than by earthquakes. (Balendra, 1993). Figure 2.4 Example frequencies of wind turbulence and earthquake (Balendra, 1993). CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 10 In a dynamic problem the structure is subjected to both external dynamic loads such as wind loads and inertia forces which act opposite the external forces, both of which vary over time. The acceleration in correspondence to the inertia forces are: 𝑓(𝑧, 𝑡) = 𝑚(𝑧) ∙ �̈�(𝑧, 𝑡) (2.1) where: 𝑓(𝑧, 𝑡) is the inertia force at height z and time t 𝑚(𝑧) is the mass of the structure at height z �̈�(𝑧, 𝑡) is the acceleration of the structure at height z and time t Figure 2.5 The structure subjected to dynamic loadings (Balendra, 1993). For the structural system: 𝑚�̈� + 𝑐�̇� + 𝑘𝜐 = 𝑃(𝑡) (2.2) where: 𝑚 is the mass 𝑐 is the vicious damping coefficient 𝑘 is the lateral stiffness 𝜐 is the lateral displacement 𝑃(𝑡) is the lateral loading The mass, damping coefficient, lateral stiffness and the structural response of lateral displacement are key factors in the dynamic problem of a building. Larger mass, lateral stiffness and damping are favorable when it comes to the dynamic behavior. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 11 2.2.3 Gravity resisting system The gravity weights can help resist the uplifting force due to the overturning moment. One of the considerations in designing gravity resisting system in relation to the lateral load resistance is to transfer as much of the gravity loads through the stabilizing system as possible. Long spans connecting the core to the perimeter structures without interior columns and walls are desired. The challenge lies in the design of the floor slabs with large spans while still meeting requirements regarding dynamic issues. The depth of the floor might be quite large. The fixed connections between floor elements and the adjacent stabilizing system can make it possible for larger span. Another strategy is to widen the effective width of the core (SOM, 2013). The uneven shortening of structural elements such as columns in the perimeter and the core walls, caused by the vertical loads, can cause problem in serviceability. This can lead to problems for floor finishing and cladding details. (MPA & fib, 2014). 2.3 Geometry and modification This section describes how the shape of the building plays a significant role in the structural design and the effect of wind load on high buildings. Due to the high speed of wind on the top of the tall building and the slender geometrical form, high-rise buildings are very subjected to wind load both in static and dynamic state. The wind force that a high-rise building subjected are along-wind, caused by the wind turbulences directly on the surfaces, and crosswind which can be induced by vortex shedding. (a) (b) Figure 2.6 The wind loads acting on a building (a) Wind flow (b) The resultant wind forces During the structural design both the along-wind and crosswind should be taken into consideration and be designed for. For a supertall building the oscillations induced by vortex in the cross direction can be more critical (Alaghmandan & Elnimeiri, 2013). When the wind speed is low it is usually the along-wind which is dominant. While the wind speed increases the crosswind will become more and more critical and can dominate the structural behavior (Xie, 2014). There are two general ways to improve the dynamic behavior of high-rise buildings. One is to optimize the building geometry to mitigate the wind loads from the source, and the other one is to choose proper structural system and improve the global stiffness during the structural design. (Alaghmandan & Elnimeiri, 2013). Since the geometry is very much dependent on the architectural design, the aerodynamic study is worth to be conducted at very beginning of the design. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 12 2.3.1 Plan shape and corner modification Buildings with square and rectangular plan section are common because they utilize the land most efficiently. However, symmetrical cross sections like square, rectangular, circular, or triangular are most susceptible to wind-induced vibration (Elbakheit, 2018). Corner modification is a very effective way to decrease the wind- induced loads on high-rise building. A rule of thumb is that the modified part should be at least 10% of the width of the building and in general the corner modification can contribute to reduce both along-wind and crosswind excitation (Xie, 2014). Figure 2.7 Examples of corner modification (Elshaer,2017) Chamfer which is 10% of the width of the building can reduce 40% along-wind response and that of 30% in the cross direction compared to a building with square section without corner modification (Holmes,2001). Among chamfer, recession and roundness, roundness is the most effective way to improve the aerodynamic and dynamic response (Kawai, 1998). 2.3.2 Modification in elevation Modification of the shapes in elevation can influence the architectural design, much more than local modification around the corners. Therefore, this process should be continuously involved in the architectural design of the building from the very beginning. Some common approaches are for example tapering, setback, twisting and opening in the building body. The common principle that makes all of these solutions work is that the change of the shape over the height make the frequency of the vortex shedding vary over the height as well to avoid the mutual excitation (Xie, 2014). Figure 2.8 Example of form modification in elevation (Elshaer,2017) CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 13 2.3.2.1 Tapering and setback Tapering means that the width of the floor plan section of the building gradually decreases over the height. Tapering is more effective in mitigating vortex shedding in the cross direction rather than the effect in the along-wind direction. But this might have a negative impact on the structure when the damping ratio is too low (Alaghmandan & Elnimeiri, 2013). This is because the narrowing to the top and the decrease in mass close to the tip can cause excessive acceleration, which is not expected with consideration to the comfort issue (Xie, 2014). Therefore, both effects need to be included. Setback involves a similar structural concept and principle as tapering and also the risk for increased acceleration. 2.3.2.2 Twisting Twisting is another effective way to reduce wind effect. Xie (2014) showed that the effectivity increases with the increase twisting degrees. Considering the difficulty for façade cladding, 120 degrees with opening on the top can be an optimal choice. 2.3.2.3 Opening Openings can help to reduce both the wind loads in the along-wind direction and vortex shedding. The opening can let the wind turbulence through the solid building body and release the surface pressure. Openings are commonly placed near the top of the building and the wind flow will be disrupted to improve the dynamic performance (Irwin, 2009). 2.3.3 Consideration regarding structural stiffness Apart from the aerodynamic effect, the geometry of buildings can influence the structural stiffness as well. Geometries with circular sections have the same stiffness in all direction unlike geometries with square sections. Tapering or setback may improve the aerodynamic response but the decrease in the structural stiffness needs to be considered as well. 2.4 Structural systems for tall buildings CTBUH describes four main types of structural systems for tall buildings: shear and frames system, interacting system, partial tubular systems and tubular systems. These can be further divided into interior and exterior structures, depending on which kind of primary lateral load resisting system a building has. Interior structures can be created by forming columns and beams in the core that are stiff enough to resist wind forces. The core is usually an elevator shaft in the middle of the building, saving a lot of open areas for each floor. Interior structures can be divided into rigid frames, shear wall hinged frames, shear wall or frame interacting systems and outrigger structures. Interior structures have two common types of lateral load resisting system, moment- resisting frames and shear trusses or shear wall system. In exterior structures, the columns and beams instead of being inside the core are placed at the perimeter of the building and form a hollow, rigid tube that is as strong as the core but much lighter. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 14 Exterior structures can be divided into tube structure, diagrid structure, space truss structures, super frames and exoskeleton structure (Ali & Moon, 2007). 2.4.1 Rigid frame A rigid frame system is an interior structure with a moment-resisting frame. This system consists of horizontal girders and vertical columns that are rigidly connected together in a planar grid form, see Figure 2.9. The dimension of the columns in this system depends mostly on the gravity loads and vertical loads. However, to ensure the lateral stability of the building the dimensions of the horizontal girders depend on the stiffness of the frame. For a concrete construction in rigid frame, the most economical span is 8-9m (MPA & fib, 2014). Figure 2.9 Typical plan of a frame structure, 2.4.2 Shear wall system The whole system behaves like vertical cantilevers that are fixed at the base. It is common that shear walls are placed in the middle of the building as the service core. Concrete core and steel-framed core are two of the most common core systems (Fu, 2018). It can also be coupled shear walls which refers to more than two shear walls being connected by beams or slabs. The overall stiffness of the system exceeds the summation of individual stiffness (Ali & Moon, 2007). Figure 2.10 Typical plan of shear wall- hinged frame structure with core CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 15 2.4.3 Shear wall and frame interaction system Shear wall and frame interactions system refers to the combination of two systems, the shear wall system, and the frame system. The frame can be a rigid frame or semi- rigid frame. One advantage of this system is that the large lateral displacement caused by moment deformation can be effectively restrained by the frame system in which the shear deformation dominates (Ali & Moon, 2007). Figure 2.11 Combined effect of braced frame and core (Ali & Moon, 2007). 2.4.4 Outrigger system The outrigger system has a core that behaves like a cantilever. By rigidly connecting the core and the exterior systems the overturning moment in the core can be decreased. Columns in the perimeter helps to resist the overturning moment by compression in the leeward and tension in the windward, see Figure 2.12. Figure 2.12 Outrigger system, with steel truss belt as outrigger Two typical outriggers are steel outrigger consisting of truss system and concrete outrigger in form of deep beams or walls. Usually, the outrigger belt can take as high as one floor and another disadvantage is that the connection of the trusses to the core walls can be very complicated to ensure the effect (Fu, 2018). CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 16 2.4.5 Tube system One type of exterior structure is the tube system. By designing the building as a hollow cantilever perpendicular to the ground it can resist lateral loads, making use of the full depth of the building and obtain maximum geometrical stiffness. There are four subcategories of this system: framed tube, braced tube, bundled tube and tube in tube. 2.4.5.1 Frame tube The hollow cantilever can be constructed by placing the columns on the exterior of the buildings closely together and rigidly connected with deep spandrel beams. This results in a rigid frame that is dense and strong along the edges of the building. The typical distance between the columns is 1.5 - 4.5m (MPA & fib, 2014). By introducing bracing members, the columns in the perimeter can be more sparsely placed than the simple frame tube, called braced tube. Buildings in braced tube can reach up to 300m economically (MPA & fib, 2014). Figure 2.13 Elevation of a braced tube CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 17 The shear lag effect is a serious issue in frame tube system which can make the exterior tube not work as a true cantilever. The columns close to the corner are subjected to larger stresses than those in the middle of the span. The introduction of the diagonal braces can help to redistribute the stresses from higher stressed corner columns to the lower stressed inner columns. (Fu, 2018). (a) (b) Figure 2.14 The illustration of shear lag effect in frame tube system (a) Frame tube works as a true cantilever; (b) The stresses in the corner columns are increased due to the shear lag effect. (Fu, 2018) 2.4.5.2 Bundled tube A smaller tube system connected together as one unit is called bundled tube. By providing cross walls or cross frames in the building the strength and stiffness of the building increases, and it is possible to have wider column spacing in the tubular walls. To further enhance the stiffness trusses could be added. (Ali & Moon, 2007) Figure 2.15 Illustration of bundled tube (Ali & Moon, 2007) CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 18 2.4.5.3 Tube in tube system Combined with an interior tube, in forms of core, the structural stiffness of tube system can further increase. With a tube in tube system, the effective height of buildings can be approximately 80 stories (Ali & Moon, 2007). 2.4.5.4 Diagrid system One structural system that is becoming popular is diagrid system. In a diagrid system most of the vertical columns are removed and given larger spacing. This system is structurally efficient as a tubular system. The diagonal components in a diagrid system can take both gravity loads and lateral loads because their triangulated arrangement is distributive and uniform. This system can resist the lateral shear by axial forces in the diagonal components, but one drawback is the complicated joints between diagonal members. Diagrid system on the façade can dominate the architectural expression. Therefore, good communication with the architects is important during the design process (Boake, 2014). Two very famous architectures in diagrid system are 30 St Mary Axe in London and Poly International Plaza in Beijing. 2.4.6 Spaces truss structure Spaces truss structure is a modified version of braced tubes which also includes diagonals that connect the exterior to the interior. The distinction of a spaces truss structure is that some diagonals penetrate inside the building, while in a braced tube system the diagonals connecting the corner columns are located on the perimeter of the building. Space truss structure is mostly used in building with large span (Fu, 2018). One typical building utilizing a space truss system is the Bank of China Tower in Hong Kong. 2.4.7 Super frame structure A super frame consists of mega columns and mega girders. This system is rigidly connected by multistory trusses at every 15 to 20 stories. “It is an ideal structure for super tall buildings” and it “resist lateral loads with minimum amount of structural material”. (Feng & Mita, 1995). Some examples of this system are the 56-storey tall Parque Central Complex towers in Venezuela and CITIC Tower in Beijing. 