Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
Vortex Shedding Induced Loads on Free Standing Structures
Author: I. Giosan, P.Eng.
Introduction
Estimating the wind-induced response of structures is a main topic in the area of the wind
engineering research over the last 40 years [1].
Analyses are usually carried out through theoretical formulations, numerical algorithms, wind
tunnel tests, full scale experiments and code provisions. A dominant property of modern wind
engineering is that accurate analyses of complex problems are seldom possible by using one of
these methods and instead they require the joint application of different techniques
For instance, wind tunnel and full-scale experiments frequently provide the input for
theoretical solutions, numerical simulations and code provisions; theoretical and numerical
methods are often the bases for organizing developing and processing measurements. The
growing importance of wind effects on structures, joined with theoretical, computational and
instrumental advances, has given great impulse to these techniques; in the meanwhile, the
evolution of each technique is linked and even more inspired by the growth of complimentary
methods.
All these powerful analytical and simulation tools are required when investigating complex
wind induced phenomena like vortex shedding.
Vortex Shedding (Fig. I.1) is the instance where alternating low pressure zones (blue colors)
are generated on the downwind side of the stack, as shown in this figure.
Fig. I.1 Vortex Shedding phenomenon induced by wind flowing over a cylinder.
These alternating low pressure zones cause the stack to move towards the low pressure zone,
causing movement perpendicular to the direction of the wind. When the critical wind speed of
the stack is reached, these forces can cause the stack to resonate where large forces and
deflections are experienced.
Every stack has a critical wind speed at which vortex shedding occurs.
Vortex shedding can be observed also at larger scale as a meteorological phenomenon.
In [Fig. I.2] is shown an image taken from space of the vortex shedding phenomenon behind
the island Juan Fernandez along the Chilean rim.
1
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
Fig. I.2 Vortex Shedding as a meteorological phenomenon.
Vortex shedding is a complex physical phenomenon, especially when it degenerates into lockin condition. As large vibrations may occur at moderate and frequent wind velocities,
structures may undergo a great number of stress cycles that lead to damage accumulation and
may determine structural failure without exceeding the ultimate limit stress.
Considering the potential vortex shedding fatigue induced damage it is very important the
design procedures to account in a realistic manner for the vortex shedding dynamic induced
loads.
Although an immense analytical and experimental effort has been made during the last fifty
years to improve the analytical vortex shedding prediction models, the design standards are
still lacking in presenting concise and easy to use analytical methodologies [3-13].
Because of the complexity of vortex shedding phenomenon at this moment there is no general
analytical method available to calculate the response of the structure to the vortex shedding
dynamic induced loads [2].
Avery detailed comparison on wind induced response of major codes and standards was made
and presented by Working Group E – Dynamic Response set up at the First International
Codification Workshop held in Bochum, Germany on September 15th 2000 [11].
A comprehensive comparison of the along wind loads and their effects on structures was
conducted on the major international codes and standards: the US Standard – ASCE 7-98, 200
[8], the Australian Standard – AS1170-2, 1989 [9], the National Building Code of Canada –
NBCC, 1995 [3], the Arhitectural Institute of Japan – AIJ-RLB, 1993 [10] and the European
Standard – Eurocode ENV1991-2-4 [6].
The comparison study concluded that there are considerable scatter in predictions among these
standards and is very important to understand the underlying differences in order to develop
unified international codes.
An historical overview in the research and simulation areas of vortex shedding phenomenon is
presented by Giovanni Solari [1].
2
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
Some of the most important pioneering research contributions in the area of wind dynamics
have come from the University of Western Ontario (UWO) in Canada and the Technical
University Aachen (RWTH) in Germany.
The Canadian research team was lead by Dr. Alan G. Davenport since 1965 when he founded
the Boundary Layer Wind Tunnel Laboratory. This laboratory has carried out innovative
design studies for major structures. For example, of the 40 tallest buildings in the world,
roughly two thirds have been studied at Western Ontario Boundary Layer Wind Tunnel
Laboratory.
Dr. Davenport has pioneered in the application of boundary layer wind tunnels to the design of
wind sensitive structures.
The mathematical spectral model proposed by the Canadian research group is used as the basis
for the Canadian National Building Code [3]. It gives accurate results for relatively stiff
structures, such as concrete chimneys, in which the aeroelastic effects are relatively small or
moderate. For more flexible structures like steel tubular towers, antenna towers, transmission
or highmast towers some of the Canadian research team results are incorporated into Canadian
Bridge Design Code [4,5].
The German team research is focused on accurate modeling of large aeroelastic effects and it
is incorporated into Eurocode [6].
As mentioned the study of dynamic response of vertical structures to vortex shedding begun in
the 60’s thanks to the pioneering contribution of Davenport. In his studies [16,17] he defined
the main parameters necessary to analyze the structural wind actions: mean static wind
pressure, mean alongwind static displacement, gust response factor and equivalent static force.
Studies developed in the 70’s by Vickery [18] and Simiu [19] perfected the method
introduced by Davenport especially with respect to wind and aerodynamic modeling. On the
other hand ESDU [20] and ECCS[21] introduced procedures to determine the maximum
values of structural effects by using influence function techniques.
Starting from Simiu’s formulation, at the beginning of 80’s, Solari derived closed for solutions
for alongwind response of structures. He also developed the Equivalent Spectrum Technique
[22], a method that schematizes the wind as an equivalent velocity field perfectly coherent in
space.
Research carried out in 90’s followed two distinct lines. The first was aimed at determining
the maximum effects due to alongwind response. The second extends original methods from
alongwind response to crosswind and torsional responses.
The first research line derives from the observation that Equivalent Static Force (ESF), as
conceived by Davenport and used by most subsequent authors, usually produces correct mean
maximum displacements but may give place to other effects like bending moments and shear
forces.
Efforts were made by Kaspersky [23] and Holmes [24,25] do develop new analytical methods
for defining the ESF in a more realistic and simplified [26] manner.
The second research line was aimed at determining the three-dimensional wind induced
response of structures.
A wide research program was carried out in Japan [27] to derive the maximum alongwind,
crosswind and torsional response of structures and their related ESF, by fitting the results of
wind-tunnel tests.
A study focused on slender structures and structural elements was initiated and coordinated by
Giovanni Solari at Department of Structural and Geotechnical Engineering from University of
3
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
Genoa in Italy. This study extended the concepts defined by Davenport for alongwinds to
crosswinds and torsional responses [28,29]. Alongwind, crosswind and torsional actions were
defined as a linear combination of the longitudinal, lateral and vertical turbulence components
by quasi-steady theory; crosswind forces and torsional moments due to vortex shedding were
superposed by considering these as independent of turbulence actions [30]. Alongwind,
crosswind and torsional responses, analyzed as uncoupled and only dependent on the related
fundamental modes of vibration, were determined in closed forms [31,32] by the generalized
spectrum technique [33] . This solution was generalized to slender vertical structures with
localized masses and linearized aeroelastic terms [33].
