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					   Wind loading and structural response
      Lecture 21 Dr. J.D. Holmes




Towers, chimneys and masts
                      Towers, chimneys and masts


• Slender structures (height/width is high)

•     Mode shape in first mode - non linear

• Higher resonant modes may be significant


•    Cross-wind response significant for circular cross-sections

    critical velocity for vortex shedding  5n1b for circular sections
                                           10 n1b for square sections
    - more frequently occurring wind speeds than for square sections
                    Towers, chimneys and masts


• Drag coefficients for tower cross-sections



                                           Cd = 2.2



                                               Cd = 1.2



                                               Cd = 2.0
                  Towers, chimneys and masts

• Drag coefficients for tower cross-sections



                                       Cd = 1.5




                                       Cd = 1.4




                                       Cd  0.6 (smooth, high Re)
                            Towers, chimneys and masts

• Drag coefficients for lattice tower sections
e.g. square cross section with flat-sided members (wind normal to face)
              4.0


   Drag       3.5
coefficient
CD (q=0O)     3.0               Australian
                                Standards                  ASCE 7-02 (Fig. 6.22) :
              2.5

                                                               CD= 42 – 5.9 + 4.0
              2.0


              1.5

                0.0   0.2     0.4     0.6      0.8   1.0
                            Solidity Ratio 

    = solidity of one face = area of members  total enclosed area

   includes interference and shielding effects between members

    ( will be covered in Lecture 23 )
                   Towers, chimneys and masts

• Along-wind response - gust response factor



                                       Shear force : Qmax = Q. Gq


                                     Bending moment : Mmax = M. Gm


                                      Deflection : xmax = x. Gx

  The gust response factors for base b.m. and tip deflection differ -
  because of non-linear mode shape

  The gust response factors for b.m. and shear depend on the height
  of the load effect, z1 i.e. Gq(z1) and Gm(z1) increase with z1
                                  Towers, chimneys and masts

• Along-wind response - effective static loads


                      160

                      140
                            Resonant              Combined
                      120
         Height (m)




                      100         Background
                       80

                       60
                                               Mean
                       40

                       20

                        0
                            0.0   0.2    0.4      0.6   0.8    1.0

                                    Effective pressure (kPa)


 Separate effective static load distributions for mean, background
 and resonant components (Lecture 13, Chapter 5)
                    Towers, chimneys and masts

• Cross-wind response of slender towers

 For lattice towers - only excitation mechanism is lateral turbulence

 For ‘solid’ cross-sections, excitation by vortex shedding is usually
 dominant (depends on wind speed)

 Two models : i) Sinusoidal excitation
               ii) Random excitation
Sinusoidal excitation has generally been applied to steel chimneys where
large amplitudes and ‘lock-in’ can occur - useful for diagnostic check of
peak amplitudes in codes and standards

Random excitation has generally been applied to R.C. chimneys where
amplitudes of vibration are lower. Accurate values are required for design
purposes. Method needs experimental data at high Reynolds Numbers.
                    Towers, chimneys and masts

• Cross-wind response of slender towers

 Sinusoidal excitation model :

 Assumptions :
 • sinusoidal cross-wind force variation with time
 • full correlation of forces over the height
 • constant amplitude of fluctuating force coefficient
‘Deterministic’ model - not random

Sinusoidal excitation leads to sinusoidal response (deflection)
                    Towers, chimneys and masts

• Cross-wind response of slender towers

 Sinusoidal excitation model :
 Equation of motion (jth mode):

                       G j a  C j a  K j a  Q j (t )
                                 

                                                       h
                                                          m(z)  j (z) dz
                                                                   2
 Gj is the ‘generalized’ or effective mass =           0


 j(z) is mode shape

                                                       h
 Qj(t) is the ‘generalized’ or effective force =   
                                                   0
                                                           f(z, t) j (z) dz
                            Towers, chimneys and masts

   • Sinusoidal excitation model

Representing the applied force Qj(t) as a sinusoidal function of time, an
expression for the peak deflection at the top of the structure can be derived :

  (see Section 11.5.1 in book)
                                        h                     h
                  y m ax(h) ρ a C  b 0  j (z) dz      C    j (z) dz
                                    2