2.4.8 Exoskeleton In exoskeleton structures the lateral-load resisting system is located outside the facade of the building. An example of this system is the Hotel de las Artes in Barcelona. One advantage of this system is that fire proofing is not a serious problem because the system is located outside the building line. However, thermal expansion, contraction and weather protection should still be considered. (Ali & Moon, 2007) CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 19 2.4.9 Classification and considerations regarding timber buildings Fazlur Khan has developed and updated the diagrams of classification of structural systems for high-rise buildings in relation to the height regarding structural efficiency, for both steel and concrete constructions in 1969, 1972 and 1973. (Ali & Moon, 2011) Supertall buildings with slenderness larger than 7 will perform very poor in resisting turning moment caused by the wind, especially for timber constructions. One potential solution for building high timber buildings is to put the stabilizing structural system on the façade, making use of the building depth to enhance the global stiffness, and at the same time try to lead as much gravity load to the exterior stabilizing system as possible. (Ramage, 2017). Taking away the interior columns and load-carrying walls can be one approach. The diagrams of exterior structures in Figure 2.16 shows also that the steel braced tube without interior columns can be built 50 stories higher than that with interior columns. Figure 2.16 Exterior structures (Ali & Moon, 2011) The exterior structures that can efficiently be built as high as 60 stories are tube structures, space truss and super frame for concrete and steel buildings. Such tube structures can be for example framed tube, braced tube, diagrid and bundled tube. Ramage (2017) argues that tubes, diagrid and mega-trusses could be considered for high-rise timber buildings. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 20 2.5 Timber as a building material In recent times, timber is one of the most common building materials in construction especially in buildings because it is environmentally friendly and has moderately high strength. Timber construction has variety in form and is very aesthetically pleasing with regards to architectural character. Despite all the advantages, timber is an anisotropic material which means it has different properties in different directions (Swedish Wood, 2016). It is also moisture dependent and is sensitive to rot and insects. Structural engineers always need to verify that the chosen timber materials, dimensions, and structural systems fulfil all the requirements regarding functionality, durability and maintenance. This section describes more about timber properties such as how it is affected by moisture and creep. Several recommended timber products and timber connections for a tall building are also introduced in this section. 2.5.1 Property of timber 2.5.1.1 Timber mechanical properties In timber the effect of the natural characteristic such as knots, spiral grain, juvenile wood, and reaction wood decides its grade and strength (Swedish Wood, 2016). Timber is an anisotropic material which means that it has different properties in different directions. These directions are defined as longitudinal, tangential, and radial. In practice, there are two main directions which are important to consider in design: parallel to the grain and perpendicular to the grain. Figure 2.17 Three directions in wood. Figure 2.13 in Design of timber structures Volume 1 (Swedish Wood, 2016) CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 21 The strength of timber depends on the angle between the load and the grain direction, see Figure 2.18. Timber has its highest strength when subject to load parallel to its grain direction. The strength of wood in compression parallel to the grain is very high however when the load is too high the fibres start to buckle, and timber will have a ductile behaviour. In compression perpendicular to the grain the shape of the wood is crushed and deformed at low forces, therefore the strength of timber in this loading direction is very low. In tension perpendicular to the grain timber has its lowest strength. Timber has high strength in tension parallel to the grain but if failure occurs it will be very brittle. Figure 2.18 Strength in compression, bending and tension Timber has a lot of advantages such as a high strength per weight ratio, it is a very light material compared to concrete and steel. In fact, the density of wood varies between 170 kg/m3 to 1200 kg/m3. Meanwhile the density of concrete is approximately 2400 kg/m3 and the density of steel is around 7850 kg/m3. The light weight of timber eases the transportation, construction and reduces the foundation work but also is a problem in building dynamics and vibration because the lower the mass, the lower the building’s inertia and stiffness. 2.5.1.2 Influence of moisture Timber is a hygroscopic material and is affected by the moisture content around it (Swedish Wood, 2016). Depending on the moisture content and environment, timber can either adsorb or desorb the moisture content. The strength of timber decreases with increasing moisture content, which is why timber needs to dry before being used as a building material, see Figure 2.19. Timber can be dried in a large kiln or solar kiln where humidity and moisture content can be controlled. If the moisture content is above the fiber saturation points (FSP) then the strength of timber is almost constant but if the moisture content is under FSP then the strength of timber increases with increasing moisture content. The compression strength of timber is most affected by the moisture content, meanwhile the tension strength of timber is almost CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 22 independent of the moisture content. Changes in moisture content in wood leads also to dimension change and the danger of being attack fungi attack may also increase. Figure 2.19 Modulus of elasticity of timber related to change of moisture content 2.5.1.3 Influence of creep Creep is defined as the extra deformation increase with time under constant load (Swedish Wood, 2016). This time dependent deformation can be divided into three parts, elastic, delayed elastic and viscous deformation. The elastic deformation occurs directly after a load is applied to a structure. After that the deformation will grow slowly under a constant load, consisting of both delayed elastic and viscous deformation. The delayed elastic deformation is reversible in time while the viscous deformation is irreversible and remains the same under loading, see Figure 2.20. There are also some external factors that affects creep such as temperature, load direction, stiffness, knots, and moisture content. Figure 2.20 Creep curve. Loading and deformation over time. Figure 2.23 in Design of timber structures Volume 1 (Swedish Wood, 2016) CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 23 2.5.2 Wood product for tall buildings There has been a huge development in the world of engineered wood products (EWP) to overcome the drawbacks of sawn timber. With EWP it is possible to produce almost all kind of shapes and sizes depending on the design demand (Swedish Wood, 2016). The largest disadvantage of sawn timber is its deviations such as knots, reaction wood and grain angle, but these deviations can be decreased with EWP. The problem with lack of old growth timber can be solved by using lower quality wood to produce EWP with desired quality. EWP are also less affected by moisture dependent deformation like shrinkage, and it is easier to control its strength and elasticity properties. High buildings are subject to larger loads therefore the primary components are usually larger and have higher strength. A building consists of many parts and each part has different requirement in terms of strength and size. Depending on the application in buildings, a combination of timber products will be better. There are three common primary components for timber buildings: solid timber, glued laminated timber (glulam) and cross laminated timber (CLT). Structural timber is also known as solid timber which normally is up to 6m long and 245 mm high. The strength grading of solid timber grades each timber into different strength class from C14 to C40. The number of the indicates the bending strength parallel to the grain of the timber. For example, C14 means that the timber product has a bending strength of 14MPa. In Sweden only C14 to C35 are available in the market. Solid timber can even be longer if finger jointing is applied. Glulam is the oldest of the engineered wood products. Glulam is made of small timber laminates bonded together with adhesives and finger joints. All the lamination is oriented in the same direction and all grains are parallel to the longitudinal axis. Glulam can be homogeneous or consist of different strength graded lamellas which is called combined glulam. Due to the gluing process, it is possible to match the lamination quality to the level of design process and therefore glulam can have bigger dimensions and more shapes than solid timber. The largest available dimension of glulam in the market is 2m x 2m and 30m length. Cross laminated timber is also called (CLT) and has lamellas oriented in different directions and often is used in stabilizing walls or slabs. Each layer of lamella in CLT is placed perpendicular to the layer above and below it. Depending on the demand and design the number of layers can be 3, 5 or 7. One important thing is the two outer layers of CLT always have the same direction and better strength. The dimension of CLT available can be up to 500 mm thick, 3 m wide and 24 m long. Laminated veneer lumber or LVL is made of veneers which are usually 20 to 90 mm thick and made from spruce or pine and glued together either in perpendicular direction or in the same direction. LVL has high bending, tension, compression strength and has high stiffness and as well as shear strength. LVL is available in dimension up to maximum 3m wide and 24m long. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 24 2.5.3 Connection in timber buildings Timber elements need to be connected to each other to form a functional structural system. Timber connections affect the structural behavior of the structural system. There are three types of timber joints: traditional timber joints, dowel joints and glued joints. The stiffness of timber joints depends on the number of fasteners, density of the timber, the fasteners diameter and type of fasteners i.e. glue, nail, screw, bolt etc. 2.5.3.1 Dowel joints This is the most common type of joint in timber (Swedish Wood, 2016). In dowel joints forces transfer through shear. Predrilled holes are required or not depending on the type of fastener and this might weaken the structure because of the reduction in the cross section. Dowels are divided in five subgroups including nails, screws, dowels, nails plates and bolts. Figure 2.21 Different types of timber connections (a) Hexagon head wood screw, (pre-drilling required), (b) Countersunk head wood screw, (c) Double threaded wood screw, (d) Dowel, (e) Nail plate, (f) Punched metal plate fastener, (g) Bolt, (h) Bolt with washer and nut (Swedish Wood, 2016). CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 25 When a dowel joint is loaded it can fail in different modes, based on Johansen’s yield theory. The assumptions in Johansen’s theory are rigid-plastic behavior of fasteners in bending and rigid-plastic behavior of the timber in embedding. Rigid-plastic behavior means that neither lateral nor axial deformation takes place at low value of loading. 2.6 Other considerations This section gives information about other consideration that are also important when working with timber material in buildings. 2.6.1 Fire safety Fire escape and fire resistance are two important aspects to ensure safety in case of fire accidents. For the former one, proper design of escape stairs and path need to be included in early design of plan layout. For the latter one the structural elements and joints need to have adequate resistance during the resistance time. 90 minutes is usually the requirement, while 120 minutes can be required for more important elements. For Oakwood Tower all the primary structural elements like columns, beams and walls are designed to have a resistance time of 120 minutes (Ramage et al., 2017). Considering the height up to 200m, the resistance time of 120 minutes might be practical for this design. For timber, a charring layer close to the surface will be formed due to exposure to the fire, which will contribute to insulate the cross section in the core. The burning rate should be regarded as constant (SIS, 2009). Fire resistance should be designed following the procedures in SS-EN1995-1-2 and SS-EN1991-1-2. The resistance time, the resulting capacity of the effective cross section and the temperature should be verified. The effective cross section can be calculated by the simplified method and be calculated by the initial cross section area subtracted by the effective charring depth. The charring depth can be calculated by expression 2.3: 𝑑𝑓 = 𝑑𝑐ℎ𝑎𝑟,𝑛 + 𝑘0𝑑0 (2.3) where: 𝑑𝑓 is the effective charring depth 𝑑𝑐ℎ𝑎𝑟,𝑛 is the design nominal charring depth 𝑘0𝑑0 is the depth of the weakened layer The values of k0 and d0 are given in Section 4.2.2 in SS-EN1995-1-2 and the calculation procedures for the nominal charring depth in 3.4.3. Since the fire design is not the focus in this project the design details will not be presented here. However, the fire design for structural elements is of great importance. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 26 1: Initial surface of the structural member 2: Border for residual cross section 3: Border for effective cross section Figure 2.22 Illustration of the residual cross section and effective cross section. Figure 4.1 in SS-EN1995-1-2 (SIS, 2009). In strategy, mega structural members are better than a great number of elements of small size since the percentages of the remaining cross section are larger for structural members of large sizes, compared to those with small sizes, if the effective depth is the same for all. It was mentioned that the large sizes mega truss elements for the timber building Oakwood Tower have great significance in fire resistance (Ramage et al., 2017). With regards to the shape of the structural members a design principle applied in the timber tower research project led by SOM (2013) is to use simple shapes with high volume to surface ratio. A square cross section is preferred rather than rectangular. The design strategy for structural members regarding the fire safety can be to choose: • Mega structural members • Members with large volume to surface area ratio, for example, members in square cross section Penetration and smoke control, sprinkle system, escape space etc. regarding fire safety should be included during the design process. Fire retardant paints and covering of the joints should be given consideration. 2.6.2 Sound and vibration issues of the floor The low mass of timber can cause acoustic problems for wooden floor elements. CLT floor elements exist on the market which, when combined with sound insulation and sound absorption layers, can handle sound issues and satisfy sound requirements for residential housing and office buildings. The investigation to ensure that the floor slabs supported by the external structure and core walls without columns can fulfill the dynamic requirements are not included in this thesis, but by study of existing timber buildings and CLT products in the market, a span up to 10m is possible. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 27 2.6.3 Daylight A simple rule of thumb regarding the daylight in design is that the window area should be at least 10% of the floor areas (Boverket, 2019). In general, when the depth of the room, i.e., the distance from the windows to the deepest walls, is smaller than 5m, adequate daylight can be expected. In the range between 5m to 10m, the daylight can partly reach. Deeper than 10m, the condition of the daylight can be regarded as bad. (Ibrahim & Hyman, 2005). To simplify the analysis, a design with maximum depth of 10m was considered as acceptable in this thesis. 2.6.4 Vertical transportation Transportation within a high-rise building relies heavily on the vertical transportation system consisting of elevator and stairs etc. A proper design of the number of elevators and what floor each of them can reach can greatly improve the efficiency of transportation. Calculation of the required number of lifts, lift speed and waiting time etc. by a lift specialist is required already in the early stage of the design process (MPA & fib, 2014). In terms of accessibility, according to Boverket, the national board of housing, building, and planning of Sweden, the main rule is that for buildings with more than one floor there must exist one elevator that provides enough space for a person with wheelchair and assistant. For buildings with more than 4 floors, at least one rescue elevator that can fit stretchers is required. The required size for this kind of elevator is 1.1x2.1m. For buildings with more than 10 floors at least two such elevators are required. (Boverket, 2011). For office buildings, there might be additional requirement on the size of elevators with consideration to rush hours. Turning Torso, the current tallest building in Sweden where most of the floors are used for residence and a few commercial usages, is provided with 5 elevators. In a previous investigation of the 200 m timber tower, various shapes of cores with 8 elevators and with minimum space of 2 m at the opening were studied. (Gyllensten & Modig, 2020). A core of approximately 10 m by 10 m should be feasible and was adopted in this thesis. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 28 2.7 Structural design The structural members should be designed both for ultimate limit state (ULS) and service limit state (SLS). 2.7.1 Design of the structure and the structural components in ULS In the ultimate limit state, the structural components should be designed to provide sufficient capacity, verified according to Eurocode and EKS11. In Appendix E the relevant procedures can be found. Some of the main ways to improve the structural performance in ultimate limit state are: • Optimize the building geometry to reduce the loads acting on the building • Optimize the geometry and structural systems to avoid shear lag for more evenly distributed stresses • Optimize the building geometry and structural systems to make full use of the building depth to improve the global stiffness • Optimize the cross section of structural components • Use materials with higher strength • Make use of the strength parallel to the grains which is usually the strongest direction for timber • Avoid buckling of the structural components to ensure full use of structural resistance 2.7.2 Design of the structure and the structural components in SLS In the design for service limit state the serviceability of the building should be controlled. The deformation and dynamic problem are two of the main issues that can cause damage of non-structural elements, comfort problems and difficulties in usability, appearance and so on. 2.7.2.1 Limitation for global deformation Horizontal deformation is crucial for high-rise buildings. The horizontal displacement of the building and the relative story displacement can affect the façade cladding. This limitation needs to be confirmed with the façade engineers. In practice values between h/300 and h/500 have been used before (MPA & fib, 2014). CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 29 u: the maximum horizontal displacement of building ui: the horizontal displacement over one floor Figure 2.23 Definition of horizontal displacement. Figure A1.2 in SS-EN1990. (SIS, 2002) For the limitation h/500, the horizontal displacement of a building of 200 m should be limited to 400 mm, where for each floor it is about 6.5 mm. The horizontal displacement of a building is a sum of that from bending deformation, shear deformation and deformation of joints. The approaches to improve the structural behavior with regards to the global deformation are for example: • Use stiffer material with higher modulus of elasticity and shear modulus • Optimize building geometries and structural systems as mentioned before to improve ULS performance, to increase the global stiffness of the structure • Stiffen joints and decrease joint slip • Decrease the loads which trigger the horizontal displacements 2.7.2.2 Limitation for deflections for beams Both the instantaneous and final deflections including creep for beams need to be controlled. Table 2.1 Limitations for deflections for beams. Table 7.2 in SS-EN 1995. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 30 The verification should be conducted according to Eurocode. In this thesis the deflection of the individual structural components is not considered, and the focus is on the global behavior. 2.7.3 Dynamic design in SLS The design and criteria to achieve satisfying vibration behaviors in the service limit state shall consider the receivers, which can be categorized as the structure, the building content, and the comfort of users. The detailed criteria are specified in the same international standards SS-ISO-10137 which is applicable for Swedish standards as well. Regarding the comfort of users, the peak acceleration is a very important variable, the users’ sensibility to it varies with the frequency of the building. (SIS, 2008). The guidance for human response to wind-induced motions is attached in Appendix C in SS-ISO-10137. Figure 2.24 shows the evaluation curves for wind-induced vibrations for buildings in horizontal directions for one-year return period, where curve 1 is for office buildings and curve 2 for residential buildings. The peak acceleration of the target floor should not exceed the evaluation curves in the structural direction (in along wind and cross wind directions) and in torsion. The peak acceleration in torsion should be recalculated to the translational torsional acceleration as 𝑟 ∙ 𝐴𝜃(𝑡), where 𝑟 is the distance to the torsional center and 𝐴𝜃(𝑡) is the angular acceleration in torsion. (SIS, 2008). A: the peak acceleration m/s2 f0: the first natural frequency Hz Figure 2.24 the evaluation curves for wind-induced vibrations. Curve 1 for office and curve 2 for residence. (SIS, 2008). CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 31 The wind velocity for 1 year return period can be determined by the following equation according to ISO 6897 (SIS,2008): 𝜐𝑇𝑎 = 0.75𝜐50√1 − 0.2𝑙𝑛(−𝑙𝑛(1 − 1 𝑇𝑎 )) (2.3) where: 𝑇𝑎 is the number of years, set to 1 year for the evaluation of comfort criteria 𝜐50 is the characteristic value of the reference wind velocity, given in EKS11 The peak acceleration can be determined by Equation (2.4) according to EKS11. �̈�𝑚𝑎𝑥 (𝑧) = 𝑘𝑝 ∙ 𝜎�̈�(𝑧) (2.4) where: �̈�𝑚𝑎𝑥 (𝑧) is the peak acceleration, see Appendix F 𝑘𝑝 is the peak factor, calculated by expression (C.7) in Appendix B 𝜎�̈�(𝑧) is the standard deviation of the acceleration, calculated by expression (G.2) in Appendix F The complete calculation procedure can be found in Appendix F. In the calculation for peak acceleration, the value for the mechanic damping of 9% was applied. The mass, damping coefficient, lateral stiffness and the structural response of lateral displacement are key factors in the dynamic problem of a building. Larger mass, lateral stiffness and damping are favorable when it comes to the dynamic behavior. For improvement of the dynamic behaviour, the most efficient ways summarised by