Super elements in high-rise buildings under stochastic wind load

(1)

(2)

(3)

Super Elements

in High-Rise Buildings

under Stochastic Wind Load

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 1 oktober 2007 om 15.00 uur

door

Raphaël Daniël Johannes Maria STEENBERGEN

(4)

Dit proefschrift is goedgekeurd door de promotoren: Prof. dr. ir. J. Blaauwendraad

Prof. ir. A.C.W.M. Vrouwenvelder

Samenstelling promotiecommissie:

Rector Magnificus Technische Universiteit Delft, voorzitter Prof. dr. ir. J. Blaauwendraad Technische Universiteit Delft, promotor Prof. ir. A.C.W.M. Vrouwenvelder Technische Universiteit Delft, promotor Prof. Dipl.-Ing. J.N.J.A. Vambersky Technische Universiteit Delft

Prof. dr. ir. J.G.M. Kerstens Technische Universiteit Eindhoven Prof. J.D. Sørensen, M.Sc., Lic.Techn., B. Com. Aalborg Universitet, Denemarken Prof. Dr.-Ing. R. Höffer Ruhr-Universität Bochum, Duitsland Dr. ir. C.P.W. Geurts TNO Bouw en Ondergrond

ISBN 978-90-5972-208-8 Uitgeverij Eburon Postbus 2867 2601 CW Delft Tel.: 015-2131484 / Fax: 015-2146888 info@eburon.nl / www.eburon.nl

Cover design: Studio Hermkens, Amsterdam.

(5)

Preface

“Mechanics is the paradise of mathematical science, because here we come to the fruits of mathematics” Leonardo da Vinci (1452-1519)

The research that is reported in this thesis was performed at Delft University of Technology, faculty of Civil Engineering and Geosciences, section Structural Mechanics. After four years of study in the ‘paradise of mathematical science’, I hope that civil engineers in the building engineering practice down on ‘this earth’ can take advantage of the new findings from this dissertation in order to improve the design of tall buildings subjected to wind load. I would like to thank dr. Wim van Horssen from the department of Applied Mathematics for his help for reaching effectively the ‘fruits of mathematics’. Mrs. Kelly Greene improved the English language in the manuscript. A word of thank to the members of my promotion committee for their comments. Especially, I express my sincere gratitude to my two promotors, prof. Johan Blaauwendraad and prof. Ton Vrouwenvelder for their devoted help. Finally I would like to express my gratitude to Anna, my dear parents and brothers for the encouragement.

(6)

(7)

List of Symbols

Latin

a width-parameter of super elements 3 and 4

ˆ_i

a the peak value of the acceleration

max

a maximum allowed acceleration

b width-parameter of super element 4

,

i j

b width transferring the wind load to stability wall i in super element j m

b building width perpendicular to the wind direction

c width-parameter of super elements 3 and 4

A

c damping term corresponding to A

B

c damping term corresponding to B

critical c critical damping f c drag coefficient , f i

c damping term corresponding to k_{f i}_,

k

c damping term corresponding to k

coh_vv the coherence of the wind speeds

d width-parameter of super elements 3 and 4

d average height of the buildings a

d width-parameter of super elements 3 and 4 b

d width-parameter of super element 4

e width-parameter of super elements 3 and 4

f static line load on the stability wall in super element 1

(12)

a

f static line load on stability wall a in super element 2 b

f static line load on stability wall b in super element 2 e

f natural frequency

end

f frequency until which the integration of a spectrum takes place i

f static line load on stability element i in super elements 3 and 4

0

f central frequency

0,d

f central frequency of the dynamic response.

0,qs

f central frequency of the quasi-static response.

1,d

f first natural frequency in the dynamic case

2,d

f second natural frequency in the dynamic case

1,qs

f first natural frequency in the quasi-static case

h height of the top of the building f

h storey height

k distributed floor stiffness along the height in super element 2

,

f i

k stiffness of the foundation in direction i p

k peak factor s

k shear factor for a non-homogeneous distribution of shear stresses

k_{ϕ rotational stiffness of the foundation} char

l characteristic length of a disturbance in the force distribution

infl

l influence length of a disturbance in the force distribution

,

nat wave

l natural wavelength of a disturbance in the force distribution

1

l height of the upper super element

2

l height of the lower super element

m width-parameter of super elements 1 and 2

m effective mass ij

m elements of the mass matrix of the floors in super element 3

n width-parameter of super elements 1 and 2 ij

n elements of the mass matrix of the floors in super element 4

1,x

n first natural frequency

p static wind pressure on the symmetric building ij

p elements of the mass matrix of the floors in super element 2

1

(13)

2

p static wind pressure on the asymmetric building in z-direction L

q wind-induced dynamic pressure at the leeward side W

q wind-induced dynamic pressure at the windward side w

q dynamic wind pressure

t thickness of the floors

w displacement

u_∗ shear velocity of the wind

v wind velocity

v mean part of the wind velocity

v variable part of the wind velocity b

v basic wind speed m

v mean wind speed s

z reference height

0

z measure for the roughness of the terrain (roughness length)

A parameter determining the stiffness of floor a in super elements 3 and 4

B parameter determining the stiffness of floor b in super elements 3 and 4

C damping term corresponding to E

F

C reduction factor to obain the effective load i

C drag coefficient x

C decay factor of the coherence in the x-direction (height) y

C decay factor of the coherence in the y-direction (width)

CI damping term corresponding to EI i

CI damping term corresponding to EI_i

D damping

E Young’s modulus

EI bending stiffness of the floor in super element 2 a

EI bending stiffness of floor a in super elements 3 and 4 b

EI bending stiffness of floor b in super elements 3 and 4 i

EI flexural rigidity of stability wall i in super elements 3 and 4

D

F reduced wind spectrum

.

