suspension bridges
Lina Thi Nguyen
Mechanical Engineering
Supervisor: Einar Norleif Strømmen, KT
Department of Structural Engineering Submission date: June 2016
Norwegian University of Science and Technology
Konstruksjonsteknikk
for
Lina Thi Nguyen
DYNAMISK RESPONS AV LANGE SLANKE HENGEBRUER
Aerodynamic response of slender suspension bridges
I Norge er det for tiden under planlegging en rekke meget slanke brukonstruksjoner, for eksempel Halsafjorden, Julsundet og Nordfjorden, alle som klassiske hengebroer, enten med enkelt eller splittet kassetverrsnitt i hovedbæreren. Disse broene har hovedspenn mellom ca.
1550 og 2050 m. De er svært utsatt for den dynamiske lastvirkningen fra vind. Halsafjorden som er den lengste med et spenn på ca. 2050 m er på grensen av det som tidligere er bygget av denne typen konstruksjoner. Prosjektene er spesielt krevende med hensyn til virvel- avløsning og bevegelsesinduserte krefter, dvs. med hensyn til å oppnå en konstruktiv utførelse som ikke medfører uakseptable virvelavløsningssvingninger ved lave vindhastigheter og tilstrekkelig sikkerhet mot en uakseptabelt lav stabilitetsgrense i koblede vertikal og torsjonssvingninger (”flutter”). Hensikten med denne oppgaven er å se på mulige utførelser av fjordkryssinger i denne spennvidden med tanke på å oppnå gunstige aerodynamiske egenskaper, og hvor det legges spesiell vekt på kryssinger i form av en eller annen variant av den klassiske hengebroen. Arbeidet foreslås lagt opp etter følgende plan:
1. Studenten setter seg inn i teorien for hengebroen som konstruksjonssystem.
2. Studenten setter seg inn i teorien for dynamisk respons og aerodynamisk stabilitet av slanke broer (se for eksempel Strømmen: Theory of bridge aerodynamics, Springer 2006).
3. For en eller flere aktuelle utførelser og spennvidder (avtales med veileder) skal det foretas en utredning med sikte på å kvantifisere de viktigste mekaniske egenskapene (dvs.
aktuelle masse- og stivhetsegenskaper). Det skal foretas beregninger av de aktuelle egenfrekvensene og tilhørende egensvingeformene som er avgjørende for broens dynamiske egenskaper. I den grad det er mulig kan beregningene baseres på regnemaskinprogrammet Alvsat (eller innhentes fra Vegdirektoratet/Bruavdelingen).
4. For de samme tilfellene som er behandlet under punkt 3 skal det foretas beregninger av vindindusert dynamisk respons. Studenten kan selv velge om hun vil legge vekt på virvelavløsning, «buffeting» eller stabilitet. For å kunne ta tilstrekkelig hensyn til bevegelsesinduserte krefter skal responsberegningene utføres i modalkoordinater i Matlab, enten i tidsplanet eller i frekvensplanet. I den grad tiden tillater det kan studenten velge å undersøke om en eller flere massedempere kan bedre systemets dynamiske egenskaper.
Studenten kan selv velge hvilke problemstillinger hun ønsker å legge vekt på. Oppgaven skal gjennomføres i samarbeid med Dr.ing. Bjørn Isaksen og Siv.ing. Kristian Berntsen i Vegdirektoratet.
NTNU, 2016-01-15
Einar Strømmen
This thesis is the final part of my Master’s degree at Norwegian University of Science and Technology (NTNU). The work has been carried out at Faculty of Engineering Science and Technology, Department of Structural Engineering.
I would like to thank my supervisor, Professor Einar Strømmen for help and guidance with this thesis. I would also like to thank Bjørn Isaksen and Kristian Berntsen from Norwegian Public Roads Administration for providing me data and help whenever needed.