Gyllensten & Modig (2020) are: - Increase mass when the eigenfrequency is smaller than 1Hz - Introduce damping when the eigenfrequency is between 1Hz and 2Hz - Increase the structural stiffness when the eigenfrequency is larger than 2Hz CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 32 3 Geometry study In this section various volume geometries were modeled, and the structural behavior, consumed material and obtained rental area were compared, among which one promising alternative was selected for further study. The research process is shown in Figure 3.1 below. The procedures labeled with dashed lines present the geometry generation method and geometries created for analysis. Those with solid lines stand for analysis conducted where those in red are finite element analysis. Figure 3.1 Work process for geometry study The general method was applied to parametrically control all geometries expect the special hyperboloid, presented in Section 3.1. By this method countless geometries can be created and it is not possible to manually study all of them. By the optimization algorithm Galapagos, see in Section 3.4.1, with its evolutionary design, the inputs parameters that govern the shape can be optimized for minimum defection. Due to the large amounts of computation that occur during the evolutionary process, the structural model was simplified to have only one point load on the top edge, which differs from the real situation where the wind load is exponentially distributed over the height. The best solution that causes minimum tip displacement was kept for further study, see Section 3.4. Apart from this, 25 geometries combined with various CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 33 façade curvature and plan sections were manually selected for study to get a deeper and more intuitive understanding, see Section 3.5. Hyperboloid geometry can in theory be a promising one due to the potential for straight load paths. It has surfaces that are doubly ruled and for each point on the surface there are always two crossing straight lines that lie on the curved surface. This was generated in its own special way and one applicable case was chosen, see Section 3.6. Thereafter all these 27 geometries will be analyzed with FEM with distributed wind loads. The result will be compared and evaluated according to a series of criteria, see Section 3.7 and Section 3.8. 3.1 Parametric volume geometry As discussed in Section 2.3, square plans with various roundness of corners were chosen to be investigated. Volume geometries were determined by the combination of shapes in plan and facade curvature. Figure 3.2 Geometries as combination of a plan shape and a facade contour, where the plan shape is chosen from squares with rounded corners. When the radius is equal to 15 the plan is circular. The facade contour is either hyperbolic or linear in the height direction. Since the geometries are controlled by the parameters it is possible to make investigations into all potential geometries within the domain, rather than the predetermined geometries presented above. The controlling parameters and geometry- generating process are illustrated in Figure 3.3. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 34 Figure 3.3 Parametrized geometry. The generation of the geometry is determined by four parameters: roundness of the corners, scales of upper two controlling curves and height of the curve in the middle. Several examples are presented as follows: Figure 3.4 Several examples of geometries and the corresponding parameters CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 35 3.2 Structural model Though in reality the structural behavior of a high-rise building, subjected to both gravity load and exponential distributed wind load, is quite complicated, several simplifications can be made in this early stage to simplify the evaluation. 3.2.1 Simplified structural model As discussed in Section 2.4, the external system is supposed to resist the lateral load as a cantilever beam. The modelling of the complex stabilizing structure was simplified to a tubal structure. Also, in this early stage, the precise load and material properties are not essential, as long as different alternatives of geometries can be sufficiently compared to each other. For a cantilever beam with identical cross section along axis and constant stiffness EI, the lateral stiffness of the structure can be reflected to a great extent by the end deflection. In the present case when the stiffness varies over the height due to the various geometries and the wind load is not evenly or linearly distributed over the height, the structural behavior can be more complex. But the global stiffness can be still reflected by the maximum horizontal displacement at the top and/or the average horizontal displacement. The wind loads act directly on the enveloping surfaces of the building by the wind pressure, varying from windward side to the leeward side. In this study it is the global structural behavior that is interesting rather than the local effect, therefore a resulting wind force on each diaphragm floor is sufficient to estimate the global structural response. Since the wind loads increase over height and the cross section of the building geometry varies, the interaction will be more complex. The result will be more reliable if the distributed wind forces over height can be more precisely estimated, rather than simplified as being uniformly distributed. The structural model was therefore simplified to a tubal shell structural connected by stiff floor slabs with distributed wind forces anchored at each floor level (see Figure 3.5(b)). (a) (b) Figure 3.5 Simplified structural model. A tubal structure (of square cross section as an example here) subjected to (a) a lateral point load. (b) Distributed point loads on each floor. The structure is fixed at the base. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 36 The evolutionary design requires significant computing resource because of the great number of models that need to go be run through. In order to avoid overly long computation times for this, the structure is further simplified to have a single point load applied on the top edge of the structure in the mass screening of parametric geometries during the evolutionary design, see Figure 3.5(a). By this all the other diaphragm floor slabs, except the one on the top where the point load acts on to efficiently activate the bending deformation, can be neglected and the number of meshes included will decrease dramatically, saving large amounts of computation time. Note that this model will only be used to get a single optimized geometry from the mass screening in the evolutionary design. The optimized geometry obtained from this process will thereafter be further analyzed with all other geometries according to the structural model with exponentially distributed wind forces, see Figure 3.5(b). The influence of the simplification in evolutionary design has been controlled, see in Appendix C4. This simplification might be a good balance between the precision and computation efficiency. 