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IN DEGREE PROJECT THE BUILT ENVIRONMENT,SECOND CYCLE, 30 CREDITS
, STOCKHOLM SWEDEN 2019
Seismic Analysis of Norra TornenA Comparison Based on the Requirements in Eurocode 8
ANDREAS BARBARANELLI
ANDREAS WALLIN
KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ARCHITECTURE AND THE BUILT ENVIRONMENT
KTH Royal Institute of TechnologyDivision of Structural Engineering and Bridges
Seismic Analysisof Norra Tornen
A Comparison Based on the Requirements inEurocode 8
AuthorsBarbaranelli, AndreasWallin, Andreas [email protected]@gmail.com
SupervisorsSwedin, PetterKaroumi, Raid
ExaminerKaroumi, Raid
Stockholm, Sweden
June 3, 2019
TRITA-ABE-MBT-19215 KTH School of ABESE-100 44 StockholmSWEDEN
c©Andreas Barbaranelli and Andreas WallinRoyal Institute of Technology (KTH)Division of Structural Engineering and Bridges
“Successful engineering is all aboutunderstanding how things break or fail”
Henry Petroski
Abstract
In Sweden, buildings are not designed to withstand earthquakes due to the rarity of an earthquake eventand its consequential damage. However, the aim of this thesis was to study the seismic performance ofsome of the highest buildings in Stockholm, called Innovationen and Helix. The purpose of the study wasto get an understanding of earthquake engineering for high rise buildings and to compare the behaviorof the two towers during seismic action.
In order to compare the two buildings and get an understanding of what will affect the seismic perfor-mance, Eurocode 8 was used. The Eurocode standard lists several properties that impacts the seismicresistance of buildings. One of the goals was to study how those factors influence the behavior of Inno-vationen and Helix and finally compare the results to each other in order to draw valid conclusions.
The method to perform the analysis was a modal analysis using a finite element analysis program. Theprogram used contains predefined response spectra’s based on Eurocode 8 which is used to define theseismic load acting on the structures.
The extracted results are listed below:
- Frequencies and mode shapes
- Modal masses
- Level and total masses
- Accelerations
- Displacements
The conclusion of the study was that Innovationen and Helix have similar properties and some pointsfrom Eurocode 8 were better fulfilled by Helix and others by Innovationen:
- Uniformity, symmetry and redundancy (Innovationen fulfills the requirements better than Helix)
- Bi-directional resistance and stiffness (Innovationen fulfills the requirements better than Helix)
- Torsional resistance and stiffness (Helix fulfills the requirements better than Innovationen)
- Adequate foundation (Helix fulfills the requirements better than Innovationen)
Of the two parameters studied, the height was the one with the most influence on seismic resistance.
Keywords: Earthquake design, High-Rise buildings, Tall buildings, Seismic Performance, Earthquake En-gineering, Modal analysis, FE-analysis, Eurocode 8, Geotechnical engineering, Design Concepts, Foun-dation
i
Preface
The interest of earthquake engineering awoke during the summer of 2018 when we discussed a subject ofthe master thesis with SWECO Structures. We got in touch with Petter Swedin at Sweco in Sundsvallwhich had earlier been researching about seismic design of buildings in Japan. In collaboration withPetter, we created a suitable topic covering earthquake resistance of buildings. A suitable buildingcomplex to study was Norra Tornen since Petter was involved in that project as a structural engineer.
We had one course about structural dynamics at KTH during the fall of 2018, the course slightly touchedthe field of earthquake engineering which was a reason why we eagerly wanted to learn more about it.
We want to thank our supervisor and examiner Professor Raid Karoumi from KTH and our supervisorPetter Swedin at SWECO Structures in Sundsvall for their valuable input and advices along the way.
We also want to thank Marco Binfare, responsible for the models at SWECO Sundsvall and our friendOscar Lonnberg. Marco for his help with the FE-models of Norra tornen and Oscar for his help with theproofreading.
A special thanks to SWECO Structures in Stockholm for providing us with office space, equipment andencouragement during the time of the work.
Thank you!
Royal Institute of Technology, may 2019Andreas Barbaranelli & Andreas Wallin
ii
Abbreviations
CAD Computer-aided Design
FEA Finite Element Analysis
FEM Finite Element Method
MDOF Multi-Degree-of-Freedom
R/C Reinforced Concrete
Rdy Dynamic Amplification Factor
SDOF Single-Degree-of-Freedom
SRSS Square Root of the Sum of the Squares
VBA Visual Basics
List of Symbols
u Acceleration
ωd Damped Natural Frequency
c Damping Constant
ξ Damping Ratio
u Displacement
p(t) Force, Time Dependent
ωp Frequency of Applied Force
P (0) Initial Force
m Mass
ωn Natural Circular Frequency
fn Natural Cyclic Frequency
Tn Natural Period
β Ratio ω/ωn
k Spring Stiffness
t Time
u Velocity
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Table of Contents
I Introduction 1
1 Background 1
1.1 Earthquake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Aim and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Assessment of Seismic Performance 5
2.1 Dynamics of Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Single-Degree-of-Freedom System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Harmonic Excitation and the Dynamic Response Factor . . . . . . . . . . . . . . . 7
2.1.3 Multi-Degree-of-Freedom System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.4 Earthquake Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Response Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Eurocode 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Structural Simplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 Uniformity, Symmetry and Redundancy . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.3 Bi-Directional Resistance and Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.4 Torsional Resistance and Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.5 Diaphragmatic Behavior at Storey Level . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.6 Adequate Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Norra Tornen Case Study 16
3.1 Norra Tornen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.1 Innovationen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.2 Helix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.3 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.1 Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
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3.2.2 Ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.3 Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
II Methodology 33
4 FEM-Design 33
5 Parameters 33
5.1 Ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.1.1 Hinged Line Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.1.2 Pile Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.1.3 Increased Pile Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.1.4 Stiffness Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Modal Analysis 40
6.1 Dynamic Calculation and Mass Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.2 Seismic Load and Design Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.3 Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.4 Results From Seismic Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.5 Result Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7 Evaluation of Cross Sectional Properties 46
7.1 Hand-Calculated Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7.2 Bi-directional Resistance From FE-Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
III Results 50
8 Frequencies and Mode Shapes 50
8.1 Innovationen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
8.2 Helix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
8.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
9 Total and Level Masses 54
v
9.1 Comparison Between Innvationen and Helix . . . . . . . . . . . . . . . . . . . . . . . . . . 54
10 Bi-directional and Torsional Stiffness 55
10.1 Innovationen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
10.2 Helix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
10.3 Bending Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
11 Accelerations and Displacements 56
11.1 Comparison Between Accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
11.2 Comparison Between Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
12 Increased Pile Stiffness 59
12.1 Frequencies and Mode Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
12.2 Accelerations and Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
13 Varying Height 61
13.1 Frequencies and Mode Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
13.2 Accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
IV Discussion and Conclusions 63
14 Discussion 63
15 Conclusions 64
A Appendix - Pile stiffness
B Appendix - Frequencies and Mode Shapes
C Appendix - Total and level masses
D Appendix - Calculation of Cross Sectional Properties of Floor Plans
E Appendix - Accelerations
F Appendix - Displacements
vi
List of Figures
1 Deaths caused by earthquakes (2000-2015) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Earth’s Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3 Plate boundaries [8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
4 Seismic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
5 Undamped system [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
6 Damped system [4][5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
7 Dynamic response factor [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
8 Multi-degree-of-freedom system [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
9 Type I elastic response spectra for ground types A - E [11] . . . . . . . . . . . . . . . . . . 12
10 Type 2 elastic response spectra for ground types A - E [11] . . . . . . . . . . . . . . . . . 12
11 Use of joints to achieve uniformity and symmetry in plan [14] . . . . . . . . . . . . . . . . 14
12 Torsional resistance and stiffness [13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
13 Different contributing roles to a diaphragm [16] . . . . . . . . . . . . . . . . . . . . . . . . 15
14 Location of Norra Tornen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
15 Innovationen on the right and Helix to the left [17] . . . . . . . . . . . . . . . . . . . . . . 17
16 Basement floor -1 [21] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
17 Entrance plan and plan 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
18 Innovationen, stair shaped appearance [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
19 Comparison between floor plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
20 Vertical load bearing system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
21 Connections between floors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
22 Wall and floor connection [30] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
23 Column connection between floors [31] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
24 Basement floor 1[32] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
25 Entrance plan and plan 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
26 Helix, stair shaped appearance [27] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
27 Comparison between floor plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
28 Vertical load bearing system, plan view [27] . . . . . . . . . . . . . . . . . . . . . . . . . . 28
29 Wall and floor connection [39] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
vii
30 Natural Period [42] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
31 Mode Shapes [42] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
32 Soil behavior due to seismic waves [43] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
33 FE-Models of Norra Tornen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
34 Hinged Line Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
35 Positions of Piles in Autocad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
36 Spring Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
37 Pile positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
38 Stiffness in x- and y-direction, R’- and S’-Piles . . . . . . . . . . . . . . . . . . . . . . . . 37
39 Different heights of Innovationen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
40 Input of coefficient of variable action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
41 Settings for eigenfrequencies calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
42 Seismic load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
43 Seismic analysis, setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
44 Finite element types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
45 Example of plan view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
46 Idealized structure into a SDOF-system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
47 Mode shapes and frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
48 Mode Shapes, Innovationen figure a-c, Helix figure d-f . . . . . . . . . . . . . . . . . . . . 53
49 Total- and level-masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
50 Accelerations from Seismic Max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
51 Deformed shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
52 Displacements from Seismic Max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
53 Varying stiffness of the piles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
54 Frequencies with varying height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
55 Max accelerations with varying height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
56 Mode Shapes, Innovationen figure a-c, Helix figure d-f . . . . . . . . . . . . . . . . . . . .
viii
List of Tables
1 Ground types [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Values of parameters describing Type 1 elastic response spectra[11] . . . . . . . . . . . . . 11
3 Values of parameters describing Type 2 elastic response spectra [11] . . . . . . . . . . . . 12
4 Dimensions and Material properties [40] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5 Region name for piles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6 Stiffness of the piles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7 Values of ϕ for calculating ΨE,i [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
8 Frequencies, mode shapes and effective masses for Innovationen . . . . . . . . . . . . . . . 50
9 Frequencies, mode shapes and effective masses for Helix . . . . . . . . . . . . . . . . . . . 51
10 Comparison mode shapes and frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
11 Cross sectional data floor plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
12 Cross sectional data floor plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
13 Bending stiffness [kN/m] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
14 Frequencies, mode shapes and effective masses for Innovationen with elastic supports . . . 60
15 Stiffness of the piles +10% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16 Stiffness of the piles +33% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17 Stiffness of the piles +46% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18 Stiffness of the piles +61% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19 Stiffness of the piles +77% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20 Stiffness of the piles +95% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21 Frequencies for Innovationen for varying height . . . . . . . . . . . . . . . . . . . . . . . .
22 Total and level masses - Innovationen . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23 Total and level masses - Helix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24 Acceleration values - Innovationen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25 Acceleration values - Helix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26 Acceleration values for Innovationen with varying height . . . . . . . . . . . . . . . . . . .
27 Displacement values - Innovationen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28 Displacements values - Helix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Part I
Introduction
1 Background
1.1 Earthquake
Earthquakes are natural phenomena causing quakes in the ground. They occur on different locationsaround the world with several thousands of deaths as a consequence. Between the year of 2000 and2016 a total amount of 801 629 deaths were reported according to the statistical web pages ”UnitedStates geological survey” (USGS) and ”Statista”, see table 1 [1][2]. Earthquakes may cause damages tostructures and humans in several ways, some examples are tsunamis, fires and landslides which may leadto structural collapses. It is not the quake itself causing dead and desolation, it is the forthcoming effectsuch as collapsing buildings. When it comes to earthquakes, there are two different priorities. The firstpriority is to reduce the loss of human life’s and the second is to reduce the economic damages whichaffects the society after an earthquake [3].
231
21357
1685
33819
298101
87992
6605708
88708
1790
226050
21942
689 1572 756
9624
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015
0
50000
100000
150000
200000
250000
300000
350000
USGS
Statista
Figure 1: Deaths caused by earthquakes (2000-2015)
When a building collapses as a consequence of an earthquake, the economic damages are usually toolarge and not defensible to repair. The buildings should be designed in a way so that the magnitude ofthe earthquake would not create large damages that harm people and the repairing costs should not betoo severe [4].
1
Earth’s Structure
The structure of the earth consists of different layers, the core, the mantle and the crust. The core consistsof two parts, the inner core which has a solid form and the outer with a liquid form, see figure 2a. Thethickness of the crust varies depending on the location, under the ocean the crust is approximately 7 kmthick and under the continents the average thickness is approximately 30km [5]. At the outer part ofthe earth, the crust and the outer part of the mantle forms the lithosphere. The lithosphere is strongcompared to the asthenosphere, which is the inner part of the mantle. The lithosphere consists of theweaker part of the mantle, those parts are the outer boundaries and are defined by stiffness and strength.The lithosphere consists of several lithospheric plates, also called ”tectonic plates”. The lithosphere isapproximately 125 km thick and floats upon the asthenosphere. Liquid under the tectonic plates are thereason why the plates move in different directions. The heat transfers from the outer core towards thesurface and starts a flow according to figure 2b, which leads to movements of the tectonic plates [6][5].
(a) The different layers of the Earth [5] (b) Heat convection, movement of tectonic plates [5]
Figure 2: Earth’s Structure
When the tectonic plates move towards each other, three different boundary conditions may occur,constructive plate boundary, destructive plate boundary and conservative plate boundary, see figure 3[7].
2
Figure 3: Plate boundaries [8]
The constructive plate boundary occurs when two plates move towards each other and collide, the partswhich separate from the plates are immediately filled by lithospheric material. The destructive plateboundary is when two plates move against each other and the more dense plate folds under the less denseplate and dive into the sub lithospheric part where it recycles. The last boundary is the conservative plateboundary, two plates move towards each other and pass without almost no crust created or destroyed inthe process. The plates move along the boundaries and cause earthquakes [8].