i j

F element force y

(14)

z

G constant depending on the mode shape

GA shear stiffness of the floor in super element 2 a

GA shear stiffness of floor a in super elements 3 and 4 b

GA shear stiffness of floor b in in super elements 3 and 4 t

GI torsional rigidity of the shaft in super elements 3 and 4 t

HI damping term corresponding to GI_t I moment of inertia

F

I influence function for force F p

I polar moment of inertia of the shaft v

I turbulence intensity

K lumped floor stiffness in super element 2

y

K constant depending on the mode shape z

K constant depending on the mode shape

L turbulence length scale

M bending moment

Q support reaction

R magnification factor

R support reaction d

R expectation of the dynamic value of a support reaction qs

R expectation of the quasi-static value of a support reaction aa

S spectrum of the accelerations

FF

S spectrum of the wind force

L

S reduced wind spectrum i

S constant qq

S spectrum of the wind pressure uu

S spectrum of the displacements vv

S variance spectrum of the wind speed

T time span in which the wind speed is averaged

,

i j

T element force

V shear force

Greek

(15)

α parameter determining l_char in super element 3

β parameter determining l_{nat wave}_, in super element 3

a

β stiffness parameter for stability wall a in super element 2

b

β stiffness parameter for stability wall b in super element 2

γ parameter determining l_char in super element 2

1

2

δ logarithmic decrement of the damping

δ power of the wind standard deviation profile

ε dimensionless frequency

ε parameter determining l_char in super element 4

ζ procentual damping

ζ parameter determining l_char in super element 4

η structural loss factor

η parameter determining l_{nat wave}_, in super element 4

foundation

η loss factor of the foundation

θ parameter determining l_{nat wave}_, in super element 4

κ curvature

κ Von Karman constant

µ mass of the structure per unit of area

µ mean value

ν Poisson ratio

ξ parameter determining l_char in super element 4

ρ specific mass of concrete

air

ρ density of the air

1

ρ mass per unit height

σ parameter determining l_char in super element 4 d

σ standard deviation of the dynamic response qs

σ standard deviation of the quasi-static response a

σ standard deviation of the acceleration v

σ standard deviation of the wind speed

(16)

ϕ rotation in the vertical plane

1

ϕ dynamic amplification factor

χ aerodynamic admittance

χ parameter determining l_{nat wave}_, in super element 4

ψ rotation in the horizontal plane

ω radial frequency e

ω natural radial frequency

0

ω central radial frequency

Φ modal shape aEI

β flexural rigidity of stability wall a in super element 2 bEI

β flexural rigidity of stability wall b in super element 2

A

ρ mass per unit length of the stability wall in super element 1 a

A

ρ mass per unit length of stability wall a in super element 2 b

A

ρ mass per unit length of stability wall b in super element 2 i

A

ρ mass per unit length of stability wall i in super elements 3 and 4

Matrices

C vector with constants

d vector with the degrees of freedom of a super element

f vector with the element forces of a super element

G matrix giving the relation between element forces and constants

H matrix with the relation between degrees of freedom and constants

K stiffness matrix

1 −

K inverse spectral stiffness matrix

1 − ∗

K complex conjugate of the inverse spectral stiffness matrix

i

F

K matrix with the foundation stiffnesses

M mass matrix of a floor aa

S matrix with the variance spectra of the accelerations

FF

S matrix with the variance spectra of the loads

RR

S matrix with the variance spectra of the support reactions uu

S matrix with the variance spectra of the displacements i

(17)

1. Introduction

The design of tall and specially shaped buildings is becoming more and more usual. Some developments in the building industry have made this possible, such as the use of high-strength materials. Modern tall buildings are relatively slender and flexible which makes them sensitive to dynamic loads. A profound understanding of the force transfer and the serviceability conditions in these types of structures is challenging; this is, among others, the case if the building plan is not constant along the height of the building. Two other important aspects are the effect of the in-plane floor stiffnesses coupling the stability elements and the influence of the torsional rigidity. The design has to ensure adequate reliability but it also has to ensure that people in the building feel comfortable. Wind-induced vibrations might cause discomfort for the occupants of a tall building. Evaluation of the dynamic behaviour is often limited to a small proportion of the total design effort. Moreover, the dynamic performance usually is not checked until the end of the design process when it is difficult and expensive to change the design.

1.1 Objectives

(18)

dynamic behaviour of the structure subjected to the in reality stochastic wind load is studied, resulting in values for the dynamic amplification and the level of comfort.

Regarding the static analysis, for structural designers, finite element programs are available; however modeling takes a lot of time, in spite of the fact that advanced input preprocessors are available. FEM programs provide a quick result for a particular building, but cannot show which structural parameters govern the response of and the force transfer in a structure. Some aspects of this force transfer are difficult to assess; such as the effect of the in-plane floor stiffnesses coupling the stability elements and the influence of the torsional rigidity of stability shafts. In the case that parts of the building or the stability elements differ in height, each abrupt change in geometry will yield a disturbance in the force distribution which can extend over several storeys. Its magnitude and influence field are difficult to estimate and it will turn out that its influence may not be disregarded. No closed-form solutions exist for acquiring insight in these phenomena.

In the dynamic stochastic case, the common structural design for wind load takes place by means of a calculation based on a simplified one mass spring system where only bending in one direction is considered. In reality combinations of bending and twisting moments occur and moreover, buildings are complex structures, such that often a single degree of freedom (sdof) analysis is unreliable. The forces in the structure can deviate from the values following from the conventional calculation and accelerations can occur which are larger than the level at which people feel comfortable. There is a need for tools that reliably assess the wind-induced behaviour of the above mentioned structures.

(19)

second one with an asymmetric ground plan, both with an abrupt change of the cross-sectional geometry along the height. A special element method will be developed, called ‘super element method’, based on closed-form solutions and a very small number of super elements to be used. Only at the heights where changes in properties of the building occur, a node is introduced between super elements. The super element method is intended as a powerful tool for the structural designer in the preliminary design stage, when the main interest is to gain insight in the behaviour of the structure for different preliminary designs. At a later stage, calculating the detailed final static structural model, standard FEM programs may be used. The computing time for a FEM analysis is not a big reason for concern, having very fast computers at our disposal nowadays. FEM provides a perfect solution per case, but does not provide an answer for the design question by which parameters the structural behaviour and the force distribution are characterised in the building type under consideration. For instance, hereafter it will appear that the ratio of a characteristic length and the building height plays an important role. To reveal this, is the very advantage of the analytical super element method. It provides deeper understanding of the structural behaviour of irregular tall buildings, which is not easily done by FEM programs.