3.2.2 Properties of elements The slabs were assumed to be so rigid that the point load will be transferred axially to the flanges and be carried mainly by the bending of the tube, not by the deformation of the slabs themselves. In order to achieve this effect, a fictive slab with a very small thickness and large modulus of elasticity was adopted. The Poisson’s constant of the slab was set to 0 to avoid the influence of transverse deformation. Figure 3.6 Floor slabs. If the thickness of the slab is small enough and the modulus of elasticity large enough, the bending stiffness EI can be sufficiently small compared to the axial stiffness E, so that the load can be regarded to be carried only by the axial resistance. For the tubal structure, usage of isotropic material is sufficient to evaluate geometries, though for a truss or frame structure the stiffness in different directions can differ. Since both shear deformation and lateral bending deformation are involved the relative bending stiffness and shear stiffness may influence how the structure behaves globally. There is uncertainty as to what extent this will affect these slender geometries, where bending dominates. The equivalent membrane Young’s modulus and shear modulus for diagrid trusses with given angles and sizes of truss members were applied in this study, by using “diagrid structure equivalent elastic orthotropic membrane method” (Shi & Zhang, 2019). CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 37 The equivalent membrane modulus for trusses can be calculated from: 𝐸 = 2𝐸𝑑𝐴𝑑𝑠𝑖𝑛 3𝜃 𝑏𝑡 (3.1) 𝐺 = 2𝐸𝑑𝐴𝑑𝑐𝑜𝑠 3𝜃𝑡𝑔𝜃 𝑏𝑡 (3.2) where: 𝐸 is the equivalent modulus of elasticity 𝐺 is the equivalent Shear modulus 𝐸𝑑 is the axial modulus of elasticity the truss beam 𝐴𝑑 is the area of cross section of the truss beam 𝑏 is the width shown in Figure 3.7 𝜃 is the angle of diagrid element, shown in Figure 3.7 t is the equivalent thickness of the membrane element The angle and height h, see Figure 3.7, for calculation are assumed to be 65 degrees and the height over two floors approximate 6.4m. The cross section of the truss element is assumed to be 1m2. Note that the actual equivalent Young’s modulus and shear modulus varies after the design of truss system. The relative bending respective shear stiffness is affected dominantly by the angles of diagrids. Figure 3.7 Illustration of deformation of diagrid element and equivalent membrane element. (a) Axial deformation (b)Shear deformation (Shi &Zhang, 2019) CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 38 3.2.3 Load Only wind loads were included in structural models for geometry study. The wind loads applied on the top was calculated according to EN 1991-1-4. The wind force acting on a structure can be calculated by the expression: 𝐹𝑤 = 𝑐𝑠𝑐𝑑 ∙ 𝑐𝑓 ∙ 𝑞𝑝(𝑧𝑒) ∙ 𝐴𝑓 (3.3) By discretization: 𝐹𝑤 = 𝑐𝑠𝑐𝑑 ∙ ∑ 𝑐𝑓 ∙ 𝑞𝑝(𝑧𝑒)𝐴𝑟𝑒𝑓 (3.4) where: 𝑐𝑠𝑐𝑑 is the structure factor, set to 1 in this study 𝑐𝑓 is the force coefficient, see Appendix A 𝑞𝑝(𝑧𝑒) is the peak velocity pressure at reference height ze, see Figure 3.8 𝐴𝑓 is the reference area of the structure or structure element, see Figure 3.9 Figure 3.8 Distribution of 𝑞𝑝(𝑧𝑒) over height CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 39 Figure 3.9 Illustration of discretization of the reference area. It was observed that the force coefficient and the reference area differ with regards to varying geometries. When the discretization method is applied to handle the peak velocity pressure that varies over height, the distribution of peak velocity and discretized strip area, as seen in Figure 3.9, together will cause different effect on wind forces on different geometries. For building in form of cylinder the peak velocity pressure can affect even the force coefficient, meaning that the force coefficient varies in height. To simplify the calculation this effect was ignored, and force coefficient was calculated with a constant building width 30m. Apart from this the structural factor was set to 1 for all geometries. The detailed calculation of wind loads can be found in Appendix B. The calculation of the force coefficient with regards to different cross sections was conducted in line with EN 1991-1-4, see Appendix A, with mentioned simplifications. The following values was adopted for wind load calculation. Figure 3.1 Force coefficient for modifications of cross section Cross section: roundness radius of corners r Force coefficient 𝑐𝑓 0-3(square with sharp ends) 1.4 3-6 1 6-14 0.7 14-15 (circle) 0.5 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 40 In the calculation with only one individual point load on the top end for evolutionary design, the magnitude of point load was calculated as: 𝐹𝑤 = 𝑐𝑓 ∙ 𝑐𝐴𝑓 ∙ 𝐹𝑏 (3.5) where: 𝐹𝑏 is a basic wind load 𝑐𝑓 is the force coefficient defined in EN 1991-1-4 𝑐𝐴𝑓 is the ratio of the reference area to that of a benchmark reference area, i.e., 30m x 200m Figure 3.10 Illustration of the reference area of a structure and the benchmark reference area 𝑐𝐴𝑓 is defined as: 𝑐𝐴𝑓 = 𝐴𝑓 𝐴𝑓0 (3.6) where 𝐴𝑓 is the actual reference value while 𝐴𝑓0 is the benchmark reference area of a building with dimension 30m x 30m x 200m, i.e., 6000m2, see Figure 3.10 above. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 41 3.3 Finite element models for the general analysis The parametric geometric models built in Grasshopper were then further developed and assembled to finite element models with the help of Karamba 3D. A convergence study was conducted. The models were verified by comparison to hand calculations. 3.3.1 Structural elements The structure includes a tubal structure and slab elements. Both are modeled as shell elements in FEM models as seen in Figure 3.11. Figure 3.11 Structural elements in FE model for the geometry analysis The features of both the tubal structure and slabs were modeled in line with the principles described in Section 3.2.2. Theoretically the stiffness of Element 2 (top slab/floor slabs) should be infinite large. However, excessively large values of this can lead to unreliable results due to numerical errors in the FE calculation. Cross sections and material properties for respective structural elements shown in Table 3.2 were applied showing a reliable structural response. Table 3.2 Cross sections and materials for structural elements Element 1 :External tubal structure Cross section Shell constant [50 cm] Material E: 10480MPa G: 2278MPa Element 2: Slab(s) Cross section Shell constant [1 cm] Material E: 105000MPa G: 52500MPa CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 42 3.3.2 Boundary condition The boundary conditions are presented as follows in Figure 3.12: Figure 3.12 Support conditions where nodes on the bottom are fixed in x-, y- and z- direction. The nodes on the bottom are constrained from displacement in all x-, y- and z- directions. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 43 3.3.3 Loads The loads acting along the x direction and in diagonal were considered. The magnitude of the load was assigned as described in Section 3.2.3. Since element 2, i.e., the roof slab on the top/floor slabs were constructed to behave like a rigid body by assigning an appropriate material, the concentrated loads can be transferred to the shell element uniformly. This was examined by the results where no evident local deformation was observed. The translation of the nodes included in the same slabs are the same. Therefore, the introduction of a single point load on the top edge/each floor slab in the FE model can be sufficient to simulate the global deformations. Figure 3.13 Loads applied in FEM model CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 44 3.3.4 Mesh The quality of the mesh is important in the FE analysis, affecting not only the speed of the calculation process, but also the precision of the result. In Karamba 3D, interaction between different parts of the model is based on the common nodes. In order for the model to perform globally as a whole, it is essential that the tube and the slabs are rigidly connected by common nodes. Therefore, the peripherical nodes on the tubal element located at each floor level should be included in the nodes for slabs as well. The shell elements for the tube and slabs were converted to meshes by the component mesh breps, making it possible to mesh from given points, common nodes that sit at the connection of elements. The size of meshes are mainly controlled by mesh resolution where 1m was used. Meshes for the tubal structural were generated by Mesh Surface which can precisely control the number of quad elements in each local direction. A convergence study for the models with the top slab and the tubal structure was carried out and can be seen in Appendix C2., showed that a mesh size of 1m is sufficient for this type of structure. Figure 3.14 Example of common nodes for the top slab and the tubal structure. The same arrangement was used for other slab elements as well to ensure good interaction between them 3.3.5 Verification of FE models From the results it was observed that the displacements for all the nodes on the top slab are identical, meaning that the slab works as a rigid body as expected. The FE models was verified by comparing end displacement to hand calculation of displacement of cantilever beam with both circular and square constant cross section, subjected to a point load. The verification shows that the result of deflection from hand calculation is slightly smaller than that from FEA. This can be explained by the fact that in hand calculation only bending deformation and not shear deformation was included. The deviation is smaller than 5% for both cases and can be regarded as within tolerance, see Appendix C3. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 45 3.4 Evolutionary design The algometric component Galapagos in Grasshopper was run to get an optimized geometry. 3.4.1 Optimizing algorithm Galapagos is a plug-in tool for generative design. There are two inputs that must be defined in order to run it: The genome: variables that generate various objects for analysis The fitness function: parameter that need to be optimized (maximized or minimized) Figure 3.15 Evolutionary computation component from Grasshopper In the present analysis the genomes are r, s1, s2 and h1, as shown in Figure 3.3 and the fitness target is the end horizontal displacement on the top. There are two different optimizing algorithms inside Galapagos. They are evolutionary solver and annealing solver, where the former one was used. In this process genetic populations in the next generation are generated mainly from a given number “best” populations in the former generation. Settings of the evolutionary computation are presented in figure 3.16. Figure 3.16 Settings for evolutionary computation in Galapagos CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 46 3.4.2 Description of the models used in geometry optimization As mentioned before, the evolutionary design requires large computational resources. The FEM model for evolutionary design, shown in Figure 3.17, consists of one tubal structure (Element 1) and a rigid top slab (Element 2). One single point load on the top (see Figure 3.18) was applied to save the computation time. Shape effect was included, and basic load value applied is 500kN. The final wind load included in calculation is according to equation (3.5). One mistake would be that the loads in the diagonal direction, supposed to be the weak direction, was not included and in theory for the building with square section should have much worse structural performance with regards to lateral displacement. On the other hand, the wind acting on the structure is smaller in the diagonal direction as well and the effect of the shape factor was supposed to be included in the optimization process. Later analysis showed that for all geometries chosen for analysis, including the optimized one from evolutionary design presented in Figure 3.23, the horizontal displacement is not significantly larger when the structure is subjected to wind forces in diagonal direction than that in x direction when the same magnitude of wind forces was applied and the omission of this may not significantly influence the result from this section. Figure 3.17 Structural elements in FEM model for evolutionary design CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 47 Figure 3.18 Loads in FEM model for evolutionary design Due to the lack of the time to update and re-run the study of this part, the materials assigned for this part differ from those in the subsequent general analysis. The material properties presented in Table 3.3 were applied. The material properties adopted for the tubal structure corresponds that of steel. This corresponds with a truss module with a size of approximate 1.6x1.6m and angles of 83 degrees. The assumed h, see in Figure 3.8, is height over 2 floors. Figure 3.3 Material properties for tubal structure and the top slab in evolutionary design Element 1 :Tube Cross section Shell constant [50 cm] Material E: 210000MPa G: 80760MPa Element 2: Slab Cross section Shell constant [1 cm] Material E: 210·1013 MPa G: 105·1013 MPa It is observed that the different assignment of material properties and loads influence the magnitude of deflections but the influence on the relative deflections between different geometries are quite minor, see Appendix C4. The result from this part can be regarded as reliable. Unlike in the general analysis of all geometries, the tubal shell element was transformed to meshed by mesh surface while there is only one set of common nodes aligned with the connection curves between the top slab and the tube, by which the pattern of meshes will be more under control and regularly placed. The density of meshes is controlled by the number of quads elements in each local direction. The quads later were divided into triangular elements. The corresponding mesh size is 1m for the tube. Convergence study can be found in Appendix D2. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 48 Figure 3.19 Mesh density: The mesh resolution was set to 1m The mesh on the top slab was generated by Mesh Breps in order to incorporate the nodes on the top edge of the tube for rigid connection between them. A mesh size of 2m was deemed to be sufficient and therefore applied to further speed up the optimization process. Figure 3.20 Mesh density of top slab. Mesh resolution 2m CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 49 3.4.3 Optimized geometry A total of 11 generations were developed in Galapagos by the evolutionary solver. Figure 3.21 Fitness over generations The fitness over generations in Figure 3.21 above showed that the fitness values among populations is approaching convergence already at generation 3. The genetic properties and the fitness are stable over generation 3- 7. In generation 8 the extreme fitness on the lower side drops dramatically while the rest of the fitness values among the same generation are still centralized. By checking the value and the visualized model in Rhino, the structure deforms in the opposite direction to the force load, which might come from the singularities, leading to invalid results. The fault source is of large probability due to the extreme large elastic modulus of the top slab. The parameters r, s1, s2 and h are fluctuating around a specific converged value. The corresponding parameters and results after a careful mesh study are: Table 3.4 Parameters and results for the optimized geometry from evolutionary design Parameter Result r [m] s1 [-] s2 [-] h1 [m] Displacement at top floor [m] Average displacement of all floors [m] Rental area [m2] 6 1 0.8 44 0.001199 0.000457 48054 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 50 Figure 3.23 The optimized geometry from evolutionary design The aim of this study is only to find optimized geometries through a mass screening of parametric geometries. For later comparison this optimized one will be further analyzed with the same material and loads and together with? all other geometries to ensure the comparability. Some of the examples extracted from different generations can be found in Appendix C5. CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 51 3.5 Typical geometries A series of geometries were created in line with the method described in Section 3.1. These geometries are numbered in Figure 3.24 where those in th