In the recent years the researchers were focused on refining the analytical models to predict
accurate results for vortex shedding at high Reynolds numbers [34], to account for mode shape
corrections [35] or to account for fatigue induced by vortex shedding phenomenon [36].
The most extensive and important experimental full scale tests to investigate the response of
structures to cyclic loads were done at University of Minnesota, Minneapolis, USA [37], and
the experimental program was focused on fatigue-resistant design on cantilevered signal sign
and light supports.
Experimental observations and numerical modeling of the vortex shedding phenomenon
linked to structural fatigue were also the focus of the research program lead by dr. Christopher
Foley at University of Marquette, Wisconsin [38].
Experimental observations, full scale tests and advanced nonlinear numerical modeling of the
dynamic induced loads on free standing structures are also a part of the research program
initiated and conducted by Ioan Giosan at West Coast Engineering Group Ltd., Vancouver,
Canada. The research program is aimed to develop standard design methodologies and
standard numerical modeling procedures accounting for dynamic induced loads. The sources
of the dynamic induced loads are simulated realistically, as a result of the impact between the
structure and the surrounding environment [39,40].
4
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
1. Vortex Shedding Phenomenon
When the wind blows across a slender structural member, vortices are shed alternately from
one side and then the other, giving rise to a fluctuating force acting at right angles to the wind
direction. This organized pattern of vortices is referred to as von Karman vortex street. A
structural member may be considered slender in this context if the aspect ratio exceeds 20 [3].
For lightly damped members, which are free to oscillate, large amplitude vibrations in the
plane normal to the wind may develop when the vortex shedding is in resonance with one of
the natural frequencies of vibration. Although this is most likely to occur for the lower modes
of vibration, vortex shedding induced effects for very flexible members may also be important
for higher modes. The character of the vortex shedding forces for circular cylinders depends
on the Reynolds number,
VD
Re =
ν
where:
V= wind speed [m/s];
D= structure diameter [m];
ν = kinematic viscosity [m2/s ].
The shedding tends to be organized at sub-critical and trans-critical Reynolds numbers
[2,3,4,5].
Sub − critical Range : Re < 3 x10 5
Critical Range : 3x10 5 ≤ Re ≤ 3x10 6
Trans − critical Range : Re > 3x10 6
In the critical range, namely for:
3x105 ≤ Re ≤ 3x106
vortex shedding tends to be irregular unless the structural motion is sufficiently large to
organize the fluctuating flow around the body. This phenomenon, referred to as “locking in”,
becomes important for lightly damped members.
When the lock-in phenomenon occurs, in some cases, severe vibration has persisted long
enough to cause fatigue cracking or structural failure (Fig.1.1).
Fig. 1.1 Antenna Tower Fatigue Failure–Pictures provided by Morrison Hershfield consulting company.
5
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
In free standing tubular structures (transmission, telecommunication, highmast or street light
poles), cracks are usually observed at the welded connection between the vertical shaft and
base plate. If there are stiffeners or gussets reinforcing the pole to base plate connection, then
the cracks will typically form at the tops of the stiffeners.
If there are hand holes, cracks may appear around the perimeter of these. Cracked anchor rods
have also occurred.
Vortex shedding tends to occur with steady continuous winds at a critical velocity. The
velocity need not be very high, but it has been found that significant vibration does not occur
unless the velocity is greater than 5 m/s. The periodic frequency of the vortex shedding can
lock in on the natural frequency of the pole, resulting in very large alternating forces acting
transverse to the wind flow direction.
Occasionally, the vibration is so severe that fatigue cracks will appear.
Although vortex shedding can "lock in" and continue as the velocity increases or decreases
slightly, if the velocity changes by more than 20 percent, the vortex shedding will stop. Gusty
variable winds, such as might occur in a severe storm typically will not cause vortex shedding.
In fact, if the wind velocity is greater than 15 m/s (35 mph), the wind is generally too turbulent
for vortex shedding to occur. In summary, the winds that are dangerous for vortex shedding
are steady winds in the velocity range 5 to 15 m/s [37].
Recent studies [38] have verified that vortex shedding can occur in tapered as well as
prismatic circular poles with almost any diameter.
Although nearly periodic in a smooth air stream, vortex shedding in turbulent boundary layer
flow conditions, which is characteristic of natural wind, tends to become less regular, with
energy distributed over a band of frequencies around ωe.
ωe = frequency at which vortex shedding occurs at a specific location [Hz].
The presence of turbulence effectively reduces the extent of the member over which the vortex
shedding forces remain correlated [3,4,5]. A reduction of the aspect ratio has somewhat
similar effects [5].
Several measures may be considered should vortex shedding induced effects prove to be
excessive.
these include:
- a strengthening or stiffening of the member, or both;
- increasing the mass;
- increasing the damping; and
- changing the aerodynamic characteristics for example, by increasing the taper, or by
the addition of aerodynamic spoilers.
Of these alternatives, increasing the damping of the member is the most desirable solution.
The effective damping can be increased using visco-elastic materials or special dynamic
absorbers [5].
6
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
2. Vortex Shedding Induced Loads - Analytical Models.
The excitation due to vortex shedding is treated as a time-varying load of frequency:
(2.1)
SV
ωe =
D
where:
S= the Strouhal Number for the cross-sectional geometry (Table 2.1).Resonant vibrations are
assumed to occur when the frequency of vortex shedding coincides with a natural frequency of
the member. The evaluation of vortex shedding-induced effects requires a dynamic analysis of
the member to determine its natural frequencies and associated mode shapes of vibration. All
modes of vibration for which vortex shedding-induced resonant vibrations occur at wind
speeds equal or less than that corresponding to the design mean hourly reference wind
pressure V, must be considered [4,5].
V ≤ 1.24(qC e )
0.5
(2.2)
Where:
q=hourly mean reference wind pressure for the design return period [Pa];
Ce=wind exposure coefficient (Table 2.3).
