Timber Footbridge in Wendelstrand An Iterative Design Process Combining Architecture and Engineering Master’s thesis in the Master’s Programme Structural Engineering and Building Technology ALEXANDER ANGRÉN MARIA BRUZELL ROLL Department of Architecture and Civil Engineering Division of Structural Engineering Lightweight Structures in collaboration with Architecture and Engineering CHALMERS UNIVERSITY OF TECHNOLOGY Master’s Thesis ACEX30 Gothenburg, Sweden 2021 MASTER’S THESIS ACEX30 Timber Footbridge in Wendelstrand An Iterative Design Process Combining Architecture and Engineering Master’s Thesis in the Master’s Programme Structural Engineering and Building Technology ALEXANDER ANGRÉN MARIA BRUZELL ROLL Department of Architecture and Civil Engineering Division of Structural Engineering Lightweight Structures in collaboration with Architecture and Engineering CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 2021 Timber Footbridge in Wendelstrand An Iterative Design Process Combining Architecture and Engineering Master’s Thesis in the Master’s Programme Structural Engineering and Building Technology ALEXANDER ANGRÉN MARIA BRUZELL ROLL © ALEXANDER ANGRÉN & MARIA BRUZELL ROLL, 2021 Examensarbete ACEX30 Institutionen för arkitektur och samhällsbyggnadsteknik Chalmers tekniska högskola, 2021 Department of Architecture and Civil Engineering Division of Structural Engineering Lightweight Structures in collaboration with Architecture and Engineering Chalmers University of Technology SE-412 96 Göteborg Sweden Telephone: + 46 (0)31-772 1000 Cover: Model in scale 1:20 of the proposed bridge design. Department of Architecture and Civil Engineering Göteborg, Sweden, 2021 I Timber Footbridge in Wendelstrand An Iterative Design Process Combining Architecture and Engineering Master’s thesis in the Master’s Programme Structural Engineering and Building Technology ALEXANDER ANGRÉN MARIA BRUZELL ROLL Department of Architecture and Civil Engineering Division of Structural Engineering Lightweight Structures in collaboration with Architecture and Engineering Chalmers University of Technology ABSTRACT Wendelstrand is a new residential area in Mölnlycke planned with sustainability and wood as overall concepts. Adjacent to the area is the lake Landvettersjön, which today is cut off by a highly travelled country road. A footbridge over the road will link the lake and Wendelstrand and create a safe crossing for both residents and visitors to the area. In this thesis a proposal for such a footbridge is developed with emphasis on the relation to the timber and sustainability-concept of Wendelstrand, as well as the integration between architectural and structural qualities. An iterative design method is used to examine and develop possible solutions for a bridge proposal. Reference projects of existing timber bridges are used in the development of feasible bridge designs. Site-specific prerequisites, intentions from the developer Next Step Group, regulations from the local municipality and the Swedish Transport Administration are considered and integrated into the design. The proposal suggests a straight bridge across the road leading to a floating structure on the lake. The bridge supports are integrated into the superstructure and designed to limit the impact on the site. Physical models are used to verify the assembly process, as well as the structural concept and spatial qualities of the design. The design corresponds with Swedish and European standards concerning structural response. Simplified calculations are performed to verify the global and detailed design of the design proposal. A solution for an efficient structure in addition to an appealing architectural appearance is achieved by applying the principles of active bending in the superstructure. Keywords: wood, timber, bridge, footbridge, conceptual, design, architecture, structure, active bending, Wendelstrand I Contents 1 INTRODUCTION 1 1.1 Aim 1 1.2 Method 2 1.3 Limitations 3 1.4 Outline 3 2 DESIGN PROCESS 4 2.1 Step I: Contextualisation 4 2.2 Step II: Conceptual design phase 4 2.2.1 Intuitive phase 4 2.2.2 Intentional phase 4 2.2.3 Evaluation phase 5 2.3 Step III: Preliminary design phase 5 2.4 Step IV: Final design phase 5 3 STEP I: CONTEXTUALISATION – REFERENCE STUDY 6 3.1 Categorisation of timber bridges 6 3.2 Reference projects 8 3.2.1 Neckartenzlingen Pedestrian Bridge 8 3.2.2 Punt Staderas 11 3.2.3 Fussgängersteg Geheidgraben 14 4 STEP I: CONTEXTUALISATION – SITE 17 4.1 About the site 17 4.2 Clients demands 19 4.3 Site-specific boundary conditions 19 4.4 Summary of contextualisation 20 5 STEP II: CONCEPTUAL DESIGN 22 5.1 Design criteria 23 5.1.1 Spatial qualities 23 5.1.2 Bridge qualities 23 5.1.3 Structural concept 24 5.2 Intuitive phase 25 5.3 Intentional design phase 28 5.4 Evaluation phase 33 6 STEP III: PRELIMINARY DESIGN 35 II CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 6.1 Final design proposal 35 6.1.1 Overall bridge design 35 6.1.2 Structural concept 36 6.1.3 Experiment with active bending 37 6.2 Model development of structural concept 40 7 STEP IV: FINAL DESIGN 42 7.1 Bridge dimensions 45 7.2 Input data 47 7.2.1 Partial factors 47 7.2.2 Material properties 49 7.2.3 Loads 50 7.2.4 Load combinations 54 7.3 Global design 56 7.3.1 Superstructure in ULS 56 7.3.2 Superstructure in SLS 60 7.3.3 Support in ULS 61 7.3.4 Dynamic analysis 64 7.3.5 Torsional stiffness 68 7.3.6 Moisture induced deformation 70 7.3.7 Floating structure 71 7.4 Local design 72 7.4.1 Prestressed connections 72 7.4.2 Lamella joints 75 7.4.3 Column to superstructure connection 78 7.5 Production 78 8 DISCUSSION 80 8.1 The proposed bridge 80 8.2 The design process 81 8.3 Suggestions for future work 82 9 CONCLUSION 84 10 REFERENCES 85 APPENDIX CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 III IV CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 Preface In this study, an investigation of structural design for a site-specific bridge has been carried out. Our aim was, in addition to fulfilling the requirements of the client, to combine principles from structural engineering and architecture. We want to aspire to rethink and challenge standard solutions to contribute to the development of timber bridge design. First, we would like to thank our examiner Robert Jockwer for supporting our ambition for the bridge design, for the enthusiasm during the process, and for contributing with important knowledge regarding structural design of timber structures. Our supervisor Prof. Karl-Gunnar Olsson is highly appreciated for his understanding and commitment to combine architecture with engineering, for emphasising on the importance of reference analysis, and for daring us to think boldly. Thanks to Emelie Silverterna and Joakim Garfvé at Next Step Group, the developer of Wendelstrand, for the opportunity to collaborate and for insight in site-specific data, as well as valuable feedback on the design proposals. We would like to give thanks to Brosamverkan for the scholarship, who financed the material usage for our physical models. The scholarship enabled us to build models to verify the structural concept, validate the assembly method, and demonstrate the relation between the bridge design and Wendelstrand. Thanks to our opponent Vera Sehlstedt for numerous discussions and for support through the process. At last, we would also like to address a special thanks to Burkard Walther, Miebach Ingenieurbüro, Camathias SA, werk1, and Wendelstrand for allowing us to use their photographs and illustrations in the thesis. Göteborg June 2021 Alexander Angrén Maria Bruzell Roll CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 V VI CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 1 Introduction Close to Mölnlycke, outside Gothenburg, the Bråta gravel pit will be phased out and changed into a new residential area named Wendelstrand. This will be Northern Europe’s largest residential area in wood and is developed by Next Step Group (E. Silverterna, personal communication, January 29, 2021). The area is situated right next to the lake Landvettersjön but the two are today separated by Boråsvägen, a heavily travelled country road (Wendelstrand, 2018). A timber bridge will facilitate a safe crossing, while the structure itself can be used to enhance the innovative ideas of Wendelstrand. Interplay between nature and the buildings is an important part in the design of Wendelstrand. An approach in the direction of environmental sustainability is taken by choosing timber as the main structural material of the buildings (Garfvé, n.d.). In 2017 the Swedish parliament decided that Sweden should reach zero net-emissions of greenhouse gases by 2045 at latest, proceeding with negative emissions (Proposition 2016/17:146 Ett Klimatpolitiskt Ramverk För Sverige, 2017). Life Cycle Analysis studies show that when comparing timber frame buildings with non-timber alternatives, including concrete and steel, the former requires lower energy and releases less greenhouse gases (Dodoo et al., 2016). In other words, one way to reduce the environmental impact of structures is to increase the use of timber. When designing timber bridges, emphasis must be laid on durable detail design preventing the superstructure from weathering, enabling the wood to dry out, and protecting the end-wood from exposure. Due to water absorption and the risk of rot, timber bridges need careful planning in order to limit the need for maintenance and frequent inspections than a bridge made out of steel or concrete (Pousette & Fjellström, 2004). Therefore, the maintenance costs of timber bridges are higher. On the other hand, timber has a high strength-to-weight ratio (Svenskt Trä, 2009). Consequently, a lighter construction can be achieved with less foundation work. However, the construction of timber bridge structures has evolved throughout the years. By learning from failure in timber bridges due to lack of careful detailing, poor choice of material, or lack of maintenance and cleaning, improved designs of modern timber bridges can be made (Pousette & Fjellström, 2004). 1.1 Aim The aim of this thesis is to develop a design proposal for a timber footbridge in Wendelstrand. Focus lies on combining structural engineering knowledge with architectural visions. The following research questions are formulated: “In what way can a bridge proposal be achieved that meets the requirements and ideas of the client Next Step Group for the planned residential area Wendelstrand?” “In what way can an iterative design method be used to develop a structural concept that enhances both engineering solutions and architectural qualities?” CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 1 1.2 Method An iterative design method consisting of four stages is applied to answer the two research questions. In the first stage of contextualisation, an overview of the task is achieved by research of reference projects and identification of site-specific requirements. The second stage is a Conceptual design phase where possible solutions are examined and developed into one suitable proposal. This design proposal is further developed in the third stage and validated in the fourth and last stage. Figure 1.1 illustrates the design process from start to end. Figure 1.1 Design process. To ensure fulfilment of the thesis aim, a set of evaluation criteria are formulated. These make the transition between the design phases, to ensure a correspondence between the final proposal and the thesis aim. The evaluation criteria include both external demands and requirements, and design criteria. The latter define expectations regarding both the visual appearance and the structural performance of the footbridge structure. Figure 1.2 illustrates the evaluation criteria, which are further explained in Chapter 5. Figure 1.2 Evaluation criteria. 2 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 1.3 Limitations The project is limited to a specific site, Wendelstrand, with real boundary conditions that the bridge design must comply with. Next Step Group is considered as the client, to establish a realistic context for the design project. They have stated clear demands regarding the bridge which will be prioritised in the design. The bridge design must consider the drastic elevation change in the landscape, as well as fulfilling non- negotiable requirements regarding shore protection, geological conditions, and standards from the Swedish Transport Administration and Eurocode. The structural response of the final bridge design is roughly estimated, and dimensions are based on conservative calculations and available product dimensions. The chosen structural concept is based on experiments with physical models, combined with understanding gained from analysis of reference projects. A thorough investigation of the theory is outside the scope. Detailed investigation for optimisation of the connections and foundation elements is not performed. Moreover, the assembly method will only be tested in physical models, while considering the limitations of production, transport, and site conditions. To limit the thesis, no Life Cycle Analysis of the bridge proposal is performed. Groundwork and stabilisation of the hill is not considered in this thesis. 1.4 Outline In Chapter 2 Design process, the design methodology with its different steps is further explained. In Chapter 3 Step I: Contextualisation – Reference study, aspects when designing a timber bridge is stated and different timber bridge structures are introduced on a general level, as well as a detailed analysis of reference projects. Chapter 4 Step I: Contextualisation – Site, contextualises the site Wendelstrand and identifies the client’s and the Swedish Transport Administration’s requirements and specifications. This will work as input for the design process. In Chapter 5 Step II: Conceptual design, the iterative design process is initiated with a divergent exploration for possible concepts. Three possible solutions are the outcome of this process, which after consultation with the client and evaluation in relation to a set of criteria, are developed into one proposal. This proposal is further refined with a suitable structural concept and rough dimensions in Chapter 6 Step III: Preliminary design. Thereafter, the structural concept is validated through calculations and the assembly method confirmed by a physical scale model in Chapter 7 Step IV: Final design . At last, Chapter 8 Discussion, gives a critical evaluation and discussion of the work. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 3 2 Design process Iterative design is applied as methodology for the design study. The method is described in fib Model Code 2010 (International Federation for Structural Concrete, 2013) and adapted in the graduate course BOM170 Structural Design (created by Björn Engström and Morten Lund). The thesis methodology is based on the latter. To initiate the design process, the task is established in a specific context. A large variation of improvised, possible solutions is developed. By an evaluation considering identified demands and contextual requirements, the proposed solutions can be narrowed down to one suitable design proposal. This is in turn developed through a precise and detailed design by the means of structural analysis and physical models. 2.1 Step I: Contextualisation The intention of a contextualisation is to create an overview over the challenge and define the limitations of the task. This is achieved in two stages: firstly, an understanding of the current task is achieved by analysing solutions of similar challenges. Secondly, the specific problem is clarified by identifying challenges related to the site. This involves identification of different aspects of demands and definition of external requirements. The aim of the contextual stage is to create a foundation for the assessment of solutions in the following design process. An analysis of reference projects helps broaden the understanding of different specific structures. These are collected in a data bank to identify possible solutions in existing structures that demonstrate their feasibility. These references support the development of design proposals in the Conceptual design phase. 2.2 Step II: Conceptual design phase The Conceptual design phase is divided into three main steps: Intuitive phase, Intentional phase, and Evaluation phase (M. Plos, personal communication, September 2, 2020). By starting off in a broad and divergent generation of improvised concepts, qualities from these can then be merged into new and more specific concepts. The ideas are gradually developed and evaluated throughout the process to result in a final concept which fulfils the initially stated project intention. 2.2.1 Intuitive phase The aim of the Intuitive phase is to generate as many different concepts as possible and investigate any idea related to the general context and demands. Every idea is possible in this phase, without any critical review or evaluation. 2.2.2 Intentional phase The next phase aims to concretise the contextual requirements and to take the client’s demands into account in the development of the structural design. This design is in turn evaluated in relation to the governing demands and requirements. As a result, the improvised concepts are developed and modified into a few, potential concepts. 4 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 2.2.3 Evaluation phase The last part of the Conceptual design phase includes a review of the contextual demands. Conflicting demands are identified and prioritised after feasibility, fulfilment of client’s intention and structural complexity. With a specified set of governing design criteria and demands, the most suitable structural concept can be defined and then brought into the next phase, the Preliminary design phase. 2.3 Step III: Preliminary design phase Based on the result in the Evaluation phase, the final proposal is developed into a complete structural concept. This includes identification of the structural behaviour, determination of material and preliminary sizing of main structural elements. Reference projects with similar challenges are used to develop the solutions. 2.4 Step IV: Final design phase The Final design phase includes a verification of the chosen structural concept. By detailed design of the structural elements, and calculations of the overall behaviour, the feasibility of the proposed concept can be proven. Physical models, in different scales, are also used to verify the structural design concept as well as the assembly method of the superstructure. This will allow for a verification of the detailed design as well as the relation between the overall structural design and its context. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 5 3 Step I: Contextualisation – Reference study As a literature study, reference projects of existing bridges are collected in a database and analysed. The reference projects will act as guidance for proper detailing, technical solutions, and ideas for structural concepts. An understanding of other qualities such structures hold is also gained from the reference analysis, such as the relation to the surroundings, consideration of pedestrians, or implementation of other functions. As these qualities are of subjective manner, the bridges are not analysed for this. Instead, it is used as inspiration in the next design step. Timber bridges make most of the studied bridges, but bridges in other materials are analysed as well. The reference data bank is illustrated in Figure 3.1, while the complete content can be found in Appendix A. Figure 3.1 An overview of the complete bridge database. 3.1 Categorisation of timber bridges The architectural appearance of a bridge is strongly linked to its structural concept. Each concept hols different benefits and limitations regarding maximum span length, superstructure dimensions, required foundation capacity, visual impact on the surroundings, material usage, and cost. The maximum span lengths vary considerably between the different bridge structures. Figure 3.2 illustrates different types of structural concepts applicable on timber bridges, categorised by span length. 6 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 Figure 3.2 Different structural bridge types categorised by length. Different structural systems carry loads in different ways. For example, a slab and beam bridge carry the load by bending, while a truss bridge globally transfers load by axial forces and bending as the local load transfer. A king post truss bridge uses bending, tension, and compression to distribute the load to the foundation, while a superstructure in a stressed-ribbon bridge works only in tension. Depending on required span length, desired visual impact and specific boundary conditions, a suitable structural concept can be applied and adapted to the specific site. Approximate span length and beam height for different structural systems are shown in Table 3.1. Table 3.1 Approximate span lengths and beam heights for different timber bridge construction types. A compilation from (Gustafsson et al., 1996) and (Svenskt Trä, n.d.). Construction type Span length, L [m] Beam height, h [m] Slab bridge 20-30 (15) L/20-L/30 Beam bridge - Simply supported beam 8 L/15-L/20 - Simply supported round wood 10 L/15-L/25 - Simply supported glulam beam 30 (40) L/15-L/20 - Underspanned beam 10-50 L/8-L/12 - Combined cross-sections 50 L/15-L/20 - Timber plated structure 50 L/8-L/12 Strut frame bridge 40 L/15-L/20 Framework - Truss 30-75 L/12-L/18 - Beam or glulam 20-50 L/20-L/30 CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 7 King post truss bridge 10-50 L/4-L/12 Arch bridge - Arched walkway 10-60 L/25 - Hung walkway 20-30 L/4-L/6 - Post-supported walkway 20-50 L/4-L/6 Truss bridge 100 L/10-L/15 Stressed-ribbon bridge 20-100 L/120 Cable-stayed bridge 20-100 L/4-L/8 Suspension bridge 20-100 L/4-L/8 3.2 Reference projects In the following sub-chapters, an analysis of three structurally different and inspiring timber bridges are presented. The different studied aspects are: - Structural concept - Vertical load distribution - Horizontal load distribution - Rotational stability - Point load - Foundation - Principle connections - Production - Assembly - Material - Durability 3.2.1 Neckartenzlingen Pedestrian Bridge In Neckartenzlingen in south-west Germany a 96 m long cycle and pedestrian bridge spans over the Neckar river. The design and structure are created by Ingenieurbüro Miebach and completed in 2017. The bridge deck is 3 m wide and there are three spans where the mid span is 44.5 m and the other two 25.7 m (Miebach, n.d.). Figure 3.3 shows the continuous superstructure with its two parallel glulam beams. 8 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 Figure 3.3 Neckartenzlingen Pedestrian Bridge (Burkhard, 2017). Used with permission. © www.hochbau-fotografie.de Structural concept The Neckartenzlingen pedestrian bridge is a cantilever bridge with Gerber joints. These hinges transfer shear forces only and are located between the supports where the bending moment is close to zero. The superstructure consists of two parallel block- glued and bent glulam beams that cantilever out from the supports. To utilise the material, the cross-section height decreases towards the mid of the span. The glulam blocks are 2.1 m wide in the top layer and tapers down to 0.8 m at the bottom (Brandt, 2018). The bridge stretches out in a gentle S-shape. Vertical load distribution Vertically distributed load is transferred in the glulam blocks through bending to the supports. Due to reduction of the cross-section height in the span midpoint, the self- weight in the middle is also reduced. The largest cross-section height is found over the supports where the bending moments are the largest. Horizontal load distribution Transversal horizontal distributed load is handled by horizontal bending of the glulam blocks, and through axial forces in the transversal beams underneath the concrete slabs. The fact that the bridge is slightly S-shaped also contributes to the horizontal load capacity. Longitudinal horizontal loads are transferred by shear in the adhesive between the glulam block layers, down to the foundation and through the columns. Rotational stability The S-shape of the bridge provides global rotational stability. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 9 Point load A vertical point load is longitudinally and transversally distributed by the concrete slab and transversal beams down to the two main parallel glulam beams. From there the load is transferred through bending to the supports and foundation. Foundation Concrete piers support the superstructure on each side over the river, while concrete foundations anchor each bridge end. The glulam beams are connected to the supports and foundations by steel profiles that penetrates the structure (Brandt, 2018). The supports are considered as simply supported. Rotational movement is allowed at each bridge end. Principle connections The continuous beam is simply supported over the supports, which results in that no bending moment must be led down to the supports. The prefabricated bridge parts are connected to each other by Gerber hinges. These are located where the bending moment along the bridge is close to zero. The glulam blocks are individually fastened to each other with adhesives. Steel profiles connects the two parallel glulam beams. An exploded view illustrates the individual bridge parts in Figure 3.4. Figure 3.4 Exploded view drawing of the Neckartenzlingen Pedestrian Bridge (Ingenieurbüro Miebach, 2017). Illustration created by Ingenieurbüro Miebach. Used with permission. Production Both the timber used for the glulam beams and the firm that manufactured the bridge components are local. Due to manufacturability the bridge cross-section has two parallel glulam beams. This creates a space where installation and electricity cables can be hidden (González, n.d.). The prefabricated wooden parts were transported to the site in different parts. The concrete slabs were pre-cast and transported to the site as well. Assembly The design allowed for sensible transport dimensions as well as a simplified assembly. The whole bridge took three days to assemble (Brandt, 2018). Two mobile cranes were used to lift the parts, standing on each side of the river. First the two outer spans were 10 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 lifted to its positions over the supports. Following, the beam over the mid span was lifted and connected with a Gerber hinge (Holzindustrie, n.d.). Material The superstructure is made of glulam beams while a pre-cast concrete slab with anti- slip surface constructs the decking. The railing and wires consist of stainless steel and the handrail is made of acetylated timber (Miebach, n.d.) Durability The 13 cm thick pre-cast concrete slab on top of the glulam beams gives a watertight protection (González, n.d.). The superstructure is protected by a 30 cm overhang of the concrete slabs. In addition, the glulam beams are tapered with 30° angle inwards which prevents rainfall from reaching the structure itself. Drainage channels made of steel are inserted under the concrete slab joints. To further protect the timber, a thin coat of glaze is applied (Brandt, 2018). 3.2.2 Punt Staderas In the municipality of Laax, Switzerland, the slender bridge Punt Staderas spans over the country road Oberalpstrasse N19 to create a safe passing for bicyclists and pedestrians. The bridge design and structure are made by Walter Bieler with help of Stephan Berni and was completed in 2015 (von Büren, 2016). The main idea was to reduce the amount of wood and use the same cross-sectional dimension in all structural elements. This resulted in a slender superstructure with a total length of 115 m with nine spans of varying lengths. The longest span is found over the road and measures 24 m. The pathway is 2.5 m wide which allows enough space for both bikers and pedestrians, and has a slope of 6% (Ekwall, 2017). Figure 3.5 shows the largest span from below and the integration between the V-shaped supports in the structure. Figure 3.5 Punt Staderas from below (Camathias SA, 2015b). Used with permission. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 11 Structural concept The superstructure is a Gerber girder built up as a Vierendeel girder. The webs in the Vierendeel frame builds up rectangular frames together with the upper and lower chords and with joints that can transfer bending moments into the webs. In contrary to a triangular truss with pinned connections to the upper and lower chord where the shear force is axially transferred through the diagonals (Wickersheimer, 1976). By using a Vierendeel girder a slender cross-section is achieved with a static height of 640 mm. The grid consists of two longitudinal layers with four beams each and an intermediate layer with transversal cross beams every 1.05 m (Ekwall, 2017). The concept is governed by the aim of using the same cross-sectional dimension for all structural elements, namely 160-by-240 mm and 14.5 m long (Guetg, n.d.). Since the span over the road is 24 m (longer than 14.5 m) glulam timber beams are used to ensure sufficient bearing capacity. The supports are V-shaped in both directions, with two considerable benefits. Firstly, the span lengths are decreased. Secondly, with an inwards V-shape, the bridge deck can provide weather protection of the support elements. In addition, the outer support beams stretch beyond the deck and enhance the railing design. The V- shaped supports improves the structural performance in two aspects. Firstly, the span lengths are decreased. Secondly, with a V-shape the V-shape allows for weather protection of the support elements by the bridge deck. The outer support beams stretch beyond the deck and enhance the railing design. Vertical load distribution A vertical, uniformly distributed load is lead through the Vierendeel girder to the supports by compression in the upper longitudinal layer and tension in the bottom longitudinal layer. At the supports the load is transferred through compression down to the foundation. Over the largest span the load is carried by bending moment through the glulam beams to the supports. Horizontal load distribution Horizontal capacity is provided by the transversal beams of the Vierendeel girder. A modest S-shape in the horizontal direction of the bridge path enhances the horizontal stability of the structure. The V-shaped supports are ordered in a 3-by-6 array with a steel bracing in the transverse direction. This geometry and additional bracing provide additional horizontal stability. For the longitudinal horizontal loads, the geometry of the V-shaped supports provide stability. Rotational stability The steel cross bracing inside the V-shaped supports provides global rotational stability (Figure 3.5). The local rotational stability comes from the stiff Vierendeel girder and connections between the members. With the use of a total of 2000 screws, each 80 cm long, the wooden elements are joined together (Oertli, 2018). Point load A vertical point load is transferred through the bridge deck down to the Vierendeel girder. The load is then transversally distributed by the cross beams to the longitudinal beams in the upper and lower layers of the girder. Then the force is carried by tension and compression through the girder to the supports, which in turn transfer the load through compression down to the foundation. 12 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 Foundation Reinforced concrete abutments are placed at the two landing points. The V-shaped supports stand on concrete blocks. Principle connections Since the same cross-sectional dimension is used for all structural members, they match very well when connecting them together. The beam elements are assembled with fully threaded screws at an angle, which provides a shear proof behaviour. The connection stiffness was tested successfully at EMPA in Dübendorf beforehand (Oertli, 2018). The connections in the superstructure are covered by the bridge deck, while all other exposed connections are covered by boards that lead water away from the connection. Figure 3.6 shows the protecting boards and their influence in the visual appearance of the bridge. Figure 3.6 Side view of Punt Staderas (Camathias SA, 2015a). Note the cover board that leads water away from the structure. Used with permission. Production The bridge consists entirely of locally cut wood. Walter Bieler personally assisted the forester in the search for suitable trees that could provide 14.5 m long beams (Guetg, n.d.). Both the Vierendeel and glulam girders were prefabricated and then transported to the site for assembly. The foundation abutments were casted on site. Assembly First the V-shaped load bearing supports were installed on the foundations. Thereafter the prefabricated girders were lifted to their position. During the assembly of the glulam beam spanning over the road, the road was closed and the beam could be lifted to its position (Standardname, 2015). The girders were then connected by screws. To make sure correct positioning of the structural parts, laser tools were used (Oertli, 2018). Material Originally the idea was to build a wooden bridge in larch with material from the immediate vicinity, but since spruce is the most widespread species in the area, it was CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 13 chosen as the main structural material. Larch was instead used in the exposed details. For the main span, glulam beams of strength class C24 are used. The paving consists of mastic asphalt. Durability The expected life span for the bridge is set to 80 years and the wood will start to turn grey until it finally becomes silverish (Oertli, 2018). Careful detailing has ensured a properly airy structure. Covering boards acts like small canopies for water where the water can drip off, and sufficient distance around allow the structure to dry out. The superstructure is protected by the covered pavement and the supports are tilted inwards. Weather-exposed surfaces consists of larch while the protected wood is spruce (Guetg, n.d.). The exposed railing and canopies that protect the superstructure are easy to replace. 