fluid - structure interaction (aeroelastic response analysis) · fluid – structure interaction...

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Fluid – Structure Interaction

(Aeroelastic Response Analysis) _________________________________________________________

JULY 2011

MAVERICK UNITED CONSULTING ENGINEERS

Fluid – Structure Interaction (Aeroelastic Response Analysis)

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TABLE OF CONTENTS

ACKNOWLEDGEMENTS ....................................................................................................................................... 3

1.1 GL, ML FLUID (WIND & WATER) - STRUCTURE INTERACTION (AEROELASTIC RESPONSE ANALYSIS) .... 4 1.1.1 Elementary Hydraulics ..................................................................................................................................................................... 5 1.1.1.1 Pipeline Design (Application of the Steady and Unsteady Bernoulli’s Equation) ..................................................................................................... 5 1.1.1.2 Elementary Wave Mechanics (Small Amplitude Wave Theory) ............................................................................................................................. 90 1.1.2 Static Aeroelastic Response ............................................................................................................................................................ 97 1.1.2.1 Along-Flow Direction Drag (Pressure Drag and Skin-Friction Drag) (Limited Amplitude Response) .................................................................... 98 1.1.2.2 Across-Flow Direction Lift (Pressure Lift and Skin-Friction Lift) (Limited Amplitude Response) ...................................................................... 100 1.1.2.3 Non-Oscillatory Torsional Divergence (Divergent Amplitude Response) ............................................................................................................. 101 1.1.3 Dynamic Aeroelastic Response ..................................................................................................................................................... 105 1.1.3.1 Along- and Across-Flow Direction Unsteady Inertial Forces on General Submerged Structures Due to Water Waves (Limited Amplitude

Response) .............................................................................................................................................................................................................................. 105 1.1.3.2 Along- and Across-Flow Direction Unsteady Inertial Forces on Cylindrical Submerged Structures Due to Water Waves (And Along-Flow

Direction Drag Forces Due to Steady Currents) (Limited Amplitude Response) ................................................................................................................... 106 1.1.3.3 Along-Flow Direction Gust Response (And Along-Flow Direction Drag Forces Due to Steady Mean Wind) (Limited Amplitude Response) .... 109 1.1.3.4 Along-Flow Direction Buffeting Response (Limited Amplitude Response).......................................................................................................... 126 1.1.3.5 Across-Flow Direction Von Karman Vortex Shedding Response (Limited Amplitude Response) ....................................................................... 127 1.1.4 Dynamic Aeroelastic Stability ...................................................................................................................................................... 142 1.1.4.1 Across-Flow Direction Galloping and Stall Flutter (Divergent Amplitude Response) .......................................................................................... 142 1.1.4.2 Across-Flow Direction Classical Flutter (Divergent Amplitude Response) ........................................................................................................... 153

BIBLIOGRAPHY ................................................................................................................................................... 161


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ACKNOWLEDGEMENTS

My humble gratitude to the Almighty, to Whom this and all work is dedicated.

A special thank you also to my teachers at Imperial College of Science, Technology and Medicine, London and my

fellow engineering colleagues at Ove Arup and Partners London and Ramboll Whitbybird London.

Maverick United Consulting Engineers


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1.1 GL, ML Fluid (Wind & Water) - Structure Interaction (Aeroelastic Response Analysis)

Flow induced vibrations include: -

(I) Static Aeroelastic Response

(i) Along-Flow Drag

(ii) Across-Flow Lift

(iii) Torsional Divergence

(II) Dynamic Aeroelastic Response

(i) Along- and Across-Flow Unsteady Inertial Force Response

(ii) Along-Flow Gust Response

(iii) Along-Flow Buffeting Response

(iv) Across-Flow Von Karman Vortex Shedding Response

(III) Dynamic Aeroelastic Stability

(i) Across-Flow Galloping

(ii) Across-Flow Classical Flutter

Apart from these, bodies within fluid are also subject to

(I) Upward buoyancy forces

(II) All direction atmospheric pressure 10kPa

(III) All direction hydrostatic forces .g.(depth in fluid)

(IV) Downward gravitational forces


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1.1.1 Elementary Hydraulics

1.1.1.1 Pipeline Design (Application of the Steady and Unsteady Bernoulli’s Equation)


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1.1.1.2 Elementary Wave Mechanics (Small Amplitude Wave Theory)

1.1.1.2.1 Introduction

Wave number, k = 2/L or 2/, L or is the wave length

Wave frequency, = 2/T, T = wave period

Wave amplitude, a measured from crest to still water level

Water depth, d

Water surface,

Wave speed, c = /T = /k

Wave height, h = 2a

Small amplitude wave theory assumes mass continuity (incompressible fluid), irrotationality (no friction hence

invalid in boundary layer near sea bed), unsteady Bernoulli, motion is periodic in x and t, a << and a << d.


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1.1.1.2.2 Basic Formulae


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1.1.1.2.3 Formulae for Special Cases


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1.1.1.2.4 Formulae for Analysis


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1.1.2 Static Aeroelastic Response

The structural load distribution on an elastic vehicle in trimmed flight is determined by solving the equations for

static equilibrium. The SOL 144 and SOL 200 processes will calculate aerodynamic stability derivatives (e.g., lift

and moment curve slopes and lift and moment coefficients due to control surface rotation) and trim variables (e.g.,

angle of attack and control surface setting) as well as aerodynamic and structural loads, structural deflections, and

element stresses.


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1.1.2.1 Along-Flow Direction Drag (Pressure Drag and Skin-Friction Drag) (Limited Amplitude Response)

The drag forces objects immersed in fluids, can be thought of as being comprised of two parts: -

Skin-friction Dτ – the forces applied by shearing as the liquid passes the solid boundaries.

Pressure-drag DP – the net forces applied to the body due to differences in the upstream and downstream

pressure fields.

Broadly speaking, both components of the total drag force increase with fluid velocity V.

nVD where n is in the range 1.2→1.8

2P VD

The relative magnitude of these two drag components depends on the geometry of the object in the fluid. For

‘stream-lined’ bodies e.g. a flat plate parallel to the flow direction, the drag will be almost entirely skin friction

while for ‘bluff’ bodies e.g. a flat plate normal to the flow direction, the pressure drag will dominate. For some

cross-sections e.g. a cylinder, the ratio of these two drag forces varies considerably depending on the flow regime.

But the skin friction drag is usually of a much smaller magnitude even for streamlined bodies.