In the case of a member with a constant cross-section, resonant vibrations for a particular
mode of vibration with natural frequency, ni, occur at a specific or critical wind speed:
(2.3)
Dω i
Vcri =
S
In the case of a tapered member, the frequency of vortex shedding at a particular wind speed
varies over the length of the member. As the wind speed increases, resonant excitation occurs
first at the smaller diameter portion of the member and then shifts to portions with larger
diameter. Consequently, vortex shedding effects associated with a particular mode of vibration
with frequency, ωi, must be examined for a range of critical wind speeds. Defining Dmin and
Dmax as the minimum and maximum cross-sectional diameters, respectively, this range is
expressed as:
(2.4)
ω i Dmin
ωD
≤ Vcr ≤ i max
i
S
S
Considering an harmonic sinusoidal model, the root mean square (RMS) time-varying vortex
shedding induced load acting at a particular location, x, along a member is expressed as:
~
FS ( x, t ) =
where:
ρV 2 C L D( x) sin[2πω e ( x)t ]
(2.5)
2
ρ= air mass density, taken as 1.29 [kg/m3];
V= the mean wind speed at location x, [m/s];
C̃L= RMS lift (across-wind) force coefficient for the cross-sectional geometry a
specified in Table: 2.1;
x= coordinate describing length along the member, [m];
D(x)= the diameter of frontal width of a member at location x, [m];
ωe(x)= frequency at which vortex shedding occurs at location x, [Hz];
t= time, [s].
7
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
For members with constant diameter or frontal width, the magnitude of the excitation is taken
to be invariant with x and proportional to the velocity pressure at the top of the member [4,5].
This is a conservative assumption since V approaches zero at the ground level. It is also
conservative to treat FS(x,t) as a spatially coherent excitation, that is, acting in phase along the
entire length of the member (Fig. 2.1 a). In reality, this begins to occur only at large
amplitudes of motion.
a) Non-tapered structure
b) Structure with taper, p
Fig. 2.1 Typical Vortex Shedding-Induced Responses in Two Modes of Vibration
The variation of the wind speed with height, the turbulent flow functions normally
experienced and the presence of signs and other accessories all tend to disrupt the spatial
correlation of the excitation.
For structures with varying diameter or frontal width, the magnitude of the excitation will vary
along the length of the member. The vortex shedding excitation at location x1, is taken to
remain in phase over the portion of the member for which the diameter of frontal width
remains within ±Ω percent of D(x1) and to be zero over the remainder of the member (Fig. 2.1
b)
A default value for Ω=±10% is prescribed in Canadian Highway Bridge Design Code [4.5]
and in the National Building Code of Canada [3].
The value of Ω=±10% is applicable for peak response amplitudes of greater than 2% of D.
8
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
For a band limited random forcing model with a Gaussian load spectrum, the induced load is
described by Davenport et al. [17], Harris and Crede [41], Vickery and Clark [42,43].
The band limited random forcing model differs from the sinusoidal model by:
- Allowing for a random (rather than harmonic) vortex lift force. This employs a
different forcing function than the sinusoidal model.
- Allowing for the energy associated with the vortex shedding to be distributed about the
dominant frequency (rather than concentrated on the dominant frequency). This
employs a bandwidth term, B, which is a measure of the distribution of the energy
(Table 2.1).
- Allowing for the three-dimensional nature of the flow – the loss of correlation of the
lift forces along the length of the member. This employs a term for the correlation
length, L, which is a measure of the length, in diameters, that the vortices remain in
phase (Table 2.1).
- Allowing for the turbulence of natural wind (turbulence leads to reduction in vortex
shedding correlation length and leads to a small reduction in the strength of the
shedding forces).
- Allowing for the variation of wind speed with height. This employs the power law
wind-velocity-profile-exponent, α, to obtain an apparent taper between the wind and
the member.
Treating vortex shedding as a sinusoidal process is an approximation leading to conservative
estimates. The variation of the wind speed with height, turbulence of the natural wind and the
presence of signs and other accessories all tend to disrupt the spatial correlation of the
excitation. It is generally accepted to be more accurate to treat the excitation as a band-limited
random process and to assume that the forcing tends to become harmonic only when the
motion of the members is sufficiently large to organize the shedding vortices [4,5,6,7]. This
tends to occur when the peak amplitude of the motion is of the order of 2% to 2.5% of the
diameter or the width of the cross-section, and greater.
For the evaluation of the vortex shedding-induced response in a particular mode of vibration,
the direction of the vortex shedding excitation at any location, x, is taken in the direction of the
motion of the member at that location. This is a simplification as the direction of the vortex
shedding force at large body amplitudes is more likely to be in the direction of the local time
derivative or velocity of the body motion. For the purpose of evaluating the generalized force,
GFi, associated with a particular mode of vibration, both assumption lead to the same RMS
and peak values.
9
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
2.2.1 Vortex Shedding Response Parameters
In table 2.1 are presented values for vortex shedding parameters, for various shapes and
Reynolds numbers.
Each parameter will be defined, and will be also presented values or empirical computation
relations.
Table: 2.1 Vortex Shedding Parameters
Cross-section
Circular
Subcritical Re<3x105
Supercritical and transcritical Re>3x105
Square
Multisided members, rolled structural shapes
S
C̃L
B
L
0.18
0.25
0.11
0.15
0.50
0.20
0.60
0.60
0.10
0.30
0.25
0.25
2.5
1.0
3.0
2.75
Where:
S= the Strouhal Number for the cross-sectional geometry.
C̃L= RMS lift (across-wind) force coefficient for the cross-sectional geometry.
B= band width, a measure of the variability of the vortex shedding frequency.
L= correlation length, the length (as a ratio of the diameter) over which the vortices act
in phase.
Strouhal Number - S can be calculated also using the methodology outlined in Canadian
National Building Code [3].
For circular and near-circular cylinders, the Strouhal number, S, is a function of the Reynolds
number, Re. Although the Reynolds number is a function of V, a trial-and-error approach [3]
to finding the critical mean speed can be avoided by examining the product, ωD2, and using
the appropriate version of equation (2.6) as follows:
⎧ϖD 2 ≤ 0.5m2 s, ⇒ Vcr = 6ϖD
⎪
if ⎨0.5 m2 s ≤ ϖD 2 ≤ 0.75m2 s, ⇒ Vcr = 3ϖD + (1.5 m2 s ) D
⎪ 2
2
⎩ϖD ≥ 0.75m s, ⇒ Vcr = 5ϖD
(2.6-1)
(2.6-2)
(2.6-3)
Equation (2.6-1) applies when Re<2x105 and S=1/6.
Equation (2.6-2) covers an intermediate region where, for computational convenience, S is
taken to increase approximately linearly as Re increases to 2.5x105.
Equation (2.6-3) usually governs, in which Re>2.5x105 and S=1/5.
The Lift Coefficient- C̃L has a very random nature which makes the evaluation of vortex
shedding response very difficult [38]
There have been studies attempting to define lift coefficients for chimneys and stacks in both
steel and concrete [45] or a general analytical model [46].