3.2.3 Fussgängersteg Geheidgraben In Olten, Switzerland, a conceptually interesting bridge is located. The design is made by the architecture firm werk1 with the engineers of Makiol Wiederkehr and was completed 2013. The bridge spans a little bit more than 7 m over a ditch which works as a retention basin, meaning that there seldom is any water in the ditch except from many days of rainfall (Makiol Wiederkehr, 2014b). The architectural idea is to have a wave-like pattern that is implemented both on the walking deck and railing, which can be seen in Figure 3.7. Figure 3.7 Fussgängersteg Geheidgraben with its wave-like superstructure and railing (Makiol Wiederkehr, 2014b). Used with permission. Structural concept Statically, the bridge works as a simply supported beam bridge where the wavy decking is the main structural part. The longitudinally beams in the superstructure are actively 14 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 bent, which creates a slab with good bending capacity both vertically and horizontally. It is assumed that the wavy handrails also to some extent contribute to the vertical bending moment capacity due to its height and actively bent members. As a simplification, the handrails can be read as a truss structure. Vertical load distribution Globally, a vertically distributed load is transferred through the wavy beams to the foundations by bending moment. The cross-sectional height of the wavy beams in the bridge deck provides the governing bearing capacity. Horizontal load distribution To create the waves, washers and nuts are used to spread and pull together the floorboards. They are mounted on two transversal steel rods above each other in a grid with 0.6 m spacing (Makiol Wiederkehr, 2014a). Together this makes a homogenous slab, which provides the horizontal capacity. Rotational stability The bridge has a relatively short span and a broad width, which provides global rotational stability. The railing with its integrated steel posts gives torsional stability. Point load A point load is transferred through one or two wavy beams towards the transversal steel rods, where the load then is transversally distributed to adjacent wavy beams and then finally longitudinally transported to the foundations. Foundation The bridge is resting on concrete foundations. Steel bearings of LNP profiles (120/120/10) connects the foundation and the timber superstructure. On one end of the bridge there are elongated holes in the connection between support and beam, which allows for swelling and shrinkage. Rotation is allowed in one bridge end. The steel bearings on respective side of the ditch have a height difference of 0.6 m (Makiol Wiederkehr, 2014b). Principle connections Washers and nuts provide the necessary distance between the thin beams to create a wave pattern in the deck. The distancers are mounted on threaded steel rods in tension. The railings are constructed by laying boards with similar wave pattern created by distances, see Figure 3.8. RRW steel profiled posts penetrate the laying boards. The distances also protect the steel posts. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 15 Figure 3.8 Railing structure with distancing (Makiol Wiederkehr, 2014a). Used with permission. Production The bridge is prefabricated, and the decking and railings are preassembled separately. The curvature of the waved boards is made by screwing the nuts on the threaded bars to wanted position, and then the nuts themselves work as distancing to the next layer board. Assembly The preassembled decking and railings are joined together, and then the bridge is lifted to its position on the concrete foundation in one piece (Makiol Wiederkehr, 2014a). Material Decking and railing is made of massive rough-sawn oak boards. Steel is used for the threaded rods, nuts, washers, posts in railing and supports. The foundation abutments are in concrete. Durability Due to the wavy character, water can drip through the structure, and the structure is also easily dried. A consequence of the weather exposed structure is an aged appearance with rust and weather torn wood (Makiol Wiederkehr, 2014a). The steel posts in the railing are protected by the distancing steels parts. An extra board is mounted on top of the railing to cover the fastenings. The fact that pedestrians walk directly on the unprotected superstructure, will shorten the service life of the bridge. It is also difficult to replace single elements without demounting the whole bridge. 16 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 4 Step I: Contextualisation – Site When designing a bridge, there are several aspects to consider. The context of the specific site needs to be analysed and studied such as required span lengths, foundation possibilities, and surrounding scenery and buildings, and connection existing paths. When these aspects are formulated, a structural concept can be explored and developed. This concept should fulfil the identified requirements connected to the context. In the following chapter, general demands based on client’s expectations are formulated. These consider both the visual appearance of the bridge, the purpose of the footbridge, and the feasibility of the proposal. Geotechnical and topographical conditions outline the site-specific context. These are in turn are affected by requirements stated by the municipality and Swedish Transport Administration. The above-mentioned aspects are identified by a study of public documents and site-specific investigations. An early visit to the site contributed to a perception of the current site concerning characteristics in the terrain, scenery, and surroundings. The project background, main challenges, and governing demands are formulated in this chapter. 4.1 About the site The chosen site is a planned development area named Wendelstrand, situated east of Mölnlycke outside of Gothenburg (Figure 4.1). The current gravel pit will be transformed into a residential area by 2026-2030, initiated by Next Step Group. Wendelstrand is planned to hold 850 new residences and public buildings, such as a nursery school, elderly home, and shops (Härryda Kommun, 2020). The community building, named Lakehouse, will be in the centre of the area, housing services such as restaurants, co-working offices, and gym. The building outlines the main square of the area, which holds the most important social services in the area (Härryda Kommun, 2020). Figure 4.1 The specific site in relation to Gothenburg (Google maps, 2021). CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 17 The general intention of Next Step Group for Wendelstrand is to develop a sustainable residential area, where the buildings adapt to the landscape and the architecture involves a high presence of timber (Härryda Kommun, 2020). A view over Landvettersjön and adjacent nature reserve is found in the southeast corner of the main square, while the roof of Lakehouse offers the most spectacular view, see Figure 4.2. The architect aims to make Landvettersjön present and visible throughout the whole area. With an increasing building height further away from the lake, the residents can see the lake from their apartments, see Figure 4.3. Figure 4.2 Birds view of Wendelstrand (Wendelstrand, n.d.-b). Photo: Snøhetta. Used with permission. Figure 4.3 Wendelstrand from Landvettersjön (Wendelstrand, n.d.-a). Photo: Snøhetta. Used with permission. 18 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 4.2 Clients demands As the architecture of Wendelstrand enables a view of the lake anywhere in the area, the only missing link to the lake is the physical one. The client, Next Step Group, envisions a bridge that connects Lakehouse and the main square of Wendelstrand with the lake, both physically and visually. The road Boråsvägen must be considered, with the priority of making the lake accessible for the residents and visitors of Wendelstrand. Furthermore, the client wishes for a bridge structure with a high presence of timber that correspond with the sustainability profile of the development project. In addition to this, the client envisions a design that offers qualities other than merely a connection between A and B, that can be combined with the expected experience of Wendelstrand and Lakehouse. 4.3 Site-specific boundary conditions The site-specific boundary conditions mainly concern the surrounding topography. Public regulations stated by the Swedish Transport Administration, and site-specific regulations regarding shoreline adds to the list of restrictions that must be considered in the bridge design. The required height clearance of 5.3 m over the road is stated by the Swedish Transport Administration (Härryda Kommun, 2019). Furthermore, the supports must be placed with a clearance of 2 m from the walkway (Trafikverket, 2021). The county administrative board in Västra Götaland has defined a shore protection area, stretching 100-200 m from the shore of Landvettersjön. The purpose of the expanded restriction is to keep the shorelines clear from permanent structures to enable public access to the water, in addition to protecting the geological conditions in and around the lake (Sektorn för samhällsbyggnad, 2020). The area between Landvettersjön and Wendelstrand lies within the shore protection and is thereby protected. Early site- investigations on behalf of Next Step Group has identified the possibility of excluding a short strip of land along the load, to make space for possible bridge foundations (E. Silverterna, J. Garfvé, personal communication, March 11, 2021). The further development of the timber footbridge will consider this explicit area as excluded from shore-protection regulations, that is, a 4.5 m wide strip along the road. As a compromise, the bridge design is limited to a maximum of two landing points on the lakeside of the road. In the initial planning process of the area of Wendelstrand, several technical investigations were performed on behalf of Härryda municipality. Important for the footbridge design is the investigation of geotechnical conditions of the site. The investigation proposes the supports to be constructed as either ground slab, drill pipe or a combination of both as the soil layers generally consist of sand with elements of silt, gravel, and solid rock (Norconsult, 2018). Furthermore, in the latest local plan, a risk of minor landslides in the steepest terrain southeast of the area right next to Boråsvägen, is identified (Härryda Kommun, 2020). The evaluation has estimated it as a non-critical situation with a ten-year risk. However, stabilisation of the terrain in question is outside the scope of the current project. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 19 Governing topographical conditions are identified as the following and illustrated together with the requirements from the Swedish Transport Administration in Figure 4.4: - 19 m height difference between Lakehouse and Landvettersjön. - Steep slope in the terrain (21°). - Boråsvägen has a varying width of 7-11 m including pavement. - Required height clearance of 5.3 m and 2 m width clearance. Figure 4.4 Governing boundaries of the site. 4.4 Summary of contextualisation From the site contextualisation, the following demands and requirements are identified. The public requirements are considered non-negotiable and must be fulfilled for the proposal to be feasible. In addition, to achieve an attractive bridge proposal, the client’s demands must be met. - Clients demands - Physical and visual relation between Wendelstrand and Landvettersjön - Make the lake accessible for residents as well as visitors - Footbridge design with a high presence of timber - Proposal for additional qualities in the design - Public regulations - The Swedish Transport Administration - Västra Götaland County Administration - Shore protection - Västra Götaland County Administration Following, the contextualisation reveals numerous unspecified design aspects that will affect the bridge design. These design aspects will guide the development of possible solutions in the next design phase. Each aspect will be investigated both separately and combined, with the aim of enabling a large variety of design proposals: - Path and landing points - As the exact landing points of the bridge are not specified by the client, the movement of the bridge path is free to investigate. - Length and inclination of the path - The length of the bridge path is related to the inclination. The inclination is regulated, thus governing the total length of the path. 20 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 - Integrated functions - A general expectation for the bridge design includes additional functions in the structure. These are not stated specifically from the client. - Architectural appearance - The client expects a bridge design that relates to the profile of Wendelstrand. No further demand is stated and is therefore part of the design investigation. - Structural concept - A free investigation of the most suitable structural concept for the bridge is allowed. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 21 5 Step II: Conceptual design When the specific site and different criteria are defined, the search for a suitable bridge concept, which is an answer to these, can begin. The search starts broad with many various suggestions and is then narrowed down into a suitable concept. To create a framework for the design proposals a set of general design criteria are formulated. These are based on qualities which are strived for and include contextual limitations and challenges, and clients demands. In addition to this, the designers aim of combining architectural qualities with structural engineering knowledge is considered in the design criteria. As stated in Chapter 2.2 the Conceptual design phase includes three sub-phases: Intuitive, Intentional and Evaluation. The Intuitive phase focus on exploring the possibilities on the site with less consideration of the design criteria. Any subjective opinion is set aside during this design process to allow for a large variety in the generation of the first ideas. Combined with a thorough review of the expected design qualities, three concepts are developed from the initial proposals in the Intentional phase. Each concept is developed to a certain level to enable a thorough comparison of the proposals and to determine the feasibility of the designs. In the last part of the Conceptual design phase, the three concepts are evaluated in relation to the stated design criteria and identified demands. The result is an appropriate solution for a footbridge in Wendelstrand. A set of defined evaluation criteria creates the transition between each design phase. The evaluation criteria are categorised into design criteria and demands, with the purpose of securing fulfilment of the contextual requirements in the design development, and inclusion of expected design qualities. To enable a large variation in the proposals, a reduction in the evaluation criteria is exercised in the transition between the first two phases to discover qualities in other solutions. The evaluation criteria used are illustrated in Figure 5.1. The proposed design will fulfil every stated evaluation criterion before being developed in the Preliminary design phase. Figure 5.1 Illustration of the governing evaluation criteria. 22 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 5.1 Design criteria The design criteria are divided into three main categories: spatial qualities, bridge qualities, and structural concept. The different aspects of the categories are explained in the following three subchapters. 5.1.1 Spatial qualities The spatial qualities consider the architectural aspect of the structure and its relation to the surroundings. This is implemented in different aspects, which are stated below. Visual guidance The client requests a visual and physical connection between Lakehouse and Landvettersjön. The design aims to establish a reachable structure with a sense of predictability for the users. In addition, the design should invite visitors from other areas and enable approach from different directions. The design criteria can be met either partly or completely, such as only a visible landing point, path extending from the city square, or a straight bridge path from Lakehouse to Landvettersjön. Relation to scenery As for all built structures, the relation to the scenery is of great importance. The architecture of Wendelstrand follows the same manner and blends in with the surroundings. It is therefore of interest to develop a bridge design with similar characteristics. As the purpose of the bridge is to connect the residential area and the lake, the design of the bridge must relate to the characteristics of both the residential area and the nature around the lake. Architectural appearance Furthermore, the structure should hold certain architectural qualities. This is not necessarily directly related to breath-taking design with large visual impact. On the contrary, such qualities can be found in careful detailing of functions such as water drainage, hand railing or torsional stability. It can also be found in a design that relates to a specific site characteristic. Although this design criterium is rather vague, the assessment is grounded on substantial arguments. The aim is to develop a structure that holds architectural qualities and simultaneously relates to its surroundings. The criteria relation to scenery and architectural appearance therefore determines a level of accepted visual impact. 5.1.2 Bridge qualities The bridge qualities consider the architectural aspect of the design in relation to its form and concern the following aspects. Presence of timber As Wendelstrand will become Northern Europe’s largest residential area in timber, it is expected that the bridge proposal aims for a high presence of timber with an obvious timber structure. Yet, a large amount of timber is not necessarily the most suitable interpretation of this design criterium but rather a careful design of the different structural members with the resulting visual expression as governing criterion. This CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 23 aspect is also related to the design category regarding Structural concept which will be elaborated further in the next subchapter. Additional functions The common function of a bridge is to be part of a traffic flow. In this case, the bridge will be part of a pathway where the landing point is the goal. In addition to the sole purpose of connecting Lakehouse and Landvettersjön, it is of interest to develop a design that adds other qualities to the site. For instance, integrated seating or permanent furniture as an extension of the superstructure, a viewing platform expanded from the bridge path, a pier stretching out over the water, or weather protection underneath the bridge. 5.1.3 Structural concept The third and last category incorporate design criteria into the structural design. Characteristic structure First and foremost, the structural design of a bridge is related to the visual appearance of the bridge. Consequently, the choice of structural concept affects the design criterion of Spatial qualities and Architectural appearance, and vice versa. The intention is to develop a bridge design where both architectural and engineering qualities are considered, with the aim of contributing to the development of timber bridge design. Therefore, a conscious consideration of these aspects is required, which ultimately will result in an enhancement of the qualities rather than compromising. Logic structure The structural concept of the bridge must be logic in the sense of the utilization and purpose of the structural members. All members will contribute to the structural performance of the bridge, which is related to both the structural and architectural aspects of the design. Form follows function, and function follows form. Whichever is governing is determined by the other evaluation criteria. Furthermore, the bridge design aims for a high presence of timber. A structural concept best suited for timber will be developed, as timber will not be chosen for the sole purpose of timber presence. An investigation of the most suitable timber product is necessary. Accessibility As Wendelstrand will lodge residents in the span from children to elderly, the bridge aims to be accessible for everyone. An inclination of 2° or less is stated as the requirement for wheelchair users (Göteborgs Stad Trafikkontoret, 2017), which is challenging to combine with the large height difference. For this reason, the bridge design will aim to meet the inclination requirements to an extent where everyone can utilize the main functions of the bridge. Whether or not the whole bridge design will meet the accessibility requirements depends on how the concept meets the other design criteria stated in this chapter. 24 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 5.2 Intuitive phase The Intuitive phase focus on generating a diversity of possible solutions. Figure 5.2 shows the evaluation criteria which are the most emphasised in this phase. Figure 5.2 Emphasised evaluation criteria in the Intuitive phase are highlighted in orange. As neither the starting- or landing points are explicitly stated by the client, an exploration of different movement patterns with corresponding qualities and consequences are explored. Three main aspects are identified as governing for the movement pattern of the bridge. Firstly, the path must connect to Lakehouse, preferably as an extension of the adjacent seating platform. Secondly, it should be possible to access the bridge from the bicycle and pedestrian lane on Boråsvägen, from both directions. As illustrated in Figure 5.3, a parking space is located along the road, which will be used by visitors to Wendelstrand. Third and last the bridge should connect to Landvettersjön. Both a physical and visual connection to the water will be explored. Figure 5.3 illustrates the three landing points to be considered. Figure 5.3 Landing points. Lakehouse, Boråsvägen with parking and Landvettersjön. Depending on the chosen path, the design criteria are fulfilled to different extents. As a result, varying spatial qualities are explored. For instance, to what extent the starting point connects to Lakehouse, whether the bridge is accessible from the road and in CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 25 which direction, or whether the bridge has a physical connection with the water. A selection of possible movement patterns is illustrated in Figure 5.4. Figure 5.4 Variation of movement patterns. The Intuitive design phase aims to explore the possibilities of the site, which is made possible with a large variation of ideas. Based on the exploration of possible patterns in Figure 5.4 different structural bridge concepts are assigned to the different paths. As the idea of a structural concept is formulated, the design is developed with principal sections and connections. The result is twelve different design concepts, illustrated in Figure 5.5. 26 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 Figure 5.5 Initial proposals. Summary of the Intuitive phase The design proposals developed in the Intuitive phase are generated through conscious consideration of: - Spatial qualities - Investigate different options of movement pattern when connecting Lakehouse, Boråsvägen and Landvettersjön. - Explore the possibilities and limitations of the site when considering the movement of the bridge. - Structural concept - Explore the possibilities of the movement patterns by applying different structural concepts, both inspired by reference projects and developed from the movement pattern itself. - Explore the possibilities and limitations of the movement patterns when considering the structural concepts. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 27 5.3 Intentional design phase The Intentional phase aims to develop the intuitive proposals and narrow them down to three structural concepts. This is done by considering the aspects of the evaluation criterions highlighted in Figure 5.6. Figure 5.6 Emphasised evaluation criteria in the Intentional phase are highlighted in orange. Focus lies on how the desired qualities can be met for different movement patterns. The study aims to determine whether one pattern is superior in meeting the design criteria, or if different movement patterns meet the desired criteria on an equal level. A variation of spatial qualities is found for different movement patterns, where different paths meet the design criteria on different levels. An exploration of different movement paths results in a large variation between the design proposals in the Intentional phase. Three different movement patterns are chosen for further development. They are illustrated in Figure 5.7 and differs in the following sense: - Accessible ramp - Direct or curved path - Ramp in one or two directions - Bridge reaching out over the water - Bridge reaching into the water Figure 5.7 Three different proposals with three different movement patterns. Next, the proposals from the Intuitive phase are evaluated for the chosen evaluation criteria of the current phase. As the criteria are not prioritised, the proposals are not 28 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 being rated in a matrix. Instead, they are evaluated in relation to each other and to which extent they satisfy the design criteria: - Does the bridge relate to the scenery? - Is the presence of timber high? - Does the design integrate additional functions in the structure? - Does the structural design add an architectural quality? - How is the impact on the surroundings? - Does the design create a physical and visual connection between Lakehouse and Landvettersjön? - Is it possible to comply with the accessibility requirement? Three keywords are formulated to give impact on the development of the design proposals: nature, embracing and sweeping. Nature as in relation to the surroundings, or resemblance of the characteristics of the site. Embracing as a quality in the bridge, which can be achieved in the design of the railings or integrated seating, and implementation of timber presence. Sweeping describes the movement of the bridge, which resembles a natural path. These keywords represent the essence of the design criteria formulated in Chapter Error! Reference source not found.. In the further development of the proposals, characteristics from these keywords are implemented to a different extent to ensure a variation in the proposals. The movement pattern governs the development of the design, where each path is assigned with a suitable structural solution. The proposals are developed to enhance a conceptual idea rather than searching for the one most suitable solution. To enable an evaluation of the proposals in the next phase, divergence is strived for. Reference projects are of great importance to support the design development. Three resulting bridge proposals are presented in the following part of this chapter. Proposal 1 The aim of this concept is to create a path that winds among the trees in two levels, where the delta on the southwest side of Landvettersjön is explored. This area is outside the local plan, but holds a lot of potential and qualities. Figure 5.8 shows a conceptual illustration of the proposal. The essence of this proposal is the relation to the scenery. The structure blends with the trees and with supports resembling tree trunks and a movement resembling a natural path. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 29 Figure 5.8 Bridge proposal 1 seen from the road. The structure consist of paths in two planes; an upper leading the visitor from Lakehouse across the road, and a lower path from the road with an accessible inclination. The upper path lands in an elevated platform above the water, while the lower path stretches out in a pier below the platform, which is seen in Figure 5.9. The two paths are connected with a staircase. Figure 5.9 Perspective of proposal 1. The load-bearing system consists of a curved glulam beam with Gerber hinges. There are inclined supports every 10th meter. The torsional stiffness comes from transverse rigid steel frames. Over the road, the span of 15 m is carried by two larger parallel glulam beams with intermediate diagonal glulam beams, creating a truss. 30 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 Proposal 2 The second proposal is a bridge that follows the straight line of sight from the road that go towards the square outside Lakehouse, across Boråsvägen, into the water, landing in a long pier, see Figure 5.10. The large height difference impose the design to include stairs to access the bridge. An accessible ramp is integrated as an extension between the pier and the road. The aim of the design is to create a clear visual and physical connection to Lakehouse, with an characteristic structure that adds an architectural quality. Figure 5.10 Bridge proposal 2 seen from the road. The structural concept of the second proposal’s superstructure is characterised by beams in a sinus pattern. The beams are rigidly connected to create interaction for vertical and horizontal bending. A perspective of the proposal is seen in Figure 5.11. Figure 5.11 Perspective of proposal 2. To emphasise the concept of curved beams, the railing consists of thin wood elements woven and stacked ontop of each other, which in turn adds to the torsional stability of CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 31 the bridge. The superstructure is protected by a bridge deck, but is exposed on the lower side. The same structural concept is applied on the stairs leading down to the pier, as well as the vertical support elements. The visible structure adds an architectural quality and experience for the visitors approaching from the road. Proposal 3 The third proposal aims to establish a clear visual and physical connection to Lakehouse, but does not have a physical connection to the water. Instead a platform is reaching out over Landvettersjön, see Figure 5.12, to create a visual connection. The pathway widens over the water, to give space for seatings on the bridge deck. Figure 5.12 Bridge proposal 3 seen from Lakehouse. The structrual design resembles reeds found along the lake, lifting the structure. The goal is to have slender columns, concentrated at few points, to achieve an airy appearance. A perspective of the bridge is shown in Figure 5.13. Figure 5.13 Perspective of proposal 3. 32 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 The structural concept is described as a beam-column system. The main load-bearing capacity is provided by two outer glulam beams with intermediate diagonal glulams beams in a zig-zag pattern. Beams of smaller dimensions are perpendicularly connected to the zig-zag beams. This will not only contribute to the horizontal stiffness, but it will create a characteristic pattern from beneath. Summary Intentional phase The design proposals developed in the Intentional phase focus on exploring the possibilities of the site outside of the local plan and investigate which additional qualities that could be included in the bridge design. The accessibility requirement played a larger role in the design proposals, as well as the intention of including visitors from every direction. The intuitive concepts are developed with emphasis on the following characteristics: - Nature: relation to scenery - Embracing: additional quality in the bridge, implementation of timber - Sweeping: movement of the bridge and relation to scenery 5.4 Evaluation phase In the last phase of the Conceptual design, the bridge proposals are evaluated in relation to the design criteria, contextual requirements, and the client’s demands. The aim is to narrow down the proposals into one, suitable concept. Client evaluation As the footbridge in Wendelstrand is requested by Next Step Group, the three intentional design proposals were pitched in a meeting on March 11, 2021. The presentation was customised to communicate the bridge concepts as a fictive design competition, considering Next Step Group as a client. Our understanding of the project Wendelstrand and analysation of the site were introduced. The three bridge proposals were presented as solutions, with their different possibilities and qualities. Already built reference projects were shown to support the arguments. From the discussion afterwards, feedback from Next Step Group focused on how to involve visitors as well as residents. As Wendelstrand aims to attract visitors from the surrounding area, a bridge structure connected to the water will be included in the overall experience of the area. It is therefore of interest to include visitors approaching the bridge and lake from Boråsvägen as well as from Lakehouse. Physical contact with the water is of greater interest than initially communicated, and the client also emphasised on including additional functions in the bridge design, to offer an experience to the public. The client also wanted suggestions for possible activities in the lake as an extension of the bridge. Moreover, the client focused on how the bridge design is affected by the Swedish Transport Administration, shore protection and private landowners of the areas outside the local plan. The demands from the Swedish Transport Administration are non- negotiable and are therefore complied with as requirements. The extent of the shore protection area was vaguely defined and was therefore not strictly incorporated as an evaluation criterion. From this meeting, it is clarified that there is a small area excepted from the shore protection regulations. A maximum of two support foundations are CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 33 allowed within this area. The impact on the seabed should be limited, where floating solutions should be aimed for. The delta area is outside the local plan of Wendelstrand, owned by private individuals and the municipality. To get permission to build something here is rather difficult to achieve. The client cannot plan for a project relying on agreement from outside parties with unpredictable interests. Based on this discussion, proposal 2 and 3 are found to be the most feasible bridge proposals to be developed, according to the client. These consider the shore protection to some extent and have a clear visual and physical connection between Lakehouse and Landvettersjön. Next Step Group emphasises on the physical connection to the water, as well as the accessibility from the road. Proposal 1 is disregarded because it is located outside the local plan. Summary of the Evaluation phase As a summarize of the client’s meeting it can be concluded that the final bridge proposal must fulfil the following: - Contextual demands: - The proposal must be within the borders of the local plan. - Respect the shore protection regulations and minimise the amount of supports along the shoreline. - Avoid supports in the lake. - Respect the demands of the Swedish Transport Administration. - Design demands: - Strong visual connection to Lakehouse, preferably in a straight line from the city square. Aim to create a sense of predictability for the users. - Establish a physical connection with the water. Not necessarily continuous from Lakehouse to the water. - The bridge design should include a proposal for additional functions. - The whole bridge does not have to be accessible from Lakehouse to the lake, since there already is a planned accessible path close to the planned bridge. However, the water must be accessible from Boråsvägen. - Design ambition: - Relate to scenery, both in aspects of nature and architecture. - Clear presence of timber. - Architectural quality in the structure. - Resemblance of a natural path. The development of the final bridge design is determined by the following aspects regarding spatial and bridge specific qualities: - Limited area for a landing point on the shoreline can complicate the stair connecting the bridge and the ground. - Visual expression of the pier in relation to the bridge structure and surrounding architecture. - Relation between the pier and pedestrians approaching both directions along Boråsvägen. - Possibility for a variation in additional functions on the pier. 34 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 6 Step III: Preliminary design The conclusions drawn from the evaluation phase specify the framework for the final design proposal. This is a tool to develop the concept to where it meets every defined evaluation criterion. Physical models are used to investigate the feasibility of the chosen structural concept. The aim of this phase is to develop the final design proposal to an extent where it can be verified and proven in the last design step: Final design. 