Seismic Waves
When plates interlock at the plate boundary high stresses emerge within the rock. When the shear stressreaches the maximum resistance of the rock, ruptures occur. The effect will be a release of energy in theform of seismic waves called P- and S-waves. The P-waves are the compression waves which will appearin form of an expanding sphere from the rupture. The S-wave is the secondary wave or shear wave whichis slower than the P-wave and can only travel through solid rock. The waves travel in both vertical andhorizontal direction. The P-waves and S-waves are both body-waves, see figure 4a. The body waveslead to surface waves, which only propagate along the surface. There are two kinds of surface waves,Rayleigh- and love- waves. Love waves propagate as horizontal motions and Rayleigh waves as verticalmotions, see figure 4b [9][6].
3
(a) Body Waves [10] (b) Surface waves [10]
Figure 4: Seismic Waves
1.2 Aim and Scope
The main objective of this thesis was to study the seismic behavior of the building complex NorraTornen consisting of Innovationen and Helix in order to compare them with respect to Eurocode 8 (SS-EN 1998-1). The aim is to see if Innovationen has a better resistance to earthquakes than Helix, sincethe two towers are different and fulfill the requirements according to Eurocode 8 differently. The studyalso investigates which parameter between height and ground has most influence on the earthquakeresistance. The study therefore focuses on six principles of conceptual design in SS-EN 1998-1 [11].
- Structural simplicity
- Uniformity, symmetry and redundancy
- Bi-directional resistance and stiffness
- Torsional resistance and stiffness
- Diaphragmatic behavior at storey level
- Adequate foundation
The study is limited to modal analysis and does not include the interaction between soil and piles.
In the comparison, earthquake load, self-weight and part of live loads were considered. No load combi-nations were performed and no check in serviceability- and ultimate- limit state was performed.
The aim was not to determine if the buildings resist earthquakes, but to investigate how large impactthe design concepts has on the earthquake resistance of the buildings.
4
2 Assessment of Seismic Performance
2.1 Dynamics of Structures
Before starting with earthquake engineering it is important to have a basic knowledge in the subject ofdynamics of structures. This chapter will explain the main principles of dynamics and provide the mostimportant equations used to solve a dynamic problem.
2.1.1 Single-Degree-of-Freedom System
Undamped Free Vibration
To explain the basic principles of a dynamic problem in a simple way, a single degree of freedom system(SDOF) will be taken as an example. Figure 5a shows a system with a lumped mass m and a masslesssupporting structure with a stiffness k. Note that the system in the figure is undamped and will thereforevibrate with the same amplitude with an increasing time t, see figure 5b [4].
(a) SDOF
Time t
Dis
pla
ce
me
nt
u
(b) Free vibration, undamped system
Figure 5: Undamped system [4]
The equation describing the motion of a SDOF system is a differential equation, see equation 1
mu(t) + ku(t) = 0 (1)
In equation 1 the u denotes the acceleration and u the displacement of the mass [4].
When the structure is disturbed from its static equilibrium, the mass gets a displacement, u(0), and avelocity, u(t), see equation 2 [4].
u = u(0) u = u(t) (2)
The solution to equation 1, having the initial condition from equation 2, can be obtained with equation3 [4].
u(t) = u(0) cos ωn t+u(t)
wnsinωn t (3)
5
The natural circular frequency is obtained from equation 4 [4].
ωn =
√k
m(4)
Once the natural circular frequency is known, the natural period and the natural cyclic frequency canbe calculated with equation 5 and 6 [4].
Tn =2π
ωn(5)
fn =1
Tn=ωn2π
(6)
Damped free vibration
All structures have some form of damping due to an energy-dissipating mechanism. Unlike the vibrationtype in figure 5b, the free vibration for an undamped system is described by equation 8, which is thesolution to the equation of motion for a damped system with the damping constant c, see equation 7 [4].
mu(t) + cu(t) + ku(t) = 0 (7)
u(t) = e−ξωnt
(u0 cosωdt+
u0 + u0ξωnωd
sinωt
)(8)
ξ, given by equation 9 is the damping ratio and ωd denotes the damped natural frequency which is givenby equation 10 [4][5].
ξ =c
2√km
(9)
ωd = ωn√
(1− ξ2) (10)
In equation 8 the damping makes the amplitude of the waves decrease exponentially with increased time.To clarify the principle see figure 6, where figure 6b represents the plot of the function in equation 8 [4].
6
(a) SDOF-dumped system
Time t
Dis
pla
cem
ent u
(b) Free vibration, damped system
Figure 6: Damped system [4][5]
2.1.2 Harmonic Excitation and the Dynamic Response Factor
To get an understanding of the response of a structure during an earthquake, it is important to compre-hend what harmonic excitation is.
During harmonic excitations the force is described by p(t) = P0 sinωt, hence the differential equationdescribing the motion of a SDOF system is not equal to 0 anymore like in equation 7 but equal toP0 sinωt [4].
In this case the solution to the differential equation becomes more complicated and the ratio between thefrequency of the applied force ωp and the natural frequency of the system ωn, denoted with the factorβ, has influence on the behavior of the structure, see equation 11 [4][6].
u(t) = e−ξωnt(Acosωdt+B sinωdt
)︸ ︷︷ ︸transient response
+
+
(static displacement)︷︸︸︷P0
k
(dynamic amplification factor Rdy)︷ ︸︸ ︷1(
1− β2)2
+(2ξβ
)2 ((1− β2
)sinωpt+ 2ξβ cos ωpt
)︸ ︷︷ ︸
steady state response
(11)
In order to illustrate why the factor β has influence on the response of the structure, the function of thedynamic amplification factor has been plotted, see figure 7 [4].
7
0 0.5 1 1.5 2 2.5 3
W/Wn
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
Rd
Damping 0.1
Damping 0.2
Damping 0.4
Figure 7: Dynamic response factor [4]
What can be seen from figure 7 is that if the ratio between the frequency of the applied force, ωp, andthe natural frequency of the system, ωn, is close to 1, an amplification of the response will occur, thisphenomenon is called resonance [4][6].
2.1.3 Multi-Degree-of-Freedom System
In structural analysis it is not always possible to reduce a structure in a SDOF system but in order torepresent the reality, it is almost always necessary to study a multi-degree-of-freedom (MDOF) system[6].
Figure 8 shows how a three storey building can be simplified. The constants m1,m2 and m3 representlumped masses concentrated in the center of each floor. The stiffness is taken into account by consideringthe contribution of the structural members involved [6].
Figure 8: Multi-degree-of-freedom system [4]
In a MDOF system the equation of motion is basically the same as for a SDOF-system. The differencebetween the systems is that the constants in a MDOF system are represented by matrices, see equation
8
12 [6].m1 0 00 m2 00 0 m3
u1
u2
u3
+
c1 + c2 −c2 0−c2 c2 + c3 −c3
0 −c3 c3
u1
u2
u3
+
k1 + k2 −k2 0−k2 k2 + k3 −k3
0 −k3 k3
u1
u2
u3
=
p1(t)p2(t)p3(t)
(12)
2.1.4 Earthquake Excitation
An earthquake can be simplified as an induced motion of the base and therefore the response of thestructures resting on the moving base is vital for the building. The displacement can be denoted byug(t) and due to the motion, inertial forces develops according to Newton’s 2nd law (F = ma). Thetotal acceleration of the mass could be represented by equation 13 where in this case utot = a, where urepresents the acceleration of the mass relative to the ground and ug the ground acceleration [5].
utot = ug + u (13)
From this simple relation, the equation of motion can be modified and expressed according to equation14 with the effective force −mug(t) [5].
mu+ cu+ ku = −mug(t) = Feff (t) (14)
2.2 Modal Analysis
A modal analysis is a calculation process that allows to solve the equation of motion for a SDOF system.Since it is difficult to predict the motion of an earthquake the study will be based on the most unfavorableground motion direction and the time period [12].
To simplify the process, Eurocode 8 prescribes predefined acceleration response spectra [12].
2.2.1 Response Spectrum
The elastic response spectrum that Eurocode 8 prescribe are shown in figure 9 and figure 10. Therepresentation is based on a single degree mass-spring system where the vibration time of the system isrepresented on the x-axis while the corresponding maximum acceleration on the y-axis [12].
The equations describing the elastic response spectra can be found in Eurocode 8, see equation 15-18[11].
0 ≤ T ≤ TB : Se(T ) = ag · S ·[1 +
T
TB·(η · 2, 5− 1
)](15)
TB ≤ T ≤ TC : Se(T ) = ag · S · η · 2, 5 (16)
TC ≤ T ≤ TD : Se(T ) = ag · S · η · 2, 5 ·[TCT
](17)
TD ≤ T ≤ 4s : Se(T ) = ag · S · η · 2, 5 ·[TC TDT 2
](18)
9
Where:
- Se(T ) is the elastic response spectrum;
- T is the vibration period of a linear single-degree-of-freedom system;
- ag is the design ground acceleration on type A ground (ag = γIagR);
- TB is the lower limit of the period of the constant spectral acceleration branch;
- TC is the upper limit of the period of the constant spectral acceleration branch;
- TD is the value defining the beginning of the constant displacement response range of the spectrum;
- S is the soil factor;
- η is the damping correction factor with a reference value of η = 1 for 5% viscous damping.
(CEN, Eurocode 8: Design of structures for earthquake resistance, 2004, 37)
Depending on the ground type, the constants presented in the expression above will have different values.Eurocode 8 classifies the ground from A to E and what defines the class is the average shear wave velocityvs,30, which can be calculated with equation 19 [11].
vs,30 =30∑
i=1,N
hi
vi
(19)
Where hi is the thickness of each layer in the range of 30 meters and vi is the shear wave velocity foreach specific layer. Table 1 shows which ground type to choose depending on ground condition [11].
10
Table 1: Ground types [11]
Ground type Description of the stratigraphic profileParameters
vs,30 (m/s)NSPT
(blows/30cm)cu (kPa)
ARock or other rock-like geological
formation, including at most 5 m ofweaker material at the surface.
> 800 - -
B
Deposits of very dense sand, gravel, orvery stiff clay, at least several tens of
metres in thickness, characterised by agradual increase of mechanical
properties with depth.
360 - 800 > 50 > 250
C
Deep deposits of dense or medium-dense sand, gravel or stiff clay withthickness from several tens to many
hundreds of metres.
180 - 360 15 - 50 70 - 250
D
Deposits of loose-to-mediumcohesionless soil (with or without some
soft cohesive layers), or ofpredominantly soft-to-firm cohesive
soil.
< 180 < 15 <70
E
A soil profile consisting of a surfacealluvium layer with vs values of type C
or D and thickness varying betweenabout 5 m and 20 m, underlain bystiffer material with vs >800 m/s.
S1
Deposits consisting, or containing alayer at least 10 m thick, of soft
clays/silts with a high plasticity index(PI >40) and high water content.
< 100(indicative)
- 10 - 20
S2
Deposits of liquefiable soils, ofsensitive clays, or any other soil profile
not included in types A – E or S1.
Once the ground type is defined the value for each parameter can be set and the exact shape of theelastic response spectra can be plotted. Table 2 shows recommended values for parameters describingType 1 elastic response spectra while table 3 shows recommended values for type 2. Type 2 shouldbe used when designing building located in regions where the risk for surface-wave magnitude with Ms
greater than 5, 5 is low, otherwise type 1 should be adopted. In some cases more than one spectra canbe used in order to achieve more accurate results [11].
Table 2: Values of parameters describing Type 1 elastic response spectra[11]
Ground type S TB(S) TC(S) TD(S)
A 1,0 0,15 0,4 2,0B 1,2 0,15 0,5 2,0C 1,15 0,20 0,6 2,0D 1,35 0,20 0,8 2,0E 1,4 0,15 0,5 2,0
11
0 0.5 1 1.5 2 2.5 3 3.5 4
T (s)
0
0.5
1
1.5
2
2.5
3
3.5
4
Se/agE
D
C
B
A
Figure 9: Type I elastic response spectra for ground types A - E [11]
Table 3: Values of parameters describing Type 2 elastic response spectra [11]
Ground type S TB(S) TC(S) TD(S)
A 1,0 0,05 0,25 1,2B 1,35 0,05 0,25 1,2C 1,5 0,10 0,25 1,2D 1,8 0,10 0,30 1,2E 1,6 0,05 0,25 1,2
0 0.5 1 1.5 2 2.5 3 3.5 4
T (s)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Se/agD
E
C
B
A
Figure 10: Type 2 elastic response spectra for ground types A - E [11]
12
2.3 Eurocode 8
In Eurocode 8 the basic design concepts are described as six different points. In this section, these sixconcepts are described.
The first four points in Eurocode 8 are used to reach maximal dissipation of seismic energy by hystereticbehavior in order to avoid a local and global collapse or instability of structures. The last two pointsare more focused on the requirements for the capacity design rules and the ductility of the differentstructural members [13].
2.3.1 Structural Simplicity
In order to resist an earthquake in a suitable way, structures should be constructed as simple as possible.The load paths should be continuous from the top of the building down to the foundation with sufficientstrength, stiffness and ductility in order to withstand external loads such as gravity- and seismic loads.If buildings satisfy those aspects, structures can more easily resist dynamic excitation. Load paths canbe complex and not direct, for example around openings where the load has to find other paths, gothrough joints between structures or down through columns supported by beams [13]. At these pointsthere will be stress concentrations at corners and breakpoints between sections. At locations with stressconcentrations it may be difficult to evaluate sufficient strength, stiffness and ductility [14]. Withoutcontinuous load paths the consequences may be significant and that is one of the reasons why this is oneof the first points stated in the design concepts of seismic design [15].
Due to increased complexity of buildings, there are difficulties to verify and rely on the results regardinglocal ductile behavior of structures and thereby satisfy the expected demands of ductile behavior afteran analysis. In order to increase the reliability of the expected ductile demands and to better understandload paths of structures, a simple structure should be accounted for in order to be on the safe side [13].