The (pressure and skin friction) drag on a body immersed in a steady fluid flow can be expressed by a

dimensionless quantity known as the drag coefficient, CD, such that the total drag force on the body is given by

Area.C.V2

1F D

2

The flow regime around a body is governed by a dimensionless quantity known as the Reynold’s Number, Re. This

dimensionless quantity is defined as the ratio of inertial to viscous forces in a fluid. The non-dimensional Reynolds

Number is defined for the flow around a cylinder as

where is the fluid density, V the stream velocity relative to the cylinder, D the diameter of the cylinder and the

dynamic viscosity of the fluid.

VDR e


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At small Re < 0.5, the inertia effects are negligible and glow pattern very similar to ideal flow, the pressure

recovery being nearly complete. Pressure drag negligible and total profile drag is nearly all due to friction drag.

Here, CD and Re exhibit a straight line from which we can conclude that drag is directly proportional to the velocity

V (Stoke’s Law).

At 2 < Re < 30, separation of boundary layers occur. The boundary layer is laminar. Two symmetrical eddies

rotating in opposition to each other are formed. They remain fixed in position and the main flow closes behind

them. The separation of the boundary layer is reflected in the graph of CD versus Re by the curvature of the line

indicating that the drag is now proportional to Vn where n 2.

Further increase in Re (90 < Re < 2 x 105) tends to elongate the fixed eddies. At Re of 90 onwards, the eddies break

away from the cylinder. The separation point moves to an earlier point. The boundary layer is laminar. The

breaking away occurs alternately from one and then the other side of the cylinder, the eddies being washed away by

the mainstream. This process is intensified by further increase in Re, whereby the shedding of the eddies from

alternate sides of the cylinder is continuous, thus forming in the wake two discreet rows of vortices. This is known

as vortex street or von Karmon vortex street. The shedding of these vortices produces circulation and hence gives

rise to lateral forces on the cylinder.

It is fortunate that at higher Re (Re > 2 x 105) values the vortices disappear because of high rates of shear and are

then replaced by a highly turbulent wake. The boundary layer changes to turbulent (instead of laminar) before

separation. The effect of this is that the separation point moves later instead of the general trend of moving to an

earlier point. Hence a smaller wake results, and thus reduced drag. Regular vortex shedding occurs, not alternating

anymore. Roughness becomes important at high Re in determining CD.

At Re > 107, the value of CD appears to be independent of Re, but there is insufficient data at this end of the range.

An explanation of the boundary layer formation and its separation is warranted. For real fluids, a thin boundary

layer grows from the stagnation point at A. The boundary layer is a layer of fluid near the surface of the body

which is affect is affected by shear forces, hence friction is prominent and ideal flow theories do not hold. A to B is

the region of velocity increasing in x (acceleration), hence the pressure decreases in x i.e. a favourable pressure

gradient as the boundary layer tends to reduce in thickness. From B to C, the velocity decreases in x (deceleration),

hence the pressure increases in x i.e. an adverse pressure gradient as the boundary layer thickens. This force which

opposes the direction of fluid motion coupled with the action of the shear forces in the boundary layer if they act

for a sufficient length will bring the fluid in the boundary layer to rest and subsequently the flow separates from the

surface of the body. When the fluid near the surface begins to move in the other direction due to the adverse

pressure gradient, this implies that the boundary layer (i.e. the point where the fluid is zero velocity) separates from

the surface. This flow separation has serious consequences in aerofoil design as once the flow separates away from

the surface, all lift is lost.

Flow in the wake is highly turbulent and consists of large scale eddies. High rate energy dissipation occurs with the

result of a pressure reduction in the wake. But the stagnation pressure in front is high and thus the resultant force

arising from pressure difference is called the drag force. The bigger the wake (i.e. the earlier the boundary layer

separation point), the smaller the pressure in the wake and so the greater the pressure drag.

A

B

C

x


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1.1.2.2 Across-Flow Direction Lift (Pressure Lift and Skin-Friction Lift) (Limited Amplitude Response)

The (pressure and skin friction) lift is analogous to the drag and acts in the direction perpendicular to the direction

of fluid motion.

Area.C.V2

1F L

2

Lift occurs when the faster moving fluid on top of the aerofoil has a lower pressure to the slower moving fluid

underneath. If the boundary layer separation occurs at the top of the aerofoil, the suction pressure is lost. This is

known as stalling. The plane will then drop from the sky. The only solution is to put the plane in a dive to regain

the boundary layer.

The two dimensional drag and lift coefficients for structural shapes are presented.

Clearly, the lift is zero for a body symmetrical about the direction of flow.


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1.1.2.3 Non-Oscillatory Torsional Divergence (Divergent Amplitude Response)

Divergence can occur if the aerodynamic torsional stiffness (i.e. the rate of change of pitching moment with

rotation) is negative. At the critical wind speed the negative aerodynamic stiffness becomes numerically equal to

the structural torsional stiffness resulting in zero total stiffness.

Torsional divergence is an instance of a static response of a structure. Torsional divergence was at first associated

with aircraft wings due to their susceptibility to twisting off at excessive air speeds (Simiu and Scanlan, 1986). Liu,

1991, reports that when the wind flow occurs, drag, lift, and moment are produced on the structure. This moment

induces a twist on the structure and causes the angle of incidence α to increase. The increase in α results in higher

torsional moment as the wind velocity increases. If the structure does not have sufficient torsional stiffness to resist

this increasing moment, the structure becomes unstable and will be twisted to failure. Simiu and Scanlan, 1986,

report that the phenomenon depends upon structural flexibility and the manner in which the aerodynamic moments

develop with twist; it does not depend upon ultimate strength. They say that in most cases the critical divergence

velocities are extremely high, well beyond the range of velocities normally considered in design.


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BD 49/01 (valid depending on the aerodynamic susceptibility parameter, Pb and satisfying some geometric

constraints) the states that a bridge is unsusceptible to non-oscillatory torsional divergence if it is stable against

self-induced instability, i.e. galloping (and stall flutter) and classical flutter.


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BD 49/01 is valid depending on the value of the aerodynamic susceptibility parameter, Pb

where ρ is the density of air, b is the overall width of the bridge deck, m is the mass per unit length of the bridge, Vr

is the hourly mean wind speed (for relieving areas), L is the length of the relevant maximum span of the bridge and

fB is the natural frequency in bending. Then

(a) If Pb < 0.04, bridge is subject to insignificant effects in respect of all forms of aerodynamic excitation.