Trondheim, June 10th 2016 Lina Thi Nguyen
This thesis studies the aerodynamic stability of suspension bridges with a streamlined single box girder or a twin box girder as cross section. The focus has been on studying how aerodynamic properties affects the response and the flutter stability limit of the bridge.
Response calculations is based on a modal approach in the frequency domain and has been carried out by use of MATLAB. Prediction of the stability limit has been based on a buffeting-response calculation instead of solving the impedance matrix in search for singularities.
For a streamlined box girder the Hardanger Bridge has been used as a case study. The results show that instability due to flutter is caused by motion induced loss of stiffness in torsion. Structural damping, motion induced stiffness coupling between the torsion and vertical displacements, and aerodynamic damping in torsion has no effect on the stability limit or the response of the system.
For a twin box girder, a proposed bridge over the Halsafjord with this cross section has been studied. If the distance between the box sections of the bridge is 20 meters or more, then the motion induced forces contributes to the stiffness in torsion. Since there’s no loss of stiffness, no flutter stability limit was found. Structural damping and motion induced stiffness coupling between the vertical and torsion modes has no effect on the stability limit. Aerodynamic damping and stiffness in torsion does affects the response values of the system, but the stability limit remains unchanged. Based on the results, crossing the Halsafjord with a suspension bridge with a main span equal to 2050 meters will be possible.
Denne oppgaven undersøker aerodynamisk stabilitet til hengebruer med et enkelt eller splittet kassetverrsnitt i hovedbæreren. Fokuset har vært p˚a hvordan de aerodynamiske egenskapene til tverrsnittet p˚avirker responsen og stabilitetsgrensen til hengebruen. Dette har blitt undersøkt ved ˚a gjøre et parameterstudium p˚a de aerodynamiske deriverte.
Responsberegningen er utført i modalkordinater i frekvensplanet ved bruk av MATLAB.
Stabilitetsgrensen er kvantifisert ved bruk av buffeting-respons-beregninger i stedet for ˚a løse singularitetproblemet for impendansfunksjonen.
For et enkelt kassetverrsnitt er Hardangerbroen blitt brukt som eksempel for ˚a foreta beregningene. Resultatene viser at instabilitet p˚a grunn av flutter er for˚arsaket av beveg- elsesindusert tap av stivhet i torsjonsretningen. Strukturell demping, bevegelsesindusert kobling av stivhet i torsjon og vertikal forskyvning, samt aerodynamisk demping i torsjon har ingen effekt p˚a stabilitetsgrensen eller responsen av systemet.
For et splittet kassetverrsnitt er en foresl˚att hengebro over Halsafjorden med dette tverrsnit- tet blitt studert. Hvis avstanden mellom kassenetverrsnittene er 20 meter eller mer, f˚ar den økt stivhet av de bevegelsesinduserte kreftene. Siden det ikke er tap av stivhet, ble det ikke funnet noen stabilitetsgrense. Strukturell demping og bevegelsesindusert kobling av stivhet mellom torsjon og vertikale forskyvninger har ingen effekt p˚a stabilitetsgrensen.
I motsetning til et enkelt kassetverrsnitt p˚avirker aerodynamisk demping og stivhet i torsjon størrelsen p˚a responsen for dette tverrsnittet, men det ble fortsatt ikke funnet en stabilitetsgrense. Basert p˚a dette vil det være mulig ˚a krysse Halsafjorden med en hengebro med et hovedspenn p˚a 2050 meter hvis et splittet kassetverrsnitt blir brukt.