6.1 Final design proposal The final design proposal is a development of design proposals 2 and 3, where a straight path from Lakehouse ends in a platform, creating visual connection to the lake. A physical connection to the lake is made possible by stairs. Figure 6.1 illustrates the final concept in its context and the following subchapters describes the concept more in detail. Figure 6.1 Conceptual model of the final design. 6.1.1 Overall bridge design The client’s wish for a strong physical and visual connection to the water is recognised by the straight sight line from the residential streets and the small square in front of Lakehouse. The bridge is accessed by stairs in the hillside. The straight path ends in a wide viewing platform, enabling a pause for the pedestrians. Stairs wind down to the ground and lands on the floating pier. A plan view of the final design proposal is seen in Figure 6.2. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 35 Figure 6.2 Plan view of the final design proposal illustrating the sight line from the city square and Lakehouse to the lake. The connection between the bridge and the pier is established by a curved staircase, which ensures a landing point within the specified area along the shoreline. An accessible ramp follows the topography and is then led parallel with Boråsvägen down under the viewing platform and then onto the floating pier. Pedestrians that approach from the other direction can access the pier by stairs down from the walkway. As the client specifically requested, the bridge must offer something more than just a physical connection to the water. A circular floating pier is chosen to be an extension of the accessibility ramp and is proposed to facilitate different possible activities to meet the client’s requests. The pier is inspired by the sculpture The Infinity Bridge by Gjøde & Povlsgaard Arkitekter in Aarhus. The circular pier offers an inner pool, as well as the possibility to anchor floating saunas, restaurant rafts and canoe rental. Additionally, the circular shape relates to the design language of the curved staircase and ensures a smooth transition for visitors approaching from three directions. Visitors can pause at any point on the circular pier, instead of being led on a straight path out in the water. 6.1.2 Structural concept The structural concept is a beam bridge with inclined columns, with a total span of approximately 27 m. The superstructure is built up by two outer straight beams and a horizontal truss action is achieved by actively bent beam in a sinus pattern. It is assumed that the internal stresses from active bending provide more horizontal stiffness to the superstructure than pre-bent beams with the same dimensions and geometrical pattern. The vertical supports are V-shaped in the longitudinal direction. Structurally this will lead to smaller span lengths, and an architecturally visually resemblance of reeds that stretches up along the shoreline. Figure 6.3 shows a side view of the bridge. 36 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 Figure 6.3 Side view of the bridge illustrating the V-shaped supports, the circular staircase, and the floating pier. To minimise the total length of the staircase, which has a height to length ratio of 1:2, the staircase winds down to an intermediate platform and is then led down straight to the ground. In addition to this, the straight staircase provides global horizontal stability to the bridge structure. The statical concept is visualised together with the bridge structure in Figure 6.4. Figure 6.4 Concept for horizontal stability of the viewing platform. The floating pier is supported by pontoons, which are anchored to the same abutment as the staircase on land. Spacing between the pontoons and the pier deck allows for sunlight to reach the seabed and ensure healthy biotope conditions in the lake. 6.1.3 Experiment with active bending As the production and assembly method is part of the bridge design the concept of using active bending in the superstructure needs to be tested and verified. The assembly of this structural concept is assumed to be rather complex compared to a simple beam bridge. Instead of performing another literature study, an investigation of physical models is used to gain understanding of active bending. First small and simple conceptual models of actively bent members are built to understand the forces and failure modes better. Then larger and more complex models are built to test the assembly method. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 37 To begin with, small strips of cardboard and paper are bent and attached to each other in different combinations. Figure 6.5 shows four models that actively bends the strips in different ways. The cardboard has much more stiffness than what paper has and is therefore referred to as the stiffer part in the observation. The bridge proposal suggests having two outer straight elements, here referred as beams, and inner bent elements, here referred to as lamellas. Figure 6.5 Experimental models of actively bent cardboard and paper strips. In Figure 6.5a, a stiffer straight beam frame is used with less stiff bent lamellas. The lamellas are connected to each other, layer after layer. Without the frame, the lamellas lay parallel to each other, and active bending is first introduced when the two outer lamellas are stretched out and connected to the stiffer straight beam frame. In Figure 6.5b, same stiffness on the outer beams as well as the inner bent lamellas is used. The curvature of the lamellas is achieved when the lamella is longitudinally pushed together and attached to the straight beam. The adjacent lamella is equally pushed and connected to the neighbouring lamella, until the outer straight beam is attached and force the deformation in the opposite direction. This method results in buckling of the lamellas. As a conclusion, the curved lamellas must have a significantly smaller stiffness than the outer beams. In Figure 6.5c, stiffer transversal beams are used with less stiff bent lamellas. The lamellas are attached together in the same manner as in Figure 6.5a, but instead of being anchored to a stiffer frame, transversal elements push the lamellas apart causing the curvature. The model in Figure 6.5d has used same stiffness on transversal beams and lamellas, in this case cardboard instead of paper. This model generates the best result, where an even curvature is achieved. As a result, the curved elements interact and create a continuous element. The resulting stiffness generated by the interaction of the curved lamellas is sufficient to maintain a desired shape for an external force and resembles the behaviour of a truss. A conclusion from this test is that a combination of a stiffer, outer frame and transverse distancing elements is considered as optimal. It is easier to control the deformation of 38 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 the lamellas with the help of transversal stiffeners. It can also be observed that the curved shape of the lamellas can be achieved in two ways: either induced by a transversal distancing force or by buckling caused by a longitudinal compressing force. Without any transversal or longitudinal force, the lamellas will automatically lay flat against each other. If looking at the boundary conditions instead, a longitudinal force can be represented in fixed connections at the ends of the curvature, while a transversal force can allow for a roller support at one of the ends. A combination of these can of course be done to reduce the residual forces required at the boundary conditions. The correlation between boundary conditions and shape is illustrated in Figure 6.6. Figure 6.6 Observations of different boundary conditions and resulting shape and forces due to active bending. The aim of using actively bent elements is to mimic a truss and thereby create horizontal stability in the superstructure. As seen in Figure 6.7 the bent elements resemble a truss, and forces can diagonally be transported. The inclination of the structs can be altered to achieve different visual appearances. Figure 6.7 Theoretical perception of the structural behaviour of beams in a sinusoidal pattern. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 39 6.2 Model development of structural concept Based on the knowledge gained from the previous small experiment, the next step is to build a concept model in wood to simulate a more realistic assembly method and to test the structural behaviour of the actively bent elements. The assembly method of the conceptual model is illustrated in Figure 6.8 and can also be described as the following: 1. A thicker element is used as outer beam and a thinner element for the curved panels. The elements are predrilled, with a few millimetres offset in the panels. 2. Threaded steel bars are inserted through one of the outer straight beams and fastened with nuts on the outside. 3. A straight panel is mounted on the threaded bars. Deformation of the panel is induced due to the offset of the holes as well as from nuts placed at desired transversal distance on the bars. 4. The rest of the panels are assembled in a similar way, creating the waved truss. 5. Lastly, the other straight beam is assembled, and secured by nuts on the outside. Figure 6.8 Diagram of the assembly method of the concept model. Evaluation of the concept model As can be seen in the second step of the assembly method in Figure 6.8, the outer beam started to deform due to the built-in longitudinal forces as described in the previous subchapter. This deformation got larger for each new panel added. To attach the last straight beam, large amount of external force was required. The predrilling of the holes had a relatively large offset, creating large amplitude of the sinus curves. As a result, the position of the nuts was adjusted thereafter, resulting in unprecise spacing. Buckling 40 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 of the threaded bars could also be observed. Ideally the bars should be in tension, but in this model, they are in compression because no spacers were used. As a conclusion from these observations, compression members should be used to secure the deformation of the panels, allowing the threaded bar to be in pure tension. Secondly, the panels should be mounted on the bars as straight elements with compression spacers at specified positions. Thereafter, as nuts in each end of the threaded bars are screwed tighter, forced compression is induced on the wood elements. With specifically placed compression distancers, the sinusoidal pattern is created. This method will ensure an even deformation and stress distribution in the structure. Regarding the pre-drilling, the position of the holes should be more accurate and measured beforehand, so the right spacing is achieved. The holes in the bent panels can preferably have larger dimension than the dimension of the threaded bars to secure some tolerance during assembly. Concerning the structural behaviour of the structure, it works well in vertical and horizontal bending. The torsional stiffness is however weak. This can be increased by using an extra layer of threaded bars, in addition to an increased height of the wooden elements. On the other hand, interaction of the curved panels is secured by pre- tensioning the superstructure. This phenomenon will increase when the threaded bars work purely in tension, and the deformation is secured by separate compression members. Applied force in horizontal direction proves that the curved panels act like a truss, which distributes the forces in compression and tension to the straight, thicker beams. Continuing from this, the concept model is developed according to the observed results. The superstructure is built in a model in scale 1:10 to verify the new assembly method as well as attempt to increase the torsional stiffness. A model of the complete bridge structure is built in scale 1:20 to verify the production method of the whole bridge structure. Moreover, this model will demonstrate the connection between the superstructure and columns as well as the horizontal stability from the straight staircase. Spatial qualities and the bridge’s context in its surrounding is demonstrated in a landscape model in scale 1:400. These models are presented in the next chapter. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 41 7 Step IV: Final design The final bridge design proposal is a straight bridge deck across the road, V-shaped column supports, a circular staircase, and a circular, floating pier. The bridge deck ends in a viewing platform, with the staircase winding around and underneath the structure. Simultaneously, an accessible ramp from the road is led through the V-shaped platform columns, underneath the staircase. Any visitor will be able to experience the bridge structure from underneath, where every path leading to the floating pier is interplayed with the bridge structure. As a result, the bridge design proposal offers both a transport route, a destination, and an experience. The overall concept together with its context is shown in Figure 7.1 and Figure 7.2. Figure 7.1 Scale model in 1:400 of the proposed design in its context. Figure 7.2 The whole bridge model in scale 1:20. 42 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 The structural concept of the bridge deck is developed to decrease the effective height of the deck, while maintaining, or increasing, the horizontal and vertical capacity. Panels of Laminated Veneer Lumber (LVL) are actively bent into a sinusoidal pattern, secured by two outer, parallel straight LVL-beams and pre-tensioned with threaded steel bars into a uniform element. Active bending is applied to achieve the wanted performance of the deck, where built in stresses allow the curved panels to interact in a truss-like behaviour giving both vertical and horizontal stiffness. The superstructure with its actively bent lamella panels can be seen in Figure 7.3. Figure 7.3 The bridge superstructure seen from the intermediate platform. V-shaped column supports decrease the span-lengths of the bridge and as a result reducing the governing forces in the superstructure. The columns go up into the superstructure and the bent panels are spread to give room to the connection. This creates a homogeneous meeting between the bridge and supports and enhances the architectural appearance of the concept. The connection is in the mid height of the superstructure, generating no extra cantilevering point. The integrated meeting between columns and superstructure is shown in Figure 7.4. Requirements on the site concerning clearance from the road and shore protected area determine the chosen landing points. The direction of the bridge is on the other hand determined by the visual connection between the lake and Lakehouse, but also by the challenges of connecting the bridge with a staircase. The design aims to limit the impact on the ground, while simultaneously achieving the required stability of the global bridge structure. The straight staircase is anchored on the shore strip in a perpendicular direction to the superstructure to achieve global horizontal stability. To reduce the total length of the straight staircase it starts at the intermediate platform, which is seen in Figure 7.5. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 43 Figure 7.4 The supports are integrated into the superstructure creating a homogeneous appearance. Figure 7.5 Viewing platform and pathway down to the ground level. The straight staircase also contributes to the horizontal stability. 44 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 7.1 Bridge dimensions A land section of the final bridge design with its correlation to Lakehouse and Landvettersjön is illustrated in Figure 7.6. An overview of the governing dimensions, and clearance is given in Figure 7.7. To ensure water drainage along the bridge, an inclination of 2% towards Lakehouse is applied. Figure 7.6 Side view of the bridge in relation to Lakehouse and Landvettersjön. Figure 7.7 Land section with main dimensions. A cross-section of the bridge is illustrated in Figure 7.8 with labels of the structural elements and governing dimensions of the bridge deck. The specific dimensions of the load-bearing elements are summarised in Table 7.1, while dimensions of the bridge deck components such as floor beams and railing are summarised in Table 7.2. The dimensions of the support columns are presented in Table 7.3. The foundation elements are suggested to be in reinforced concrete but required dimensions are not calculated. A rough estimation of the required capacity of these elements is presented in Table 7.22. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 45 Figure 7.8 Cross-section of the superstructure and bridge deck with corresponding materials and dimensions. Table 7.1 Dimensions of the load-bearing elements. Element Material Dimension Value [mm] Straight beams LVL, Kerto-S w x h 108 x 600 Curved panels LVL, Kerto-S w x h 54 x 600 Compression spacers CHS d, t 76.1, 8 Prestressing bar Dywidag, 26WR d 26.5 Table 7.2 Dimensions of bridge deck components. Element Material Dimension Value [mm] Transverse floor beams Solid timber w x h, c-c 75 x 90, 600 Longitudinal floor beams w x h, c-c 90 x 45, 600 Plank deck w x h, c-c 195 x 22.5, 210 Railing w x h, c-c 120 x 70, 1200 Solid board Plywood t 9 Bitumen felt YEP 2500 t 2 Table 7.3 Dimensions of support elements. Element Material Dimension Value [mm] Support columns intermediate LVL w x h 216 x 260 Support columns platform w x h 270 x 260 46 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 The design of the staircase is performed as a simply supported beam subjected to vertical load according to the load cases calculated for the superstructure. The structure is built up similarly as the bridge superstructure, with two outer straight beams and actively bent LVL panels. The final dimensions of the staircase structure are presented in Table 7.4. Table 7.4 Dimensions of the load-bearing elements of the staircase. Element Material Dimension Value [mm] Straight, outer beams LVL, Kerto-S w x h 108 x 400 Curved beams LVL, Kerto-S w x h 54 x 400 Compression distancers CHS d, t 76.1, 8 Prestressing bar Dywidag, 18WR d 17.5 7.2 Input data To verify the bridge design with its cross-sections hand calculations and a simplified FE analysis are performed. The overall bridge design is based on European Standards and Swedish Standards. General requirements for timber footbridges are formulated by the Swedish Transport Administration. Overall bridge design concerning the capacity is covered in Bärighetsberäkning av broar and states which European standards that are used to determine the capacity of the structure (Ronnebrant, s2020). Detailed design of the bridge is covered in Krav Brobyggande (Krona, 2019). To ensure a more concise writing, EC5-1 will be used as abbreviation for Eurocode SS-EN 1995-1 etc. The following standards are used for the different calculation aspects: - Design material properties for LVL: EC5-1-1 (SS-EN 1995-1-1:2004) - Bridge specific properties: EC5-2 (SS-EN 1995-2:2004) - Dimensioning with the partial factor method: EC0 (SS-EN 1990:2010) - General loads - Wind load: EC1-1-4 (SS-EN 1991-1-4:2005) - Snow load: EC1-1-3 (SS-EN 1991-1-3:2003) - Bridge specific loads: EC1-2 (SS-EN 1991-2:2003), with corresponding partial factors - Load combinations: EC1-1 (SS-EN 1991-1-1:2005) - Bridge details: The Swedish Transport Administration, Krav Brobyggande (TDOK 2016:0204) 7.2.1 Partial factors The timber bridge is dimensioned according to EC0 where the partial coefficient method is applied. On the load side partial factors are used to consider exposure, load duration, load situations and -combinations. The material related partial factors are taken from EC5. Material properties The characteristic material properties are adjusted with partial factors that considers exposure, load duration, material, and element size according to EC5-1, which gives the design material properties: CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 47 𝑓 𝑓 = 𝑘 [𝑀𝑃𝑎] 7. 1 𝛾 The correlation factor kmod considers the load duration and moisture content of the structural material. The bridge is classified as service class 3 according to EC5-1. However, if the main structural members can be sufficiently protected against rain, service class 2 can be applied. In this case, a conservative approach is chosen and thereby service class 3 is used. Self-weight is classified as permanent load while imposed loads such as wind load (EC5-1) and traffic load from pedestrians (EC5-2) are considered short-term loads. According to EC5-1, combination of loads of different duration should be determined for the shortest load duration, which in this case is short- term load duration. For LVL exposed to short-term load duration in service class 3 (EC5-1), the correlation factor is set according to 𝑘 = 0.7 7. 2 The partial factor 𝛾 is determined by the specific material and differs for control of capacity and control of deformation. For LVL, the partial factor according to EC5-1 is: 𝛾 = 1.2 7. 3 For LVL with rectangular cross-section and majority of the veneers are oriented in the same direction, the size effect in bending and tension must be considered. For LVL in bending, the reference height is 300 mm. For any element height in bending other than 300 mm, the characteristic bending strength fm.k should be multiplied with the factor kh according to EC5-1. For LVL in tension, the reference length is 3000 mm. For any element length other than 3000 mm, the characteristic value ft.0.k should be multiplied with a factor kl according to EC5-1. The design capacity of the LVL beams and panels are determined for veneers in the same direction. The dimensions determining the size effect factors for LVL in bending and tension, and the resulting factors are summarised in Table 7.5. Table 7.5 LVL size effect. Load action Value Element height in bending hlvl 0.6 [m] Element length in tension L 27.3 [m] Size effect for bending kh.lvl 0.92 Size effect for tension kl.lvl 0.876 Load combinations As the risk of personal injury in the case of structural collapse is considered serious, the bridge is categorized as safety class 3 according to 13 § in Boverket EKS 10 (Boverket, 2016). The partial coefficient considering the safety of a load bearing structure is set according to 14 § EKS 10: 𝛾 = 1.0 7. 4 48 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 The timber bridge is verified in Ultimate Limit State (ULS) for deformation of the structure, STR: “Internal failure or excessive deformation of the structure or structural members, including footings, piles, basement walls, etc., where the strength of construction materials of the structure governs” (Chapter 6.4.1, EC0). Load combinations in ULS considering the capacity of the structure is calculated according to Equation 6.10a and b in EC0. For unfavourable permanent loads G: 𝐿𝐶 = 𝛾 ∙ 𝛾 ∙ 𝐺 7. 5 For variable loads Qk.i, the loads are combined as follows: 𝐿𝐶 . = 𝛾 ∙ 𝛾 ∙ 𝑄 . 7. 6 For interacting variable loads Qk.i and Qk.j, where Qk.i is the main load, the loads are combined accordingly: 𝐿𝐶 . = 𝛾 ∙ 𝛾 ∙ 𝑄 . + 𝛾 ∙ 𝛾 ∙ 𝜓 . . ∙ 𝑄 . 7. 7 The partial factors for load combinations are found in Table 7.6 according to EC0 and Boverket EKS 10. Relevant load factors are found in Table 7.7, extracted from Table A2.2 in EC0. Table 7.6 Partial factors for load combinations. Type of load Characteristic Frequent Quasi-permanent Permanent load 𝛾 1.2 1.0 1.0 Variable load 𝛾 1.5 1.0 1.0 Risk class 3 𝛾 1.0 1.0 1.0 Table 7.7 Load factors for load combinations. Type of load Characteristic Frequent Quasi-permanent 𝜓 . 𝜓 . 𝜓 . Traffic load 0.4 0.4 0 Wind load 0.3 0.2 0 Snow load 0.7 0.5 0.2 7.2.2 Material properties Laminated Veneer Lumber, LVL, is chosen as the main structural material of the load- bearing elements in the bridge. Kerto-S beam is chosen for both the outer straight beams and the curved beams, with all lamellas oriented with the same fibre direction. Usually the majority of the lamellas in Kerto-S are oriented with fibre direction along the beam and the rest in vertical direction. The stiffness can be increased with a few lamellas perpendicular to the length, but this is not accounted for in the following capacity calculations. Characteristic material properties for Kerto-S beam are extracted from the dominating European Supplier of LVL, Mestä Wood, according to manufacturer’s declaration of CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 49 performance (Eurofins Expert Services Oy, 2020). The straight beams are built up by 9 times 3 mm veneer sheets glued together from 4 lamellas. The curved panels are built up by 2 lamellas. The characteristic material properties of the Kerto-S beam, and the resulting design strength properties adjusted with corresponding partial factors, are listed in Table 7.8 according to producer’s certificate (Eurofins Expert Services Oy, 2020). Table 7.8 Design properties of LVL. Design strength properties Value [MPa] Bending edgewise fm.0.edge.d 16.9 Bending flatwise fm.0.flat.d 28.8 Tension parallel to grain ft.0.d 12.8 Compression parallel to grain fc.0.d 14.6 Compression perpendicular to grain, edgewise fc.90.edge.d 2.50 Compression perpendicular to grain, flatwise fc.90.flat.d 0.92 Shear edgewise fv.0.edge.d 1.71 Shear flatwise fv.0.flat.d 0.96 Characteristic material properties Value Size effect parameter s 0.12 Elastic modulus, parallel to grain E0.k 11600 [MPa] Elastic modulus, parallel to grain, mean E0.m 13800 [MPa] Elastic modulus, perpendicular to grain, edge, mean E90.edge.m 430 [MPa] Elastic modulus, perpendicular to grain, flat, mean E90.flat.m 130 [MPa] Density, characteristic ρk 480 [kg/m3] Density, mean ρm 510 [kg/m3] 7.2.3 Loads 7.2.3.1 Permanent loads The self-weight of the bridge deck components is calculated according to the following equation. Li is the length of the element and ni is the number of elements on the bridge: 𝐿 𝐺 . = 𝜌 . ∙ 𝑔 ∙ 𝑛 ∙ 𝑤 ∙ ℎ [𝑘𝑁/𝑚] 7. 8 𝐿 The resulting permanent loads are listed in the table below, where non-load-bearing elements include floor beams, plywood and construction boards, plank deck, epoxy resin cover, and railing. Table 7.9 Permanent loads Characteristic permanent load Value [kN/m] Straight beams Gk.lvl.s 0.61 Curved beams Gk.lvl.c 2.14 Distancers Gk.dis 0.50 Non-load-bearing elements Gk.deck 1.14 Total permanent load Gk 4.39 50 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 7.2.3.2 Variable loads The variable loads working on the bridge structure are identified as: - Traffic load from pedestrians only, as no service vehicle can access the bridge. - Wind load on the bridge deck and on the support columns. - Snow load as the bridge must be manually cleared. This load will only be included in the load combinations if it exceeds the permanent load. Temperature deformations are discharged from the calculations as the expansion coefficient for timber is small. Traffic load, vertical Load Model 4: “Crowd loading”, is applied according to EC1-2. As the bridge is likely to be subjected to an even flow of pedestrians rather than a continuous dense crowd, the characteristic value of uniformly distributed load is set according to: 120 𝑄 = 2.0 + [𝑘𝑁/𝑚 ] 7. 9 𝐿 + 30 𝑘𝑁 𝑘𝑁 2 ≤ 𝑄 ≤ 5 7. 10 𝑚 𝑚 According to EC1-2, the uniformly distributed load should be applied in the most unfavourable parts of the surface, in both longitudinal and transverse direction. A conservative approach is chosen where the load is applied as uniformly distributed over the cross-section. Traffic load, horizontal For footbridges only, a horizontal force Qflk should be considered, acting along the bridge deck axis at the pavement level, according to EC1-2: 𝑄 = 0.1𝑞 [𝑘𝑁] 7. 11 The horizontal force is considered to act simultaneously with the corresponding vertical load. Wind load on the bridge structure The transverse, horizontal wind load is calculated according to EC1-1-4. The vertical wind load is determined for both uplift and downwards. According to EC1-1-4 does the vertical wind load only have significant effect on the structure if the force exceeds the permanent load of the bridge. In this case, the downward vertical wind load will not be of the same order as the dead load and is therefore disregarded in the capacity calculations of the bridge. The uplifting force will be considered in the detailed design of the bridge connections. The force in transverse direction Fw.x is determined by the simplified method according to EC1-1-4, 𝜌 being the air density: CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 51 1 𝐹 . = 𝜌 ∙ 𝑣 ∙ 𝐶 ∙ 𝐴 . [𝑘𝑁] 7. 12 2 For structures in terrain category II, the wind load factor Cx can be defined with interpolation of table 8.2 in EC1-1-4, determined by the reference height ze which is altitude above sea level plus bridge height. For load combinations without traffic load and plain webs, the reference area is calculated as: 𝐴 . = 𝐿 ∙ 𝑑 [𝑚 ] 7. 13 The reference height d for an open parapet and open safety barrier is set accordingly, dtot being the total height of the bridge deck: 𝑑 = 0.6𝑚 + 𝑑 [𝑚] 7. 14 The wind velocity is defined according to EC1-1-4: 𝑚 𝑣 = 𝑐 ∙ 𝑐 ∙ 𝑣 . = 26.0 7. 15 𝑠 The fundamental values used to calculate peak velocity pressure depends on the geographical conditions of the bridge in question. The fundamental value for Wendelstrand is 𝑣 . = 25.0 𝑚/𝑠 (Boverket, 2019b). The directional factor cdir and seasonal factor cseason are set to 1.0. The basic velocity pressure is calculated as 1 𝑘𝑁 𝑞 = 𝜌 ∙ 𝑣 . = 0.39 7. 16 2 𝑚 Longitudinal wind forces depend on the horizontal wind forces and the geometry of the bridge. For plated bridges, the longitudinal wind force is 25% of the transverse wind forces (Chapter. 8.3.4 in EC1-1-4). The wind force along the bridge is set to: 𝐹 . = 0.25𝐹 . [𝑘𝑁] 7. 17 Wind load on column supports Wind load on columns are calculated according to Chapter 7.6 in EC1-1-4. The design situation is assumed to be wind blowing in the transverse direction of the bridge. The wind force on a rectangular column is calculated as 𝐹 = 𝑐 ∙ 𝑐 ∙ 𝑐 ∙ 𝑞 ∙ 𝐴 [𝑘𝑁] 7. 18 The seasonal factors cs and cd are set to 1.0. The peak velocity pressure is calculated as follows, with ce(z) depending on the terrain category for the bridge structure. 𝑞 = 𝑐 (𝑧) ∙ 𝑞 [𝑘𝑁] 7. 19 The force coefficient for rectangular cross-sections with sharp corners is determined as 52 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 𝑐 = 𝑐 . ∙ 𝜓 7. 20 The end-effect factor 𝜓 is determined by Figure 7.36 in EC1-1-4 and depends on the solidity ratio 𝜑 and the slenderness 𝜆. These in turn depend on the dimensions of the projected column. The force coefficient 𝑐 . is determined by linear interpolation of Figure 7.23 in EC1-1-4. Snow load Snow load on bridge decks is calculated with the same method as for roof structures, according to Chapter 5 in EC1-1-3. The snow load is calculated as: 𝑠 = 𝜇 ∙ 𝐶 ∙ 𝐶 ∙ 𝑠 [𝑘𝑁] 7. 21 The exposure coefficient Ce is conservatively set to 1.0 for normal conditions. The thermal coefficient Ct is not relevant for pedestrian bridges and is set to 1.0. The shape coefficient 𝜇 depends on the inclination of the bridge structure, and is set to 0.8 according to Table 5.2 in EC1-1-3. The characteristic value of snow on the ground is determined by the geographical conditions on the site and is given by Boverket EKS as 𝑠 = 1.5 𝑘𝑁/𝑚 (Boverket, 2019a). Total variable load The resulting variable loads are listed in Table 7.10. Table 7.10 Variable loads Characteristic variable load Value Traffic load, vertical Qfk.z 10.2 [kN/m] Traffic load, horizontal Qfk.x 27.9 [kN] Wind load, horizontal Fw.x 2.32 [kN/m] Wind load, along Fw.y 15.8 [kN] Snow load qsnow 3.0 [kN/m] 7.2.3.3 Accidental load Due to traffic under the bridge, accidental design situations must be included in the dimensioning of the support columns according to EC1-2. For stiff columns and roads in urban areas, the impact force is set to: 𝐹 = 500 𝑘𝑁 7. 22 𝐹 = 250 𝑘𝑁 7. 23 The impact forces are assumed to not occur simultaneously. The impact height is determined to ℎ = 1.25 𝑚 with the impact area depending on the vehicle. This cannot be wider than the width of the support, according to EC1-2. The column capacity is tested for accidental actions without the influence of variable loads. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 53 7.2.4 Load combinations Load model 4 is applied to calculate the bridge capacity. This includes characteristic values, frequent values, and quasi-permanent values for a uniformly distributed load according to Table 2.1 in EC1-2. The following subchapters present the design load combinations used to calculate the capacity of the bridge superstructure and columns in ultimate limit state and serviceability state. Design load combinations – bridge superstructure The governing load combinations on the bridge superstructure are illustrated in Figure 7.9. The design load combinations used to calculate the capacity of the bridge superstructure in ultimate limit state are summarised in Table 7.11. Snow load is disregarded in the load combinations as it does not exceed the permanent load on the bridge. Figure 7.9 Governing load combinations on the superstructure. Table 7.11 Design loads for capacity calculations Load combination Value Vertical load qd.z.uls 20.6 [kN/m] Self-weight + traffic load Transverse load qd.x.uls 3.48 [kN/m] Wind load Longitudinal load qd.y.uls 4.11 [kN] Wind load + traffic load The design loads used to calculate the instantaneous and final deformations of the bridge in serviceability state are summarised in Table 7.12. 