(b) Bridges having 0.04 ≤ Pb ≤ 1.00 shall be considered to be within the scope of the BD 49/01 rules.

(c) Bridges with Pb > 1.00 shall be considered to be potentially very susceptible to aerodynamic excitation

and are thus beyond the scope of BD 49/01. Further analytical and/or wind tunnel tests must be performed.

Normal highway bridges of less than 25m span should generally be found to be category (a). Bridges of spans

greater than 250m are likely to be category (c). Covered footbridges, cable supported bridges and other structures

where any of the parameters b, L or fB cannot be accurately derived shall be considered as category (c). For the

purposes of initial/ preliminary categorisation, the following may be used to given an indicative range for Pb:

Vr between 20 and 40 m/s;

m/b between 600 and 1200 kg/m2;

fB between 50/L0.87 and 100/L0.87, with L in metres..

The geometric constraints for the applicability of BD 49/01 are detailed.

(i) Solid edge members, such as fascia beams and solid parapets shall have a total depth less than

0.2d4 unless positioned closer than 0.5d4 from the outer girder when they shall not protrude above

the deck by more than 0.2d4 nor below the deck by more than 0.5d4. In defining such edge

members, edge stiffening of the slab to a depth of 0.5 times the slab thickness may be ignored.

(ii) Other edge members such as parapets, barriers, etc., shall have a height above deck level, h, and a

solidity ratio, φ, such that φ is less than 0.5 and the product hφ is less than 0.35d4 for the effective

edge member. The value of φ may exceed 0.5 over short lengths of parapet, provided that the total

length projected onto the bridge centre-line of both the upwind and downwind portions of parapet

whose solidity ratio exceeds 0.5 does not exceed 30% of the bridge span.

(iii) Any central median barrier shall have a shadow area in elevation per metre length less than 0.5m2.

Kerbs or upstands greater than 100mm deep shall be considered as part of this constraint by

treating as a solid bluff depth; where less than 100mm the depth shall be neglected, see figure

below.

In the above, d4 is the reference depth of the bridge deck. Where the depth is variable over the span, d4 shall be

taken as the average value over the middle third of the longest span.


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1.1.3 Dynamic Aeroelastic Response

Aeroelastic analysis concerns the interaction of aerodynamic and structural (inertial, damping and stiffness) forces.

It is utilized in the design of airplanes, suspension bridges, missiles, power lines, tall chimneys etc.

1.1.3.1 Along- and Across-Flow Direction Unsteady Inertial Forces on General Submerged Structures Due

to Water Waves (Limited Amplitude Response)

Based on small amplitude wave mechanics, the unsteady (time varying) pressure distribution under the surface of

the fluid on a general submerged structure due to waves is

)kxtsin(kdcosh

)dy(kcoshgap

where = density of fluid

a = amplitude of waves

k = 2/L, where L is the wavelength

y = height above fluid surface, measure positive upwards

d = total depth of fluid

= 2/T, where T is the wave period

This pressure expression is integrated with respect to y from –d to 0 for the total force.


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1.1.3.2 Along- and Across-Flow Direction Unsteady Inertial Forces on Cylindrical Submerged Structures

Due to Water Waves (And Along-Flow Direction Drag Forces Due to Steady Currents) (Limited

Amplitude Response)

The unsteady inertial force on a cylinder is given by

length)per Force(t

u.C.d.

4

1F M

2

CM is the inertial coefficient which varies depending upon the degree of flow separation and wake formation.

Experimental measurements have shown that the amplitude of the fluid motion relative to the diameter of the

cylinder has an important effect on the nature of the induced loading. The ratio is defined as the Keulegan-

Carpenter number. The KC number is really the drag versus inertia ratio.

KC = uT/d

where u is the velocity amplitude, T the period of oscillatory motion and d the diameter of the cylinder. This

characterizes the Morrison’s equation which includes the steady drag force and the unsteady inertial force.

length)per Force(t

u.C.d.

4

1d.C.V

2

1F M

2D

2

For small KC, the velocity amplitude is small relative to the diameter of the cylinder. Flow does not separate fully

and thus in the limiting case where KC is very small the flow becomes potential flow. Hence the drag coefficient

reduces to 0.0 and the inertial coefficient becomes 2.0 (idealized case). For large KC, the velocity amplitude is

larger than the cylinder diameter. Both inertial and drag forces are important. However since the drag is

proportional to V2, this will produce the dominant contribution to Morison’s equation.

KC

Coefficients

1.0

0.0

2.0

10 20 30 40 50

CM

CD


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1.1.3.3 Along-Flow Direction Gust Response (And Along-Flow Direction Drag Forces Due to Steady Mean

Wind) (Limited Amplitude Response)

The total wind velocity is

Utotal (z, t) = Umean (z) + Ugust (z, t)

The mean velocity is defined

Umean (z) = U10 . (z / 10)

where U10 is the reference mean wind speed at 10m above ground and is the roughness coefficient

= 0.16 for open country

= 0.28 for woods, villages and towns

= 0.40 for large city centres

Note that U10 usually lies between 24 and 34ms-1. The mean velocity gives rise to the steady drag force.

The gust gives rise to the dynamic gust response. The spectral density of the gust is defined as follows.

Gust spectral density = Gust spectrum x (Aerodynamic admittance function)2

The gust spectrum is the wind speed spectra at a particular point. The aerodynamic admittance function defines the

relationship between the gust frequency and its area of influence. Gusts of higher frequency have a smaller area of

influence and vice versa. The gust spectral density function usually exhibit peak frequencies between 0.05 and

0.1Hz.

Passive tuned mass dampers offer little to reducing gust response due to the impulsive nature of the excitation.

Stiffening the structure is the usual effective approach.