1 Introduction 1
2 Theory 2
2.1 Response Types . . . 3
2.2 Motion Induced Instabilities . . . 4
2.3 The buffeting load . . . 5
2.3.1 Aerodynamic Derivatives . . . 7
2.3.2 Wind field cross spectral densities . . . 8
2.4 Buffeting Response . . . 10
3 Results 17 3.1 Response calculation with Matlab . . . 18
3.2 Frequeny ratio ωθ/ωz . . . 19
3.3 Hardanger Bridge . . . 20
3.3.1 Dynamic response of the Hardanger Bridge . . . 21
3.3.2 Parametric study on the Aerodynamic Derivatives . . . 23
3.4 Halsa Bridge . . . 25
3.4.1 Aerodynamic Derivatives . . . 26
3.4.2 Dynamic response . . . 28
3.4.3 Parametric study of the aerodynamic derivatives . . . 32
4 Discussion 34
5 Conclusion 35
Appendix A: Fourier constants and mode shapes 36
Appendix B: Statistical concepts in wind engineering 39
Appendix C: Matlab scripts 48
List of Figures 63
List of Tables 65
References 66
Introduction
Modern bridges are becoming longer and more slender, especially in Norway where the traffic volumes is low. Building bridges with many traffic lanes is therefore unnecessary resulting in very slender bridges. Aerodynamic stability of such bridges is therefore very important and needs to be investigated.
As part of a project to eliminate the ferry connections along the coastal highway E39 (proposed by Norwegian Public Roads Administration (NPRA)), prospect of crossing the Halsafjord is currently being investigated. A suspension bridge with a main span equal to 2050 meters with a twin box girder as cross section is proposed as a solution. The twin box girder is a cross section consisting of two box girder sections bound together by transverse girder at certain distances.
The overall aim of this thesis is to study the aerodynamic stability of the twin box girder cross section in order to see if it’s possible to cross the Halsafjord with a sufficient flutter stability limit. This will be done by investigating the aerodynamic stability for a proposed bridge over Halsafjorden. Comparison will be done with the aerodynamic stability of a streamlined box girder by studying the Hardanger Bridge. The purpose is to look at similarities and differences between the two cross sections. In addition, a parametric study on how the aerodynamic properties of the cross sections affects the stability limit will also be investigated for the two cases.
The theory for a buffeting-response-calculation in order to quantify the flutter stability limit will be presented in Chapter 2. In Chapter 3, the flutter stability limit will be in- vestigated for the two different cross section mentioned. How the aerodynamic properties affects the stability limit will also be investigated in this chapter. Chapter 5 discusses the validity of the results, and Chapter 6 concludes the thesis.
Theory
In this chapter the theory necessary for a response calculation will be presented. The theory is based on a more comprehensive description that can be found in Theory of Bridge Aerodynamics [1]. As the solution will be carried out in the frequency domain, it will be necessary to establish description of the wind field as well as structural properties in the frequency domain. In order to do this, the wind field will be regarded as a stochastic process and it’s characteristics can then be described by statistical properties such as variance and covariance. In the frequency domain the statistical properties are represented by spectral densities. (An introduction to statistical concepts in wind engineering can be found in Appendix B). Since structural displacement is caused by the wind field, the response will also be regarded as a stochastic process. The focus will then be on finding the variance of the fluctuating displacement components.
Response calculation in the frequency domain will require establishment of transfer func- tion from the wind field cross-spectral densities to the modal load spectrum, and finally the desired response spectrum can be found. The steps in time domain and frequency domain is shown in Figure 2.1.
Throughout this chapter, it will be taken for granted that the structure is a bridge and thus horizontal. The main flow direction of the wind is perpendicular to the direction of the bridge span. The wind flow can be split into three fluctuating components. In the along-wind direction, the component isU =V+u(x, t), where V is the mean wind velocity that only varies with the height, and u(x, t) is the fluctuating turbulent component that varies in time and space. In the across wind horizontal and vertical direction only the turbulent component v(x, t) and w(x, t) is considered.
Figure 2.1: Time and frequency domain representations [1]
2.1 Response Types
The wind field give raise to forces when interacting with a structure. Wind forces that arise from pressure fluctuation or turbulence in the oncoming flow is known as buffeting.