54 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 Table 7.12 Design loads for deformation calculations Load combination Value [kN/m] Vertical load qd.sls.G 4.39 Self-weight Vertical load qd.sls.Q 10.2 Traffic load Design load combinations – support columns The largest reaction force in the support columns depend on how the variable load is distributed on the bridge. The load response of the bridge in terms of reaction forces is illustrated in Figure 7.10. Figure 7.10 Load response from distributed load along the bridge. The most critical load case is applied for the largest reaction force in each column, which is summarised in Table 7.13. Table 7.13 Critical load cases for each column support Column Load Reaction force, Reaction force, combination vertical [kN/m] horizontal [kN/m] Span Column B 3 -260 -44.8 Column C 2+3 459 78.4 Column D 1+3+4 308 52.1 Column E 1+3 -82.9 -15.8 These reaction forces are combined with wind load acting on the columns. Furthermore, the support columns are verified for the accidental load due to traffic under the bridge. These are only combined with the permanent loads on the bridge. The design loads used to calculate the capacity of the support elements in Ultimate Limit State are summarised in table Table 7.14 and illustrated in Figure 7.11. Table 7.14 Design loads for capacity calculations Design load combination Value Transverse load, column B, C qw 1.45 [kN/m] Transverse load, column D, E qw 1.89 [kN/m] Accidental load, column C qdx 741 [kN/m2] Longitudinal vs transverse direction qdy 308 [kN/m2] Accidental load, column D qdx 593 [kN/m2] Longitudinal vs transverse direction q 2dy 308 [kN/m ] CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 55 Figure 7.11 Variable load on the supports. Hill-side columns to the left, and lake-side to the right. 7.3 Global design In the following subchapters the calculated response in ULS and SLS of the superstructure and columns are presented. 7.3.1 Superstructure in ULS Maximum load effects The maximum load effects on the bridge structure is calculated for a continuous beam in four unequal spans. The maximum load effects calculated for load in vertical direction are summarised in Table 7.15 and illustrated in Figure 7.12. 56 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 Figure 7.12 Load response due to uniformly distributed load Table 7.15 Load effects, vertical load. Load effects Critical load case Value Span Bending moment, largest span 1+3 Mf.max 294 [kN/m] Bending moment, intermediate support 2+3 Ms.max 477 [kN/m] Shear force, intermediate support 2+3 V4.max 281 [kN] Reaction force, platform support 2+3 R5.max 459 [kN] Maximum deflection, largest span 1+3 δmax -25.1 [mm] The maximum load effects on the bridge structure in transverse direction are summarised in Table 7.16. Table 7.16 Load effects, transverse load. Load effects Critical load case Value Span Bending moment, largest span 1+3 Mf.max 49.7 [kN/m] Bending moment, intermediate support 2+3 Ms.max 80.8 [kN/m] Shear force, intermediate support 2+3 V4.max 48.2 [kN] Reaction force, platform support 2+3 R5.max 78.4 [kN] Bending stresses due to induced deformation The induced deformation of the LVL panels leads to bending stresses parallel to the grain, between the lamellas. The stresses depend on the lamella thickness, radius of the curve and stiffness of the material. The radius is decided through iterative parametric design, based on lamella thickness, desired curvature length and interaction with bending and axial compression. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 57 The bending stresses are verified for the bending capacity of the LVL according to EC5: 𝜎 . . ≤ 𝑘 ∙ 𝑓 . . . 7. 24 In bent glulam elements, the reduced capacity is considered by applying a reduction factor kr. As a conservative approach, the capacity of the bent LVL elements is reduced in the same manner. The reduction factor is determined as follows, where t is the thickness of the bent panel and rin is the inner radius of the curve: 𝑟 𝑟 0.76 + 0.001 , < 240 𝑘 = 𝑡 𝑡𝑟 7. 25 1, ≥ 240 𝑡 The stresses in the curved zoned for bent lamellas are calculated according to the following equation, t being the lamella thickness: 𝐸 . 𝑡 𝜎 . . = 7. 26 2𝑟 The chosen geometry and resulting bending stresses are presented in the table below: Table 7.17 Curvature of the LVL panels and resulting bending stresses. Factor Value Inner radius of curved panels rin 14.5 [m] Bending stresses in lamellas 𝜎 . . 10.8 [MPa] Verify capacity The bending, compression and shear capacity of the bridge is verified stepwise to monitor the structural response of the load-bearing element. The design bending stresses in the whole cross-section are verified for the design bending capacity in the respective direction according to Chapter 6.1.6 in EC5 for km = 0.7 for LVL. The compression stresses parallel to the grain are verified according to Chapter 6.1.4 in EC5 for the whole cross-section, the straight beams, and the curved beams respectively. The effect of combined bending and axial compression is controlled for both the whole cross-section and the straight beams according to: 𝜎 . . 𝜎 . . 𝜎 . . + + 𝑘 ≤ 1 7. 27 𝑓 . . 𝑓 . . 𝑓 . . 𝜎 . . 𝜎 . . 𝜎 . . + 𝑘 + ≤ 1 7. 28 𝑓 . . 𝑓 . . 𝑓 . . As the curved panels are designed for the bridge deck to work as a horizontal truss, the combined bending and compression capacity of the panels are checked in z-direction only. Due to induced deformations in the panel, the compression strength is reduced with a buckling factor kc.z determined according to Chapter 6.3.3 in EC5. Partly 58 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 interaction is assumed for the induced bending stresses in the panels, with an interaction factor kcurve set to 0.7. As the design of the curved panels fulfils the criteria “lateral displacement of its compressive edge is prevented throughout its length and torsional rotation is prevented at its supports” (EN 1995-1-1, Chapter 6.3.3 (5)), the buckling factor kcrit is set to 1.0. The curved panels are checked according to the following equation: 𝜎 . . 𝜎 . . 𝜎 . . + + 𝑘 ≤ 1 7. 29 𝑘 . ∙ 𝑓 . . 𝑘 ∙ 𝑓 . . 𝑓 . . The bridge deck is verified for the combined effect of induced bending stresses, bending in two directions and axial compression. The buckling factors kc.z and kc.y are defined conservatively for the whole bridge deck, where the buckling length in vertical direction is set equal to the length of the largest bridge span and transverse direction equal to the spacing between the compression distances. Slenderness ratios larger than 0.3 reveal that the stresses in each direction will increase due to deflection, and should therefore be verified according to Equation 6.23-24 in EC5: 𝜎 . . 𝜎 . . 𝜎 . . 𝜎 . . + + 𝑘 + 𝑘 ≤ 1 7. 30 𝑘 . ∙ 𝑓 . . 𝑓 . . 𝑓 . . 𝑓 . . 𝜎 . . 𝜎 . . 𝜎 . . 𝜎 . . + 𝑘 + + 𝑘 ≤ 1 7. 31 𝑘 . ∙ 𝑓 . . 𝑓 . . 𝑓 . . 𝑓 . . The shear stresses are verified according to Chapter 6.1.7 in EC5. The resulting utilisation ratios in ULS are presented Table 7.18. Table 7.18 Utilisation ratios in ULS Design stress Utilisation [%] Superstructure Bending, vertical main 49 Bending, transverse main 35 Axial compression 0.8 Bending + compression, vertical main 49 Bending + compression, transverse main 35 Bending + compression + induced deformation, vertical main 86 Bending + compression + induced deformation, transverse main 72 Shear, vertical load 26 Shear, transverse load 8 Straight beam Bending capacity ratio, vertical load 27 Bending capacity ratio, transverse load 23 Axial compression 0.4 Bending + compression, vertical main 40.2 Bending + compression, transverse main 35.1 Curved panel Bending capacity ratio, vertical load 53.9 Bending capacity ratio, transverse load 93.5 CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 59 Induced deformation 51.8 Axial compression 0.85 Bending + compression, main direction 34.0 Bending + compression + induced deformation, vertical 70.2 Observation of the results show that the curved panels handle most of the bending stresses in the superstructure and are more utilised than the straight beams. Second order effects of the panels are not considered in the calculations. 7.3.2 Superstructure in SLS The maximum deflection is calculated for traffic load and permanent load only, according to Chapter 7.2 in EC5-2. Deformations The deformations in a structure subjected to loads and moisture must be verified according to Chapter 2.2.3 in EC5. The final deformations are calculated for quasi permanent load combinations, while the instantaneous deformation is calculated for characteristic load combinations in SLS. As the bridge deck is composed by members with equal modulus of elasticity and shear modulus, the final deformation is calculated as 𝑢 = 𝑢 . + 𝑢 . . 7. 32 The final deformation due to permanent load G and variable load Q1 are calculated as 𝑢 . = 𝑢 . 1 + 𝑘 [𝑚𝑚] 7. 33 𝑢 . . = 𝑢 . . 1 + 𝜓 . ∙ 𝑘 [𝑚𝑚] 7. 34 Creep deformations in timber structures are influenced by moisture content and are accounted for with the creep factor kdef. For LVL in climate class 3, the creep factor is set to 2.0. The resulting infinite and final deformations for permanent and variable load are presented in Table 7.19. Table 7.19 Infinite and final deformations. Deformations Value [mm] Infinite deformation, permanent load uinst.G -12.5 Infinite deformation, variable load uinst.Q -5.27 Final deformation ufin 28.3 Instantaneous and final deformations of a pedestrian bridge are limited according to Chapter 7 in EC5-2: 1 𝛿 = 7. 35 200 The largest deformation is found in the span across the road, and is verified accordingly: 60 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 𝑢 ≤ 𝛿 ∙ 𝐿 [𝑚𝑚] 7. 36 Deflection due to shear is neglected as the deck is assumed to interact like a horizontal truss. 7.3.3 Support in ULS The following subchapter presents the final design of the support columns and the required capacity of the support foundation. The capacity of the columns is verified for ULS load combinations. Maximum load effects on the columns The reaction force from vertical load on the bridge is divided by the number of columns at the support. The horizontal reaction force is translated to a bending moment at the foundation and a resulting vertical force in the support column. As the columns are inclined in longitudinal direction, the vertical load on one column is divided into axial and perpendicular components. The load response in the columns are illustrated in Figure 7.13. Figure 7.13 Load response in the supports due to load on the bridge. The reaction force from horizontal load on the bridge is translated by shear through the column to the foundation. The columns are tested for the axial load component combined with bending due to wind load, in addition to shear stresses in the top and bottom. The design of the bridge deck enables the perpendicular load component to be taken by the LVL panels between the inclined columns according to the principles of strut and tie. Furthermore, the support columns under the viewing platform are tested for an additional vertical load from the staircase. Table 7.20 summarises the load effects on the support columns in the intermediate support and platform support respectively. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 61 Table 7.20 Load effects on columns at the supports Load effects Value Intermediate support Axial load N0.B -92.7 [kN] N0.C 163 [kN] Bending moment from wind load Mx 0.96 [kNm] Deflection from wind load δw.x 0.14 [mm] Shear force VB -11.2 [kN] VC 19.6 [kN] Deflection from accidental load δw.x 77.9 [mm] δw.y 136 [mm] Platform support Axial load N0.D 200 [kN] N0.E -30.5 [kN] Bending moment from wind load Mx 11.3 [kNm] Deflection from wind load δw.x 12.2 [mm] Shear force VD 17.9 [kN] VE -3.95 [kN] Deflection from accidental load δw.x 172 [mm] δw.y 154 [mm] Staircase load Pk 0.64 [kN] Capacity verification The compression capacity of the columns is verified according to Chapter 6.3.2 in EC5. The columns are designed as pinned-pinned with an effective buckling length lef equal to the column length. The design compression stresses are verified according to EC5: 𝜎 . . ≤ 𝑁 7. 37 The critical buckling load of the column is calculated for the respective direction as 𝑁 . = 𝑘 . ∙ 𝑓 . . [𝑘𝑁] 7. 38 𝑁 . = 𝑘 . ∙ 𝑓 . . [𝑘𝑁] 7. 39 Combined transverse and axial load in the columns are verified according to EC5: 𝜎 . . 𝜎 . . 𝜎 . . + + 𝑘 ≤ 1 7. 40 𝑘 . ∙ 𝑓 . . 𝑓 . . 𝑓 . . 𝜎 . . 𝜎 . . 𝜎 . . + 𝑘 + ≤ 1 7. 41 𝑘 . ∙ 𝑓 . . 𝑓 . . 𝑓 . . The bending stresses are due to initial deformation in the columns, combined with transverse load from wind or accidental load: 𝑀 = 𝑁 (Δ + δ ) + 𝑀 [𝑘𝑁𝑚] 7. 42 62 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 The initial out of straightness is applied on the LVL columns is conservatively assumed as the value for glulam (Swedish Wood, 2016): 𝐿 Δ = [𝑚𝑚] 7. 43 300 Tension stresses in the columns are verified according to EC5, where the size effect is accounted for in the design capacity of the column: 𝜎 . ≤ 𝑓 . . 7. 44 The effect of combined transverse load and tension is verified according to EC5: 𝜎 . 𝜎 . . 𝜎 . . + + 𝑘 ≤ 1 7. 45 𝑓 . . 𝑓 . . 𝑓 . . 𝜎 . 𝜎 . . 𝜎 . . + 𝑘 + ≤ 1 7. 46 𝑓 . . 𝑓 . . 𝑓 . . Shear stresses in the top and bottom part of the column are verified according to EC5: 𝜏 ≤ 𝑓 . . . 7. 47 The applied vertical load on the platform columns lead to an additional moment in the column, which is calculated as: 𝑀 = 𝑁 (Δ + δ ) + 𝑀 + 𝑃 ∙ e [𝑘𝑁𝑚] 7. 48 With P being the point load from the staircase and ep the eccentricity of the load in relation to the column end. The resulting bending stresses combined with an axial load are verified in the same manner. The resulting utilisation ratios in of the support columns are presented in Table 7.21. These are due to permanent and variable load on the superstructure combined with specific loads on the columns. Table 7.21 Support column utilisation ratios Design stress Utilisation [%] Intermediate columns Axial compression 19.1 Axial compression and bending, wind load 27.0 Axial compression and bending, accidental load 62.6 Shear 30.6 Tension 10.4 Tension and bending 18.2 Platform columns Axial compression 53.6 Axial compression and bending, wind load 90.3 Axial compression and bending, wind load + staircase 88.1 CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 63 Axial compression and bending, accidental load x-dir 95.5 Axial compression and bending, accidental load + staircase 130 Shear 14.9 Tension 3.1 Tension and bending 39.8 Note that the accidental loads require much larger capacity of the column elements than variable and permanent loads combined. The columns close to the road that also supports the staircase requires larger dimension than the rest, preferably with a width of 324 mm instead of 270 mm. This results in a utilisation ratio of 89%. Loads on abutments Summary of the load on the abutments is presented in Table 7.22. Table 7.22 Required capacity of the abutments. Load response Load [kN] Hill-side support Vertical 54.7 Transverse 9.28 Intermediate support Vertical 216 Transverse 364 Horizontal 438 Shear 78.4 Platform support Vertical 305 Transverse 455 Horizontal 365 Shear 71.6 7.3.4 Dynamic analysis In addition to capacity and deformation analysis, the dynamic response of the main bridge structure must be verified. An exact response of the structure will not be carried out, but an estimation of the response based on predicted periodic forces for similar load situations is considered sufficient. Resonance may occur if one of the natural frequencies of the bridge is equal to the frequency from an imposed force. If the imposed frequency is larger than the natural frequency of the bridge, vibrations may also occur. The natural frequency depends on the stiffness, mass, and span length of the structure. In EC5-1, the natural frequency for a simply supported, rectangular floor structure is calculated as 𝜋 (𝐸𝐼) 𝑓 = 7. 49 2𝑙 𝑚 64 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 For a bridge deck with several spans and varying stiffness along the length, this equation is too conservative. Instead, a FE model is built to simulate a more accurate dynamic response of the main structure of the bridge deck. In addition to the natural frequency, the corresponding deformation mode is of interest. This indicates what type of induced force that will cause resonance. Finite Element model Abaqus is used to simulate the dynamic response of the main structure of the bridge deck. As laminated veneer lumber is the main material of the load-bearing elements, the model must consider the orthotropy of the material. This is ensured by using plane stress elements to build up the section. The bridge deck is modelled as a shell element with length corresponding to the total length of the bridge and width set to the distance between the outer straight beams. To simulate the behaviour of the real bridge structure, the stiffness must be accounted for. As the model computes the stiffness based on the dimensions of the element combined with material properties, the real stiffness can be accounted for by adjusting the geometry of the FE model. An effective height is calculated from the moment of inertia in the real bridge deck while the width is kept equal to bridge dimensions. As the stiffness of the bridge deck varies in the span and over the support, two separate models are built. 12 ∙ 𝐼 ℎ . = [𝑚𝑚] 7. 50𝑊 12 ∙ 𝐼 ℎ . = [𝑚𝑚] 7. 51 𝑊 Material properties are assigned according to the mean values for Kerto-S beams for elastic behaviour and material type lamina. As the producer does not state the shear modulus in transverse direction, this can be estimated to one tenth of the longitudinal direction. In the model, the property of Kerto-Q is conservatively chosen. Material orientation is assigned to the model according to the bridge design. Table 7.23 present the chosen material properties used in the FE model. Table 7.23 Material properties in FE model. Mean values Value Density ρm 5.10e-10 [ton/mm3] Modulus of elasticity, parallel to grain, along E0.m 13800 [MPa] Modulus of elasticity, perpendicular to grain, E90.edge.m 430 [Mpa] edgewise Shear modulus, edgewise G0.edge.m 600 [Mpa] Shear modulus, parallel to grain, flatwise G0.flat.m 600 [Mpa] Shear modulus, perpendicular to grain, flatwise G90.flat.m 22 [Mpa] The shell element is partitioned at the support locations and assigned with corresponding boundary conditions, according to Figure 7.14. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 65 Figure 7.14 Boundary conditions assigned to the FE model. The frequency analysis is performed with linear perturbation and limited for a maximum frequency of 50 Hz. A structural, quadratic mesh is chosen with element size of 10 mm. Limitations of the model The purpose of the FE model is to estimate the dynamic response of the bridge deck, to exclude the risk of resonance due to an induced force. Therefore, the complexity of the bridge deck is disregarded in the model, as an estimation of the natural frequency can be determined for an adjusted stiffness. These are performed conservatively and does not represent the accurate response of the structure. However, if the estimated natural frequency of the structure is smaller than or close to the predicted vibrations from pedestrians or wind, the model would need to be refined to exclude the risk of resonance. Otherwise, adjustments to the structure is necessary with the aim of increasing the structural stiffness. As a result of this, the FE model cannot be used to simulate the torsional stiffness, as this largely depend on the geometry of the deck. Result of frequency analysis The first natural frequency is found for vertical vibration in the span-model and support- model at 7 Hz and 8 Hz respectively. The two models show similar response in the following modes for type of vibration but the intervals between each mode are larger for the support-model, which is associated with the larger stiffness of the model. Owing to this, only the results of the span-model are presented and verified, as these are the more critical and more representative. The first eight modes are presented in Table 7.24. The modes indicate which direction the deck will deform for different frequencies. This behaviour is relevant to identify to predict what type of external forces that can cause resonance in the structure. Table 7.24 Eigenmodes, model of span. Increment Frequency [Hz] Type Mode 1 7 Vertical Mode 2 14 Torsional Mode 3 19 Vertical Mode 4 20 Horizontal Mode 5 30 Torsional Mode 6 36 Vertical Mode 7 47 Base state Mode 8 48 Horizontal 66 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 However, a comparison of the results of the two models reveal that a shared natural frequency is found for 20 Hz. Both models experience horizontal deformation, but for different mode numbers (Mode 4 and 3). It is also observed that the two models deform in the opposite direction for the second and third vertical vibration modes, and for the horizontal vibration modes. However, this is assumed to be a feature in Abaqus as the models are symmetric. Figure 7.15 to Figure 7.17 illustrate the vertical, torsional, and horizontal vibration modes for the span-model. Figure 7.15 Vertical deformation for eigenmode 1 and 3. Figure 7.16 Torsional deformation for eigenmode 2 and 5. Figure 7.17 Horizontal deformation for eigenmode 4 and 8. Verification of dynamic response Vertical accelerations of a timber bridge caused by one pedestrian depends on the first natural frequency for vertical deformation of the bridge (EC5-2): CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 67 200 ⎧ , 𝑓 < 2.5 𝐻𝑧 𝑀 ∙ 𝜉 𝑎 . = 7. 52 ⎨ 200 , 2.5 𝐻𝑧 < 𝑓 < 5.0 𝐻𝑧 ⎩𝑀 ∙ 𝜉 For a group of joggers crossing the bridge, the vertical vibration is expected at 3 Hz (Chapter 5.7, EC1-2). As the natural frequency for vertical deformation is larger than 5 Hz for both stiffness-models, the bridge deck structure will not resonate for pedestrian induced acceleration in vertical direction. Horizontal accelerations of a timber bridge caused by one pedestrian is calculated as (EC5-2): 50 𝑎 . = , 0.5 𝐻𝑧 < 𝑓 < 2.5 𝐻𝑧 7. 53 𝑀 ∙ 𝜉 As the frequency for horizontal deformation is larger than 2.5 Hz, the bridge deck structure will not resonate for pedestrian induced acceleration in horizontal direction. Limitations on transverse wind-induced vibrations are not considered in EC5-2 and are also disregarded in these verifications. 7.3.5 Torsional stiffness When looking at torsional stiffness, there are two aspects to consider. The first aspect is whether the cross-section has a closed or open force path, and the second aspect is how the force is transported in the lengthwise direction towards the supports. For closed domains, the torsional stresses all go in the same direction and the rotational stability is determined by the length of the lever arm from the rotation centre. For an open domain, the torsional stresses reverse and go in both directions in the cross-section, and the rotational stability is more dependent of the material. As a result, a closed cross-section has a higher torsional stiffness compared to an open domain. The difference of the force paths in an open and closed cross-section is illustrated in Figure 7.18. Figure 7.18 Torsional stress flow of a closed versus open cross-section. For the second aspect of torsional stiffness, the force should also be able to travel in a helix-pattern in the lengthwise direction to the supports. The schematically force path is illustrated in Figure 7.19. 68 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 Figure 7.19 Lengthwise force flow of a tube. The torsional stiffness of the proposed bridge design is achieved by the closed force path created by the two layers of tension bars as well as the straight LVL beams. A principle force flow is presented in Figure 7.20. Figure 7.20 Principle torsional stress flow of the superstructure. When looking at the lengthwise force flow, the torsional stiffness is mainly achieved by the outer straight beams, which carries the torsion with help of shear forces. It is also possible to see a helically force path through the superstructure. If only looking at the superstructure from above, the top part of the bent panels carries the forces in tension and compression through the diagonals in the built-up truss to the opposite straight beam. Then the forces diagonally go down vertically in the straight beams, and then horizontally again through the truss diagonals. This force flow is not as governing as the forces carried through shear in the straight beams, resulting in quite a poor torsional stiffness of the whole superstructure. A schematically lengthwise force flow through the diagonals of the superstructure is shown in Figure 7.21. The truss system of the actively bent beams is statically determined when the edges are transversely fixed at the boundaries. In this case, a transverse beam secures the rotation at the edges of the outer straight beams and curved panels. At the supports, the columns are integrated into the deck. By this, the deck is fixed in each end of every span, and consequently rotationally secured. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 69 Figure 7.21 Helically force flow through the superstructure. The bent panels create a truss system enabling forces in the horizontal as well as the vertical plane. The exact torsional stiffness of the cross-section is not calculated but is instead tested in a physical model in scale 1:10. Induced rotations on the bridge deck prove a weaker torsional stiffness compared to vertical and horizontal bending stiffness, which indicate that the governing failure mode of the superstructure will be torsion. If required, the torsional stiffness of the superstructure can be increased by inserting a third layer of tension bars, creating additional path for the stresses. Another possibility would be to make the connection between the superstructure and the supporting beams for the bridge deck more rigid. This will also create a new force path for the stresses. A larger lever arm will result in larger torsional stiffness, so the tension bars should be as close to the upper and lower edges as possible. As a general comment, the capacity of the drilled holes for the tension bars needs to be checked. 7.3.6 Moisture induced deformation Moisture dependent deformations must be considered if the moisture content in the entire bridge increase from factory to site climate. According to manufacturer, the LVL products are delivered with a moisture content of 8-10%. As the bridge is designed for service class 3, with assumed moisture content of 10-16%, moisture dependent deformations must be considered (Eurofins Expert Services Oy, 2020). The capacity of the bridge is verified for larger moisture content by the factor kmod. The dimensional changes due to moisture change with resulting stresses are calculated below. According to the manufacturer, the moisture induced dimensional change Δ𝐿 is calculated as Δ𝐿 = Δ𝜔 ∙ 𝛼 ∙ 𝐿 7. 54 Where Δ𝜔 is the change of moisture content, 𝛼 is the dimensional variation coefficient for each direction, provided by the manufacturer, and L is the dimensional length in the respective direction. 70 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 The total dimensional change in the transverse direction is the sum of the dimensional change of each panel. The resulting compression stresses due to dimensional change in transverse direction are taken by the compression steel tubes and tension bar. Change in vertical and lengthwise direction will lead to shear stresses in the connection between panel and steel element. Resulting stresses due to dimensional change are calculated for each direction respectively Δ𝐿 σ = E ∙ 𝜀 = E 7. 55 𝐿 The resulting dimensional change and corresponding stresses in the respective direction are listed below. As the moisture induced stresses are small related to stresses due to applied load, these are not considered in further detail design. The moisture induced stresses are presented in Table 7.25. Table 7.25 Resulting dimensional change and stresses due to moisture change Direction Length change [mm] Stresses [MPa] Transverse 0.19 0.08 Vertical 0.15 0.03 Lengthwise 0.22 0.11 7.3.7 Floating structure The floating pier is built up by transverse panel deck of solid timber and longitudinal IPE-beams with diagonal bracing. The beams are simply supported by pontoon elements with a spacing of 5 m, see Figure 7.22. Figure 7.22 Cross-section of the floating platform structure. The dimensions of the pontoons are governed by the load on the deck, which is the self- weight of the deck elements combined with pedestrian load. The principles of the partial CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 71 factor method are not used to determine the design load as only an estimated dimension of the pontoons is searched for. 𝑞 = 𝐺 + 𝑄 [𝑘𝑁/𝑚] 7. 56 The total load on each pontoon is calculated accordingly, where Cm is the mean circumference of the deck and n is the number of pontoons: 𝐶 𝑞 . = 𝑞 [𝑘𝑁/𝑚] 7. 57 𝑛 Archimedes principle of uplift force in water is used to determine the required volume of the pontoons, where the uplift force FL must be equal to the total load on each pontoon: 𝐹 = 𝑉 ∙ 𝜌 ∙ 𝑔 [𝑘𝑁] 7. 58 The required dimension of one pontoon is 2 x 3.5 x 5 m (h x w x L). 7.4 Local design 7.4.1 Prestressed connections The prestressing between the lamellas should be sufficient to compensate for the bending stresses due to transverse bending and prevent slip due to transverse shear. The required compression force is calculated for the Service Limit State. A steel bar is threaded through the cross-section to enable pre-tensioning of the cross- section. Compression steel tubes secure the forced curvature of the LVL panels, and steel plates distribute the compression force to the panels. Curvature force The force required to create the sinusoidal shape is calculated as a point load on a continuous beam to the desired deflection. The load-situation can be translated into a simply supported beam with counteracting bending moments at the supports, as illustrated in Figure 7.23. Figure 7.23 Load model of the curved panels. To achieve a sinusoidal pattern, a certain deflection p is required and is defined in Chapter 7.3.1 about induced bending. The corresponding point load P to achieve this 72 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 deflection can be solved from the equation for a fixed beam with midspan deflection p and bending moment M1 and M2 𝑀 𝐿 𝑀 𝐿 𝑃𝐿 𝑝 = + + [𝑚𝑚] 7. 59 16𝐸𝐼 16𝐸𝐼 48𝐸𝐼 The flexural rigidity EI is modulus of elasticity for flatwise bending times the modulus of elasticity of one LVL panel. The span length is equal to the frequency of the sinus- curves. The required compression on the LVL-panels must be larger than the required point load P. Required compression on the pre-tensioned lamellas The friction between the LVL panels must be larger than the transverse load effect. This is secured by a threaded prestress bar. Dywidag prestress bars are usually used in stress- laminated decks in Sweden. In this case, a threaded bar of diameter 26.5 mm is chosen (OIB, 2018). The minimum prestressing force is the largest value of transverse bending stress or transverse shear stress on the bridge deck. This is illustrated in Figure 7.24 and stated in the following equation. Figure 7.24 Minimum prestress force on the lamellas. 1.5𝑉 6𝑀 𝑁 = 𝑚𝑎𝑥 , [𝑘𝑁/𝑚] 7. 60 ℎ ∙ 𝜇 ℎ The transverse load effects due to uniformly distributed load across the deck, which is in this case is traffic load, are calculated as: 𝑞 𝐿 𝑀 = [𝑘𝑁𝑚] 7. 61 8 𝑞 𝐿 𝑉 = [𝑘𝑁] 7. 62 2 CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 73 For planed wood to planed wood, moisture content > 16% and load perpendicular to the grain, the friction coefficient 𝜇 is set to 0.40 according to table 6.1 in EC5-2. The area of the shear stress is taken as the effective area of the steel plate, where the forces are assumed to be distributed in a 45° angle from the edge of the plate. The bending stresses are calculated over the height over the panels times the effective width of the panels. In this case, the bending stresses are dimensioning for the required friction force. The corresponding compression force on the contact area between the LVL- panels are calculated as following, 𝜎 being the resulting bending stresses and wef times hef the effective height and width: 𝑁 . = 𝜎 ∙ 𝑤 ∙ ℎ [𝑘𝑁] 7. 63 The required tension force in the steel bars is calculated as: 𝑁 . = 𝜎 ∙ 𝐴 [𝑘𝑁] 7. 64 The spacing between the contact points in the bridge deck are not considered in these calculations, which therefore leaves a rather conservative result. If the holes for the bar is 20 percent less than the height of the lamellas, or if the hole size is less than 50 mm (EC5-2) the holes can be disregarded from capacity calculations. For the chosen dimension of prestress bar, these calculations can be disregarded. Verifications The corresponding compression stress from the steel bars must be larger than the required curvature point load P on the panels: 2𝑁 ≥ 𝑃 [𝑘𝑁] 7. 65 Furthermore, the LVL panels must have sufficient capacity to withstand the compression forces. The required compression stresses on the panels are distributed from the steel plates to the panels and are verified for the design compression capacity perpendicular to the grain, h being the height of the panels: ℎ 𝜎 ≤ 𝑓 ℎ . . [𝑀𝑃𝑎] 7. 66 The required tension force Nc.bar must be smaller than the maximum initial stressing force of the bars, Pm.0.max. In addition, maximum allowed pretension of the bar is 70% of the ultimate tension capacity: 𝑁 . ≤ 𝑃 . [𝑘𝑁] 7. 67 2 𝑁 . ≤ 0.7𝑓 ∙ 2𝐴 [𝑀𝑃𝑎] 7. 68 Finally, the compression tubes must have sufficient capacity to withstand the compression forces. The compression tubes must be checked for buckling if the 74 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 slenderness ratio 𝜆 is larger than 0.2. The minimum compression stresses distributed by the steel tubes is verified as: 𝑁 . 𝜎 . = [𝑀𝑃𝑎] 7. 69 2 ∙ 𝐴 𝜎 . ≤ 𝑓 7. 70 The required forces on the cross-section are summarised in Table 7.26, with resulting utilisation ratios of the compressed cross-section in Table 7.27. Table 7.26 Required prestress forces Force Value [kN] Point load for deflection P 186 Compression force on the panels Nt 880 Tension force in the steel rods Nc.bar 6.3 Compression force in the steel tubes Nc.tube 3.1 Table 7.27 Resulting utilisation ratios of the prestressed connection Stresses Utilisation [%] Compression stress on the panel 16.8 Max initial tension in the prestress bar 1.0 Utilised tension capacity 0.54 Compression of the steel tubes 0.7 7.4.2 Lamella joints As the superstructure is designed to be continuous, the LVL straight beams and the bent panels need to be joined into separate pieces before assembled to the whole superstructure. For the straight beams, slotted-in steel plates are used and located where the bending moment is as small as possible. For the total superstructure length 27.3 m joints are required, and the joint design is illustrated in Figure 7.25. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 75 Figure 7.25 Detail of joint for the straight LVL beams. For the bent panels, the same principle with slotted in steel is used, but here steel hooked plate strips are used instead to increase the shear friction between the elements. To assure that there is enough capacity for the bent panels the joint of two LVL panels is placed at the connection to another continuous panel, see Figure 7.26. Another steel hooked plate strip is inserted between the two panels to achieve even more friction between two bent panels. This extra shear friction results in less required prestressing force in the threaded bars. 76 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 Figure 7.26 Detail of joint for the bent LVL panel as well as the connection to an adjacent panel. To make sure that the superstructure does not have a local weakness, the joint of a bent panel is never at the same location as the joint of the adjacent panel. Consequently, the joints are phase shifted to distribute the weakness of a joint. Figure 7.27 shows in an illustrative way the phase shifting of the bent panel joints and the location of the joints of the straight beams. Figure 7.27 Phase shifted joints for the bent LVL panels as well as the joint locations for the straight LVL beams. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 77 7.4.3 Column to superstructure connection The support columns are connected to the inside of the LVL panels with the use of a steel cap and bolts. The steel cap is integrated in the superstructure during the assembly and connected to the columns when the structure is lifted into place. A detail of the connection is illustrated in Figure 7.28. Figure 7.28 Connection between superstructure and support. 7.5 Production To produce the bridge, prefabricated parts are transported to the site where they then are assembled on ground before lifted to their final positions. The schematical plan for the production on site is as following. First ground preparation is done before concrete foundations are casted on their locations. The two column supports on each side of the road are built, with temporary supporting scaffolding. Elements of the superstructure are preassembled into longer pieces before being bent and prestressed in the assembling of the continuous superstructure. This also applies for the straight staircase. The straight staircase is lifted to its position and attached to the platform support, providing horizontal support to the platform and bridge superstructure. Then the bridge superstructure is lifted by cranes onto the supports and connected to the pinned connections. After that, the curved steel stair is lifted to its position and connected to the cantilevering parts of the platform tower as well as supported from underneath. At last the railing and decking is built on top of the superstructure as well as the platform and stairs. For the superstructure, prefabricated beams and panels shorter than 12 m are transported to the site. As a result, no specific allowance for transport must be applied for. On site the straight lamella beams and the bent lamella panels are joined into one long continuous piece. A schematic sketch of the joint positions is shown in Figure 7.27 as previously described. 78 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 The beams and panels are pre-drilled for better precision in addition to a simplified assembly process. One straight beam is used as a reference position with threaded bars put through. Thereafter each lamella panel is mounted one by one onto the threaded bars with distancing pieces in specific positions. Figure 7.29 shows the assembly method of the superstructure in a schematic way. When all panels and distances are at the right positions, prestressing of the threaded bars is done by fastening the nuts. This will reduce the distance between lamella panels, and the distancing pieces will prevent shrinking which creates bending of the lamella panels. The nuts are fastened until the gap between the panels is closed and the required prestressing force is achieved. Figure 7.29 Schematic assembly method of the superstructure. When executing the assembly method in a model in scale 1:10, it was found important to screw the nuts in the same pace along the whole superstructure to avoid that one end shrinks faster than the other end. An uneven tensioning caused deformation on some of the tension bars. The curved lamella panels are held together by friction created by the prestressing force in the threaded bars. Worth of noting is that a bent lamella with a radius of 14 m and a threaded bar centre-to-centre distance of 2.1 m the offset of drilled hole between the straight beam and the lamella is only 1.4 mm! CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 79 8 Discussion The aim of the thesis can be summarised in the following two research questions: “In what way can a bridge proposal be achieved that meets the requirements and ideas of the client Next Step Group for the planned residential area Wendelstrand?” and “In what way can an iterative design method be used to develop a structural concept that enhances both engineering solutions and architectural qualities?”. The challenge was to design a bridge that meets the ideas of Next Step Group, that corresponds with the profile of Wendelstrand and has a customised design for the actual site. The design proposals were developed according to a set of design criteria and evaluated for specified demands. The intention of integrating architectural qualities with engineering solutions was specified as a governing design criterion to secure a development of the design proposals within the specified framework of contextual criteria and requirements. 8.1 The proposed bridge The bridge design answers to the request of the client Next Step Group on several levels. Firstly, the straight path establishes both the physical and visual connection from Wendelstrand to Landvettersjön. Secondly, the strong wood-profile is recognised through the materiality of the structural concept and is included in the detailing. An example is how the columns stretch up into the superstructure, where the wave-pattern is straightened out to provide enough space for the connection. Lastly, the bridge has additional qualities, rather than just being a connection between point A and B. The viewing platform enables pedestrians to pause and look out over Landvettersjön. In addition, the circular floating pier enables a variety of possible activities in the lake such as the possibility to swing, attach floating restaurants, sauna boats and add a canoe rental. This will be available for the public as well as residents in Wendelstrand. In addition, pedestrians can experience the bridge structure from underneath as both the stairs and accessible ramp are led underneath the platform structure. Attention for architectural qualities are reflected in the design choices regarding the visual expression of the bridge. A challenge with the design proposal was to solve the connection between the viewing platform and the lake, which has a height difference of approximately 8 m. An idea was to lead the staircase out in the lake and connect it to the pier. Since only floating structures could be used in the water, it would be difficult to provide horizontal stability to the viewing platform without creating a large anchoring to the land. Therefore, the rational decision was to lead the staircase directly down on the shore strip. In that way, the shore protection is respected and permanent impact in the lake is limited. The requirements from the Swedish Transport Administration are met in the final bridge design, in terms of clearance as well as general detailing and dimensioning of the bridge structure. However, only the main connection between the superstructure and columns are thoroughly designed, in addition to the connection between the bent LVL panels. As many connections are yet to be solved, the bridge design must be further investigated before construction is possible. For sustainability aspects, the structural height of the superstructure is strived for to be as small as possible, to reduce the amount of structural material. Further optimisation of reducing the amount of timber 80 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 can be done in the rest of the bridge structure, for example in the viewing platform or staircase. Moreover, a strong integration between architectural and engineering qualities are found in the bridge proposal, where the structural concept is a big part of the visual appearance. Curved beams in a sinusoidal pattern mimic the behaviour of a horizontal truss, which is determined by the geometry of the beams. Active bending and pre- tensioning of the deck provides required horizontal and vertical stiffness. Consequently, a characteristic visual appearance is assigned to the bridge structure. 8.2 The design process No examination of different design methods was performed in the beginning of the thesis, but the chosen method was adapted without any further ado. The purpose of the iterative design method was to investigate, develop and evaluate to result in the most suitable bridge design for the current site. Relating to the stated research questions, the iterative process aimed to include expectations of the client combined with efficient engineering solutions and architectural qualities. A spatial investigation was concentrated on possible paths, affected by aspects concerning topography conditions and contextual requirements, in addition to the visual and conceptual relation to Wendelstrand. Structural concepts were then applied to these paths. An examination of how or if these structures affected the above-mentioned aspects was then iteratively performed. Furthermore, the possibilities of including other functions in the bridge design were investigated in relation to the structural concept and contextual limitations. Qualities of the intentional concepts were combined and developed into more refined designs, which in turn were evaluated and compared to each other. To evaluate the different aspects separately and in relation to each other, and ensure a broad investigation of possible qualities, every contextual demand was, intentionally, not followed strictly in each phase. By doing so, the design task was looked at from different angles and a large and thorough variety in the design concepts were achieved. The different perspectives resulted in a conscious development of the final design proposal. Lastly, the final bridge proposal aimed to meet every defined contextual requirement, as well as fulfilling the intention of the design criteria. Hand calculations were used to obtain the dimensioning forces and bending moments in the structure, which in turn determined the dimensions of the load-bearing parts. Accurate simplifications were used to determine the effect of active bending of the LVL panels. The bending was integrated in the calculations by applying initial deformations in combination with requirements for stressed skin panels. As a limitation, only the most important connections for the structural concept were calculated and designed. That is, the connection between the LVL panels and required friction force obtained from the prestress bar. Consequently, several details were not validated or optimised, but preliminary designed based on reference projects. By building physical model an understanding of the consequences and possibilities of active bending of the LVL panels were achieved. It was a quick and easy way to understand how the built-in forces and stresses behave. There was no time for an extra literature study of active bending, so physical models were a great tool. The insight of achieving the same deformed shape by either longitudinally push the lamella together CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 81 or by pressing it outwards using transversal spacers was useful when developing the design and the detailing of the lamellas. As a result, a feasible assembly method was developed and proven in scaled physical models. Time was spent on building physical models to verify the structural stability and assembly method. Consequently, less time was given to create a detailed FE-model to analyse the accurate structural response. The general limitation of the thesis was determined with the chosen site, and the intention of designing a site-specific bridge for a real client. Clear boundary conditions and specific requirements were defined, which established a limitation for the bridge design. Following, as the scope concerned the conceptual design of the bridge structure, less time was given to design the floating pier structure in the lake. A suggestion for cross-section and load-bearing concept was given, in addition to a proposal for different activities on the pier. 8.3 Suggestions for future work As a result of the limitations of the thesis, and the obtained answers to the research questions, new questions arose. The following listed topics would contribute to developing the final design proposal into a more feasible bridge design. Torsional stability Physical models proved that the bridge structure was likely to fail due to lack of torsional stability above other structural responses. As accurate simulations of the rotational capacity were excluded from the design verifications, this would be prioritised in further development of the design. If the torsional stability proved to be insufficient, it would be suggested to add another pair of prestressing bars adjacent to the existing ones. This would create a squared geometry, which would allow for an additional horizontal path for the helicoidal load transportation in lengthwise direction. Another solution would be to secure a stiffer connection between the superstructure and the transverse beams on the bridge deck, as well as adding cross-bracings underneath. Active bending Active bending in the bridge design needs further research. As the solution is based on understanding from physical experiments, a thorough literature study on the topic would be beneficial for the development of the bridge proposal. A simulation using Finite Element tools could also contribute for further development of the concept. This would allow for a quick comparison between actively bent beams with built-in stresses and pre-bent elements. As the geometry would still mimic a horizontal truss, could the same element dimensions be used, how would the vertical capacity be affected, and how would the required friction force from the prestress bar be affected? With the use of computer simulations, an investigation and optimisation of the relation between wave amplitude and frequency, number of elements and element dimensions could be performed. In addition to this, the torsional behaviour could be monitored for the solutions presented above. It would also be easier to obtain quick results when comparing the structural solution for different timber materials, or a combination of several. In addition to this, a couple of interesting aspects concerning the combination of active bending, LVL and bridge structures is proposed for further research: 82 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 - Consequences of applying the method concerning material and connections. How are actively bent LVL beams affected and what are the long-term effects? - What are the exact structural benefits, and do they exceed the complexity of assembly and construction? Connections The superstructure is designed as a continuous element. Consequently, the joints of the curved LVL panels must enable continuous behaviour. A design for these joints and their locations are proposed, but should ultimately be further investigated, optimised, and verified before being applied to a real bridge structure. It should also be mentioned that the thinness of the LVL panels may complicate such a solution, as the amount of effective material in the connection is small. Floating structure and foundation elements To achieve a complete bridge design proposal, the floating structure must be solved. Firstly, the design of the pontoons should be further developed to obtain a structure with small visual and physical impact on the lake. Different load cases resulting from the various activities proposed for the client must be considered. Secondly, as the floating structure is suggested to be anchored to the staircase abutment, this foundation element requires careful detailing. Following from this, a design solution for the bridge foundation elements is outside of the thesis scope, which should be prioritised in further development of this design proposal. Horizontal stability The global horizontal stability of the platform support was secured with the use of the staircase as a structural member. Future work should include further investigation into other solutions to the stability challenge, preferably with the aim of reducing the amount of structural material. CHALMERS Architecture and Civil Engineering, Master’s Thesis ACEX30 83 9 Conclusion When designing and building a timber bridge many and various aspects must be considered. Common for all bridges, structural aspects must be included when developing the bridge design. The bridge is in turn defined by site-specific conditions, which in this case is Wendelstrand in Mölnlycke. Unique preconditions and limitations require unique solutions, which are developed through thorough investigations and identification of design possibilities. As a result, a site-specific bridge proposal can be developed to fulfil the identified contextual requirements. The iterative design process generated a large variation of possible solutions and provided a wide perspective for the design proposals. It was a very useful tool to see the task from different perspectives and to move forward in the design process. Clearly stated boundary conditions and design requirements define the framework for the design development, and a direction for the evaluation of the proposals. Governing design factors, contextual requirements, and site limitations were prioritised in relation to the research question. This proved to be helpful when comparing different proposals or evaluating conflicting designs choices. Simultaneously, challenging the weight of the defined evaluation criteria proved to be necessary to broaden the perspective on possible solutions. As a result, the final design proposals were developed consciously, where every possible aspect was considered. At last, the outcome of the whole thesis could be summarised in the following: A specific site implies unique preconditions and requires a unique solution. 84 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 10 References Boverket. (2016). Boverkets konstruktionsregler, EKS 10. Boverket. Boverket. (2019a). 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Calculations – Global dimensioning and Detailed design 88 CHALMERS, Architecture and Civil Engineering, Master’s Thesis ACEX30 APPENDIX A - Bridge database PRINCIPLE DESCRIPTION STRUCTURAL CONCEPT VERTICAL LOAD HORIZONTAL LOAD POINT LOAD ROTATIONAL STABILITY FOUNDATION PRODUCTION ASSEMBLY MATERIAL DURABILITY CONNECTIONS Slab bridge Beam bridge Pedestrian Bridge Main structure consisting of Design load of 5 kN/m² Due to its S-shape, extra Point load is transferred The bridge gets rotational The bridge stands simply Gerber hinge between beam Local wood engineering firm The continuous beam Laminated glulam beam The prefabricated concrete Crossing the Neckar River trepped block laminated Bending of beams to help against horisontal down to the foundation by stability due to its S-shape supported at two concrete parts. manufactured the bridge creates moment diagram Railing including horizontal slabs has an overhang of 30 Location: Neckartenzlingen, glulam beams that follow a compression of pylons forces is gained. The bending in the beam. columns in the middle. The The beams have hinged components. with zero crossings in the stainless steel wires and cm, which works as a roof for Germany gentle S-shape. For an horisontal force is taken by ends are also considered connections at the support. The cantilever bridge timber main span, which are handrail of acetylated timber. the structure. The glulam Designer: Ingenieurbüro efficient utilization of material the glulam blocks as well as pinned with rotational parts was prefabricated and designed as so-called Decking made of pre-cast beams gradually angling 30 Miebach the cross-section is graded the transversal smaller allowance. transported to the site in Gerber hinge. In this way the concrete slabs with anti-slip degrees in beneath the Engineer: Engineering office according to the stress beams. three parts. design achieves sensible surface. bridge helps to keep rain Blankenhorn, Gottlob usage. Prefabricated concrete slab transport dimensions and a from reaching the structure Brodbeck GmbH & Co. KG with anti-slip surface simplified assembly. itself. Completed: 2017 In addition, there are steel Length: 96 m drainage channels fitted Width: 3 m below the joints between the concrete slabs. As an extra precaution, the wood is also treated with a thin glaze to protect it against damp and air pollution. The cable railing is made from steel and stainless steel, Bow River Pedestrian The primary structural Uniformly load distribution Horisontal steel bracing 40-50 pedestrians would The horizontal steel trussing Tension rods tie the propped The central drop span sits on 3 segments: two haunch Extensive prefabrication Glulam girder and timber 75 year design life Bridge system is simple: Propped transferred down to the handles horisontal load. cause lateral "lock-in" and provides both the diaphragm cantilevers down to neoprene bearing pads on glulam girders on either side allowed the pices to be deck, horisontal steel truss, Location: Banff, Canada by drilled piers located just foundation through bending. increase the resonant and support for the service Rundlestone-faced concrete notches in the receiving ends and a removable, modular shipped to site and concrete foundation. Designer: StructureCraft outside the normal river accelerations. Two mass- pipes concealed just below abutments at either end of of the cantilevered glulam timber deck (allowing for assembled on the shoreline. Engineer: Fast + Epp channel, 40m haunched dampers placed under the the bridge deck. The bracing the bridge. girders. The neoprene pads ease of replacement and All cutting, drilling, sanding A visually minimal stainless Completed: 2013 glulam girders cantilever deck (visually exposed) were is configured such that only Control of vibration due to transfer the compression access to the service pipes and finishing was performed cable guardrail system Length: 113 m from either side to support a tuned to address footstep the timber chords are small width: two uniquely- across the halving joint. Long running beneath) indoors under controlled involving 135m long Width: 4 m 34m suspended span. and jogging excitation continuous, resulting in very tuned mass dampers were timber screws either side of conditions so that members continuous cables, required little length expansion, one of suspended beneath the the connection reinforce the were protected from the fine-tuned pretension The bridge cross section the bonus features of wood. bridge to reduce dynamic joint to prevent splitting elements both in transit and analysis to ensure adequate comprises twinned sets of forces. perpendicular to grain, and a on site. Jigs were built to tension in the summer, and glulam girders stepped to steel drag strap atop the ensure accurate assembly of avoid overtension in the follow the flow of forces, beams connect them and the main bridge components winter. which range in depth from transfer axial loads. in the field. Erected in three 2.6m at the piers to 0.9m at lifts over two days. the suspended span. The 4m wide deck is made of pre- stressed solid timber panels, removable to provide access to the service pipes hidden below, and to allow for simple replacement, if required. Stuttgart Wooden Bridge This family of mass timber Load is carried through The load is transferred in Extra thick in mid span (at The rigid intregral joint Reinforced concrete The superstructure is The block glued laminated The girder is lifted to its final Spruce block-glued The superstructure is Location: Remstal, Germany bridges are the world's first bending moment, cross- shear by the stiff cross- largest bending moment) provides rotational stability abutments which takes up monolithically integrated into timber beam is connected location. Very important that laminated timber beam. protected against direct Designer: Knippers Helbig + integral wooden bridges and section also varies in line section of the girder. It due to point load? due to the extensive the shape from the girder. the reinforced concrete into one big piece. Then 78 there is the exact right angle Decking of 8 cm thick weathering by the bridge Cheret Bozic have therefore no joints and with the moment distribution. seems that there are distribution of the long The rebars frin the girder is abutmends. There is no rebars with a length of 2.3- and location at the precast carbon fiber deck and a diffusion barrier Engineer: - bearings. The bridge transversal steel bars for the rebars. casted to the abutments. expansion joint at the 3m were pressed into the abutments since the reinforcement concrete. cover on top of the timber Completed: 2019 superstructure, a block-glued hand rails which could aslo transition to the abutment. girder. 1.2 m of the rebars concrete is going to be deck. Sensors for permanent Length: 30 m laminated spruce timber contribute to the horisontal are glued in the girder. casted on site and that there moisture monitoring are Width: 3.2 m beam is connected rigidly to capacity. should be a tight connection. installed in the solid wood the reinforced concrete After rebar connection, and structure at relevant points. abutments. The stepped, Use shear locks at the installation of sensors, it is laminated timber beam is abutments? Looks like so. transported to construction The end faces of the girder is formed corresponding to the site. protected with a special end- moment force diagram. grain seal to permanently revent poisture from beign transported over the end- grain surfaces into the girder. "The carbon fiber reinforcement is extremely durable, which promises a significantly longer life span compared to conventional solutions of the highly stressed bridge deck." Castle Moat Bridge The world's first composite The plexiglas-wood structure Steel frames in the bottom Point loads from pedestrians The high composite girders The forces goes down to The PMMA is mounted with The whole superstructure is First the foundation is casted Glued laminated timber The wooden parts are Location: Darmstadt, bridge made of wood and on each side of the walkway distributes the horisontal are led directly through the provides rotational stability vertically into the ground, several bolts(?) to the upper prefabricated in one piece. on site. The bridge is then Thermoplastic PMMA protected with steel sheets Germany plexiglas. functions as beams where forces. steel frame to the two due to shear. through two I-beams and and lower glulam girders. transported to the site in one Steel covering as well as that cantilevers over the Designer: TU Darmstadt, FB It is a temporary model the lower timber handles composite girders as well as then into concrete The composite structure is piece, and lifted to its final steel frame beams. The bolt/screw holes Architektur und FB Statik structure for research tension and compression in it also is distributed The bridge is 4m wide and foundations at each end. fastened to I-beams over position. are covered as well. Engineer: Fast + Epp purposes. the upper one, while the longitudinally by the steel the supports are placed as both supports. Completed: 2007 plexiglas transfers shear. frames. wide as possible. This gives No horisontal force goes into The PMMA is completely Length: 26 m Composite-beam bridge. better rotational stability than the wall to protect the recycable and could be Width: 4 m To maintain the cultural and if the composite-beams castle's historical aspect. dismantled into smaller parts historical apsects of the would be as narrow as the and used again in other castle surrounding, the walkway. projects when the research bridge stands on two phase is finished. supports in the moat and is Due to the cantilevering at cantilevering the last part to both ends of the bridge, the the pavement. span shortens which also increase rotational stability. Aurland Lookout, Aurland, Bending. Load bearing steel Vertical support of the deck Horisontal stiffness of the Point load is transferred to Rotational/lateral stability in Two concrete foundations; Wooden joints on the side of Prefabricated and Cast-in-situ foundation. Load bearing galvanised Pressure treated pine makes Norway frames of rectangular section in inclined steel "legs", bridge is provided by the the edge beams through the the bridge deck is provided upper foundation deals with the rail: semi-circle joints customised steel beams Framework and on-site steel, covered with pressure the wood more rot resistance Location: Aurland, Norway covered with pressurised connected to the lower upper foundation. transversal framework, to the by transversal steel horisontal wind forces (invisible from a distance) (walkway). Prefabricated and assembly of the deck treated pine. Two parallell (needs continuous Designer: Saunders pine wood. foundation element on the foundation elements. framework in the deck (shaped as a lying U, bent massive wood panels (crane). Steel beams lifted frames with a rectangular maintenance to avoid arkitektur, Wilhelmsen mountain side. Tension in fastened to the rock with ("railing") on place by cranes, section; 0.3m wide, 1.1m degradation). Will show arkitektur the curve of the ramp tension bolds. Lower framework welded on site. high cross section. Floor: signs of weathering over Engineer: Node Rådgivende foundation supports the two Covered with wood panels steel trusses with c-c 1 m time. Ingeniører AS steel legs, bolted to the and massive wood. covered with 0.65 m massive Completed: 2005 mountain-side. Cross- wood, connected with Length: 30 m section of the steel legs: screws from the underside of Width: 4 m 0.3x0.5 m the steel frame. Fussgängersteg Simple supported beam The pavement beam (also Threaded rods in a grid of A point load is firstly The hand rails also gives Concrete foundation. There Two threaded rods in the The bridge was After prefabrication, the The decking, railing and Due to the wavy shape, Gheidgraben bridge. wavy) takes care of the 600mm handles the distributed transversally stability for bending in are elongated holes at one of walkway wavy beam prefabricated. bridge was lifted to its handrail consists of massive water can fall off the Location: Gheidgraben, vertical load. horisontal forces. through the rods to the two longitudinal way. the support, to allow elements. The boards are position. rough-sawn oak boards. structure, and it can be dried. Schweiz outer beams, and secondly Relatively short span and shrinkage and swelling. spread apart or pulled Steel threaded rods. The intention is to let the Designer: Werk1 Architekten longitudinally through the wide bridge, so that helps together with washers and bridge have a aged und Planer AG wavy beam elements. with the rotational stability. LNP 120x120x10mm steel nuts as well as sleeves. appearance with rust and Engineer: Makiol Wiederkehr profiles over the supports. wather torn wood. AG Completed: 2013 Length: 7.45 m Width: 1.97 m The Claude Bernard Continuous arch with metal Consists of three variable Horisontal steel beams (with The load is transferred The bridge is curved, which Branch-like steel columns. Bolted connection between Pile-mounted abundments Foundations and connecting Three steel tube "arches" The two steel arches are Overpass framework. The three steel dimensional arches. Allows bracing) provides horisontal transversally to the steel provides rotational stiffnes. Probably concrete the lifted steel truss and the and stairways. Prefabricated stairs assembled, thereafter connected by metal protected by wood cladding, Location: Paris, France tubes in an arch shape are for larger intertia but lesser stability. frames with the help of the The perpendicular ramp also foundation below. rest of the structure. timber the framework, covered by to prevent damage. The Designer: DVVD connected by the metal weight of the structure; transversal steel beams. can provide rotational The wooden parts works arch including cladding and prefabricated deck was lifted timber cladding. Timber wood can easily be replaced. Engineer: DVVD framework. allowing the structure to be When the load reach one stability (depending on more like cladding. deck. into place and connected to fretwork on each side of the Water can easily run through Completed: 2015 Rest areas and viewpoints in lifted by crane, as well as steel frame it can be connection with the bridge) the deck that acts as protective the gaps between the wood. Length: 100 m the middle. lowering the height of the transferred to the foundation supports in one night. railing. Width: - arch. through one of three Counterweights was used to It seems a bit difficult to In order to reduce the height longitudinal steel beams. increase the maintain the steel structure. of the structure and its load-bearing capacity of the Maybe it is possible to impact upon the landscape, crane during assembly remove larger segments of and in the interests of a more the wood cladding? subtle outline, the load- bearing structure has been conceived as two variable- inertia three-dimensional arches. This solution enhances the inertia of the structural beams and reduces weight. Yusuhara Wooden Bridge A 4.4 m wide bridge girder The load is transferred The yellow marked girder is A point load is transferred The homogenuous girder Steel frames and steel RHS There are cut-outs in the The cedar pieces are locally The foundation is casted and Local Japanese laminated The strucutre is protected by Museum consisting of 11 cypress through steel frameworks on able to take the horisontal through the overlapping and (marked in yellow in picture in mid column leads the beams to make perfect fit produced, and they are steel frames constructed. red cedar. the upper roof as well as the Location: Takaoka-Gun, japanese laminated lumber the sides and through the forces. decreasing grid. The beams for horisontal load forces down to concrete and give stability. But on assembled on site. From the center pillar (with form of the structure narrows Japan by 180x700mm, are resting 'Tokio' structure down to the are taking the point load by distribution) also helps with foundations. some places the beams are steel core) the cantilevering down the closer to the Designer: Kengo Kuma and on a center pier by traditional mid pillar. bending, through the rotational stability as well also connected by screws. The