global analysis of a tubed structural system for an ...1113844/fulltext01.pdf · system for an...

Global analysis of a tubed structural system for an inclined slender tall building

Paulina Chojnicka and Lydia-Foteini Marantou

June 2017

TRITA-BKN. Master Thesis 519, 2017

ISSN 1103-4297

ISRN KTH/BKN/EX--519--SE

KTH School of ABE

SE-100 44 Stockholm

SWEDEN

Royal Institute of Technology (KTH)

Department of Civil and Architectural Engineering

Division of Concrete Structures

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Abstract Building engineering is called upon to keep up with the pace and challenges of modern design, which aims not only to build higher and greener, but also to fulfill the demands of the growing population and simple human curiosity.

The main purpose of this study was to examine the global behavior of a slender and inclined (V-shaped) 300 m high rise building with different structural systems applied. In order to properly evaluate them, four different parametric studies were conducted. These included determining the appropriate inclination angle and the geometry of a simple beam system and later comparing fourteen different structural systems, namely trusses, diagrids, Tubed Mega Frames and moment frames. Parallel to this, a further investigation was made on a shell and beam element model, in order to assess the simplifications made and to control the obtained results.

This study was based on various simulations in Finite Element Analysis programs, primarily ETABS, but also SAP2000 and Autodesk Robot Structural Analysis. The modelling included the definition of geometry and applied loads and results in extracting the desirable forces and deformations. Additionally, the automatic design for structural members was used for the purpose of a comprehensive study of the chosen structural systems. The designed structures were subjected to static analysis (dead, live, wind, seismic load), dynamic analysis (response spectrum and time history function) and nonlinear P-delta effect. A buckling analysis was also performed to determine the modes and associated load factors for buckling. In the end, the structural response in terms of displacement and acceleration was compared.

The inclination angle study set the defining angle at 10° from vertical, with respect to the serviceability limit deflection. Comparing alternative truss geometries in a 2D parametric study resulted in the choice of four different systems (X, N, K and W trusses). In the 3D analysis, the chosen truss systems, together with three variations of diagrid systems, and seven Tubed Mega Frames with two moment frame structures were further analyzed. In both groups, the mass and the material of the systems were kept similar and the comparison was basically based on the obtained maximum displacement and natural periods of the buildings. The shell and frame model comparison gave a difference in displacements between 0 and 12%. Finally, the comprehensive study of the Tubed Mega Frame, X truss and diagrid structures showed that these buildings were performing similarly to vertical buildings with a top story displacement within the suggested limits (less than 673 mm). Further investigation should be made concerning the acceleration under synthetic earthquake, which exceeded the suggested norms, as well as the connecting nodes between the trusses and the inclined columns.

The outcome of this study implied the possibility of construction and usability of inclined, slender, tall buildings with respect to the Ultimate Limit State and the Service Limit State, as specified in the American standard, ASCE 7-10, and opened new possible issues for further research.

Keywords: inclined tall building, Tubed Mega Frame, truss system, diagrid, concrete, composite, P-delta

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Sammanfattning Byggingenjörskonst ställs idag inför många utmaningar inom modern design som syftar till att inte endast bygga högre och miljövänligare, men även uppfylla kraven från den ökande populationen och den mänskliga nyfikenheten.

Huvudsyftet med denna studie var att undersöka det globala beteendet av ett 300 m slank och lutande (V-formad) höghus genom att tillämpa olika strukturella system. För att utvärdera dem så har fyra olika parametriska studier utförts. I dessa bestämdes vinkeln av byggnaden och geometrin genom ett enkelt balk-system. Därefter jämfördes 14 olika strukturella system, huvudsakligen fackverk, diagrids, Tubed Mega Frames och momentramar. Samtidigt genomfördes en analys på en skal- och balkelementmodell i syfte att bedöma de förenklade antaganden och resultaten.

Denna studie har baserats på olika simuleringar i Finita Element Analys program, främst i ETABS men även SAP2000 samt Autodesk Robot Structural Analysis. I modelleringen definierades geometrin och laster för att erhålla önskvärda resultat. Dessutom har den automatiserade designfunktionen för de strukturella delar använts för att ge en uppfattning om vilka dimensioner byggnaden kan tänkas ha. Undersökningen har utförts genom en statisk analys (egenvikt, nyttig-, vind- och seismiska laster), en dynamisk analys (response spectrum och time history function) och även med hänsyn till olinjära P-delta analyser. En knäckningsanalys utfördes för att bestämma moderna och tillhörande lastfaktorer för knäckning. Slutligen jämfördes det strukturens förskjutning och acceleration.

Den bestämda vinkeln sattes som 10° med hänsyn till bruksgränstillstånd. Genom att jämföra olika fackverkssystem i en 2D parametrisk studie resulterade valet i fyra olika system av X, N, K och W fackverk. I 3D analysen undersöktes de valda fackverkssystemen tillsammans med tre olika diagrid system, och sju olika Tubed Mega Frame system med två olika momentramsstrukturer. I samtliga analyser valdes systemets massa och material så lika som möjligt för att få en representativ jämförelse som baserades på maximala förskjutningar och de naturliga perioderna. Jämförelsen mellan skal- och balkmodellen gav en skillnad i förskjutning mellan 0-12%. Slutligen visade studien med Tubed Mega Frame, X-fackverk och diagrids liknande tendenser som vertikala byggnader med maximal förskjutning inom de föreslagna gränserna på 673 mm. Ytterligare forskning bör göras med hänsyn till accelerationer under syntetiska jordbävningar som överstiger de föreslagna normerna, och de anslutande noderna mellan avstyvningarna och de lutande pelarna.

Resultaten från denna studie tyder på möjligheterna av tillämpning och byggandet av lutande, slanka höghus med hänsyn till brottgränstillstånd och bruksgränstillstånd enligt de amerikanska standarderna, ASCE 7-10, och öppnar upp nya möjligheter för ytterligare forskning.

Nyckelord: lutande hög byggnad, Tubed Mega Frame, fackverk, diagrid, betong, komposit, P-delta

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Preface This thesis has been conducted during the spring of 2017, as a final part of our two-year studies at the department of Civil and Architectural Engineering at KTH. The research was performed at the headquarters of Tyréns company in Stockholm and it treats the analysis and design of an inclined high rise building.

We would like to thank Fritz King, our supervisor in Tyréns, for sharing his knowledge, guidelines and support along the whole process of this work. He gave us the freedom to decide upon research questions and at the same time supervised us, ready to help with any difficulties we came across.

We also appreciate the help of our KTH supervisor, Mikael Hallgren, who provided us with comments and advice throughout the project.

We are also grateful to Tyréns, which provided us with the necessary software and great working conditions to work on our thesis.

Stockholm, June 2017

Chojnicka Paulina & Marantou Lydia-Foteini

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Notations Latin letters

A Cross section area ASD Allowable strength design C Damping of a structure D Outside diameter of pipes {D} Displacement vector E Modulus of elasticity fcr Flexural buckling stress fe Elastic critical buckling stress fy Yield strength of material G Shear modulus Gf Gust factor gQ Peak factor for background response gR Peak factor for resonance response gv Peak factor for wind response I Moment of inertia Ic Moment of inertia of column section Ig Moment of inertia of girder section Iz Intensity of turbulence i Radius of gyration [K] Global stiffness matrix K Stiffness of a structure K Effective length factor K1 Effective length factor for braced condition K2 Effective length factor for unbraced condition K33, K22 Effective length K-factors in the major and minor directions for appropriate

braced (K1) and unbraced (K2) condition Lb Laterally unbraced length of member Lcr Critical buckling length LRFD Load and resistance factor design or limit state design l Length l0 Buckling length M Mass of a structure Mx Moment around x axis My Moment around y axis Pa Maximum axial force pcr Critical buckling stress pel Elastic critical buckling resistance of the member in the plane of bending,

calculated based on the assumption of zero side-sway Pn Nominal compressive force based on the controlling buckling mode Q Background response factor R Resonant response factor {R} Global load vector RLLF Reduced live load factor for an element r22 Radius of gyration about 2-2 axis

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r33 Radius of gyration about 3-3 axis r(t) Load function Sa Design earthquake spectral response acceleration SD1 Design earthquake spectral response acceleration parameter at 1 s period SDS Design earthquake spectral response acceleration parameter at short period SLS Serviceability Limit State SM1 MCER spectral response acceleration parameter at 1 s period adjusted for site

class effects SMS MCER spectral response acceleration parameter at short period adjusted for

site class effects T Fundamental period of the structure T0 Period starting the horizontal part of the response spectrum TL Long-period transition period Tr Torsional moment of the section TS Period ending the horizontal part of the response spectrum t Thickness tc Nominal torsional strength ULS Ultimate Limit State u(t) Displacement function u’(t) Velocity function u’’(t) Acceleration function

Greek letters

λ Slenderness parameter λ1 Slenderness value to determine the relative slenderness φ Resistance factor for compression (LRFD) Ω Safety factor for compression (ASD)

Symbols

⎕ Rectangular cross section ⌶ I-beam shaped cross section

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Contents

1 Introduction ............................................................................................................................................. 1

1.1 Background ...................................................................................................................................... 1

1.2 Aim and scope ................................................................................................................................. 1

1.3 Assumptions and limitations ..................................................................................................... 2

2 Method ....................................................................................................................................................... 3

2.1 Literature study .............................................................................................................................. 3

2.2 Parameter study ............................................................................................................................. 3

2.3 Comprehensive study of TMF and braced systems .......................................................... 4

3 State of the art ......................................................................................................................................... 5

3.1 High rise buildings ......................................................................................................................... 5

3.1.1 General information ............................................................................................................. 5

3.1.2 Slenderness .............................................................................................................................. 5

3.1.3 Inclined structures ................................................................................................................ 5

3.2 MULTI rope-free elevator ........................................................................................................... 5

3.3 Structural systems and solutions for high rise buildings ............................................... 6

3.3.1 Tubed Mega Frames ............................................................................................................. 6

3.3.2 Diagrids ..................................................................................................................................... 7

3.3.3 Trusses ...................................................................................................................................... 8

3.3.4 Moment frames ...................................................................................................................... 9

3.3.5 Composite columns .............................................................................................................. 9

3.3.6 High strength concrete ..................................................................................................... 10

3.4 Codes and specifications .......................................................................................................... 11

3.4.1 Wind load .............................................................................................................................. 11

3.4.2 Seismic action ...................................................................................................................... 11

3.4.3 P-delta effect ........................................................................................................................ 13

3.4.4 Buckling ................................................................................................................................. 14

3.4.5 Shear lag ................................................................................................................................. 18

3.5 Finite Element Analysis ............................................................................................................ 20

3.5.1 Procedure .............................................................................................................................. 20

3.5.2 Element types ...................................................................................................................... 21

4 Parameter study .................................................................................................................................. 23

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4.1 Inclination angle .......................................................................................................................... 23

4.1.1 Model ....................................................................................................................................... 23

4.1.2 Loads ....................................................................................................................................... 24

4.1.3 Results .................................................................................................................................... 24

4.1.4 Discussion ............................................................................................................................. 25

4.2 2D truss models ........................................................................................................................... 25

4.2.1 Model alternatives ............................................................................................................. 26

4.2.2 Loads ....................................................................................................................................... 27

4.2.3 Results .................................................................................................................................... 27

4.2.4 Discussion ............................................................................................................................. 28

4.3 3D study of structural systems .............................................................................................. 28

4.3.1 Model ....................................................................................................................................... 28

4.3.3 Loads ....................................................................................................................................... 38

4.3.4 Results .................................................................................................................................... 39

4.3.5 Discussion ............................................................................................................................. 40

4.4 Comparison of shell and frame elements within Tubed Mega Frame structures in ETABS ........................................................................................................................................................... 41

4.4.1 Models: ................................................................................................................................... 41

4.4.2 Loads: ...................................................................................................................................... 43

4.4.3 Results: ................................................................................................................................... 44

4.4.4 Discussion ............................................................................................................................. 46

5 Design of TMF and braced systems .............................................................................................. 47

5.1 Models ............................................................................................................................................. 47

5.1.1 Geometry ............................................................................................................................... 48

5.1.2 Materials ................................................................................................................................ 48

5.1.3 Cross sections ...................................................................................................................... 48

5.1.4 Boundary conditions ......................................................................................................... 50

5.1.5 Element types ...................................................................................................................... 50

5.1.6 Assumptions ......................................................................................................................... 50

5.2 Loads ................................................................................................................................................ 50

5.2.1 Dead loads ............................................................................................................................. 51

5.2.2 Live loads ............................................................................................................................... 51

5.2.3 Wind ........................................................................................................................................ 51

5.2.4 Seismic action ...................................................................................................................... 52

5.2.5 Combinations ....................................................................................................................... 53

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5.2.6 Modal case ............................................................................................................................. 54

5.2.7 Construction sequence ..................................................................................................... 54

5.2.8 Buckling ................................................................................................................................. 55

5.3 ETABS design and results ........................................................................................................ 55

5.3.1 Design ..................................................................................................................................... 55

5.3.2 Response spectrum ........................................................................................................... 61

5.3.3 Time history analysis ........................................................................................................ 64

5.3.4 Comparison of the response spectrum and time history analysis .................. 68

5.3.5 Construction sequence ..................................................................................................... 71

5.3.6 Buckling ................................................................................................................................. 72

5.4 Model verification ....................................................................................................................... 73

5.5 Discussion ...................................................................................................................................... 76

6 Conclusion and further research .................................................................................................. 79

6.1 Conclusion...................................................................................................................................... 79

6.2 Further research.......................................................................................................................... 79

Bibliography ................................................................................................................................................... 81

Appendices ...................................................................................................................................................... 85

Appendix A Results from inclination angle study ................................................................. 85

Appendix B Geometry of trusses analyzed in the 2D study ............................................... 89

Appendix C Results from the 2D study ...................................................................................... 93

Appendix D Base reactions and column forces from the 3D study ................................. 97

Appendix E Analytical calculations for the models............................................................... 99

Appendix F Results from Design study ...................................................................................109

1 Introduction

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1 Introduction

1.1 Background

Throughout the centuries, mankind has always been fascinated by building higher. This resulted both from the need to provide more space to the increasing urban population, as well as from the competitive urge to create a new symbol of growth and an iconic representation of the place of origin.

Tall buildings represent nowadays a new form of city planning, the so called vertical cities, which aim to accommodate residential, office or hotel space or a combination of them. With the highest existing building reaching 828 m (Burj Khalifa) and the Jeddah Tower setting the barrier to 1008 m upon its completion in 2019, the expected question is how tall can we build, followed by what kind of structural system would be the most efficient one.

Contrary to the central core stabilized building, which has been the most widely used structural system for high rise buildings so far, the Swedish company Tyréns has proposed the Tubed Mega Frame structure concept. This structural innovation uses mega hollow columns and perimeter walls as the main load bearing system, allowing for a maximum floor space utilization ratio (King et al. 2016).

Nowadays, the most popular method to construct tall buildings is to use a core with elevator shafts in the center or on the sides. It is also possible to use parallel bearing walls, self-supporting boxes, suspension and frames. Those elements are commonly in use, which makes them well designed, reliable and easy to erect and maintain. However, they are not equally usable with the new generation of elevators. The new innovative system of elevators is about to settle the look of the new tall buildings in the most unexpected shapes.

This thesis studied the Tubed Mega Frame system as well as the 3D truss system for the case of an inclined high rise building, in order to evaluate which of these systems performs better and in what ways it should be further developed.

1.2 Aim and scope

The aim of this master thesis was to investigate the effects of different inclination angles on the global behavior of a structure, as well as to determine the suitable structural system with regard to the greatest stiffness and structural performance, but also serviceability efficiency.

The study was performed on a 300 m tall V-shaped model (see Figure 1) with triangular floor plans along the towers and square floor plans at the base of the building. The inclination angle of the towers and the structural system for further investigation (see chapter 5) was to be determined in the preliminary study (see chapter 4).

1 Introduction

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Figure 1. Shape of the building

1.3 Assumptions and limitations

This thesis studied the global performance of the V-shaped building, in order to assess the structural systems used. The adequate design of the foundation of the building was excluded from the analysis. The different structural systems were subjected to linear and nonlinear static and dynamic loadings. In the case of wind loading, the vortex shedding effects were neglected.

In the preliminary study of the inclination angle, a simplified analysis was conducted. This included a simple beam model (equivalent columns) subjected only to self-weight load. A linear analysis was performed and preliminary results were extracted (reactions, stresses, displacements). Its goal was to understand how the inclination angle affects the behavior of a simple structure under vertical loading.

In the parametric study, different 2D truss systems were examined. The load case included the dead load along with a simplified wind load (uniformly distributed along one inclined tower). The analysis was linear and only some of the results were presented. The design of the cross section of the members and the connections in the truss system are excluded.

The 3D analysis of the truss system and the Tubed Mega Frame system was carried out in ETABS software and included the superstructure model with the applied dead, live and wind loads. The same mass of the models of the same material was kept in order to qualify for a valid comparison of the different structural systems. The results were presented and compared.

In the comprehensive analysis, the best performing structural systems were examined under a set of loads (dead load, super dead load, live load, wind load, seismic action). The analysis included static load calculations, P-delta effect and dynamic study.

This analysis did not account for the constructability of the building, in terms of considering the construction methods, however a construction sequence analysis was performed. The analysis was conducted in the finite element analysis software with a guidance of Tyréns and KTH.

2 Method

3

2 Method

2.1 Literature study

This report starts with a literature study, based on the studies made so far within the concepts of tall buildings and used structural systems, together with the new Tubed Mega Frame system. This way a sufficient background was provided as a reference, contributing to the reliability and the precision of the current study.

Therefore, a better understanding of current solutions was gained, along with their advantages and disadvantages as to their application and possible problems as well as more suitable solutions which were preliminary judged as appropriate for future consideration.

A complete literature study would include a summary of relevant journals and information on the studied issue together with an evaluation of the results in context of technical knowledge. Due to the shortage of time, this study was not performed to that extent. Instead, all sources are listed in bibliography with detailed information about the author, title, year of publication etc.

2.2 Parameter study

The parameter study was carried out by performing multiple simulations of models in finite element analysis software while changing one or more parameters and comparing the results, followed by conclusions. These parameters concerned the geometry of the structure, the chosen cross section and the materials assigned.

Our study was conducted in a software which enables a correct modelling analysis in terms of load definition and determination of the internal forces, stresses and strains and structural behavior.

Creating a model for finite element analysis includes:

defining a geometry defining the materials and their properties defining preliminary cross sections defining restrains and boundary conditions inclusion of loads

The desirable output in our case contained:

Internal forces Stresses Strains Deformations

This output helped to model the structure, evaluate its suitability and identify the critical points.

2 Method

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2.3 Comprehensive study of TMF and braced systems

This study treats the 3D simulation and design of 3 models chosen as the best fit in the preliminary analysis under increased number of loads. A finite element analysis software was used (the background theory is described in chapter 3.4.5) to design structural elements and receive a reliable response of the system. Ultimate Limit State and Service Limit State were checked for chosen elements of the system using static and dynamic load combinations and including nonlinear effect (P-Delta, creep, shrinkage) in specific cases. Construction sequence was also conducted to account for the real deflections due to dead load, as this acts sequentially during the construction. Simple check of the model was done with the aid of analytical methods.

Specifications for this study including geometrical details, cross section properties and applied loads are described in chapter 5.

3 State of the art

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3 State of the art

3.1 High rise buildings

3.1.1 General information

The council of tall buildings identifies a building as “tall” with respect to its comparatively bigger height than that of its neighbor buildings, but also to its proportions and the transport and technical solutions that are being used. Another criterion to classify a “tall building” is to attribute at least 50% of its height to usable floor area (CTBUH 2013).

3.1.2 Slenderness

The most impressive buildings are not only the tall, but also those that keep the right proportion between width and height. This proportion is called slenderness ratio and the overall feeling of the skyscraper depends on it in a significant level. A building is considered to be slender if its height to floor span ratio is at least 1:10, which has a meaningful influence on the behavior of a structure, primarily increasing the deflection (Awida 2011).

The slenderest existing building is Highcliff in Hong Kong with the slenderness ratio of 1:20 and the height of 252 m. On second place we find the pencil-thin 432 Park Avenue in New York with the ratio of 1:15 and height of 425.5 m, which is far greater than any other slim building, followed by Sky house and One Madison, also in New York, with a slenderness of 1:12. It is worth mentioning that the 111 West 57th Street, also situated in the “city of skyscrapers” will win the world-record breaking slenderness ratio of 1:24 reaching the height of 438 m (Marcus et al. 2015).

3.1.3 Inclined structures

Traditionally, high rise buildings are built vertical. The design and construction of inclined buildings is still a very recent architectural phenomenon, on the grounds of which not much research has been conducted.

Usually the leaning of existing buildings is unintentional and can be caused by poor ground conditions (tower of Pisa, Tiger Hill Pagoda in China), badly designed foundation or another structure completed later (Suurhusen Church Tower in Germany, Big Ben) or even an extraordinary, but sometimes unknown event (The Leaning Temple of Huma) (SA Rogers n.d.).

Some of the most representative examples of modern tilted buildings are the Veer towers in Las Vegas, the inclined towers in Madrid and the Capital Gate Tower in Abu Dhabi. All the above are built with a central core structure.

3.2 MULTI rope-free elevator

Along with the increasing height of tall buildings rises the need to efficiently move the building’s inhabitants up higher. This leads to using more shafts, thus wasting more space

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of the useable floor area. Elevators’ design has not advanced over the last 160 years, after they were invented by Elisha G Otis. A call for an innovative solution is therefore created.

This call is now answered by ThyssenKrupp Company, which proposes a new elevator design, called MULTI (Thyssenkrupp 2015). The MULTI revolutionary technology is based on omitting hoist ropes and movement of cars in shafts in the same way that trains move in rail systems, with several cabins per shaft and permitting vertical and horizontal movements in the buildings. Powerful magnets are used in this new elevator system, instead of the old rope system, and combined with new elements, such as new and lightweight carbon composite materials for cabins and doors, add up to a mere 50 kg weight. Comparing this to the 300 kg weight in standard elevators, ends up to an overall 50% reduction in the total elevator system weight (Jetter & Gerstenmeyer 2015).

The exchange between the cars is now possible at every shaft position, allowing for more than two cars moving horizontally at the same time as well as longer travel distances. Another benefit of the MULTI elevator system is the reduction of peak currents by up to 60%, reducing the elevator footprint and augmenting the energy savings of the building. One of the most important advantages, however, is the decrease in the overall size and the outer surface area of the building where the MULTI is introduced. A percentage of up to 40% of more useable floor space is to be gained, according to ThyssenKrupp. This way, tall buildings can be designed in various forms, accommodating elevators that are no longer confined to one vertical path, as well as the potential to create new transportation networks that will shape the cities of the future.

3.3 Structural systems and solutions for high rise buildings

In the context of high rise buildings, a great number of new structural systems have been developed. These include frames, core, shear walls, belt truss systems and tubed systems which have also led to using higher performance steel and concrete to carry the large forces.

3.3.1 Tubed Mega Frames

The Tubed Mega Frame concept implies that rigid moment tubed frames are placed in the perimeter of the building. It is an efficient structural system that makes use of the whole building’s width to resist the lateral load overturning moment, and maximizes the gravity loads that are taken by the outside columns, increasing the building’s stability. Another advantage of the tubular structure system is that it provides a bigger plan area.

The exterior tubular structure which takes also all the lateral forces, is composed by vertical columns in the building’s periphery which are in part connected by horizontal elements (see Figure 2). Both the vertical and the horizontal elements usually have large dimensions. The efficiency of this system is acquired by taking all the material to the periphery columns which in combination with strong horizontal members will work as a cantilever structure.

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Figure 2. Examples of Tubed Mega Frame structures (King n.d.)

3.3.2 Diagrids

The diagrid, or else diagonalized grid structure, is a triangulated member system that creates resistance to gravity and lateral loads for a building. As the structural elements are mostly located at the exterior of the building, a diagrid can be characterized as a tubed system. Although only recently invented (early 2000), diagrids aim to discard the vertical columns in the perimeter of the building. This is achieved by using an intersection point, a nodal connection, which is the basis of a 3D triangular frame of a single thickness. Within the framework of tubed systems, the diagrid system can contribute to the lateral stability of the building, eliminating the need of a central core system (Boake 2016). This attribute was thoroughly studied in this thesis, in order to efficiently move all the loads of the building’s structure and floors to the outside diagrid system.

The efficiency of the diagrid lies on its triangular shape, which resists both gravity and lateral forces. Each member of the diagrid is either tensed or compressed. Kyoung-Sun Moon (Moon 2005) studied the optimal diagrid angle to be between 53o and 76o, with corner columns and 70o without, for a 60-story building. He also found that the higher the building, the bigger should be the angle of the diagrid bracing members, due to the fact that shear deformation is more eminent in a fewer story building. These results would change in the case of an inclined building (Moon 2005), so they cannot be verified in this study.

The diagrid system seems to be a well-performing system for high rise buildings when the fabrication of the connecting hub is also well considered and it can give new ways to diversify a building’s visual expression. As seen in Figure 3 below, the Swiss Re building uses small triangular windows, while the Hearst Tower uses large diamond-shaped windows.

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Figure 3. Diagrid structures: Swiss Re (2004), Hearst Tower (2006)

3.3.3 Trusses

Truss in its classical meaning is a plane or spatial system of bars connected with hinges transferring nodal forces to pinned supports. This results in having only axial internal forces and achieving efficient performance with minimum use of material (usually steel or wood, but also concrete). However, the design, construction and further maintenance might be inconvenient.

Truss systems can be divided into many groups according to geometry (see Figure 4).

Figure 4. Exemplary types of trusses

There is no easy way to unequivocally decide which type of truss is the most applicable for a given structure. The X type trusses can be analyzed in two ways: either compressed diagonals are counted in and designed, or they can be skipped, which is profitable especially for long elements. The N type trusses (Pratt) is a simpler structure, though not optimal for certain kinds of loads, since compression takes place in diagonals and an increased cross section is required. The K type stands out by decreasing compression and buckling length in vertical elements. It can be stiffer, but on the other hand, it requires more complex design and construction. The last alternative is W type with a simpler design, but a good performance only under a distributed load action (SkyCiv Cloud Engineering software n.d.).

Nevertheless, the construction of pinned connections in the optimal place, namely on the axis of connected members, is extremely difficult, especially in the case of enormous building systems (see the John Hancock Center). Also, nodal forces are rarity. In practice, the trusses are subjected to distributed loads (e.g. wind load, dead load) and these cause additional moments and shear forces in the members.

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Bracing systems tend to reduce the shear lag effect (see chapter 3.4.5) and therefore, increase the building’s stiffness to withstand lateral loads.

3.3.4 Moment frames

Moment frames consist of columns and beams connected in rigid joints, without the need of supplementary bracing system. The buckling length of the comprising elements is reduced due to the moment-resisting joints, which lowers the dimensions of the cross sections. This system is not as efficient as the braced frames, especially under lateral loads’ action. It is prone to sway, which is a less stable condition compared to no sway buckling state (Yura 2015)(see Figure 5).

Figure 5. Sway and no-sway buckling modes (Yura 2015)

Moment frames are used for office and other buildings, where the large, undivided area is under big concern. The floor area can be easily adapted within a regular net of columns. This compensates for the higher costs needed to fulfill the serviceability requirements.

3.3.5 Composite columns

In high rise buildings, the gravity of the building creates a large axial load in the columns, reducing also their ductility. If the section dimensions of the column are widened, then again the axial load will be larger, leading to a limited bearing capacity of the columns. In accordance to the need to optimize the construction materials and reduce the size of elements required within the structural systems of high rise buildings, using composite materials seems to be the best solution.

Composite sections contribute to minimizing the size and number of structural elements, along with the use of higher grade materials. It seems that this is a viable solution with respect to cost, structural behavior and performance within the tall buildings design, especially for buildings of 300 meters height and more (Trabucco et al. 2016).

The composite action of concrete and steel collaboration lies on the steel’s high bearing capacity and good seismic behavior and on the concrete’s contribution on the rigidity, durability and fire resistance of the column (CABR 2006). In order to achieve the shear resistance and the bonding at the concrete-steel interface, shear connectors are used, such as shear studs, perfo-bond ribs and T-connectors (see Figure 6). Since the use of composite columns implies the welding of large sections on site, the steel sections are placed in a way to serve a connection to another structural element.

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Figure 6. Shear connectors (CABR 2006)

In Figure 7 below, some composite sections are depicted.

Figure 7. Encased composite member sections (CABR 2006)

There are no available design standards which can provide information on the proper design of reinforced column sections with more than one embedded steel profile (Bogdan et al. 2012).

The development of a method of calculation for concrete sections with several encased steel sections, requires the calculation of section characteristics, including the moment of inertia, the plastic moment, the elastic neutral axis and the plastic neutral axis of huge mega column sections (ACI Committee 2005).

3.3.6 High strength concrete

Concrete with a characteristic cube strength between 60 and 100 MPa (recently even higher strengths are being used) is defined as a high strength concrete (The Concrete Society n.d.). Increasing the strength and elastic modulus results in reduced size of bearing elements and/or lowered reinforcement steel content, which leads to a minimized construction cost.

In order to achieve a better performance, it is necessary to use different mineral admixtures like pozzolans (silica fume, fly ash) which react with hydration products of

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Portland cement. Also, chemical admixtures are used, namely superplasticizers and water-reducing retarders (PCA America’s Cement Manufacturers n.d.). The water-cement ratio should in any case be less than 0.35 and aggregates’ sizes should be uniformly distributed.

3.4 Codes and specifications

In this thesis, the study was conducted according to the American code ASCE 7-10 for load definition and global response check and building code requirements ACI 318-05 and AISC 360-10 were used for concrete and steel design respectively.

3.4.1 Wind load

Both the structural system and the façade of a tall building should be designed to withstand the wind loads to which they are subjected. The direction of the wind and the change in the wind speed is not a two-dimensional case, as it is dependent on the surface roughness, topography and surrounding landscape. In order to identify the wind-induced structural loads in the case of a high-rise building, a wind tunnel test on a scale model should normally take place. This would account for the wind acceleration in both along-wind and across-wind direction, causing the vortex shedding effect, as well as the aerodynamic interactions caused between the neighboring buildings and the geometry of the analyzed building (Irwin et al. 2013).

In the current study, however, such a test is not performed. Instead, the wind load is considered as a two-dimensional wind flow and is applied in ETABS program according to the ASCE 7-10 specifications. Later the comfort criteria are verified by studying the top story acceleration.

3.4.2 Seismic action

The seismic performance of the building is studied in order to check the building’s response to ground movement.

Until the 1970s, the seismic design of high rise buildings was based on the plastic rotation capacity of steel frame beam ends. In case of a stronger earthquake however, this could lead to residual deformations in frames. In order to overcome this problem, a lot of seismic-isolation techniques have been invented over the last fifty years contributing to the elastic behavior of the building. Parallel to that, the structural performance of high rise buildings has been improved by introducing higher strength and more ductile steel materials in new shapes and forms which can enhance significantly the seismic resistance of the building.

In Figure 8 below, a combination of different structural technologies is used in Otemahi tower to provide high seismic resistance, using oil dampers, buckling-restraint braces and friction dampers to reduce the story drift ratio, a rigid high strength steel frame structure, using CFT (concrete filled tubular) columns and a moment control frame structure with braces at the core of the building (Nakai 2015).

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Figure 8. Otemahi tower structural framing (Nakai 2015)

According to ASCE 7-10, the design earthquake spectral response acceleration parameters SDS, at short period and SD1 at 1 s period are determined from equations 3.1 and 3.2, respectively (ASCE 2005).

𝑆𝐷𝑆 =2

3𝑆𝑀𝑆 3.1

𝑆𝐷1 =2

3𝑆𝑀1 3.2

The design response spectrum curve is taken from equation 3.3 and is depicted in Figure 9.

𝑆𝑎 = 𝑆𝐷𝑆(0.4 + 0.6𝑇

𝑇0) 3.3

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Figure 9. Design response spectrum (ASCE 2005)

3.4.3 P-delta effect

P-delta effect is a second order effect which occurs when the first order deformations cause eccentricity for the external forces. This phenomenon changes the internal forces and moments leading to a difference in the deformations. A simple case which explains this effect is shown in Figure 10 below:

Figure 10. P-Delta effect about column (Computers & Structures Inc. n.d.)

P-delta (P-δ) is defined as the effect along the members (local deformation), whereas P-Delta (P-Δ) is considered only at the end, which simplifies calculations and shortens the computational time in numerical calculations (Computers & Structures Inc. n.d.).

P-delta might have a huge impact on the structure, especially in tall and slender buildings. For inclined structures, however, this effect is not fully investigated. In this study, P-delta was considered for all the different load cases and combinations, due to the fact that even the self-weight of the structure causes deformations which change the geometry and the axis position.

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3.4.4 Buckling

Concentrated compressive forces or stresses acting on a beam or column can cause the lateral or out of plane displacement of the member. This phenomenon is called buckling.

When a member is subjected to axial compression, there can be three forms of buckling:

1. Flexural buckling 2. Torsional buckling 3. Flexural-torsional buckling

When it is subjected to bending, there can be lateral torsional buckling.

Flexural buckling is in general the buckling mode, which governs the design of a member in pure compression. The maximum buckling load is reached when the combined axial and flexural stresses reach the yield stress, or when the moment reaches the plastic moment resistance. The slenderness for flexural buckling is taken from equation 3.4.

𝜆 =𝐿𝑐𝑟

𝑟𝑖 ·

1

𝜆1 3.4

Where ri is the radius of gyration about the relevant axis and

𝜆1 = 𝜋√𝐸

𝑓𝑦

Steel Buckling

For steel frame design, the flexural buckling stress is given by equations 3.5 and 3.6.

𝑝𝑐𝑟 = 0.877𝑝𝑒𝑙 3.5

when

𝑝𝑒𝑙 < 0.44𝑓𝑦

or

𝑝𝑐𝑟 = [0.658𝑓𝑦

𝑝𝑒𝑙]𝑓𝑦 3.6

when

𝑝𝑒𝑙 ≥ 0.44𝑓𝑦

where

𝑝𝑒𝑙 =𝜋2𝐸

(𝐾1𝐿/𝑟𝑖)2 3.7

pel – the elastic critical buckling strength of the member in the plane of bending, calculated based on the assumption of zero side-sway (N/m2)

E – the modulus of elasticity of steel, i.e. 210000 MPa

K1 – the effective length factor in the plane of bending, when there is no lateral translation. Also, the ratio KL/r is called the slenderness ratio and is taken as the maximum value, taking the minimum radius of gyration ri. In the AISC specification there is no limit on KL/r, but it is recommended that it does not exceed the practical limit of 200, based on professional judgement and construction economics.

There are two types of K-factors in the AISC 360-05 code. The first type of K-factor is used for calculating the Euler axial capacity if all the beam-column joints are held in place, i.e.,

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no lateral translation is allowed. The resulting axial capacity is used in calculation of the B1 factor. This K-factor is named as K1 in the code. The other K-factor, K2 is used for calculating the Euler axial capacity if all the beam-column joints are free to sway, i.e. lateral translation is allowed. There is also another K-factor, Kltb, for lateral torsional buckling. By default, Kltb is taken as equal to K2minor. However, the user can overwrite all these values on a member by member basis (AISC 360-05/IBC 2006).

A fixed connection shortens the buckling length to half in comparison to a pinned connection (EN 1992-1-1 2004) which has a huge impact on the capacity of the compressed columns. Buckling lengths for different types of supports are shown in Figure 11 below.

Figure 11. Buckling length according to EN 1992-1-1

In the case of framed structures, when diagonal bracing or rigid shear walls are used, the columns are prevented from side-sway and the effective lengths for buckling are calculated differently. K factor is calculated according to AISC for columns braced against and columns subjected to side-sway, by the chart seen in Figure 12 (Al-Ghalibi 2014).

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Figure 12. K factor calculation in framed structures (Al-Ghalibi 2014)

G factor is defined as:

G =𝛴(

𝐼𝑐

𝑙𝑐)

𝛴(𝐼𝑔

𝑙𝑔)

Where Ic is the moment of inertia and lc the unsupported length of a column section and Ig and lg the respective lengths of a girder or other restraining member. For instance, for the frame section shown in Figure 13, the G factor for columns A and B is calculated from equations 3.8 and 3.9:

GA =

𝐼𝑐𝐴𝑙𝑐𝐴

+𝐼𝑐𝐵𝑙𝑐𝐵

𝐼𝑔𝐴

𝑙𝑔𝐴+

𝐼𝑔𝐵

𝑙𝑔𝐵 3.8

GB =

𝐼𝑐𝐵𝑙𝑐𝐵

+𝐼𝑐𝐶𝑙𝑐𝐶

𝐼𝑔𝐶

𝑙𝑔𝐶+

𝐼𝑔𝐷

𝑙𝑔𝐷 3.9

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Figure 13. G factor calculation in framed structures

The unsupported lengths of the columns should account for both flexural buckling and lateral torsional buckling. ETABS determines automatically the unsupported length ratios, which are specified as a fraction of the frame object length. This ratio, when multiplied with the frame object length gives the unbraced length of the member, which, when multiplied by the adequate K factor, gives the real buckling length. Both of those parameters can be overwritten by the user (AISC 360-05/IBC 2006).

The two unsupported lengths, l33 and l22, as shown in Figure 14 should be considered for flexural buckling. These lengths are taken between the support points of the member in the corresponding directions. The length l33 corresponds to instability about the 3-3 axis (major axis), and l22 corresponds to instability about the 2-2 axis (minor axis). The length lLTB is also used for lateral-torsional buckling caused by bending in the major direction (i.e. about the 3-3 axis).

The program locates the member support points automatically, namely at the member connectivity, diaphragm constraints and support points and evaluates the corresponding unsupported lengths. It is possible for the unsupported length of a frame object to be evaluated by the program as greater than the corresponding member length. This can happen in the case that a column has a beam supporting it in one direction, but not the other, at a floor level. Then, the column is assumed to be supported in one direction only at that story level, and its unsupported length in the other direction will exceed the story height. Therefore, the lengths l22 and l33, lLTB can be overwritten and be given greater values.

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Figure 14. Unsupported lengths l33 and l22

Concrete Buckling

The critical buckling force in the case of concrete frame design, is taken from equation 3.10 (ACI 318-08/IBC 2009)

𝑃𝑐𝑟 =𝜋2𝐸𝐼

(𝑘𝑙𝑢)2 3.10

E – modulus of elasticity of steel, i.e. 210000 MPa

I – moment of inertia in the plane of bending, mm4

k – conservatively taken as 1, but can be overwritten by the user

lu – the unsupported length of the column for the considered direction of bending

The two unsupported lengths l22 and l33 for the minor and major direction are specified as before.

K factor is now calculated with the use of the Jackson-Moreland Alignment Chart for columns in braced frames, as seen in and G factors are taken from equation 3.11 for the non-side-sway case and from equation 3.12 for the side-sway permitted case.

G𝐴GB

4(

𝜋

𝐾)

2

+G𝐴+GB

2(1 −

𝜋

𝐾

tan(𝜋

𝐾)

) + (2 tan(

𝜋

2𝐾)

𝜋

𝐾

) − 1 = 0 3.11

((

𝜋

𝐾)

2𝐺𝐴𝐺𝐵

36− 1) ∗ tan (

𝜋

𝐾) −

G𝐴+GB

6(

𝜋

𝐾) = 0 3.12

Where GA and GB the relative stiffnesses of the column at upper and lower joints, calculated as the ratios of the sum of stiffnesses of columns meeting at the same joint as given in equation 3.9 (Tikka & Mirza 2014). K factor is again taken from Figure 12.

3.4.5 Shear lag

In tubed systems, the lateral load is carried by the corner columns and the self-weight is carried by the interior columns or walls. In lateral loading, however, tubed frames seem to suffer from shear lag, leaving the mid columns less stressed than the corner columns.

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This phenomenon happens because of the difference in deflection direction for columns and beams, leading to a different rotation in the shared joints. The axial force distribution along the flange and the web of the column has a parabolic shape and is assumed to be either parabolic or hyperbolic cosine function (see Figure 15).

Figure 15. Shear lag effect in axial stress distribution in the building's columns in web and flanges (Gaur & Goliya 2015)

The shear lag effect can be described by comparing the shear lag ratio for two systems. This is the ratio between the axial forces at the corner columns to the axial forces at the middle of the panel. Shear walls can be used to resist shear lag, followed by rigidly connected beam-column frames (Leonard 2007). Another practical way to decrease shear lag is by introducing an inclined bracing system, thus increasing the stiffness of the columns. As seen in Figure 16, the angle of the bracing system can significantly change the shear lag ratio (Gaur & Goliya 2015).

Figure 16. Shear lag change with varying angle of bracing system for different heights (Gaur & Goliya 2015)

In a 120-story high building, the shear lag ratio starts changing from positive to negative value at around the 100-th story (see Figure 17).

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Figure 17. Shear lag effect in high rise buildings with tubed frame system and shear lag ratio vs number of stories (Gaur & Goliya 2015)

The trussed tubed system, also known as diagonalized tubed system (see John Hancock center in Chicago), is a type of tubed frame system with diagonals in the façade, which acts as a truss in a plane and together with the trusses on the perpendicular façades contribute to the tubed frame action. In this case, the tube gets rid of the closely spaced columns and becomes more resistant to wind forces (Gaur & Goliya 2015).

3.5 Finite Element Analysis

Finite Element Analysis (FEA) is a numerical, approximate solution for Partial Differential Equations. It was first fully introduced in the 1950s, but due to the insufficient performance of computers it was not popularized until the 1980s (Budzyński 2006).

FEA has a lot of advantages; a wide application in many disciplines, no geometrical, material, load or element type restrictions and easy improvement of approximation. However, a negligent application might lead to convergence problems or false results (Cook et al. 2002).

3.5.1 Procedure

The Finite Element Method starts with dividing an idealized model to smaller (finite) elements of simple geometry (discretization). It is assumed that they are connected only in nodes creating a mesh. After that, the model is ready for solving a group of equations and a convergence check. A detailed outline is presented in Figure 18.

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Figure 18. Procedure for Finite Element Analysis (Cook et al. 2002)

Consistent nodal loads are the point loads located in the nodes of the mesh and are calculated based on the real load state of the model.

Shape functions are used to approximate the displacement in between the nodes. They can improve the accuracy of the solution, but can also extend the computation time.

A global stiffness matrix is applied to the whole model and can be created by transformation of local matrices of all the elements. It is singular and symmetric.

For structural calculations, the system of equations contains the global stiffness matrix [K], the global load vector {R} (composed of consistent nodal loads) and the displacement vector {D}. The system can be solved from the equilibrium equation 3.13 after it is reduced due to the boundary conditions:

[𝐾]{𝐷} = {𝑅} 3.13

The boundary conditions are taken at the points or surfaces, where constraints (known value points) are specified or known loads (point forces or pressures) are applied.

3.5.2 Element types

The most common element types can be divided in 3 groups, according to their dimensions:

1D elements (e.g. spring element) 2D elements (e.g. beam, bar, plane elements) 3D elements (e.g. beam, bar, plate, shell or solid elements)

In our analysis, the elements that were used the most were 2D and 3D beam and 3D shell elements.

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Beam elements

Beam elements are defined by their axis with assigned cross section and characterized by 3 degrees of freedom per node in 2D analysis and 6 in 3D analysis. They can be studied using 2 different theories, Euler Bernoulli and Timoshenko (see Figure 19), which differ significantly in terms of shear strain inclusion. For beams with height to length ratio higher than 1/10, shear strain should be included according to Timoshenko theory. In a different case, the Euler-Bernoulli theory is more applicable due to the phenomenon called shear locking occurring in thin beams if Timoshenko theory is used (due to the fact that in the strain energy equation, the ratio GA/EI is used and it becomes larger for thinner beams)(Pacoste 2016a).

Figure 19. Euler-Bernoulli and Timoshenko theory (Pacoste 2016a).

Shell elements

Shell elements are defined by mid-surface and the assigned thickness. Similar to beam elements, it can be solved with an aid of two theories, Mindlin and Kirchoff (see Figure 20), which differ in the approach with regard to the shear strain in the same manner as for the beam elements.

Figure 20. Mindlin and Kirchoff theory (Pacoste 2016b)

Shell elements are usually modelled as three or four-noded elements with 5 degrees of freedom (“drilling” rotation is skipped).

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4 Parameter study A preliminary study was performed in the current research in order to determine and describe the effects of different parameters. Due to the lack of literature within inclined buildings, a preliminary study regarding inclination angle, truss models and three-dimensional structural systems was carried out.

4.1 Inclination angle

In this section, the study of stresses with regard to different inclination angles was conducted. For a constant length and varying angle of every 5°, the reactions, displacements and stresses at the base of the building are checked including nonlinear effects (P-delta).

4.1.1 Model

The model was used in this simplified analysis consisted of 2 beams connected at the fixed support (see Figure 21). The model remained symmetrical in the whole study.

In order to investigate how the same load affects the structures when a different inclination angle was assigned, it was decided to keep the constant length of 296.05 m and a circular, hollowed beam cross section.

Figure 21. Model used for the inclination study

Cross section

D = 30 m

t = 0.25 m

Area: A = 23.36 m2

Moment of inertia: I = 2585.2 m4

Material: steel S355

Figure 22. Cross section of the beam model

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4.1.2 Loads

In this study, only the self-weight of the structure was included. The wind load was applied in further study with the selected inclination angle and height of the structure.

4.1.3 Results

The results of the inclination study were extracted from the linear and nonlinear, static analysis performed in SAP2000 and presented in Table 1 and Table 2 below. The inclination angle was measured from the vertical axis.

Stresses and reactions were measured on the bottom of the model.

Table 1. Reactions and stresses from the inclination study

The inter-story drift of the building was defined as the lateral deflection of a floor relative to the one immediately below it divided by the distance between floors ((δn - δn-1)/h). According to the serviceability limit state criteria, specified by ASCE 7, the lateral inter-story drift for the wind load should not exceed the height of the floor in vertical direction divided by 400. The drift limits that are commonly used for building design are on the order of 1/500-1/200 of the story height that designers use as general indices of non-structural damage for the building (Ellingwood 1989).

For a hypothetical story height of 3.5 m, the inter-story drift ratio is given in Table 2 and the serviceability limit was taken as 1/400.

In Table 2, UX and UZ are the maximum displacements obtained at the top end of the beams.

Reactions Stress

Angle Moment Force Tensile Compressive

⁰ kNm kN kPa kPa

40 50 661 976 1 064 901 276 499 -311 412

35 45 207 026 1 064 901 243 637 -280 971

30 39 408 022 1 064 901 208 922 -248 391

25 33 309 099 1 064 901 172 616 -213 921

20 26 956 674 1 064 901 134 997 -177 824

15 20 399 093 1 064 901 96 350 -140 372

10 13 686 262 1 064 901 56 970 -101 853

5 1 770 800 1 064 901 17 624 -63 026

0 0 1 064 901 0 -22 787

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Table 2. Displacements from the inclination study

4.1.4 Discussion

From Table 1, there is noticeable difference in tensile stresses when the inclination angle changed from 10° to 15° (stress rises about 69%). Also, an even bigger change can be observed in transition from 5° to 10° (increase of 223%).

The serviceability limit was not exceeded for an angle equal and lower than 10°. The limit displacement concerned not only the self-weight, but also all the loads which are to be imposed on the structure but were not included in this study. Although it was not noticeably exceeded now, it might have been critical for the improved structural system.

Another interesting observation was that with an increasing angle and deflection, the exclusion of the P-delta effect resulted in smaller error. In this case, only dead load was included, therefore P-delta was not of concern and any vertical displacement could be compensated by constructing each story to a theoretical elevation which would incorporate an increase in the typical floor to floor height. The results of this study are shown in the figures in Appendix A.

Taking all the above into consideration together with the need of analyzing the most extreme case possible, we decided to proceed with 10° inclination angle of each tower.

4.2 2D truss models

In order to find the most suitable structural system for the leaning towers, different types of trusses were examined. The possible combinations were divided into 6 groups due to their geometry:

1. Non-truss group – non-bracing models, with a tying beam placed every 30 m or a belt truss placed in 2 or 3 positions along the model.

2. X group – truss models with X bracing system, 30 m wide 3. N group – truss models with N bracing system, 30 m wide 4. K group – truss models with K bracing system, 30 m wide 5. V group – truss models with V bracing system, similar to K group, but rotated

by 90°

Displacement (excl. P-delta) Serviceability

Limit ratio Displacement (incl. P-delta)

Angl

e UZ UX

Inter-story

drift ratio UZ

Error

UZ UX

Error

UX

⁰ m m m (1/400) m % m %

40 1.43 1.68 0.006 0.003 1.45 1.35% 1.70 1.37%

35 1.14 1.60 0.006 0.003 1.16 1.44% 1.63 1.46%

30 0.87 1.48 0.005 0.003 0.88 1.52% 1.50 1.55%

25 0.63 1.31 0.004 0.003 0.64 1.60% 1.33 1.64%

20 0.42 1.10 0.003 0.003 0.42 1.63% 1.12 1.70%

15 0.25 0.85 0.003 0.003 0.25 1.63% 0.87 1.75%

10 0.12 0.58 0.002 0.003 0.12 1.54% 0.59 1.78%

5 0.04 0.30 0.001 0.003 0.04 1.10% 0.30 1.80%

0 0.02 0.00 0.000 0.003 0.02 - 0.00 -

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6. W group – truss models with bracing system similar to V group, but with a width of 60 m (in not inclined trusses each member is tilted by 45°)

Due to different geometry, number and length of members and then mass of models we decided to choose one system, the one with the lowest deflection, from each group to continue with the more detailed 3D study in ETABS. This procedure was to ensure that due to mismatched cross section none of the well performing systems would be excluded because of the increased self-weight and then deflection.

4.2.1 Model alternatives

Geometry

In the figure below the global geometry is presented. The height of the structure was 300 m, the length was 305.94 m and the inclination was 1:5, which gave the angle of around 11.31°. There were 2 fixed supports on the bottom of the structure. Detailed geometries of each truss types are shown in Appendix B. The connection between the members were taken as fixed to shorten the buckling length.

Figure 23. Global dimensions of 2D truss models

Cross section

Due to the differences in the geometry of the models, one way to compare the performance of each type of truss was to assign the same cross section. This procedure gave the most reliable results, since it allowed to omit the effect of different distribution of inner forces because of varying stiffness. A round, hollowed profile seemed to be the most suitable due to its sufficient moment of inertia along with economical use of material and uniform torsion.

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Dimensions:

D = 2000 mm

t = 200 mm

Material: steel S355

Figure 24. Cross section of the 2D truss systems

This cross section was assigned to all members in each model within the context of this preliminary study. The trusses with the best performance were to be further investigated in the 3D model analysis, including the design of the most suitable cross sections and the comparison of their deflections and masses.

4.2.2 Loads

The self-weight was assigned automatically in X direction and was based on the cross section and the material.

The wind load was chosen as 30 kN/m of horizontal distributed not projected load acting along Z axis on one of the longitudinal members which corresponds to 90 mph wind speed acting on a 30 meters wide wall. A hurricane of 90 mph is within category 1, according to Saffir-Simpson Hurricane Wind scale (National Weather Service n.d.).

4.2.3 Results

The results from the static, linear analysis, as it was conducted in Robot Structural Analysis are presented in Table 3 below. Displacements UX and UZ were the highest displacements obtained on the top of the structure. The given stresses were global maximum (compression) and minimum stresses (tension). The truss with the lowest displacement in each group is shaded in the table.

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Table 3. Results from the 2D truss analysis

Results including reactions and inner forces are attached in Appendix C. In Robot, the compressive forces were assigned with plus and the tensile forces with minus.

4.2.4 Discussion

It can be noticed that in most groups the displacement values related to the mass of the model. For an inclined structure like this, the self-weight seemed to significantly affect the horizontal displacement. In the further study, we decided to proceed with the structures with the best deflection performance despite their weight (shaded in gray) from groups X, K, N and W (numbers 7, 9, 16 and 18).

4.3 3D study of structural systems

In order to determine the optimal inclination angle and to reject less suitable systems, a more detailed 3D analysis was conducted. In this study, the live load and a more accurate wind load were included, where the comparison of the models were based on the same mass and materials.

4.3.1 Model

The 3D study was handled in ETABS software. Each model consists of 94 floors including the mechanical floors.

Displacement Stress Mass

No Name Min UX Min UZ Max Smax Min Smin g

cm cm MPa MPa t

1 Truss -2 557.1 -12 752.8 2 502.0 -2 519.2 5 967

2 Belts -107.2 -494.1 652.8 -659.0 8 365

3 Belt truss 1 -619.4 -3 037.6 1 905.3 -1 892.3 10 872

4 Belt truss 2 -233.6 -1 102.8 1 317.3 -1 216.3 13 458

5 X not full -62.6 -257.2 748.4 -632.6 12 152

6 X + belts -40.5 -137.4 244.4 -129.6 15 939

7 X -36.9 -122.8 232.0 -102.5 13 541

8 +N + belts -38.6 -128.8 253.4 -109.2 12 527

9 -N + belts -32.0 -110.4 194.2 -108.2 11 777

10 +N -N -35.8 -120.6 245.7 -105.6 12 152

11 -N +N -34.6 -117.9 200.7 -111.8 12 152

12 +V + belts -39.3 -131.6 299.6 -117.7 14 328

13 +V -34.4 -115.8 256.7 -105.4 11 930

14 -V + belts -39.4 -132.0 252.5 -112.3 14 328

15 W + belts -35.0 -117. 251.7 -110.7 12 152

16 W -30.0 -101.7 208.8 -103.7 9 754

17 +K + belts -46.0 -158.1 296.2 -158.5 14 399

18 -K + belts -32.9 -107.7 198.6 -88.1 14 399

19 +K -K + belts -36.2 -127.1 170.4 -102.5 14 399

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29

Geometry

The height of the structure was 296.05 m and the inclination angle was at 10°. The floors at the base were square and increased in dimensions from 19.5x19.5 m to 29x29 m. The floors in each tower were in the shape of a 30-meter wide equilateral triangle.

Model alternatives

There were 2 types of modeling alternatives:

1. Steel truss structures – includes 4 variants of typical truss systems and 3 variants of diagrid systems.

2. Concrete frame structures – includes 5 variants of tubed mega frame systems and 2 variants of moment frame systems

Both groups were compared separately due to different materials used in bearing elements.

Steel truss structures

In this type, structures differed only in the bracing system (except diagrid 3, where all columns and bracing elements had the same cross section). They had steel columns (rectangular, hollowed, see Figure 25) on the sides and composite mega columns (with 4 encased Jumbo profiles, see Figure 26) on the outside compression side. The cross sections of bracing elements were chosen as hollowed rectangular or I beam with dimensions matched to keep the similar mass of each model.

They were divided as follows (for bracing models see Figure 27 to Figure 33):

1. Truss X type a. Cross sections:

i. Composite columns 3.072x3.072 m, HD400x1299 ii. Steel columns ⎕1x1x0.1 m

iii. Braces ⌶1x0.5x0.1 m b. Material:

i. Steel A992Fy50 ii. Concrete C90/105

2. Truss K type a. Cross sections:


Figure 25. Steel column cross section in the truss structures

Figure 26. Composite mega column with 4 encased steel profiles cross section

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iii. Braces ⌶1x0.5x0.1 m b. Material:


3. Truss N type a. Cross sections:


iii. Braces ⌶1x0.5x0.13 m, ⌶1x0.5x0.1 m b. Material:


4. Truss W type a. Cross sections:

i. Composite columns 3.072x3.072 m, HD400x1299 ii. Steel columns ⎕ 1x1x0.1 m

iii. Braces ⌶1x0.5x0.1 m, ⎕0.9x0.9x0.1 m b. Material:


5. Diagrid with 3 mesh in the row (Diagrid 1) a. Cross sections:


iii. Braces ⎕0.5x0.5x0.047 m b. Material:


6. Diagrid with 4 mesh in the row (Diagrid 2) a. Cross sections:


iii. Braces ⎕0.5x0.5x0.044 m b. Material:


7. Diagrid with 4 mesh in the row and without corner columns (Diagrid 3) a. Cross sections:

i. Columns ⎕0.8x0.8x0.087 m ii. Braces ⎕0.8x0.8x0.087 m

b. Material: i. Steel A992Fy50

ii. Concrete C90/105

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Figure 27. Truss X

Figure 28. Truss K

Figure 29. Truss N

Figure 30. Truss W

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Figure 31. Diagrid 1



Concrete frame structures

The tubed mega frames consisted of 3, 6 or 9 hollowed, rectangular, concrete mega columns (see Figure 34) placed on the perimeter of the building and concrete belt walls tying them together every 7th floor in the upper part of the building or every 6th floor in the bottom part of the building. This system was chosen as a better performing option than tying every 14th floor with a belt wall of the height of 2 floors (Partovi & Svärd 2016).

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The moment frame consisted of 10 or 15 columns similar to those in tubed mega frame structures and concrete beams (hollowed, rectangular) across the floors creating a triangular net.

The cross sections of the columns, walls and beams were chosen to give the same model mass with the additional condition that the mass of the columns as well as walls and beams in each model are kept similar for reasons of comparison. Keeping the same denominator allowed to investigate the structural performance of the various systems having the same amount of material contributing to the axial and shear forces and moment conditions.

Figure 34. Cross section of a concrete mega column in the tubed frame model

The reinforcement properties for column sections are presented in Figure 35 below:

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Figure 35. Reinforcement properties for hollow mega columns

Properties of each model (shown in Figure 36 to Figure 42):

1. TMF with 3 columns in the corners (TMF 1) a. Cross sections:

i. Mega columns 3.75x3.75 m, t = 0.75 m ii. Belt walls t = 0.75 m

b. Material: i. Concrete C90/105

ii. Steel rebar A615Gr60

2. TMF with 3 columns in the middle of the external walls (TMF 2) a. Cross sections:




3. TMF with 6 columns on the external walls (TMF 3) a. Cross sections:

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4. TMF with 6 columns on the corners and external walls (TMF 4) a. Cross sections:




5. TMF with 9 columns on the corners and external walls (TMF 5) a. Cross sections:




6. Moment frame with 10 columns (Frame 1) a. Cross sections:

i. Columns 2.00x2.00 m, t = 0.410 m ii. Beams 0.50x0.50 m, t = 0.145 m



7. Moment frame with 15 columns (Frame 2) a. Cross sections:

i. Columns 1.50x1.50 m, t = 0.39 m ii. Beams 0.50x0.50 m, t = 0.105 m



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Figure 36. TMF 1

Figure 37. TMF 2

Figure 38. TMF 3

Figure 39. TMF 4

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Figure 40. TMF 5

Figure 41. Frame 1

Figure 42. Frame 2

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4.3.3 Loads

For the 3D study, each load type was considered separately:

Figure 43. Load cases used in the 3D study of structural systems

Dead load

The dead load as applied as the self-weight of the mega structure and slabs. This load was generated automatically and did not require any further step. The slabs had a thickness of 250 mm and the rest of the elements are specified in chapter 4.2.1. The installation loads and the non-structural parts of the building (e.g. façade) were not included in this study.

Live load

The live load was dependent on the use and the occupancy of the building and it included human occupants, furnishings, nonfixed equipment, storage, construction and maintenance activities. In this study, the live load was taken as 2.5 kN/m2 uniformly distributed at each floor according to ASCE 7-10 specifications as applied for a mixed combination of residential and office building (ASCE 2010). The live load used in the case of office and residential buildings are presented in Table 4:

Table 4. Minimum uniformly distributed live loads and minimum concentrated live loads

Occupancy or use Uniform Concentrated

psf (kN/m2) psf (kN/m2)

Office buildings File and computer rooms shall be designed for heavier loads base on anticipated occupancy

Lobbies and first-floor corridors 100 (4.79) 2,000 (8.90) Offices 50 (2.40) 2,000 (8.90) Corridors above first floor 80 (3.83) 2,000 (8.90)

Residential Dwellings (one- and two- family) Uninhabitable attics without storage 10 (0.48) Uninhabitable attics with storage 20 (0.96) Habitable attics and sleeping areas 30 (1.44) All other areas except for stairs and balconies 40 (1.92) Hotels and multifamily houses

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Private rooms and corridors serving them 40 (1.92) Public rooms and corridors serving them 100 (4.79)

Figure 44. Live load pattern for 3D study of structural systems

Wind load

The wind load was defined in ETABS as an automatically generated wind load according to ASCE 7-10 and was applied using rigid diaphragms on the floor surface. This way, the wind load was transferred to each floor and then to the structure.

Figure 45. Wind load pattern for 3D study of structural systems

4.3.4 Results

The results of linear, static analysis are presented in the tables below. For additional information about the base reactions and column forces, see Appendix D.

Mass comparison plays a major role in judging the efficiency of the structural systems using the same material (see Table 5 below).

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Table 5. Mass source for each model

With regard to the serviceability of the building, the most important factor is the floor to floor displacement (drift) and the periods of vibrations shown, for each system, in Table 6.

Table 6. Results from 3D study for displacement and period

4.3.5 Discussion

From Table 5 it is clear, that the mass of the models was identical within each group. This maked the structural systems comparable and the study valid.

As shown in Table 6, the displacements and vibration periods differed for the different systems and for each imposed load. Therefore, it was not easy to make a clear comparison among all these systems at the same time. For an inclined building the most urgent issue

Mass [tons]

All Slabs Concrete

columns

Steel

columns Bracing

Tubed

columns

Walls or

beams

Truss X 68 233 41 910 15 922 3 010 7 392 - -

Truss K 68 365 41 910 15 922 3 010 7 523 - -

Truss N 68 213 41 910 15 922 3 010 7 371 - -

Truss W 68 205 41 910 15 922 3 010 7 363 - -

Diagrid 1 67 951 41 673 15 922 3 010 7 346 - -

Diagrid 2 68 066 41 673 15 922 3 010 7 460 - -

Diagrid 3 67 985 41 673 - 3 247 26 311 - -

TMF 1 94 827 41 673 - - - 38 246 14 908

TMF 2 94 830 41 673 - - - 38 245 14 911

TMF 3 94 809 41 673 - - - 38 222 14 914

TMF 4 94 814 41 673 - - - 38 235 14 906

TMF 5 94 839 41 673 - - - 38 260 14 906

Frame 1 94 628 41 673 - - - 38 110 14 844

Frame 2 95 261 41 673 - - - 38 255 15 332

Displacement at the top [m] Period [s]

Dead load Live load Wind load Mode 1 Mode 2 Mode 3

Truss X 1.40 0.22 0.75 13.26 7.57 7.33

Truss K 1.65 0.26 0.77 13.12 7.48 7.43

Truss N 1.56 0.25 0.87 14.49 8.65 7.98

Truss W 1.54 0.25 0.76 13.62 7.82 7.44

Diagrid 1 1.70 0.32 1.18 10.72 9.35 8.07

Diagrid 2 1.65 0.31 1.15 11.50 9.37 8.97

Diagrid 3 3.34 0.80 1.48 10.51 9.90 7.24

TMF 1 1.37 0.23 0.71 8.77 7.11 5.85

TMF 2 4.10 0.71 1.20 11.03 10.17 9.66

TMF 3 3.14 0.54 1.08 10.52 9.24 8.01

TMF 4 2.00 0.35 0.88 9.63 8.12 6.70

TMF 5 2.17 0.37 0.92 9.85 8.27 6.94

Frame 1 8.64 1.53 4.63 22.52 16.96 15.21

Frame 2 6.73 1.19 3.27 18.82 15.94 12.79

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is to minimize the deflections due to wind and live loads and for tall and slender ones, to obtain the optimal periods of vibrations. Although the top story displacement might by limited by increasing the stiffness of the structure (i.e. by enlarging the bearing elements), the modes of vibrations are more complex to handle – increasing stiffness and mass at the same time may give the opposite results. The choice of the system which was to be further investigated was based on balanced criteria.

In the case of the bracing systems, the X truss behaved better than the rest of the bracing systems and mainly within deflection (14-18% less live load deflection and 1-16% less wind load deflection). On the other hand, the diagrid can be an attractive solution due to the considerably lower vibration periods (14-25% less than the lowest period of truss systems). Based on these results, an optimization study was performed further in this study, in order to judge which system performs better with respect to analysis and design.

The selection was simple within concrete frame systems – Tubed Mega Frames performed better than moment frames with respect to deflection and vibration modes, which was anticipated (Partovi & Svärd 2016). The TMF 1 system (3 columns in the corners of the building tied with belt walls every 7th or every 6th floor) seemed to be the best alternative due to the lowest deflection and periods of vibrations.

4.4 Comparison of shell and frame elements within Tubed Mega Frame structures in ETABS

Using beam or shell elements in the studied structural system may have influenced the performance of the model in the FEA software. This section aims to show the study of the difference in using these kinds of elements in the analysis.

4.4.1 Models:

The studied model was a Tubed Mega Frame system with tubed columns in the corners of the building and belt walls connecting them every 7th floor. Three alternative models were analyzed.

1. Shell elements model (belt walls, slabs and columns as shell elements) 2. Frame model (belt walls and slabs as shells, columns as frames) 3. Modified Frame model (same as in case 2, with cross section modifiers in order to

obtain the same mass and stiffness as in the shell elements model)

The slabs were the same for all variants (thickness 250 mm, C30/37). The models are seen in Figure 46 and Figure 47.

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Shell model:

Figure 46. Geometry of the Shell model

Wall section:

C90/105 Thickness 750 mm Height of the whole floor Shell-thin Connected to the corners of the columns

Column section:

3000x3000 mm in axis Thickness 750 mm Shell-thin Center in the corners of the building

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Frame model:

Figure 47. Geometry of Frame model

Wall section:

C90/105 Thickness 750 mm Height of the whole floor Shell-thin

Column section:

C90/105 3750x3750 mm 750 mm thick

Table 7. Cross section modifiers for the Modified Frame model

The modifiers’ values came from the comparison between shell element properties and frame element properties. The differences came from the model simplifications.

4.4.2 Loads:

Dead, live and wind load were modelled exactly as in the 3D study of structural systems.

Seismic:

Within this study, seismic action was included in the analysis to fully investigate and compare the performance of both models.

Element Feature Modifier

Slab Mass 0.9670

Wall Mass 0.7913

Wall Shear v13, v23 2.1718

Column Mass 1.0038

Column Moment of inertia about axis 2 and 3 0.9560

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Figure 48. Seismic load for the shell and Frame model comparison study

4.4.3 Results:

The base reactions in tables 8, 9 and 10 refer to the global axis directions and are a sum of reactions at the base for each mega column.

Table 8. Base reactions for the Shell model

Table 9. Base reactions for the Frame model

Load case FX FY FZ MX MY MZ

kN kN kN kNm kNm kNm

Dead -0.17 0.00 887389 204.91 -48.62 0.77

Live -0.03 0.00 151804 -0.02 -8.07 0.13

Wind 1 -19463 0.00 0.01 -0.20 -3340046 -1.21

Seismic 1 -16256 0.00 0.01 -0.23 -3645746 4.97



Dead 0.01 0.00 929937 -0.08 1.41 0.11

Live 0.00 0.00 163519 -0.01 0.24 0.02

Wind 1 -21762 0.00 0.00 -0.10 -3734447 -0.03

Seismic 1 -17010 0.00 0.00 -0.11 -3823929 -0.03

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Table 10. Base reactions for the Modified Frame model

The differences in base reactions for the Shell and Frame models for the wind load case were due to the different geometry of the models’ slabs, since the slabs’ geometry in the Shell model were adapted to the shell mega columns in the corners (see Figure 46), thus taking a non-triangular shape as they were in the Frame models. This way the diaphragms taking the wind load were not the same in the compared models.

The maximum displacements in all cases were obtained in X direction. The load types, as before, were considered separately.

Table 11. Displacement comparison for Shell and Frame models

The 12 modes of vibration for the different models are presented in Table 12 below. As previously, the error is given in relation to the Shell model.

Table 12. Period comparison for Shell and Frame models

The mass differences occurred due to the different geometry of columns, floors and walls. They were minimized by using the cross section modifiers (see Table 7).



Dead 0.01 0.00 929937 -0.09 1.35 0.09

Live 0.00 0.00 163519 -0.02 0.23 0.02

Wind 1 -21761 0.00 0.00 -0.12 -3734447 -0.01

Seismic 1 -16230 0.00 0.00 -0.12 -3641517 -0.01

Load case Direction Shell Frame Error

Modified

Frame Error

m m % m %

Dead X 1.458 1.366 6.27 1.373 5.78

Live X 0.242 0.231 4.54 0.232 4.05

Wind 1 X 0.640 0.709 10.87 0.718 12.22

Seismic 1 X 0.728 0.748 2.82 0.721 1.01

Mode Shell Frame Error

Modified

Frame Error

s s % s %

1 8.55 8.77 2.61 8.60 0.66

2 7.17 7.11 0.82 6.94 3.14

3 5.87 5.85 0.39 5.74 2.27

4 4.80 4.69 2.33 4.58 4.62

5 2.41 2.13 11.62 2.10 12.78

6 2.07 1.86 10.43 1.83 11.49

7 1.71 1.60 6.09 1.57 7.91

8 1.55 1.53 1.35 1.50 3.41

9 1.24 1.12 9.30 1.10 11.00

10 1.06 1.02 4.42 1.00 6.30

11 0.83 0.74 11.00 0.73 12.21

12 0.82 0.73 10.67 0.72 12.27

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Table 13. Mass comparison for Shell and Frame models

4.4.4 Discussion

As seen from the above comparison, the Frame model deflected less under vertical loads (dead, live loads) and more under lateral loads (wind, seismic action). This difference tended to decrease after considering the cross section modifiers. A similar situation was observed for the periods of vibration.

The base reactions differ more noticeably, but for the significant reactions the difference was not larger than 12%. Those values did not show a tendency to decrease after including cross section modifiers.

It is worth mentioning that modelling a belt wall with frame elements would not give the same results, since those frames would connect at a single point on the top and bottom of the wall, whereas a shell element belt wall is connected to the columns along the height of the whole story. However, if we were to use frame elements for a belt wall, a bracing would be required due to the fact that a single beam does not provide a sufficient restrain, even if rigids links are assigned at the connecting joints. This would be called a belt truss (see Figure 49).

Figure 49. Belt truss system

In the context of the parameter study and analysis, these simplifications and errors might be acceptable, but within the comprehensive design of the structural system, a detailed modelling should take place.

Group

Shell Frame Error Modified

Frame Error

t t % t %

All 90 488 94 827 4.80 90 486 0.00

Slabs 40 300 41 673 3.41 40 298 0.00

Walls 11 797 14 908 25.97 11 796 0.01

Tubes 38 392 38 246 0.38 38 391 0.00

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5 Design of TMF and braced systems This chapter treats the design of the best performing models in ETABS. The design of the cross sections resisting possible load combinations followed by the evaluation of their structural properties was conducted. The dynamic response of the systems was checked together with a nonlinear analysis of construction sequence. The models were checked and verified.

5.1 Models

Three models were investigated in the design phase:

Tubed Mega Frame TMF 1 (Figure 50) X truss (Figure 51) Diagrid (Figure 52)

Figure 50. TMF 1 design

Figure 51. X truss design


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Figure 52. Diagrid design

5.1.1 Geometry

The geometry of the whole structure remained the same as in the 3D preliminary study (see chapter 4.3.1). Only the cross sections of elements and bracing density changed after the design and optimization of the three structural systems.

5.1.2 Materials

In Table 14 below the material properties for the used elements are presented:

Table 14. Materials used in comprehensive study

5.1.3 Cross sections

The slabs were modelled as in chapter 4.3.1.

Tubed mega frame

The cross sections of the concrete columns remained tubed and rectangular as in the previous study. Along the height of the building the columns were divided into 3 groups as shown in Table 15 below:

Element Material f’c / fy E ν ρ

MPa GPa - kg/m3

Mega columns, belt

walls, bracing C90/105 90 44 0.2 2548.5

Slabs C30/37 30 33 0.2 2548.5

Rebar A615Gr60 413.7 200 - 7849.0

Bracing S355 355 210 0.3 7849.0


49

Table 15. Cross sections of tubed columns in the TMF structure

The belt walls have a thickness of 1500 mm and they are not designed in this thesis.

X truss

In the X truss, the two outside inclined columns were tubed concrete columns, with the same section along the height of the building. The inside corner columns were also concrete with a rectangular tubed section, but were divided into 2 groups along the vertical columns, which extend from the base to the 22nd floor and another 2 groups of tubed concrete columns were assigned along the inclined part of the high-rise building. The X trusses were comprised by steel trusses of 2 different tubed sections and the distinction was made between the inner side of the building, where the two towers face each other, and the two outer sides.

Table 16. Cross sections for the X truss structure

Diagrid

In this case, the outside columns comprised a composite section with 4 encased “Jumbo” steel profiles. The inside columns were rectangular, hollowed steel tubes filled with high strength concrete. The braces were chosen according to the demanded capacity and consist of circular concrete sections.

Table 17. Cross sections for the diagrid structure

Group Placement External dimensions Thickness

# story m m

Bottom columns 1 - 32 6.80x4.60 1.30

Middle columns 33 - 58 5.50x3.50 1.00

Top columns 59 - 94 5.00x3.00 0.80

Group Material Placement

External

dimensions Thickness

# story m m

Outside Mega

columns concrete 1 - 94 4.50x6.70 1.20

Inside columns concrete 1 - 6 3.00x5.00 1.00




Outside trusses steel 1 - 94 2.00x1.50 0.20

Inside trusses steel 21 - 94 1.50x1.50 0.20

Group Placement

External

dimensions Thickness

# story m m

Outside columns 1 - 94 3.00x3.00 solid

Bottom inside columns 1 - 24 2.00x2.00 0.15

Top inside columns 25 - 94 1.50x1.50 0.10

Bracing 1 - 20 2.00 solid





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5.1.4 Boundary conditions

The supports in the bottom of the columns were modelled as fixed. The type of foundation or ground conditions were not specified or considered in this design.

5.1.5 Element types

The belt walls as well as the slabs were modelled as shell-thin with rectangular (4-noded quadrilateral) automatically meshed elements with dimensions not exceeding 1.5 m. For frame (beam) elements ETABS is using a division at each floor.

5.1.6 Assumptions

The models were simplified in the way that they included only slabs and main columns with bracing or belt walls. Neither primary beams were considered to connect the columns, nor secondary beams to transfer the floor loading to the columns. The slabs were modelled as rigid diaphragms, which allowed to use the autogenerated wind load and provided rigid in-plane stiffness of the slabs.

Installation loads were modelled as distributed load on the slabs and the building’s façade was modelled as concentrated point loads on the columns. Any intermediate columns, necessary to support the slabs in the appropriate interspaces, were not included in this design.

The buckling lengths of the members did not include the effect of the slabs rigidity and were equal to the distance between the bracing connections or the belt walls.

Material nonlinearities were neglected if not specified otherwise (see chapter 5.2.7).

The wind and seismic loads were defined for Shanghai (Zhang 2014) and may vary for different locations. The design was performed for tubed concrete columns, steel and concrete braces and the strength capacity of the belt walls was not checked. Also, steel connections as well as concrete joints should have been separately investigated and designed. In the design, the joints were considering the depth of the members in ETABS, which shortens calculations, but this could not assure the validity of the automatically designed sections in such critical points.

This study was completed for a prototype building and it aimed to provide information about suitable systems for inclined, slender structures. In terms of the real design, a more detailed analysis with all the suitable loads, as well as additional actions such as wind tunnel test, design of intermediate columns and study of material nonlinearities should have taken place.

The constructability of the structure was not investigated. Construction sequence used here determines more accurate deformations of the structure under the dead load.

5.2 Loads

The loads in the design study fulfilled the following criteria:

They were applied and combined according to the American code ASCE 7-10. Each static load case was modelled in 2 cases, with and without nonlinear analysis

of global P-Delta effect.


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The dynamic analysis was performed by determination of the 100 first natural frequencies and periods which were used in the response spectrum and time history analysis.

5.2.1 Dead loads

The dead loads included the self-weight of the structural members (columns, slabs, walls, bracing). Nonstructural members (façade, installations) were assigned as 0.8 kN/m2 uniformly distributed load on slabs and 90 kN point load at each floor and column. The direction of the load was determined by gravity (-Z).

5.2.2 Live loads

The live loads were defined as 2.5 kN/m2 uniformly distributed load on slabs (see chapter 4.3.2). The ETABS program uses live load reduction function and attribute area method according to ASCE 7-10. It is not common to have maximum live load on every floor and for that reason columns are going to carry less load. This was taken into account when the program automatically assigned a live load reduction factor, RLLF, to each element. According to ASCE7-10, the formula is taken from equation 5.1 (ASCE 2010).

𝑅𝐿𝐿𝐹 = 0.25 +15

√𝐴1 5.1

RLLF – the reduced live load factor for an element, unitless. It was multiplied with the unreduced live load to get the reduced live load.

A1 – the influence area of the element in ft2. For a column, this was taken as four times the tributary area. The influence area for a beam, brace or wall was taken as twice the tributary area. A1 must be greater than 37.16 m2.

5.2.3 Wind

The wind load according to ASCE 7-10 was taken for a 50-year return period wind for ultimate limit state (95 mph) and a 10-year return period wind for service limit state (85 mph).

The recommended gust effect factor, taken from ASCE 7-10, depends on basic wind speed V, exposure category, building natural frequency and damping and was obtained from equation 5.2.

𝐺𝑓 = 0.9251+1.7I𝑧−√𝑔𝑄

2 Q2+𝑔𝑅2 𝑅2

1+1.7gv𝐼𝑧 5.2

Where gQ and gv are taken as 3.4

gQ – the peak factor for background response

gR – the peak factor for resonance response

gv – the peak factor for wind response

Q – the background response factor

R – the resonant response factor

Iz – the intensity of turbulence


52

The calculated Gust factor was 0.91 (see Appendix E). The other parameters remained the same as specified in the 3D preliminary study (see Figure 45).

5.2.4 Seismic action

The seismic action parameters were the same as described in chapter 4.4 (see Figure 48).

In this chapter, 2 kinds of dynamic analysis were performed: response spectrum and time history (referred also as synthetic earthquake).

Response spectrum analysis is a linear-dynamic statistical method measuring maximum displacement and acceleration for each mode of vibration according to design response spectrum (specified in Figure 53). In this analysis, the time evolution of the structure’s response is not considered.

Time history analysis considers time dependent loading and is based on measuring the structural response for each time step during and after the load application (might also consider nonlinear effects, but in this case linear analysis is performed) (Computers & Structures Inc. n.d.). The ETABS program uses the dynamic equilibrium equation 5.3.

𝐾 · 𝑢(𝑡) + 𝐶 · 𝑢′(𝑡) + 𝑀 · 𝑢′′(𝑡) = 𝑟(𝑡) 5.3

The response spectrum as well as the synthetic earthquake were defined based on the parameters used in static seismic analysis. Time function for time history analysis was matched to the design response spectrum in the frequency domain (Figure 54).

From this dynamic analysis, accelerations, displacements and story drifts were extracted to determine the serviceability of the building and allowed for comparisons.

Figure 53. Design parameters for response spectrum


53

Figure 54. Time function matched to the response spectrum

The damping of the structure varied depending on type and height of the structure, used material and other factors. The common practice is to assume a damping ratio of 5%. For reinforced concrete, high rise buildings, damping ratio is taken as 3% in Japan (Arakawa & Yamaoto 2004) and measured damping ratio of tall buildings in California reaches values lower than 2,5% (Cruz & Miranda 2017). In this study, the damping ratio was assumed as 3% of critical damping.

5.2.5 Combinations

According to ASCE 7-10 the following static load combinations were used for Ultimate Limit State (rain and snow loads are omitted):

1. 1.4D 2. 1.2D + 1.6L 3. 1.2D + 1L 4. 1.2D + 0.5W 5. 1.2D + 1W + 1L 6. 1.2D + 1E + 1L 7. 0.9D + 1W 8. 0.9D + 1E

Combinations 9 and 10 were used for Service Limit State check. Combination 9 is proposed in literature (Taranath 2016) and combination 10 is commonly used to check the story drift and building deflection (it omits the dead load because deflection under this load is minimized during construction).


54

9. 1D + 0.5L + 1WSLS 10. 1L+ 1WSLS

Symbols:

D – dead load (including super dead load) L – live load W – wind load WSLS – wind load for Service Limit State E – seismic action (earthquake load)

All combinations were considered both with and without P-Delta effect.

A dynamic analysis concerning response spectrum and time history functions was performed separately from the load cases.

5.2.6 Modal case

Natural frequencies and periods were modeled as eigen modal case (for undamped free vibration modes). P-Delta effect was not considered. For the static analysis, the 12 first modes were calculated, but for dynamic investigation (response spectrum and time history functions), the first 100 modes were determined to receive the full response of the structure.

The second was based on the aim to obtain the cumulative effective mass participation (energy contained in each resonant mode) in all response directions close to 100%. In all three models, the mass contribution converged to a percentage of 98.9%~99.3% with respect to the total mass of the models, after the 96th mode. The typical requirement is that for the response spectrum analysis, at least 90% of the model's mass must participate in the solution.

5.2.7 Construction sequence

Construction sequence was performed in ETABS in order to simulate the actual condition during the construction and check the force distribution and deformations of the structure under dead load (including installation and façade). Auto construction sequence case was performed twice, with and without accounting for material nonlinearity, such as creep and shrinkage. For this case P-Delta effect was included to achieve as precise result as possible. Following to this, a design of concrete columns was performed to check if the assumed cross sections are appropriate. Also, the limit deformation and maximum deformation coming from the serviceability limit state combination, including construction sequence instead of just the dead load, were compared.

The auto construction sequence load case was performed one step per story starting from the unstressed state. It was assumed that the concrete for the columns and floors was placed one story at a time. To investigate the influence of material nonlinearity, the duration of each construction step was estimated to 5 days, based on the construction methodology of Tubed Mega Frames (Dahlin & Yngvesson 2014) and the loads were assumed to be added after 3 days of curing the concrete. This gave 475 days of construction in total.


55

5.2.8 Buckling

The buckling of the whole structure was modelled in ETABS using 2 sets of loads combined as buckling load cases:

Dead load Dead, live and wind load

This analysis was on eigenvalues and P-Delta effect was not considered. This means that the structure was assumed elastic, with no nonlinearities or imperfections.

5.3 ETABS design and results

5.3.1 Design

The design can be performed automatically in ETABS for concrete, steel or composite structures. However, special attention should be paid in joints and column-wall connections.

In the figures below, story displacement values in X direction are marked with blue and in Y direction with red.

The base reactions for all the load combinations can be found in Appendix F.

Tubed Mega Frame

Auto design in ETABS assigns appropriate reinforcement for a given column cross section fulfilling the demand of 1 to 6%. The nodes of the inner columns, on the 21st floor where the two towers connect, should have been designed separately.

The displacement under the Serviceability Limit State combination (number 9) is shown in Figure 55 below (maximum value is 1.17 m, which was going to be compensated while performing a more detailed analysis of construction sequence in chapter 5.3.4).


56

Figure 55. Displacement under SLS combination (number 9) for the Tubed Mega Frame structure

The displacement under the live and wind combination (number 10) was significantly lower and reached 0.34 m in X direction (see Figure 56).

Figure 56. Displacement under live and wind load combination (number 10) for the Tubed Mega Frame structure


57

The first 12 modes of vibration with their direction factors are presented in Table 18 below.

Table 18. Modes of vibration for the Tubed Mega Frame system

Mode Period

UX UY UZ RZ sec

1 5.705 1 0 0 0

2 5.313 0 1 0 0

3 4.172 0 0 0 1

4 3.565 0.003 0.007 0 0.991

5 1.677 0 0 0 1

6 1.458 0 1 0 0

7 1.385 0 1 0 0

8 1.251 1 0 0 0

9 1.028 0 0 0 1

10 0.825 0.002 0.012 0 0.986

11 0.692 0 1 0 0

12 0.667 0 0 0 1


58

X truss

The Serviceability Limit State load combination (number 9) for the X truss system, gave a maximum deformation of 0.93 m, as shown in Figure 57.

Figure 57. Displacement under SLS combination (number 9) for the X truss structure

The displacement under the live and wind combination (number 10) was significantly lower as shown in Figure 58 (0.26 m in X direction)

Figure 58. Displacement under live and wind load combination (number 10) for the X truss structure


59

In Table 19, the first 12 modes of vibration are presented.

Table 19. Modes of vibration for the X truss system

Diagrid

As previously, the deflection under Serviceability Limit State combination (number 9) exceeded the suggested limits (see Figure 59) and reached 1.49 m for the top floor.

Figure 59. Displacement under SLS combination (number 9) for the diagrid structure

The displacement under the live and wind combination (number 10) was 0.48 m in X direction (see Figure 60).

Mode Period

UX UY UZ RZ sec

1 5.925 0 1 0 0

2 5.040 1 0 0 0

3 3.707 0 0 0 1

4 3.176 0.071 0.106 0 0.823

5 1.364 0 1 0 0

6 1.301 0 0 0 1

7 1.075 1 0 0 0

8 1.045 0 1 0 0

9 0.953 0 0 0 1

10 0.711 0.086 0.33 0 0.584

11 0.576 0 1 0 0

12 0.495 0 0 0 1


60

Figure 60. Displacement under live and wind load combination (number 10) for the diagrid structure

The first 12 modes of vibration with their directions are presented in Table 20 below.

Table 20. Modes of vibration for the diagrid system

Mode Period

UX UY UZ RZ sec

1 5.840 1 0 0 0

2 5.388 0 1 0 0

3 4.550 0 0 0 1

4 3.938 0 1 0 0

5 1.708 0 0 0 1

6 1.592 0 1 0 0

7 1.358 0 1 0 0

8 1.252 1 0 0 0

9 1.062 0 0 0 1

10 0.887 0 1 0 0

11 0.658 0 1 0 0

12 0.651 0 0 0 1


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5.3.2 Response spectrum

The response spectrum analysis results in maximum displacement and story drift, which are shown in the figures below.

Global direction X is marked with blue and Y with red.

Tubed Mega Frame

As it can be seen in Figure 61, the maximum story displacement for the Tubed Mega Frame structure was 262 mm in both X direction and Y direction. The maximum acceleration reached 1769 mm/s2 in X direction.

Figure 61. Maximum story displacement for TMF system – response spectrum

In Figure 62, the maximum story drifts are shown. As predicted, the drifts were lower on the floors with belt walls. The maximum story drift did not exceed the value of 0.0015, which was below the limit for risk category III (ASCE 2010).


62

Figure 62. Maximum story drifts for TMF system – response spectrum

X truss

In the case of the X truss system, the maximum displacement for the top floor was 272 mm in X direction and 284 mm in Y direction (see Figure 63). The maximum acceleration reached 2251 mm/s2 in X direction.

Figure 63. Maximum story displacement for X truss system – response spectrum

The maximum story drifts are shown in Figure 64. It is obvious that the drifts were constrained in the points where the X trusses are connected. The maximum story drift was 0.0019 in Y direction and 0.0013 in X direction.


63

Figure 64. Maximum story drifts for the X truss system – response spectrum

Diagrid

The maximum story displacement for the diagrid structure was 273 mm for X direction and 279 mm for Y direction (see Figure 65). The calculated maximum acceleration reached 2600 mm/s2 in Y direction.

Figure 65. Maximum story displacement for diagrid system – response spectrum

As previously, the maximum story drift was below the limit for risk category III (0.015). Drifts for stories 36, 46 and 56 were close to 0 as shown in Figure 66. It was caused by braces connected to the vertical part of column, which kept the stability of the structure. On other floors where braces were connected, a significant drift reduction was observed.


64

Figure 66. Maximum story drifts for diagrid system – response spectrum

5.3.3 Time history analysis

In this case, the time function was applied separately and the response of the structure is plotted in the figures below.

Tubed Mega Frame

As presented in Figure 67, the time history function gave less conservative results than the response spectrum analysis. The maximum displacement was around 210 mm and the story drift did not exceed the ratio of 1.5·10-3.

Figure 67. Deformations on X direction under time history matched to the response spectrum for TMF


65

In Figure 68 the shear force and overturning moment for the structure are shown. The maximum base reactions were 50 MN for shear force in X direction and 3793 MNm for overturning moment MY. The maximum acceleration was measured at the top story and reached 1782 mm/s2 in Y direction.

Figure 68. Reactions on X direction under time history matched to the response spectrum for TMF


66

X truss

From time history analysis in X truss system, the maximum displacement was 226 mm at the top in X direction and 248 mm in Y direction, which was less than in the response spectrum analysis. The story drift ratio was higher, at 1.58·10-3.

Figure 69. Deformations on X direction under time history matched to the response spectrum for X truss

The shear force and overturning moment diagrams can be seen in Figure 70. The base reaction had now a maximum value of 64 MN in X direction and the maximum MX was 3797 MNm. At the top story the acceleration reached 3395 mm/s2.

Figure 70. Reactions on X direction under time history matched to the response spectrum for X truss system


67

Diagrid

As shown in Figure 71, the maximum displacement for the diagrid structure was around 230 mm and the story drift ratio did not exceed 1.9·10-3.

Figure 71. Deformations on X direction under time history matched to the response spectrum for the diagrid system

In Figure 72, the shear force and overturning moment for the diagrid structure are shown. The maximum base reactions were 45 MN for shear force in X direction and 2884 MNm for overturning moment MX.

Figure 72. Reactions on X direction under time history matched to the response spectrum for the diagrid system


68

The maximum acceleration was measured in the top story and reached 4272 mm/s2 in X direction.

5.3.4 Comparison of the response spectrum and time history analysis

Due to the fact that the response spectrum and time history were separate analyses conducted with different methods, the comparison of the obtained accelerations showed differences reaching 50%. Note that these differences were calculated in reference to response spectrum values.

Table 21. Acceleration obtained in response spectrum and time history analyses

The following sections contain figures (Figure 73 to Figure 78) comparing displacements in X and Y directions which resulted from response spectrum and time history analyses.

Tubed Mega Frame

Figure 73. Displacement on X direction obtained in response spectrum and time history analyses for the Tubed Mega Frame structure

0

10

20

30

40

50

60

70

80

90

0 50 100 150 200 250 300

# St

ory

Displacement [mm]

Response spectrum Xdirection

Time history X drection

X direction [mm/s2] Y direction [mm/s2]

Response

spectrum

Time

history Difference

Response

spectrum

Time

history Difference

TMF 1769.25 1379.75 22.0% 1590.08 1781.78 12.1%

X truss 2250.56 3394.67 50.8% 2236.72 2912.90 30.2%

Diagrid 2592.95 1900.32 26.7% 2599.66 1778.32 31.6%


69

Figure 74. Displacement on Y direction obtained in response spectrum and time history analyses for the Tubed Mega Frame structure

X truss

Figure 75. Displacement on X direction obtained in response spectrum and time history analyses for the X truss structure

0

10

20

30

40

50

60

70

80

90

0 50 100 150 200 250 300

# St

ory

Displacement [mm]

Response spectrum Ydirection

Time history Ydirection

0

10

20

30

40

50

60

70

80

90

0 50 100 150 200 250 300

# St

ory

Displacement [mm]




70

Figure 76. Displacement on Y direction obtained in response spectrum and time history analyses for the X truss structure

Diagrid

Figure 77. Displacement on X direction obtained in response spectrum and time history analyses for the diagrid structure

0

10

20

30

40

50

60

70

80

90

0 50 100 150 200 250 300

# St

ory

Displacement [mm]


Time history Y direction

0

10

20

30

40

50

60

70

80

90

0 50 100 150 200 250 300

# St

ory

Displacement [mm]




71

Figure 78. Displacement on Y direction obtained in response spectrum and time history analyses for diagrid structure

5.3.5 Construction sequence

As seen in the tables below, the AutoSeq load case (construction sequence) gave much lower deflections than when the dead load was applied after the end of construction (not the real situation). This resulted from adjusting the column height and the height of the story at each construction stage.

Tubed Mega Frame

After substituting the dead load with the AutoSeq load case considering material nonlinearities in nonlinear Serviceability Limit State combination (including P-Delta effect), the deflection acquired was below the strictest limit (h/600, where h is the height of the floor) – see Table 22. The design of the columns was checked also for the nonlinear AutoSeq case to prove a good performance of the building under construction. The results confirmed that the design was conducted properly.

Table 22. Displacement of the 94th floor of the TMF structure considering construction sequence

0

10

20

30

40

50

60

70

80

90

0 50 100 150 200 250 300

# St

ory

Displacement [mm]


Time history Y direction

Story Load Case/Combo Direction Max

Direction Max Max drift

ratio mm mm

94 Dead+Sdead nl Z -278 X 871 0.0046

94 AutoSeq Z -38 X 123 0.0038

94 AutoSeq nl Z -45 X 145 0.0020

94 1D 0,5L 1Wsls nl Z -301 X 1171 0.0019

94 AutoSeq 0,5L 1Wsls nl Z -68 X 446 0.0007


72

X truss

For the X truss system, considering the staged construction and the time dependent properties of the concrete material resulted in less top story displacement than in the case when the dead load was applied in the whole structure. This displacement was also lower than the limit displacement.

Table 23. Displacement of the 94th floor of the X truss structure considering construction sequence

Diagrid

In Table 24 the results of a similar analysis, performed for the diagrid structure is shown. The final deflection was under the conservative limit h/600 (0.00775).

Table 24. Displacement of the 94th floor of the structure considering construction sequence

5.3.6 Buckling

The design also accounted for the global and local buckling of the structural members. The load cases were taken as defined in chapter 5.2.8. The six first buckling modes were considered in this analysis and can be seen in Appendix F.

As shown in Table 25 the scale factors for Tubed Mega Frames started from 19 which means that the structure was prone to buckling under a load 19 times bigger than the applied load.

The X truss system was more prone to global than local buckling (which occured in 4th mode) and the scale factor was also on the safe side.

In the case of the diagrid structure, buckling occured as a local failure for the most loaded brace members from the 3rd mode.



ratio mm mm

94 Dead+Sdead nl Z -222 X 695 0.0034

94 AutoSeq Z -16 X 37 0.0768

94 AutoSeq nl Z -18 X 45 0.0001

94 1D 0,5L 1Wsls nl Z -234 X 932 0.0046




ratio mm mm

94 Dead+Sdead nl Z -295 X 1090 0.0047

94 AutoSeq Z -13 X 50 0.0103

94 AutoSeq nl Z -17 X 64 0.0072

94 1D 0,5L 1Wsls nl Z -386 X 1492 0.0070



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Table 25. Buckling factors for Tubed Mega Frame, X truss and diagrid design

A hand calculation was performed to check the buckling resistance for a compressed X truss brace. This check included control of flexural buckling and flexural and axial compression of the member for the maximum load given by load combinations. The analytical calculations can be found in Appendix E.

5.4 Model verification

The verification of the analyzed models was also performed. The analysis results were hereby studied and included the check of the total mass and the model symmetry. The mass of the models was calculated in Mathcad software (see Appendix E) and compared with the ETABS output for each model.

As shown in Table 26 and Figure 79, the axial forces and joint reactions showed symmetry in the models, as expected, which is a good indication that the restraints, constraints and loading were correctly modelled.

TMF X truss Diagrid

Case Mode Scale Factor Scale Factor Scale Factor

Buckling under dead

loads

1 23.71 21.26 21.60

2 26.68 29.22 26.16

3 27.66 30.39 28.19

4 42.32 34.83 30.41

5 46.14 35.14 30.41

6 50.66 35.27 30.75

Buckling under dead,

live, wind loads

1 19.56 17.39 16.98

2 21.84 24.05 19.99

3 22.76 24.52 20.95

4 35.06 28.25 21.33

5 37.91 28.51 21.74

6 41.57 30.54 21.77


74

Figure 79. Axial forces due to self-weight in the models

Table 26. Base reactions due to self-weight in the models

Base joint coordinates FX FY FZ

m kN kN kN

TMF

x=-12.3771, y=0, z=0 -62842.32 -0.06 415242.94

x=12.3771, y=0, z=0 62842.34 -0.06 415242.49

x=0, y=-15, z=0 -0.02 155.28 245443.10

x=0, y=15, z=0 0.01 -155.16 245441.79

X truss

x=-12.3771, y=0, z=0 -41067.43 0.01 476908.72

x=12.3771, y=0, z=0 41067.43 0.00 476908.72

x=0, y=-15, z=0 0.00 40025.18 186274.29

x=0, y=15, z=0 -0.00 -40025.19 186274.29

Diagrid

x=-12.3771, y=0, z=0 -33252.59 0.06 198270.27

x=12.3771, y=0, z=0 33253.04 0.00 198270.86

x=0, y=-15, z=0 0.05 -2.62 12530.56

x=0, y=15, z=0 0.07 2.65 12531.68


75

Tubed Mega Frame

In Table 27 below the mass comparison for the TMF system is made. The received difference was neglectable and the biggest difference occurred for slabs, which were approximated in Mathcad calculations.

Table 27. Mass control of the Tubed Mega Frame system

X truss

Table 28. Mass control of the X truss system

Diagrid

In the case of the diagrid structure, only the mass of the columns and slabs was compared:

Table 29. Mass control of the diagrid system

ETABS Hand calculations

Group Mass Mass Error

kg kg %

All 134 742 272 135 421 394 0.50

Slabs 41 673 492 42 345 879 1.61

Walls 29 815 382 29 823 031 0.03

Tubes 63 253 398 63 252 484 0.00

Bottom columns 27 198 507 27 198 107 0.00

Middle columns 17 545 427 17 545 184 0.00

Top columns 18 509 463 18 509 193 0.00



kg kg %

All 135 251 692 134 372 639 0.91

Slabs 41 673 487 42 345 879 1.61

Outside columns 32 406 990 32 568 934 0.49

Inside columns 24 929 072 23 435 975 1.50

Columns 57 336 061 56 004 909 0.93

Braces 36 242 144 36 021 851 0.07



kg kg %

All 110 143 317 - -

Slabs 41 673 492 42 345 879 1.61

Outside columns 15 922 213 15 922 397 0.00

Inside columns 10 165 315 10 165 253 0.00

Columns 26 087 528 26 087 649 0.00

Braces 42 382 297 - -


76

5.5 Discussion

The Tubed Mega Frame structure was proven to perform well under the applied loads. The designed columns’ and walls’ dimensions did not limit the constructability of the building or the access to façade openings apart from the case where the belt walls were assigned. This structural system was quite flexible and enabled geometrical changes to the structure, such as adjusting the floor shape. The Ultimate Limit State conditions were fulfilled and the dynamic load behavior was sufficient. The Serviceability Limit State displacement was also fulfilled and so was the story drift limit in the seismic time history analysis. Global buckling was not probable to occur under the given loading. The structure weighed around 135 kt. Compared to the Petronas Twin Towers which weighs 385 kt with a height of 452 m or to the central core, outrigger and perimeter frame system of 432 Park Avenue (83 kt and 264 m high), the studied inclined TMF system showed a good structural behavior.

The X truss system was flexible and did not limit the window space significantly. As most of the outer bracing systems, these enormous X trusses were placed on the façade. The ULS and SLS conditions were fulfilled and the system performed better than TMF and diagrid for deflection under static loads. The scale factor for buckling in this case was large (17.4). However, the mass of the system (135 kt) together with a high amount of steel used in the bracings led to increased construction costs for this structure.

The performance of the diagrid structure was worse compared to the Tubed Mega Frames and X truss system. The cross sections of the braces might limit the open window access, but the geometry of the building is still possible to adjust. The Ultimate Limit State and Serviceability Limit State conditions were fulfilled; however, deflection and local buckling were critical issues for this system and should be controlled. The mass of the structure was lower than in the two other cases (110 thousand tones) which means lower construction cost.

The resulting displacements from time history and response spectrum analysis satisfied the serviceability limit displacement, however the induced maximum story accelerations were beyond the suggested comfort limit criteria, as seen in Table 30.

Table 30. Comfort criteria against acceleration (Balushi & John 2014)

It is necessary to study further these peak acceleration values in order to evaluate the building’s dynamic behavior. Considering that the along-wind accelerations depend on the surface roughness and the structural dumping of the building (Balushi & John 2014),

Level Acceleration

Effect (m/sec2)

1 < 0.05 Humans cannot perceive motion

2 0.05 – 0.1 Sensitive people can perceive motion. Hanging objects may

move slightly

3 0.1 – 0.25 Level of motion may affect desk work. Long term exposure

may produce motion sickness

4 0.25 – 0.4 Desk work becomes difficult or almost impossible

5 0.4 – 0.5 Difficult to walk naturally and standing people may lose balance

6 0.5 – 0.6 Unable to walk naturally

7 0.6 – 0.7 People cannot tolerate motion or walk

8 > 0.85 Objects begin to fall


77

in the case of a slimmer and lighter system, the structure could be more susceptible to vibrations. Apart from changing the structural stiffness of the building, the damping should be also taken into consideration. Increasing the damping can be achieved by installing damping devices, such as visco-elastic and friction dampers, tuned mass dampers and tuned liquid dampers. Adding such devices comes always hand in hand with a life-cycle cost analysis. However, damping increase measurements were not treated in this study.

6 Conclusion and further research

79

6 Conclusion and further research

6.1 Conclusion

This project has studied different structural systems and their behavior on an inclined high rise building. The preliminary study has shown that for a small inclination of the building (around 10°) a choice of structural system which performs within the set limits of bearing capacity and serviceability was possible.

The chosen angle seemed to be a good compromise between functionality, structural performance and technological innovation. Also, investigating different truss systems resulted in the best choice for the needs of this project. The analysis provided distinct and comparable results.

The design of certain elements showed that only steel cross sections were not sufficient to fulfill the capacity requirements both for the X truss and the diagrid structure. Therefore, concrete or composite members were assigned and designed for the defined loading. The use of steel cross sections in the case of the X truss bracings are not commonly produced, so a special research as to their design, welding and production might be necessary. A thorough design should also be performed concerning the critical connection points between all the members, especially the joint where the two inclined columns come together to a vertical one at the base of the building.

The deflections of the three structural systems did not exceed the limits according to ASCE 7-10 after considering construction sequence, nevertheless the X truss system performs better in this regard.

Overall, all structural systems had a similar behavior under dynamic seismic load, however accelerations did not satisfy the comfort criteria. This could be solved by using damping systems, which is common practice for tall buildings.

Every system had a sufficient buckling factor, which was over 15. The comparison showed, that for the first six buckling modes, local failure occured only in braced systems.

From an architectural aspect, the Tubed Mega Frame system could be more easily adapted to different forms and shapes. The two braced systems required big sections in the members on the façade and corner columns, which could limit the open window space.

6.2 Further research

Our proposition for further research within inclined and slender high rise buildings is to revise the assumptions made in this study, for instance the geometry of the structure, material nonlinearity (such as cracking of concrete, creep and shrinkage) as well as to provide a more detailed design of belt walls and connections.

Intermediate columns should be designed to support the floor slabs and meet the requirements of the inclined structure. The interaction between inside columns, slabs and cross walls would also be interesting to study and interpret. Another issue to be taken into account would be the detailed study of the dynamic response of the structure (for example time history analysis for wind function) and testing it in a wind tunnel.

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Tikka, T.K. & Mirza, S.A., 2014. Effective Length of Reinforced Concrete Columns in Braced Frames. International Journal of Concrete Structures and Materials, 8(2), pp.99–116.

Trabucco, D. et al., 2016. Composite Megacolumns. Council on Tall Buildings and Urban Habitat (CTBUH) in conjunction with ArcelorMittal, pp.8–11.

Yura, J.A., 2015. The effective length of columns in unbraced frames. Engineering Structures, 102, pp.132–143. Available at: http://linkinghub.elsevier.com/retrieve/pii/S0141029615004824.

Zhang, H., 2014. Global Analysis and Structural Performance of the Tubed Mega Frame, ISSN 1103-4297 ; 426. KTH Royal Institute of Technology.

Appendix A Results from inclination angle study

85

Appendices


0

50

100

150

200

250

300

0 5 10 15 20 25 30 35 40

Stre

ss [

MP

a]

Angle [°]

Tensile stress

-350

-300

-250

-200

-150

-100

-50

0

0 5 10 15 20 25 30 35 40

Stre

ss [

MP

a]

Angle [°]

Compressive stress


86

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0 5 10 15 20 25 30 35 40

Def

lect

ion

[m

]

Angle [°]

Vertical deflection

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

0 5 10 15 20 25 30 35 40

Def

lect

ion

[m

]

Angle [°]

Horizontal deflection

0.000.200.400.600.801.001.201.401.601.802.002.202.40

0 5 10 15 20 25 30 35 40

Def

lect

ion

[m

]

Angle [°]

Deflection


87

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%

1.40%

1.60%

1.80%

5 10 15 20 25 30 35 40Dif

fere

nce

in in

clu

din

g an

d n

ot

incl

ud

ing

P-d

elta

eff

ect

Angle [°]

P-delta effect for vertical deflection

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%

1.40%

1.60%

1.80%

2.00%

5 10 15 20 25 30 35 40

Dif

fere

nce

in in

clu

din

g an

d n

ot

incl

ud

ing

P-d

elta

eff

ect

Anlge [°]

P-delta effect for horizontal deflection

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%

1.40%

1.60%

1.80%

2.00%

5 10 15 20 25 30 35 40Dif

fere

nce

in in

clu

din

g an

d n

ot

incl

ud

ing

P-d

elta

eff

ect

Angle [°]

P-delta effect

Appendix B Geometry of trusses analyzed in the 2D study

89


1 – Truss

2 – Belts

3 – Belt truss 1

4 – Belt truss 2

5 – X not full

6 – X + belts

7 – X

8 – +N + belts


90

9 – -N + belts

10 – +N -N

11 – -N +N

12 – +V + belts

13 – +V

14 – -V + belts

15 – W + belts

16 – W


91

17 – +K + belts

18 – -K + belts

19 – +K -K + belts

Appendix C Results from the 2D study

93


Reactions

No Name 5 FX 5 FZ 5 MY 6 FX 6 FZ 6 MY

kN kN kNm kN kN kNm

1 Truss 64 294 -4 801 -910 543 -5 776 13 979 -1 170 694

2 Belts 153 323 -17 920 -229 098 -71 288 27 098 -239 501

3 Belt truss 1 168 002 -19 251 -808 378 -61 380 28 429 -856 512

4 Belt truss 2 221 645 -26 907 -513 537 -89 664 36 086 -531 068

5 X not full 211 245 -25 584 -254 016 -92 073 34 762 -259 537

6 X + belts 279 982 -15 503 2 754 -123 672 24 681 -13 988

7 X 244 640 -24 408 -3 040 -111 846 33 586 -10 232

8 +N + belts 228 121 -44 023 -16 249 -105 272 53 201 -45 060

9 -N + belts 218 669 -12 313 -991 -103 173 21 491 -12 988

10 +N -N 220 807 -42 582 -15 712 -101 635 51 761 -44 411

11 -N +N 226 006 -13 045 -1 344 -106 834 22 223 -13 105

12 +V + belts 254 452 -47 085 -26 808 -113 944 56 264 -32 206

13 +V 219 445 -40 718 -21 513 -102 454 49 897 -29 474

14 -V + belts 256 437 -13 817 -3 833 -115 930 22 996 -9 634

15 W + belts 222 099 -42 610 -18 001 -102 926 51 789 -47 504

16 W 187 096 -35 976 -12 712 -91 440 45 154 -44 650

17 +K + belts 257 498 -32 279 3 021 -116 290 41 457 -43 793

18 -K + belts 256 162 -30 593 4 868 -114 954 39 771 -16 482

19 +K -K + belts 257 744 -17 476 11 339 -116 536 26 654 -10 120


94

Inner forces

No Name Max FX Max FZ Max MY Min FX Min FZ Min MY

kN kN kNm kN kN kNm

1 Truss 62 706 1 907 622 564 -34 015 -36 342 -1 164 161

2 Belts 152 579 1 306 304 160 -76 499 -21 185 -292 192

3 Belt truss 1 167 234 9 141 273 352 -67 044 -25 339 -849 980

4 Belt truss 2 221 336 10 851 180 373 -96 281 -17 544 -524 536

5 X not full 210 879 4 099 88 573 -98 383 -16 085 -260 548

6 X + belts 241 302 1 663 13 081 -128 841 -2 040 -14 402

7 X 222 248 1 889 22 468 -105 051 -2 896 -16 460

8 +N + belts 231 044 1 787 18 026 -80 412 -2 234 -22 781

9 -N + belts 188 992 1 306 7 250 -106 665 -1 402 -12 559

10 +N -N 223 589 1 784 17 645 -77 406 -2 209 -22 244

11 -N +N 195 728 1 610 11 203 -110 399 -1 734 -12 819

12 +V + belts 257 463 1 522 14 811 -103 104 -3 475 -33 341

13 +V 221 888 1 478 13 086 -93 351 -2 853 -28 046

14 -V + belts 239 890 1 591 8 360 -119 469 -1 478 -18 725

15 W + belts 224 861 1 882 13 799 -79 398 -1 874 -24 533

16 W 189 237 1 763 10 595 -72 001 -1 582 -19 544

17 +K + belts 256 480 2 614 39 501 -121 547 -5 635 -32 201

18 -K + belts 217 195 2 387 6 532 -88 033 -1 306 -16 873

19 +K -K + belts 190 399 1 681 11 591 -115 789 -1 741 -23 253


95

Displacement Stress Mass

No Name Min UX Min UZ Max Smax Min Smin g

cm cm MPa MPa t

1 Truss -2 557.10 -12 752.80 2 502.0 -2 519.2 5 967

2 Belts -107.20 -494.10 652.8 -659.0 8 365

3 Belt truss 1 -619.40 -3 037.60 1 905.3 -1 892.3 10 872

4 Belt truss 2 -233.60 -1 102.80 1 317.3 -1 216.3 13 458

5 X not full -62.60 -257.20 748.4 -632.6 12 152

6 X + belts -40.50 -137.40 244.4 -129.6 15 939

7 X -36.90 -122.80 232.0 -102.5 13 541

8 +N + belts -38.60 -128.80 253.4 -109.2 12 527

9 -N + belts -32.00 -110.40 194.2 -108.2 11 777

10 +N -N -35.80 -120.60 245.7 -105.6 12 152

11 -N +N -34.60 -117.90 200.7 -111.8 12 152

12 +V + belts -39.30 -131.60 299.6 -117.7 14 328

13 +V -34.40 -115.80 256.7 -105.4 11 930

14 -V + belts -39.40 -132.00 252.5 -112.3 14 328

15 W + belts -35.00 -117.90 251.7 -110.7 12 152

16 W -30.00 -101.70 208.8 -103.7 9 754

17 +K + belts -46.00 -158.10 296.2 -158.5 14 399

18 -K + belts -32.90 -107.70 198.6 -88.1 14 399

19 +K -K + belts -36.20 -127.10 170.4 -102.5 14 399

Appendix D Base reactions and column forces from the 3D study

97


Forces in columns for steel truss systems

Composite column Vertical part of steel column

System Load case

Force in the

base

Stress in the

base

Force in the

bottom

Force in the

top

kN kPa kN kN

X truss

Dead -303764 -26420.56 -14148.7547 71389.9495

Live -71287.1861 -6200.35 -5066.673 8967.8139

Wind -132714 11543.10 81252.0056 59272.2913

K truss

Dead -330545 -28749.89 -3004.2427 59352.9705

Live -75984.5954 -6608.97 -4370.9397 5271.0636

Wind 121246.064 -10545.64 93651.4523 58706.794

N truss

Dead -332383 -28909.76 -3599.6224 78430.6185

Live -77572.7901 -6747.09 -3430.9601 10759.8621

Wind 120512.2667 10481.80 95848.7249 66672.6787

W truss

Dead -319747 -27810.71 -13576.8521 78381.0973

Live -74287.9611 -6461.37 -5943.9195 11397.8433

Wind 120329.3198 10465.88 96100.1167 63968.0878

Diagrid 1

Dead -318023 -27660.764 12148.737 67383.749

Live -69634.415 -6056.610 -1069.6978 10628.6685

Wind -127215 -11064.810 60303.5923 -29568.2867

Diagrid 2

Dead -338828 -29470.326 17656.9791 68917.6371

Live -76135.254 -6622.035 -34.6402 11220.9385

Wind 124186.906 10801.435 72363.7343 46774.697

Diagrid 3

Dead -75754.641 -6588.930 -19472.9541 86460.999

Live -18566.507 -1614.864 -4766.8633 20757.0918

Wind 70768.494 6155.249 47865.23 22307.857


98

Base reactions for concrete frame systems

System Load

case

FX FY FZ MX MY MZ


TMF 1

Dead 0.0057 0.0002 929937 -0.0828 1.4081 0.1108

Live 0.001 3.49E-05 163519.2 -0.0138 0.2383 0.0188

Wind -21761.5 0.0004 4.43E-05 -0.1035 -3734447 -0.0288

TMF 2

Dead 0.0032 -0.0049 929961.4 1.256 1.4446 -0.1876

Live 0.0005 -9.00E-04 163519.2 0.2189 0.2507 -0.0327

Wind -21761.5 -0.0004 0.00E+00 0.11 -3734448 -0.0803

TMF 3

Dead -0.0042 -2.04E-05 929758.1 0.0003 -0.7782 0.0307

Live -0.0007 -3.56E-06 163519.2 0.0001 -0.1358 0.0053

Wind -21761.5 0.0002 1.00E-04 -0.0342 -3734444 0.0008

TMF 4

Dead -0.0094 6.30E-03 929812.2 -25940.1 -1.0878 0.0329

Live -0.0016 1.10E-03 163519.2 -0.156 -0.1904 0.0054

Wind -21761.6 0.0001 2.27E-05 -0.0569 -3734453 0.2971

TMF 5

Dead 0.0495 -2.40E-03 930052.7 0.6738 12.458 0.1604

Live 0.0085 -4.00E-04 163519.2 0.1162 2.1479 0.0277

Wind -21761.5 0.0004 -2.00E-04 -0.0615 -3734447 0.0703

Frame

1

Dead 0.1356 2.30E-02 918443.8 219.0623 -1297.43 0.9159

Live 0.0243 4.10E-03 163519.2 -0.9692 5.2537 0.1639

Wind -21761.3 7.20E-03 4.82E-05 -1.9373 -3734392 0.726

Frame

2

Dead 0.0121 3.90E-03 934193.2 41.3795 -62.6498 -0.1827

Live 0.0021 7.00E-04 163519.2 -0.0972 0.4164 -0.0323

Wind -21761.6 -1.20E-03 -2.24E-05 0.2054 -3734451 0.0959

Appendix E Analytical calculations for the models

99


Gust factor calculations


100


101

Mass calculations for TMF model


102


103

Mass calculations for X truss model


104


105

Mass calculations for diagrid structure – columns


106

Buckling check of X truss braces


107


108

Appendix F Results from Design study

109


Base reactions under load combinations for the Tubed Mega Frame structure

Load Case/Combo

FX FY FZ MX MY MZ

kN kN kN kN-m kN-m kN-m

1,4D 0.0067 0.0007 1989199 -0.1989 1.6408 0.0884

1,2D 1,6L 0.0067 0.0007 1966659 -0.1984 1.6389 0.0883

1,2D 1L 0.0064 0.0007 1868547 -0.188 1.5517 0.0836

1,2D 0,5W Max 0.0058 9484.671 1705028 2089569 1.4269 0.1035

1,2D 0,5W Min -14550.9 -12646.2 1705028 -1567177 -2497054 0.0403

1,2D 1L 1W Max

0.0065 18969.34 1868547 4179138 1.5926 0.1391

1,2D 1L 1W Min

-29101.9 -25292.5 1868547 -3134354 -4994109 0.0127

1,2D 1L 1E Max 0.0066 0.0009 1868547 5364264 1.6011 0.0868

1,2D 1L 1E Min -24126.8 -24126.8 1868547 -0.2425 -5364258 -0.0165

0,9D 1W Max 0.0045 18969.3 1278771 4179138 1.0957 0.1123

0,9D 1W Min -29101.9 -25292.5 1278771 -3134354 -4994110 -0.0141

0,9D 1E Max 0.0045 0.0007 1278771 5364264 1.1041 0.06

0,9D 1E Min -24126.8 -24126.8 1278771 -0.1824 -5364259 -0.0433

0,9D 1E nl Max 1.61E-05 0 1278447 5364261 -0.9255 4.0619

0,9D 1E nl Min -24126.8 -24126.8 1278447 1.4858 -5364261 3.9774

0,9D 1W nl Max

0.0001 18969.3 1278447 4179137 -0.9059 754.7

0,9D 1W nl Min -29101.9 -25292.4 1278440 -3134350 -4994112 -746.7

1,2D 0,5W nl Max

0.0001 9484.7 1704596 2089570 -1.2334 380.6

1,2D 0,5W nl Min

-14551.0 -12646.2 1704592 -1567174 -2497057 -370.04

1,2D 1,6L nl 1.62E-05 9.05E-06 1966227 2.0489 -1.2738 5.3842

1,2D 1L nl 1.45E-05 8.61E-06 1868115 2.0299 -1.2644 5.3408

1,2D 1L 1E nl Max

2.18E-05 2.25E-06 1868115 5364262 -1.2534 5.4514

1,2D 1L 1E nl Min

-24126.8 -24126.8 1868115 2.017 -5364261 5.367

1,2D 1L 1W nl Max

0.0001 18969.3 1868115 4179137 -1.2338 756.057

1,2D 1L 1W nl Min

-29101.9 -25292.4 1868108 -3134349 -4994112 -745.275

1,4D nl 1.37E-05 9.18E-06 1988695 2.3312 -1.4568 6.1464

1D 0,5L 1Wsls Max

0.0052 13451.9 1502616 2963599 1.2737 0.1064

1D 0,5L 1Wsls Min

-20637.4 -17935.9 1502616 -2222700 -3541529 0.0167

1D 0,5L 1Wsls nl Max

1.09E-05 13451.9 1502256 2963551 -1.0451 244.59


110

1D 0,5L 1Wsls nl Min

-20637.4 -17935.9 1502256 -2222631 -3541390 -235.687

Buckling modes for the Tubed Mega Frame structure

1st mode – X direction

2nd mode – torsion

3rd mode – Y direction

4th mode - torsion


111

Modes of vibration for the Tubed Mega Frame structure


2nd mode – Y direction

3rd mode - torsion

4th mode – X direction


112

5th mode - torsion

6th mode – torsion

Base reactions under load combinations for the X truss structure

Load Case/Combo

FX FY FZ MX MY MZ


1,4D 6.51E-06 0 1996193 0.0003 -0.0016 -3E-05

1,2D 1,6L 6.36E-06 0 1972653 0.0003 -0.0026 -3E-05

1,2D 1L 6.07E-06 0 1874542 0.0003 -0.0021 -2.8E-05

1,2D 0,5W Max 5.46E-06 9484.667 1711023 2089568 -0.0014 0.0002

1,2D 0,5W Min -1.46E+04 -12646.2 1711023 -1567176 -2497056 -0.0002

1,2D 1L 1W Max

5.82E-06 18969.33 1874542 4179135 -0.0022 0.0004

1,2D 1L 1W Min

-2.91E+04 -25292.4 1874542 -3134351 -4994111 -0.0003

1,2D 1L 1E Max 5.63E-06 0 1874542 5239012 -0.0022 0.0003

1,2D 1L 1E Min -2.42E+04 -24205.6 1874542 0.0003 -5239012 -0.0005

0,9D 1W Max 3.94E-06 18969.33 1283267 4179135 -0.0011 0.0004

0,9D 1W Min -2.91E+04 -25292.4 1283267 -3134351 -4994111 -0.0003

0,9D 1E Max 3.75E-06 0 1283267 5239012 -0.0011 0.0003

0,9D 1E Min -2.42E+04 -24205.6 1283267 0.0002 -5239012 -0.0005

0,9D 1E nl Max -8.42E-07 0 1282943 5239003 0.0014 -0.0129

0,9D 1E nl Min -2.42E+04 -24205.6 1282943 -0.0034 -5239012 -0.0134

0,9D 1W nl Max

-6.50E-07 18969.44 1282943 4179132 0.0014 629.6195

0,9D 1W nl Min -2.91E+04 -25292.4 1282943 -3134352 -4994111 -629.646

1,2D 0,5W nl Max

-6.60E-07 9484.719 1710591 2089566 0.0019 314.7988


113

1,2D 0,5W nl Min

-1.46E+04 -12646.2 1710591 -1567176 -2497056 -314.834

1,2D 1,6L nl -6.10E-07 0 1972221 -0.0049 0.0008 -0.0187

1,2D 1L nl -5.82E-07 0 1874110 -0.0048 0.0013 -0.0183

1,2D 1L 1E nl Max

-1.02E-06 0 1874110 5239003 0.0012 -0.018

1,2D 1L 1E nl Min

-2.42E+04 -24205.6 1874110 -0.0047 -5239012 -0.0185

1,2D 1L 1W nl Max

-8.31E-07 18969.44 1874110 4179132 0.0012 629.6144

1,2D 1L 1W nl Min

-2.91E+04 -25292.4 1874110 -3134352 -4994111 -629.651

1,4D nl -6.25E-07 0 1995689 -0.0054 0.0023 -0.0205

1D 0,5L 1Wsls Max

4.72E-06 13451.94 1507612 2963597 -0.0016 0.0003

1D 0,5L 1Wsls Min

-2.06E+04 -17935.9 1507612 -2222698 -3541530 -0.0002


-6.88E-07 13451.94 1507252 2963463 0.0013 333.5098


-2.06E+04 -17935.9 1507252 -2222601 -3541462 -333.54

Buckling modes for the X truss structure

1st mode – Y direction 2nd mode – X direction


114

3rd mode – torsion

4th mode – local buckling

Modes of vibration for the X truss structure

1st mode – Y direction

2nd mode – X direction


115

3rd mode - torsion


5th mode - torsion



116

Base reactions under load combinations for the diagrid structure

Load Case/Combo

FX FY FZ MX MY MZ


1,4D 0.0008 -8.82E-06 1651472 0.0476 0.0546 -0.0062

1,2D 1,6L 0.0008 -9.02E-06 1677179 0.0502 0.0558 -0.0063

1,2D 1L 0.0008 -8.48E-06 1579067 0.0467 0.0524 -0.006

1,2D 0,5W Max 0.0007 9484.67 1415548 2089569 0.0464 -0.0013

1,2D 0,5W Min -14550.95 -12646.23 1415548 -1567176 -2497055 -0.0068

1,2D 1L 1W Max

0.0008 18969.34 1579067 4179137 0.0517 0.002

1,2D 1L 1W Min

-29101.91 -25292.45 1579067 -3134353 -4994111 -0.009

1,2D 1L 1E Max 0.0008 -8.39E-06 1579067 4222778 0.0517 0.0014

1,2D 1L 1E Min -19685.50 -19685.40 1579067 0.0466 -4222777 -0.0091

0,9D 1W Max 0.0005 18969.34 1061661 4179137 0.0343 0.0039

0,9D 1W Min -29101.91 -25292.45 1061661 -3134353 -4994111 -0.007

0,9D 1E Max 0.0005 -5.58E-06 1061661 4222778 0.0344 0.0034

0,9D 1E Min -19685.50 -19685.50 1061661 0.0305 -4222777 -0.0071

0,9D 1E nl Max 5.93E-06 -6.77E-06 1061337 4222771 -1.0623 0.0517

0,9D 1E nl Min -19685.61 -19685.59 1061337 -6.0342 -4222778 0.0501

0,9D 1W nl Max

5.94E-06 18969.48 1061337 4179130 -1.0622 600.6413

0,9D 1W nl Min -29102.13 -25292.57 1061337 -3134358 -4994112 -600.5413

1,2D 0,5W nl Max

8.005E-06 9484.74 1415116 2089560 -1.4164 300.3632

1,2D 0,5W nl Min

-14551.06 -12646.29 1415116 -1567184 -2497057 -300.2281

1,2D 1,6L nl 8.049E-06 -9.32E-06 1676747 -8.0888 -1.5343 0.0716

1,2D 1L nl 8.052E-06 -9.31E-06 1578635 -8.09 -1.4901 0.0703

1,2D 1L 1E nl Max

7.942E-06 -9.11E-06 1578635 4222769 -1.49 0.0709

1,2D 1L 1E nl Min

-19685.61 -19685.59 1578635 -8.0552 -4222779 0.0693

1,2D 1L 1W nl Max

7.95E-06 18969.48 1578635 4179128 -1.49 600.6605

1,2D 1L 1W nl Min

-29102.13 -25292.57 1578635 -3134360 -4994113 -600.5221

1,4D nl 9.399E-06 -1.09E-05 1650968 -9.4408 -1.6526 0.0794

1D 0,5L 1Wsls Max

0.0006 13451.94 1261383 2963599 0.0413 0.0009

1D 0,5L 1Wsls Min

-20637.36 -17935.92 1261383 -2222699 -3541530 -0.0069


6.444E-06 13451.94 1261023 2963457 -1.2172 310.9702


-20637.36 -17935.92 1261023 -2222563 -3541279 -310.8579


117

Buckling modes for the diagrid structure



3rd mode – torsion and local buckling

4th mode – X direction and local buckling


118

Modes of vibration for the diagrid structure



3rd mode - torsion



119

5th mode - torsion


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