Vortex shedding is due to forces that stems from additional generation of vortices and turbulence at the surface of the body due to friction. Lastly, motion induced loads effects are caused by oscillation of the body due to the flow, and interaction between the body and flow may generate additional forces. The different effects along with the wind velocity region where they are most likely to occur is shown in Figure 2.2. Vortex shedding tend to arise at low wind speeds, buffeting response at intermediate wind speeds, whereas motion induced effects are present at high wind velocities.
Figure 2.2: Response variation with mean wind velocity. Static response to the left.
Dynamic response to the right [1].
2.2 Motion Induced Instabilities
Instability of a system develops when a small increase in the mean wind velocity causes the response to become infinitely large. The source for instability is due to change in structural damping and stiffness properties of the system. There are four different instability that are usually dealt with, static divergence, galloping, instability in pure torsion and lastly flutter, a coupling of torsion and vertical modes. One way to detect instability limits is by looking for singularities in the impedance matrix given in equation (2.1), if the absolute value of the determinant of this matrix is zero, then an instability is found.
Eˆη(ω, V) =
(
I−κae−
ω·diag
1 ωi
2
+ 2iω·diag
1 ωi
·(ζ −ζae)
)
(2.1) The impedance matrix is a function of the mean wind velocity V and the frequency ω. The stability problem of interest is found through conditions related to these two parameters. When solving |detEˆη(ωr,Vcr)|= 0 for the flutter instability limit, the following assumption is made
ωr=ωz(Vcr) =ωθ(Vcr) (2.2) the resonance frequencyωr is assumed to be the same for the vertical and torsional mode when the critical wind velocityVcr is reached as motion of the two modes are merged into one. This assumption may be incorrect as the stability limit can already be reached when the resonance frequency of these two modes have not merged [2,3]. Therefore, instead of solving the impedance matrix searching for singularities, a buffeting response calculation will be carried out in order to quantify the flutter stability limit. The theory for this will be presented in the next section.
2.3 The buffeting load
The buffeting wind load consist not only of loads due to turbulence in the coming wind flow, but also motion induced contribution. The motion induced forces stems from inter- action between the flow and the oscillating body. The buffeting theory is based on the assumption that loads can be calculated from the instantaneous velocity pressure along with load coefficients for drag, lift and moments. These coefficient are obtained from wind tunnel tests with a section model.
Just like the wind flow, cross sectional displacements can be split into a time invariant mean and zero mean fluctuating part. Thus the cross sectional displacementry, rz and rθ
becomes
ry rz rθ
=
¯ ry
¯ rz
¯ rθ
+
ry(x, t) rz(x, t) rθ(x, t)
(2.3)
Figure 2.3: Instantaneous flow and displacement quantities [1]
Figure 2.3 shows the displacement of a cross section with the wind components. The cross section is first given the displacement ¯ry(x), ¯rz(x) and ¯rθ(x). Further dynamic displacement ry(x, t), rz(x, t) andrθ(x, t) are caused by fluctuating wind components. In the final position, the cross sectional drag, lift and moment forces are given by the use of Bernoulli’s equation with flow dependent coefficients as
qD(x, t) qL(x, t) qM(x, t)