2.3.2 Uniformity, Symmetry and Redundancy
When a building gets excited the inertia forces that occur in the mass of the building need to bedistributed. When it comes to uniformity, Eurocode 8 mentions two kind of uniformity: in plan andalong the height of the building [11].
In plan uniformity is when the main structural parts are even distributed and therefore the torsionaleffect are reduced. An efficient way to increase the dynamic performance under this point of view is touse joints in order to divide the shape of the plan in several uniform parts, see figure 11 [11][14].
13
Figure 11: Use of joints to achieve uniformity and symmetry in plan [14]
Uniformity along the height of the building means that the strength and stiffness in elevation are evenlydistributed. Following this rule can help reducing stress concentrations and the development of sensitivezones in form of weak storeys [11][14].
In order to eliminate or decrease eccentricities between mass and stiffness, the mass of a structure shouldbe closely related to its strength [11][14].
Redundancy is the capacity of the structure to distribute the load in several load paths. Having alterna-tive ways to transmit seismic loads will decrease the risk of failure. It is therefore important to place theplastic hinges of the structure in strategical points in order to facilitate this phenomenon [11][13][14].
2.3.3 Bi-Directional Resistance and Stiffness
An earthquake motion can affect a structure from any direction in the ground which means that thephenomenon is bi-directional. To increase stability in structures it is therefore advisable to have similarresistance in any direction [11][14]. When designing the structure the initial principle is that the structuralmembers shall be equally arranged along two orthogonal axis in order to reach equal resistance in formof strength, stiffness and ductility when excited from seismic load [13].
2.3.4 Torsional Resistance and Stiffness
When designing a structure, engineers try to avoid torsional effect since those effects are difficult tohandle and stresses develop in a non-uniform way. An effective way to reduce such effect is to designstructures with the center of resistance and center of mass as close as possible, see figure 12 for theprinciple [11][13] [14].
14
Figure 12: Torsional resistance and stiffness [13]
2.3.5 Diaphragmatic Behavior at Storey Level
In order to reach a good seismic resistance according to diaphragmatic behavior of a structure, thereare some important aspects to consider. To provide a good seismic resistance there has to be a seismiclateral load resisting system which withstand the lateral loads from an earthquake. The following pointsare important in order to reach a good diaphragmatic behavior [16]:
- Resist gravity loads
- Provide lateral support to vertical element
- Resist out-of-plane forces
- Resist thrust from inclined columns
- Transfer inertial forces to vertical elements of the seismic force-resisting system
- Transfer forces through the diaphragm
- Support soil loads below grade
(Moehle, Hooper, Kelly and Meyer 2012, 6)
In figure 13, these points are illustrated [16].
Figure 13: Different contributing roles to a diaphragm [16]
15
To reach a good diaphragmatic behavior at storey level, the slab should be rigid connected to the verticalmembers in order to have a minimized displacement in the horizontal direction. A rigid connection tothe walls over and under the slab contributes to a small difference in displacement between the structuralmembers of the vertical lateral load system [14]. Rigid connected slabs lead to better resistance againstlateral loads acting on the vertical members. When all diaphragms are rigid connected to the seismicresisting system of each floor, the diaphragm will counteract lateral buckling and second order forces[16].
One of the main tasks for diaphragms are to distribute the inertial forces that may occur in the slabs, tothe members included in the seismic load resisting system. The inertial forces also include contributionfrom pieces of walls and columns [16]. In order to succeed with satisfying diaphragmatic behavior atstorey level, the slabs have to be stiff enough in their own plane. To avoid losing stiffness, openings inthe slab should be avoided as well as high aspect ratios (proportional relationship between height andwidth) [13].
2.3.6 Adequate Foundation
Eurocode 8 states that the design of a superstructure, the foundation and their connection, should ensurea uniform seismic excitation [11]. It is important that all members contributing to the seismic resistanceare tied with the foundation at the same level. If the members are tied to different levels, there maybe distinguishing displacements as an effect after seismic excitation. In case of multi-level foundations,there should be a strong connection between the levels, such as strong reinforced concrete (R/C) walls[13]. For soft soils, micro piles can be used in order to support the foundation, to achieve an adequatefoundation it is also necessary to design the foundation as rigid cellular [14]. The foundation needs tobe stiff and strong enough in order to guarantee a good structural response to earthquake excitations.What is also important is how the foundation distributes the lateral seismic forces, called base shears,in an uniformed way [13].
3 Norra Tornen Case Study
3.1 Norra Tornen
Norra Tornen is a future building complex in the central of Stockholm at Torsplan, see figure 14. NorraTornen consists of two towers, Innovationen and Helix, see figure 15. The total heights of Innovationenand Helix are 125 and 111 meters respectively. The actual space the buildings occupy on the ground areapproximately 800m2 [17].
16
(a) View Central of Stockholm (site location) [18]
(b) View Torsplan (site location) [18]
Figure 14: Location of Norra Tornen
Figure 15: Innovationen on the right and Helix to the left [17]
17
Innovationen and Helix consist of prefabricated concrete walls and cast in site concrete on prefabricatedconcrete slabs. The bodies are not equal and differ in the foundations. The foundation for Innovationenconsists of a concrete slab resting on piles while Helix’s bottom slab consists of concrete resting on rockground [19][20].
3.1.1 Innovationen
Innovationen is 125m tall. The actual height over the ground is 116m because of a two level basement,see figure 16 [17][21].
Figure 16: Basement floor -1 [21]
Innovationen occupies approximately 800m2 on the ground, see figure 17a, that is less than the squarearea of the floors higher up since plan 4 to 15 are approximately 920m2, see figure 17b. Plan 4 to 15have an overhang unlike the floors below [22][23].
18
(a) Entrance floor [22]
(b) Plan 4 [23]
Figure 17: Entrance plan and plan 4
Innovationen has a stair looking appearance, see figure 18, and has its largest plans between level 4 to15, see figure 17b [17][23].
19
Figure 18: Innovationen, stair shaped appearance [17]
The plan size gets smaller from plan 16 to 21, see figure 19a, plan 22 to 29, see figure 19b, and the laststep from floor 30 to 36, see figure 19c, with a terrace above and a final roof plan [24][25][26].
20
(a) Plan 16 [24]
(b) Plan 22 [25]
(c) Plan 30 [26]
Figure 19: Comparison between floor plans
21
The load bearing system consists of reinforced concrete with a stabilizing core to withstand externallateral forces, see figure 20a and 20b. Shear walls are connected to the concrete core, the building has aslender look and the shear walls contribute to increased stiffness in the weak direction [27].
(a) Vertical load bearing system, plan view [27]
(b) Vertical load bearing system, 3D-view [27]
Figure 20: Vertical load bearing system
Except the stabilizing core and shear walls there are internal- and perimeter columns in order to providea vertical load path. The columns are arranged with a distance of 4800mm, see figure 20a. The facadehas a box shaped design with cantilever walls, the perimeter walls acts as supports for the cantileverwalls in the facade. The floor slabs consist of pre-fabricated slabs with reinforced concrete at the bottomand cast in site concrete on top of it. The floor slabs have their spans in one of the directions along oracross the building according to the grid system, see figure 20a [27].
22
The core is the stabilizing structure of the building which goes from the first floor to the top floor withconnected pre-fabricated concrete walls. The walls are connected in different ways through the floorsaccording to figure 21a. One solution is with a steel beam within the wall with a welded rod on the top,while the overlaying wall is threaded over the rod [28]. The walls are stabilized with a cast of concretebetween the walls. The floors are resting on a suspender solution with a welded steel plate on top of theprefabricated part of the floor and the steel profile in the underlying wall. Another type of solution, seefigure 21b, is when the walls are connected with re-bars of type A, without connected floors [29].
(a) Wall and floor connection [28] (b) Wall connection [29]
Figure 21: Connections between floors
Another type of wall connection which is not connected to the core is, according to figure 22, where thewalls are connected to each other with shear studs. The shear studs are pre-cast into the underlying wallwith a rod attached at the top of the shear stud. The overlaying walls are then threaded over the rods.The rod are stabilized with a casting after the wall is put in place, see figure 22. At figure 22, the detailis also showing the connection between the floor and the walls. The floor is placed 80mm upon the wallwith connected steel plate and associated shear studs [30].
Figure 22: Wall and floor connection [30]
The load bearing columns are connected through the floors by anchor bolts, see figure 23 for one of thesolutions. When the walls are connected, the cavity which is left to be able to reach the bolts is filledwith concrete. The details differ depending on floor but the concept with anchor bolts remain the same[31].
23
Figure 23: Column connection between floors [31]
3.1.2 Helix
Helix is 111m high. Figure 24 shows a plan of the basement of the building. Helix covers a smaller areaon the ground than Innovationen since the width is approximately 14m and length 43m, see figure 24[17][32].
Figure 24: Basement floor 1[32]
Like Innovationen, the floor area in the superstructure is smaller than the floor area of the basement,see figure 25 which shows an overview of the entrance floor and plan 4 [33][34].
24
(a) Entrance floor [33]
(b) Plan 4 [34]
Figure 25: Entrance plan and plan 4
In elevation plan, Helix is similar to Innovationen with its stair shape appearance, see figure 26. However,there are some differences: height, number of floors and the overall shape [35].
25
Figure 26: Helix, stair shaped appearance [27]
The cross section of Helix becomes gradually smaller with increasing height. The plans appear differentlyto each other, even if the core of the building is in the same position for every floor, see figure 27[36][37][38].
26
(a) Plan 19 [36]
(b) Plan 24 [37]
(c) Plan 30 [38]
Figure 27: Comparison between floor plans
Same construction principles as for Innovationen are used for the design of Helix. To withstand lateralforces, Helix has a core composed by concrete walls going from the bottom floor to the top floor of thebuilding. Shear walls, inner column and perimeter column are also present to increase the stiffness in alldirections, see figure 28 [27].
27
Figure 28: Vertical load bearing system, plan view [27]
An important difference between Helix and Innovation when it comes to the load bearing system isthat Helix has several cast in place walls and slabs while Innovationen has the main bearing structuralcomponents prefabricated. For a typical connection between walls and slab, see figure 29. For connectionsbetween prefabricated components, see section 3.1.1 [39].
Figure 29: Wall and floor connection [39]
3.1.3 Materials
The materials used for Innovationen and Helix are listed in table 4 [40].
28
Table 4: Dimensions and Material properties [40]
Structural part Dimension Concrete class Exposure class Creep coefficient Shrinkage
Core/Basementwalls t=400mm C50/60 XC1 1.38 0.49hBasement outer walls t=400mm C50/60 XC4 1.11 0.32hSandwich Inner walls t=400mm C50/60 XC1 1.5 0.56hSandwich front layer t=400mm C30/37 XC1 1.5 0.65hSandwich outer layer t=70mm C30/37 XC4+XF1 1.5 0.50h
Core t=300mm C50/60 XC1 1.43 0.50hInner walls (K1-K2) t=200mm C50/60 XC1 1.5 0.56hInner walls (others) t=200mm C40/50 XC1 1.81 0.58h
Inner walls (cor t=200mm C40/50 XC1 1.81 0.58hInner walls elevator t=150mm C40/50 XC1 1.88 0.63h
Corridor walls t=200mm C40/50 XC1 1.5 0.56hPiles 400x400 C50/60 XC1 1.39 0.56h
3.2 Parameters
3.2.1 Height
The effects of an earthquake on buildings varies depending on the height combined with the wave lengthspropagating from the quake. Tall buildings are not necessarily taking more damage than short buildings,it depends on the periods (wavelengths). For short buildings, more severe damage may occur than for tallbuildings if the period is short (high-frequency) than if the period is long (low-frequency). Tall buildingsmore probably collapses when excited by low frequency ground motions. Two examples to explain thosedifferences are by looking at wave propagation against boats and ships. Small boats have more troublehandling small rapid waves which will cause it to rock back and forth and may cause overturning orcapsizing while it will not affect the big ship substantially. Big waves will instead cause more damage toa ship than a small boat, in the same way small- and tall buildings will be affected [41].
When a building is excited by a horizontal push, it will start moving back and forth at its naturalfrequency. The time it takes for one wave or cycle to be completed is called natural period, see figure 30[42].
29
Time t
Dis
pla
cem
ent u
Natural Period (T)
Figure 30: Natural Period [42]
To find the natural frequency of a building, the inverse of the natural period is taken. The result iscalculated in rad/s, see equation 20 [42].
ω =1
T(20)
One rule of thumb to calculate the natural period of a building is to divide the number of storeys by ten.Other factors will also affect the natural period of a building such as geometry and material propertiesbut the most crucial factor to consider is the height [42].
Buildings may sustain the most severe damage when resonance is reached. Resonance occurs when thevibration of the ground have the same (or near) natural period as the natural period of the building. Thenatural period is the period of the natural frequency when the first mode occurs, see figure 31. Buildingsare vibrating at its natural period when they are excited by a horizontal push. Several horizontal pushesleads to an acceleration of the building with an increasing displacement as a consequence. The ground,just as buildings, vibrates at its natural period when they are excited by an earthquake. The naturalperiod of the ground varies between 0.2 s to 2 s depending on soil properties, which are in the span ofregular buildings. This means that when buildings are excited by earthquake ground motion, it is mostlikely that resonance may occur. The acceleration of the building may be several times higher than theacceleration of the ground. Since the natural frequency depends on equation 20, buildings suffer mostsevere damage at ground motions corresponding to their natural frequency [42].
30
Figure 31: Mode Shapes [42]
3.2.2 Ground
The ground is a parameter that correlates to seismic waves. Depending on ground properties, a quakemay cause ground motions and destruction on buildings. The correlation between ground propertiesand destruction due to earthquakes is a fact known long through history. The knowledge about soilled to more suitable choices for settlements. When it comes to the soil and behavior of structures dueto ground motions, those motions varies depending on the soil condition. A building which is foundedon soils with different conditions tends to response differently to those motions. Thereby the possibledamage a building may be exposed to does not only depend on structural properties. The interactionbetween the soil and the foundation, soil-structure-interaction (SSI), also matters [43].