girder might come in 'Tokiyo' system is built up column it gets. Associates Japanese and Chinese compression to the next as the "moment stiff" column larger pieces. with a temporary framework. Engineer: Kengo Kuma and cantilever structure. layer, bending, compression connection at the ground. The framework is removed Associates and then finnally down to the when the 'Tokiyo' and girder Completed: 2012 "The overall structure is column. The central column is is connected to the other two Length: 47 m created utilizing the crossed shaped and has a steel frames. Width: 8 m overlapping wood member large cross shaped system called 'Tokiyo' used foundation, to prevent so it in traditional Japanese starts to rotate. temple architecture, creating a presence (materiality) and abstractness which should be called 'wood masonry' that cannot be obtained with a framework type of structure. Strut frame bridge Frame bridge King post truss bridge Krúsrak bridge Truss arch bridge. Distributed in the curved Transversal horisontal steel Can support the heaviest Steel ties in the upper part Concrete pylons Block-glued laminated timber Completely prefabricated Transported to site in one Accoya wood: acetylated Due to chemical treatment Location: Sneek, The two wooden trusses are truss elements, through the beams in deck transfer load load class of 60 tons. helps with the rotational from accoya piece and lifted to its place softwood (locally produced in the bridge can withstand Netherlands linked together in the middle vertcal deck to the concrete to the outer longitudinal Lifts point load and transfer stability. Arnhem). Road-surface of insects, fungi and weather. Designer: OAK Architects for stability. Optimising of the foundations wooden beams. through truss. As well as steel (the original design Life expactancy 80 yeards Engineer: Lüning cross-section for traffic directly through the bottom intension of wooden deck Completed: 2010 resulted in the curved shape beam. required a thickness of 2m) Length: 32 m and triangular framework. Width: 12 m Height (clearance): 4.6 m King post truss bridge with struts Erdberger Footbridge in Strutted frame and Bending in bridge deck. Bracing in the legs, takes A point load in mid of the Cross bracing on top to give Concrete foundation surface The foundation is considered Prefabricated and In-situ casted concrete Main structure: glulam of A steel plate is covering the Vienna suspension structure. Two Load is transferred either to horisontal load. bridge is ransferred through horisontal and rotational meets legs perpendicular. pinned. The connection preassembled sections foundations + four pairs of domestic larch. Deck: end cut (and thus fiber Location: Vienna, Austria main girders in glulam (0.36 the tension cable and then bridge deck, through the stability. between leg and bridge deck (legs) supports with wind bracing. plywood boards + 6 cm opening) for where the Designer: Arch. Johannes m width, cc 3.34m). Span down in compression tension cable up to the consists of steel plates Thereafter the two blacktop. Foundations: in- inclined legs meets the Zeininger lengths: 14.8m - 52.8m - through the tilted legs, or diagonal legs and then down through the leg and then prefabricated and situ concrete. Steel bridge deck. Engineer: Ingenieurbüro 14.8m. Top point 6m above directly to a tilted leg and to the foundation. bolts holding the wood and preassembled sections secondary beams There is also a wider plate Alfred Pauser, deck. then down to foundation. More at the sides, the load steel together. (approx 45 m long) lifted into (galvanised), suspensions, on the legs, making the Ziviltechnikergesellschaft für Inclination: 6% goes through bending in the place and connected to the railings and masts water run off and away from Bauwesen bridge deck, and then down supports. Wind bracing structure. Bridge deck is a bit Completed: 2003 by compression in the tilting above the deck installed (maybe too small) Length: 85.2 m legs or directly to the before the middle, cantilevering out over the Width: 3.7 m foundation. suspended section is lifted structure to prevent water into place. from rinning down the superstructure. Arch bridge Pedestrian Bridge in It has the shaped section of Main structure in bottom of Transverse beams in upper The frame connecting the The curvature of the bridge Concrete foundations dug Bolts into metal sheets Prefabrication of the The superstructure is lifted to Glulam beams, galvanized There is a small gap Zapallar a boat, but corresponds to a bridge handles the bending part of the superstructure three beams helps to together with the frame of deep into the clay on both superstructure in Santiago. its final position. After that metal net between the lower beam and Location: Zapallar, Chile triangular inverted arch, moment. provides horisontal stability. distribute point loads as well the three beams provides sides of the road. electrical equipment, and the covering "råspont" to let Designer: Enrique Browne which is longitudinally as meeting the required rotational stability. pathway and railings are it dry out. But other than that, Engineer: Alfonso Larrain curved, which reduces deflection criterias. added. As well as a no documented durability Completed: 2008 distortion and enhances the safetynet to prevent people methods. Length: 20 m structure. from throwing rocks on the Width: 2 m cars. Main structure consists of a curved beam in the bottom. Secondary structure is two parrallel beams to the main structure located as the two vertices of an upside down triangle. The beams are joined as a frame with the boardwalk and the hand rail skeleton. Kintai Bridge Wooden arch bridge. Vertical load is transferred The structural weight of the Point loads are distributed in The timber walkway consists Original stone piers was later The bridge was initially In the later eras of rebuilding, - - - Location: Iwakuni, Japan Five spans (three arch through the arches to the bridge provides horisontal the grid of timber elements of layers of crossed wood changed to concrete constructed without the use standard rulers were used in Designer: Kikkawa Hiroie spans, two girder birdge foundation elements. The stability when subjected to and transferred through the elements, which provides foundations. On dry riverbed, of nails (the numerous small combination with drawings Engineer: Kikkawa Hiroie spans), four stone piers and girder bridge at the bridge floods. The width of the logitudinal elements of the rotational stiffness to the the bridge is supported by pieces was fitted carefully which simplified the Completed: 1673 (2004) two wooden piers (at ends work mainly in bending, supports provides horisontal walkway to the foundation bridge. The structural height braced timber columns. together), but copper sheets rebuilding significantly. Length: 175 m endspan, on dry riverbed). and has a shorter span stability. elements. of the arch varies accross were laid over the main Width: 5 m The wooden deck was between the vertical the length (thicker at the wooden parts after placed "floating" on top of supports than the arches. supports, thinner in the top), construction (for protection the frame using mortise and The timber supports at the making the arches as and additional stiffness to the tenon joints, which allowed dry riverbed are diagonally material efficient as possible. connection). the rising flood water to lift braced up the pathway and carry it off down stream while the main structure would be spared. Rebuilt locally. Three midspans: 35.1 m, endspans 34.8 m Fussgängerbrücke im Arch bridge. Two three- This walking area is A steel cross-bracing is used A point load is transferred to The cross bracing and the Reinforced concrete The arch bridge is formed The arches, the middle fields The bridge was transported Main structure: glued The pavement is sealed Wildpark Langenberg hinged arches made of glued designed as a tension band between the arches. the arch beams through the two arches gives rotational foundations where the arch from the primary structural of the carriageway, the to the site in five large parts, laminated timber in spruce / watertight at the sides with T- Location: Langnau am Albis, laminated timber in the main and connected to the cross cross bracing as well as stability. meets ground as well at the parts of the two three-hinged railing and the stiffening which were assembled in fir. steel prifiles which protects Schweiz structure. member in the middle of the rectangular steel profiles in two ends of the bridge. The arches with cross-sections of crosses were assembled in one night while Albisstrasse Pavement: stone mastic the main structure from Designer: Makiol Wiederkehr Arches in compression and arch. the middle of the arch. foundation meets the arches 240 x 1080-940 mm as well the factory and delivered to was closed to traffic. This asphalt (Gussasphalt) with weather. AG walking area as a tension perpendicular to the as the two strut rows with the construction site in five was followed by the final fine grit scattering to prevent Engineer: Makiol Wiederkehr band. compression forces. cross-sections of 360x120- large elements. assembly of the railing, slipping when wet Drainage pipes along the AG Arch span: 44m 160mm and 280x120- which then also served as Railing: untreated larch. longitudinal slope, and then Completed: 2009 140mm, each consisting of fall protection for all led away from the structure Length: 78 m glued laminated timber in subsequent work. at the foundations. Width: 2.6 m spruce / fir. The handrail battens are Ribbed panels made of screwed in from the outside laminated veneer lumber so they can be easily with a thickness of 69mm replaced. and ribs in glued laminated timber with a cross-section in the ramp area of 120x640mm and in the bridge center area of 120x320mm run over the steel crossbeam in the longitudinal direction. Kingsway Pedestrian Arch bridge, cable stayed The post-tensioned walkway As the arch is tied to the post-Point load is transferred to Tension cables on each side Large buttresses was The joints and details were Six thin glulam beams (30 m The glulam arches was Timber arch, post-tensioned Precasted concrete planks Bridge deck. is suspended from the arch, tensioned walkway, the the edge of the walkway, of the arch/walkway provies intentionally avoided (due to carefully analysed as such long, 150 mm thick) was bi- prefabricated in a parking ot concrete deck, steel tension was chosen instead of steel Location: Burnaby, BC, US Simple arch with post- which distributes the load to outward, horisontal forces in through the tension rods and lateral stability, as well as econony) by tying the arch to hybrid wood-steel arch has axially curved and anchored adjacent to the site. The rods and steel roof cover. plate with concrete fill, to Designer: Busby Perkins + tensioned concrete walkway the foundation at the edges. the longitudinal direction is down through the arch to the lateral stiffening between the the walkway. The foundation not been done before. at both edges by a steel entire timber arch + steel Glass guards connected to minimise long-term Will ssupended on steel tension Uniform stress distribution in minimised. The pinched edges of the bridge. continuous arch elements. elements only take vertical Consideration: construction haunch. The two outer roof cover was prefabricated the tension rods (2.4 m tall) maintenance. The mass of Engineer: Fast + Epp rods. The arch is tied to the the timber elements was shape (provided by tension The bi-axial curve of the arch loads from the bridge. tolerance, difference in beams (which are required to close to the site and lifted the concrete deck also Completed: 2008 walkway itself to resist the provided by using plocking rods in the top) provies (wider at the edges, thinner shrinkage, uniform stress bend the furthest) was into place using two cranes. mitigate vibrations. Length: 44 m outward forces and avoid pieces to transfer load form horisontal (transverse) in the middle) provies distribution, etc. thinner (75 mm thick, 600 The road was closed for only Width: 3 m large and costly buttresses. the edge of he arch to the stability at the edges. The horisontal/rotational stability The steel haunch-to- mm deep). The individual a day. more interior wood glass guards connected to at the edges. concrete platform connection beams are connected and The glulam elements were members. the steel rods result in large includes a sliding connection forms a double curved solid post-tensioned on-site using Projected HSS beams from horisontal wind forces to avoid large thrust forces arch. Also called a timber steel rods glued into the the glulam arches are imparted to the tension rods, on the souuth concrete drop-in span cross holes. connected to steel rods, which required detailed structure. which support the concrete analysis. walkway. These sections, together with the internal wood blocking between the glulam beams, transfer some of the vertical loads away from the outermost arches. Richmond Olympic Oval Depth of one truss: 660 mm Vertical load (self weight of The traingular arches was A point load on the roof will CNC'd lywood bulkheads The internal tension rod of Every second 2x4 "strand" in The panels were designed After the main arches and Timber V-shaped roof panels The roof timber trusses are Location: Richmond, Canada The roof panels connect and the roof) is distributed in the hydraulically bent and be distributed among the hold the V-shape in place, the panels are bolted to the the cross-section is parametrically, where temporary bracing was made out of small pieces of made out of lumber from Designer: Cannon Design stabilise the primary arches, truss arches to the main tensioned with a steel rod. three trusses of each panel providing rotational stability top of the main arches in six continuous, connected with dimensions a structral erected, the roof panels were pine. Covered with stressed- forests killded by the Engineer: Fast + Epp, which span about 100 m timber arches on each side, The end plates of the arches (that works rigidly due to the of each truss (the black connection points (three on clip angles to the design of each unit lifted in place and bolted to skin plywood creating a rigid mointain pine beetle. The StructureCraft accross the building. and down to the foundation. transfer the compression in continuous, stressed, sheet is a part of the fire each side). neighbouring strand. optimised for the unique the main arches (6 panel. Hybrid main arch of project is said to have stored Completed: 2008 500 panel units (one unit=3 The outward forces of the the timber arch into tension plywood skin), down to the safety and acoustic The foundation elements of Additionally screwed position in the global model. connection points). glulam and steel. Steel over 2500 tonnes of carbon Length: 3-12.8 m truss arches), 55 unique panels was minmised due to in the tie rods. main arches and further to installment) the main arches are concrete connections. Several rules embedded in Thereafter the continuous tension rods, concrete by utilising these trees. The Width: - panels forms the tension rod tying the the foundation. buttresses. The triangular the vode frove the geometric plywoodlayer was stiched to buttresses. structure itself is protected The transversal roof panels trusses together. shape of the buttresses layout of the openings and the panels. by a continuous sheet of consist of bent and post- provides both longitudinal splice lodations (important Since the panels were plywood, which again is tensioned V-shaped timber (outward forces) and for the acoustic performance optimised and controlled covered by a thin metal truss arches composed of transverse stability to the of the panels). One panel parametrically in the digital sheet. small wood elements. structure, where the height unit consist of three trusses. model, the manual re-work The main arches are and selfweight of the The unit was post-tension was minimised. composed of timber and buttresses provides the and anchored by steel steel, which due to the low necessary vertical stability. tension rods. height works as a hybrid of beam and arch (both bending and shear). Underspanned bridge Langlaufbrücke Samedan Compression arch with a Compression in the arch and The pathway consisting of Point load is led transversally Pretension gives rotational Concrete foundation Steel parts distribute high The bridge is prefabricated in The compression beams are Superstructure: The walkway has wear and Location: Samedan, tension cable, which gives struts, and tension in tension transversal beams (into a to the two compression stability. compression forces at timber parts and then transported to lifted to their position, then Gluelaminated timber with tear planks, which can be Switzerland "two extra supports" in the cable "slab"), leads the horisontal arches and then down to connections. The V- the site. the bridge deck is mounted. steel tension band. replaced. As well as extra Designer: - middle. forces through shear to the foundation. connection has a cover Lastly the railing is applied. planks for vehicle tires. Engineer: - supports. sheet to prevent water to The superstructure is Completed: 2005 weaken the structure. protected with help of that Length: - the bridge deck cantilevers Width: - out on both sides. Pedestrian and Cycle Timber-steel framework Tension member 350mm In the level of the glue Top glulam chord distributes The diagonals in the truss The concrete abutment piers Extra bracing system for Superstructure prefabricated The two superstructure parts The glulam beams are made Douglas fir, a species that is Bridges Over the Gave construction. a bottom chord wide steel plate with varying laminated beams there is an the point load down to the helps with the rotational are dressed with stones from horisontal wind forces. in two halves. was transported to site and of laminated Douglas wood, naturally resistant to d'Aspe and Gave d'Ossau in steel and a top chord of thickness between 20 to additional round steel steel truss system. Tension stability. the torrents. Prestressed 30 mm diameter Lower steel tension member then spliced together. which are on the upper side biological attacks (class 3, Location: Oloron, France several glulam beams 60mm according to the diagonal brace to carry the in bottom steel plates, down The last two cross-beams steel rods. welded on site. Steel tension member protected with rhepanol foil. as per EN 355). Designer: RFR placed together. The glulam bending moment and thus wind loads. to foundation. are reinforced to carry the Elimination of overstressed Decking was placed in situ welded on site (post- The planks of oak timber are The deck beam structure are Engineer: Ingenieurbüro beams are attached to the axial force distribution. neoprene and steel bridge joint design by adding after the bridge was on site. tensioning), before lifted to fluted to ensure adequate protected by the decking. Miebach steel cross beams. This The struts join the cross- bearings that transfer the stiffening plates. it's final position by crane. resistance for walkers. Rephanol waterproofing foil, Completed: 2009 network enables a low beams next to their ends and reaction loads to the Finally decking was placed in sandwiched between the Length: 50 m superstructure height with a therefore transfer higher abutments. situ due to lack of lifting The oak decking forms the longitudinal beams and Width: 2.5 m large span of 48 meters. Due axial loads to the timber capacity from cranes. wear surface for pedestrians decking gives further to the combination of timber beams situated at the outer and cyclists. It can be easily protection against moisture. and steel the bridge is very edges of the deck. The stiffer replaced in time, along with light and slender. the cross-beams in the waterproof membrane. horizontal direction the more Each decking plank re- even the load distribution ceived two of anti-sliding between timber beams. The strips, made of an epoxy geometry of the cross-beams resin and sand filled groove. was therefore optimized for their horizontal inertia, to minimize their visual appearance and take advantage of the reduced total structural depth. Truss bridge Punt Staderas Rigid frame bridge. Vertical load is distributed in The width of the support Point loads are transferred The structural height of the Piers: 3x6 rows of inclined Shear tests verified the load- Locally produced wood (no information) Wood deck and piers - Location: Laax, Switzerland The brigde girder is designed the layered grating and to the elements provies horisontal through the layered grating deck grider adds to the beams that creates a fan- bearing capacity and rigidity elements of the same Consists exclusively of Designer: Walter Bieler AG as a beam grating, which bottom longitudinal beams stability in the transverse to the bottom longitudinal rotational stability of the shape. of the bolded connections dimension, except the wooden members of 160/240 Engineer: Walter Bieler AG, acts as a Vierendeel girder in (through distribution of direction of the bridge, while beams, to the deck, as well as the At the ends, the bridge between the side and cross- longitudinal glulam beams in mm except from the main Stephan Berni the longidutinal direction. bending forces), to the the fanned shape of the of supports/foundation. intermediate layer of stands on reinforced members. the main span across the span across the road, which Completed: 2015 The grating consists of two support elements. the supports stabilises in the transverse elements. The concrete abutments road. consists of glulam beams, Length: 115 m vertical layers of four A Vierendeel girder is longitudinal direction. Vierendeel truss stiffenes the C24. Width: 2.5 m longitudinal beams each, characterised by the bridge. and an intermediate layer rectangular frame (compared with horisontal cross beams to triangular truss frames). every 1.05 m. The layered Moment joints are used to structure enables a static resist substantial bending height of 0.64 m. forces. Enables a larger Bicycle and pedestrian span than a trangular truss bridge. structure but has a more Different length of the nine complicated distribution of spans. stresses. Slope: 6% (accessible) Thalkirchner Brücke Continuous space The distributed load is Due to the dense space Point load on asphalt Since the bridge is 13 m Concrete foundations in river Casted steel nodes with one All parts were standardised Pre-assembled parts which The framework consists of The main structure of the Location: München, framework over 15 spans, transferred in the truss the horisontal load can distributed down on wide and that there is a wide as well as on land. common center node for all and manufactured in a large were later assembled on glued spruce and larch wood bridge is well protected Germany each 13.5 m long. compression space truss easy find its way down to the corrugated steel sheet. The contact with the concrete connecting bars. amount. They were pre- site. and is held together by cast against weather by a Designer: Dietrich Ingenieur The bridge is carrying both arch down to the concrete foundation through the load then goes down through foundations the bridge has a assembled in larger steel knots. complete covering from Architektur cars and pedestrians. foundations. diagonals. With the 'small' the short trusses through good rotational capacity. Each bar head is screwed assembly units. At last they above, under the walkway Engineer: Dietrich Ingenieur spans the buckling lengts one of the four longitudinal Also the many spans helps with one screw each to the where assembled together The railings are made of and roadway. The main Architektur Three different cross- are small which also helps arches and then down to the with the rotational stability node ball. High-strength on the construction site. larch wood. The pavements structure also inlines inwards Completed: 1992 sections and four different with the horisontal stability. concrete foundations. (compared to longer spans). screws are used with have been renewed due to under the bridge deck, so Length: 197 m lengths of the spruce bars. (The asphalt also gives extra pretensioned so they could Three different cross- wear and tear (sidewalk in that it is even further Width: 13 m dead-weight.) handle the absorb the sections and four different oak and asphalt on protected. dynamic loads from the road lengths of the spruce bars roadway). traffic. The ends of the bars are sealed with epoxy resin. Mathematical Bridge Arch bridge, consisting of The tangential and radial A horisontal bracing system Poin load on deck transfers The arch form gives The stone foundation surface Today, there are bolted Assumption. Built on site on Assumption. A temporary Original bridge: Oak Due to the straight Location: Cambridge, UK straight timbers arranged beams forms a triangulated under the walkway through horisontal bracing to rotational stability, as well as meets the bridge connections (former iron temporary scaffolding in the scaffold in the water, and the superstructure, iron screws tangentials running through Designer: William Etheridge radially and tangentially. truss. Compression in the distributes the horisontal the two side trusses. Then the rigid supports. perrpendicular. screws and oak pins). The water. bridge was built on that. The and oak pins. the radial parts, water Engineer: William Etheridge tangential beams, and the loads. the load goes in picture also shows how the two longitudinal trusses copy 1905: Rebuilt in teak, with gathered there starting to Completed: 1649 (1905) force is lifted up through the compression in the tangents tangential beams goes of each other and then bolted connections instead. degenerate the bridge. Length: 12 m vertical beams. and lifted up by tension in through the radial beams added the horisontal bracing. Width: - the radials, and zigzaging with cut outs to get a better The oak superstructure of down to the abutments. match as well as a rigid Etheridge's bridge has had connection. to be refurbished twice. In 1904-5, the entire timber structure was rebuilt in teak. Bolted connections replaced the original iron screws and oak pins. La passerelle de la Paix Combined truss arch and The triangular truss The wide triangular shape Point loads transferred to the Triangular steel elements Soil diapgragm walls in the Due to the efficient assembly Steel elements for the 3 stages of assembly: Steel truss elements (S355). The epoxy rasin coat of the Location: Lyon, France cantiveler bridge. Two transferres the vertical load provides horisontal stability triangular truss elements and that links and stiffenes the foundation: 10 m by 14 m, 20 method, the welding and superstructure fabricated in 1st: foundation, and Light wooden deck (oak wooden deck increases the Designer: Dietmar cantivelers built of a 3D from the box girder and the to the bridge. Also, the longitudinally to the top and bottom chords. m deep coating operations were Switzerland. The two ends of production and assembly of planks, 50 mm thickness) durability of the wood (and Feicgtinger Architects assymetric tube structure. arch to the foundation cantilever box girder foundation elements. Damping system was simplified the main arch were the super-elevated geometry with anti-slipping (epoxy provies anti-slipping). Engineer: Schlaich Two walkways (upper and elements. contributes to horisontal subjected to dynamic testing assembled and welded on on temporary supports (at resin bands with silex). The welds connecting the Bergermann & Partner lower riverbank) join and stability in the longitudinal before installment. site after the single tube the side of the river bank). Railing: inox integrating LED truss and box girder is Completed: 2014 form a large public space. direction. The "eccentricity" elements, diagonals and 2nd: installation of the lights and cable-net filling. hidden under the girder Length: 157 m Two arches formed of tube of the varying cross-section vertical diaphragm structures temporary cable support deck. Width: 5 m, 8 m sections = bottom chord. Box- allows the two paths to and upper box girder units system on top of the bridge. girder = top chord. counterweight eachother. were brought to site by tryck. 3rd: placing the bridge in its Top and bottom chords are The central third of the arch final position. The whole linked and stiffened by was brought to site in a structure of 160 m long andd triangular steel elements and single piece. 8 m tall was put to place at diagonals to create a truss The three sections were the same time (transported tructure. The relative assembed on site by boat). 3rd: insallation of positions between the the upper chord box girder, elements varied across the damping system etc. length of the bridge. Goal: minimise the time for Bicycle lane + footpath and which the river had to be footpath. 8 + 5 m closed to shipping traffic, and 220 m total, 160 m span, best solution for achieving 60m appriach bridge on the geometrical accuracy during park side. Height above assembly on the site. water: 8 m L'Estellier Footbridge Steel truss arch bridge. The inclined truss elements Horisontal stiffness is Point load is transferred from The V-shape of the bridge The bridge lands on concrete The elements are either (no documented information (no documented information Steel Regular maintenance of the Location: Aiguines, France Truss elements assembles distributes the vertical load provided by the rectangular the walkway to the provides the rotational foundations on each side. bolted or welded. was found) was found) steps, bolt tightening, anti- Designer: the structure. from the path to the upper foundation elements (placed transverse elements and stability. The walkway is The rectangular shape of the rust paining and guardrails. Engineer: Can only take 10 people at and lower steel continuous perpendicular to the bridge), inclined/diagonal elements, placed on transverse foundation provides Completed: 2004 the time chords, down to the and by the V-shape of the to the upper and lower elements on the lower chord, horisontal stability to the Length: foundation. bridge. chords. that are longitudinally structure. As the bridge is Width: connected to eachother on tied in tension, outward the outer side of the bridge, forces are minimised. as well as diagonally connected to the lower chord. The upper and lower chord are stiffened with vertical and diagonal steel members. This together provides rotational stability to the bridge, athough it is said to oscillate when you walk on it. Truss Bridge in Traunreut Truss construction with Load is transferred through Steel cross bracing in top Point load is distributed From the lower transverse Abutments in reinforced Centerline extension of Truss is most likely produced After that the truss structure Truss structure mainly of The roof cantilevers so the Location: Traunreut, round columns and copper the truss system in tension and bottom of the truss gives transversally through the gray steel tube, dagonals go concrete. glulam truss and horisontal in a workshop and is assembled into one piece, glulam beams of spruce. rain fall down far away from Germany covering. and compression. The truss horisontal stability. The load deck out to the two main up to the upper transverse Connections are pinned on steel bracing meets in one transported to site in pieces. the bridge is lifted to it's final Cross beams in steel. The the structure. Water of rain Designer: Dietrich Ingenieur The covered truss structure system itself is precambered is transferred through the truss frame walls. Then tubes to give rotational one side (and roller on other node. position. Then wooden longitudinal beams are also cannot reach neither the Architektur, Traunstein consists mostly of glulam to reduce final deflection horisontal bracing in roof tansferred through the truss stability? side?) decking and roof with copper made of glue laminated structure nor the decking. Engineer: Köppl, Rosenheim beams of spruce that are position. down through the down to the abutments. plates are added. And so the wood. The decking made of Completed: 2008 standardized in length and longitudinal truss, and finally steel balustrade and wooden larch planks and the steel Length: 50 m cross-section. to the abutments. handrail. railing contribute to the Width: 3 m aesthetic appeal of the bridge. The roof construction with copper plates. Steel balustrade with wood handrail. Rasteplass i Lillefjord Steel truss frame Is transferred through the Top view. A point load is transferred The 3D truss frame is U Conrete foundation on both Welded connections of RHS Prefabricated steel frame Steel frame transported by Steel truss frame. Deck and The wooden parts are meant Location: Lillefjord, Norway 3D frame in the curve due to trus system. There is one Since the truss frame is U- transversally to the truss, shaped, and has a wide sides on the shore. On the steel profiles. The wood truck and lifted on its place. benches in wood to be torn from hikers. They Designer: Pushak torsion forces. upper and one lower truss. shaped, there are horisontal down to the abutments. connection point on the side with the higher slope, plank structure is screwed to Then the wood cladding was can easily be replaced since Engineer: Pushak diagonals underneath the shore, which contributes to rocks are placed to prevent the steel frame, which added together with the toilet the wood is only cladding Completed: 2006 bridge deck, which handles the rotational stability. soil movement. makes wooden parts house. and the main structure is in Length: - horisontal forces. replaceable. steel. Width: - Purpose to let the wood turn grey and age naturally. Railway Crossing in Covered truss structure of Through tension and Timber bracing as well as Transferred longitudinally Rigid steel frames at the two Assymetrical supporting Diagonals change angles (no Whole bridge prefabricated Installed at night in 1.5 h. Larch timber and larch Covered roof to protect Nettersheim larch timber and larch compression in truss. The horisontal steel beams through bridge deck beams, end points of the whole truss structure. Two steel support one with the same angle) Lifted with crane glulam. Deck of larch planks. structure. Location: Nettersheim, glulam. diagonals does not have the out through transversal steel beam structures = staircases Bolted connection into steel Staircase and parapet of Germany Integrated steel frame at the same angles, but varying beams to the two main truss plate. galvanized steel with Designer: Wollenweber two end points angles. walls. Then down to anthracite coating Architectur foundation through steel Engineer: Schaffitzel column/frame Holzindustrie Completed: 2014 Length: 29.5 m Width: 3 m Bostanlı Bridge and Turss girder bridge. Vertical load distributed by Concrete bulkheads Point loads are transferred Secondary truss-beams The bridge girder rests on Thermo ash-wood surface Thermowood (ash) has Sunset