= 1 2ρVrel2
D·CD(α) B·CL(α) B2·CM(α)
(2.4)
In structural axis these forces are given as
qtot(x, t) =
qy qz qθ
tot
=
cosβ −sinβ 0 sinβ cosβ 0
0 0 1
·
qD qL qM
(2.5)
It’s taken as an assumption for linearization that the turbulence components u(x, t) and w(x, t) are small compared to V. In addition, cross sectional rotations is also assumed to be small. Thus cos(β)≈1 and sin(β)≈ tan(β)≈(ω−r˙z)/V. The relative velocity Vrel2 can be found by use of Pythagora’s theorem. The relative velocity and the angle of flow incident α as seen in the Figure 2.3 then becomes
Vrel2 = (V +u−r˙y)2+ (w−r˙z)2 ≈V2+ 2V u−2Vr˙y (2.6)
α = ¯rθ+rθ+β ≈r¯θ+rθ+ ω V − r˙z
V (2.7)
The curves obtained for the load coefficientsCD, CL and CM from static tests are nonlin- ear. In order to use them, they are replaced by the linearization version below
CD(α) CL(α) CM(α)
=
CD( ¯α) CL( ¯α) CM( ¯α)
+αf ·
CD0 ( ¯α) CL0( ¯α) CM0 ( ¯α)
(2.8)
where ¯α = ¯rθ and αf = rθ +ω/V −r˙z/V. CD0 , CL0 and CM0 is the slope of the curves for the load coefficients at ¯α. Combining equation (2.4) - (2.8), then the total load qtot(x, t) =q(x) +¯ q(x, t) can be divided into a static part given by equation (2.9)
¯ q(x) =
¯ qy
¯ qz
¯ qθ
= ρV2B 2 ·
(D/B) ¯CD C¯L BC¯M
= ρV2B 2
ˆbq (2.9)
and a dynamic part given as
q(x,t) =
qy qz qθ
=Bqv+Cae˙r+Kaer (2.10)
where
v(x, t) = [u w]T (2.11)
r(x, t) = [ry rz rθ]T (2.12)
Bq(x) = ρV B 2
2(D/B) ¯CD (D/B)CD0 −C¯L 2 ¯CL CL0 + (D/B) ¯CD 2BC¯M BCM0
= ρBV
2 ·Bˆq (2.13)
Cae(x) = −ρV B 2
2(D/B) ¯CD (D/B)CD0 −C¯L 0 2 ¯CL CL0 + (D/B) ¯CD 0
2BC¯M BCM0 0
(2.14)
Kae(x) = ρV2B 2
0 0 (D/B)CD0 0 0 CL0 0 0 BCM0
(2.15)
The matrix Bqvis the load associated with turbulence or the buffeting load. WhileCae˙r andKaeris the motion induced load associated with structural velocity and displacement.
Note that in the Cae matrix, all terms related to the torsional velocity ˙rθ is equal to zero. Thus if equation (2.14) is used in the response calculation, then there will be no aerodynamic structural damping in torsion.
The equations above is valid in both the time domain as well as in the frequency domain.
Since calculation will be done in the frequency domain, a more general and improved expression in terms of so called aerodynamic derivatives forCaeandKaewill be introduced in order to provide more accurate results. For the general case, the structural damping in torsion will be taken into account.
2.3.1 Aerodynamic Derivatives
The general frequency domain versions of Cae and Kae is given below as
Cae=
P1 P5 P2 H5 H1 H2 A5 A1 A2
and Kae=
P4 P6 P3 H6 H4 H3 A6 A4 A3
(2.16)
These coefficients are functions of the frequency of motion, the mean wind velocity and the cross section. Wind tunnel tests with a section model is required in order to determine these coefficients. Cae and Kae are often normalised as shown in equation (2.17)
Cae = ρB2
2 ·ωi(V)·Cˆae and Kae= ρB2
2 ·[ωi(V)]2·Cˆae (2.17)
Cˆae=
P1∗ P5∗ BP2∗ H5∗ H1∗ BH2∗ BA∗5 BA1 B2A∗2
and Kˆae =
P4∗ P6∗ BP3∗ H6∗ H4∗ BH3∗ BA∗6 BA∗4 B2A∗3
(2.18)
The coefficient contained in Cˆae and Kˆae are called the aerodynamic derivatives. These are extracted from section tests in wind tunnels as function of the reduced velocity ˆV = V /(Bωi(V)). The reduced velocity is dependent on the mean wind velocity V and the resonance frequency which is a function of V. The total stiffness of the combined system is determined by the structural stiffness K and the motion induced stiffness Kae. Since the stiffness changes with V, this will cause a change in the resonance frequency ωi. As a results, response calculation will often demand iterations.