Structures built on soft soil may suffer greater damage then structures built on rock. The amplitude ofthe seismic waves tends to increase depending on the thickness of the layer down to the bedrock. Theamplification factor varies between 1.5 − 6 with a layer thickness varying between approximately one-to thirty meter [42]. Depending on ground conditions, the ground is divided in two different groupsdepending on the average shear velocity of the site, for more information about different ground types,see section 2.2.1 [11].
Different fails of the ground may occur due to liquefaction, site amplification and slope failure, see figure32 [43]. Liquefaction is when a building has parts or all of its foundation on saturated ground. Whenthe structure is excited by the seismic ground motions, the pore water pressure will be induced, whichmay lead to a flow slide of the ground [44]. Site amplification is the amplification which is decided bythe layers of the soil as well as the properties, which are above the engineering bedrock, see figure 32and section 3 in Eurocode 8 (SS-EN 1998-1:2004) [11][42][45]. Slope failure may occur on slopes withboth saturated and un-saturated soil. This may happen when the slope is excited by cyclic loading. Theshear resistance of the soil may be reduced and the consequence of lowered shear resistance may lead tolarge movement of soil masses [43].
31
Figure 32: Soil behavior due to seismic waves [43]
3.2.3 Mass
The mass of a building is a parameter affecting the inertia forces. With a large mass the inertia forces areincreased compared to building with less mass. In order to achieve significant resistance against inertiaforces, the mass of the system may be reduced by decreasing the thickness of the floor slabs. It is alsopossible to change the components, for instance, concrete floor slabs into composite materials in order todecrease the mass [46]. According to Newton’s second law, see equation 21, the product of the mass (m)and acceleration (m/s2) creates a force that makes the building move or rotate. Every building has itsown equilibrium position, which is reached when the sum of all forces are equal to zero. When buildingsare displaced from their equilibrium position there is a force compelling the building to move back to theequilibrium position. The force compelling the building to sway back is dependent on the stiffness andthe displacement. When the equilibrium is reached, the building will continue swaying in the oppositedirection because of the velocity of the returning mass. In order to understand the theory of motion, seesection 2.1 [14].
F (t) = ma (21)
Irregularities in buildings lead to inertia forces such as un-symmetrical floor plans and unequal distribu-tion of mass and stiffness may lead to that the center of mass (CM) and the center of rigidity (CR) donot end up in the same location of the building[47]. When CM and CR do not end up together there isa risk of both twisting and swaying, see figure 12 in section 2.3.4. Due to lateral torsional effects whena building is excited by earthquake, the structural components at the edges of the building may sustaingreat stress concentrations and severe deformation, which may lead to collapsed buildings [47][48].
32
Part II
Methodology
4 FEM-Design
FEM-Design is a software developed by the Swedish company StruSoft. The program is based on finiteelement analysis and is used in the civil engineering field in order to design bearing structures accordingto Eurocode with National Annex (NA) [49].
FEM-Design allows to perform static-, dynamic- and seismic analysis. The seismic analysis is howeverlimited since the only calculation process supported is the modal analysis [12].
5 Parameters
Innovationen and Helix were modeled in FEM-Design, mainly by Sweco Structures in Sundsvall. Inorder to carry through analysis of Innovationen and Helix, models used for the project were provided bySweco, see figure 33. In this chapter the method of extracting results from existing models are explained.The only modeling work done in this thesis was the removing of floor plans and the modeling of piles forInnovation, in order to extract results depending on height- and foundation-properties.
(a) 3D-view, Helix (b) 3D-view, Innovationen
Figure 33: FE-Models of Norra Tornen
33
5.1 Ground
Boundary conditions for the connection between the foundation and the rock ground were created intwo different ways. The first way was with line supports were the stiffness of the supports were set toinfinity in x- and y-direction while the rotational stiffness was set to zero, this means that the buildingswere free to rotate but rigid in x-and y-direction at the connection to the ground. The other way was tomodel the piles with correct pile stiffness.
5.1.1 Hinged Line Support
In order to model the hinged line support, the first step was to select K2-floor plan, which was thesecond basement plan, then the top view was selected. Next step was to select the ”line support group”under the tab labeled ”Structure” in FEM-design and choose rigid support, see figure 34. The hingedline supports were modeled in the same way for both Innovationen and Helix.
(a) Foundation, Innovationen (b) Line Support Stiffness
Figure 34: Hinged Line Supports
5.1.2 Pile Support
The piles were represented by spring supports with stiffness in x- y- and z-direction in order to catchthe behavior of the building when it was not locked by infinite stiffness at the ground. The position ofeach pile was provided by SWECO as a dwg-file according to figure 35. The blue dots represent eachpile, red lines represent the center of each wall and the black lines show where the piles were modeledin FEM-Design. Since the model did not contain a slab, the piles could not be modeled at their originalpositions and the stiffness of each pile was added together in order to model the piles as spring supportsto the cellar walls.
34
Figure 35: Positions of Piles in Autocad
In order to model the piles as supports, a dwg-file according to figure 35 was imported into the FE-Model and the dwg-drawing was positioned according to the model. To model the piles, point supportgroup under the structure label was used. The default settings were changed according to the calculatedstiffness of the piles, see figure 36. It was important to make sure that the supports were fully connectedto the center lines of the model.
(a) Foundation, Innovationen (b) Spring Stiffness
Figure 36: Spring Supports
5.1.3 Increased Pile Stiffness
To investigate the effect of the piles, their stiffness where increased gradually by 10%, 21%, 33%, 61%and 100%. The original stiffness in horizontal and vertical direction where increased each time by 10%,k(n) = kn · (10%)n where n is the number of steps and k is the stiffness.
5.1.4 Stiffness Calculation
In order to model the spring supports, the stiffness of the piles were calculated. To calculate the stiffness,a drawing of the piles was used in order to see where the piles were located and the inclination of eachpile, see figure 37. The letters in the figure were matched with cross sectional properties given in table
35
5. The vertical piles were not fastened at the foundation plate. This means that the vertical piles wereexposed to pressure but not tension. The inclined piles were fastened at the foundation plate and therebywere assumed to withstand both pressure and tension. It was assumed that the weight of the buildingwas enough to avoid significant results in vertical direction.
Figure 37: Pile positioning
Table 5: Region name for piles
Core diameter[m]
Pile lengthdown to rock ground
Vertical piles[m]
Pile lengthdown to rock ground
Inclined piles[m]
3.0 5.0 7.0 9.2 3.0 5.0 7.0 9.0180 A B C D M N O P150 E F G H Q R S T80 I J K L - - - -
To calculate the stiffness of the piles, Euler-Bernoulli theory for 2D Beam Elements was used. Thestiffness for each pile was calculated in horizontal direction (x and y) and in vertical direction (z). Thetorsional stiffness was not calculated since it was set to infinite in the FE-program.
The first step to calculate the stiffness of the piles was to decide the local coordinates for each pile. Thevertical piles had the following coordinates:
- 3.0m −→ (x1, y1) = (0.0, 0.0), (x2, y2) = (0.0, 3.0)
- 5.0m −→ (x1, y1) = (0.0, 0.0), (x2, y2) = (0.0, 5.0)
- 7.0m −→ (x1, y1) = (0.0, 0.0), (x2, y2) = (0.0, 7.0)
- 9.2m −→ (x1, y1) = (0.0, 0.0), (x2, y2) = (0.0, 9.2)
36
The inclined piles were modeled with the proportion 1.5:1. The horizontal stiffness was set for x- ory-direction depending on which side of the building the piles were positioned. The piles labeled S andR, positioned on the short side of the building, had a stiffness in x- direction while the piles labeled N,positioned on the long side of the building, had a stiffness in y-direction, see figure 37. The Inclined pileshad the following coordinates:
- 5.0m −→ (x1, y1) = (0.0, 5.0), (x2, y2) = (3.33, 0.0)
- 7.0m −→ (x1, y1) = (0.0, 7.0), (x2, y2) = (4.67, 0.0)
The piles labeled R’ and S’, positioned on the long side of the building had, a stiffness both in x- andy-direction, see figure 37. The cross sectional properties were the same as for R- and S-piles accordingto table 5. In order to calculate the stiffness in both directions, a different inclination was considered forx- and y-direction for the piles. To calculate the different horizontal stiffness, R’-piles were consideredin two ways according to figure 38a, and the S’-piles according to figure 38b. u is the stiffness in thehorizontal direction, v in the vertical direction and ϕ is the rotational stiffness.
(a) R’-Pile, stiffness considered in x- and y-direction (b) S’-Pile, stiffness considered in x- and y-direction
Figure 38: Stiffness in x- and y-direction, R’- and S’-Piles
The R’-piles were calculated with an x- and y-stiffness according to the inclination properties of figure38a. The S-piles were calculated with an x- and y-stiffness according to figure 38b. The stiffness inz-direction for R’- and S’-piles were assumed to be the same as for the stiffness in z-direction for R- andS-piles. The piles depending on x-and y-direction had the following coordinates.
- 5.0m (x-direction) −→ (x1, y1) = (0.0, 5.0), (x2, y2) = (0.58, 0.0)
- 5.0m (y-direction) −→ (x1, y1) = (0.0, 5.0), (x2, y2) = (3.28, 0.0)
- 7.0m (x-direction) −→ (x1, y1) = (0.0, 7.0), (x2, y2) = (0.81, 0.0)
- 7.0m (y-direction) −→ (x1, y1) = (0.0, 5.0), (x2, y2) = (4.60, 0.0)
Second step was to calculate the transformation matrix (T), see matrix 22, and its transpose (TT ), seematrix 23. To calculate nxx, nxy, nyy and nyx, the lengths of the piles had to be calculated according toPythagorean theorem, see equation 24.
37
T =
nxx nyx 0 0 0 0nxy nyy 0 0 0 00 0 1 0 0 00 0 0 nxx nyx 00 0 0 nxy nyy 00 0 0 0 0 1
(22)
TT =
nxx nxy 0 0 0 0nyx nyy 0 0 0 00 0 1 0 0 00 0 0 nxx nxy 00 0 0 nyx nyy 00 0 0 0 0 1
(23)
L =√
(x2 − x1)2 + (y2 − y1)2 (24)
In order to calculate nxx, nxy, nyy and nyx, equations 25a and 25b were used.
nxx =x2 − x1
L(25a)
nyx =x2 − x1
L(25b)
nxx = nyy
nyx = −nxy
Third step was to calculate the local stiffness k′ according to matrix 26. The elastic modulus (E) of thesteel piles was set to 210GPa. The area (A) and the moment of inertia (I) were varying depending onthe cross section of the piles. For cross sectional properties, see table 5.
k′ =
EA
L0 0 −
EA
L0 0
012EI
L3
6EI
L20 −
12EI
L3
6EI
L2
06EI
L2
4EI
L0 −
6EI
L2
2EI
L
−EA
L0 0
EA
L0 0
0 −12EI
L3−
6EI
L20 −
12EI
L3−
6EI
L2
06EI
L2
2EI
L0 −
6EI
L2
4EI
L
(26)
Last step was to calculate the global stiffness matrix (k) according to equation 27.
k = TT k′ T (27)
Matrix 28 represents the stiffness of the piles located in region A. Matrix 29 represents the stiffness ofpiles located in region N. The stiffness are presented in kN/m/m for vertical and horizontal directionand kNm/m/◦ for rotational stiffness.
38
k =
4.81E + 06 0 −7.21E + 06 −4.81E + 06 0 −7.21E + 06
0 1.78E + 09 0 0 −1.78E + 09 0−7.21E + 06 0 1.44E + 07 7.21E + 06 0 7.21E + 06−4.81E + 06 0 7.21E + 06 4.81E + 06 0 7.21E + 06
0 −1.78E + 09 0 0 1.78E + 09 0−7.21E + 06 0 7.21E + 06 7.21E + 06 0 1.44E + 07
(28)
k =
2.74E + 08 −4.10E + 08 1.50E + 06 −2.74E + 08 4.10E + 08 1.50E + 06−4.10E + 08 6.16E + 08 9.97E + 05 4.10E + 08 −6.16E + 08 9.97E + 051.50E + 06 9.97E + 05 7.21E + 06 −1.50E + 06 −9.97E + 05 3.60E + 06−2.74E + 08 4.10E + 08 −1.50E + 06 2.74E + 08 −4.10E + 08 −1.50E + 064.10E + 08 −6.16E + 08 −9.97E + 05 −4.10E + 08 6.16E + 08 −9.97E + 051.50E + 06 9.97E + 05 3.60E + 06 −1.50E + 06 −9.97E + 05 7.21E + 06
(29)
In table 6, the complete stiffness of each pile in horizontal-, vertical and rotational-direction are presented.For the piles with increased stiffness, 10%, 21%, 33%, 46%, 61% and 100% see appendix A.
Table 6: Stiffness of the piles
u v fi
A 4.81E+06 1.78E+09 1.44E+07B 1.04E+06 1.07E+09 8.66E+06C 3.79E+05 7.63E+08 6.18E+06D 1.67E+05 5.81E+08 4.70E+06E 2.32E+06 1.24E+09 6.96E+06F 5.01E+05 7.42E+08 4.17E+06G 1.83E+05 5.30E+08 2.98E+06H 8.04E+04 4.03E+08 2.27E+06I 1.88E+05 3.52E+08 5.63E+05J 4.05E+04 2.11E+08 3.38E+05K 1.48E+04 1.51E+08 2.41E+05L 6.51E+03 1.15E+08 1.84E+05N 2.74E+08 6.16E+08 7.21E+06R 1.90E+08 4.28E+08 3.47E+06
R’x 1.02E+07 7.28E+08 4.15E+06R’y 1.87E+08 4.34E+08 3.49E+06S 1.36E+08 3.05E+08 2.48E+06
S’x 7.13E+06 5.20E+08 2.96E+06S’y 1.34E+08 3.10E+08 2.49E+06
5.2 Height
Height was the second parameter that was evaluated. The height parameter was only investigated forInnovationen. The total height was changed six times, first the ”Crane” level was deleted, then the”Top” level, ”Roof” level and level 36. When only 35 levels remained, the building had the same amountof floors as Helix. The last two steps were to delete all floors down to level 27 and then to level 20, seefigure 39. In order to erase the different floors, ”Storey” was selected under the ”Structure” tab andthen the current floor was simply deleted.