Lounge Composite girder consisting the outer steel I-beams, provides horisontal stability through the transverse steel adds rotational stability to the concrete bulkheads that lies on a steel frame. better durability compared to Location: Karşiyaka, Turkey of steel I-beams + concrete supported by the steel frame. on each end of the main frame to the outer steel bridge. The asymmetric on untreated wood. Designer: Studio Evren surface. Transversal steel beam. beams, to the concrete shape of the cross-section is an artificial hill made out of a Başbuğ frame; 10 girders, 4 m supports. handled by the bow shape in resin-bound natural stone Engineer: Novawood, YDÇ spacing. longitudinal direction. The mixture. The material is Completed: 2016 Assymmetric shape. Slightly concrete cover of the girder porous, which allows for Length: 35 m bow-shaped in longitudinal and transerse wood cover mainaing Width: 9.5 m direction. provides additional rotational a better stormwater startegy. stability to the bridge. Stressed-ribbon bridge Essinger Bridge Stressed ribbon bridge The parallell glulam girder The triangular shaped A point load is distributed to The railing design with Concrete foundation. Steel The triangulated supports Prefabricated glulam beams, Cast in situ of concrete Spruce glulam beams On top of the bracing system Location: Essing, Germany The span consists of 9 beams in tension distributes bracing connected to the the three sets of longitudinal outgoing double triangles, rods connected to the bridge have pinned connection to lengths of 40-45m and foundation. Prefabricated Casted steel joints zinc sheets are placed to Designer: Büro für Ingenieur- parallell spruce glulam the vertical loads to the bridge deck and the rail glulam girder beams. Then helps with the local rotational deck for horisontal forces. the glulam beams transported by trucks. Joined and pre-assembled pillars protect the main structure Architectur Dipl. Ing. Richard beams of 0.4 m lentgth triangular foundation increase the horisontal down in the large stability. together by finger joints. were put on the foundation. from rainfall. J. Dietrich (0.22x0.65 m) elements. stability. Horisontal stability triangulated frame down to The girder beams are At the abutments the glulam Prefabricated pillars and pre- The prefabricated glulam Engineer: Heinz Brüninghoff Main span 73 m. in the longitudinal direction is the concrete foundation. tensioned, also preventing beams have bolted assembled. beams were lifted by a crane There is a problem with Completed: 1986 Static height: ; 7 m above provided in the triangular rotation. connections. It is allowed to and they were later on joined brown rot, especially at the Length: 190 m water lever at lowest shaped foundation elements. rotate there as well. All connections, joints, drive separately. The fingerjoints supporting frames. Width: 3.2 m midpoint (3 m downwards) shafts, coupling elements were pulled together and Live load 5 kN/m² The horisontal timber and nailed plates have been treated. A tent was used bracing carries load to steel made in cast steel. All around the joining, to give tie rods, which brings the fasteners had the same size, right weathering conditions. force down to the concrete so only one size was needed footing. to create. Cable-stayed bridge Anaklia-Ganmukhuri Cable-stayed deck (60 m , The cables carries the bridge There are transversal beams Transported to either of the The pylon is standing in an Y-columns in concrete. "Hess Limitless" adhesive All wooden components The prefabricated framework Timber truss deck. Steel A translucent polycarbonate Pedestrian Bridge 84 m span) combined with in tension in the larges under the bridge deck. The three main longitudinal upside down V, creating connection: prefabricated. is lifted ontop of the casted cables, concrete pylon cladding is applied as Location: Anaklia, Georgia simply supported truss deck spans. For the shorter triangular shaped truss (with beams in the truss system. equilibrium. Every support is Finger joint glued on site. concrete pillars. Pylon is built covering, to protect the wood Designer: Leonhardt, Andrä (48 m) spans, the load is transferred the inclination) also provides And then transported Y shaped and there is a steel Scarfs in glugam top and and then the bridge parts are structure. und Partner Spatial triangular framework. through tension and horisontal stability. through the cables to the profile attached at the bottom faces. connected from the pylon Engineer: Fast + Epp GmbH compression in the truss. pylon, or directly to the supports preventing the outwards and then hung by Completed: 2011 foundation. wood truss to rotate. the steel cables. Length: 506 m Width: 9 m Suspension bridge Punt Ruinalta Suspension bridge in wood The suspension cables bear The walking deck consists of A point load is transported The suspension cables are The concrete pillars are There is air gaps between all First the pylons and The suspension cables are Steel cables, wood in bridge There is a small airgap Location: Bonaduz, and steel, and concrete the self weight in tension. transversal horisontal from transversal walkway tilted outwards, this gives integrated in the design. elements to make sure that foundations are casted. After mounted on site. The deck and reinforced concrete between all components, so Switzerland pylons/foundation. beams, which are densely beams into the longitudinal extra rotational stability They should compensate for moist and water can dry out. that the suspension cables prefabricated bridge deck pylons and abutments. rain and dirt can fall off and Designer: Walter Bieler AG spread. This provides beams, and then distributed compared to cables a tensile force of 100 ton The transversal pathway are mounted on site. The parts are then lifted on their so that the structure is able Engineer: Walter Bieler AG horisontal stability. Each to the cables. The perpendicular to the walking (981 kN). beams har hung in two rods bridge deck is prefabricated right position with the help of to dry. Completed: 2010 beam is connected by two longitudinal beams are deck. going up in longitudinal in smaller pieces and then a cableway and then Length: 105 m longer rods at each end. relatively stiff, so the bridge beams. This longitudinal assembled on site. connected to the tension Width: - deck does not deflect locally beams are lifted up by the cables. There is an overlay at the point load. suspension cables. on the bridge parts so they can be connected together. Iya Vine Bridge Suspension bridge made out The bridge is carried by the The bridge sways a bit. Point load from a human is The suspension cables helps The supports are using Originally woven connections The vines are voven around Assumption. The hand rail Actinidia arguta vines, The bridge is rebuilt every 3 Location: Iya Valley, Japan of mountain vine. suspension cables. As well The tension in the hand rails transferred transversally with the rotational stability. nearby trees as further or tied connections. But the main steel cables. As cables is straightened over harvested from the year with new vines. Designer: - Built from actinidia arguta as by the upper hand rail together with the suspension through the thin "beams" to support. The hand rails are since a steel cable is hidden well as fastening the walk the span. Then the foot mountains during the harsh Engineer: - vines amassing a total of 5 cables thuroughly connected cables gives horisontal the hand rails, and then further raised by the use of beneath, we cannot see the way "beams". platforms are added from winter, were woven together Completed: 12th Century tons to the supports at both ends. stability. through the suspension lever arms. steel connections. But most each side and then the to form the vine bridge. Length: 45 m cables. There are also likely clamps. suspension cables added Width: - To strengthen the bridge, longitudinal cables/vines and then tensioned. steel cables are hidden under the thin "beams" as inside the vine. Due to safety well (but not as high force reasons, the bridge is rebuilt are going through here. every three years. "Today's bridges are more sturdy than their predecessors: steel cables are hidden beneath the vines, the gaps between planks measure seven inches, and each bridge is rebuilt every three years." APPENDIX B - Global dimensioning and Detailed design Global dimensioning Calculate for six straight, continuous beams with an intermediate support. The span is built up by two parallel, continuous LVL beams with curved LVL panels in between, working in active bening. Additional bridge parts: transverse floor beams, plywood + construction board, longitudinal floor beams and transverse panel deck with epoxy resin and quartz sand cover. The outer LVL beams are protected with prebent LVL panel. The railing is fastened in steel L-profiles to the transverse floor beams with longitudinal timber panels. The detailed design of the railing connection will be performed at a later stage. The bridge is designed for pedestrians only as any service veichle or cyclists cannot access the bridge. The bridge is categorized for safety class 3 The bridge is calculated for load case 4 (LM4) according to EN1991-2 4.3.2 (2) (d). Site geometry L1 := 4.2m Length of the left span L2 := 2.1m Length of the left support span L3 := 16.8m Length of the right span L4 := 4.2m Length of the right support span L := L1 + L2 + L3 + L4 = 27.3 m Total length of the bridge B := 2.5m Width of deck Sinus pattern geometry 4.2 ssin := m = 2.1 m Length of half sinus wave = cc distance threaded bar2 bsin := 130mm Distance between sinus waves rsin := 14.5m Radius of the sinus waves bsin.sup.m := 476mm Mid CC distance between panels over the support bsin.sup.e := 443mm Outer CC distance bt panels over the support Permanent load - load bearing elements Straight LVL beams LVL dimensions tveneer := 3mm Nominal thickness of veneer (Mestä Wood) nveneer.s := 9 Number of veneers in one lamination tlvl.s := tveneernveneer.s = 27mm Lamella thickness, standrard dimension Beam dimensions hlvl.s := 600mm Height of outer beams blvl.s := 4tlvl.s = 108mm Width of outer beams nlvl.s := 2 Number of straight, outer beams M aterial properties, LVL S grade beams fm.0.edge.k.lvl := 44MPa Bending edgewise Eurofins certificate fm.0.flat.k.lvl := 50MPa Bending flatwise ft.0.k.lvl := 35MPa Tension II to grain fc.0.k.lvl := 35MPa Compression II to grain fc.90.edge.k.lvl := 6MPa Compression edgewise, perpendicular to grain fc.90.flat.k.lvl := 2.2MPa Compression flatwise perpendicular to grain fv.0.edge.k.lvl := 4.1MPa Shear edgewise fv.0.flat.k.lvl := 2.3MPa Shear flatwise E0.k.lvl := 11600MPa Elastic modulus, II to grain kg ρk.lvl := 480 Density 3 m E0.mean.lvl := 13800MPa Elastic modulus, II to grain E90.edge.mean := 430MPa Elastic modulus, perp to grain, edge E90.flat.mean := 130MPa Elastic modulus, perp to grain, flat kg ρmean.lvl := 510 Mean density3 m sk := 0.12 Size effect parameter Size effect Size effect LVL beam subjected to bending, if the majority of the layers are oriented in the same direction  sk  300mmk  h.lvl := min  , 1.2 if hh  lvl.s  300mm = 0.92 EN1995-1-1 eq. 3.3  lvl.s   1 otherwise Size effect LVL beam subjected to tension, assume that the LVL beam is continuous  sk   2 3000mm kl.lvl.s := min  , 1.1 if L  3000mm = 0.876 L   EN1995-1-1 eq. 3.4 1 otherwise Partial factors kmod.p := 0.5 Permanent load, service class 3 EN1995-1-1 tab. 3.1 kmod.s := 0.7 Short term load, service class 3 γM.lvl := 1.2 LVL Design strength fm.0.edge.k.lvl EN1995-1-1 eq. 2-14 fm.0.edge.d.lvl := kmod.pkh.lvl = 16.87MPaγM.lvl fm.0.flat.k.lvl fm.0.flat.d.lvl := kmod.p = 20.833MPaγM.lvl ft.0.k.lvl ft.0.d.lvl.s := kmod.pkl.lvl.s = 12.774MPaγM.lvl fc.0.k.lvl fc.0.d.lvl := kmod.p = 14.583MPaγM.lvl fc.90.edge.k.lvl fc.90.edge.d.lvl := kmod.p = 2.5MPaγM.lvl fc.90.flat.k.lvl fc.90.flat.d.lvl := kmod.p = 0.917MPaγM.lvl fv.0.edge.k.lvl fv.0.edge.d.lvl := kmod.p = 1.708MPaγM.lvl fv.0.flat.k.lvl fv.0.flat.d.lvl := kmod.p = 0.958MPaγM.lvl Self-weight of all straight beams kN Gk.lvl.s := ρk.lvlgnlvl.s blvl.shlvl.s = 0.61 m Curved LVL panels LVL dimensions nveneer := 9 Number of veneers in one lamination tlvl := tveneernveneer = 27mm Standard dimension Stora Enso S grade P anel dimensions hlvl := hlvl.s = 600mm Height of panels blvl := 2tlvl = 54mm Width oh panels llvl := 12000mm Length of LVL panel elements, not relevant for calculations nlvl := 14 Number of LVL panels Llvl := L Total length of the LVL panels Size effect As the height of the curved panels are the same as the straight beams, k.h is the same. Check size effect in tension for the total length of the continuous, curved panels.  sk     2 3000mm kl.lvl.curve := min  , 1.1 if LL  lvl  3000mm = 0.876 EN1995-1-1 eq. 3.4  lvl   1 otherwise Design strength ft.0.k.lvl EN1995-1-1 eq. 2-14 ft.0.d.lvl.curve := kmod.pkl.lvl.curve = 12.774MPaγM.lvl Self-weight of all curved panels Llvl kN Gk.lvl.curve := ρk.lvlgnlvl blvlhlvl = 2.135L m Distancers The distancers are steel tubes in compression, threaded with a steel rod in tension. A steel plate distributes the compression forces from the steel tube to the LVL panel. Two levels Steel quality: S275, structural hollow sections Dimensions steel plates tplate := 20mm Thickness of the steel plate bplate := 150mm Width of the steel plate hplate := 150mm Height of the steel plate Aplate := hplatetplate Cross-sectional area D imensions steel tubes dtube := 76.1mm Diameter, outer ttube := 8mm Thickness of the steel tube D1 := dtube Outer diameter d1 := dtube - 2ttube = 60.1mm Inner diameter Number of elements, one layer Ltube := bsin - 2tplate = 90mm Length of steel tubes nlvl ntube.x := = 7 Number of steel tubes across2 stube := ssin = 2.1 m Spacing tubes = distance between half amplitudes  L - D1  ntube.y := floor  = 12 Number of tubes along the bridge  stube  π 2 2D1 - d  1  - 3 2Atube := = 1.712  10 m4 nplate.x := 2ntube.x Number of steel plates across nplate.y := ntube.y Number of steel plates along the bridge Total number of elements, n layers nlayer := 2 Number of layers Material properties of the steel fy := 275MPa For t < 40 mm EN1993-1-1:2005 tab. 3.1 fu := 430MPa Esteel := 210GPa kg ρk.steel := 7850 EN1993-1-13 m The tube is considered as very stocky/intermediate slender column, i.e. buckling capacity is not considered. Self-weight Self-weight of the steel tubes, total Ltubentube.y kN Gk.tube := ρk.steelgntube.xAtube nL layer = 0.073 m Self-weight of the steel plates, total bplatenplate.y kN Gk.plate := ρk.steelgnplate.x Aplate nL layer = 0.426 m Total self-weight of the load-bearing elements, on the length of the beams kN Gk.bridge := Gk.lvl.s + Gk.lvl.curve + Gk.tube + Gk.plate = 3.245 m Permanent load - non-load bearing elements Transverse floor beams Material properties, C24 kg ρk.c24 := 420 Mean density DOTS2 tab. 3.33 m Dimensions stfl := 600mm Spacing Ltfl := B = 2.5 m Length ttfl := 70mm Assumed width and thickness based on similar designs btfl := 90mm  L - btfl  ntfl := floor  = 45 Number of transverse floor beams along the length of the bridge  stfl  S elf-weight of the floor beams on the length of the bridge btflntfl kN Gk.tfl := ρk.c24gLtflttfl = 0.107L m Plywood board The plywood and construction board protects the load-bearing structure from moisture and dirt accumulation. The board is continuous over the whole area of the bridge deck. Material properties, plywood K20/70 kg ρk.pl := 420 Mean density "Tillhörande handling BKR3 m Vänderply" tab. 2 Dimensions tpl := 9mm Thickness bpl := B = 2.5m Lpl := L = 27.3 m Self-weight of the panel board Lpl kN Gk.pl := ρk.plgbpltpl = 0.093L m Longitudinal floor beams Solid timber, C24 Dimensions Llfl := 2stfl = 1.2m The plank deck spans over two floor beams tlfl := 45mm blfl := 90mm slfl := 120mm Spacing longitudinal floor beams  B - btfl  nlfl := floor  = 4 Number of longitudinal floor beams over the width of the bridge  stfl  Self-weight of longitudinal floor beams along the bridge kN Gk.lfl := ρk.c24gnlflblfltlfl = 0.067 m Transverse plank deck Solid timber, C24 Dimensions Lpd := B = 2.5m tpd := 70mm Assumed dimensions based on similar bridge designs bpd := 195mm spd := bpd + 15mm = 210mm Recommended spacing between planks  L - bpd  npd := floor  = 129 Number of transverse floor beams along the length of the bridge  spd  S elf-weight of the floor beams on the length of the bridge bpdnpd kN Gk.pd := ρk.c24gLpdtpd = 0.664L m Railing Solid timber, C24 Total height of railing pole, fastened to the floor beams. Required height over bridge deck: 1.2 m Dimensions Lra := 1.2m + tpl + tlfl + tpl + ttfl = 1.333 m 3 sra := 2stfl = 1.2  10 mm Center-to-center distance railing poles hra := 120mm Assumed dimensions based on similar bridge designs bra := 70mm nra := 2ntfl = 90 Total number of railing poles Horizontal railing planks hra.pl := 45mm tra.pl := 22.5mm nra.pl := 13 Number of planks on the height of the railing sra.pl := 92mm Self-weight of two rail poles branra kN Gk.ra := ρk.c24ghraLra = 0.152L m Total weight from non-load bearing elements, on the main beams Add weight from epoxy resin + sand kN Gk.cover := 0.06 m T otal weight from non-load bearing elements kN Gk.deck := Gk.tfl + Gk.pl + Gk.lfl + Gk.pd + Gk.ra + Gk.cover = 1.143 m Sectional constants clvl.s := 200mm Distance from deck edge to outer beam edge slvl := bsin + 2blvl = 238mm Center-to-center distance between the panels Cross-sectional area 2 Alvl.s := blvl.shlvl.s = 0.065m One beam 2 Alvl := blvlhlvl = 0.032 m One panel 5 2 Atot := nlvl.sAlvl.s + nlvlAlvl = 5.832  10 mm Point of gravity Simplified calculations without the steel plates and tubes Point of gravity from the t op of the load-bearing beams, z-direction  h  lvl.s   hlvl  nlvl.sAlvl.s  + n A  2 z :=   lvl lvl  2  tp = 0.3 mnlvl.sAlvl.s + nlvlAlvl Symmetry as the height of the glulam beams and LVL-panels are the same Point of gravity from the o uter edge of the outer straight beam, x-direction Midpoint distance between the outer beams B - 2clvl.s - 2blvl.s le := = 0.942 m2 nlvl nspan := = 7 Number of openings2   n   span  ns.half := floor  - 1 = 2  2  xtp := le + blvl.s = 1.05 m x.tp is in the middle of the deck due to symmetry M oment of interia in the span 3 3 blvl.shlvl.s hlvl.sblvl.s Ix.lvl.s := Iz.lvl.s := Straight beams12 12 3 3 blvlhlvl hlvlblvl Curved panels Ix.lvl := I :=12 z.lvl 12  2  h   2 lvl.s     hlvl    4 Ix.tot := nlvl.sIx.lvl.s + Alvl.s - ztp  + nlvlIx.lvl + Alvl - ztp  = 0.017m  2     2   2 ns.half+1  2 blvl  4 Iz.tot := n   lvl Iz.lvl + 22Alvl  (xtp - blvl.s - slvli) + 2A lvlxtp - blvl.s -  ... = 0.436 m 2   i = 1   2  b lvl.s   + nlvl.sI z.lvl.s + Alvl.sx -  tp 2   Iz.tot 3 Ix.tot 3 Wz := = 0.416m Wx := = 0.058mxtp ztp Variable load 1. Traffic load Pedestrian trafic is considered as short term actions according to EN 1995-2 2.3.1.2 (1) kN qfk := 5 Recommended, characteristic load EN1991-2 5.3.2.1 (1)2 m Vertical load on the bridge surface for LM4 (EN1991-2 4.3.2 (2) (d)) 120 kN kN Q fk := 2.0 +    = 4.094 EN1991-2 eq. 5.1L 2 2  + 30  m m  m  kN kN Controlq.fk := "OK" if 2.5 < Qfk < 5 = "OK"2 2 m m "Not OK" otherwise Vertical load on all beams kN Qfk.z := QfkB = 10.236 m Vertical load on each beam B kN Qfk.n.z := Qfk = 0.64nlvl.s + nlvl m Horizontal load along the bridge Horizontal load acting along the bridge in level with the upper surface of the decking. The characteristic load is set to 10% of the vertical traffic load (EN1991-2 chp. 5.4 (2)) Horizontal load along all beams Qfk.y := 0.10BLQfk = 27.943kN Point load Horizontal load along each beam Qfk.y Qfk.n.y := = 1.746kN Point loadnlvl.s + nlvl This load work together with the vertical load 2. Wind load EN1991-1-4 chp. 8 Wind actions on bridges Design dimenisons dtot := hlvl.s + tpl + tlfl + tpl + ttfl = 0.733m EN1991-1-4 fig. 8.2 b := B = 2.5 m L = 27.3 m Force coefficient for bridges b = 3.411 dtot 1.3 - 2.4 ax := = -0.275 EN1991-1-4 fig. 8.34 - 0 b cfx.0 := 2.4 + ax = 1.462 Force coefficientdtot Horizontal wind load For an open parapet and open safety barrier d := dtot + 0.6m = 1.333m Reference area EN1991-1-4 tab. 8.1 For load combinations without traffic load, plain (web) beams 2 EN1991-1-4 8.3.1 (4)(a) Aref.x := Ld = 36.391 m m vb.0 := 25.0 Fundamental value for Wendelstrand (Boverket)s cdir := 1.0 Directional factor EN1991-1-4 4.2 (2) N2 cseason := 1.0 Season factor EN1991-1-4 4.2 (2) N3 m vb := cdircseasonvb.0 = 25 EN1991-1-4 eq. 4.1s Basic wind velocity kg ρ := 1.25 Air density 3 m 1 2 kN qb := ρvb = 0.391 Basic velocity pressure EN1991-1-4 4.102 2 m Terrain parameters Terrain category II EN1991-1-4 tab. 4.1 c0 := 1.0 Orography factor EN1991-1-4 4.3.1 (1) kl := 1.0 Turbulence factor EN1991-1-4 4.4 (1) z := 58m + 5.3m Altitude at site + bridge height kr := 0.24 National annex tab. D2 z0 := 0.05m Roughness length EN1991-1-4 tab. 4.1 zmin := 2m Minimum height zmax := 200m Maximum height EN1991-1-4 4.3.2 (1) Exposure factor 2   z   7ce.z :=  krln  1 + = 5.82 National annex tab. D2   z    0  z ln      z0   Peak velocity pressure 3 EN1991-1-4 eq. 4.8 qp.z := ce.zqb = 2.273  10 Pa Reference height from lowest ground level to the centre of the bridge structure (hlvl.s + tpl + tlfl + tpl + ttfl) ze := 5.3m + = 5.667 m EN1991-1-4 8.3.1 (6))2 b 0.5 < < 4 = 1 dtot ze  20m = 1 F orce factor  b  3.6 - 6.7 EN1991-1-4 tab. 8.2Cx := 6.7 +  - 0.5  = 4.122  dtot  4 - 0.5 Force in x-direction, Simplified method (EN1994-1-4 8.3.2) 1 2 Fw.x := ρvb CxAref.x = 58.595kN EN1991-1-4 eq. 8.22 Total horizontal wind load on the long side of the bridge, on all beams Fw.x kN Qw.x := = 2.146L m Total horizontal wind load on the long side of the bridge, on one beam Fw.x kN Qw.n.x := = 0.134L(nlvl.s + nlvl) m Horizontal wind load, along the bridge Fw.y := 0.25Fw.x = 14.649kN EN1991-1.4 8.3.4 (1) Total horizontal wind load along each beam Fw.y Point load Qw.y := = 0.916kNnlvl.s + nlvl 3. Snow load EN 1991-1-3 chp. 5.1 μi := 0.8 Flat "roof" Ce := 1.0 Ct := 1.0 kN sk.swe := 1.5 Boverket EKS2 m kN ssnow := μiCeCtsk.swe = 1.2 2 m T otal snow load on the bridge kN qsnow := ssnowB = 3 m The snow load is considered in the load combinations if it exceeds the permanent loads Controlsnow := "Consider snow" if qsnow > Gk.bridge + Gk.deck = "Don't consider snow" "Don't consider snow" otherwise ULS Load combinations EKS10 13§ Risk class 3 γd := 1.0 EKS10 14§ Vertical loads (x) Total permanent load on all beams, vertical (z) kN Gk := Gk.bridge + Gk.deck = 4.387 m EN1990 eq. 6.10 Total variable load on all beams, vertical (z) kN Qfk.z = 10.236 m Total variable load on one beam, vertical kN Qfk.n.z = 0.64 m Partial factors Traffic load are considered short term load duration. kmod.s = 0.7 Partial factors for load combinations according to STR-2 γG := 1.2 γQ := 1.5 Load factors for load combinations ψ0.fk := 0.4 Traffic load EN1990 tab. A2.2 Load cases kN LC1z := γGGk = 5.265 Load case 1: self-weightm kN LC2z := γGGk + γQQfk.z = 20.618 Load case 2: self-weight + traffic loadm kN LC3z := γGGk + γQψ0.fkQfk.z = 11.406 Load case 2: self-weight + characteristic traffic loadm Critical load case, vertical direction kN qd.z := max(LC1z , LC2z , LC3z) = 20.618 m Transverse loads (x) Total variable load on all beams, transverse (x) kN Qw.x = 2.146 m Partial factors for load combination Load combination factors for variable loads on bridges ψ0.w := 0.3 Wind load main load EN1990 tab. A2.2 L oad combinations, x-direction Wind load is the only load in x-direction kN LC1x := γQQw.x = 3.22 m kN LC2x := γQψ0.wQw.x = 0.966 m kN qd.x := max(LC1x, LC2x) = 3.22 m Longitudinal loads (y) T otal variable load on one beam, longitudinal (y) Qk.n.y := Qfk.n.y + Qw.y = 2.662kN Partial factors for load combination Load combination factors for variable loads on bridges ψ0.w = 0.3 Wind load main load ψ0.fk = 0.4 Traffic load main load Load combinations, y-direction LC1y := γQ(Qw.y + Qfk.n.y) = 3.993kN LC2y := γQ(Qw.y + ψ0.wQfk.n.y) = 2.159kN Wind main load LC3y := γQ(ψ0.fkQw.y + Qfk.n.y) = 3.169kN Traffic main load Critical load case, y-direction qd.y := max(LC1y, LC2y, LC3y) = 3.993kN Point load on one beam Load case 2, traffic main load is the critical load case. Load response - vertical load in ULS Span lengths Ltot := L1 + L2 + L3 + L4 = 27.3 m Critical load cases kN qd.ULS.z := qd.z = 20.618 Design load (traffic load + self-weight)m kN Gd.ULS.z := γGGk = 5.265 Self-weight onlym Distributed loads, vertical load Distributed load in four, unequal spans. Vertical load working on all beams in the cs. Load in each span: kN kN q1.ULS := qd.ULS.z = 20.618 q3.ULS := qd.ULS.z = 20.618m m kN kN q2.ULS := qd.ULS.z = 20.618 q4.ULS := qm d.ULS.z = 20.618 m Support rotation 3 3 MCL M L q LM 3 D 3 3 3AL1 MBL1 q1L1 θ θ = + + C2 = + + A 3EI 6EI 24EI3EI 6EI 24EI 3 3 MAL1 MBL1 q1L1 MCL3 MDL3 q3L3 θB1 = + + θ = + +6EI 3EI 24EI D1 6EI 3EI 24EI 3 3 MBL2 MCL2 q2L2 MDL4 MEL4 q4L4 θB2 = + + θ3EI 6EI 24EI D2 = + + 3EI 6EI 24EI 3 3 MBL2 MCL2 q2L2 MDL4 MEL4 q2L4 θC1 = + + θ = + +6EI 3EI 24EI E 6EI 3EI 24EI Continuum condition gives: θB1 = - θB2 -θC1 = θC2 θD1 = - θD2 Support moment MA.ULS := 0kNm End support hill-side ME.ULS := 0kNm End support lake-side MC.guess := 1000kNm Initial guess  3 3 MA.ULSL1 q  1.ULSL1 MC.ULSL2 q2.ULSL2  - + + +6 24 6 24  M  B.ULS(MC.ULS) :=  L1 + L2    3   3 3 MBL2 q2L2 MDL3 q3L  3  - + + + 6 24 6 24 MC =  L2 + L  3   3   3 3 MC.ULSL3 q3.ULSL3 ME.ULSL4 q4.ULSL  4  - + + + 6 24 6 24 M D.ULS(MC.ULS) :=  L + L  3 4    3    3 M   B.ULS(MC.guess)L2 q2.ULSL2  + ...   6 24    3    M  D.ULS(MC.guess)L3 q3.ULSL3  + +  6 24   MC.ULS := root M    C.guess + , M L + L  C.guess = -471.453kNm   2 3     3   MB.ULS := MB.ULS(MC.ULS) = 44.479kNm MD.ULS := MD.ULS(MC.ULS) = -402.431kNm Mmax.s.ULS := min(MA.ULS , MB.ULS, MC.ULS, MD.ULS , ME.ULS) = -471.453kNm Reaction forces -MA.ULS MB.ULS q1.ULSL1 RA.ULS := + + = 53.888kNL1 L1 2 MA.ULS -MB.ULS q1.ULSL1 RB1.ULS := + + = 32.707kNL1 L1 2 -MB.ULS MC.ULS q2.ULSL2 RB2.ULS := + + = -224.033kNL2 L2 2 RB.ULS := RB1.ULS + RB2.ULS = -191.326kN MB.ULS -MC.ULS q2.ULSL2 RC1.ULS := + + = 267.331kNL2 L2 2 -MC.ULS MD.ULS q3.ULSL3 RC2.ULS := + + = 177.299kNL3 L3 2 RC.ULS := RC1.ULS + RC2.ULS = 444.63kN MC.ULS -MD.ULS q3.ULSL3 RD1.ULS := + + = 169.082kNL3 L3 2 -MD.ULS ME.ULS q4.ULSL4 RD2.ULS := + + = 139.115kNL4 L4 2 RD.ULS := RD1.ULS + RD2.ULS = 308.197kN MD.ULS -ME.ULS q4.ULSL4 RE.ULS := + + = -52.519kNL4 L4 2 Rtot.ULS := q1.ULSL1 + q2.ULSL2 + q3.ULSL3 + q4.ULSL4 = 562.869kN Rcheck.ULS := RA.ULS + RB.ULS + RC.ULS + RD.ULS + RE.ULS - Rtot.ULS = 0kN Rmax.c.ULS := max(RA.ULS , RB.ULS, RC.ULS, RD.ULS , RE.ULS) = 444.63kN Rmax.t.ULS := min(RA.ULS , RB.ULS, RC.ULS, RD.ULS , RE.ULS) = -191.326kN Shear force distribution V1.ULS(x) := RA.ULS - q1.ULSx V2.ULS(x) := RA.ULS + RB.ULS - q1.ULSL1 - q2.ULS(x - L1) V3.ULS(x) := RA.ULS + RB.ULS + RC.ULS - q1.ULSL1 - q2.ULSL2 - q3.ULSx - (L1 + L2) V4.ULS(x) := RA.ULS + RB.ULS + RC.ULS + RD.ULS ... + -q1.ULSL1 - q2.ULSL2 - q3.ULSL3 - q4.ULSx - (L1 + L2 + L3) VULS(x) := V1.ULS(x) if x  L1 V2.ULS(x) if L1 < x  L1 + L2 V3.ULS(x) if L1 + L2 < x  L1 + L2 + L3 V4.ULS(x) if L1 + L2 + L3 < x  L1 + L2 + L3 + L4  0m  0   0  VA.x.ULS :=   V 0m A.y.ULS :=   =  kN  RA.ULS   53.888   L1   4.2 R V :=   =   m V :=   B2.ULS   -224.033  B.x.ULS  L   4.2 B.y.ULS = kN  1      -RB1.ULS   -32.707   L1 + L2   6.3  RC2.ULS   177.299  VC.x.ULS := =   m V    L + L   6.3 C.y.ULS := =  kN  1 2   -RC1.ULS   -267.331   L1 + L2 + L 3   23.1  R  D2.ULS   139.115 VD.x.ULS := = m V := = kN     L1 + L2 + L3  D.y.ULS   23.1   -RD1.ULS  -169.082   L1 + L2 + L3 + L 4   27.3   -RE.ULS  52.519 VE.x.ULS := = m V := =       kN  L1 + L2 + L + L   27.3  E.y.ULS 3 4   0   0  Vmax.ULS := max(VA.y.ULS, VB.y.ULS, VC.y.ULS, VD.y.ULS, VE.y.ULS) = 177.299kN Vmin.ULS := min(VA.y.ULS, VB.y.ULS, VC.y.ULS, VD.y.ULS, VE.y.ULS) = -267.331kN Moment distribution x M1.ULS(x) := RA.ULSx - q1.ULSx 2  L1   x - L1  M2.ULS(x) := RA.ULSx + RB.ULS(x - L1) - q1.ULSL1x -  - q2.ULS(x - L1)  2   2  M3.ULS(x) := RA.ULSx + RB.ULS(x - L1) + RC.ULSx - (L1 + L2) ...  L1    L2  x - (L1 + L2) + -q1.ULSL1x -  - q 2  2.ULS L2x - L1 +  - q  2  3.ULS x - (L1 + L2)  2  M4.ULS(x) := RA.ULSx + RB.ULS(x - L1) + RC.ULSx - (L1 + L2) + RD.ULSx - (L1 + L2 + L3) ...  L1    L2    L  + -q1.ULSL1x -  - q2.ULSL2x - L1 +  - q3.ULSL3x -  3 L + L +  ...  2    2    1 2 2  x - (L1 + L2 + L3) + -q4.ULSx - (L1 + L2 + L3)  2  MULS(x) := M1.ULS(x) if x  L1 M2.ULS(x) if L1 < x  L1 + L2 M3.ULS(x) if L1 + L2 < x  L1 + L2 + L3 M4.ULS(x) if L1 + L2 + L3 < x  L1 + L2 + L3 + L4 Field moment x00 := 0 x0.1.ULS := Maximize(M1.ULS, x00) = 2.614 m Point where shear force is zero x0.2.ULS := Maximize(M2.ULS, x00) = -6.666 m x0.3.ULS := Maximize(M3.ULS, x00) = 14.899 m x0.4.ULS := Maximize(M4.ULS, x00) = 29.847 m x0.1.ULS := x0.1.ULS if 0m  x0.1.ULS  L1 Check if zero shear force point is within the span, otherwise it is replaced with a dummy number. 0m otherwise x0.2.ULS := x0.2.ULS if L1  x0.2.ULS  L1 + L2 L1 otherwise x0.3.ULS := x0.3.ULS if L1 + L2  x0.3.ULS  L1 + L2 + L3 L1 + L2 otherwise x0.4.ULS := x0.4.ULS if L1 + L2 + L3  x0.4.ULS  L1 + L2 + L3 + L4 Ltot otherwise Mf1.ULS := M1.ULS(x0.1.ULS) = 70.422kNm Mf2.ULS := M2.ULS(x0.2.ULS) = 44.479kNm Mf3.ULS := M3.ULS(x0.3.ULS) = 290.867kNm - 12 Mf4.ULS := M4.ULS(x0.4.ULS) = -1.63  10 kNm Mmax.f.ULS := max(Mf1.ULS, Mf2.ULS, Mf3.ULS, Mf4.ULS) = 290.867kNm Maximum field moment Zero moment positions  L1  xM0.1.ULS := rootM ULS (x) , x, 0 ,  = 0 m 2   L1 L2  xM0.2.ULS := rootM (x) , x, , L +  = 4.397 m ULS 2 1 2   L2 L3  xM0.3.ULS := rootMULS(x) , x, L1 + , L1 + L2 +  = 9.587 m 2 2   L3 L4  xM0.4.ULS := rootM ULS (x) , x, L1 + L2 + , L1 + L2 + L3 +  = 20.211 m2 2   L4  xM0.5.ULS := rootMULS(x) , x, L1 + L2 + L3 + , Ltot = 2  Deflection 4 EI := E0.mean.lvlIx.tot = 241.445MPam Flexural Rigidity MULS(x) κULS(x) := Curvature EI L  1  κULS(x)(L1 - x) dx  0 - 4 θA.ULS := = 3.926  10L1 L  1  κULS(x)x dx  0 - 4 θB1.ULS := = 5.215  10L1 L1+L 2  κULS(x)(L1 + L 2 - x) dxL1 - 4 θB2.ULS := = -5.215  10L2 L1+L 2  κULS(x) (x - L1) dxL1 - 3 θC1.ULS := = -1.269  10L2 L1+L2+L 3  κULS(x)(L1 + L2 + L - x 3 ) dxL1+L2 - 3 θC2.ULS := = 1.269  10L3 L1+L2+L 3  κ  ULS (x)(x - L1 - L2) dx L1+L2 - 3 θD1.ULS := = 2.07  10L3 L +L  1 2 +L3+L4  κULS(x)(L1 + L2 + L3 + L4 - x ) dxL1+L2+L3 - 3 θD2.ULS := = -2.07  10L4 L1+L +L 2 3 +L4  κULS(x)(x - L1 - L2 - L3) dx L1+L2+L3 - 4 θE.ULS := = -9.031  10L4 x x f1.ULS(x) := θA.ULSx -   κULS(x) dx dx   0 0 x x f2.ULS(x) := θB2.ULS(x - L ) -  1 κ (x) dx dx  ULSL1 L1 x x f3.ULS(x) := θC2.ULS(x - L1 - L2) -   κULS(x) dx dx L +L 1 2 L1+L2 x x f  4.ULS(x) := θD2.ULS(x - L1 - L2 - L3) - κULS(x) dx dx L1+L +L 2 3 L1+L2+L3 fULS(x) := f1.ULS(x) if x  L1 f2.ULS(x) if L1 < x  L1 + L2 f3.ULS(x) if L1 + L2 < x  L1 + L2 + L3 f4.ULS(x) if L1 + L2 + L3 < x  L1 + L2 + L3 + L4 Maximum deflection L1 xθ0.1.guess := = 2.1 m Guess of where curvature is zero2 L2 xθ0.2.guess := L1 + = 5.25 m2 L3 xθ0.3.guess := L1 + L2 + = 14.7 m2 L4 xθ0.4.guess := L1 + L2 + L3 + = 25.2 m2  x θ0.1.guess   xθ0.1.ULS := root κULS(x) dx - θA.ULS , xθ0.1.guess = 2.214 m Position where curvature 0  is zero  x θ0.2.guess  x θ0.2.ULS := root κ (x) dx - θ ULS B2.ULS , xθ0.2.guess = 5.449 m  L1   x θ0.3.guess  xθ0.3.ULS := root  κ  ULS (x) dx - θC2.ULS , xθ0.3.guess = 14.866 m   L1+L2   x θ0.4.guess  x θ0.4.ULS := root κULS(x) dx - θ D2.ULS, xθ0.4.guess = 24.799 m  L1+L2+L3  fmax.1.ULS := -f1.ULS(xθ0.1.ULS) = -0.551mm Maximum field deflection fmax.2.ULS := -f2.ULS(xθ0.2.ULS) = 0.485mm fmax.3.ULS := -f3.ULS(xθ0.3.ULS) = -24.744mm fmax.4.ULS := -f4.ULS(xθ0.4.ULS) = 1.552mm fmax.up.ULS := max(fmax.1.ULS , fmax.2.ULS , fmax.3.ULS , fmax.4.ULS) = 1.552mm fmax.down.ULS := min(fmax.1.ULS , fmax.2.ULS , fmax.3.ULS , fmax.4.ULS) = -24.744mm Summary of design load response - vertical load in ULS Maximum support moment in support C Maximum field moment in field 3 Mmax.z := max( Mmax.s.ULS , Mmax.f.ULS ) = 471.453kNm Maximum reaction force (compression) in support C Maximum reaction force (tension) in support B Rmax.c.z := Rmax.c.ULS = 444.63kN Rmax.t.z := Rmax.t.ULS = -191.326kN Maximum shear force in support C Design shear force = negative shear Vmax.z := max( Vmax.ULS , Vmin.ULS ) = 267.331kN Maximum deflection in span 3 δmax.uls := fmax.down.ULS = -24.744mm Load response - transverse load in ULS Distributed loads, transverse load kN kN q1.ULS.x := qd.x = 3.22 q3.ULS.x := qd.x = 3.22m m kN kN q2.ULS.x := qd.x = 3.22 qm 4.ULS.x := qd.x = 3.22 m Support rotation Continuum condition gives: θB1 = - θB2 -θC1 = θC2 θD1 = - θD2 Support moment MA.ULS.x := 0kNm End support hill-side ME.ULS.x := 0kNm End support lake-side MC.guess := 1000kNm Initial guess  3 3 MA.ULS.xL1 q1.ULS.xL1 M  C.ULS.xL2 q2.ULS.xL2  - + + + M (M ) :=  6 24 6 24  B.ULS.x C.ULS.x  L  1 + L2   3   3 3 MBL2 q2L2 MDL3 q3L3   - + + +6 24 6 24  M =  C  L2 + L3    3   3 3 M C.ULS.xL3 q3.ULS.xL3 ME.ULS.xL4 q4.ULS.xL4  - + + + ( )  6 24 6 24  M D.ULS.x MC.ULS.x :=  L + L   3 4   3    3 M   B.ULS.x(MC.guess)L2 q2.ULS.xL2 + ...    6 24    ( ) 3    MD.ULS.x M  C.guess L3 q3.ULS.xL3  + +  M := root M +  6 24   C.ULS.x  C.guess , M = -73.618kNmL + L   2 3  C.guess      3   MB.ULS.x := MB.ULS.x(MC.ULS.x) = 6.945kNm MD.ULS.x := MD.ULS.x(MC.ULS.x) = -62.84kNm Mmax.s.ULS.x := min(MA.ULS.x, MB.ULS.x , MC.ULS.x , MD.ULS.x, ME.ULS.x) = -73.618kNm Reaction forces -MA.ULS.x MB.ULS.x q1.ULS.xL1 RA.ULS.x := + + = 8.415kNL1 L1 2 MA.ULS.x -MB.ULS.x q1.ULS.xL1 RB1.ULS.x := + + = 5.107kNL1 L1 2 -MB.ULS.x MC.ULS.x q2.ULS.xL2 RB2.ULS.x := + + = -34.983kNL2 L2 2 RB.ULS.x := RB1.ULS.x + RB2.ULS.x = -29.876kN MB.ULS.x -MC.ULS.x q2.ULS.xL2 RC1.ULS.x := + + = 41.744kNL2 L2 2 -MC.ULS.x MD.ULS.x q3.ULS.xL3 RC2.ULS.x := + + = 27.685kNL3 L3 2 RC.ULS.x := RC1.ULS.x + RC2.ULS.x = 69.429kN MC.ULS.x -MD.ULS.x q3.ULS.xL3 RD1.ULS.x := + + = 26.402kNL3 L3 2 -MD.ULS.x ME.ULS.x q4.ULS.xL4 RD2.ULS.x := + + = 21.723kNL4 L4 2 RD.ULS.x := RD1.ULS.x + RD2.ULS.x = 48.125kN MD.ULS.x -ME.ULS.x q4.ULS.xL4 RE.ULS.x := + + = -8.201kNL4 L4 2 Rtot.ULS.x := q1.ULS.xL1 + q2.ULS.xL2 + q3.ULS.xL3 + q4.ULS.xL4 = 87.893kNTotal reaction force - 14 Rcheck.ULS.x := RA.ULS.x + RB.ULS.x + RC.ULS.x + RD.ULS.x + RE.ULS.x - Rtot.ULS.x = 1.455  10 kN Rmax.c.ULS.x := max(RA.ULS.x, RB.ULS.x , RC.ULS.x , RD.ULS.x, RE.ULS.x) = 69.429kN Rmax.t.ULS.x := min(RA.ULS.x, RB.ULS.x , RC.ULS.x , RD.ULS.x, RE.ULS.x) = -29.876kN Shear force distribution V1.ULS.x(x) := RA.ULS.x - q1.ULS.xx V2.ULS.x(x) := RA.ULS.x + RB.ULS.x - q1.ULS.xL1 - q2.ULS.x(x - L1) V3.ULS.x(x) := RA.ULS.x + RB.ULS.x + RC.ULS.x - q1.ULS.xL1 - q2.ULS.xL2 - q3.ULS.xx - (L1 + L2) V4.ULS.x(x) := RA.ULS.x + RB.ULS.x + RC.ULS.x + RD.ULS.x ... + -q1.ULS.xL1 - q2.ULS.xL2 - q3.ULS.xL3 - q4.ULS.xx - (L1 + L2 + L3) VULS.x(x) := V1.ULS.x(x) if x  L1 V2.ULS.x(x) if L1 < x  L1 + L2 V3.ULS.x(x) if L1 + L2 < x  L1 + L2 + L3 V4.ULS.x(x) if L1 + L2 + L3 < x  L1 + L2 + L3 + L4  0m   0   0  VA.x.ULS.x :=   V 0m  A.y.ULS.x :=  = kN  R    A.ULS.x  8.415  L 1   4.2  RB2.ULS.x   -34.983  VB.x.ULS.x := =    m V   L  4.2 B.y.ULS.x := =    kN  1   -RB1.ULS.x  -5.107   L1 + L 2   6.3  R  C2.ULS.x   27.685 VC.x.ULS.x := =   m V :=  =  kN  L1 + L2   6.3 C.y.ULS.x  -RC1.ULS.x  -41.744   L1 + L 2 + L3   23.1   R  D2.ULS.x   21.723 VD.x.ULS.x := =   m V  D.y.ULS.x := =  kN  L1 + L2 + L3   23.1   -RD1.ULS.x  -26.402   L1 + L2 + L 3 + L4   27.3   -RE.ULS.x  8.201VE.x.ULS.x := = m V := = kN  L1 + L2 + L3 + L4    E.y.ULS.x     27.3   0   0  Vmax.ULS.x := max(VA.y.ULS.x , VB.y.ULS.x, VC.y.ULS.x, VD.y.ULS.x , VE.y.ULS.x) = 27.685kN Vmin.ULS.x := min(VA.y.ULS.x , VB.y.ULS.x, VC.y.ULS.x, VD.y.ULS.x , VE.y.ULS.x) = -41.744kN Moment distribution x M1.ULS.x(x) := RA.ULS.xx - q1.ULS.xx 2  L1   x - L1  M2.ULS.x(x) := RA.ULS.xx + RB.ULS.x(x - L1) - q1.ULS.xL1x -  - q2.ULS.x(x - L1)  2   2   L  M3.ULS.x(x) := RA.ULS.xx + RB.ULS.x(x - L1) 1 + RC.ULS.xx - (L1 + L2) - q1.ULS.xL1x -  ... 