2.3.2 Wind field cross spectral densities
As mentioned in the beginning of this chapter. The statistical properties of the wind field is represented by spectral densities in the frequency domain. Response calculation will therefore require the cross spectral density of the turbulence components. The loading related to the turbulence was given as Bqv (see equation 2.10). The steps to obtain the wind field cross spectral densities consist of Fourier transforming Bqv in order to obtain the Fourier amplitudes. The following is then obtained
aq(x, ω) =
aqy aqz aqθ
=
ρV B 2
·Bˆq·av (2.19)
where
av(x, ω) = [au aw]T (2.20)
The cross spectrum is defined through general definition by the Fourier amplitudes of the load as
Sqq(x1, x2, ω) = lim
T→∞
1 πT
haq∗(x1, ω)·aTq(x2, ω)i
=
ρV B 2
2
·bˆq(x1)· lim
T→∞
1 πT
hav∗(x1, ω)·aTv(x2, ω)i·bˆTq(x2)
=
ρV B 2
2
·bˆq(x1)·Sv(x1, x2, ω)·bˆTq(x2)
(2.21)
where
Sv(∆x, ω) = lim
T⇒∞
1 πT
"
a∗uau a∗uaw a∗wau a∗waw
#
=
"
Suu Suw Swu Sww
#
=
"
Suu 0 0 Sww
#
(2.22) It’s has been assumed that the cross spectra between flow components u and w is neg- ligible. The cross spectrum is often defined by a normalized co-spectrum and a single point spectra describing the turbulence components in the frequency domain (see equa- tion (2.23)). In the time domain these properties are describes by the auto covariance functions or coefficients.
Snn(ω,∆x) = Sn(ω)·Coˆ nn(ω,∆x) where n=u, v, w (2.23) The Kaimal and von K´arm´an spectra of turbulence components can be used in calculation.
Equation (2.24) shows the Kaimal auto spectra that’s the chosen spectra for response calculation in Chapter 3.
Sn(ω) σ2n =
An
2πωˆn
1 + 1.5· A2πnωˆn5/3
where n =u, v, w (2.24)
where ˆωn=ω·xf Ln/V and n = u, v or w. xfLn is the integral length scale, a measure of the size of the vortices in the wind. The length scales usually depend on the roughness of the terrain and the height above the ground. These length scale can be approximated from full-scale measurements [6]. Unless full scale measurements indicates otherwise, the values for Ancan be taken asAu = 6.8 andAv =Aw = 9.4. Under homogeneous conditions, the co-spectra can be estimated as
Coˆ nn(∆s, f) = exp −cnsω·∆s V(zf)
!
n=u, v, w s=xf, yf, zf
∆s = ∆xf,∆yf,∆zf
(2.25)
where ∆s=|s1−s2|, and the constant cns can be taken as
cns=
cuyf =cuzf ≈9/2π
cvyf =cvyf =cwyf ≈6/2π cwyf ≈3/2π
(2.26)
Since the buffeting load and cross spectra has been defined. The next section will develop the modal load spectrum and the response spectrum so that the response due to the wind field can be found.
2.4 Buffeting Response
The response of a structure r = [ry rz rθ]T can in general be found by solving the equation of motion (in matrix form)
M¨r+C˙r+Kr=q (2.27)
whereMis the mass, Cdamping andKthe stiffness of the structure. These matrices are all assumed to be diagonal, and q is the external load vector. An approximate solution to the equation of motion can be found by assuming thatr can be represented by a linear combination of eigenmodes φi and time dependent generalized coordinate ηi(t).
r(x, t) =Φ(x)·η(t) (2.28) The components in the two matrices are
Φ=hϕ1. . .ϕn. . .ϕN
mod
i
η(t) = [η1. . . ηn. . . ηNmod]T
(2.29)
whereϕn = [φy φz φθ]T (see Figure 2.4). The eigenmodesϕi along with corresponding eigenfrequencies can be found by solving the eigen-value problem of the undamped and unloaded equation of motion given by
K−ω2Mϕ= 0 (2.30)
Details on the solution process for this equation will not be presented, but can be found in textbook such as Structural Dynamics[4], or Dynamics of Structures [5]. It will be assumed that the eigenfrequencies and eigenmodes of the system is known.