For design purposes, as described in codes of practise, the wind force due to both mean speed and gust can be

obtained using the equivalent static wind force

W = . cf . q . A

where

= gust factor

cf = aerodynamic resistance coefficient

q = wind drag pressure = Umean2/2

A = area loaded

The gust factor considers both the stochastically varying aerodynamic properties of the natural wind and the

vibration behaviour of the structure in its fundamental mode as is given by

= 1 + R (B + s.F/)0.5

where

R = terrain factor = function ()

B = basic gust factor = function (d, h)

d = width of structure

h = height of structure

s = size factor = function (f.h/Uh,d/h)

F = gust energy factor = function (f/Uh)

f = structural fundamental frequency in wind direction

Uh = mean wind speed at height h

= damping (expressed as logarithmic decrement) of the fundamental mode

In BS 6399-2, the pressure and skin friction drag is respectively

where Cf is the friction drag coefficient, As is the area swept by the wind and Cr is the dynamic augmentation factor


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where Sg is the gust factor. The pressure drag force is

The (internal and external) pressures, p applied on the structure are calculated from

p = qs . Cp . Ca

where Cp is the (internal or external) pressure coefficient, Ca is the size effect factor and the dynamic wind pressure,

qs is

qs = 0.613Ve2

where the (structure dependent) effective wind speed, Ve is

Ve = Vs . Sb

where

Sb = the terrain factor

Vs = (site dependent) site wind speed

= Vb . Sa . Sd . Ss . Sp

where

Vb = (area dependent) basic hourly mean wind speed at 10m height above sea level

Sa = altitude factor

Sd = direction factor

Ss = seasonal factor

Sp = probability factor

The skin friction drag is ignored. Only the more significant pressure drag is accounted for. For the site of Metz

which is in Department Moselle, i.e. Zone 2, corresponding basic wind speed (reference EC1) is

Vb = 26 m/s

The site wind speed

Vs = Vb . Sa . Sd . Ss . Sp

26 m/s where

the altitude factor, Sa 1.0 (non significant topographical effects)

the direction factor, Sd = 1.0 (wind direction III – 210)

the seasonal factor, Ss = 1.0 (since it is a permanent building and hence exposed for at least 6 months)

the probability factor, Sp = 1.0 (for standard 2% probability of exceedance)

The structure dependent effective wind speed, Ve is

Ve = Vs . Sb

= 48 m/s

where the terrain and building factor (taking effective height, He = 30m conservatively as the whole height and

assuming site in town >100km to closest distance to sea) is

Sb = 1.85


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The dynamic wind pressure, qs is thus

qs = 0.613Ve2

= 1.42 kPa

The pressure, p applied on the structure is thus

p = qs . Cp . Ca

= qs.(Cpe-Cpi).Ca

= 1.106(Cpe-Cpi)

where the size effect factor (taking the loaded area diagonal, a = (902 + 302)0.5 = 95m and assuming site in town

>100km to closest distance to sea) is

Ca = 0.78


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where the external pressure coefficient Cpe for duo-pitch roofs is obtained from as follows


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The scaling lengths bL and bW where bL = L or bL = 2H, whichever is the smaller, and bW = W or bW = 2H,

whichever is the smaller. External pressure can be negative for low-pitched roofs while it becomes positive for

higher pitches. For a +60° roof, on average we have +0.8 on the side facing the wind and –0.65 on the side not

facing the wind. The asymmetric load case must be applied.

External pressure coefficients for mansard roofs and other multi-pitch roofs should be derived for each plane

face by the procedure given for duo-pitch roofs.


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The external pressure coefficient Cpe for flat roofs is obtained from


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The coefficients for hipped roofs are presented.


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The external pressure coefficient Cpe for building walls is obtained from


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The internal pressure coefficient Cpi for roofs and buildings is obtained from the following for buildings with

dominant openings

The internal pressure coefficients for enclosed buildings is

Pressure coefficients are considered positive when pressure is acting on the surface (i.e. normal to the surface) of

structure and negative when the pressure is acting away from the surface (still normal to the surface). Hence

positive external pressure acts inward on structure and positive internal pressure acts outward on structure, and vice

versa. It is important then to apply pressure loads in the local coordinate system of the element in a finite element

program and not in the global axes system.

Internal pressure occurs due to openings such as windows, doors, vents and cladding. In general if the windward

panel has a greater opening than the leeward panel, more wind comes in than out and hence the interior is subject to

positive (outward of building) pressure, and vice versa.

Note that the net pressure coefficient, Cp for buildings is

Note that the net pressure coefficient, Cp on a freestanding signboard is 1.8.

Note that the net upward pressure coefficient, Cp on a canopy is 1.8.


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Note that the net pressure coefficients, Cp on long cylinders and sharp edged elements such as rolled sections,

plate girders and boxes are

Finally, the design pressure is

pdesign = 0.85[1.106(Cpe-Cpi)].(1+Cr)

where the dynamic augmentation factor Cr is given by

The gust factor is

Sg = 1 + gtStTt

= 1.52

obtained from


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gust peak factor, gt = 2.30 (for He ~ 30m, a ~ 95m) (Table 24)

turbulence factor, St = 0.159 (for He ~ 30m, site >100m from sea) (Table 22)

turbulence adjustment factor, Tt = 1.43 (for He ~ 30m, upwind distance from edge of town > 30m)

(note Tt = 1 for country site) (Table 23)

Now,

where terrain correction factor, Kt = 0.75 (since town site; 1.33 if sea coast) and the terrain and building factor


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SO = ScTc(1+Sh) = 0.996

obtained from

the fetch factor, Sc = 1.2 (Table 22)

the fetch adjustment factor, Tc = 0.83 (Table 23)

topographic increment, Sh = 0 (for flat sites)

Hence,

Kh x Kb = 0.75 x (20 x 0.996 / (no2 x 95))2/3 x (26/(24))

= 0.287 (1/no2)2/3 (1/)

where no is the fundamental natural frequency in Hz and the the viscous damping of critical.

Thus the dynamic augmentation factor becomes

Cr = 0.567 x [(1+(0.287 (1/no2)2/3 (1/))/60)0.5 1]

Note that the structure is mildly dynamic for Cr < 0.25, but susceptible to dynamic amplifications in response to the

gust if Cr > 0.25 in which case this simplified equivalent static procedure would be less accurate, but generally

more conservative.