Difficulty in defining the lift coefficient is enhanced by the fact that vortices tend tom have
Correlation Lengths (L) over which vortices are assumed to act in phase [2,45].
10
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
There have been several simplified approaches that can be used for determining the lift
coefficient and correlation lengths for vortex shedding response. The Canadian Highway
Bridge Design Code (CAN/CSA-S6-2000) [4,5] has a relatively concise approach to
consideration of vortex shedding deformations and the corresponding stresses.
This standard [4.5] suggests for C̃L and L the values presented in Table: 2.1.
B= Frequency Band Width.
It is assumed that there is a range of wind speeds over which the vortex shedding frequency
and the structure’s natural frequency lock-in with one another and result large across-wind
vibrations. The difficulty with lock-in phenomenon arises from finding the range of natural
frequency of the structure, for each mode shape, over which lock-in can occur [2].
It has been recommended that the tendency of vortex shedding to cause across-wind vibrations
be computed using a range of frequencies around the shedding frequency and comparison to
modal vibration frequencies of the structure. Mathematically, this can be described as [2,38],
(2.7)
ωiL ≤ ωi ≤ ωiU
Where:
ωi – is the ith natural frequency of the structure;
ωLi – is the ith natural frequency of the structure lower bound;
ωUi – is the ith natural frequency of the structure upper bound;
These values are calculated using
⎡ Vz' ⎤
SVz SVz'
'
L
ωi = ωi − ωi =
−
= ωi ⎢1 − ⎥
Dz
Dz
⎣ Vz ⎦
⎡ Vz' ⎤
SVz SVz'
ω = ωi + ω =
+
= ωi ⎢1 + ⎥
Dz
Dz
⎣ Vz ⎦
U
i
'
i
(2.8)
(2.9)
Where:
Vz – is the mean wind speed at elevation z;
V΄z – is the wind speed turbulent component at elevation z;
If the structure’s natural frequency for mode of vibration i lies within these lower and upper
bounds, vortex shedding could excite mode of vibration i.
The turbulent component of the wind speed is often assumed to be zero-mean random variable
with Gaussian distribution [2,45] and its magnitude depends upon height above earth’s
surface. As a result a roughness length, z0, is often utilized to estimate this effect.
There have been many experimental attempts to categorize the surface roughness parameter
based upon ground exposure. The ground surface roughness parameter describes the frictional
effect of the underlying surface geometry. It is also described as the surface momentum sink
for the atmospheric flow [11]. It is well defined only locally for homogeneous terrain and
neutral atmospheric conditions.
Table.2.2 shows the best estimations of roughness as depicted by Wieringa [47] and is a
reasonable basis for determining appropriate roughness lengths for mainland areas.
11
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
Table: 2.2 Roughness Lengths of homogeneous surfaces
Surface Type
Roughness Length (m)
Sea, loose sand, and snow
0.0002
Concrete, flat desert, tidal flat
0.0002-0.0005
Flat snow field
0.0001-0.0007
Rough ice field
0.001-0.012
Fallow ground
0.001-0.004
Short grass and moss
0.008-0.03
Long grass and heather
0.02-0.06
Low mature agricultural crops
0.04-0.09
High mature crops (grain)
0.12-0.18
Continuous bush land
0.35-0.45
Mature pine forest
0.8-1.6
Tropical forest
1.7-2.3
Dense low buildings (suburbs)
0.4-0.7
Regularly-built large town
0.7-1.5
Typical roughness coefficient magnitudes for various terrain topologies are also given by
Dyrbye and Hansen [2]:
⎧0.01m for open land with little vegetation and houses;
⎪
z 0 = ⎨0.05m for agricultural areas with few houses and wind breaks;
⎪0.3m for villages and agricultural areas with manny wind breaks;
⎩
Surface roughness can be also determined empirically. Lettau [48] formulated z0 from field
experiments as
(2.10)
HS
z 0 = 0.5
A
Where:
H – is the average height of obstacles within the area A.
S – is the total projected frontal area of the obstacles.
A - is the surface area.
The 0.5 factor corresponds to the average drag coefficient of the area based upon the primary
obstacle’s projected frontal area. For a relatively large variety of S values, the equation (2.10)
produces z0 that are within 25% of measured values from wind profiles.
Based on the above values can be concluded that mostly of the transmission, antenna or
highmast towers are located in the areas that may be considered to have few wind breaks and
trees. Therefore, a roughness length of z0= 0.3m can be used for computations.
Experimental results have allowed the empirical definition of the standard deviation for
turbulent horizontal wind component to be determined. An expression for the standard
deviation that has been validated through experiment is [2]:
12
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
σ V = AV f
(2.11)
Where A≈2.5 for a roughness length z0=0.3m. Vf is the friction velocity which can be
computed with formula:
Vz K
Vf =
(2.12)
ln(z z 0 )
Where:
K – is von Karman constant and can be approximated as 0.4.
Equation (2.11) combined with (2.11) becomes:
(σ V )z =
AVz K
ln(z z 0 )
(2.13)
It will be assumed that the frequency boundaries given by equations (2.8) and (2.9) can be
based upon the maximum turbulent wind speed variation from the mean speed, V. This
maximum can be defined as V΄z,max . It will be assumed that the peak variation in turbulent
component from the mean will be δ(σv) (above or below). Equations (2.8) and (2.9) can be
rewritten as:
⎡ Vz',max ⎤
⎡ δ (σ V )z ⎤
L
ωi = ωi ⎢1 −
(2.14)
⎥ = ωi ⎢1 −
Vz ⎦
Vz ⎥⎦
⎣
⎣
⎡ Vz',max ⎤
⎡ δ (σ V )z ⎤
ω = ωi ⎢1 +
⎥ = ωi ⎢1 +
Vz ⎦
Vz ⎥⎦
⎣
⎣
U
i
(2.15)
where the standard deviation of the turbulent component is computed using equation (2.13) at
the given height above the ground.
The Reynold Number – Re can be calculated for ν = 1.5x10-5 m2/s.
VD
Re =
× 10 5
1.5
(2.16)
Wind exposure coefficient - Ce can be estimated based on values presented in Table.2.3 [4].
Table: 2.3 Wind Exposure Coefficient
Height H [m]
Exposure Coefficient, Ce
0 to 10
1.0
Over 10 to 16
1.1
Over 16 to 25
1.2
Over 25 to 37
1.3
Over 37 to 54
1.4
Over 54 to 76
1.5
Over 76 to 105
1.6
13
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
An analytical formulation for estimating the wind exposure coefficient is presented in
ANSI/ASCE7-2005 [8].