(20)

1.2 Relation to previous work

In common finite element analysis, a super element is a grouping of finite elements that, upon assembly, may be regarded as an individual element for computational purposes. These purposes may be driven by modeling or processing needs. The super element is drawn up by the condensation of the internal degrees of freedom. This kind of sub-structuring was invented by aerospace engineers in the early 1960s. Przemieniecki’s book (Przemieniecki, 1968) contains a fairly complete bibliography of early work. The generic term super element was coined in the late 1960s (Egeland and Araldsen, 1974). In the present study the super elements are not composed by a process of condensation. Instead, for each super element a set of simultaneous differential equations is derived and closed-form solutions are obtained for the displacement functions and used to formulate the super elements. In the dynamic case, the super elements are based on the equations of motion describing the behaviour of super elements. The super elements are now spectral super elements. To determine the stress state within a super element, a considerable amount of calculations no longer need to be performed to know the stresses in all the finite elements inside a super element. The stress state follows directly from the closed-form solutions. This saves calculation time and provides a good insight in the structural behaviour of the building. The result is a smart symbiosis of an analytical approach and the strategy of the finite element method. The analytical approach is based on the use of a system of differential equations describing the behaviour of a super element. For solving the problem of including the boundaries and transitions between super elements the scheme of the finite element method is used.

(21)

Research of wind-induced dynamic behaviour can generally be divided into three fields.

(a) Wind engineering; the stochastic properties, the determination of wind speeds, assessment of the spatial correlation between pressures on a structure, the along-wind and crosswind dynamic loads.

(b) Serviceability limits states; human perception of motion and the acceleration as a measure of it.

(c) Structural mechanics; formulation and solution of the differential equations, drawing up the super elements and spectral super elements, determination of the force distribution and accelerations in structures.

The project will mainly focus on the last item, existing theories on the first two items will be adapted into the new structural model.

The three items (a), (b) and (c) will be clarified below.

(22)

(23)

(24)

However for wind no operational FEM programs exist using a stochastic wind load.

In order to determine the accelerations, many of the above mentioned studies are linked together in the present study. Fields (a), (b) and (c) and integrated. New spectral elements are developed; these are inserted in the (slightly adapted) stochastic approach which is applied in order to determine the response to along-wind forces. And finally human perception of the horizontal vibrations is analysed. This results into a new approach to the wind-induced behaviour of high-rise buildings. No longer a single degree of freedom approximation is used, but a highly accurate dynamic stochastic model, well appropriated for design.

1.3 Scope

Four types of super elements will be discussed, which can arbitrarily be combined to a certain building. The discussed super elements are typical examples of building structures with plans that occur frequently in Western Europe. The presented super element method is universally valid, such that, other future types of super elements can be derived and inserted into the program in order to obtain more freedom in designing a building. For other building plans only the stiffness matrix of the new super element needs to be derived on the basis of the then applying differential equations. The overall method, which is systematically drawn up in this study, remains the same. This study is restricted to four super elements and two structure types, one with a symmetric plan, and a second with an asymmetric plan.

(25)

of the floors is expected to be important; floors will be included in the model as an elastic connection between the stability elements. As a result, at an abrupt change in geometry along the height of the building, the moments and shear forces in the stability elements are expected to suddenly redistribute along the height of the structure. Disturbances are expected at the transitions between different super elements, extending along a certain number of storeys, depending on the in-plane floor stiffness. The effect of the torsional rigidity of the shaft is investigated. We will compare the obtained results to the case in which we take an infinitely large in-plane floor stiffness and an infinitely large torsional rigidity. The influence of the foundation stiffness is also studied.

In the dynamic case, a spectral super element is developed, based on the set of equations of motion. The wind load on the structure varies in time and space and its properties can only be described in statistical terms. Therefore, a full stochastic description of the wind load is used in combination with the spectral super element approach. Both the Ultimate Limit State and the Serviceability Limit State are considered. Because of the fluctuation of wind load, dynamic responses of the structure occur which may be larger than the static response. Therefore, in the ULS the dynamic amplification is determined and compared to the dynamic amplification factor from NEN 6702 and Eurocode NEN-EN 1991-1-4.

(26)

The theory developed in this study will be implemented into a Fortran program, calledICAROS: Integrated Closed-form Analysis of high-Rise buildings

On the basis of Super elements. The great benefits of ICAROS are easy applicability relative to existing methods and more realistic values. This because the building is modelled more accurately via the use of super elements and the wind is considered as a stochastic load. It makes possible the analysis in an early design stage of a large number of alternatives. In today’s practise hardly any dynamic structural models are used. Instead a factor from the NEN or Eurocode norm is applied. The new expedient and high-quality calculation tool will lead to better designs.

(27)

2. Structural Model | Super Elements

Two types of buildings will be discussed, one with a symmetric plan, and a second with an asymmetric plan, both with an abrupt change of the cross-sectional geometry along the height. The symmetric and asymmetric building are shown in respectively Figs. 2.1 and 2.2.

Figure 2.1: Ground plan, side view and front view of the symmetric structure.

A B A B A-A 1 2f B-B 1 2 fa 1 2 f 1 2fa 1 2 fb 1 2 fb wall a 1 2 2× βbEI + 1 2 2× βaEI 1 2 2× βaEI

(a) front view (b) side view

(28)

Figure 2.2: Ground plan, side view and front view of the asymmetric structure.

Slender tall walls and shafts assure the stability of both buildings. In the symmetric building two stability walls have a reduced height relative to the rest of the building. In the asymmetric building a complete wing of the building has a reduced height. In the case of the symmetric building, wind in direction p (see Fig. 2.1) will be studied. Wind load in the other direction is not subject of this study. In the case of the asymmetric building, wind in both directions p1 and p2 (see Fig. 2.2) will be studied. In the static part of this

study, we assume uniformly distributed wind surface loads on the façades. These surface loads are translated into line loads f, f and_a f (symmetric _b

(c) front view

(a) plan (b) side view

(29)

building) and f to ₁ f (asymmetric building) on the stability elements. In the ₅ dynamic stochastic part of this study the façades are loaded by randomly distributed wind surface loads; these are translated into load vectors for the structures. The loads are treated more in detail in parts I (chapter 4) and II (chapter 7) of this study for the static and dynamic stochastic case respectively. 2.1 Definition of the buildings

We consider the symmetric and asymmetric structure as shown in respectively Fig. 2.1 and Fig. 2.2. Both structures have a discrete change of the cross-sectional geometry along the height. Only flexural deformation of the stability elements in the main directions is considered; in the case of the asymmetric structure, the torsional deformation of the shaft is also accounted for. Because of the slenderness of the structure, shear deformation in the walls can be disregarded. Deformation by normal force is neglected. In Fig. 2.1, the lower part of the symmetric structure has four stability walls, whereas in the upper part of the structure only the two outer walls remain. Both outer walls have a joint flexural rigidity β_aEI and both inner walls have a joint flexural rigidity

β_bEI , where β_a+β_b = 1. In the lower part of the symmetric structure the

floors between the stability walls have a length m and a width n (see Fig. 2.1). In Fig. 2.2, the stability elements in the asymmetric structure have been indicated; they consist of two walls and one shaft. The lower part of the structure has an L-shaped ground plan; whereas the upper part has a rectangular plan. The shaft transfers moments in two directions and additionally a twisting moment. The flexural rigidities EI_{1 2}₋ and EI_{4 5}₋ of the stability elements and the torsional rigidity GI (the third rigidity _t

3 t

EI GI ) of the shaft have been assigned in Fig. 2.2. The two floor parts in

the asymmetric structure have lengths +a c and +b d respectively. Their widths are d and _a d respectively (see Fig. 2.2). _b

(30)

We denote the floor spacing as h_f (see Fig. 2.1 and Fig. 2.2). In general floors will act in two ways; they can transfer forces in-plane (membrane action) and forces out-of-plane (bending action). In the model of this study only the membrane action is considered. This choice stems from a widely spread construction process for tall buildings in Western Europe. The stability elements are cast in situ and around and between these stability elements the rest of the building is erected in most cases with mainly prefabricated elements. Prefab columns and prefab beams are installed and on these beams prefab prestressed floor elements are placed. The nodes in this structure must be considered as hinges, no moments can occur in the connections between the prefab elements. At the edges of the floor elements reinforcement is applied with concrete in situ, resulting in a tension girder all around. On top of the prefab floor elements a top layer is cast in situ to connect them structurally. In these ways the in-plane action of the floors is ensured. This is necessary because the floors have to transfer the wind load from the façade to the stability elements. For this reason the floors are designed as stiff flat horizontal plates. The in situ cast top layer assures a seamless connection to the stability elements, but this connection does not have any structural value in bending of the floors. So, moments from the stability elements can never be transferred to the floors, resulting in out-of-plane moments in the floors. As a consequence the stability is merely ensured by the stability elements. In some cases also in situ casted flat plate floors are used, but also here the contribution of the floors and columns to the overall stiffness is small. In conclusion, the out-of-plane floor stiffness is not important in the model and is therefore not considered. Only the in-plane stiffness of the floors is taken into account; floors act as an elastic connection between the stability elements.

(31)

Figure 2.3: Reduced structural model of the symmetric building

The choice of this schematization has some important implications. No asymmetric wind load on the façade can be analysed and therefore no torsion can occur in the schematization. Also the stochastic model will not be entirely correct. The cross-spectra between the loads on two identical long walls cannot be accounted for. This also holds for the two identical short walls. This limits the reliability of the use of this model in the stochastic analysis. In the asymmetric building we do not deal with this problem and the results will be more reliable.

Elastic supports at the base of the structures can be introduced. These are the boundary conditions of the systems. Introducing an elastic support in the direction of a degree of freedom u takes place in the global stiffness matrix by _i adding the foundation spring stiffness to the main diagonal term which corresponds to that degree of freedom. We define the foundation spring stiffness in the direction of u as _i k_{f i}_,.

(32)

2.2 Definition of the super elements

The super elements are defined in a orthogonal x-y-z coordinate system. The x-axis is chosen along the height of the structure. The building plan is in the y-z plane as shown in Figs. 2.1 and 2.2. Both the symmetric and the asymmetric structure are built up out of two super elements. Within a super element all floors have the same thickness and stability elements do not change within a super element either.

Super element 1 is the upper part of the symmetric structure, it is a single bending beam with length l (see Fig. 2.3). We describe the structural ₁ behaviour of super element 1 with four degrees of freedom as shown in Fig. 2.4; the corresponding element forces are defined accordingly. Four section forces and their positive signs are defined at both ends of this super element. These forces are functions of the vertical x-coordinate; their positive direction of action at both ends of the super element is defined according to the coordinate system in Fig. 2.4.

Figure 2.4: Super Element 1 with four degrees of freedom, section forces and element forces.

The vector with degrees of freedom of super element 1 is defined as follows:

(33)

Super element 2 is the lower part of the symmetric building with length l ₂ (see Fig. 2.3); it consists of two elastically coupled bending beams. Super element 2 is shown in Fig. 2.5. We describe the structural behaviour of super element 2 with eight degrees of freedom as shown in Fig. 2.5a; the corresponding element forces are defined accordingly in Fig. 2.5c. Eight section forces are defined at both ends of the super element according to the coordinate system in Fig. 2.5b.

Figure 2.5: Super element 2 with 8 degrees of freedom (a), 8 section forces (b) and 8 element forces (c).

{

ϕ ϕ ϕ ϕ

}

= w_a_,1 _a_,1 w_b_,1 _b_,1 w_a_,2 _a_,2 w_b_,2 _b_,2

2

d (2.2)

(34)

Figure 2.6: Super element 3.

For super element 3, 14 degrees of freedom are defined in Fig. 2.7.

Figure 2.7: Super element 3 with 14 degrees of freedom.

The element forces are defined corresponding to these d.o.f.’s in Fig. 2.9.

(35)

Figure 2.8: Super element 3 with 14 section forces.

Figure 2.9: Super element 3 with 14 element forces.

(36)

Apart from the element forces, 14 section forces and their positive signs in super element 3 are introduced in Fig. 2.8. These forces are functions of the vertical x-coordinate; their positive direction of action at both ends of the super element is defined in Fig. 2.8.

Super element 4 is the lower part of the asymmetric structure with length l ₂ (two walls and one shaft). Its plan, side view and front view are shown in Fig. 2.10. We describe the structural behaviour of super element 4 with 18 degrees of freedom as shown in Fig. 2.11. The element forces are defined corresponding to these d.o.f.’s and shown in Fig. 2.13. Apart from the element forces, 18 section forces and their positive signs in super element 4 are introduced in Fig. 2.12. These forces are functions of the vertical x-coordinate; their positive direction of action at both ends of the super element is defined in Fig. 2.12.

Figure 2.10: Super element 4.

(37)

Figure 2.11: Super element 4 with 18 degrees of freedom.

Figure 2.12: Super element 4 with 18 section forces.

(38)

Figure 2.13: Super element 4 with 18 element forces.

{

}

ϕ ϕ ψ ϕ ϕ ϕ ϕ ψ ϕ ϕ = _1,1 _1,1 _2,1 _2,1 ₁ _4,1 _4,1 _5,1 _5,1 1,2 1,2 2,2 2,2 2 4,2 4,2 5,2 5,2 w w w w w w w w 4 d (2.4) We assume the lumped in-plane floor stiffnesses with spacing h_f to be equally distributed along the height of a super element. The so calculated response has a smooth course through the in reality step-wise pattern. The in-plane floor stiffness will be presented as a matrix in section 2.3. In the dynamic case we also deal with the mass of the structure. The mass matrices of the different floors will be calculated in section 2.4. In this case we also have to take into account the damping of the system; it will be treated in section 2.5.

The distributed stiffness and mass of the floors allows for a description of the behaviour of a super element with a set of coupled differential equations in terms of the displacements that correspond to the line loads f . In super _i element 1 we have load f , in super element 2 loads f and _a f (see Fig. 2.3). _b

(39)

In super element 3 we have loads f_{1 4}₋ and in super element 4 we have loads

− 1 5

f (see Fig. 2.2). It is remarked that distributing the in-plane floor stiffness along a half storey height below and a half story height above the floor is not possible at the top of the building, the bottom of the building and at the height where the geometry changes. Here only half of the stiffness matrix of the floor can be distributed. The other half of the discrete floor stiffness matrix is added directly to the relevant node in the global spectral stiffness matrix, for both the symmetric and the asymmetric structure. The same applies for the mass matrix of the floors. The action of the stability elements and the distributed floors is of a parallel nature. Therefore the needed set of differential equations for each super element is an assemblage of the differential equations of the stability elements and the stiffness (and in the dynamic case the mass) of the distributed floors. From the set of differential equations describing the (dynamic) behaviour, an exact (spectral) stiffness matrix can be derived for each super element. This is the subject of the chapters 3 and 6. The standard assembling procedure then yields a global (spectral) stiffness matrix.

2.3 Stiffness matrices of the floors

In this section we derive the stiffness matrix of the floors in the four super elements.

2.3.1 Super Element 1

In super element 1 the floors will not deform and therefore do not add a stiffness to the system.

(40)

homogeneous distribution of the shear stresses. The in-plane stiffness K of one floor is defined in Fig. 2.14.

Figure 2.14: Definition of the stiffness K of one floor in super element 2.

Taking into account flexural and shear deformation, for the stiffness K is found: = ⋅ + 3 2 12 1 12 5 ₁ 5 _s EI K EI m m k GA (2.5)

For the distributed floor stiffness k (see Fig. 2.3) results:

= _f

k K h (2.6)

In case of an isotropic floor of constant thickness t and casted in situ, EI and GA are easily calculated: EI =₁₂1 Etn3 and GA=5₆Gnt. In case of a prefab floor with a top layer casted in situ, the structural engineer has to make a realistic estimate of the stiffnesses; however it does not affect the main scheme of the theory.

2.3.3 Super elements 3 and 4

In Fig. 2.15 one floor of super element 4 has been drawn and the displacements of the stability elements have been defined. In each stability wall we have one displacement w in its strong direction and in the shaft we have two displacements and one rotation. The central axes of the ground plan are indicated by dash-dot lines.

(41)

Figure 2.15: Ground plan of the asymmetric building with the relevant displacements.

In Fig. 2.16 we draw the ground plan with the centre lines, the displacements and the corresponding forces. Rigidities EI and _a GA apply for the floor part _a with length a and EI and _b GA for the floor part with length b . _b

Figure 2.16: Schematisation one floor of the asymmetric building with displacements and forces.

To derive the stiffness matrix of one floor, in Fig. 2.17, the stiffnesses A and

B are defined.

(42)

The stiffness matrix for the floor part with length a has been determined in Appendix A: ψ ⎡ − ⎤⎧ ⎫ ⎧ ⎫ ⎢₋ ₋ ⎥ ⎪ ⎪ ⎪ ⎪₌ ⎨ ⎬ ⎨ ⎬ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎢ ₋ ⎥_{⎩ ⎭ ⎩ ⎭} ⎣ ⎦ 1 1 2 2 2 A A aA w F A A aA w F T aA aA a A (2.7) with: = + 3 2 3 1 3 1 a a s a EI A EI a a k GA (2.8)

The floor part with length b has a similar stiffness matrix:

ψ ⎡ − − ⎤⎧ ⎫ ⎧ ⎫ ⎢₋ ⎥ ⎪ ⎪ ⎪ ⎪₌ ⎨ ⎬ ⎨ ⎬ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎢₋ ⎥_{⎩ ⎭ ⎩ ⎭} ⎣ ⎦ 5 5 4 4 2 B B bB w F B B bB w F T bB bB b B (2.9) with: = + 3 2 3 1 3 1 b b s b EI B EI b b k GA (2.10)

We distribute the discrete floor stiffness matrix equally along a storey height f

(43)

= / _f

A A h (2.12)

= / _f

B B h (2.13)

The zero-terms in the stiffness matrix in (2.11) indicate that there is no relation between w w and 1, 2 w w . In (Dusseldorp, 2000) this was confirmed 4, 5

by a FEM analysis for the floor. It was found that the elements in the upper right and lower left of the stiffness matrix are a factor 10 smaller than the 3 other elements. So it is justified to assume these terms are zero in (2.11). In the same study it was also shown that the non-zero elements in the stiffness matrix (2.11) differ less than 3 % from the FEM results. Therefore we conclude that the analytical stiffness matrix in (2.11) is reliable.

For the stiffness matrix of the distributed floors in super element 3 (left floor in Fig. 2.16 and Fig. 2.17) we obtain:

ψ ⎡ ₋ ⎤_{⎧ ⎫ ⎧ ⎫} ⎢₋ ₋ ⎥ ⎪ ⎪ ⎪ ⎪₌ ⎨ ⎬ ⎨ ⎬ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎢ ₋ ⎥_{⎩ ⎭ ⎩ ⎭} ⎣ ⎦ 1 1 2 2 2 A A aA w F A A aA w F T aA aA a A (2.14)

2.4 Mass matrices of the floors

The mass of the main support structure of the building consists of two parts, the mass of the stability elements itself and the mass of the floors. The mass of the floor is taken into account by dividing it over the stability elements and by then distributing it along the height of these elements. So we get a mass per unit length in which the mass of the stability elements and the floors is included. The distribution of the floor mass over the different stability elements is done by means of a mass matrix.

(44)

1 1 3 f SE _h M = ρmnt (2.15) with: f

h : height of one storey ρ : specific mass of concrete

m : length of a floor part (see Fig 2.1) n : width of a floor part (see Fig 2.1) t : thickness of the floor

In Fig. 2.18, the floor in super element 2 is shown with the degrees of freedom.

Figure 2.18: Floor in super element 2 with degrees of freedom.

The mass matrix for the structure with degrees of freedom from Fig. 2.18 is derived in Appendix B.1 on basis of a virtual work equation including inertia forces. The floor of super element 2 in Fig. 2.18 is schematized as a beam with bending and shear stiffness. The floors parts are almost squares (usually m will not differ much from n), so the influence of the shear deformation will be included. In order to determine the mass matrix we have to determine the displacement field in the floor in relation to the two degrees of freedom of the stability walls. We calculate the static deformation field; however we ought to use the dynamic deformation field which is different for each eigen mode. The first eigen mode will not differ much from the static deformation field.

(45)

Since the first eigen mode is by far the most important, the proposed method is a reasonable approximation.

Here, the mass matrix is shown for the special case m=n and the shear factor = 1

s

k , otherwise the expressions become too long; the mass matrix is

distributed along the height of one storey and we obtain:

ρ ⎡ ⎤ = _⎢ _⎥ ⎣ ⎦ 2 1 0.65 0.19 0.19 1.97 f m t h M (2.16)

where ρ is the specific mass of the floors. In order to reduce the length of the expressions in the differential equation (see chapter 6), we use the parameters p_ij and the mass matrix of super element 2 (SE2) then becomes:

⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ 11 12 2 21 22 SE p p p p M (2.17) with: ρ = 2 11 0.65 f p m t h ρ = = 2 12 21 0.19 f p p m t h ρ = 2 22 1.97 f p m t h

The vector with degrees of freedom of the floor is:

{

}

= ₁ ₂ T

u u

u (2.18)

2.4.3 Super Elements 3 and 4

In Fig. 2.19, one floor in super element 4 is shown. The floor in super element 3 is the left hand part of that floor.

(46)

Figure 2.19: One floor in super element 4.

Figure 2.20: Floor in super element 4 with degrees of freedom.

Determining the mass matrix, in Appendix B.2, two cases are discussed. 1) The shear deformation of the floors is included in the interpolation functions and both translation and the rotation inertia part of the mass matrix are included. 2) Only bending in the floors is considered and therefore only the translation inertia part of the mass matrix is included. Also here we remark that we calculate the static deformation field as a reasonable approximation of the dynamic deformation field. From the calculations in Appendix B.2 it appears that including the shear deformation of the rectangular floors and the rotational inertia in the mass matrix of super elements 3 and 4 is unnecessary precision work for the first eigen mode. We commit already an error in the

(47)

model by assuming a static deformation field instead of the dynamic one. For higher eigen modes, including the shear deformation of the floors and the rotational inertia in the mass matrix becomes necessary. However the lowest eigen mode is by far dominant in the system, so we will leave the shear deformation of the floors and the rotational inertia in the mass matrix out of the equations of motion of super elements 3 and 4.

In this way for the mass matrix of the floor part with length a is obtained:

2 2 2 2 3 33 39 11 140 280 280 39 17 3 280 35 35 11 3 2 280 35 105 a a a a t d a a a a a a ρ ⎡ ₋ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⋅ ⋅ _⎢ − _⎥ ⎢ ⎥ ⎢− − ⎥ ⎢ ⎥ ⎣ ⎦ M_a (2.19)

For the floor part with length b we have a similar mass matrix:

3 2 2 2 2 2 3 11 105 35 280 3 17 39 35 35 280 11 39 33 280 280 140 b b b b t d b b b b b b ρ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⋅ ⋅ _⎢ _⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ M_b (2.20)

For the floor part with length c we obtain:

2 1 2 2 3 1 1 2 3 a c c t d c c ρ ⎡⎢ ⎤⎥ = ⋅ ⋅ ⎢ ⎥ ⎣ ⎦ M_c (2.21)

For the floor part with length d we have a similar mass matrix:

(48)

The total mass matrix of the floor in super element 4 (Fig. 2.19) is composed from the mass matrices of the four floor parts; in addition to this, in displacement w the floor part with length +₂ b d is involved and in displacement w the floor part with length +₄ a c is involved. The total mass matrix is distributed along the height of the stability elements to obtain a uniformly distributed mass and the result is:

(

)

4 2 2 1 2 2 33 39 140 280 39 17 280 35 11 3 280 35 0 0 0 0 a a a a b SE f a a d a d a d a d a c d b d t h d a d a c ρ ⎡ ⎢ ⎢ ⎛ ⎞ ⎢ _{+ +} ₊ ⎜ ⎟ ⎢ _⎝ _⎠ ⋅ ⎢ = ⎛ ⎞ ⎢− _⎜− + _⎟ ⎢ _⎝ _⎠ ⎢ ⎢ ⎢ ⎣ M

(

)

2 2 1 2 2 3 1 3 3 1 3 2 1 2 2 3 3 2 2 1 2 2 2 11 0 0 280 3 0 0 35 2 2 3 11 105 105 35 280 3 17 39 35 35 280 11 39 33 280 280 140 a a a b b b b b a b b b b d a d a c d a c d b d d b d d b d b d d b d d a c d b d b d b d b ⎤ − _⎥ ⎥ ⎛₋ ₊ ⎞ ⎥ ⎜ ⎟ _⎥ ⎝ ⎠ _⎥ ⎛ ₊ ⎞₊ ⎛ ₊ ⎞ ⎛ ₋ ⎞ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ _⎥ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ _⎥ ⎛ ₋ ⎞ ⎛ ₊ ⎞₊ ₊ ⎥ ⎜ ⎟ ⎜ ⎟ _⎥ ⎝ ⎠ ⎝ ⎠ _⎥ ⎥ ⎥⎦ (2.23) The vector with degrees of freedom of the floor is:

{

ψ

}

= ₁ ₂ ₄ ₅ T

w w w w

u (2.24)

(49)

⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 11 12 13 21 22 23 31 32 33 34 35 4 43 44 45 53 54 55 SE m m m m m m m m m m m m m m m m m M (2.25)

The parameters m_ij can be found in expression (2.23).

In Fig. 2.21a, the floor of super element 3 has been drawn and in Fig. 2.21b the floor is indicated with the centre lines and the degrees of freedom of the stability elements.

(a) (b)

Figure 2.21: Floor from super element 3 (a) with central lines and degrees of freedom (b).

(50)

(

)

(

)

(

)

3 2 2 1 2 2 2 2 1 2 2 3 1 3 3 33 39 140 280 39 17 280 35 11 3 280 35 0 0 11 0 280 3 0 35 2 0 105 a a a a b SE f a a a a a a b d a d a d a d a c d d e t h d a d a c d a d a c d a c d a c d d e ρ ⎡ ⎢ ⎢ ⎛ ⎞ ⎢ ₊ ₊ ₊ ⋅ _⎢ ⎜ ⎟ = _⎢ ⎝ ⎠ ⎛ ⎞ ⎢− _⎜− + _⎟ ⎢ _⎝ _⎠ ⎢ ⎢⎣ ⎤ − _⎥ ⎥ ⎛₋ ₊ ⎞ ⎥ ⎜ ⎟ _⎥ ⎝ ⎠ ⎛ ₊ ⎞ ⎜ ⎟ ⎝ ⎠ + + _{+ ⎦} M ⎥ ⎥ ⎥ ⎥ ⎥ (2.26) The vector with degrees of freedom of this floor is:

{

ψ

}

= ₁ ₂ ₄ T

w w w

u (2.27)

In order to reduce the length of the expressions in the differential equation (see chapter 6), we use the parameters n_ij and the mass matrix becomes:

⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 11 12 13 21 22 23 3 31 32 33 44 SE n n n n n n n n n n M (2.28)

The parameters nij can be found in expression (2.26).

2.5 Damping

(51)

Figure 2.22: Dynamic model of a flexural beam.

In order to formulate the differential equation for the flexural beam with damping, modelled in Fig. 2.22c, the kinematic, constitutive and equilibrium equations are formulated. In Fig. 2.23 the positive displacement w , the moments M and distributed load q are defined.

Figure 2.23: Sign conventions for the flexural beam.

In the following equations, κ is the curvature and V is the shear force.

κ = − 2 2 d w dx (kinematics) κ κ = + d M EI CI dt (constitution) + = ⇔ − 2 = 2 0 dV d M q q dx dx (equilibrium)

(a) Flexural Beam

(b) Model with stiff elements and the elasticity modeled in the springs

(c) Model with stiff elements, the elasticity modeled in the springs and the damping in the dashpots.

x

z

φ _w

(52)

The constitutive equation needs some explanation. Hook’s law is:

σ = ⋅E ε

but now we have to insert damping: σ = ⋅ + ⋅ E ε C ε

with C the damping coefficient and ε the time derivative of the strain ε. It holds: σ =

∫

⋅ A M z dA with: σ = ⋅ + ⋅ E ε C ε , ε κ= ⋅ z and ε κ= ⋅ z it yields:

(

κ κ

)

= ⋅ + ⋅ ⋅

∫

2 A M E C z dA κ κ = + M EI CI with: =

∫

2 A I z dA

EI is the bending stiffness related to the elasticity of the beam; CI is an analogous quantity for the damping in the beam. Successive substitution provides the fourth-order differential equation for a beam with damping:

∂ ₊ ∂ ∂ ₌ ∂ ∂ ∂ 4 4 4 4 w EI CI w f t x x

For a twisting rod we analogously obtain:

ψ _ψ ∂ ₊ ∂ ∂ ₌ ∂ ∂ ∂ 2 2 2 2 t t GI HI f t x x

with ψ being the in-plane rotation of the beam, GI the torsional rigidity _t

and HI the corresponding damping term. _t

(53)

We define them as follows:

For super element 2: c with k (see expression 2.6) and CI with EI. _k

For super elements 3 and 4: c with A (expr. 2.12), _A c with B (expr. _B 2.13), CI with ₁ EI , ₁ CI with ₂ EI , ₂ CI with ₄ EI , ₄ CI with ₅ EI and ₅

t

HI with GI (see Fig. 2.2). _t

For the foundation stiffness the damping is defined as c_{f i}_, with k_{f i}_, (see section 2.1).