Thus, the design pressure is

pdesign = 0.85[1.106(Cpe-Cpi)].(1+Cr)

= 0.85[1.106(Cpe-Cpi)].(1+0.567 x [(1+(0.287 (1/no2)2/3 (1/))/60)0.5 1])

To ensure that structure not susceptible to dynamic amplifications from gust, need Cr at most 0.25. For assumed

damping of critical of = 0.04, thus no must be at least 0.19Hz. At this value of Cr of 0.25,

pdesign = 0.85[1.106(Cpe-Cpi)].(1+0.25)

= 1.18 (Cpe-Cpi) kPa

For comfort criteria, the peak acceleration at the top of a building for resonance in a fundamental bending mode

can be estimated from

2

0

1

res

2

0res

n2M

MG

n2xx

where

Gres = gust factor for resonant component = g2(v/V)h(SE/)0.5

M = mean base overturning moment; for a square building, it can be approximated by 0.6 ½ Vh2bh2

M1 = inertial base bending moment for unit displacement at top of building; for constant density and linear

mode shape, = 1/3 bdh2 (2n0)2

g = peak factor

n0 = first bending mode natural frequency; can be approximated by 46/h where h is height in metres

(v/V)h = longitudinal turbulence intensity at height h

T = period under consideration, sec; usually 600 sec for acceleration criteria

h = height of building

b = width of building

d = depth of building

Vh = hourly mean wind speed at height h

S = size factor = 1/[(1+3.5n0h/Vh)(1+4n0b/Vh)]

E = longitudinal turbulence spectrum = 0.47N/(2+N2)5/6

N = reduced frequency = nLh/Vh

Lh = measure of turbulence length scale = 1000 (h/10)0.25

= air density

S = building density

= critical damping ratio

Note that tall buildings have generally SLS and ULS values of reduced velocity (Vh/(n0b)) within the range 2 to 10.

Bachmann gives acceleration limits for buildings as follows.


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Perception Acceleration Limits

Imperceptible a < 0.005g

Perceptible 0.005g < a < 0.015g

Annoying 0.015g < a < 0.05g

Very Annoying 0.05g < a < 0.15g

Intolerable a > 0.15g


constraints) states that provided the fundamental frequencies in both bending and torsion calculated are greater

than 1Hz, the dynamic magnification effects of turbulence may be ignored. The dynamic magnification effects of

turbulence may also be neglected if PT < 1.0 where (using consistent set of units)

and is the density of air, b is the overall width of the bridge deck, m is the mass per unit length of the bridge and

fB the natural frequency of bridge in bending.

Vs is the site hourly mean wind speed (10m above ground level). σflm is the peak stress in the structure per unit

deflection in the first mode of vibration, derived for the most highly stressed location in the relevant element (Units

in N/mm²/Unit deflection). σc is a reference stress as follows

for steel beam elements, σc = 600 N/mm2 for the longitudinal flange bending stress;

for truss bridges, σc = 750 N/mm2 for the chord axial stress;

for concrete elements (composite or concrete bridges), σc = 80 N/mm2 for primary bending concrete stress

for cable-stayed bridges the peak stay axial stress should additionally be examined, with σc = 1200 N/mm2.

The MSC.NASTRAN aeroelastic dynamic response problem determines the response of the aircraft to time or

frequency-varying excitations. Atmospheric turbulence is the primary example of this type of excitation, but wind

shear and control surface motion can also have an aeroelastic component. Methods of generalized harmonic

(Fourier) analysis are applied to the linear system to obtain the response to the excitation in the time domain. The

gust response analysis may be regarded either as a stationary random loading or as a discrete gust. The gust

analysis capability computes the response to random atmospheric turbulence, discrete one-dimensional gust fields,

and control surface motion and other dynamic loading. The random response parameters calculated are the power

spectral density, root mean square response, and mean frequency of zero-crossings. The response to the discrete

gust and control surface motion is calculated by direct and inverse Fourier transform methods since the oscillatory

aerodynamics are known only in the frequency domain. The time histories of response quantities are the output in

the discrete case.

One subsonic and three supersonic lifting surface aerodynamic theories are available in MSC.Nastran, as well as

Strip Theory. The subsonic theory is the Doublet-Lattice method, which can account for interference among

multiple lifting surfaces and bodies. The supersonic theories are the Mach Box method, Piston Theory, and the

ZONA51 method for multiple interfering lifting surfaces. MSC.Nastran has implemented six aerodynamic theories:

1. Doublet-Lattice subsonic lifting surface theory (DLM)

2. ZONA51 supersonic lifting surface theory

3. Subsonic wing-body interference theory (DLM with slender bodies)

4. Mach Box method

5. Strip Theory

6. Piston Theory

The coupling with aerodynamic loads has also been added to the existing MSC.Nastran structural modal frequency

response capability, SOL 146. Analyses of frequency response to arbitrarily specified forcing functions can be

carried out using the oscillatory aerodynamic loads from many of the available aerodynamic theories. Frequency


125

response to a harmonic gust field can be calculated at subsonic speeds using the Doublet-Lattice method for

wing/body interference, and by the ZONA51 method for interfering lifting surfaces at supersonic speeds. Because

unsteady aerodynamic loads are obtained only for steady-state harmonic motion, they are known only in the

frequency- and not the time-domain. In SOL 146, Inverse Fourier Transform techniques provide the appropriate

methods by which transient response is obtained from the frequency response. Both forward and inverse Fourier

transforms are provided so that the time-varying forcing function or the gust profile can be transformed into the

frequency domain. Then, after convolution with the system frequency response, the inverse transform leads to the

transient response of the system to the specified forcing function or gust profile.

Stationary random response of the system, is available in SOL 146 from specified loadings and the power spectral

densities of loads. Loads may be either specified force distributions or harmonic gust fields. The statistical

quantities of interest in the response are A, the ratio of standard deviations (rms values) of the response to that of

the input loading, and N0, the mean frequency of zero crossings (with a positive slope) of the response. The

capability to compute these quantities was added to MSC.Nastran by modifying the existing random response

module to include options to generate various atmospheric turbulence power spectra and to perform the calculation

of N0. Dynamic aeroelasticity differs from the flutter analysis in that the right-hand side of the equation is no longer

zero. Instead, loading, which can be in either the frequency or the time domain, is applied. For both types of

loading, MSC.Nastran performs the primary analyses in the frequency domain. If the user has supplied loadings in

the time domain, Fourier Transform techniques are used to convert the loadings into the frequency domain, a

frequency response analysis is performed, and the computed quantities are transformed back to the time domain

using Inverse Fourier Transform techniques. Aeroelastic frequency response analysis in MSC.Nastran is performed

in modal coordinates and has a basic equation of the form

The right-hand side provides the loading in modal coordinates, which can be aerodynamic or nonaerodynamic in

nature and is a function of the analysis frequency. Nonaerodynamic generalized loads, designated PHF(w), are

obtained in the standard fashion from the loadings applied to physical coordinates. The solution of the equation

entails solving for the generalized displacements by decomposition/forward-backward substitution techniques

applied to the coupled set of complex equations. Because modal reduction techniques have been applied, the

solution costs are typically modest. Once the generalized displacements have been computed, standard data

recovery techniques can be used to determine physical displacements, velocities, stress, etc.