The Ce coefficient is calculated with formula:
⎛ z
C e = 2.01⎜
⎜z
⎝ g
2
⎞α
⎟
⎟
⎠
(2.17)
Where:
z – height above the ground or 5m, whichever is greater;
zg – is a constant which varies with the exposure conditions. Based on information
presented in [8] , zg should be taken 274.3m for exposure C.
α - is a constant which varies with the exposure conditions. Based on information
presented in [8] , α should be taken 9.5 for exposure C.
14
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
3. Analytical Methodologies
Before calculating the vortex shedding induced loads based on one of the analytical models
presented above one has to determine the structure’s locations where the vortex shedding
loads potentially lock in. Resonance in a particular mode of vibration with a natural frequency
ωi occurs when ωe=ωi.
A)
Members with constant diameter or frontal width
Will be calculated the wind speeds Vcri which induce vibration frequencies ωe close to
structure’s natural frequency modes ωi:
(3.1)
Dω i
Vcri =
S
ωi = ωe =
SVcri
(3.2)
D
Vcri ≤ 1.24(qC e )
0.5
(3.3)
The height above ground used to compute Ce shall correspond to the height above ground of
location of coordinate x. The location at which ωe is computed shall be taken as the top of the
member.
All the parameters in above equations were previously defined.
B)
Members with tapered diameter or frontal width.
In the case of a tapered member, the frequency of vortex shedding at a particular wind speed
varies over the length of the member. As the wind speed increases, resonant excitation occurs
first at the smaller diameter portion of the member and then shifts to portions with larger
diameter. Consequently, vortex shedding effects associated with a particular mode of vibration
with frequency, ωi, must be examined for a range of critical wind speeds. Defining Dmin and
Dmax as the minimum and maximum cross-sectional diameters, respectively, this range is
expressed as:
(3.3)
ω i Dmin
ωD
≤ Vcr ≤ i max
i
S
S
Vcr ≤ 1.24(qCe )
0.5
(3.4)
i
The wind speeds Vcri will induce vibration frequencies ωe(x) close to structure’s natural
frequency modes ωi at location x:
(3.5)
SVcr i
ω i = ω e (x ) =
D( x )
ωe(x) shall be computed at sufficient locations along the member to determine at which
location vortex shedding excitation can occur.
15
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
All the parameters in above equations were previously defined.
3.1 Analytical Methodology for the Sinusoidal Model
The time-varying across-wind or lift excitation resulting from the shedding of vortices shall be
determined as follows.
3.1.1 Non-tapered Structures
The vortex shedding induced load per unit length acting at any location x along the structure
shall be taken as:
~
FS ( x, t ) =
ρV 2 C L D sin[2πω e t ]
(3.1.1.1)
2
The maximum force for each vibration mode will be determined - FsiMax, multiplied with the
value of the corresponding mode shape amplitude – µi(x) and applied on the structure at
location x.
The resulting generalized forces will be applied on the structure alternately from one side to
the other with a frequency equal with ωi.
Will be calculated stresses at each location x and compared with the endurance limit value for
the material used as well as for each critical connection.
For the first natural frequency mode the equivalent static force per unit length can be well
approximated with this formula [3]:
FS =
C1
⎛
ρD 2 ⎞
⎟
H D ⎜⎜ ζ 1 − C2
M ⎟⎠
⎝
qD
(3.1.1.2)
where:
M= average mass per unit length over the top one-third of the structure, [kg/m];
For most situations:
3 H D
C1 = 3 for H D > 16; C1 =
for H D < 16
4
C2 = 0.6
If:
ρD 2
ζ < C2
M
then large amplitude motions up to one diameter may result [9].
If V is low, temperature gradients may produce very low turbulence levels, and in such cases
vortex induced motions are significantly increased, particularly for very slender structures. If
V is less than 10m/s and H/D>12, then:
C1=6 and C2=1.2.
16
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
3.1.2 Tapered Structures
~
FS ( x, t ) =
ρV 2 C L D( x) sin[2πω e ( x)t ]
(3.1.2.1)
2
The maximum force for each vibration mode will be determined - FsiMax, multiplied with the
value of the corresponding mode shape amplitude – µi(x) and applied on the structure at
location x.
The resulting generalized forces will be applied on the structure alternately from one side to
the other with a frequency equal with ωi.
Will be calculated stresses at each location x and compared with the endurance limit value for
the material used as well as for each critical connection.
For tapered structures for the first natural frequency mode the equivalent static force per unit
length (3.1.1.2) can be used, if the variation in diameter over the top third is less than 10% of
the average diameter of the top third.
If the diameter variation exceeds 10%, then the effective static load need be applied only over
that part of the structure over which the diameter is within 10% of the average for that part.
For tapered structures with a diameter variation exceeding 10% over the top third:
C1=3 and C2=0.6
and no increase in these coefficients required for low values for V.
3.2 Analytical Methodology using the Band Limited Random Forcing Model
Treating vortex shedding as a sinusoidal process is an approximation leading to conservative
estimates. The variation of the wind speed with height, turbulence of the natural wind and the
presence of signs and other accessories all tend to disrupt the spatial correlation of the
excitation. It is generally accepted to be more accurate to treat the excitation as a band-limited
random process and to assume that the forcing tends to become harmonic only when the
motion of the members is sufficiently large to organize the shedding vortices [4,5,6,7]. This
tends to occur when the peak amplitude of the motion is of the order of 2% to 2.5% of the
diameter or the width of the cross-section, and greater.
The below methodology simulates the vortex shedding induced effects for amplitudes smaller
than 2.5D% using a band-limited random approach.
To apply this methodology one has to perform a forced vibration analysis.
The maximum stresses in a member due to vortex shedding excitation shall be computed by
loading the member with the peak inertia loads acting statically.
The magnitude of the peak inertia load per unit length at any location x along the member for
mode of vibration i shall be taken as:
Fi ( x ) = (2πω i ) 2 y i ( x )m( x )
17
(3.2.1)
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
where:
Fi(x)= member peak inertia load at location x for mode of vibration i, [N/m]
m(x)= mass per unit length of member at location x, [kg/m]
yi(x)= peak member displacement due to vortex shedding excitation at location x for
member mode of vibration i, [m]
yi ( x ) = ai µ i ( x )
(3.2.2)
where:
ai(x)= modal coefficient of magnitude of the oscillatory displacement for member
mode of vibration i.
µi(x)= amplitude of the member mode shape at location x for mode of vibration i.
The modal coefficient of magnitude for a particular mode of vibration due to vortex shedding
excitation shall be computed as follows.