39
(a) 36 floors, Roof- and Top-level (b) 36 floors, Roof level (c) 36 floors
(d) 35 floors (e) 27 floors (f) 20 floors
Figure 39: Different heights of Innovationen
The height of each deleted floor and the total height of the superstructure:
- Crane Level −→ 0.75m, Total height −→ 121.2m
- Top Level −→ 3.45m, Total height −→ 120.45m
- Roof Level −→ 3.15m, Total height −→ 117m
- Level 36 −→ 3.15m, Total height −→ 113.85m
- Level 27 −→ 3.15m, Total height −→ 85.5m
- Level 20 −→ 3.15m, Total height −→ 66.3m
6 Modal Analysis
Modal analysis is the most accepted method all over the world in order to design structure againstearthquake loads [12].
A modal analysis on booth Innovationen and Helix was performed in the finite element program FEM-Design.
6.1 Dynamic Calculation and Mass Definition
Before doing a seismic analysis, a dynamic calculation was performed. This step was necessary since thecalculation of the seismic effect requires the vibrations shapes and the corresponding periods.
40
To calculate the mode shapes and the corresponding period, the mass of the structure is crucial. Theself-weight of the structure was converted into mass. The same procedure was performed for the variableloads in order to consider some live loads inside the building during the earthquake.
The conversion has been performed according to EC8 3.2.4(2) where the inertial effects were evaluatedconsidering all the gravity loads, see equation 30 [11].
∑Gk,j” + ”
∑ΨE,i ·Qk,i (30)
Where the Gk,j is the self-weight of the structure, Qk,i is the variable loads and ΨE,i is a coefficient forvariable action which was defined with equation 31 [11].
ΨE,i = ϕ ·Ψ2,i (31)
The factor ϕ is set to 0.8 according to table 7 and ΨE,i is set to 0, 3 according to EN 1991-1-1:2002 [11].
Table 7: Values of ϕ for calculating ΨE,i [11]
Type of variableaction
Storey ϕ
Categories A-C*Roof 1,0Storeys with correlated occupancies 0,8Independently occupied storeys 0,5
Categories D-F*and Archives
1.0
Following the procedure above, the coefficient for variable action was calculated to be 0.24. This meansthat only 24% of the live loads were considered to act on the building during the earthquake, see figure40.
Figure 40: Input of coefficient of variable action
Before running the eigenfrequencies analysis some settings in the program were changed, see figure 41.The maximum number of shapes was set to 20 in order to reduce the calculation time. The option ”tryto reach 90% of the horizontal effective mass” was marked since it was important for the seismic analysis
41
to have the contribution of all the dominant mode shapes. The directions considered in the analysis wereonly in the x- and y-direction, the z-axis was neglected because the results in that direction were smallsince the mass of the total structure was relatively large.
Figure 41: Settings for eigenfrequencies calculation
6.2 Seismic Load and Design Spectrum
In FEM-design, the first step in order to define the seismic load was to decide what structure type theanalysis should be performed on. Both Helix and Innovationen were building structure, hence that optionwas marked in the program, see figure 42a.
The next parameter to define was the damping factor, also called viscous damping, which was set to 5%.That value was decided according to Eurocode 8. In the same window, a value of 1 was set as ”behaviorfactor for the displacements”, see figure 42a.
(a) Structure information (b) Input data for horizontal spectra
Figure 42: Seismic load
FEM-Design contains predefined design spectrum that are built up according to Eurocode 8, see section2.2.1. Figure 42b shows how the horizontal spectra was defined in the program. In this analysis, the
42
ground type A was chosen and the elastic response spectra type 2 was used. Ground type A resultsin predefined values for the parameters describing the shape of the curve, see table 3. For explanationof the parameters, see section 2.2.1. The last two values, q and β in the dialog box in figure 42b arethe behavior factor, depending on the material, type of structure and the lower bound factor for thehorizontal design spectrum respectively.
The reason for choosing ground type A was because the foundation was resting on rock ground, seetable 1. One assumption made by taking that decision was that both the building had the same groundcondition since the same curve was used for both analysis. In reality, the ground type that the twobuildings were resting on differed a little from each other. Due to that the spectra could have beendifferent but since the aim of the study was to analyze and compare the performance of the two towersunder seismic action the same input data was used.
When the spectra is defined, some setup for the seismic analysis need to be done, see figure 43. Fem-Design proposes three methods of calculation:
- Static, linear shape
- Static, mode shape
- Modal analysis
In this case, the method of calculation chosen was the modal analysis which can be used in all nationalcodes. This method requires sufficient summation of vibration shape in order to calculate the totalresponse of the structure. This was why it was important to run the eigenfrequencies calculation forenough vibration shapes. A modal analysis permits to consider all three directions, x, y and z, but in thisstudy, as mentioned before, the z-axis was neglected, hence the mass contribution in that direction waszero for all the mode shapes, see figure 43a. The number marked in yellow in the figure below representsthe mass contribution for that specific mode shape. The number in orange represents the mass of the twodominant modes. According to EC8 4.3.3.3.1(3) the sum of the modal masses in each column, marked inyellow and orange in figure 43a should be at least 90% of the total mass of the structure. When selectingthe masses, all the values above 2% were considered and the sum in percentage became 92, 3% in thex-direction and 91, 8% in the y-direction.
The summation rules in the program to choose between were two: SRSS (square root of the sum of thesquares) or CQC (complete quadratic combination). The one selected for this study was the SRSS sincethe vibration modes for individual direction were not dependent on each other according to equation 32.
TjTi
> 0, 9 and Tj ≤ Ti (32)
In equation 32 the Tj represents the period of the torsional mode while Ti is the period of the translationalmode.
43
(a) Seismic analysis, method (b) Seismic analysis, combination rule
Figure 43: Seismic analysis, setup
The dialog box on the right, in figure 43b, shows two options for the combination rule that FEM-Designcan adopt in order to solve the problem. Even in this case the SRSS was selected and the contributionof all the directions can be taken into account. In figure 43, EE is the seismic action effect while EEi isthe seismic action effect due to vibration mode i.
6.3 Finite Elements
The choice of the finite elements is a crucial step in a FE-Analysis. Depending on the element type andintegration rules, the analysis can yield different results.
For this study the ”standard” elements have been adopted, which means that the generated mesh wascomposed by 4/3/2 nodes, figure 44a, in contrast to the ”accurate” element type which contains 9/6/3nodes, figure 44b.
(a) Standard element types (b) Accurate element types
Figure 44: Finite element types
The reason for choosing the ”standard” element type is that the calculation time is considerably shorterwhen selecting this option.
6.4 Results From Seismic Calculation
After the analysis was run, nodal accelerations, displacements, modal masses etc. were extracted fromthe model. The seismic results are similar to normal static results and they can be combined usingnormal load combination.
In this study no load combinations were performed since the interesting part was the behavior underseismic action.
44
6.5 Result Extraction
In order to extract the results into graphs and tables, different tools were used besides FEM-Design:
- Text document files
- Excel with Visual Basics (VBA)
- Matlab (Plots)
- Latex (Tables)
The first step was to extract results from FEM-Design. The extracted results were:
- Acceleration (Seismic analysis)
- Displacement (Seismic analysis)
- Modal Mass (Seismic analysis)
- Frequency and Mode Shape (Eigenfrequencies analysis)
- Total and Level Mass (Eigenfrequencies analysis)
In order to extract the results, the ”List” button under the ”Analysis” tab was selected and then thecurrent information. The next choice was to save the results into text- or excel-files. In ”list” mode thechoice for which parts the results should be extracted for was ”Current selection” or ”Visible objects”.
For Accelerations and displacements, the results were imported into text files and then sorted in excelwith VBA-macro. In FEM-Design there was an error in seismic analysis. When the results were exportedit was impossible to extract the nodes for each floor separately, all nodes in the model were exported,which resulted in more than 200 000 nodes (varying depending on simulation). That was an error whichaccording to ”StruSoft” was not going to be handled until future updates.
When extracting acceleration and displacement, the maximum values for x- and y-direction were theexported results, not the total maximum value (SRSS) which was the result shown in the FE-Modelwhen checking them visually. In order to pair the correct acceleration/displacement with SRSS for eachfloor, equation 33 was used.
E =√E2x + E2
y + E2z (33)
To sort the nodes for acceleration and displacement, all nodes for each floor were imported to text filesdepending on ”Load cases” instead of ”Seismic analysis” under the ”List” tab. In that way it was possibleto sort the nodes by the correct floor in excel. When the nodes were sorted by each floor it was possibleto pair the node for each floor to the node with the correct SRSS value depending on the seismic analysis.After the sorting of SRSS values to the correct floor, the maximum value of each floor was found. Thiswas made by a VBA-code.
The maximum values were then extracted from Excel to Matlab where plots were created.
The frequencies and level masses were extracted to Excel from the eigenfrequecies analysis.
All tables were created in latex.
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7 Evaluation of Cross Sectional Properties
7.1 Hand-Calculated Values
As mentioned in section 2.3 it is important for a structure to have a bi-directional and torsional resistance.In order to evaluate the resistance of the two buildings, some geometrical properties were calculated.The following list shows what parameters have been calculated to compare the two buildings.
- Center of gravity
- Center of resistance
- Eccentricity
- Second area moment of inertia
- Torsional constant
- Torsional rigidity
- Torsional stiffness
All this parameters were evaluated for each floor of the two buildings and mean values were comparedbetween Innovationen and Helix.
The first step to calculate the center of gravity for each floor was to find relevant floor plans and handlethem like cross sections where walls and columns are the stiffening parts of the body.
To estimate the locations of the center of gravity, a reference point in the xy-plane was fixed. Figure45 illustrate a typical plan view, while equations 34a and 34b show how the center of gravity has beencalculated.
XG =
n∑i=0
bihixi
n∑i=0
bihi
(34a)
YG =
n∑i=0
bihiyi
n∑i=0
bihi
(34b)
46
Figure 45: Example of plan view
Once the center of gravity was calculated, the second area moment of inertia was estimated using theclassic formula for the inertia combined with Steiner’s theorem, see equations 35a and 35b.
Ix =
n∑i=0
bih3i
12+ bihi(YG − yi)2 (35a)
Iy =
n∑i=0
hib3i
12+ bihi(XG − xi)2 (35b)
For the center of resistance, the calculation procedure was basically the same as for the center of gravity.Instead of the area, the stiffness of each structural component (walls and columns) was used, see figure 45.What could have been important in this case if different materials or concrete classes had been used wasthe Young Modulus of the material but since it was assumed that the same concrete was used for wallsand columns, the factor was simplified. The same principles were valid for the boundary conditions whichhad influence on the stiffness of each structural component, but a simplification was used. Equation 36a- 37b shows the formula for the moment of inertia and the stiffness for each component present in theplan.
Ix,i =bih
3i
12(36a)
Iy,i =hib
3i
12(36b)
47
Kx,i =12EcIy,iL3
(37a)
Ky,i =12EcIx,iL3
(37b)
In equations 37a and 37b, the Ec is the Young modulus and the factor 12L3 is due to the boundary
condition, in this case fixed-fixed end condition with a height L. The location of the center of resistancewas calculated with equation 38a and 38b.
XR =
n∑i=0
Ky,ixi
n∑i=0
Ky,i
=
n∑i=0
Ix,ixi
n∑i=0
Ix,i
(38a)
YR =
n∑i=0
Kx,iyi
n∑i=0
Kx,i
=
n∑i=0
Ix,ixi
n∑i=0
Ix,i
(38b)
As already mentioned, it was assumed that the same boundary conditions and material properties wereused. So instead of stiffness Ki, the moment of inertia of each structural component was inserted inequations 38a and 38b.
Next stage was to calculate the eccentricity e which was simply the difference between the center of mass(also called center of gravity) and the center of resistance according to figure 12 and equation 39a and39b.
ex = |CGx − CSx| (39a)
ey = |CGy − CSy| (39b)
The next three factors that were calculated describe the torsional resistance of the structures. Thetorsional constant J is a geometry dependent parameter and it was estimated with equation 40. Oncethe torsional constant was calculated, the torsional rigidity was simply the product of the constant andthe shear modulus of the materials, see equation 41.
J =
n∑i=0
hit3i
3(40)
GJ =E
2(1 + ν)︸ ︷︷ ︸Shear modulus
· J (41)
In equation 41, E is the young modulus and the ν is the Poisson’s ratio. To finally get the torsionalstiffness, GJ
L the torsional rigidity was divided by the entire height of the structures.
7.2 Bi-directional Resistance From FE-Model
The bi-directional (bending-stiffness) was calculated from the eigenfrequencies and the active modal massin each mode. Equation 4 was used and the stiffness k was evaluated since the frequency and the mass
48
were known.
Figure 46 shows the principle of the model and how the whole structure could be idealized as a SDOFsystem where the stiffness k is calculated using equation 42.
Figure 46: Idealized structure into a SDOF-system
k = (2πf)2m (42)
49
Part III
Results
8 Frequencies and Mode Shapes
This section will provide the results obtained from the eigenfrequencies analysis performed on Innova-tionen and Helix. In section 8.3, the results are summarized in a table in order to compare the data.
8.1 Innovationen
Table 8 shows the analysis results for Innovationen. The first column indicates the mode shapes of thestructure, the second column gives the eigenfrequencies in Hz while the third shows the period in s. Thenext four columns show values on how much of the mass is contributing to the mode shapes respectively.Those values are given both in percent and in tons, for each mode there is an x- and a y-direction. Theunderlined numbers represent the most dominant mode shapes, the values above 2% have been takeninto account for the summation in the last row.