126

1.1.3.4 Along-Flow Direction Buffeting Response (Limited Amplitude Response)

The airflow behind obstacles exhibit increased turbulence which can be of a stochastic nature or a period regular

shedding of (von Karman) vortices (defined by the Strouhal number). These vortices give rise to periodic dynamic

forces acting on structures that lie within such an air flow. This phenomenon is called buffeting. Note that we are

not talking about the response on the obstacle (which may suffer from vortex shedding excitations) but instead on

the structures behind, i.e. within the turbulent wake. The ratio of the distance a between the obstacle and the

structure and the width d of the obstacle is important. Considerable dynamic effects are obtained from a/d 2 to 16,

peaking at about 8.

Buffeting can be avoided by stiffening the downwind structure in the air flow. Alternatively, since the wind effect

is periodic, its influence can be reduced by increasing damping or using tuned mass dampers, or as with vortex

shedding control, Scruton helical stabilizing devices can be used.


127

1.1.3.5 Across-Flow Direction Von Karman Vortex Shedding Response (Limited Amplitude Response)

When a body is subjected to wind flow, the separation of flow around the body produces forces on the body, a

pressure force on the windward side and a suction force on the leeward side. The pressure and suction forces result

in the formation of vortices in the wake region. These vortices tend to be shed from alternate sides of the body at

Reynold’s number of around 90 to 2 x 105. The large radius (small curvature) vortices produce higher pressures

whilst the small radius (large curvature) vortices produce lower pressures, giving rise to alternating lateral forces.

Vortex shedding is responsible for the whistling of hanging telephone or power cables. When the frequency of the

vortex shedding (aka wake frequency) is close to a prominent natural frequency of the structure, the resulting

condition is called lock-in, so-called because the wake frequency remains locked to the natural frequency for a

range of wind speeds. During lock-in, the structural member oscillates with increased amplitude but rarely exceeds

half the across wind dimension of the body. As the velocity further increases, the wake frequency will again

break away from the natural frequency.

The vortex shedding phenomena is describable in terms of a non-dimensional Strouhal Number defined as

where NS is the frequency of the vortex shedding, D is the characteristic dimension of the body projected on a plane

normal to the mean flow velocity and V the stream velocity. If the shedding frequency is approximately

proportional to V, then S is constant.

The Strouhal number for a cylinder is approximately 0.2 but more accurately

Re

19.7-10.198

The Strouhal number for other noncircular shapes are presented.

Wind velocity

Vort

ex S

hed

din

g F

req

uen

cy

Lock-in region around natural

frequency of structure

Number sReynold' andgeometry of functionV

DNS S


128

The dynamic response at resonance of a natural structural frequency (with the vortex shedding frequency) can be

obtained by multiplying the static response to the lift force

Area.C.V2

1F L

2

by the dynamic amplification 1/(2). In other words the dynamic equilibrium equation is

tN2sin.Area.C.V2

1kuucum sL

2

Unfortunately, CL is not constant instead varying with amplitude.

Alternatively, the following equation may be used

y0/d = 0.123.CL.(1/S2).(1/Sc)

where y0 is the displacement amplitude, S the Strouhal number and Sc the Scruton number given by

Sc = 2.m.(2)/(.D2)

where is the (of critical) damping, m the mass per unit length and D the across flow dimension.

Other expressions are presented, here r is another notation for the Scruton number Sc.

Graphs showing estimates of displacement are presented.


129

As an example, consider a steel circular pile driven vertically into the bed of a river of depth 20m and stream

velocity V. The pile dimensions are

Outer diameter, D = 0.7m

Wall thickness, t = 0.01m

Mass of pile in air, mp = 169kg/m

I = 1.29E-3 m4

E = 200E9 N/m2

Density of water, = 1020 kg/m3

Added mass of water per unit length = ¼ D2

Damping ratio, = 0.015

Firstly, the critical stream velocities, V at which vortex shedding locks-in to the first two fundamental frequencies

are required.

Total pile mass per unit length = mass in air + water in hollow pile + added mass of water

= 169 + 1020 x ¼ (0.7-0.02)2 + 1020 x ¼ (0.7)2

= 931.9 kg/m

Effective length of pile adding a fixing length of 6D say = 20 + 6 x 0.7 = 24.2m

1st natural bending frequency = 4mL

EI

2

52.3

≈ 0.50Hz

2nd natural bending frequency = 4mL

EI

2

22

≈ 3.15Hz

Lock in for the 1st mode occurs at V = ND/S = 0.50 x 0.7 / 0.2 = 1.75m/s

Lock in for the 2nd mode occurs at V = ND/S = 3.15 x 0.7 / 0.2 = 11.03m/s

Secondly, the amplitude of the cross-flow vibrations need to be estimated. This can be performed using the graph

presented above.

Scruton number, Sc or r = 2.m.(2)/(.D2) = 2 x 931.9 x (2 x x 0.015) / (1020 x 0.72) = 0.35


130

From graph, (A/D)/ = 1.1

Hence, A1 = 1.1 x 0.7 x 1.305 = 1.0m

Hence, A2 = 1.1 x 0.7 x 1.499 = 1.2m

The pile is clearly unacceptable due to its susceptibility to vortex shedding for low velocities and due to the large

amplitude vortex induced motion. The above proceedings can be most readily applied to a steel chimney stack in a

steady wind stream.

To avoid vortex shedding, increase structural frequency to avoid resonance, increase structural damping

inherent or by using tuned mass dampers or add a vortex suppression device such as a Scruton helical device to

ensure vortex shedding lines are no longer vertical lines but spirals and hence reducing the dynamic effect.

BD 49/01 (valid depending on the aerodynamic susceptibility parameter, Pb) states that any bridge with

fundamental vertical bending or torsional frequency greater than 5Hz shall be stable with respect to vortex

shedding. Truss bridges with solidity ratio, < 0.5 are also stable against vortex shedding excitations. The solidity

ratio is defined as the ratio of the net total projected area of the truss components to the front face of the windward

truss over the projected area of encompassed by the outer boundaries of the truss, excluding depth of the deck.

Plate girder bridges shall be considered unsusceptible to excitation of the vertical bending or torsional modes if

the critical wind speed, Vcr is greater than the reference wind speed, Vvs = 1.25Vr where Vr is the hourly mean wind

speed. Otherwise susceptible. The critical wind speed is obtained from below.

Note that d4 is the depth of the bridge and b* the effective width. Where the depth is variable over the span, d4 shall

be taken as the average value over the middle third of the longest span; is either fB or fT as appropriate, i.e. the

natural frequencies in bending and torsion respectively (Hz) calculated under dead and superimposed dead load.


131

Truss bridges with > 0.5 can be evaluated using the plate girder bridge equation by taking d4 as d4.

Eurocode 1-2.4:1995 presents rules for vortex shedding as follows.


132


133


134


135


136


137


138


139


140


141


142

1.1.4 Dynamic Aeroelastic Stability

In this self-induced vibration, the aerodynamic excitation forces depend on the motion of the structure itself.

Circular cylinders are not affected by this kind of vibration, but all other sectional shapes are.

1.1.4.1 Across-Flow Direction Galloping and Stall Flutter (Divergent Amplitude Response)

Across-wind galloping causes a crosswise vibration in the bridge deck. As the section vibrates crosswise in a steady

wind velocity U, the relative velocity changes, thereby changing the angle of attack (α). Due to the change in α, an

increase or decrease on the lift force of the cylinder occurs. If an increase of α causes an increase in the lift force in

the opposite direction of motion, the situation is stable. But on the other hand if the vice versa occurs, i. e., an

increase of α causes a decrease in lift force, then the situation is unstable and galloping occurs. A classical example

of this phenomenon is observed in ice covered power transmission lines. Galloping is reduced in these lines by

decreasing the distance between spacing of the supports and increasing the tension of the lines.

Under certain conditions of profile shape and incidence angle, the so-called aerodynamic damping can be negative

and where structural damping is small, galloping instability occurs. It depends on the characteristics of the variation

of drag, lift and pitching moment with the angle of incidence. The Glauert-den Hartog instability criterion for

galloping is

dCL/dCD < 0 or since dCy/d = [dCL/dCD] thus dCy/d > 0

where CL is the lift coefficient, is the angle of incidence of the airflow and Cy the force coefficient. This is proven

from the mathematics as presented below.

From the mathematics, the critical wind speed is given by

Ugallop/nD = 2.Sc/(dCy/d)=0


143

where n is the natural structural frequency and the Scruton number (or so-called reduced damping) Sc =

2.m.(2)/(.D2). Note that U/nD is the reduced velocity. Hence the critical galloping wind velocity can be

calculated using the above equation with the gradient (dCy/d) where is in radians obtained from graphs of Cy

versus . Likewise the natural frequency n of the structure can be adjusted to increase the critical velocity.


144

As an example, it may be necessary to determine the fundamental natural frequency of a structure so that it should

not flexurally gallop at wind speeds less than U = 30m/s with the flow direction at = 0. Some parameters are =

1.23 kg/m3, D = 0.1m, m = 25kg/m and = 0.015.


Sc = 2.m.(2)/(.D2) = 2 x 25 x (2 x 0.015) / (1.23 x 0.12) = 383

From the above Cy vs graph for square sections, (dCy/d)=0 = 2.7 (Note calculated in radians!)

Hence 2.Sc/(dCy/d)=0 = 2 x 383 / 2.7 = 284

Natural frequency for critical wind speed of 30m/s, n = 30 / (284 x 0.1) = 1.05 Hz.

Another example involves the flexural galloping susceptibility assessment of a solid Rhendex pile with a

submerged depth of 15.8m in a steady stream. Some additional parameters are D = 0.7m, mass = 607.5kg/m, I =

2.2E-3m4, E = 200E9 N/m2, = 0.01 and = 1020kg/m3. Added water mass = 1020 x ¼ (0.7)2 = 393kg/m.

Total mass = 607.5 + 393 = 1000.5 kg/m

Effective length of pile adding a fixing length of 6D say = 15.8 + 6 x 0.7 = 20.0m

1st natural bending frequency = 4mL

EI

2

52.3

≈ 0.93Hz


Sc = 2.m.(2)/(.D2) = 2 x 1000.5 x (2 x 0.01) / (1020 x 0.72) = 0.25

From the above Rhendex section CD and CL vs graph, we see that dCL/d is negative for 25° <

< 50° and 82° < < 98° for which dCy/d = CD + dCL/d is 1.2 and 2.4 respectively.

Hence 2.Sc/(dCy/d)25°<<50° = 2 x 0.25 / 1.2 = 0.42

and 2.Sc/(dCy/d)82°<<98° = 2 x 0.25 / 2.4 = 0.24

Thus, U = 0.42 x 0.93 x 0.7 = 0.27 m/s for 25° < < 50°

and U = 0.24 x 0.93 x 0.7 = 0.14 m/s for 82° < < 98°

Another galloping example involves the torsional galloping of a bridge deck based on experimental motion

derivatives as opposed to the static derivatives dCy/d. Consider a bridge deck with the following parameters, I =

540000kgm2, vertical stiffness per unit length k = 21060N/m, torsional damping ratio T = 0.007, = 1.2kg/m3 and

width of deck B = 18m. Equating the torsional equation of motion to the Scanlan and Tomko moment and denoting

the critical condition to be the onset of negative total damping, the following presents the results.


145

Estimates of amplitude for a square section can be made from the following graph.

Galloping can be avoided by changing the cross section, changing the flow direction, increasing the Scruton

number or by increasing damping.

Galloping is a 1 DOF system whilst classical flutter is a 2 DOF (bending and torsion) system. Galloping differs

from vortex shedding in a few ways.

Vortex Shedding Galloping

Caused by instability of the fluid motion Caused by motion of structure

Critical stream velocity depends on shape

of body, but not damping

Critical stream velocity depends on shape of body AND

damping

Amplitude is limited Amplitude is not limited

Damping affects amplitude Damping has little effect on amplitude


constraints) states that a bridge will be stable against galloping and stall flutter if the critical wind speed for

bending and torsional motion, Vg is greater than wind speed Vwo

Vr is the hourly mean wind speed. Vd is the maximum wind gust speed for the relevant maximum span; K1A is a

coefficient selected to give an appropriate low probability of occurrence of these severe forms of oscillation. For

locations in the UK, K1A = 1.25. Note that a higher value of K1A is appropriate for other climatic regions, eg

typically K1A = 1.4 for a tropical cyclone-prone location. The critical galloping and stall flutter wind speed is

considered for vertical bending and torsional motion. Vertical motion need be considered only for bridges of types

3, 3A, 4 and 4A as shown before, and only if b < 4d4. The critical velocity for vertical motion is

where the reduced velocity is


146

Note that is the density of air, m is the mass per unit length of the bridge and fB the natural frequency of bridge in

bending. Cg is 2.0 for bridges of type 3 and 4 with side overhang greater than 0.7d4 or 1.0 for bridges of type 3, 3A,

4 and 4A with side overhang less than or equal to 0.7d4; d4 is the reference depth of the bridge; δs is the logarithmic

decrement of damping. The following values of δs shall be adopted unless appropriate values have been obtained

by measurements on bridges similar in construction to that under consideration and supported on bearings of the

same type. If the bridge is cable supported the values given shall be factored by 0.75.