3.1.1 Non-tapered Structures
~
ai =
3.5 C L ρD 2π 0.25C
Bζ i (4πS ) GM i
2
(3.1.1.1)
Except that if yi(x) exceeds 0.025D, ai shall be taken as
H
~
ai =
2 C L ρD 3 ∫ µ i (x )dx
0
2
(3.1.1.2)
ζ i (4πS ) GM i
where:
C=
(H D )2
x 3α µ i2 ( x )
dx
1 + H 2 LD ∫0 H 1+3α
H
(3.1.1.3)
GMi= member generalized mass for mode of vibration i, [kg]
H
GM i = ∫ m( x )µ i2 ( x )dx
(3.1.1.4)
0
ζI= structural damping for the ith mode, expressed as a ratio of critical damping.
α= wind velocity profile exponent (Table: 3.1.1.1).
18
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
Table: 3.1.1.1
Area
City Centers or Industrial Areas
Suburban or Well-Wooded Areas
Open Country with Scattered Trees
α
0.36
0.25
0.15
3.1.2 Tapered Structures
For a member with a tapered diameter or frontal width
yi ( x ) = ai ( x1 )µ i ( x )
where:
~
πL C L ρD 4 ( x1 )
µi ( x1 )
ai ( x1 ) =
2ζ iψ ( x1 ) (4πS )2 GM i
(3.1.2.1)
(3.1.2.2)
Except that if yi(x)>0.025D(x), ai(x1) shall be taken as:
~
2ρ C L D
ai ( x1 ) =
x1 +b
2
(x1 ) ∫ D( x ) µi (x1 ) dx
x1 −b
2
i
(3.1.2.3)
(4πS ) ζ GM i
where:
x1= location along the tapered member at which vortex shedding excitation potentially occurs
b= the length of the member above or below location x1 for which D(x) is within Ω of D(x1).
The quantity Ω shall be taken as 10% unless a smaller value can be justified.
ψ ( x1 ) =
dD ( x1 ) αD ( x1 )
+
x
dx
(3.1.2.4)
For a tapered member ai(x1) shall be computed for all locations, x1, along the member at which
vortex shedding excitation can occur for mode of vibration, i, as determined by 3.B. The
largest value of ai(x1) computed shall be used for determining yi(x) and the peak inertia loads.
In the case of a member with a uniform taper, p, the limits of integrations specified in equation
3.1.2.3 become:
ΩD( x1 )
p
ΩD( x1 )
x1 − b = x1 −
p
x1 + b = x1 +
19
(3.1.2.5)
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
In the case of a non-uniformly tapered member, the local variation of D around x=x1 can be
approximated by a linear taper p.
A good approximation for tapered members is to neglect the variation of D over the limits of
integration. With this assumption the modal coefficient of magnitude for a member with a
taper p for the ith mode becomes:
~
2 2 ρ C L D 4 ( x1 )
µ
ai ( x1 ) =
2
p(4πS ) ζ iGM i ave
(3.1.2.6)
where:
|µ|ave= average of the absolute values of the mode shape over the portion of the member
centered on x1 for which D(x) is within ±Ω percent of D(x1). Except near the node points,
which do not contribute to maximum values:
µ ave ≈ µ (x1 )
(3.1.2.7)
For a free-standing tapered member, the region of maximum excitation for the fundamental
mode is at approximately ¾ of the height and moves downwards for higher modes of
vibration.
Although the evaluation of ai for a particular mode of vibration must be carried out over the
entire member, the response is normally governed by the excitation from its main components.
For example, in the case of street light poles, the response in particular modes of vibration is
dominated by shedding from the pole, with the excitation on the luminaire bracket being of far
lesser significance. As a good first approximation, the evaluation of various modal coefficients
can thus be confined to locations along the pole.
Unless experimentally determined values are available, the damping ratio values - ζi for
members in all modes of vibration shall be taken as
ζi = 0.0075 for steel and aluminum members;
ζi = 0.015 for concrete members.
To determine the damping ratios for higher vibration modes can be used the methodology
presented in the first part of this thesis.
4. Proposed Design Procedure.
The design procedure proposed to be used as Standard Design Procedure (SDP) are based on
the sinusoidal formulation of the vortex shedding phenomenon. This procedure will be used in
conjunction with a commercial general purpose finite element program (ALGOR) which has
the provisions to apply time-varying (sinusoidal) loads on the finite element model nodes. The
structural finite element model will be built using beam finite elements.
20
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
4.1 Design Procedure for Non -Tapered and Tapered Structures
1.) Build a linear beam finite element model which will simulate geometrically and
structurally the investigated structure.
2.) Perform a Frequency Finite Element Simulation to obtain the first eight natural
frequency modes ωi, i=1 to 8.
3.) Transform the linear finite element model into a nonlinear one, with isotropic material
formulation for beam elements. If using ALGOR prepare the model to perform a
Mechanical Event Simulation
4.) With equation (2.2) will be determined the maximum critical wind speed at the top of
structure.
5.) Considering the variation of Wind Exposure Coefficient Ce with height (Table: 2.3 or
equation (2.17)) and the variation of the wind speed at the top of structure in
increments of 1m/s (from 1m/s to Vcr m/s), calculate the wind speeds in the center of
each beam finite element.
6.) Based on the wind speeds calculated at point 5 and the diameter of structure (or across
to flats dimension) calculate Reynolds Number (with equation (2.16)) in the center of
each finite element. Find the Reynolds sub-critical (Re < 3x105) and trans-critical
(Re > 3x106) intervals.
7.) Based on equation (2.3) calculate the frequency in the center of each finite element for
wind speeds from 1m/s to Vcr m/s. Find the natural frequencies, ωi, i=1 to 8, locations
for sub-critical and trans-critical Reynolds ranges.
8.) For each natural frequency mode locations calculate the maximum sinusoidal vortex
shedding induced force (using equation (2.5)) and the moment generated by this force
at the base of the structure. For each natural frequency mode consider the location with
the highest over-bending moment computed at the base of structure. Consider only the
locations which fall in the sub-critical or trans-critical Reynolds range.
9.) Using equations (2.13, 2.14, 2.15) calculate the lower and upper natural frequency
bounds for each natural frequency mode location found at point 8.
10.) For each natural frequency mode at each considered location define the frequency
domain and calculate the sinusoidal vortex shedding induced force in the center of
each finite element which fall in that frequency domain.
11.) For each natural frequency mode and considered frequency domain apply the
sinusoidal load (force vs. time) in the center of each finite element considered and run
the model.
12.) Compare the stress in the critical areas with the endurance limit.
13.) For the areas with stress distribution higher than endurance limit build a more detailed
finite element model and follow the design procedures presented in the first part of this
thesis or in reference [40].