Figure 2.4: A general mode with three components [1]
The external loads of the system are assumed to be divided into
qtot =q(x, t) +qae(x, t,r,˙r,¨r) (2.31) where q(x, t) = [qy qz qθ]T is the flow induced part, and qae = [qy qz qθ]Tae is the motion induced part. The modal equilibrium equation can then be written as
M˜0·η¨+C˜0·η˙+K˜0·η=Q˜tot (2.32) where Q˜tot = Q˜ +Q˜ae. The index zero specify the structural properties in vacuum or in still air (V=0). The elements in the modal mass, damping and stiffness matrices are given by
M˜0 =diag[ ˜Mi] M˜i =R
L
ϕTi ·M·ϕidx C˜0 =diaghC˜ii C˜i = 2 ˜Miωiζi
K˜0 =diaghK˜i
i K˜i =ωi2M˜i
(2.33)
where ζi is the structural damping ratios also assumed as known quantities. The values can be chosen from experience or a code of practice. The flow induced part of the total modal load vector in equation (2.32) is then
Q˜ = [ ˜Q1 . . . Q˜i . . . Q˜Nmod]T (2.34)
where the elements are
Q˜n=
Z
Lexp
ϕTn ·qdx (2.35)
Bringing the modal equilibrium equation (eq.(2.32)) into the frequency domain is done by a Fourier transformation rendering
−Mω˜ 2+Ciω˜ +K˜·aη(ω) =aQ˜(ω) +aQ˜ae(ω, η,η,˙ η)¨ (2.36) where aη,aQ˜ and aQ˜ae is the Fourier amplitudes of η, Q˜ and Q˜ae. It is taken as an as- sumption that the motion induced load part is proportional to the structural displacement, velocity and acceleration. Thus
aQ˜ae =−M˜aeω2+C˜aeiω+K˜ae·aη(ω) (2.37) The motion induced massM˜aeis often negligible in wind engineering [1], and will therefore be omitted from this point on. C˜aeandK˜ are allNmodbyNmodmatrices and non-diagonal, and the elements are given by
C˜aeij K˜aeij
=
Z
Lexp
ϕTi ·Cae·ϕj ϕTi ·Kae·ϕj
dx (2.38)
the index Lexp indicate integration over the wind exposed part of the structure. The content of Cae andKae was defined in section 2.3.1. Inserting equation (2.37) into (2.36) and collecting terms related to structural motion on the left hand side gives
h−Mω˜ 2+C˜ −C˜aeiω+K˜ −K˜aei·aη(ω) = aQ˜(ω) (2.39) multiplying this equation withK˜ =diag[ω2iM˜i] and introducing the reduced modal vector as
aQˆ(ω) =K˜−1· aQ˜(ω) =
"
· · ·
R
Lexp
ϕTn(x)·aq(x,ω)dx ω2nM˜n · · ·
#T
(2.40)
where aq(x, ω) =haqy aqy aqθiT. Equation (2.39) can then be solved for aη(ω)
aη(ω) =Hˆη(ω)·aRˆ(ω) (2.41)
where
Hˆη(ω) =
(
I−κae−
ω·diag
1 ωn
2
+ 2iω·diag
1 ωn
·(ζ −ζae)
)−1
(2.42) is the non-dimensional frequency response matrix. I is the identity matrix, and κae, ζae and ζ are
κae =K˜−1K˜ae ζae = 1
2diag[ωn]·K˜−1·C˜ae ζ =diag[ζi]
(2.43)
Recalling from section 2.3.1 that Kae=ρB2/2·ωi·Kˆae and Cae=ρB2/2·ωi·Cˆae, then the elements in these Nmod by Nmod matrices above are
κaeij = K˜aeij
ω2iM˜j = ρB2 2 ˜mi ·
R
Lexp
ϕTi ·Kˆae·ϕj
R
L
ϕTi ϕjdx (2.44)
ζaeij = ωi 2
C˜aeij ω2iM˜i
= ρB2 4 ˜mi ·
R
Lexp
ϕTn ·Cˆae·ϕj
R
L
ϕTi ϕjdx (2.45)
here ˜mi = ˜Mi/R
L
ϕTi ϕidx has been introduced, and ωi is assumed to be independent of the wind velocity V.