2    L2  x - (L1 + L2) + -q2.ULS.xL2x - L1 +  - q  2  3.ULS.x x - (L1 + L2)  2  M4.ULS.x(x) := RA.ULS.xx + RB.ULS.x(x - L1) + RC.ULS.xx - (L1 + L2) + RD.ULS.xx - (L1 + L2 + L3)  L1    L2    L3  + -q1.ULS.xL1x -  - q2.ULS.xL2x - L1 +  - q3.ULS.xL3x - L + L +  2    2    1 2 2  x - (L1 + L2 + L3) + -q4.ULS.xx - (L1 + L2 + L3)  2  MULS.x(x) := M1.ULS.x(x) if x  L1 M2.ULS.x(x) if L1 < x  L1 + L2 M3.ULS.x(x) if L1 + L2 < x  L1 + L2 + L3 M4.ULS.x(x) if L1 + L2 + L3 < x  L1 + L2 + L3 + L4 Field moment x00 := 0 x0.1.ULS.x := Maximize(M1.ULS.x , x00) = 2.614 m Point where shear force is zero x0.2.ULS.x := Maximize(M2.ULS.x , x00) = -6.666m x0.3.ULS.x := Maximize(M3.ULS.x , x00) = 14.899 m x0.4.ULS.x := Maximize(M4.ULS.x , x00) = 29.847 m x0.1.ULS.x := x0.1.ULS.x if 0m  x0.1.ULS.x  L1 Check if zero shear force point is within the span, otherwise it is replaced with a dummy number. 0m otherwise x0.2.ULS.x := x0.2.ULS.x if L1  x0.2.ULS.x  L1 + L2 L1 otherwise x0.3.ULS.x := x0.3.ULS.x if L1 + L2  x0.3.ULS.x  L1 + L2 + L3 L1 + L2 otherwise x0.4.ULS.x := x0.4.ULS.x if L1 + L2 + L3  x0.4.ULS.x  L1 + L2 + L3 + L4 Ltot otherwise Mf1.ULS.x := M1.ULS.x(x0.1.ULS.x) = 10.996kNm Mf2.ULS.x := M2.ULS.x(x0.2.ULS.x) = 6.945kNm Mf3.ULS.x := M3.ULS.x(x0.3.ULS.x) = 45.419kNm Mf4.ULS.x := M4.ULS.x(x0.4.ULS.x) = 0kNm Mmax.f.ULS.x := max(Mf1.ULS.x, Mf2.ULS.x, Mf3.ULS.x, Mf4.ULS.x) = 45.419kNMmaximum field moment Zero moment positions  L1  xM0.1.ULS.x := rootMULS.x(x) , x, 0 ,  = 0 m 2   L1 L2  xM0.2.ULS.x := rootMULS.x(x) , x, , L1 +  = 4.397 m 2 2   L2 L3  xM0.3.ULS.x := rootMULS.x(x) , x, L1 + , L1 + L2 +  = 9.587 m 2 2   L3 L4  xM0.4.ULS.x := rootMULS.x(x) , x, L1 + L2 + , L 2 1 + L2 + L3 +  = 20.211 m2   L4  xM0.5.ULS.x := rootMULS.x(x) , x, L1 + L2 + L3 + , Ltot = 27.3 m 2  Summary of design load response - transverse load in ULS Maximum support moment in support C Maximum field moment in field 3 Mmax.x := max( Mmax.s.ULS.x , Mmax.f.ULS.x ) = 73.618kNm Maximum reaction force (compression) in support C Maximum reaction force (tension) in support B Rmax.c.x := Rmax.c.ULS.x = 69.429kN Rmax.t.x := Rmax.t.ULS.x = -29.876kN Maximum shear force in support C Design shear force = negative shear Vmax.x := max( Vmax.ULS.x , Vmin.ULS.x ) = 41.744kN Load response - longitudinal load in ULS Distributed axial load on all beams in the cross-section qd.y = 3.993kN qd.y kN qd.y.beams := = 1.901 Design loadB - 2clvl.s m Design stresses - ULS Bending stress calculated in the center of gravity of the beams over the third support Mmax.z σm.d.z := = 8.084MPa Vertical laodWx Mmax.x σm.d.x := = 0.177MPa Transverse loadWz km := 0.7 LVL E dgewise bending in z-direction Verify according to EN 1995-1-1 eq. 6.11, 6.12. σm.d.z σm.d.x ηm.zx := + k  = 0.485f mm.0.edge.d.lvl fm.0.flat.d.lvl σm.d.z σm.d.x ηm.xz := km + = 0.344fm.0.edge.d.lvl fm.0.flat.d.lvl M oment capacity of the straight LVL beam 2 2 hlvl.sbb lvl.slvl.shlvl.s - 3 3- 3 3 W W := = 6.48  10 m lvl.s.z := = 1.166  10 m lvl.s.x 66 MRd.lvl.s.z := Wlvl.s.xfm.0.edge.d.lvl = 109.318kNm MRd.lvl.s.x := Wlvl.s.zfm.0.edge.d.lvl = 19.677kNm Utilisation of the bending capacity of the straight beams Mmax.z Mmax.x MEd.z := = 29.466kNm M := = 4.601kNmn + n Ed.xlvl.s lvl nlvl.s + nlvl MEd.z MEd.x ηM.s.z := = 0.27 ηM M.s.x := = 0.234 Rd.lvl.s.z MRd.lvl.s.x Design bending stresses in one straight LVL beam MEd.z σm.d.s.z := = 4.547MPaWlvl.s.x MEd.x σm.d.s.x := = 3.945MPaWlvl.s.z Moment capacity of the curved LVL panel 2 2 blvlhlvl - 3 3 hlvlblvl - 4 3 Wlvl.x := = 3.24  10 m W := = 2.916  10 m6 lvl.z 6 MRd.lvl.z := Wlvl.xfm.0.edge.d.lvl = 54.659kNm MRd.lvl.x := Wlvl.zfm.0.edge.d.lvl = 4.919kNm Utilisation of the bending capacity of the straight beams MEd.z MEd.x ηM.curve.z := = 0.539 ηM.curve.x := = 93.532%MRd.lvl.z MRd.lvl.x Design bending stresses in one curved LVL panel MEd.z σm.d.curve.z := = 9.094MPa Still assuming that the panels are straightWlvl.x MEd.x σm.d.curve.x := = 15.779MPaWlvl.z Check bending stresses due to induced deformation According to EN1995-1-1 chp. 6.4.3 (for glulam and LVL) σm.90.d  krfm.d EN 1995-1-1 eq. 6.41 tlvl = 0.027 m Thickness one lamella rsin = 14.5 m Radius of the curved panels Stresses in the curved zone of the bent panels, between the lamellas E0.k.lvltlvl σm.90.d := = 10.8MPa2rsin rsin rsin kr.lvl := 0.76 + 0.001 if < 240 = 1 EN 1995-1-1 eq 6.41blvl blvl 1 otherwise Verify σm.90.d ηm.90.d := = 0.518kr.lvlfm.0.flat.d.lvl Combine the induced bending stresses with stresses from applied load and axial compression forces Design compression stresses Fc.0.d := qd.y(nlvl + nlvl.s) = 63.888kN Compression force in all elements Fc.0.n.d := qd.y = 3.993kN Compression force in one element Fc.0.d σc.y := = 0.11MPa Design compression stress, all elementsAtot Fc.0.n.d Design compression stress, one straight LVL beam σc.y.s := = 0.062MPaAlvl.s Fc.0.n.d σc.y.curve := = 0.123MPa Design compression stress, one LVL panelAlvl Utilisation in compression only Verify according to EN 1995-1-1 eq. 6.2 σc.y - 3 ηc.y := = 7.512  10fc.0.d.lvl σc.y.s - 3 ηc.y.s := = 4.225  10fc.0.d.lvl σc.y.curve - 3 ηc.y.curve := = 8.451  10fc.0.d.lvl Control of bending in two directions and axial compression - straight LVL beam The straight LVL beam is subjected to bending and axial compression and should be verified according to EN1995-1-1 6.2.4 (combined bending and axial compression). Check interaction according to EN1995-1-1 6.2.4 Edgewise orientation in z-direction km = 0.7 EN1995-1-1 6.1.6 (2) 2  σc.y.s  σm.d.s.z σm.d.s.x ηm.d.s.xz :=   + km + EN1995-1-1 eq. 6.19  fc.0.d.lvl fm.0.edge.d.lvl fm.0.flat.d.lvl 2  σc.y.s  σm.d.s.z σm.d.s.x ηm.d.s.zx :=   + + km EN1995-1-1 eq. 6.20  fc.0.d.lvl fm.0.edge.d.lvl fm.0.flat.d.lvl Control of bending and axial compression - curved LVL panels The LVL panels are initially deformed and subjected to bending and axial compression, and should be verified according to EN1995-1-1 6.3.3 (stability). As the LVL panels are designed to work as a horizontal truss, the bending in x-direction will not be checked for one panel only. Bending in z-direction + compression + induced deformation will be checked for one panel. Distance between the distancers ssin = 2.1 m lef.curve := 0.9ssin Simply supported beam, uniformly distributed EN1995-1-1 tab. 6.1 load Critical bendning stress, for softwood with solid rectangular cross-section 2 0.78blvl σm.crit.curve := E0.k.lvl = 23.266MPa EN1995-1-1 eq. 6.32hlvllef.curve Relative slenderness for bending fm.0.edge.k.lvl λrel.m.curve := = 1.375 EN1995-1-1 eq. 6.30σm.crit.curve Factor considering reduced bending strength due to lateral buckling kcrit := 1 Calculate instability factor for the curved panel Iz.lvl iz.curve := DTS 5.2 p. 26Alvl le.curve := ssin = 2.1 m Slenderness ratio le.curve λz.curve := = 134.715iz.curve λz.curve fc.0.k.lvl λrel.z.curve :=  = 2.355 EN1995-1-1 eq. 6.22π E0.k.lvl Controlλ.rel.z.curve := "Eq. 6.19, 6.20" if λrel.z.curve < 0.3 = "Eq. 6.23, 6.24" "Eq. 6.23, 6.24" otherwise The stresses in z-direction increase due to deflection. Check capacity according to EN1995-1-1 eq. 6.24. βc := 0.1 EN1995-1-1 eq. 6.29 2 kz.curve := 0.5  1 + βc(λrel.z.curve - 0.3) + λ rel.z.curve  = 3.377 EN1995-1-1 eq. 6.28 1 kc.z.curve := = 0.173 2 2 EN1995-1-1 eq. 6.26 kz.curve + kz.curve - λrel.z.curve Panel in bending in one direction and compression 2 σc.y.curve  σm.d.curve.z  ηm.d.curve.cz := +   = 0.34 EN 1995-1-1 eq. 6.35kc.z.curvefc.0.d.lvl  kcritfm.0.edge.d.lvl P anel in bending inone direction, compression and induced deformations Assume partly interaction kcurve := 0.7 2 σc.y.curve  σm.d.curve.z  σm.90.d ηm.d.curve.czd := +   + k  = 0.702k curvec.z.curvefc.0.d.lvl  kcritfm.0.edge.d.lvl  kr.lvlfm.0.flat.d.lvl Control of bending and axial compression - whole cross-section Largest unsupported span L3 = 16.8 m EN1995-1-1 tab. 6.1 Simply supported beam, uniformly distributed load lef.deck := 0.9L Critical bendning stress, for softwood with solid rectangular cross-section 2 0.78(B - 2clvl.s) 4 σm.crit.deck := E0.k.lvl = 3.519  10 MPa EN1995-1-1 eq. 6.32hlvllef.curve Relative slenderness for bending fm.0.edge.k.lvl EN1995-1-1 eq. 6.30 λrel.m.deck := = 0.035σm.crit.deck kcrit = 1 Calculate instability factor for the whole bridge deck Ix.tot Iz.tot ix.deck := = 0.173 m iz.deck := = 0.865 m DTS 5.2 p. 26Atot Atot le.x.deck := ssin = 2.1 m le.z.deck := L le.x.deck le.z.deck λx.deck := = 12.124 λz.deck := = 31.561ix.deck iz.deck λz.deck fc.0.k.lvl EN1995-1-1 eq. 6.22 λrel.z.deck :=  = 0.552π E0.k.lvl λx.deck fc.0.k.lvl λrel.x.deck :=  = 0.212π E0.k.lvl Controlλ.rel.z.deck := "Eq. 6.19, 6.20" if λrel.z.deck < 0.3 = "Eq. 6.23, 6.24" "Eq. 6.23, 6.24" otherwise Controlλ.rel.x.deck := "Eq. 6.19, 6.20" if λrel.x.deck < 0.3 = "Eq. 6.19, 6.20" "Eq. 6.23, 6.24" otherwise Both slenderness ratios must be <0.3 to use eq. 6.19-20. The stresses increase due to deflection. Check capacity according to eq. 6.23, 24. 2 kz.deck := 0.5  1 + βc(λ rel.z.deck - 0.3) + λrel.z.deck  = 0.665 EN1995-1-1 eq. 6.28 2 kx.deck := 0.5  1 + β (λ c rel.x.deck - 0.3) + λrel.x.deck  = 0.518 1 kc.z.deck := = 0.966 2 2 EN1995-1-1 eq. 6.26 kz.deck + kz.deck - λrel.z.deck 1 kc.x.deck := = 1.009 2 2 kx.deck + kx.deck - λrel.x.deck Verify z-direction according to EN1995-1-1 eq. 6.24 σc.y σm.d.z σm.d.x ηm.d.cz := + + km = 0.493 EN1995-1-1 eq. 6.24kc.z.deckfc.0.d.lvl fm.0.edge.d.lvl fm.0.flat.d.lvl σc.y σm.d.z σm.d.x ηm.d.cx := + k  + = 0.351 EN1995-1-1 eq. 6.23k f mc.x.deck c.0.d.lvl fm.0.edge.d.lvl fm.0.flat.d.lvl Control of bending, axial compression and induced deformation - whole cross-section Mainly bending in x-direction, eq. 6.23 σc.y σm.d.z σm.d.x σm.90.d ηm.d.cxd := + k  + + k  = 0.714k m curvec.x.deckfc.0.d.lvl fm.0.edge.d.lvl fm.0.flat.d.lvl kr.lvlfm.0.flat.d.lvl Mainly bending in z-direction, eq. 6.24 σc.y σm.d.z σm.d.x σm.90.d ηm.d.czd := + + km + kcurve = 0.856kc.z.deckfc.0.d.lvl fm.0.edge.d.lvl fm.0.flat.d.lvl kr.lvlfm.0.flat.d.lvl Design shear stresses Shear stresses over the third support kcr := 1.0 LVL EN1995-1-1 eq. 6.13a Assume interaction between the LVL panels and beams, verify for the whole cross-section area 2 Acr := Atotkcr = 0.583m Vmax.z τd.z := = 0.458MPa Vertical laodAcr Vmax.x τd.x := = 0.072MPa Transverse loadAcr Verify according to EN 1995-1-1 eq. 6.13 τd.z ητ.z := = 0.268fv.0.edge.d.lvl τd.x ητ.x := = 0.075fv.0.flat.d.lvl Summary of utilisation ratios in ULS Bending W hole cross-section Bending, z main Bending, x main ηm.zx = 0.485 ηm.xz = 0.344 O ne straight beam Capacity ratio, bending z-dir Capacity ratio, bending x-dir ηM.s.z = 0.27 ηM.s.x = 0.234 One assumed straight panel Capacity ratio, bending z-dir Capacity ratio, bending x-dir ηM.curve.z = 0.539 ηM.curve.x = 0.935 Induced deformation One curved panel ηm.90.d = 0.518 Compression Whole cross-section ηc.y = 0.751% One straight beam ηc.y.s = 0.423% One assumed straight panel ηc.y.curve = 0.845% Bending + compression Whole cross-section Bending, z main Bending, x main ηm.d.cz = 0.493 ηm.d.cx = 0.351 O ne straight beam Bending, z main Bending, x main ηm.d.s.zx = 0.402 ηm.d.s.xz = 0.378 O ne assumed straight panel Bending, z-direction only ηm.d.curve.cz = 0.34 Bending + compression + induced deformation Whole cross-section Bending, z main Bending, x main ηm.d.czd = 0.856 ηm.d.cxd = 0.714 One curved panel Bending, z main ηm.d.curve.czd = 0.702 Shear Whole cross-section ητ.z = 0.268 ητ.x = 0.075 SLS load combinations Service class 3 kdef := 2.0 EN1995-1-1 tab. 3.2 Load combinations, vertical load Partial factors for load combinations in SLS according to STR-2 γsls := 1.0 LH3 tab. 5.3 Load combination factors for variable loads on bridges in SLS ψ1.w := 0.2 Wind load frequent load EN1990 tab. A2.2 ψ1.fk := 0.4 Traffic load frequent load ψ2.w := 0 Wind load quasi permanent load EN1990 tab. A2.2 ψ2.fk := 0 Traffic load qasi permanent load Load combinations, vertical load, permanent Self-weight kN qd.sls := γslsGk = 4.387 m Load combinations, vertical load, variable Traffic load kN LC1q.f := γslsQfk.z = 10.236 m kN LC2q.f := γslsψ2.fkQfk.z = 0 m kN qd.sls.q := max(LC1q.f , LC2q.f ) = 10.236 m Calculate the deflection due to traffic load only (EN1995-2 7.2) Use man values of density (EN1995-2 7.1 (1)). Load response - permanent load in SLS Distributed loads Load distribution: self-weight is uniformly distributed in all spans. q1.SLS := qd.sls q3.SLS := qd.sls q2.SLS := qd.sls q4.SLS := qd.sls Support moment MA.SLS := 0kNm ME.SLS := 0kNm MC.guess := 1000kNm Initial guess  3 3 M L q L M L q L A.SLS 1 1.SLS 1 C.SLS 2 2.SLS 2  - + + + 6 24 6 24 M B.SLS(MC.SLS) :=  L + L   1 2   3   3 3 MBL2 q  2L2 MDL3 q3L3  - + + + 6 24 6 24 M = C  L + L   2 3   3   3 3 MC.SLSL3 q3.SLSL3 ME.SLSL4 q4.SLSL  4  - + + + ( )  6 24 6 24  M D.SLS MC.SLS :=  L + L   3 4   3    3   MB.SLS(MC.guess)L q   2 2.SLSL2   + ...     6 24 3     MD.SLS(M  C.guess)L3 q3.SLSL3    + +  6 24   MC.SLS := root   MC.guess + , MC.guess = -100.316kNm L  2 + L3     3   MB.SLS := MB.SLS(MC.SLS) = 9.464kNm MD.SLS := MD.SLS(MC.SLS) = -85.63kNm Mmax.s.SLS := min(MA.SLS , MB.SLS, MC.SLS, MD.SLS , ME.SLS) = -100.316kNm Reaction forces -MA.SLS MB.SLS q1.SLSL1 RA.SLS := + + = 11.466kNL1 L1 2 MA.SLS -MB.SLS q1.SLSL1 RB1.SLS := + + = 6.96kNL1 L1 2 -MB.SLS MC.SLS q2.SLSL2 RB2.SLS := + + = -47.67kNL2 L2 2 RB.SLS := RB1.SLS + RB2.SLS = -40.71kN MB.SLS -MC.SLS q2.SLSL2 RC1.SLS := + + = 56.883kNL2 L2 2 -MC.SLS MD.SLS q3.SLSL3 RC2.SLS := + + = 37.726kNL3 L3 2 RC.SLS := RC1.SLS + RC2.SLS = 94.609kN MC.SLS -MD.SLS q3.SLSL3 RD1.SLS := + + = 35.977kNL3 L3 2 -MD.SLS ME.SLS q4.SLSL4 RD2.SLS := + + = 29.601kNL4 L4 2 RD.SLS := RD1.SLS + RD2.SLS = 65.578kN MD.SLS -ME.SLS q4.SLSL4 RE.SLS := + + = -11.175kNL4 L4 2 Rtot.SLS := q1.SLSL1 + q2.SLSL2 + q3.SLSL3 + q4.SLSL4 = 119.768kN Total reaction force - 14 Rcheck.SLS := RA.SLS + RB.SLS + RC.SLS + RD.SLS + RE.SLS - Rtot.SLS = -1.455  10 kN Rmax.c.SLS := max(RA.SLS , RB.SLS, RC.SLS, RD.SLS , RE.SLS) = 94.609kN Rmax.t.SLS := min(RA.SLS , RB.SLS, RC.SLS, RD.SLS , RE.SLS) = -40.71kN Shear force distribution V1.SLS(x) := RA.SLS - q1.SLSx V2.SLS(x) := RA.SLS + RB.SLS - q1.SLSL1 - q2.SLS(x - L1) V3.SLS(x) := RA.SLS + RB.SLS + RC.SLS - q1.SLSL1 - q2.SLSL2 - q3.SLSx - (L1 + L2) V4.SLS(x) := RA.SLS + RB.SLS + RC.SLS + RD.SLS ... + -q1.SLSL1 - q2.SLSL2 - q3.SLSL3 - q4.SLSx - (L1 + L2 + L3) VSLS(x) := V1.SLS(x) if x  L1 V2.SLS(x) if L1 < x  L1 + L2 V3.SLS(x) if L1 + L2 < x  L1 + L2 + L3 V4.SLS(x) if L1 + L2 + L3 < x  L1 + L2 + L3 + L4  0m   0   0  VA.x.SLS :=   V :=   =  kN 0m  A.y.SLS  RA.SLS   11.466   L 1   4.2  RB2.SLS   -47.67 VB.x.SLS := =    m V :=   =  kN  L1   4.2 B.y.SLS  -RB1.SLS   -6.96   L1 + L 2   6.3  R  C2.SLS   37.726 VC.x.SLS := =   m VC.y.SLS := =   kN  L1 + L2   6.3  -RC1.SLS   -56.883   L1 + L2 + L 3   23.1   RD2.SLS   29.601  VD.x.SLS := =   m VD.y.SLS :=   = kN  L1 + L2 + L3   23.1     -RD1.SLS  -35.977   L1 + L 2 + L3 + L4   27.3   -RE.SLS  11.175 VE.x.SLS := =   m VE.y.SLS :=   =  kN L1 + L2 + L3 + L4   27.3   0   0  Vmax.SLS := max(VA.y.SLS , VB.y.SLS, VC.y.SLS, VD.y.SLS , VE.y.SLS) = 37.726kN Vmin.SLS := min(VA.y.SLS , VB.y.SLS, VC.y.SLS, VD.y.SLS , VE.y.SLS) = -56.883kN Moment distribution x M1.SLS(x) := RA.SLSx - q1.SLSx 2  L1   x - L1  M2.SLS(x) := RA.SLSx + RB.SLS(x - L1) - q1.SLSL1x -  - q 2  2.SLS(x - L1)  2  M3.SLS(x) := RA.SLSx + RB.SLS(x - L1) + RC.SLSx - (L1 + L2) ...  L1    L2  x - (L1 + L2) + -q1.SLSL1x -  - q2.SLSL2x - L1 +  - q 2    2  3.SLS x - (L1 + L2)  2  M4.SLS(x) := RA.SLSx + RB.SLS(x - L1) + RC.SLSx - (L1 + L2) + RD.SLSx - (L1 + L2 + L3) ...  L1    L2    L3  + -q1.SLSL1x -  - q 2  2.SLS L2x - L1 +  - q3.SLSL3x - L1 + L2 +  ...  2    2  x - (L1 + L2 + L3) + -q4.SLSx - (L1 + L2 + L3)  2  MSLS(x) := M1.SLS(x) if x  L1 M2.SLS(x) if L1 < x  L1 + L2 M3.SLS(x) if L1 + L2 < x  L1 + L2 + L3 M4.SLS(x) if L1 + L2 + L3 < x  L1 + L2 + L3 + L4 Field moment x00 = 0 x0.1.SLS := Maximize(M1.SLS, x00) = 2.614 m Point where shear force is zero x0.2.SLS := Maximize(M2.SLS, x00) = -6.666 m x0.3.SLS := Maximize(M3.SLS, x00) = 14.899 m x0.4.SLS := Maximize(M4.SLS, x00) = 29.847 m x0.1.SLS := x0.1.SLS if 0m  x0.1.SLS  L1 Check if zero shear force point is within the span, otherwise it is replaced with a dummy number. 0m otherwise x0.2.SLS := x0.2.SLS if L1  x0.2.SLS  L1 + L2 L1 otherwise x0.3.SLS := x0.3.SLS if L1 + L2  x0.3.SLS  L1 + L2 + L3 L1 + L2 otherwise x0.4.SLS := x0.4.SLS if L1 + L2 + L3  x0.4.SLS  L1 + L2 + L3 + L4 Ltot otherwise Mf1.SLS := M1.SLS(x0.1.SLS) = 14.984kNm Mf2.SLS := M2.SLS(x0.2.SLS) = 9.464kNm Mf3.SLS := M3.SLS(x0.3.SLS) = 61.891kNm - 14 Mf4.SLS := M4.SLS(x0.4.SLS) = -9.459  10 kNm Mmax.f.SLS := max(Mf1.SLS, Mf2.SLS, Mf3.SLS, Mf4.SLS) = 61.891kNm Maximum field moment Zero moment positions  L1  xM0.1.SLS := rootMSLS(x) , x, 0 ,  = 0 m 2   L1 L 2  xM0.2.SLS := root MSLS(x) , x, , L1 +  = 4.397 m 2 2   L2 L3  xM0.3.SLS := rootM SLS (x) , x, L1 + , L1 + L2 +  = 9.587 m2 2   L3 L4  xM0.4.SLS := rootMSLS(x) , x, L1 + L 2 + , L1 + L2 + L3 +  = 20.211 m2 2   L4  xM0.5.SLS := rootMSLS(x) , x, L1 + L2 + L3 + , L  = 2 tot Deflection 4 EI = 241.445MPam Flexural Rigidity MSLS(x) κSLS(x) := Curvature EI L  1  κSLS(x)(L1 - x) dx  0 - 5 θA.SLS := = 8.353  10L1 L  1  κSLS(x)x dx  0 - 4 θB1.SLS := = 1.11  10L1 L +L  1 2  κSLS(x)(L1 + L 2 - x) dxL1 - 4 θB2.SLS := = -1.11  10L2 L  1 +L2  κSLS(x)(x - L1) dxL1 - 4 θC1.SLS := = -2.701  10L2 L1+L +L 2 3  κSLS(x)(L1 + L2 + L3 - x) dx L1+L2 - 4 θC2.SLS := = 2.701  10L3 L1+L2+L 3  κSLS(x)(x - L1 - L 2) dxL1+L2 - 4 θD1.SLS := = 4.404  10L3 L  1 +L2+L3+L4  κSLS(x)(L1 + L2 + L3 + L4 - x) dx L1+L2+L3 - 4 θD2.SLS := = -4.404  10L4 L1+L2+L +L 3 4  κSLS(x)(x - L1 - L2 - L 3) dxL1+L2+L3 - 4 θE.SLS := = -1.922  10L4 x x f1.SLS(x) := θA.SLSx -   κSLS(x) dx dx   0 0 x x f2.SLS(x) := θB2.SLS(x - L ) -  1 κ  SLS(x) dx dxL 1 L1 x x f3.SLS(x) := θ   C2.SLS(x - L1 - L2) - κSLS(x) dx dx  L1+L  2 L1+L2 x x f4.SLS(x) := θD2.SLS(x - L1 - L2 - L3) -   κSLS(x) dx dx L   1+L2+L3 L1+L2+L3 fSLS(x) := f1.SLS(x) if x  L1 f2.SLS(x) if L1 < x  L1 + L2 f3.SLS(x) if L1 + L2 < x  L1 + L2 + L3 f4.SLS(x) if L1 + L2 + L3 < x  L1 + L2 + L3 + L4 Maximum deflection xθ0.1.guess = 2.1 m Guess of where curvature is zero xθ0.2.guess = 5.25 m xθ0.3.guess = 14.7 m xθ0.4.guess = 25.2 m  x θ0.1.guess   xθ0.1.SLS := root κSLS(x) dx - θA.SLS , xθ0.1.guess = 2.214 m Position where curvature 0  is zero x  θ0.2.guess  xθ0.2.SLS := root  κSLS(x) dx - θ B2.SLS , xθ0.2.guess = 5.449 m  L1   x θ0.3.guess  xθ0.3.SLS := root  κSLS(x) dx - θC2.SLS , xθ0.3.guess = 14.866 m  L1+L2   x θ0.4.guess  x θ0.4.SLS := root κSLS(x) dx - θD2.SLS, x θ0.4.guess = 24.799 m  L1+L2+L3  fmax.1.SLS := -f1.SLS(xθ0.1.SLS) = -0.117mm Maximum field deflection fmax.2.SLS := -f2.SLS(xθ0.2.SLS) = 0.103mm fmax.3.SLS := -f3.SLS(xθ0.3.SLS) = -5.265mm fmax.4.SLS := -f4.SLS(xθ0.4.SLS) = 0.33mm fmax.up.SLS := max(fmax.1.SLS, fmax.2.SLS, fmax.3.SLS, fmax.4.SLS) = 0.33mm fmax.down.SLS := min(fmax.1.SLS, fmax.2.SLS, fmax.3.SLS, fmax.4.SLS) = -5.265mm Summary of design load response - permanent load in SLS Maximum support moment in support C Maximum field moment in field 3 Mmax.sls.z := max( Mmax.s.SLS , Mmax.f.SLS ) = 100.316kNm Maximum reaction force (compression) in support C Maximum reaction force (tension) in support B Rmax.c.sls.z := Rmax.c.SLS = 94.609kN Rmax.t.sls.z := Rmax.t.SLS = -40.71kN Maximum shear force in support C Design shear force = negative shear Vmax.sls.z := max( Vmax.SLS , Vmin.SLS ) = 56.883kN Maximum deflection in span 3 δmax.sls := fmax.down.SLS = -5.265mm Load response - variable load in SLS Distributed loads Maximum deflection for variable load in span 1+3 q1.SLS.q := qd.sls.q q3.SLS.q := qd.sls.q q2.SLS.q := 0 q4.SLS.q := 0 Support moment MA.SLS.q := 0kNm ME.SLS.q := 0kNm MC.guess := 1000kNm Initial guess  3 3 MA.SLS.qL q  1 1.SLS.qL1 MC.SLS.qL2 q2.SLS.qL2  - + + + 6 24 6 24 M B.SLS.q(MC.SLS.q) :=  L1 + L2    3  3 3  MBL2 q  2L2 MDL3 q3L3  - + + + 6 24 6 24 M = C  L  2 + L3   3   3 3 MC.SLS.qL3 q  3.SLS.qL3 ME.SLS.qL4 q4.SLS.qL4  - + + + 6 24 6 24 MD.SLS.q(MC.SLS.q) :=  L  3 + L4   3    3   M   B.SLS.q(MC.guess)L2 q2.SLS.qL2   + ...   6 24    3   MD.SLS.q(MC.guess)L3 q3.SLS.qL3   + +   6 24  M  C.SLS.q := rootMC.guess + , MC.guess = -235.874kNmL + L    2 3     3   MB.SLS.q := MB.SLS.q(MC.SLS.q) = 24.266kNm MD.SLS.q := MD.SLS.q(MC.SLS.q) = -194.54kNm Mmax.s.SLS.q := min(MA.SLS.q, MB.SLS.q , MC.SLS.q , MD.SLS.q, ME.SLS.q) = -235.874kNm Reaction forces -MA.SLS.q MB.SLS.q q1.SLS.qL1 RA.SLS.q := + + = 27.272kNL1 L1 2 MA.SLS.q -MB.SLS.q q1.SLS.qL1 RB1.SLS.q := + + = 15.717kNL1 L1 2 1 1 -MB.SLS.q MC.SLS.q q2.SLS.qL2 RB2.SLS.q := + + = -123.876kNL2 L2 2 RB.SLS.q := RB1.SLS.q + RB2.SLS.q = -108.159kN MB.SLS.q -MC.SLS.q q2.SLS.qL2 RC1.SLS.q := + + = 123.876kNL2 L2 2 -MC.SLS.q MD.SLS.q q3.SLS.qL3 RC2.SLS.q := + + = 88.439kNL3 L3 2 RC.SLS.q := RC1.SLS.q + RC2.SLS.q = 212.316kN MC.SLS.q -MD.SLS.q q3.SLS.qL3 RD1.SLS.q := + + = 83.519kNL3 L3 2 -MD.SLS.q ME.SLS.q q4.SLS.qL4 RD2.SLS.q := + + = 46.319kNL4 L4 2 RD.SLS.q := RD1.SLS.q + RD2.SLS.q = 129.838kN MD.SLS.q -ME.SLS.q q4.SLS.qL4 RE.SLS.q := + + = -46.319kNL4 L4 2 Rtot.SLS.q := q1.SLS.qL1 + q2.SLS.qL2 + q3.SLS.qL3 + q4.SLS.qL4 = 214.948kNTotal reaction force - 14 Rcheck.SLS.q := RA.SLS.q + RB.SLS.q + RC.SLS.q + RD.SLS.q + RE.SLS.q - Rtot.SLS.q = -2.91  10 kN Rmax.c.SLS.q := max(RA.SLS.q, RB.SLS.q , RC.SLS.q , RD.SLS.q, RE.SLS.q) = 212.316kN Rmax.t.SLS.q := min(RA.SLS.q, RB.SLS.q , RC.SLS.q , RD.SLS.q, RE.SLS.q) = -108.159kN Shear force distribution V1.SLS.q(x) := RA.SLS.q - q1.SLS.qx V2.SLS.q(x) := RA.SLS.q + RB.SLS.q - q1.SLS.qL1 - q2.SLS.q(x - L1) V3.SLS.q(x) := RA.SLS.q + RB.SLS.q + RC.SLS.q - q1.SLS.qL1 - q2.SLS.qL2 - q3.SLS.qx - (L1 + L2) V4.SLS.q(x) := RA.SLS.q + RB.SLS.q + RC.SLS.q + RD.SLS.q ... + -q1.SLS.qL1 - q2.SLS.qL2 - q3.SLS.qL3 - q4.SLS.qx - (L1 + L2 + L3) VSLS.q(x) := V1.SLS.q(x) if x  L1 V2.SLS.q(x) if L1 < x  L1 + L2 V3.SLS.q(x) if L1 + L2 < x  L1 + L2 + L3 V4.SLS.q(x) if L1 + L2 + L3 < x  L1 + L2 + L3 + L4  0m  0   0  VA.x.SLS.q :=   V 0m A.y.SLS.q :=   =   kN  RA.SLS.q  27.272   L 1   4.2 R  B2.SLS.q   -123.876 VB.x.SLS.q := =   m VB.y.SLS.q := =  kN  L1   4.2  -RB1.SLS.q  -15.717   L1 + L 2   6.3  R  C2.SLS.q   88.439 VC.x.SLS.q := =   m V := = kN L1 + L2   6.3 C.y.SLS.q    -RC1.SLS.q  -123.876   L1 + L2 + L 3   23.1   RD2.SLS.q   46.319  VD.x.SLS.q := =   m V :=   =  kN  L + L + L   23.1  D.y.SLS.q 1 2 3   -RD1.SLS.q   -83.519   L1 + L2 + L3 + L 4   27.3   -RE.SLS.q  46.319 VE.x.SLS.q := =   m V :=   =   kN  L1 + L2 + L3 + L4   27.3  E.y.SLS.q   0   0  Vmax.SLS.q := max(VA.y.SLS.q , VB.y.SLS.q , VC.y.SLS.q , VD.y.SLS.q , VE.y.SLS.q) = 88.439kN Vmin.SLS.q := min(VA.y.SLS.q , VB.y.SLS.q , VC.y.SLS.q , VD.y.SLS.q , VE.y.SLS.q) = -123.876kN Moment distribution x M1.SLS.q(x) := RA.SLS.qx - q1.SLS.qx 2  L1   x - L1  M2.SLS.q(x) := RA.SLS.qx + RB.SLS.q(x - L1) - q1.SLS.qL1x -  - q 2  2.SLS.q(x - L1)  2   L  M3.SLS.q(x) := RA.SLS.qx + RB.SLS.q( ) 1 x - L1 + RC.SLS.qx - (L1 + L2) - q1.SLS.qL1x -  ... 2    L  x - (L + L ) + -q2.SLS.qL2x -  2 L1 +  - q3.SLS.qx - (L1 + L2) 1 2      2    2  M4.SLS.q(x) := RA.SLS.qx + RB.SLS.q(x - L1) + RC.SLS.qx - (L1 + L2) + RD.SLS.qx - (L1 + L2 + L3) ...  L1    L2    L3  + -q1.SLS.qL1x -  - q2.SLS.qL2x - L1 +  - q 2    2  3.SLS.q L3x - L  1 + L2 +  ...2  x - (L ( )  1 + L2 + L3) + -q4.SLS.qx - L1 + L2 + L3   2  MSLS.q(x) := M1.SLS.q(x) if x  L1 M2.SLS.q(x) if L1 < x  L1 + L2 M3.SLS.q(x) if L1 + L2 < x  L1 + L2 + L3 M4.SLS.q(x) if L1 + L2 + L3 < x  L1 + L2 + L3 + L4 Field moment x00 = 0 x0.1.SLS.q := Maximize(M1.SLS.q , x00) = 2.664 m Point where shear force is zero x0.2.SLS.q := Maximize(M2.SLS.q , x00) = x0.3.SLS.q := Maximize(M3.SLS.q , x00) = 14.94 m x0.4.SLS.q := Maximize(M4.SLS.q , x00) = x0.1.SLS.q := x0.1.SLS.q if 0m  x0.1.SLS.q  L1 Check if zero shear force point is within the span, otherwise it is replaced with a dummy number. 0m otherwise x0.2.SLS.q := x0.2.SLS.q if L1  x0.2.SLS.q  L1 + L2 L1 otherwise x0.3.SLS.q := x0.3.SLS.q if L1 + L2  x0.3.SLS.q  L1 + L2 + L3 L1 + L2 otherwise x0.4.SLS.q := x0.4.SLS.q if L1 + L2 + L3  x0.4.SLS.q  L1 + L2 + L3 + L4 Ltot otherwise Mf1.SLS.q := M1.SLS.q(x0.1.SLS.q) = 36.333kNm Mf2.SLS.q := M2.SLS.q(x0.2.SLS.q) = kNm Mf3.SLS.q := M3.SLS.q(x0.3.SLS.q) = 146.201kNm Mf4.SLS.q := M4.SLS.q(x0.4.SLS.q) = kNm Mmax.f.SLS.q := max(Mf1.SLS.q, Mf2.SLS.q, Mf3.SLS.q, Mf4.SLS.q) = kNm Maximum field moment Mmax.f.SLS.q := Mf3.SLS.q = 146.201kNm Overwritten Zero moment positions  L1  xM0.1.SLS.q := rootMSLS.q(x) , x, 0 ,  = 0 m 2   L1 L2  xM0.2.SLS.q := rootMSLS.q(x) , x, , L1 +  = 4.396 m 2 2   L2 L3  xM0.3.SLS.q := rootMSLS.q(x) , x, L 1 + , L1 + L2 +  = 9.596 m2 2   L3 L4  xM0.4.SLS.q := rootM SLS.q (x) , x, L1 + L2 + , L + L + L +  = 20.285 m2 1 2 3 2   L4  xM0.5.SLS.q := rootMSLS.q(x) , x, L1 + L2 + L3 + , Ltot = 2  Deflection 4 EI = 241.445MPam Flexural Rigidity MSLS.q(x) κSLS.q(x) := Curvature EI L  1  κSLS.q(x)(L1 - x) dx  0 - 4 θA.SLS.q := = 2.012  10L1 L  1  κSLS.q(x)x dx  0 - 4 θB1.SLS.q := = 2.716  10L1 L  1 +L2  κSLS.q(x)(L1 + L2 - x) dx L1 - 4 θB2.SLS.q := = -2.716  10L2 L1+L 2  κSLS.q(x)(x - L 1) dxL1 - 4 θC1.SLS.q := = -6.487  10L2 L +L +L  1 2 3  κSLS.q(x)(L1 + L 2 + L3 - x) dxL1+L2 - 4 θC2.SLS.q := = 6.487  10L3 L +L  1 2 +L3  κSLS.q(x) (x - L1 - L2) dxL1+L2 - 3 θD1.SLS.q := = 1.128  10L3 L1+L +L +L 2 3 4  κ  SLS.q (x)(L1 + L2 + L3 + L4 - x) dx L1+L2+L3 - 3 θD2.SLS.q := = -1.128  10L4 L1+L2+L3+L 4  κ  SLS.q (x)(x - L1 - L2 - L3) dx L1+L2+L3 - 4 θE.SLS.q := = -5.64  10L4 x x f1.SLS.q(x) := θA.SLS.qx -   κSLS.q(x) dx dx   0 0 x x f2.SLS.q(x) := θ   B2.SLS.q(x - L1) - κ  SLS.q(x) dx dxL 1 L1 x x f3.SLS.q(x) := θC2.SLS.q(x - L1 - L2) -   κ  SLS.q(x) dx dxL 1+L2 L1+L2 x x f4.SLS.q(x) := θD2.SLS.q(x - L1 - L2 - L3) -   κSLS.q(x) dx dxL +L +L 1 2 3 L1+L2+L3 fSLS.q(x) := f1.SLS.q(x) if x  L1 f2.SLS.q(x) if L1 < x  L1 + L2 f3.SLS.q(x) if L1 + L2 < x  L1 + L2 + L3 f4.SLS.q(x) if L1 + L2 + L3 < x  L1 + L2 + L3 + L4 Maximum deflection xθ0.1.guess = 2.1 m Guess of where curvature is zero xθ0.2.guess = 5.25 m xθ0.3.guess = 14.7 m xθ0.4.guess = 25.2 m  x θ0.1.guess   xθ0.1.SLS.q := root κSLS.q(x) dx - θA.SLS.q, xθ0.1.guess = 2.221 m Position where curvature 0  is zero  x θ0.2.guess  x θ0.2.SLS.q := root κSLS.q(x) dx - θ B2.SLS.q, xθ0.2.guess = 5.443m  L1   x  θ0.3.guess  xθ0.3.SLS.q := root  κSLS.q(x) dx - θ C2.SLS.q , xθ0.3.guess = 14.898 m  L1+L2   x θ0.4.guess  x θ0.4.SLS.q := root κSLS.q(x) dx - θD2.SLS.q , xθ0.4.guess = 24.875 m  L1+L2+L3  fmax.1.SLS.q := -f1.SLS.q(xθ0.1.SLS.q) = -0.284mm Maximum field deflection fmax.2.SLS.q := -f2.SLS.q(xθ0.2.SLS.q) = 0.251mm fmax.3.SLS.q := -f3.SLS.q(xθ0.3.SLS.q) = -12.537mm fmax.4.SLS.q := -f4.SLS.q(xθ0.4.SLS.q) = 0.912mm fmax.up.SLS.q := max(fmax.1.SLS.q, fmax.2.SLS.q, fmax.3.SLS.q, fmax.4.SLS.q) = 0.912mm fmax.down.SLS.q := min(fmax.1.SLS.q, fmax.2.SLS.q, fmax.3.SLS.q, fmax.4.SLS.q) = -12.537mm Summary of design load response - variable load in SLS Maximum support moment in support C Maximum field moment in field 3 Mmax.sls.z.q := max( Mmax.s.SLS.q , Mmax.f.SLS.q ) = 235.874kNm Maximum reaction force (compression) in support C Maximum reaction force (tension) in support B Rmax.c.sls.z.q := Rmax.c.SLS.q = 212.316kN Rmax.t.sls.z.q := Rmax.t.SLS.q = -108.159kN Maximum shear force in support C Design shear force = negative shear Vmax.sls.z.q := max( Vmax.SLS.q , Vmin.SLS.q ) = 123.876kN Maximum deflection in span 3 δmax.sls.q := fmax.down.SLS.q = -12.537mm SLS capacity Limited deflection Limitation of deflection for pedestrian load 1 δlim := EN1995-2 tab. 7.1200 Deflection due to shear Shear deflections should be checked for slender web panels. As the bridge deck is assumed to interact like a horizontal truss, the shear deflections can be neglected. L3 Controlw.s := "Check shear deflection" if < 10 = "Ignore shear deflection"hlvl "Ignore shear deflection" otherwise Instantaneous deformation, variable load Deformation due to frequent load uinst.Q := fmax.down.SLS.q = -12.537mm Instantaneous deformation, permanent load Deformation due to frequent load uinst.G := fmax.down.SLS = -5.265mm Final deformation ufin = uinst.Q(1 + ψ2kdef ) + uinst.G(1 + kdef ) ψ2.w = 0 ψ2.fk = 0 ufin := uinst.Q + uinst.G(1 + kdef ) = -28.332mm C ontrol final deformation, span 3 Span 1 Controlu.fin := "OK" if ufin < δlimL3 = "OK" "Not OK" otherwise Detailed design Indata bbeams := B - 200mm2 = 2.1 m Distance between outer beams Global load response The load reactions are extracted for critical load responses, and corresponds to different load cases (see Chapter 7 in report). Vertical load (permanent+variable, uls) T ransverse load (wind) Shear force, support C Vz := 281kN Vx := 48.2kN Reaction force, support C Rz := 459kN Rx := 78.4kN Support moment, support C Ms.z := 477kNm Ms.x := 80.8kNm Field moment, span 3 Mf.z := 294kNm Mf.x := 49.7kNm Prestressing force The prestressing between the lamellas must be large enough to hold the bending stresses due to transverse bending, and hinder "glidning" due to transverse shear. Calculate in ULS. The force required to create the sinusoidal shape is calculated as a point load pushing down a continuous beam to the desired deflection. This can also be transferred into a simply supported beam with counteracting bending moments at the supports. D imensions, sectional constants p1 := 70mm Wanted deflection Lspan := ssin2 = 4.2 m Distance between blvl = 54mm Lamella thickness hlvl = 600mm Lamella height 2 Alvl = 32400mm Lamella area 4 Iz.lvl = 7873200mm Second moment of area, around weak axis Em.0.flat := 130MPa Modulus of elasticity for flatwise bending 2 EIP := Em.0.flatIz.lvl = 1.024kNm Flexural rigidity Calculate required point load for defleciton p 2 2 3 M1L M2L P1L p1 = + + Deflection in mid span due to point load in middle of16EI 16EI 48EI span and counteracting bending moments  2 2 48EI  M1L M2L   P1 = p1 - -  Point load in middle of span for a specific deflection3  16EI 16EIL  P ( ) 1 Lspan M1 P1 := -  0.5 Left support moment due to pointload in middle of span2 2 P ( ) 1 Lspan M2 P1 := -  0.5 Right support moment due to pointload in middle of span2 2 P1.guess := 20N Guess of required point load to achieve wanted deflection  2 2  48EI    P  M1(P1.guess)Lspan M2(P1.guess)Lspan  P1 := root P   1.guess - p1 - - , P3  16EI 16EI  1.guess  L P P span   P1 = 185.672 N Required point load to reach wanted deflection Compressive stress due to pre-stressing The distancers are steel tubes in compression, threaded with a steel rod in tension. A steel plate distributes the compression forces from the steel tube to the LVL panel. The long-term pre-stressing forces of a stress-laminated bridge deck is verified according to SS-EN 1995-2 Chp. 6.1.2 Dimensions Dywidag bar dbar := 26.5mm Diameter pre-stressing bar fu.bar := 1050MPa Hight-strength steel bar, Dywidag normally used in Sweden Pmax := 646kN Max initial stressing force nlayer.bar := 2 Number of bars over the height of the panels 2 πdbar - 4 2 Abar := = 5.515  10 m4 dbar2 checkhole := "Disregard capacity check of holes" if < 20% = "Disregard capacity check of holes"hlvl "Check capacity of holes" otherwise Steel plate (between panel and compression tube) Aplate.cs := hplatetplate Cross-sectional area 2 Aplate.flat := bplatehplate = 0.023 m Face area Effect of prestressing Required presressing force  1.5V t 6Mt  N = max ,   hμ 2 h  V.t = shear force between lamellas M.t = bending due to uniformly distributed load on the deck (across) h = height of lamella panel L oad effect load on the deck, transverse uniformly distributed load Consider vertical load: self-weigt + traffic load (pedestrians) Total permanent load across the deck Gk.bridge + Gk.deck kN Gk.across := L = 47.907B m Total variable load across the deck kN Qfk.z.across := QfkL = 111.773 m Consider the deck as simply supported on the two outer, straight beams. Calculate the load effect due to uniformly distributed load across the deck Calculate in SLS kN qt := γGGk.across + γQQfk.z.across = 225.148 m 2 qtbbeams Mt := = 124.113kNm8 qtbbeams Vt := = 236.405kN2 Shear stress between lamellas Consider the shear force in one bar. Vt Vt.bar := = 118.203kN2 Friction coefficient μd := 0.40 Coefficient of friction, EC5-2 Table 6.1: planed timber to planed timber (MC >16%), perpendicular to grain. Effective height of the cross-section: The height of the shear connection correspond to the height of the steel plate, including the load path effect (45 deg) blvl hef.v := hplate + 2 = 258mm Obs overlap with the other plate!tan(45deg) blvl bef.v := bplate + 2 = 0.258 mtan(45deg) 2 Aef.v := hef.vbef.v = 0.067 m 2hef.v < hlvl = 1 OBS the effective height of two plates are larger than the height of the LVL panels 1.5Vt.bar τv := = 6.659MPaAef.vμd Bending stress in the upper part of the lamellas Consider the bending stresses at the edge of the panels (conservative, more exact would be to use the bending stress in the point of the tension rod). hef.b := hlvl = 0.6 m bef.b := bef.v = 258mm 2 bef.bhef.b Wef.b := 6 Mt σb := = 8.018MPaWef.b Required compression force on the lamellas Minimum required compression stress σc.t := max(τv, σb) = 8.018MPa Bending stresses are dimensioning for the required compression force Minimum required compression force on the panels 3 Nc := σc.tbef.bhef.b = 1.241  10 kN Corresponding required tension force from two steel bars Nc.bar := σc.t2Abar = 8.844kN Minimum required compression force in one steel tube Nc.bar Nc.tube := = 4.422kN2 V erify the compression capacity in LVL Corresponding compression stress from the steel plate to the panel hef.vbef.v σc.panel := σc.t = 3.448MPahef.bbef.b σc.panel ησ.panel := = 23.641%fc.0.d.lvl Verify the tension capacity of the steel bar Nc.bar ησ.bar := = 1.369%Pmax Maximum allowed pretension of the tension bar is 70% of the ultimate tension capacity. Nc.bar ηN.bar := = 0.764%fu.bar2Abar Verify that the compression force is larger than the required point load on the panels checkN.c.bar := "OK" if P1 < Nc.bar = "OK" "NOT OK" otherwise Check buckling of compression tubes Sectional constants π 4 4 - 6 4 I  x.tube := D1 - d64 1  = 1.006  10 m π  4 4 - 6 4Iz.tube := D1 - d  1  = 1.006  10 m64 l0.tube := 0.5Ltube = 45mm Effective length Check slenderness l0.tube λtube := = 1.856 Slenderness Ix.tube Atube 2 π EsteelIx.tube 6 Ncr := = 1.03  10 kN2 l0.tube Atubefy λbar := = 0.021Ncr checkslenderness := "NO Buckling occurs" if λbar  0.2 = "NO Buckling occurs" "Buckling needs to be checked" otherwise V erfiy Minimum required compression force on the panels = compression force on the steel tubes Nc.tube σtube := = 2.584MPaAtube checkσ.tube := "OK" if σtube  fy = "OK" "NOT OK" otherwise σtube ηtube := = 0.9%fy