14.) Knowing the location of the structure the frequencies and wind speeds which generate
stresses over endurance limit, find the annual number of hours for the wind blowing in
that range and calculate the annual number of load cycles. The structure’s life can be
estimated using available “Cycle Life – Stress Range “ design curves [49].
21
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
5. Mechanical Event Simulation of Vortex Shedding Structural Response
The analytical experiments will apply the above outlined simulation procedure on two types of
free standing structures: a non-tapered antenna tower and a tapered highmast pole.
5.1. Non-Tapered Antenna Tower
The antenna tower (Fig.5.1.1) is 40.8m tall and has four
sections flanged together.
The diameter of each section is 0.6m and the thickness is:
12.7mm for bottom section, 9.52mm for second section,
7.94mm for third section and 6.35mm for top section.
At the top of tower will be clamped-on a frame which will
hold the telecommunication antennas.
Above and below each flanged connection there is an access
opening with a removable cover plate and the base plate is
bolted using 38mm diameter anchor bolts.
Lets follow step by step procedure 4.1.
1) A beam finite element model was built. Were
used 30 linear beam finite element with the
proper cross-sectional properties. The model
was fully constrained at the base. Were also
simulated the masses of all attachment at the
proper elevation.
2) The natural frequency modes were calculated:
Mode
#
1
2
3
4
5
6
7
8
Frequency
[Hz]
1.040
4.340
11.260
20.810
34.170
49.120
67.880
87.100
3) A nonlinear finite element model was built.
4) The maximum critical wind speed at the top of
pole was calculated for:
q=450 Pa (N/m2)
Ce=1.4 (At top of pole)
and Vcr=31.12m/s
Fig.5.1.1 40.8m Antenna Tower
22
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
5) The wind speed at the top of pole was increase in 1m/s increments until reached a value close to Vcr (32m/s in our case).
Considering the variation of coefficient Ce with height was calculated the wind speed for center of each finite element
(Table 5.1.1).
Table: 5.1.1 Wind speed variation with height.
23
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
6) Reynolds number was calculated at the center of each finite element for wind speed increments of 1m/s from 1m/s to
32m/s. The gray area is the domain where Reynolds Number is within sub-critical or trans-critical range. (Table 5.1.2).
Table: 5.1.2 Reynold Number versus wind speed and height.
It can be observed that with the increase of wind speed the areas where potentially the vortex shedding phenomenon occur, moves
towards the bottom of the pole.
24
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
7) Were calculated the frequencies in the center of each finite element for wind speed increments of 1m/s from 1m/s to
32m/s.With red color is marked up the domain where Reynolds Number is within sub-critical or trans-critical range.
Were marked up with red the cells where the pole frequency matches the natural frequencies. The gray area is the
domain where the pole frequency is within upper and lower frequency boundaries. (Table 5.1.3).
Table: 5.1.3 Reynold Number versus wind speed and height.
It can be observed that only the first natural frequency (1.04 Hz) can be found in the domain of interest. The vortex shedding
phenomenon can potentially appear at the top of pole for 4m/s wind speed, at 19.72m above ground level for 5m/s wind speed and at
3.4m above ground level for 6m/s wind speed.
25
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
8) Were calculated the vortex shedding induced forces and base overturning moments for potential locations (Table 5.1.4).
Table: 5.1.4 Vortex shedding induced forces and base overturning moments for potential locations.
ω1=1.04
Elevation
[m]
40.120
21.080
3.400
D1
[m]
0.600
0.600
0.600
Hz
V1
[m/s]
4.0
4.1
4.2
F1
[N]
10.1
10.8
11.0
M1
[Nm]
405.4
228.6
37.4
The top location produces the highest bending moment at the base of pole and was marked up with grey color but the largest
frequency domain is for middle location (elevation 21.08m) and the overall bending moment will be higher. Therefore, this location is
considered to produce the highest stress at the base of pole.
9) Were calculated the upper and lower boundary limits for the first natural frequency mode (Table 5.1.5).
Table: 5.1.5 Upper and lower boundary limits for first natural frequency mode.
Mode
Frequency
Vicr
#
1
[Hz]
1.040
[m/s]
4.14
A
z0
K
z
A*Vcr*K
ln(z/z0)
2.5
0.4
0.05
40.12
-0.62
Upper
Limit
Lower Limit
[Hz]
1.20
[Hz]
0.88
The frequency domain for the first natural frequency mode was marked up with grey color in table 5.1.3
26
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
10) For the frequency domain are calculated the vortex shedding induced
forces in the center of the finite elements (Table 5.1.6).
Table: 5.1.6 Vortex shedding induced forces for first natural frequency mode.
The vortex shedding induced forces are sinusoidal time varying and the values in table: 5.1.6
are the maximum values.
11) The time-varying vortex shedding induced forces will be applied in the center
of corresponding finite elements.
27
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
12) The time-varying vortex shedding induced bending stress at the bottom of pole
(Fig: 5.1.2) is in the range: -2 MPa to 2 MPa which is less than 31MPa which is
the endurance limit for the critical welded connections at the bottom of pole,
therefore the vortex shedding phenomenon will not produce fatigue damage at
the bottom of this investigated antenna structure. Thus steps 13 and 14 are not
required for this structure.
Time Varying Stress Induced by Vortex Shedding
for First Natural Frquency Mode
2500000
2000000
1500000
Stress (Pa)
1000000
500000
0
-500000
0
10
20
30
40
50
-1000000
-1500000
-2000000
-2500000
Time (s)
Fig. 5.1.2 Time-varying stress induced by vortex shedding phenomenon.
Time Varying Top Pole Displacement Induced by Vortex Shedding
for First Natural Frquency Mode
0.025
0.02
Displacement (m)
0.015
0.01
0.005
0
-0.005
0
10
20
30
40
50
-0.01
-0.015
-0.02
-0.025
Time (s)
Fig. 5.1.3 Time-varying displacement induced by vortex shedding phenomenon.
28
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
5.2. Tapered Highmast Tower
In fig 5.2.1. is shown a 38m highmast pole
which will be investigated for dynamic
response to vortex shedding induced loads.
The structure has four sections which overlap
each other. The thickness of bottom section is
7.94mm, second and third sections 6.35mm
and top section 4.76mm.
The structure is 12 sided with 850mm across
to flats dimension for bottom and 200mm
across to flats for top.
The structure is bolted to foundation using
38mm diameter anchor bolts.
At the bottom there is a large winch servicing
cutout with a 16mm thick reinforcing ring.
Inside the pole, at the bottom, there is a winch
mounted on a angle profile.
At the top of pole will be docked a frame
which will hold 12 floodlights with their
ballasts.