Table 8: Frequencies, mode shapes and effective masses for Innovationen
ShapeFrequency
[Hz]Period
[s]mx’[%]
my’[%]
mx’[t]
my’[t]
1 0,280 3,567 3 52,7 1130 197002 0,448 2,231 0,9 7,6 321 28223 0,487 2,052 57,8 3,9 21592 14644 1,132 0,883 1,1 10 418 37435 1,193 0,838 1,9 7,3 707 27176 1,383 0,723 22,3 2,5 8318 9347 1,973 0,507 0,1 1,1 39 4208 2,507 0,399 0,4 5,5 151 20419 2,809 0,356 3,5 0,2 1296 8210 2,853 0,351 3,4 0,2 1281 5711 3,745 0,267 0 2,4 0 89612 4,125 0,242 0 0,8 0 29213 4,328 0,231 2,4 0,2 886 6914 4,823 0,207 0 0,2 0 7215 5,288 0,189 0 1,3 0 48016 5,922 0,169 0 0,5 0 175
Sum 92,3 91,8 34502 34317
8.2 Helix
The same results are given for Helix as for Innovationen, see table 9.
50
Table 9: Frequencies, mode shapes and effective masses for Helix
ShapeFrequency
[Hz]Period
[s]mx’[%]
my’[%]
mx’[t]
my’[t]
1 0.288 3.472 0.5 55.5 151 184312 0.438 2.285 54.8 2.7 18200 8863 0.472 2.118 9 6.7 2998 22364 1.108 0.903 0.7 6.6 218 22055 1.232 0.812 1 12.2 320 40516 1.56 0.641 19.8 0 6575 07 2.051 0.488 0.7 2.5 231 8198 2.679 0.373 0.1 4.3 40 14179 2.91 0.344 0.6 0.1 196 3510 3.312 0.302 5.9 0 1954 011 3.728 0.268 0 0 0 012 3.73 0.268 0 0 0 013 3.731 0.268 0 0 0 014 3.731 0.268 0 0 0 015 3.732 0.268 0 0 0 016 3.732 0.268 0 0 0 0
Sum 90,5 90,5 30047 30045
8.3 Comparison
In order to compare the frequencies for each mode, the results from table 8 and 9 are summarized intable 10. The last two columns of the table give the kind of mode shape which has been mainly evaluatedfrom the percentage of the active modal masses and by analyzing the deformed plot of the FE-Model.
Table 10: Comparison mode shapes and frequencies
Shape nrFrequency
[Hz]Mode
Innovationen Helix Innvoationen Helix
1 0.280 0.288 Bending y Bending y.2 0.448 0.438 Torsional Bending x.3 0.487 0.472 Bending x Torsional4 1.132 1.108 Torsional Torsional5 1.193 1.232 Torsional Bending y6 1.383 1.56 Bending x Bending x7 1.973 2.051 Torsional Torsional8 2.507 2.679 Torsional Bending y9 2.809 2.91 Torsional Torsional10 2.853 3.312 Torsional Bending x
To illustrate table 10 graphically, a plot was created, see figure 47. The x-axis represents the number ofthe mode shape while the y-axis represents the frequency in Hz. In addition to the frequency of eachmode, the plot gives information of the mode shape, see legend in figure 47.
51
1 2 3 4 5 6 7
Mode
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Fre
quency (
Hz)
Innovationen
Helix
Bending in x-dir.
Bending in y-dir.
Torsional
Figure 47: Mode shapes and frequencies
The last part of this section consist of the illustration of the deformed shape of the two structures. Figure48a - 1st bending mode in y-direction for Innovationen. Figure 48b - 1st torsional mode for Innovationen.Figure 48c - 1st bending mode in x-direction for Innovationen. Figure 48d - 1st bending mode in y-direction for Helix. Figure 48e - 1st bending mode in x-direction for Helix. Figure 48f - 1st torsionalmode for Helix. For modes shape 4− 6 see Appendix B.
52
(a) Mode 1 - f = 0.280Hz (b) Mode 2 - f = 0.448Hz. (c) Mode 3 - f = 0.487Hz
(d) Mode 1 - f = 0.288Hz (e) Mode 2 - f = 0.438Hz (f) Mode 3 - f = 0.472Hz
Figure 48: Mode Shapes, Innovationen figure a-c, Helix figure d-f
53
9 Total and Level Masses
In this section, the level masses of the buildings are plotted in graphs in order to illustrate how the massof the structures is distributed in the x- and y-direction. A relevant result in addition to the level massesare the total lumped mass of the two structures, which can also be seen in the same plot.
9.1 Comparison Between Innvationen and Helix
Figure 49 illustrates the distribution of the mass in the two main directions for both Innovationen andHelix. The x-axis represents the dislocation in m from the local coordinate system of the models whilethe y-axis indicates the height of the building. The small markers represent the lumped mass of eachstorey level while the large markers are the total lumped mass of the buildings. For values, see AppendixC.
10 15 20 25
x-coordinate (m)
0
20
40
60
80
100
120
He
igh
t (m
)
Level masses - Innovationen
Total lumped mass - Innovationen
Level masses - Helix
Total lumped mass - Helix
10 11 12 13 14
y-coordinate (m)
0
20
40
60
80
100
120
He
igh
t (m
)
Figure 49: Total- and level-masses
54
10 Bi-directional and Torsional Stiffness
This section presents the results obtained from the calculation of the bi-directional- and torsional stiffness.For full calculation procedure of one floor see Appendix D. The table in this section presents mean valuesof cross sectional properties.
10.1 Innovationen
Table 11 gives the relevant cross sectional properties of Innovationen.
Table 11: Cross sectional data floor plans
CG
[m]
CS
[m]
e
[m]
I
[m4]
J
[m4]
GJ
[kNm2
rad ]
GJL
[kNmrad ]
x y x y x y x y
17.16 12.22 14.69 12.87 2.95 0.69 1902 5479 2.63 4.11E+07 3.40E+05
In order to understand the factors presented in table 11, see list below.
- CG: Center of gravity
- CS: Center of resistance
- e: Eccentricity
- I: Second area moment of inertia
- J: Torsional constant
- GJ: Torsional rigidity (G is the shear modulus)
- GJ/L: Torsional stiffness (L is the total height of the building)
10.2 Helix
Table 12 gives the relevant cross sectional properties of Helix. For explanations of all the factors, seesection 10.1.
Table 12: Cross sectional data floor plans
CG
[m]
CS
[m]
e
[m]
I
[m4]
J
[m4]
GJ
[kNm2
rad ]
GJL
[kNmrad ]
x y x y x y x y
20.29 9.68 19.77 8.59 0.78 0.34 1112 6214 2.29 3.59E+07 3.24E+05
55
10.3 Bending Stiffness
The bending stiffness was estimated from the eigenfrequencies and the modal masses, see table 8 insection 8.1. The results from the rough estimation of the stiffness k is presented in table 13.
Table 13: Bending stiffness [kN/m]
x-direction y-direction
Innovationen Helix Innovationen Helix
1.834E+05 1.250E+05 5.531E+05 5.475E+05
11 Accelerations and Displacements
11.1 Comparison Between Accelerations
Accelerations obtained from the seismic analysis are plotted in figure 50. The y-axis represents the totalheight of the building in m while the x-axis represents the acceleration in m/s2. In order to obtain aclear graph, the considered accelerations are the maximum at each floor represented by dots in the plots.
56
0 0.5 1
ax (m/s2)
0
20
40
60
80
100
120H
eig
ht
(m)
Innovationen
Helix
0 0.5 1 1.5
ay (m/s2)
0
20
40
60
80
100
120
He
igh
t (m
)
0 0.5 1 1.5
SRSS (m/s2)
0
20
40
60
80
100
120
He
igh
t (m
)
Figure 50: Accelerations from Seismic Max
For acceleration values, see table 24 and 25 in Appendix E, where the first two columns indicate the planand the height respectively while the last three columns represent the acceleration values on each floorin the x-direction, y-direction and the SRSS.
57
11.2 Comparison Between Displacements
(a) Innovationen (b) Helix
Figure 51: Deformed shape
Displacements obtained from the seismic analysis are plotted in figure 52. The y-axis represents the totalheight of the building in m while the x-direction represents the displacements in mm. In order to obtaina clear graph, the considered displacements are the maximum at each floor represented by dots in theplots.
58
0 20 40
dx (mm)
0
20
40
60
80
100
120H
eig
ht
(m)
Innovationen
Helix
0 20 40 60 80
dy (mm)
0
20
40
60
80
100
120
He
igh
t (m
)
0 20 40 60 80
SRSS (mm)
0
20
40
60
80
100
120
He
igh
t (m
)
Figure 52: Displacements from Seismic Max
For displacement values, see table 27 and 28 in Appendix F, where the first two columns indicate theplan and the height respectively while the last three columns represent the displacement values on eachfloor in the x-direction, y-direction and the SRSS.
12 Increased Pile Stiffness
In this section the results obtained after having increased the stiffness of the piles in the model arepresented. The extracted results are frequencies, mode shapes, modal masses, accelerations and dis-placements.
12.1 Frequencies and Mode Shapes
Table 14 shows the frequencies, mode shapes and the effective masses obtained when having modeledthe piles as elastic supports.
59
Table 14: Frequencies, mode shapes and effective masses for Innovationen with elastic supports
ShapeFrequency
[Hz]Period
[s]mx’[%]
my’[%]
mx’[t]
my’[t]
1 0.274 3.646 2.7 54.6 1022 204102 0.439 2.278 0.2 7.2 89 26893 0.481 2.081 59.6 3 22286 11164 1.118 0.895 1.4 10.3 511 38655 1.186 0.843 1.9 6.7 710 25126 1.365 0.733 22.1 2.7 8266 9947 1.942 0.515 0 1.2 0 4618 2.483 0.403 0.4 5.4 161 20229 2.778 0.36 4.8 0.3 1800 12010 2.815 0.355 1.8 0.1 675 4611 3.71 0.27 0 2.5 0 92612 4.105 0.244 0 0.7 0 27213 4.279 0.234 2.1 0.2 799 7014 4.779 0.209 0 0.2 0 7315 5.242 0.191 0 1.3 0 48016 5.883 0.17 0 0.4 0 158
Sum 90,3 92,4 34173 34535
No significant differences in frequencies and mode shapes were observed after having increased the stiffnessof the piles.
12.2 Accelerations and Displacements
Figure 53 shows the maximum acceleration and displacement in the building when increasing the stiffnessof the piles. As can be seen in the plots, both the accelerations and displacements remained pretty muchunchanged.
0% 10% 21% 33% 46% 61% 100%1.24
1.25
1.26
1.27
1.28
1.29
1.3
1.31
1.32
1.33
1.34
SR
SS
(m
/s2)
(a) Acceleration
0% 10% 21% 33% 46% 61% 100%68
68.5
69
69.5
70
70.5
71
71.5
72
SR
SS
(m
m)
(b) Displacement
Figure 53: Varying stiffness of the piles
60
13 Varying Height
The last parameter that was changed in order to analyze the behavior of the building was the height.
13.1 Frequencies and Mode Shapes
The frequencies were extracted in order to see how the values varied when decreasing the height. Figure54 shows the frequency for each mode when decreasing the number of floors in the structure. For values,see table 21 in Appendix B.
2 4 6 8 10 12 14 16
Mode
0
1
2
3
4
5
6
7
8
9
10
Fre
qu
en
cy (
Hz)
39 floors
38 floors
37 floors
36 floors
35 floors
27 floors
20 floors
Figure 54: Frequencies with varying height
13.2 Accelerations
Figure 55 shows a plot of the maximum acceleration on each floor when decreasing the height of thebuilding. The x-axis represents the acceleration and the y-axis represents the height of the building.
61
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
SRSS (m/s2)
0
20
40
60
80
100
120H
eig
ht
(m)
39 floors
38 floors
37 floors
36 floors
35 floors
27 floors
20 floors
Figure 55: Max accelerations with varying height
62
Part IV
Discussion and Conclusions
14 Discussion
Bi-Directional- and Torsional-Stiffness
A comparison was performed between Innovationen and Helix in Section 8.3, see table 10 and figure 47.As we can see in the table, the frequencies of the first mode are similar, 0.280Hz for Innovationen and0.288Hz for Helix, and both of them are bending modes in the y-direction. This is reasonable since bothof the buildings are thinner in that direction and similar.
When it comes to the second mode, the difference in frequencies remains relatively small. What we cansee is that the second mode of Innovationen (0.488Hz) is a torsional mode while Helix second mode(0.438Hz) is a bending mode in the x-direction. What can also be observed from figure 48e is that sometorsion occurs in the displayed shape and makes it more similar to Innovationen.
The third mode is, like the first two modes, similar in frequencies, 0.487Hz for Innovationen and 0.472Hzfor Helix but different in mode shapes. Innovationen resulted in a bending mode in the x-direction whileHelix reached its first torsional mode.
According to the hand calculations there are some contradictions since the torsional stiffness is lower forHelix (3.24E + 5 kNm/rad) than for Innvoationen (3.40E + 5 kNm/rad), even though they are close toeach other, see table 11 and 12. The same contradictions are valid when it comes to the bending stiffness.Table 13 shows higher values in both x-direction and y-direction for Innovationen and lower values forHelix, in contrast to the moment of inertia presented in table 11 and 12, where Helix has higher valuein the one about the y-axis (6214m4). This gap in the results could be due to approximations and notfully correct assumptions when calculating the torsional resistance from the floor plans. It could also bedue to a different modeling approach for Helix compared to Innovationen in the software FEM-Design.
What can also be discussed is the eccentricity (e), see table 11 and 12, calculated by taking the differencein meters between the center of resistance and the center of mass. Those values turn out to be smallerfor Helix than for Innovationen which is favorable for the torsional resistance according to figure 12 insection 2.3.4.
From figure 47 it can be seen that the curve describing the frequencies and the mode of the two buildingsfollow the same shape for the first four modes. After the 4th mode, Innovationen has lower frequenciesthan Helix. That can be due to the smaller amount of active modal mass in Helix, which leads to astructure more sensible to higher frequencies, in contrast to Innovationen which is more sensible to lowerfrequencies.