                                                            0
                      b         16π 2 G jη jSt 2                  h
                                                      4π Sc St 2   j (z) dz
                                                                      2
                                                                  0

                                                                           Cj
where j is the critical damping ratio for the jth mode, equal to
                                                                       2 GjK j
         nsb    n jb
   St                           Strouhal Number for vortex shedding
        U(ze ) U(ze )             ze = effective height ( 2h/3)

           4mη j
    Sc                     (Scruton Number or mass-damping parameter)
           ρa b   2
                            m = average mass/unit height
                       Towers, chimneys and masts

   • Sinusoidal excitation model

This can be simplified to :   y max      k.C
                                    
                                b     4 .Sc.St2

                                                      
                                                             (z) dz 
                                                               h

   where k is a parameter depending on mode shape                    j
                                                      
                                                            0
                                                            h         
                                                            (z) dz 2
                                                          0
                                                                   j
                                                                      
   The mode shape j(z) can be taken as (z/h)

 For uniform or near-uniform cantilevers,  can be taken as 1.5; then k = 1.6
                       Towers, chimneys and masts

   • Random excitation model (Vickery/Basu) (Section 11.5.2)
   Assumes excitation due to vortex shedding is a random process

    ‘lock-in’ behaviour is reproduced by negative aerodynamic damping
   Peak response is inversely proportional to the square root of the damping

   In its simplest form, peak response can be written as :

                     ˆ
                     y                 A
                       
                     b [(Sc / 4 )  K (1  y 2            )]1 / 2
                                      ao               2
                                                  yL

A = a non dimensional parameter constant for a particular structure (forcing terms)

Kao = a non dimensional parameter associated with aerodynamic damping
yL= limiting amplitude of vibration
                      Towers, chimneys and masts

   • Random excitation model (Vickery/Basu)

   Three response regimes :
                          Maximum tip 0.10
                           deflection /
                            diameter


                                              ‘Lock-in’
                                              Regime

                                      0.01


                                             ‘Transition’
                                               Regime

                                                                              ‘Forced
                                                                             vibration’
                                     0.001                                    Regime
                                              2              5      10       20
                                                            Scruton Number


Lock in region - response driven by aerodynamic damping
                    Towers, chimneys and masts

• Scruton Number

The Scruton Number (or mass-damping parameter) appears in peak response
calculated by both the sinusoidal and random excitation models

                                 4mη
                          Sc 
                                 ρa b2
                                                                      mη
Sometimes a mass-damping parameter is used = Sc /4 = Ka =
                                                                     ρa b2

 Clearly the lower the Sc, the higher the value of ymax / b   (either model)


 Sc (or Ka) are often used to indicate the propensity to vortex-
 induced vibration
                    Towers, chimneys and masts

• Scruton Number and steel stacks

Sc (or Ka) is often used to indicate the propensity to vortex-induced
vibration
e.g. for a circular cylinder, Sc > 10 (or Ka > 0.8), usually indicates low
amplitudes of vibration induced by vortex shedding for circular cylinders

American National Standard on Steel Stacks (ASME STS-1-1992) provides
criteria for checking for vortex-induced vibrations, based on Ka

Mitigation methods are also discussed : helical strakes, shrouds, additional
damping (mass dampers, fabric pads, hanging chains)

A method based on the random excitation model is also provided in ASME
STS-1-1992 (Appendix 5.C) for calculation of displacements for design
purposes.
                      Towers, chimneys and masts

  • Helical strakes

  For mitigation of vortex-shedding induced vibration :
                                                                   h/3

                                                                           h
                                                          0.1b



                                                           b




Eliminates cross-wind vibration, but increases drag coefficient and along-wind
vibration
                   Towers, chimneys and masts

 • Case study : Macau Tower


Concrete tower 248 metres (814 feet) high
Tapered cylindrical section up to 200 m (656 feet) :
     16 m diameter (0 m) to 12 m diameter (200 m)


‘Pod’ with restaurant and observation decks
   between 200 m and 238m
Steel communications tower 248 to 338 metres (814 to 1109 feet)
                  Towers, chimneys and masts
• Case study : Macau Tower

aeroelastic
model
(1/150)
                   Towers, chimneys and masts

  • Case study : Macau Tower


• Combination of wind tunnel and theoretical
  modelling of tower response used

 • Effective static load distributions
       • distributions of mean, background and resonant wind loads
         derived (Lecture 13)


 • Wind-tunnel test results used to ‘calibrate’
   computer model
                Towers, chimneys and masts

 • Case study : Macau Tower


Wind tunnel model scaling :


 • Length ratio Lr = 1/150

 • Density ratio r = 1

 • Velocity ratio Vr = 1/3
                Towers, chimneys and masts

• Case study : Macau Tower


 Derived ratios to design model :

• Bending stiffness ratio EIr = r Vr2 Lr4

• Axial stiffness ratio EAr = r Vr2 Lr2

• Use stepped aluminium alloy ‘spine’ to model
  stiffness of main shaft and legs
                 Towers, chimneys and masts
• Case study : Macau Tower

                                                         Wind-tunnel
Mean velocity                                            AS1170.2
profile :                                                Macau Building Code
                                             350


                     Full-scale Height (m)
                                             300
                                             250
                                             200
                                             150
                                             100
                                             50
                                              0
                                                   0.0   0.5   1.0   1.5
                                                          Vm /V240
                 Towers, chimneys and masts
• Case study : Macau Tower

                                 Wind-tunnel
 Turbulence             MACAU TOWER - Turbulence
                                 AS1170.2
 intensity                       Macau Profile
                             Intensity Building Code
 profile :                       350
                                 300
                    Height (m)   250
                    Full-scale

                                 200
                                 150
                                 100
                                 50
                                  0
                                       0.0   0.1        0.2   0.3
                                                   Iu
                   Towers, chimneys and masts
Case study : Macau Tower
Wind tunnel test results - along-wind b.m. (MN.m) at 85.5 m (280 ft.)

                           R.m.s.
                           MACAU TOWER Mean
                            0.5% damping Minimum
                           Maximum

                 2000
                 1500
                 1000
                  500
                    0
                 -500 0       20     40     60     80     100
                   Full scale mean wind speed at 250m (m/s)
                 Towers, chimneys and masts
Case study : Macau Tower
Wind tunnel test results - cross-wind b.m.(MN.m) at 85.5 m (280 ft.)


                         MACAU
                        R.m.s.  TOWER Mean
                         0.5% damping Minimum
                        Maximum

                 2000
                 1500
                 1000
                  500
                    0
                 -500 0      20     40    60     80     100
                -1000
                -1500
                -2000
                  Full scale mean wind speed at 250m (m/s)
                 Towers, chimneys and masts
Case study : Macau Tower


• Along-wind response was dominant
• Cross-wind vortex shedding excitation not strong because
  of complex ‘pod’ geometry near the top
• Along- and cross-wind have similar fluctuating components
  about equal, but total along-wind response includes mean
  component
                 Towers, chimneys and masts
Case study : Macau Tower
Along wind response :
• At each level on the structure define equivalent wind loads
  for :
   – mean wind pressure
   – background (quasi-static) fluctuating wind pressure
   – resonant (inertial) loads
• These components all have different distributions

• Combine three components of load distributions for
  bending moments at various levels on tower

• Computer model calibrated against wind-tunnel results
                 Towers, chimneys and masts
Case study : Macau Tower
Design graphs



                         cracked concrete 5% damping

                                      Mean           Maximum
                          500
            Along-wind
             bending      400
             moment       300
               at 200
                          200
              metres
              (MN.m)      100
                            0
                              0      20    40    60    80    100
                           Full scale mean wind speed at 250m (m/s)
                               Towers, chimneys and masts
Case study : Macau Tower
Design graphs
                         Macau Tower Effective static loads
                                     (s=0 m)
                                   U m ean = 59 7m/s; 5% damping


                         350
                         300                                       Mean
            Height (m)




                         250
                                                                   Background
                         200
                                                                   Resonant
                         150
                         100                                       Combined
                          50
                           0
                               0            100         200
                                        Load (kN/m)
 End of Lecture 21

       John Holmes
225-405-3789 JHolmes@lsu.edu

				
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