Inside the pole are ran the steel cables required
to lower or rise the frame with lamps.
Lets follow step by step procedure 4.2.
1) A beam finite element model
was built. Were used 30 linear
beam finite element with the
proper
cross-sectional
properties. The model was fully
constrained at the base. Were
also simulated the masses of all
attachment at the proper
elevation. The slip joint
connections cross sectional
properties were calculated
considering the change in
thickness because of the
adjacent sections overlapping.
Fig. 5.2.1. 38m Highmast Pole
29
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
2) The natural frequency modes were calculated:
Mode
#
1
2
3
4
5
6
7
8
Frequency
[Hz]
0.750
2.570
6.280
11.420
18.660
27.250
37.590
49.440
3) A nonlinear finite element model was built.
4) The maximum critical wind speed at the top of pole was calculated for:
q=450 Pa (N/m2)
Ce=1.4 (At top of pole)
and Vcr=31.12m/s
5) The wind speed at the top of pole was increase in 1m/s increments
until reached a value close to Vcr (32m/s in our case). Considering the
variation of coefficient Ce with height was calculated the wind speed
for center of each finite element (Table 5.2.1).
30
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
Table: 5.2.1 Wind speed variation with height.
6) Reynolds number was calculated at the center of each finite element for wind speed increments of 1m/s from 1m/s to
32m/s. The gray area is the domain where Reynolds Number is within sub-critical or trans-critical range. (Table 5.2.2).
31
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
Table: 5.2.2 Reynold Number versus wind speed and height.
It can be observed that with the increase of wind speed the areas where potentially the vortex shedding phenomenon occur, moves
towards the top of the pole.
32
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
7) Were calculated the frequencies in the center of each finite element for wind speed increments of 1m/s from 1m/s to
32m/s.With red color is marked up the domain where Reynolds Number is within sub-critical or trans-critical range.
Were marked up with colors the cells where the pole frequency matches the natural frequencies (Table 5.2.3).
Table: 5.2.3 Reynold Number versus wind speed and height.
33
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
8) Were calculated the vortex shedding induced forces and base overturning moments for potential locations (Table 5.2.4).
Table: 5.2.4 Vortex shedding induced forces and base overturning moments for potential locations.
9) Were calculated the upper and lower boundary limits for the first natural frequency mode (Table 5.2.5).
Table: 5.2.5 Upper and lower boundary limits for natural frequency modes.
34
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
10) For the frequency domains are calculated the vortex shedding induced forces in the center of the finite elements (Tables
5.2.6 to 5.2.9).
Table: 5.2.6 Vortex shedding induced forces – 1st frequency mode. Table: 5.2.7 Vortex shedding induced forces – 2nd frequency mode.
35
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
Table: 5.2.8 Vortex shedding induced forces – 3rd frequency mode. Table: 5.2.9 Vortex shedding induced forces – 4th frequency mode.
36
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
The vortex shedding induced forces are sinusoidal time varying and the values in tables: 5.2.6 to
5.2.9 are the maximum values.
11) The time-varying vortex shedding induced forces will be applied in the center of
corresponding finite elements.
12) The time-varying vortex shedding induced bending stress range at the bottom of pole
(Table 5.2.10) is less than 31MPa, which is the endurance limit for the critical
welded connections at the bottom of pole, only for the first natural frequency mode
therefore the vortex shedding phenomenon will potentially produce fatigue damage
at the bottom of pole for second, third and forth frequency modes. Thus steps 13 and
14 are required for this structure [40].
37
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
6. Conclusions
As demonstrated, based on the above simulations, the tubular, multisided or round, tapered or
nontapered free standing structures may be subject to considerable dynamic induced stress by
vortex shedding phenomenon.
Also could be seen that the vibrations induced by vortex shedding may occur at relatively
moderated wind speeds, therefore the structures may undergo a considerable number of stress
cycles.
Considering these aspects the risk of structural fatigue must be taken into account at the design
stage.
Even if fatigue analysis is not explicitly demanded in the design code which is applied, the risk of
fatigue should be kept in mind when designing slender free standing structures.
For this the design engineer must have a good understanding of the vortex shedding phenomenon
and follow a logic design procedure to investigate the response of the structure to vortex shedding.
The design standards lack in presenting easy to use calculation methodologies for investigating
vortex shedding phenomenon and its impact on
structures, and usually the design procedures require the
induced forces to be applied statically.
Vortex shedding is a dynamic phenomenon and to get
realistic structural responses to this phenomenon it must
be simulated dynamically. The dynamic structural
response is the worst case scenario because of the
realistic account for inertial induced loads.
As seen in the analytical models the wind speed is the
main parameter and good wind speed estimations are
critical.
Based on the above outlined design procedure one can
easily estimate the critical wind speed which potentially
trigger the vortex shedding phenomenon as well as the
upper and lower boundaries for that critical wind speed.
After defining the areas where potentially the vortex
shedding occurs will be calculated the time-varying
induced forces which will act on the structure.
The design procedure can be easily automated using MS
EXCEL program in conjunction with a general purpose
Fig. 6.1 High Stress Locations – Pole Bottom
finite element software.
The interpretation of the stress range results should be made correctly considering for each critical
connection the proper fatigue category.
To evaluate correctly the stress in the investigated connection a detailed finite element model should
be built.
As seen in fig. 6.1 is very hard to predict the high stress locations for complex geometries without
numerical simulations, loading the structure with the vortex shedding induced forces in all possible
directions to get a good feeling about the structure’s response.
The design procedure presented above (4.1) brings few interesting and original things:
38
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
-
it is an easy to follow procedure;
all parameters are clearly defined;
the vortex shedding induced loads are applied dynamically like in the real situation;
the procedure can be fully automated.
The results obtained testing this design procedure on transmission, antenna and highmast towers are
in the realistic range and comparable with experimental observations.
Considering that computational fluid dynamics simulation or wind tunnel tests to investigate the
vortex shedding effects for these type of structures will be very time consuming and expensive this
easy to use methodology represents an efficient tool for the numerical analyst involved into the
design process.
39
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
References
1. Solari, G. – The role of analytical methods for evaluating the wind-induced response of
structures, Journal of Wind Engineering in Industrial Aerodynamics, 90, 1453-1477, 2002.
2. Claës Dyrbye, Svend O. Hansen – Wind Loads on Structures, John Wiley & Sons, 1999.
3. NBC 1995 National Building Code of Canada -1995.
4. CAN/CSA-S6-00 Canadian Highway Bridge Design Code -2005.
5. CAN/CSA-S6.1-00 Commentaries to Canadian Highway Bridge Design Code-2005.
6. