Global stability of high-rise buildings on foundation on piles

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STOCKHOLM SVERIGE 2018,

Global stability of high-rise

buildings on foundation on piles

HUSEIN DHORAJIWALA AGNIESZKA OWCZARCZYK

KTH

SKOLAN FÖR ARKITEKTUR OCH SAMHÄLLSBYGGNAD

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Global stability of high-rise buildings on foundation on piles

Authors:

Husein D^HORAJIWALA Agnieszka OWCZARCZYK

Supervisors:

Kent A^RVIDSSON Bert Gunnar N^ORLIN

June 11, 2018

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In Sweden as well as other countries the trend of building higher is more and more popular.

The global stability of tall buildings is a very important aspect that has to be taken into account while designing. Foundation on piles, that is common in Sweden, reduces the global stability compared to foundation directly on bedrock. Using inclined piles in the foundation is inevitable for high-rise buildings, because they are essential for transferring the horizontal loads into the bedrock. The aim of this paper is to look into the influence that foundation on piles has on global stability and investigate two simple methods to asses global stability. In this thesis the influence of the stiffness of the substructure (foundation), length and inclination of the piles on the global stability were investigated. It was also looked into how does the pile center affect the rotation and thus global stability. One method that was presented was based on the equivalent stiffness. Displacement at the top of the wall is used in order to calculate the bending stiffness that is reduced due to foundation on piles and further calculate buckling load on the basis of Euler buckling. In the other method that was proposed rotation at the foundation level was taken into account so as to calculate rotational spring stiffness and later buckling load due to combined flexural and rotational buckling.

The analysis was conducted on a simple two dimensional problem, namely stabilizing wall as well as a building stabilized by two towers. Three different configurations of piles were investigated for single wall as well as for the structure.

The investigation showed that the position of pile center has its effect on the global stability.

The closer the pile center is to the foundation on piles the better the global stability of a structure. The length of the piles plays a role in stability as well. The longer the piles are the worse the stability is. With longer piles the overall stiffness of a structure decreases and thus the global stability. The analysis showed that the foundation of piles significantly lowers the stability of high rise building. The investigated methods showed that the one based on rotation at the base gave better results compared to the method based on the equivalent bending stiffness.

But to use this first method, the position of the pile center is required to be known in order to get correct results which in a complex structure is hard to estimate. In an analysis of a building stabilized by two towers it was seen that when the inclined piles that are inclined opposite to each other in a pile group and are positioned far from rotation center of a structure it increases the global safety and rotational stiffness as well. It is recommended to use such configuration of piles that the pile center is at the foundation level in order to increase global stability.

Key words: High-rise buildings, Tall buildings, Buckling, Global Stability, Foundation, Straight piles, Inclined piles;

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I Sverige och andra länder är trenden att bygga högre alltmer populärt. Den globala stabiliteten hos höga byggnader är en viktig aspekt som måste beaktas vid byggnadens utformning.

Den vanligaste grundläggningsmetoden i Sverige är grundläggning med pålar. Denna typ av grundläggning minskar den globala stabiliteten jämfört med grundläggning direkt på berggrunden. Att använda sneda pålar i grundläggningen är oundviklig för höghus, eftersom de är nödvändiga för att överföra horisontella laster till berggrunden. Syftet med detta examensarbete är att se hur grundläggning på pålar påverkar den globala stabiliteten och undersöka två enkla metoder för global stabilitet. I detta examensarbete undersöktes hur styvheten på- verkar grundläggning med pålar med olika längder och lutningar, med hänseende på den globala stabiliteten. Pålcentrumets påverkan av rotation och den globala stabiliteten har även studerats. En metod som presenterades i examensarbetet är baserades på ekvivalent styvhet.

Där utböjning på toppen av väggen togs för att beräkna böjstyvheten som reduceras på grund av grundläggning med pålar och ytterligare beräknades knäcklasten baserat på Eulers knäck- ning. I den andra metoden som föreslogs togs rotationen vid grundläggningsnivån med i be- räkningen för att beräkna rotationsfjäderns styvhet och senare knäckningslasten på grund av kombinerad böjnings- och rotationsknäckning.

Analysen genomfördes på en enkel tvådimensionell vägg och en tredimensionell byggnad som är stabiliserad av två torn. Tre olika konfigurationer av pålar undersöktes för enkel vägg och även för byggnaden.

Utredningen av examensarbetet visade att positionen av pålcentrum har en stor påverkan på den globala stabiliteten. Ju närmare pålcentrumet är till grundläggningsnivån desto bättre är den globala stabiliteten hos en konstruktion. Längden på pålarna har även en betydelse när det gäller stabiliteten. Ju längre pålarna är desto värre blir stabiliteten. Med längre pålar minskar den totala styvheten hos hela konstruktionen och därmed minskar även den globala stabiliteten. Utredningen visade även att metoden med rotation vid grundläggningsnivån gav mer noggrannare resultat än metoden för ekvivalent styvhet. Men för att kunna använda den först- nämnda metoden behöver man ha kännedom om vart pålcentrum ligger för konstruktionen och detta kan vara svårt att uppskatta. I en analys av en byggnad stabiliserad av två torn visa- des det att när pålarna är placerade långt från rotationscentrum av en konstruktion ökar den globala säkerheten och rotationsstyvheten. Det rekommenderas att använda sådan konfigura- tion av pålar att pålcentrum ligger på grundnivå för att öka den globala stabiliteten.

Nyckelord: Höga hus, flervåningshus, Knäckning, Global stabilitet, Grundläggning, Raka pålar, Sneda pålar;

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This paper was written as a termination of Master program Civil and Architectural Engineer- ing at KTH Royal Institute of Technology. It was a part of Degree Project in Structural En- gineering and Bridges that was conducted for four months at WSP at Structural Engineering department in Stockholm.

We are grateful to our supervisor at WSP Tech.Dr Kent Arvidsson for providing us with this interesting subject, huge engagement and guidelines throughout the work.

We would like to thank WSP for giving us opportunity to write our thesis in a nice atmosphere and providing us with necessary equipment.

We would like to thank our supervisor at KTH Tech.Dr Bert Gunnar Norlin for his guidance and always making it easier for us to look at the complex problems in a simplified way.

Stockholm June 2018

Husein Dhorajiwala & Agnieszka Owczarczyk

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1 Introduction 1

1.1 Background . . . . 1

1.2 Aim and scope . . . . 1

1.3 Limitations . . . . 2

1.4 Outline . . . . 2

2 High-rise buildings 3 2.1 Loading . . . . 3

2.2 Stabilization . . . . 3

2.2.1 Principles of situation of stabilizing units . . . . 4

2.3 Global stability . . . . 6

2.3.1 Shear buckling . . . . 6

2.3.2 Flexural buckling . . . . 8

2.3.2.1 Euler Buckling . . . . 8

2.3.2.2 Flexural buckling for multi-storey structures . . . . 9

k₁-value for distributed line load . . . 11

2.3.3 Buckling due to rotation at the foundation . . . 12

2.3.3.1 Critical load due to point load . . . 12

2.3.3.2 Critical load due to vertical distributed loading. . . 13

2.3.3.3 Critical load due to multiple point loads . . . 14

2.3.4 Combined buckling . . . 14

2.3.4.1 Flexural and rotation buckling . . . 15

2.3.5 Stability in Eurocode 2 . . . 15

2.3.6 Torsion . . . 16

2.3.7 Flexural-torsional buckling . . . 17

2.3.7.1 Characteristic moment of inertia c₁ . . . 18

Example: Double symmetric floor plan . . . 19

Example: Single symmetric floor plan . . . 20

2.3.8 The global safety factor . . . 21

3 Foundation 23 3.1 End bearing piles . . . 23

3.2 Slender straight and inclined piles . . . 24

Example . . . 25

3.3 Warping resistance of piles . . . 28

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4.1 Parametric study on wall . . . 29

4.1.1 Influence of pile center . . . 32

4.1.2 Influence of stiffness of the piles . . . 32

4.1.2.1 Vertical stability . . . 32

4.1.2.2 Top deflection . . . 33

4.1.3 Method evaluation . . . 33

Equivalent stiffness method (ESM) . . . 33

Combined flexural and rotational buckling (F&R) . . . 33

4.1.4 Utilization of the piles . . . 34

4.2 Structure stabilised by two towers . . . 34

5 Results 38 5.1 Parametric study on wall . . . 38

5.1.1 Influence of pile center . . . 38

5.1.2.1 Vertical stability . . . 43

Case 1 . . . 43

Case 2 . . . 45

Case 3 . . . 47

Comparison between Case 1, 2 and 3 . . . 48

5.1.2.2 Top deflection . . . 50

Case 1 . . . 51

Case 2 . . . 52

Case 3 . . . 53

Equivalent stiffness method (ESM) . . . 55

Combined flexural and rotational buckling (F&R) . . . 57

5.2 Analysis of a structure stabilised by two towers . . . 59

6 Discussion and Conclusion 62 6.1 Discussion . . . 62

6.1.1 Infuence of pile center . . . 62

6.1.5 Structure stabilised by two towers . . . 64

6.2 Conclusion . . . 64

6.3 Further research . . . 65

References 66

Appendix A Appendix B Appendix C Appendix D

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Appendix E Appendix F Appendix G Appendix H Appendix I Appendix J Appendix K

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2.2.1 Examples of stabilizing systems that are unstable. (Johann Eisele (2003) and

Sundquist (2016)) . . . . 5

2.2.2 Examples of stabilizing systems that are stable. (Sundquist 2016) . . . . 5

2.3.1 a) bending b) shear deformation d) both bending and shear deformation. . . . . 6

2.3.2 A simple model for pure shear. . . . 7

2.3.3 A column subjected to shear deformation. . . . 7

2.3.4 Point loads acting on the structure, a) indirectly through hinged connected slabs, b) directly to the shear wall/tower. Remade from Lorentsen (1985) . . . . 9

2.3.5 Coefficient for multi-storey structures k₁for buckling. Remade from Lorentsen (1985) . . . 10

2.3.6 Distributed point loads on a cantilever with a fixed base. Remade from Algers et al. (1961) . . . 11

2.3.7 Cantilever with elastic support subjected to point load . . . 12

2.3.8 Cantilever with elastic support subjected to vertical distributed loading. . . 13

2.3.9 Cantilever with elastic support subjected multiple point loads. Remade from Sundquist (2016) . . . 14

2.3.10 Torsion. a) warping torsion b) pure torsion. . . 17

2.3.11 Displacement u and v and rotation ϕ of a floor plan. Remade from Lorentsen (1985). . . 17

2.3.12 Dimensions and distances of a floor plan. . . 18

2.3.13 Primary bending stiffness EI of shear walls in a floor plan. . . 19

2.3.14 Double symmetric floor plan with two stabilizing towers. Remade from Sundquist (2016). . . 20

2.3.15 Single symmetric floor plan with three stabilizing shear walls. Remade from Sundquist (2016). . . 21

3.1.1 Axial load F and resistance R of end bearing pile. Remade from Pålkommissio- nen (2007) . . . 23

3.2.1 Two cases with piles to take into account. Small horizontal force and vertical force acting in the piles a) and large horizontal force acting on the piles b). Remade from Svahn & Alén (2006) . . . 25

3.2.2 FE model for straight and inclined piles with a horizontal force. . . 25

3.2.3 FE models with spring and without spring supports. . . 27

3.3.1 Warping resistance of inclined piles on a foundation slab in top view. . . 28

4.1.1 2d wall model supported by piles . . . 29

4.1.2 Pile configuration for a) Case 1 b) Case 2 c) Case 3. . . 30

4.1.3 Pile configuration for a) Case 1 b) Case 2 c) Case 3 with dimension in meters. . . 30

4.1.4 a) vertical distributed line-load b) horizontal distributed load of the wall. . . 31

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4.1.5 Pile configuration and lengths of the wall. . . 32

4.2.1 Floor plan of the 3D model. Dimensions in meters. . . 34

4.2.2 a) equally distributed live load applied on the slab, b) equally distributed line load that represents wind applied on the edge of the slabs and c) point loads at the two opposite corner-edges of the slab of every storey representing torsional loads. . . 35

4.2.3 Reaction forces in kN in z-axis for piles. . . 36

4.2.4 Reaction forces in kN in z-axis for piles. . . 37

5.1.1 Global buckling for different ρ-values for Case 1. . . 42

5.1.2 Influence of stiffness of the piles to global safety for 90 meter wall for Case 1. (Legend: Inclination of the piles and their length) . . . 44

5.1.3 Influence of of stiffness of the piles to global safety for 45 meter wall for Case 1. (Legend: Inclination of the piles and their length). . . 45

5.1.4 Influence of of stiffness of the piles to global safety for 90 meter wall for Case 2 with 10 meter long piles (Legend: Inclination of the piles). . . 46

5.1.5 Influence of of stiffness of the piles to global safety for 45 meter wall for Case 2 with 10 meter long piles (Legend: Inclination of the piles). . . 46

5.1.6 Influence of stiffness of the piles to global safety for 90 meter wall for Case 3 with 10 meter long piles (Legend: Inclination of the piles). . . 47

5.1.7 Influence of stiffness of the piles to global safety for 45 meter wall with Case 3 for 10 meter long piles (Legend: Inclination of the piles). . . 47

5.1.8 Influence of stiffness of the piles to global safety for 90 meter wall with 3:1 inclination of piles for Case 1, 2 and 3 with 10 meter long piles. . . 48

5.1.12 Influence of stiffness of the piles to overall stiffness for 90 meter wall for Case 1. (Legend: Inclination of the piles and their length) . . . 51

5.1.13 Influence of stiffness of the piles to overall stiffness for 45 meter wall for Case 1. (Legend: Inclination of the piles and their length) . . . 51

5.1.14 Influence of stiffness of the piles to overall stiffness for 90 meter wall for Case 2 for 10 meter long piles (Legend: Inclination of the piles). . . 52

5.1.18 Compressive section forces for all three cases. . . 59

5.2.1 Torsional shape of the structure. The red lines are the original positions of the structure. . . 60

5.2.2 Displacement (dashed lines) of piles, tower and foundation slab for pile configuration 2 a) and 3 b). . . 61

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2.3.1 Euler’s four buckling cases. . . . 8

3.2.1 Horizontal reaction forces for straight and inclined piles with varying length and design shear strength. . . 26

3.2.2 Ratio between reaction forces for straight and inclined piles. . . 26

5.1.1 Global buckling shapes for different ρ-values. . . 39

5.1.2 Ratio between S_crfrom analytical methods and FEM for different ρ-values. . . . 42

5.1.3 Global safety factor for pile lengths of 10 meters S_cr,10and 20 meters S_cr,20from FEM. . . 43

5.1.4 Decrease in critical loading for different bending stiffness of pile group for Case 1 with 10 meter long piles. ( α-value is a ratio between the bending stiffness of the foundation to bending stiffness of a wall) . . . 44

5.1.5 Safety factors for Case 1, 2,3 and wall founded on bedrock for 90 and 45 meter wall for the same pile dimensions. . . 50

5.1.6 Deflections [mm] at the top for ase 1, 2,3 and wall founded on bedrock for 90 and 45 meter wall for the same pile dimensions due to horizontal distributed loading. . . 50

5.1.7 Ratio of ScrESM to FEM for Case 1 with pile inclination 3:1. . . 55

5.1.8 Ratio of S_crESM to FEM for Case 1 with pile inclination 4:1. . . 55

5.1.10 Ratio of ScrESM to FEM for case 2 with pile inclination 4:1. . . 56

5.1.13 Ratio of ScrF&R to FEM for Case 1 with pile inclination 3:1. . . 57

5.1.14 Ratio of S_crF&R to FEM for Case 1 with pile inclination 4:1. . . 57

5.1.15 Ratio of S_crF&R to FEM for Case 2 with pile inclination 3:1. . . 57

5.1.18 Ratio of S_crF&R to FEM for case 3 with pile inclination 4:1. . . 58

5.2.1 Different outputs from FEM for different cases. All piles are the same in the pile group. . . 60

5.2.2 Ratio of S_crfor analytical methods to FEM. . . 61

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α Ratio between the bending stiffness of the foundation to bending stiffness of a wall β Effective length factor for Euler’s Buckling

δ Virtual vertical displacement γ Shear strain

φ Angle of twist

ρ Relation between pile center and total length of the building τ Shear stress

ϕ Rotation angle at the base A_s Area of the slab

c₁ Characteristic moment of inertia C_ud Design shear strength of the soil e Eccentricity

G Shear modulus

I_p Polar moment of interia

I_x Moment of inertia about x-axis I_y Moment of inertia about y-axis I_{e f f} Effective moment of inertia

k Relative flexibility of the restrained moment at the base of the structure k₁ Coefficient for multi-storey structures for buckling

k_d Modulus of subgrade for soil K_v Torsional constant

K_w Warping constant K_ϕ Rotational stiffness Lcr Critical length

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Mϕ Bending moment due to rotation at the base of the structure p Vertical distributed loading

P_cr,ϕ Critical load due to rotation P_cr,b Critical load due to bending P_cr,s Critical load due to shear Pcr Total critical load

Q Vertical load applied on the structure R_s Section forces

T_t Pure torsion T_w Warping torsion V Shear force

w Width of the cross-section of the pile W_e Internal work

W_i Internal work

x_T Distance between gravity center GC and rotation center RC in x-axis yT Distance between gravity center GC and rotation center RC in y-axis λ Rotation at the base of the structure due to a moment equal to one

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Introduction

1.1 Background

Today more than half of the world’s population lives in urban cities and that trend is growing every year. When new dwellers need places to live and work, the cities are expanding out- wards and upwards. Building high-rise structures decreases the growth rate of cities when it comes to the city’s area, as well as shortens distance for people to travel from e.g. work to home. High prices of land also forces developers to build higher and higher. Nowadays in Sweden as well as other countries there is a trend to build taller buildings that are around 10 to 30 storeys high.

It is very important to design foundation, so that it has sufficient capacity to transfer loads into the bedrock and provide stability. In Sweden foundation on piles or directly on bedrock are common. When the foundation is not properly designed then it can lead to a catastrophe.

In high-rise buildings external vertical forces can be transferred to the straight piles. However, only straight piles are not sufficient when horizontal forces in form of e.g. wind occur. In- stead, horizontal forces can be transferred through inclined piles. Soil could also account for horizontal support, but in Sweden there is mostly clay which does not give much of horizontal support, that is why it is essential to have inclined piles when wind forces are significant.

Therefore, it is of vital importance to have a good knowledge about the behaviour of straight and inclined piles in order to design high-rise buildings.

Stability of a structure will significantly decrease if founded on piles compared to foundation directly on bedrock. Therefore, it is essential for engineer to understand buckling phenomena for high-rise buildings for different foundation types.

1.2 Aim and scope

The purpose of this thesis is to investigate the influence of foundation on piles on the global stability of a superstructure as well as to propose and investigate simple analytical methods to asses global buckling. Those analytical methods will be compared with Finite Element (FE) simple two dimensional and three dimensional models.

During the research the following questions will be answered:

• How does the stability of structure change for different ratio between stiffness of superstructure and substructure?

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• How does the length of the piles and height of the building influence global stability?

• How does the position of the pile center affect the rotation and thus global stability?

To limit the scope of thesis, a constant modulus of elasticity of concrete is going to be chosen for every storey and stabilizing structures in the building.

1.3 Limitations

The research in this thesis is based only on linear analysis, e.g cracking of concrete is not considered. Second order effects were not taken into account for stability analysis. Local buckling of members was not considered, only global buckling of the whole structure. The analysis was conducted only for concrete structures and one type of stabilizing system, namely shear wall structure. The piles that were taken into account in the analysis were end-bearing piles with constant stiffness of piles, so it is both valid for steel and concrete. It was not considereted that the piles have increasing moment of inertia I the closer it gets to the bedrock, instead the lowest distance to the center line was chosen to underestimate the bending stiffness of a inclined pile.

1.4 Outline

In the first part of this paper a theory part is presented which is essential to understand the problem of global stability related to foundation on piles. The next following chapters cover method and results for parametric study on the wall and simple analysis for a structure stabilized by two towers with symmetric plan. Then, the results are followed by discussion and conclusions.

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High-rise buildings

There is no specific definition of high-rise building or tall building, i.e it is subjective. For example a structure that is not significantly high, but because of its proportion is slender may be considered as a tall building. In European cities a building that is significantly higher (for instance 14 storeys) than the surrounding buildings may be seen as tall building, but a building with the same height situated in other metropolis such as New York, Chicago or Hong Kong compered to surrounding structures might be considered low-rise. (on Tall Buildings &

CTBUH)

According to Smith & Coull (1991), a building can be classified as high rise when the impact of lateral forces, i.e. wind on a structure, is significant.

2.1 Loading

High rise buildings have to be designed to withstand the most severe possible loads acting on the structure during their life as well as construction time. Loads that act on a tall structure will differ from the ones acting on a low structures mostly by lateral loading, e.g. wind load.

Wind load is not only significant for high structures because it acts on a larger surfaces, but its magnitude is greater in higher latitudes (Smith & Coull 1991). Wind load can give rise to big deflections as well as sway and its magnitude depends on rigidity, slenderness and height of the structure. The design of the structures is not much influenced by wind forces for the typical structures that are up to 10 storeys (Smith & Coull 1991). The magnitude of wind forces will depend not only on the size and shape, but also on the location of the building and should be assessed with the help of design manuals. Vertical loads that act on a high-rise building are self-weight, live load and snow load.

2.2 Stabilization

There are different possible structural system of high - rise buildings, such as shear walls, coupled shear walls, frames, trusses and many more. When choosing appropriate structural form a designer must consider different aspects, such as:

• type of loading acting on the structure

• material - concrete, steel, wood or combinations

• method of construction

• dimensions of the building

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• purpose of the building - office or residential which will differ regarding the floor plan In Sweden it is frequent that shear walls and columns are stabilizing components of high-rise buildings (Sundquist (2016)). In this thesis only this structural system are considered.

Beams, slabs, columns and walls are load-bearing elements that have to transfer the load to the foundation. Columns and walls transfer vertical loads, whereas slabs and beams are respon- sible of taking horizontal forces and further redistribute them to walls and columns. Columns are not used as stabilizing components in high-rise building only to provide support for beams and slabs.

The function of shear walls is to provide resistance against horizontal loading. Stabilization with shear walls and towers is a good solution for residential buildings, because the floor plan is the same on every storey and the stabilizing walls can be continuous on the whole height and can act as a cantilever. The location of the bearing walls is not problematic, because the walls can function as separator of apartments providing acoustic insulation. As well as be situated at the position of elevator, stairways or installations shaft. When choosing this way of stabilizing, it is of vital importance that the floors act as rigid plates or beams, because they have to transfer horizontal forces to the walls. But normally the floors are concrete floors which are thick. Stabilization with shear walls is suitable for buildings up to 35 storyes(Smith & Coull (1991)). The shear wall sections are not only planar, but can have different shapes such as U, H, L, T or a tower.

2.2.1 Principles of situation of stabilizing units

Stabilization with one stabilizing unit, for example tower is not appropriate, because the structure might be susceptible to torsional buckling. One tower can be sufficient if it is a central core with significant size compared to the size of the floor plan. The principles of stabilization require at least two towers, or one tower with a wall. If only walls are used, at least three have to be used and cannot be placed on the same extension line. And additionally two of walls have to be placed perpendicular to the longer side of the building. Examples of stabilizing systems that are unstable are shown on figure 2.2.1 and stable on figure 2.2.2.

If the plan of a structure is symmetrical in the direction of loading, except for eccentric wind load, the structure will not suffer from twisting. When a plan of structure is asymmetric in the direction of loading the structure will twist and translate around rotation center.

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Figure 2.2.1: Examples of stabilizing systems that are unstable. (Johann Eisele (2003) and Sundquist (2016))

Figure 2.2.2: Examples of stabilizing systems that are stable. (Sundquist 2016)

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2.3 Global stability

Stability refers to resistance of a structure or member to buckling. The critical load determines the limit for the total vertical load a structure or a member can withstand. It is not only of vital importance to check the local stability of separate elements constituting the structure, but also to check the global stability of a structure as one. In this paper only the global stability for tall buildings is covered. Behaviour of high rise buildings as a whole, can be seen as cantilever column that can undergo bending deformation or shear deformation or combination of these two (Smith & Coull (1991)), see figure 2.3.1. For slender structures (tall buildings) shear deformation is of a less importance compared to low-rise buildings, because bending deformation is dominating for tall structures. Transverse, as well as torsional buckling might include both bending and shear contribution. Rotation at the foundation as well contributes to global instability.

Figure 2.3.1: a) bending b) shear deformation d) both bending and shear deformation.

2.3.1 Shear buckling

In order to check the materials response due to shear, the shear modulus G has to be defined.

According to EN (2004) the Poisson’s ratio v for an uncracked concrete is equal to 0.2 and the modulus of elasticity E depends on the concrete class. Then shear modulus becomes:

G= ^E

2(1+v) =0.42E≈0.4E (2.3.1)

A simple model in figure 2.3.2 presents pure shear. Shear forces V cause shear stress τ. Then the deformation angle that is called shear strain γ can be expressed as:

γ= ^τ

G (2.3.2)

The shear stress τ with shear force V and a constant area A can be expressed as:

τ= ^V

A (2.3.3)

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By inserting eq. 2.3.3 into eq. 2.3.2, the following expression for shear strain γ can be written as:

γ= ^V

GA (2.3.4)

Figure 2.3.2: A simple model for pure shear.

A column subjected to fictitious shear force V, having a constant cross sectional area A, length L will cause deflection y at the top, see figure 2.3.3.

Figure 2.3.3: A column subjected to shear deformation.

The fictitious force V causes a deflection y at the top of the column by:

y=γL (2.3.5)

Bending moment at the fixed support due to an axial force P_cr,sis:

M_P = Pcr,sy (2.3.6)

By inserting eq. 2.3.4 and eq. 2.3.5 into eq. 2.3.6 gives:

MP = ^V

GA LPcr,s (2.3.7)

A bending moment at the fixed support caused by the force V can be expressed as:

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M =VL (2.3.8) by setting equations equal(2.3.7) = (2.3.8)the critical load due to shear can be derived:

P_cr,s=GA (2.3.9)

2.3.2 Flexural buckling 2.3.2.1 Euler Buckling

Euler’s buckling method can be used in order to calculate the critical buckling load P_cr for a column that have constant bending stiffness EI. The four general Euler’s buckling cases are presented in table 2.3.1. (Sundquist 2016)

Table 2.3.1: Euler’s four buckling cases.

Case 1 Case 2 Case 3 Case 4

Fixed at the base and

Pinned at both ends Fixed at the base and

Fixed at both ends

free at the top pinned at the top

β=2.0 β=1.0 β=0.7 β=0.5

With help of Euler’s second buckling case (see table 2.3.1) a general equation can be derived for critical buckling load for which the critical length of the column can be written as L_cr= βL.

The β-value is the effective length factor that depends on a certain buckling case, see table 2.3.1 and the derivation of the buckling load can be seen in Sundquist (2016). The general equation for Euler’s buckling for critical load can be expressed as:

P_cr= ^π

2EI L²_cr = ^π

2EI

(βL)² ^(2.3.10)

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2.3.2.2 Flexural buckling for multi-storey structures

In the multi-storey shear wall structures, the vertical load from dead load and live load can be taken by columns or directly by stabilizing elements, see figure see figure 2.3.4 a) and b) respectively. Lorentsen (1985) have shown that the moment distribution is different for these two cases when number of storeys is below six, but when it is above six then the moment distribution becomes equal. (Lorentsen 1985)

If a structure have equal bending stiffness EI, equally distributed point loads and have fixed support at the base, the critical load for flexural buckling P_cr,bcan be calculated according to:

P_cr,b=k₁EI

L²₁ (2.3.11)

Approximate values for the coefficient for multi-storey structure k₁can be taken from the graph in figure 2.3.5. More accurate values can be obtained from Vianello’s method. The conditions of using the diagram in figure 2.3.5 for k₁-value is that:

• the structure has a constant bending stiffness EI over its entire length

• structure is fixed at the foundation

• all slabs take equal load

• all storeys have equal height

• columns are hinged to the slabs in order to function as stiff slabs

In figure 2.3.5 the coefficient k₁is depended on the load case (if the load is acting on columns or stabilizing element) and number of storeys n. If the number of storeys n goes to infinity, then the coefficient for multi-storey structure k₁is (Lorentsen 1985):

k₁ =7.83 (2.3.12)

Figure 2.3.4: Point loads acting on the structure, a) indirectly through hinged connected slabs, b) directly to the shear wall/tower. Remade from Lorentsen (1985)

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When the number of storeys are above four then the difference between k₁-value for between these cases is very small, see figure 2.3.5. For structures with four storeys and below, it is important to distinguish between load cases and take k₁-value form appropriate curve. Linear interpolation between the curves in figure 2.3.5 can be performed if the load is acting on both columns (case A) and the stabilizing element (case B). (Lorentsen 1985)

For more complex structure shear walls and towers will have different configurations on the floor plan. This will cause that the structure is not only susceptible to flexural buckling, but torsional as well and will influence the characteristic moment of inertia c₁ that is described more thoroughly in section 2.3.7.1. Then the critical load Pcrcan be expressed as:

P_cr,b=k₁E c₁

L²₁ (2.3.13)

Figure 2.3.5: Coefficient for multi-storey structures k₁ for buckling. Remade from Lorentsen (1985)

See section 2.3.5 to see how EN 1992-1-1 interpretate global stability and the coefficient k₁.

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k₁-value for distributed line load

The k₁-value can be calculated with the help of ready equation for effective length factor β in Algers et al. (1961) in chapter 157:352 for a cantilever with fixed base, see figure 2.3.6.

Figure 2.3.6: Distributed point loads on a cantilever with a fixed base. Remade from Algers et al. (1961)

The conditions for this example is that the cantilever have a constant moment of interia and is applied with constant vertical point loads. The β-value can be calculated as:

β= v u u u t

1+2.176 P0

P₁

0.794 (2.3.14)

Then the coefficient for multi-storey structures k₁is:

k₁ =

π β

2

(2.3.15) For more cases to calculate the β-value for different structures, see Algers et al. (1961).

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2.3.3 Buckling due to rotation at the foundation

When the structure is founded directly on bedrock then it can be considered as a fully clamped cantilever structure. However, when the building is situated on the soil then the elastic restraint support should be assumed in order to account for rotation of the foundation. Nonethe- less, this is only the approximation for the reason that the data about soil properties are never certain and the interaction between the foundation and the soil is hardly ever elastic. (Sundquist 2016). The effective stiffness of the structure due to elastic clamping is decreased and thus global stability is lowered as well (Smith & Coull (1991). As a consequence of rotation at the base the deflection due to horizontal loading will become greater.

If the structure is supported on foundation on piles, both straight and inclined piles, it will rotate with respect to its rotation center with a certain angle due to loading. The configuration as well as degree of inclination will have impact on the degree of rotation at the support.

The global stability of a structure founded on piles will significantly decrease compared to the same structure founded on bedrock.

Elastic restraint can be described as rotational spring at the base with a rotational stiffness K_ϕ. Critical loading for the structure that is founded on elastic ground or on piles is derived below.

It is assumed that bending stiffness (EI) as well as torsional stiffness (GA) is infinite, i.e. the structure displaces only due to the rotation at the base. For two cases the derivation will be shown, for the structure that is subjected to the point load at the top, see figure 2.3.7 (derived in Gambhir (2007)) and the structure that is subjected to vertical distributed loading p, see figure 2.3.8. The case when the structure is subjected to multiple loads (see figure 2.3.9) will also be presented.

2.3.3.1 Critical load due to point load

The structure shown in figure 2.3.7 rotates with angle ϕ and displace vertically with∆ distance.

External work of the system is then:

W_e=P∆ (2.3.16)

Figure 2.3.7: Cantilever with elastic support subjected to point load

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The displacement was found form the geometry as:

∆= L(1−cos ϕ) By using only first two terms of Taylor series of cos ϕ then:

∆= L(1−cos ϕ) =L

1−

1− ^ϕ

2

∼= L ϕ²

2 (2.3.17)

If the moment at the base is M=K_ϕ ϕthen the internal work of the rotational spring is:

W_i = ¹

2 M_ϕ ϕ= ¹

2 K_ϕ ϕ² (2.3.18)

From the condition that the external work (done by external load) is equal to the internal work (done by internal forces) then:

PL ϕ² 2 = ¹

2 K_ϕ ϕ² (2.3.19)

Giving:

P_cr,ϕ = ^K^ϕ

L (2.3.20)

2.3.3.2 Critical load due to vertical distributed loading.

Figure 2.3.8: Cantilever with elastic support subjected to vertical distributed loading.

In the case when vertical distributed load is acting along the length of the cantilever (see figure 2.3.8) the resultant force P = pl will be acting at the half of the height. This case differs from the one above by the external work:

We= P ∆

2 (2.3.21)

Thus:

PL ϕ² 4 = ¹

2 K_ϕ ϕ² (2.3.22)

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Giving:

Pcr,ϕ= ^2K^ϕ

L (2.3.23)

2.3.3.3 Critical load due to multiple point loads

The equation for critical loading for the cantilever shown in figure 2.3.9 was derived in Sundquist (2016). In this case n is a number of point loads and l stands for distance between each loading.

The total length of the cantilever is L= nl.

Figure 2.3.9: Cantilever with elastic support subjected multiple point loads. Remade from Sundquist (2016)

The critical loading for the single load is:

N_cr= ²

n(n+1)lλ (2.3.24)

The total critical load for the system is then:

P_cr,ϕ=∑^N^cr⁼ ₍_n₊²₁₎_lλ ⁼ ₍_n₊²ⁿ₁₎_Lλ ⁼ ₍_n²ⁿ₊₁₎ ^K_L^ϕ ^(2.3.25)

For a infinite number of point loads eq. 2.3.25 becomes:

P_cr,ϕ = ²ⁿ (n+1)

Kϕ

L ≈ ^2K^ϕ L

Which yields the same equation as (2.3.23) for distributed loading.

2.3.4 Combined buckling

According to Algers et al. (1961) a structure that undergoes both bending and shear deformation can be expressed as:

Pcr= ^P^cr,b

1+λP_cr,b (2.3.26)

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The shear strain γ can be written as:

γ= ¹

P_cr,s (2.3.27)

By inserting eq. 2.3.27 into eq. 2.3.26 it becomes Pcr = ^P^cr,b

1+ ^P^cr,b Pcr,s

(2.3.28)

In a simplified form of eq. 2.3.28 it can be seen that there is both bending and shear contribution to buckling, see section 2.3.2 and section 2.3.1 respectively.

1

P_cr = ¹

P_cr,b+ ¹

P_cr,s (2.3.29)

If elastic rotation at he base of a structure is present, then additional term Pcr,ϕcan be added into eq. 2.3.29:

1 Pcr

= ¹

P_cr,b + ¹ Pcr,s

+ ¹

Pcr,ϕ (2.3.30)

2.3.4.1 Flexural and rotation buckling

In Carlsson (1969) it is suggested that if the structure rotates at the base, the moment of inertia I in equation for critical loading due to bending (eq. 2.3.11) should be reduced by the following factor:

χ=1+ ¹+n 2n

k₁EI λ

L (2.3.31)

The eq. 2.3.31 was derived from combined flexural and rotation buckling. In the equation n stands for the number of storeys in the building.

Where λ is a rotation at the base of the structure due to a moment equal to 1 and Kϕ is the rotational stiffness at the base, then:

λ= ^M Kϕ

= ¹ Kϕ

(2.3.32) The rotational stiffness Kϕ can be written as below, for which the rotation angle at the base is ϕcasue by applied moment M.

K_ϕ = ^M

ϕ (2.3.33)

2.3.5 Stability in Eurocode 2

The critical flexural buckling load P_cr,baccording to EN (2004) can be calculated as:

P_cr,b=k₁ ∑ EI

L² (2.3.34)

Where the ∑ EI is the summation of the bending stiffness of all stabilizing elements in the structure in a specific direction and L is the length of the structure. The coefficient k1is based

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on that the structural elements have a constant bending stiffness on the total height of the structure. The vertical load in a single storey needs to have the same load pattern on every storey on the structure. Then k₁is:

k₁=7.8 n n+1.6

1

1+0.7k (2.3.35)

Where n is the number of storeys in structure and k is the relative flexibility of the restrained moment at the base of the structure can be defined as:

k = ^ϕ M

EI

L (2.3.36)

Second order effect according to EN (2004) can be neglected if:

Q=_0.1P_cr,b _(2.3.37)

Where Q is the vertical applied load on the stabilizing elements of the structure. The condition for neglecting the second order effect is that the structural system does not have significant shear deformations. Therefore, the structural elements is without any openings.

Where the ϕ is the rotation angle at the base for a bending moment M. If the base of the structure is rigidly restrained, then k=0. See Annex H in EN (2004) when considering second order effects.

2.3.6 Torsion

The phenomenon of torsion occurs when a beam or structure twist. When the axial external force is applied in a point other than center of twist CT of the cross section then the structure or a beam will be subjected to torque or torsional moment and will twist. (Norlin (2015)). There- fore, the torsional stiffness is an important factor for stabilizing elements of a building to resist twisting. The torsional behaviour of a building depends on height, length and thickness of stabilizing elements as well as their configuration on the plan. (Smith & Coull 1991)

Torsional moment can be taken up in two ways, by plane torsion (Saint Venant torsion) or warping torsion (Vlasov torsion). The structure can be subjected solely to pure torsion, warping torsion or mixed torsion (both pure and warping). Circular cross sections will carry torque through pure torsion. Pure torsion occurs when a plane section remains plane and rotates without out-of-plane displacement , see figure 2.3.10 b). For any other sections that are non circular, e.g. rectangular, the warping will occur. In that case plane sections will not remain plane after twisting, , see figure 2.3.10 a). Torsion can be expressed as:

T =Tt+Tw=GKvφ⁰−EKwφ⁰⁰⁰ (2.3.38) Where Tt is a pure torsion, Twis a warping torsion, φ is angle of twist, Kvis torsional constant and Kwis warping constant.

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Figure 2.3.10: Torsion. a) warping torsion b) pure torsion.

2.3.7 Flexural-torsional buckling

The flexural-torsional buckling occurs when the floor plan of the building have combination of displacements u, v and rotation ϕ, due to applied axial forces, see figure 2.3.11. The slabs are assumed to be stiff and the displacements can occur in x- and y-direction. Double symmetric floor plans where the gravity center GC and rotation center RC is in the same position will undergo buckling in flexural or torsional modes and buckling will occur for the lowest of these modes. Single symmetric and asymmetric floor plans does not have gravity center GC and rotation center RC in the same position and will buckle in flexural-torsional mode. (Trahair 1993)

Figure 2.3.11: Displacement u and v and rotation ϕ of a floor plan. Remade from Lorentsen (1985).

A cantilever column at the fixed base is restrained against bending and torsional modes, but the free top of the column does not have any restraints. Therefore, a cantilever column’s resistance depends on the its flexural and torsional modes at the free top. If a column has low torsional GJ and warping EI_w stiffness, then it is more likely that the column will buckle in torsional mode. If a column has low flexural stiffness EI on its weak direction, then it will buckle in flexural mode. (Trahair 1993)