In a response calculation to find the flutter stability limit in chapter 3, it will be assumed that
Φ= [ϕ1 ϕ2] =
"
φz 0 0 φθ
#
(2.46) thus the expression for theκaeij andζaeij with use of the aerodynamic derivatives becomes
κae11 = ρB2 2 ˜mz ·
R
Lexp
φ2z
1 ·H4∗dx
R
L
φ2z
1dx = ρB2H4∗
2 ˜mz , κae12 = ρB2 2 ˜mz ·
R
Lexp
φz1φθ2BH3∗dx
R
L
φ2z
1dx (2.47)
κae21 = ρB2 2 ˜mθ ·
R
Lexp
φθ2φz1BA∗4dx
R
L
φ2θ
2dx , κae22 = ρB2
2 ˜mθ·
R
Lexp
φ2θ
2 ·B2A∗3dx
R
L
φ2θ
2dx = ρB2H4∗
2 ˜mz (2.48)
ζae11 = ρB2 4 ˜mz·
R
Lexp
φ2z1H1∗dx
R
L
φ2z
1dx = ρB2H1∗
4 ˜mz , ζae21 = ρB2 4 ˜mz·
R
Lexp
φz1φθ2BH2∗dx
R
L
φ2z
1dx = ρB3
4 ˜mzH2∗
R
Lexp
φz1φθ2dx
R
L
φ2z
1dx (2.49)
ζae21 = ρB2 4 ˜mθ·
R
Lexp
φθ2φz1BA∗1dx
R
L
φ2z1dx = ρB3 4 ˜mθA∗1
R
Lexp
φθ2φz2dx
R
L
φ2θ2dx , ζae22 = ρB2 4 ˜mθ·
R
Lexp
φ2θ2B2A∗2dx
R
L
φ2θ2dx = ρB4A∗1 4 ˜mθ (2.50) where index 1 indicates the z-direction, and 2 indicates the θ-direction. The content of the non-dimensional frequency response function is then
Hˆη(ω) =
(
I−κae−
diag
ω ωn
2
+ 2i·diag
ω ωn
·(ζ −ζae)
)−1
=
("
1 0 0 1
#
−
"
κae11 κae12 κae21 κae22
#
−ω2
"
ωz−2 0 0 ωθ−2
#
+ 2iω
"
ω−1z 0 0 ωθ−1
# "
ζz 0 0 ζθ
#
−
"
ζae11 ζae12 ζae21 ζae22
#!)−1
(2.51)
The final step that remains is to define the response spectra Srr. This is found by first finding Sη through the Fourier amplitudes in equation (2.41)
Sη(ω) = lim
T→∞
1 πT
a∗η·aηT= lim
T→∞
1 πT
HˆηaQˆ
∗
·HˆηaQˆ
T
=Hˆ∗η· lim
T→∞
1 πT
aQ∗ˆ ·aTQˆ·HˆTη
=Hˆ∗η·SQˆHˆTη
(2.52)
where the normalised modal load matrix SQˆ is
SQˆ(ω) = lim
T→∞
1 πT
aQ∗ˆ ·aTQˆ= lim
T→∞ (2.53)
The elements in the the rows and columns of this matrix is given by