Torsional motion shall be considered for all bridge types. The critical velocity shall be taken as follows.

and

but if bridges of type 3, 3A, 4 and 4A have b < 4d4, Vg shall be taken as the lesser of 20fTd4 or 5fTb. Note that fT is

the natural frequency in torsion in Hz and b is the total width of bridge.

Eurocode 1-2.4:1995 presents rules for galloping as follows.


147


148


149


150


151


152


153

1.1.4.2 Across-Flow Direction Classical Flutter (Divergent Amplitude Response)

Flutter occurs under combined (coupled) torsional and bending modes, whereby the reactions caused by the

torsional vibrations predominate. It occurs when for a particular phase between torsion and bending, vibrational

energy is extracted by the structure from the constant flow of air. Flutter is the oscillatory aeroelastic instability that

occurs at some airspeed at which energy extracted from the airstream during a period of oscillation is exactly

dissipated by the damping of the structure. The aerodynamic damping is a function of the reduced speed which is

defined

bf

UU

T

speed reduced flutter

where U is the wind speed, fT the torsional frequency and b the width of the bridge. Instability occurs when the

wind velocity is greater than the reduced speed AND the aerodynamic damping, A is greater than the structural

damping. This results in the critical flutter speed relationship. The motion is divergent in a range of speeds above

the critical flutter speed, here given for fT/fB > 1.2 as follows.

b.f..2.b..

r.m.72.0.5.0

f

f1.U B2

B

Tspeed critical flutter

where is the bridge profile shape factor, m is the mass/length of bridge, r the radius of gyration, b the effective

width of the bridge along the wind direction, the air density. Important for avoiding flutter is the ratio of torsional

to bending natural frequencies (fT/fB). This should be as large as possible (about 3).


154

Experimental based expressions for lift FL and moment FM are presented.


155

Aerodynamic coefficients Hi* and Ai* for a thin airfoil (i = 1,2,3) and three streamlined box decks (i = 1,2,3,4) are

presented as follows.

Flutter can be avoided by choosing a suitable section. Increasing damping does not achieve the same

improvement as for vortex shedding or galloping.


156

The Tacoma Narrows Suspension Bridge at Washington, which opened on the 1st of July 1940, failed at a wind

speed as low as 67km/h due to flutter on the 6th of November 1940. The Tacoma Narrows Bridge consisted of two

126m towers, two 330m side spans, and a 840m main span stiffened with 2.4m deep girders with span/depth of 350

and span/width of 72. The center of the deck was rising and falling vertically by 0.9m and deflecting laterally by

0.61m. Suddenly, the bridge started twisting violently, with the deck appearing to be twisting by almost 45 degrees.

At its ultimate, 180m of the main span tore away from the suspenders, the side spans sagged and the towers tilted

3.6m towards each shore.


constraints) states that a bridge will be stable against classical flutter if the critical wind speed for bending and

torsional motion, Vf is greater than wind speed Vwo

Vr is the hourly mean wind speed.Vd is the maximum wind gust speed for the relevant maximum span; K1A is a

coefficient selected to give an appropriate low probability of occurrence of these severe forms of oscillation. For

locations in the UK, K1A = 1.25. Note that a higher value of K1A is appropriate for other climatic regions, eg.

typically K1A = 1.4 for a tropical cyclone-prone location. The critical wind speed for classical flutter is

where the reduced velocity VRf is

but not less than 2.5. Note that is the density of air, b is the overall width of the bridge deck, m is the mass per

unit length of the bridge, fB the natural frequency of bridge in bending, fT the natural frequency of bridge in torsion

and r is the polar radius of gyration of the effective bridge cross section at the centre of the main span (polar second

moment of mass/mass)0.5.

Flutter analysis utilizes complex eigenvalue analysis to determine the combination of airspeed and frequency for

which neutrally damped motion is sustained.

Three methods of flutter analysis are provided in MSC.NASTRAN SOL 145, the American flutter method (called

the K method in MSC.NASTRAN), an efficient K method (called the KE method) for rapid flutter evaluations, and

the British flutter method (called the PK method) for more realistic representation of the unsteady aerodynamic

influence as frequency-dependent stiffness and damping terms. Complex eigenvalue analysis is used with the K

method, and the QR transformation method is used with the KE and PK methods.


157

Eurocode 1-2.4:1995 presents rules for flutter as follows.


158


159

1.1.4.2.1 American K-method

The basic equation for modal flutter analysis by the K-method is


160

For the K-method of solution, the aerodynamic term is converted to an equivalent aerodynamic mass

1.1.4.2.2 British PK-method

1.1.4.2.3 KE-method


161

BIBLIOGRAPHY

1. DESIGN MANUAL FOR ROADS AND BRIDGES. Design Rules for Aerodynamic Effects on Roads and

Bridges BD 49/01. Highways Agency, UK.

2. TRL Limited. Background to the Development of BD 49/01: Design Rules for Aerodynamic Effects on Roads

and Bridges BD 49/01. TRL Report 528, TRL Limited, United Kingdom, 2002.

3. SELVAM, Dr. R.P & GOVINDASWAMY, Suresh. Aeroelastic Analysis of Bridge Girder Section Using

Computer Modelling. University of Arkansas, May 2001.

4. SIMIU and SCALAN. Wind Effects on Structures, An Introduction to Wind Engineering.

5. Dr LLYOD, Smith. Aerodynamic Forces on Structures. Imperial College of Science Technology and

Medicine, London, 2000.

6. Dr SWAN. Wave Mechanics and Wave Loading Lectures. Imperial College of Science Technology and

Medicine, London, 2000.

7. Dr Hardwick, J.D. Fluid Mechanics Lectures. Imperial College of Science Technology and Medicine,

London, 1997-1998.

8. MACNEAL-SCHWENDLER CORP. MSC.NASTRAN Aeroelastic Analysis User Guide 2001. MacNeal-

Schwendler Corp., Los Angeles, 2001.

fluid - structure interaction (aeroelastic response analysis) · fluid – structure interaction...

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