Total- and Level-Masses
The total- and level- masses describe the distribution of the mass along the x-, y- and z-axis. As wecan see in figure 49, Innovationen has a more uniformed mass distribution than Helix in both the x-and y-direction. This means, according to Eurocode 8 and section 2.3.2, that Innovationen will probablydistribute the inertia forces in a more suitable way.
63
Accelerations and Displacements
Accelerations and displacements are important results since they describe how the comfort in the buildingwill be.
In figure 50 the comparison between Innovationen and Helix shows that Innovationen has higher accel-erations in x-direction and SRSS, however the maximum acceleration is found at the top floor of Helix.Innovationen has also slightly higher acceleration in y-direction in general but the maximum accelerationis found at the top floor of Helix. From this results we can deduce that Innovation has a stiffer structurein the x-direction, which is also proved from the results in table 13. However, the difference in stiffnessis not that large in the y-direction.
To further strengthen that theory, the displacement plot can also be observed in figure 52, where Inno-vationen has smaller values in both the x- and y-direction. According to the relationship F = k · u →u = F/k a stiffer structure gives smaller displacements and since the seismic load applied is the same forboth structures, results show that the stiffness k is higher for Innovationen.
Increased Pile Stiffness on Innovationen
The results obtained after having modeled the foundation of Innovationen with elastic supports repre-senting the piles resulted in a decreased stiffness of the whole structure. Table 8, which represents hingedline supports, shows a frequency of 0.280Hz while table 14, elastic point supports, shows 0.274Hz. Thatwas observed from the extracted eigenfrequencies, which were lower for all the modes.
When the stiffness of all the piles were increased proportionally by a multiplication factor, we found thatthe frequencies, accelerations and displacements remained unchanged. The maximum acceleration anddisplacement in Innovationen were approximately 1.29m/s2 and 70mm respectively, see figure 53a and53b.
Decreased Height on Innovationen
The last investigated parameter turned out to be the one with most influence on the behavior of Inno-vationen under seismic action. From the plots in figure 55 it is obvious that a decreased height leads toincreased accelerations along the height of the building.
According to the graph in figure 54 the frequencies of the structure increase with a decreasing height.What can also be seen from the graph is that the derivative, thus the inclination of the plots, increaseand the gap between the mode will be larger when decreasing the number of floors.
15 Conclusions
The following points describe the conclusions of this study.
- According to the hand calculations Innovationen has a higher torsional stiffness and a higherbending stiffness in the y-direction than Helix. However the bending stiffness in the x-direction ishigher for Helix than for Innovationen.
- The modal analysis shows that Innovationen has a lower torsional stiffness but a higher bendingstiffness in both the x and y-direction.
64
- The mass of Innovationen is more evenly distributed than for Helix even though the center ofstiffness and the center of mass are closer to each other in Helix.
- The accelerations are in general higher in Innovationen along the height of the building, howeverthe maximum acceleration is on the top floor for Helix.
- Helix has larger displacements than Innovationen along the whole height of the structure.
- Modeling the piles as elastic supports instead of rigid line supports leads to decreased eigenfre-quencies and therefore the stiffness of the building is reduced.
- Increasing the stiffness of the piles has no considerable effect on frequencies, accelerations anddisplacements in the building.
- Decreasing the height of the building increase the eigenfrequencies and leads to increased acceler-ations but decreased displacements.
Innovationen and Helix have similar resistance when comparing our results against the six principles ofconceptual design. Both buildings fulfill the requirements differently.
- Structural simplicity (No conclusion)
- Uniformity, symmetry and redundancy (Innovationen fulfills the requirements better than Helix)
- Bi-directional resistance and stiffness (Innovationen fulfills the requirements better than Helix)
- Torsional resistance and stiffness (Helix fulfills the requirements better than Innovationen)
- Diaphragmatic behavior at storey level (No conclusion)
- Adequate foundation (Helix fulfills the requirements better than Innovationen)
Our last conclusion is that the height parameter has more impact on the earthquake resistance than theground parameter (stiffness of piles).
Future Work
The Following list consists of suggestions for further work
- Comparison of different types of parameters, such as mass and soil.
- Other methods of calculations, such as static linear- or mode-shape.
- Non-linear static analysis pushover.
- Usage of other FEM-programs to compare the results.
- Perform a field experiment installing accelerometers on the buildings in order to extract relevantdata. In such a way it is possible to measure the reliability of the FE-Model.
- Time-History analysis.
65
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[38] SWECO Structures(2019). Helix, NYBYGGNAD BOSTADSHUS - VANINGSPLAN 30[Internal Document]. Sundsvall: Sweco Structures
[39] SWECO Structures(2018). Helix, NYBYGGNAD BOSTADSHUS -PLATSGJUTNINGSDETALJER [Internal Document]. Sundsvall: Sweco Structures
[40] Binfare M(2015). Berakningsrapport - NORRA TORNEN, INNOVATIONEN, BH PREFAB[Internal Document]. Sundsvall: Sweco Structures
[41] IRIS Incorporated Reaserch Institutions for Seismology. Buildings and earthquakes - Whichstands? Which falls? [Internet]. New York: IRIS; [Date unknown][cited 2019 February 12].Available from:https://www.iris.edu/hq/files/programs/education and outreach/retm/tm 100112 haiti/BuildingsInEQs.pdf
[42] Taranath Bungale S. TALL BUILDING. 1 ed. Boca Raton: Taylor & Francis Group; 2017
[43] Kokusho T. INNOVATIVE EARTHQUAKE SOIL DYNAMICS. 1 ed. London: CRC Press; 2017
[44] Cakmak A.S. Soil Dynamics and Liquefaction: 42 ed. Southampton: Elsevier Science PublishersB.V.; 1987
[45] Junbo J. Soil Dynamics and Foundation Modeling - Offshore and Earthquake Engineering. 1 ed.Bergen: Springer International Publishing AG; 2018
[46] Manohar S, Madhekar S. Seismic Design of RC Buildings - Theory and Practice. 1 ed.Maharashtra: Springer India; 2015
[47] Yiu C, Chan C, Huang M, Li G. Evaluation of lateral-torsional coupling in earthquake responseof asymmetric multistory buildings [Internet].Struct. Design Tall Spec. Build. 2014; 23: 1007–1026
[48] Hejal R, Chopra Anil K. Lateral-torsional coupling in earthquake response of Frame buildings. J.Struct. Eng. 1989; 115(4): 852-867
[49] Strusoft. FEM-Design [Internet]. Malmo: Strusoft; 2019. [cited 2019 May 17]. Available from:https://strusoft.com/products/fem-design
A Appendix - Pile stiffness
Table 15: Stiffness of the piles +10%
u v fi
A 5.29E+06 1.96E+09 1.59E+07B 1.14E+06 1.18E+09 9.52E+06C 4.16E+05 8.40E+08 6.80E+06D 1.83E+05 6.39E+08 5.18E+06E 2.55E+06 1.36E+09 7.65E+06F 5.51E+05 8.16E+08 4.59E+06G 2.01E+05 5.83E+08 3.28E+06H 8.85E+04 4.44E+08 2.50E+06I 2.06E+05 3.87E+08 6.19E+05J 4.46E+04 2.32E+08 3.72E+05K 1.62E+04 1.66E+08 2.65E+05L 7.16E+03 1.26E+08 2.02E+05N 3.01E+08 6.78E+08 7.93E+06R 2.09E+08 4.70E+08 3.82E+06
R’x 1.13E+07 8.00E+08 4.56E+06R’y 2.06E+08 4.77E+08 3.84E+06S 1.49E+08 3.36E+08 2.73E+06
S’x 7.85E+06 5.72E+08 3.26E+06S’y 1.47E+08 3.41E+08 2.74E+06
Table 16: Stiffness of the piles +33%
u v fi
A 5.82E+06 2.16E+09 1.75E+07B 1.26E+06 1.29E+09 1.05E+07C 4.58E+05 9.24E+08 7.48E+06D 2.02E+05 7.03E+08 5.69E+06E 2.81E+06 1.50E+09 8.42E+06F 6.06E+05 8.98E+08 5.05E+06G 2.21E+05 6.41E+08 3.61E+06H 9.73E+04 4.88E+08 2.75E+06I 2.27E+05 4.26E+08 6.81E+05J 4.90E+04 2.55E+08 4.09E+05K 1.79E+04 1.82E+08 2.92E+05L 7.87E+03 1.39E+08 2.22E+05N 3.31E+08 7.46E+08 8.72E+06R 2.30E+08 5.17E+08 4.20E+06
R’x 1.24E+07 8.80E+08 5.02E+06R’y 2.26E+08 5.25E+08 4.22E+06S 1.64E+08 3.69E+08 3.00E+06
S’x 8.63E+06 6.29E+08 3.58E+06S’y 1.62E+08 3.75E+08 3.02E+06
Table 17: Stiffness of the piles +46%
u v fi
A 7.02E+06 2.60E+09 2.11E+07B 1.52E+06 1.56E+09 1.26E+07C 5.53E+05 1.11E+09 9.03E+06D 2.43E+05 8.48E+08 6.87E+06E 3.39E+06 1.81E+09 1.02E+07F 7.31E+05 1.08E+09 6.10E+06G 2.67E+05 7.74E+08 4.35E+06H 1.17E+05 5.89E+08 3.31E+06I 2.74E+05 5.14E+08 8.22E+05J 5.92E+04 3.08E+08 4.93E+05K 2.16E+04 2.20E+08 3.52E+05L 9.50E+03 1.68E+08 2.68E+05N 4.00E+08 9.00E+08 1.05E+07R 2.78E+08 6.24E+08 5.07E+06
R’x 1.50E+07 1.06E+09 6.05E+06R’y 2.73E+08 6.33E+08 5.10E+06S 1.98E+08 4.46E+08 3.62E+06
S’x 1.04E+07 7.59E+08 4.32E+06S’y 1.95E+08 4.52E+08 3.64E+06
Table 18: Stiffness of the piles +61%
u v fi
A 6.40E+06 2.37E+09 1.92E+07B 1.38E+06 1.42E+09 1.15E+07C 5.04E+05 1.02E+09 8.22E+06D 2.22E+05 7.73E+08 6.26E+06E 3.08E+06 1.65E+09 9.25E+06F 6.66E+05 9.87E+08 5.55E+06G 2.43E+05 7.05E+08 3.97E+06H 1.07E+05 5.36E+08 3.02E+06I 2.50E+05 4.68E+08 7.49E+05J 5.39E+04 2.81E+08 4.49E+05K 1.96E+04 2.01E+08 3.21E+05L 8.65E+03 1.53E+08 2.44E+05N 3.64E+08 8.20E+08 9.58E+06R 2.53E+08 5.69E+08 4.62E+06
R’x 1.36E+07 9.68E+08 5.52E+06R’y 2.49E+08 5.77E+08 4.64E+06S 1.81E+08 4.06E+08 3.30E+06
S’x 9.49E+06 6.91E+08 3.94E+06S’y 1.78E+08 4.12E+08 3.32E+06
Table 19: Stiffness of the piles +77%
u v fi
A 7.02E+06 2.60E+09 2.11E+07B 1.52E+06 1.56E+09 1.26E+07C 5.53E+05 1.11E+09 9.03E+06D 2.43E+05 8.48E+08 6.87E+06E 3.39E+06 1.81E+09 1.02E+07F 7.31E+05 1.08E+09 6.10E+06G 2.67E+05 7.74E+08 4.35E+06H 1.17E+05 5.89E+08 3.31E+06I 2.74E+05 5.14E+08 8.22E+05J 5.92E+04 3.08E+08 4.93E+05K 2.16E+04 2.20E+08 3.52E+05L 9.50E+03 1.68E+08 2.68E+05N 4.00E+08 9.00E+08 1.05E+07R 2.78E+08 6.24E+08 5.07E+06