The damping quantities will be used in the equations of motion, however it is necessary to convert them into physical parameters which are measurable. First we look to the case in which we would model the structure as a single degree of freedom system. The constitutive relationship in the time-domain is (with k the stiffness, c the damping and F the force):

( )

+

( )

=

( )

ku t cu t F t

We transform it into the frequency domain:

( )

ω + ω ω

( )

=

( )

ω

i

ku cu F or

(

k+ciω

) ( )

u ω = F

( )

ω

Now we have a (frequency dependent) complex valued stiffness in the frequency domain:

( )

ωω ω ∗ ₌ _{= +} i F k k c u

Here it holds that:

Real stiffness k of the element = Re k

( )

* Real damping c of the element = Im k

( )

* ω

We now make a comparison between the stiffness in the static situation k and the (complex) stiffness in the dynamic situation k+ icω. Within this framework the dimensionless loss factor η is defined as:

( )

ω η= = ⋅ * * Im Re k _c k k

We can also speak about an amplification factor R :

(

ω

)

= ⋅

(54)

In engineering applications, the loss factor η turns out to be a constant material parameter, independent of the frequency. Values of this loss factor η can be found in literature. So then the damping is defined by:

η ω

⋅

= k

c

Another damping quantity which is often used in the dynamic analysis of single degree of freedom systems is ζ defined by:

ζ = ⇔ = ⋅ζ _critical

critical

c

c c

c

We want to know the relation between η and ζ . We consider the situation that the frequency ω is equal to the first natural frequency ω_e:

2 2 2 2 critical e m k k c c km k k k m ζ ζ ζ ζ ζ ω ⋅ ⋅ = ⋅ = ⋅ = ⋅ = = e k c η ω =

Because damping only has real physical meaning for lightly damped structures at the natural frequency, we conclude:

η= 2 ζ

We now study a beam on elastic foundation; this is the general case for all stability elements being elastically coupled by the in-plane floor stiffnesses. The constitutive relationship in the time-domain is:

( )

( ) ( )

_{( )}

( )

_{( )}

∂ ∂ ∂ ∂ + + + = ∂ ∂ ∂ ∂ 4 4 4 4 u t u t u t u t EI CI ku t c f t t t x x

We transform it into the frequency domain:

(

+ ω

)

∂

( ) (

ω + + ω

) ( )

ω =

( )

ω ∂ 4 4 i u i EI CI k c u f x

Now we have new (frequency dependent) complex valued stiffnesses in the frequency domain:

ω + i

(55)

Here also we get analogous loss factors: ω η= ⋅C E ω η = ⋅c k

Because in the whole structure (stability elements and floors) we deal with (more or less the same) pre-stressed concrete, we can make the assumption

that η= 0.02 (CUR report 17: = 55R or η= 0.018 ) along the whole

structure. This corresponds with a damping factor ζ of 1% in the case of a single degree of freedom system. This is realistic for the main support structure. If we include the overall damping (‘dressed’ structure) we have to use ζ = 0.02 or η= 0.04. So the damping quantities in the equations of motion become: η ω ⋅ = k k c and η ω ⋅ = E

C for super element 2 and

η ω ⋅ = A A c , η ω ⋅ = B B c , η ω ⋅ = E C , η ω ⋅ = G

H for super elements 3 and 4.

For the foundation we obtain: η ω = , , foundation f i f i k c

The damping of the foundation η_foundation is larger than the structural damping described above. It will be of the order O

( )

0.1 .

(56)

(57)

(58)

(59)

3. Stiffness Matrices Super Elements

In this chapter we derive the stiffness matrices of the different super elements from the sets of differential equations describing the behaviour of these super elements.

3.1 Outline of the general procedure

The degrees of freedom of the super element are collected in the vector d and the corresponding element forces in the vector f. The stiffness matrix K establishes the relation between the two vectors:

f =K d (3.1)

To determine the exact stiffness matrix of a super element, we proceed as follows. Let the number of degrees of freedom be n. First we derive a set of differential equations describing the behaviour of the super element. The homogeneous solution of this set contains n unknown constants C. On the basis of this homogeneous solution a relation can be formulated between the degrees of freedom d and the unknown constants C; thus a set of n equations appears:

HC = d (3.2)

The n n× - matrix H is merely determined by geometric parameters.

Inverting this relation gives:

(60)

From the solution of the differential equations the section forces can be calculated. For a super element with length l , these section forces can be expressed in the element forces at x =0 and x = . l

This results in the following relation:

f = GC (3.4)

The matrix G is also n n× . Substitution of (3.3) into (3.4) gives:

f = GH –1d (3.5)

Hence K is:

K =GH –1 (3.6)

The final result will be a symmetric matrix. The above indicated procedure will be used to draw up the stiffness matrices of the four super elements. In sections 3.2-3.5 for the super elements we only present the differential equations and their homogeneous solution. For elaborations Appendix C is referred to.

3.2 Derivation and solution of the set of differential equations; super element 1

Super element 1 is a flexural beam, the differential equation describing its behaviour and its stiffness matrix are well known and are therefore not presented here.

3.3 Derivation and solution of the set of differential equations; super element 2

Super elements in high-rise buildings under stochastic wind load

Super Elements

in High-Rise Buildings

under Stochastic Wind Load

Preface

Table of Contents

List of Symbols

1. Introduction

2. Structural Model | Super Elements

{

}

{

}

{

}

(

)

(

)

{

}

(

)

(

)

(

)

{

}

∫

(

)

∫

∫

( )

( )

( )

( )

( )

( )

(

) ( )

( )

( )

( )

( )

( )

( )

( )

(

)

( )

( ) ( )

( )

( )

( )

(

)

( ) (

) ( )

( )

( )

3. Stiffness Matrices Super Elements

_{( )}

_{( )}