R’x 1.50E+07 1.06E+09 6.05E+06R’y 2.73E+08 6.33E+08 5.10E+06S 1.98E+08 4.46E+08 3.62E+06
S’x 1.04E+07 7.59E+08 4.32E+06S’y 1.95E+08 4.52E+08 3.64E+06
Table 20: Stiffness of the piles +95%
u v fi
A 6.40E+06 2.37E+09 1.92E+07B 1.38E+06 1.42E+09 1.15E+07C 5.04E+05 1.02E+09 8.22E+06D 2.22E+05 7.73E+08 6.26E+06E 3.08E+06 1.65E+09 9.25E+06F 6.66E+05 9.87E+08 5.55E+06G 2.43E+05 7.05E+08 3.97E+06H 1.07E+05 5.36E+08 3.02E+06I 2.50E+05 4.68E+08 7.49E+05J 5.39E+04 2.81E+08 4.49E+05K 1.96E+04 2.01E+08 3.21E+05L 8.65E+03 1.53E+08 2.44E+05N 3.64E+08 8.20E+08 9.58E+06R 2.53E+08 5.69E+08 4.62E+06
R’x 1.36E+07 9.68E+08 5.52E+06R’y 2.49E+08 5.77E+08 4.64E+06S 1.81E+08 4.06E+08 3.30E+06
S’x 9.49E+06 6.91E+08 3.94E+06S’y 1.78E+08 4.12E+08 3.32E+06
B Appendix - Frequencies and Mode Shapes
(a) Mode 4 - f = 1.132Hz (b) Mode 5 - f = 1.193Hz (c) Mode 6 - f = 1.383Hz
(d) Mode 4 - f = 1.108Hz (e) Mode 5 - f = 1.232Hz (f) Mode 6 - f = 1.56Hz
Figure 56: Mode Shapes, Innovationen figure a-c, Helix figure d-f
Table 21: Frequencies for Innovationen for varying height
ModeFrequencies [Hz] for respective number of floors
39 38 37 36 35 27 20
1 0.280 0.283 0.291 0.304 0.318 0.460 0.5922 0.448 0.448 0.451 0.456 0.462 0.527 0.7613 0.487 0.492 0.505 0.528 0.551 0.769 1.0754 1.132 1.138 1.153 1.183 1.213 1.507 1.9415 1.193 1.201 1.232 1.286 1.345 1.973 3.0006 1.383 1.397 1.437 1.505 1.578 2.338 3.3407 1.973 1.975 1.992 2.028 2.068 2.528 3.4308 2.507 2.525 2.589 2.690 2.806 3.816 5.1319 2.809 2.825 2.860 2.916 2.981 4.035 5.76310 2.853 2.872 2.938 3.052 3.179 4.434 6.49211 3.745 3.747 3.802 3.913 4.064 4.998 7.20812 4.125 4.169 4.279 4.447 4.613 6.039 8.04313 4.328 4.365 4.476 4.660 4.872 6.368 8.26014 4.823 4.825 4.856 4.933 5.057 6.755 9.24915 5.288 5.307 5.399 5.568 5.810 7.478 9.49416 5.922 5.934 5.984 6.084 6.217 7.888 9.558
C Appendix - Total and level masses
Table 22: Total and level masses - Innovationen
Mass[t]
x-coor[m]
y-coor[m]
z-coor[m]
223 19.4 13.1 0.0933 19.9 13.1 3.61079 19.6 13.0 6.81098 19.4 13.1 9.91246 21.1 13.4 13.11329 21.6 13.7 16.21303 21.4 13.5 19.41274 21.1 13.6 22.51256 21.0 13.5 25.71279 21.1 13.6 28.81258 21.0 13.5 32.01280 21.1 13.7 35.11249 20.9 13.5 38.31278 21.2 13.7 41.41254 20.9 13.5 44.61279 21.2 13.6 47.71190 20.1 13.5 50.91037 18.1 13.1 54.01063 18.2 13.1 57.21037 18.2 13.2 60.31062 18.2 13.1 63.51044 18.2 13.1 66.61007 17.3 12.9 69.8871 15.2 12.5 72.9789 13.8 12.4 76.1792 13.9 12.3 79.2800 14.0 12.4 82.4788 13.9 12.3 85.5801 13.9 12.4 88.7792 13.9 12.2 91.8741 13.1 12.1 95.0632 11.5 12.1 98.1645 11.7 11.8 101.3644 11.7 12.1 104.4650 11.7 11.8 107.6633 11.7 12.1 110.7653 12.1 12.0 113.9640 11.7 12.2 117.0308 11.3 12.0 120.5130 14.0 12.7 121.2
Total 37369 18,0 13,0 53,6
Table 23: Total and level masses - Helix
Mass[t]
x-coor[m]
y-coor[m]
z-coor[m]
269 21.2 13.0 0.01084 22.1 12.8 4.11019 22.3 12.9 7.31074 22.5 12.9 10.41220 23.8 12.9 13.61257 23.9 13.0 16.71238 23.7 12.9 19.91219 23.5 12.9 23.01212 23.3 12.8 26.21216 23.5 12.9 29.31213 23.3 12.8 32.51209 23.4 12.9 35.61228 23.5 12.8 38.81170 22.6 12.6 41.91115 21.4 12.5 45.11107 21.3 12.4 48.21176 21.0 12.5 51.41081 20.9 12.4 54.51055 20.0 12.1 57.7923 17.6 11.7 60.8870 16.5 11.8 64.0860 16.3 11.6 67.1867 16.5 11.8 70.3863 16.4 11.5 73.4814 15.5 11.4 76.6757 14.3 11.3 79.7761 14.4 11.3 82.9758 14.3 11.3 86.0757 14.4 11.3 89.2761 14.3 11.3 92.3636 13.0 11.5 95.5575 11.3 10.9 98.6538 11.2 11.3 101.8572 11.3 10.9 104.9480 11.4 11.5 108.1257 11.6 11.3 111.2
- - - -- - - -- - - -- - - -
Total 33212 19.7 12.3 49.3
D Appendix - Calculation of Cross Sectional Properties of FloorPlans
E Appendix - Accelerations
Table 24: Acceleration values - Innovationen
Plan Height [m]Seismic Max [m/s2]
ax ay SRSS
0 0 0.091 0.056 0.1121 3.6 0.18 0.229 0.3022 6.75 0.266 0.453 0.5263 9.9 0.296 0.577 0.6494 13.05 0.461 0.716 0.8575 16.2 0.502 0.608 0.8126 19.35 0.493 0.534 0.7527 22.5 0.467 0.577 0.7448 25.65 0.408 0.657 0.7769 28.8 0.452 0.632 0.77910 31.95 0.385 0.668 0.77411 35.1 0.377 0.64 0.74612 38.25 0.381 0.613 0.72613 41.4 0.505 0.514 0.73814 44.55 0.553 0.548 0.79515 47.7 0.617 0.577 0.85516 50.85 0.511 0.697 0.88417 54 0.53 0.642 0.83518 57.15 0.47 0.727 0.86819 60.3 0.55 0.687 0.88220 63.45 0.458 0.733 0.86721 66.6 0.519 0.646 0.83122 69.75 0.508 0.595 0.78823 72.9 0.466 0.566 0.74124 76.05 0.524 0.495 0.73325 79.2 0.567 0.509 0.77326 82.35 0.63 0.545 0.84027 85.5 0.598 0.655 0.89028 88.65 0.692 0.592 0.91429 91.8 0.582 0.651 0.87930 94.95 0.594 0.541 0.81331 98.1 0.297 0.552 0.69932 101.25 0.281 0.452 0.59933 104.4 0.292 0.439 0.63434 107.55 0.344 0.516 0.70535 110.7 0.421 0.686 0.87636 113.85 0.508 0.842 1.041
Roof 117 0.83 0.855 1.197Top 120.45 0.744 0.999 1.293
Crane Level 121.2 0.795 0.657 1.049
Table 25: Acceleration values - Helix
Plan Height [m]Seismic Max [m/s2]
ax ay SRSS
0 0 0.066 0.017 0.0801 4.125 0.131 0.147 0.1992 7.275 0.183 0.269 0.3303 10.425 0.258 0.376 0.4574 13.575 0.304 0.481 0.5705 16.725 0.343 0.468 0.5816 19.875 0.348 0.500 0.6117 23.025 0.436 0.448 0.6258 26.175 0.463 0.470 0.6619 29.325 0.434 0.548 0.70010 32.475 0.374 0.599 0.70711 35.625 0.382 0.593 0.70712 38.775 0.321 0.614 0.69513 41.925 0.334 0.582 0.67314 45.075 0.289 0.577 0.64715 48.225 0.339 0.532 0.63316 51.375 0.309 0.518 0.60517 54.525 0.375 0.472 0.60718 57.675 0.407 0.429 0.59519 60.825 0.398 0.480 0.63020 63.975 0.368 0.540 0.66021 67.125 0.386 0.582 0.70822 70.275 0.359 0.657 0.75723 73.425 0.346 0.675 0.77124 76.575 0.301 0.710 0.78325 79.725 0.290 0.673 0.74926 82.875 0.218 0.655 0.70927 86.025 0.243 0.576 0.64928 89.175 0.207 0.524 0.59129 92.325 0.318 0.349 0.56530 95.475 0.347 0.358 0.56931 98.625 0.439 0.377 0.69632 101.775 0.517 0.653 0.85433 104.925 0.662 0.819 1.07134 108.075 0.712 1.147 1.36235 111.225 0.791 1.307 1.538- - - - -- - - - -- - - - -- - - - -
Table 26: Acceleration values for Innovationen with varying height
Plan Height [m]SRSS from Seismic Max [m/s2]
39 38 37 36 35 27 20
0 0 0.112 0.113 0.118 0.124 0.108 0.146 0.1671 3.6 0.302 0.293 0.278 0.295 0.287 0.408 0.4712 6.75 0.526 0.516 0.485 0.513 0.502 0.586 0.7973 9.9 0.649 0.627 0.583 0.616 0.603 0.787 0.9334 13.05 0.857 0.844 0.778 0.807 0.795 0.966 1.1875 16.2 0.812 0.844 0.766 0.787 0.765 1.073 1.3026 19.35 0.752 0.809 0.731 0.746 0.719 1.102 1.3557 22.5 0.744 0.742 0.737 0.743 0.697 1.073 1.3318 25.65 0.776 0.780 0.780 0.793 0.760 1.022 1.2929 28.8 0.779 0.792 0.807 0.816 0.791 0.964 1.23810 31.95 0.774 0.794 0.812 0.825 0.805 0.967 1.24011 35.1 0.746 0.791 0.797 0.812 0.788 1.001 1.26012 38.25 0.726 0.770 0.782 0.797 0.768 1.067 1.29013 41.4 0.738 0.753 0.771 0.787 0.741 1.094 1.23914 44.55 0.795 0.786 0.772 0.791 0.735 1.116 1.16415 47.7 0.855 0.844 0.789 0.812 0.745 1.100 1.06816 50.85 0.884 0.906 0.831 0.850 0.780 1.088 1.07017 54 0.835 0.853 0.854 0.878 0.818 1.075 1.02818 57.15 0.868 0.883 0.886 0.909 0.863 1.072 1.26919 60.3 0.882 0.894 0.898 0.919 0.886 1.061 1.53420 63.45 0.867 0.894 0.891 0.915 0.899 1.036 1.75321 66.6 0.831 0.879 0.857 0.886 0.898 1.026 -22 69.75 0.788 0.844 0.835 0.877 0.910 0.988 -23 72.9 0.741 0.806 0.810 0.872 0.917 0.836 -24 76.05 0.733 0.760 0.821 0.900 0.929 0.944 -25 79.2 0.773 0.795 0.861 0.940 0.935 1.101 -26 82.35 0.840 0.849 0.912 0.975 0.916 1.234 -27 85.5 0.890 0.891 0.930 0.961 0.876 1.392 -28 88.65 0.914 0.900 0.919 0.901 0.776 - -29 91.8 0.879 0.902 0.846 0.815 0.695 - -30 94.95 0.813 0.857 0.751 0.719 0.659 - -31 98.1 0.699 0.789 0.702 0.705 0.720 - -32 101.25 0.599 0.699 0.630 0.731 0.862 - -33 104.4 0.634 0.654 0.715 0.891 1.052 - -34 107.55 0.705 0.729 0.835 1.065 1.238 - -35 110.7 0.876 0.904 1.021 1.265 1.408 - -36 113.85 1.041 1.084 1.206 1.465 - - -
Roof 117 1.197 1.247 1.377 - - - -Top 120.45 1.293 1.351 - - - - -
Crane Level 121.2 1.049 - - - - - -
F Appendix - Displacements
Table 27: Displacement values - Innovationen
Plan Height [m]Seismic Max [mm]
ax ay SRSS
0 0 0.39 0.16 0.581 3.6 0.96 1.03 1.672 6.75 1.44 2.19 2.653 9.9 1.83 3.11 3.654 13.05 2.06 4.54 5.235 16.2 3.35 4.66 6.096 19.35 2.78 6.12 6.987 22.5 4.94 6.25 8.208 25.65 5.68 7.58 9.719 28.8 6.43 9.02 11.3010 31.95 5.70 11.57 12.9511 35.1 6.30 13.24 14.7112 38.25 6.91 14.93 16.5013 41.4 7.54 16.68 18.3514 44.55 8.16 18.44 20.2215 47.7 8.80 20.25 22.1216 50.85 9.45 22.02 24.0117 54 10.04 23.78 25.8618 57.15 10.69 25.57 27.7719 60.3 11.37 27.42 29.7320 63.45 12.06 29.28 31.7221 66.6 12.78 31.19 33.7622 69.75 13.51 33.11 35.8223 72.9 14.24 35.01 37.8524 76.05 15.01 36.90 39.8925 79.2 15.80 38.77 41.9326 82.35 16.60 40.63 43.9527 85.5 17.43 42.50 45.9928 88.65 18.27 44.34 48.0129 91.8 19.12 46.19 50.0430 94.95 19.91 47.98 52.0031 98.1 20.71 49.81 54.0032 101.25 21.58 51.67 56.0433 104.4 22.45 53.52 58.0934 107.55 23.34 55.33 60.1035 110.7 24.22 57.13 62.1036 113.85 25.11 58.88 64.06
Roof 117 25.99 60.57 65.95Top 120.45 29.16 61.02 67.67
Crane Level 121.2 25.71 55.68 61.38
Table 28: Displacements values - Helix
Plan Height [m]Seismic Max [mm]
ax ay SRSS
0 0 0.336 0.237 0.5511 4.125 0.853 0.758 1.5602 7.275 1.208 2.138 2.5153 10.425 2.027 3.349 4.1904 13.575 1.921 5.271 5.6815 16.725 2.529 6.344 6.8686 19.875 2.778 7.510 8.0727 23.025 3.419 8.913 9.5898 26.175 3.891 10.298 11.0669 29.325 4.447 11.714 12.56510 32.475 5.229 13.177 14.23211 35.625 7.657 13.817 15.88512 38.775 9.246 14.990 17.80813 41.925 9.449 17.579 20.04514 45.075 11.192 18.647 21.93415 48.225 11.237 21.419 24.27516 51.375 13.097 22.319 26.06017 54.525 12.819 25.236 28.39318 57.675 14.810 26.178 30.25519 60.825 14.546 29.310 32.80920 63.975 16.644 30.181 34.63821 67.125 16.345 33.520 37.37922 70.275 18.603 34.349 39.22823 73.425 18.202 37.867 42.09824 76.575 20.599 38.618 43.92425 79.725 20.017 42.253 46.83626 82.875 22.572 42.911 48.63327 86.025 21.904 46.652 51.61728 89.175 24.556 47.186 53.33129 92.325 23.759 50.993 56.33130 95.475 26.471 51.395 57.94431 98.625 25.587 55.252 60.96232 101.775 28.358 55.467 62.41933 104.925 27.376 59.339 65.41734 108.075 30.147 59.568 66.87535 111.225 30.925 61.518 68.962- - - - -- - - - -- - - - -- - - - -
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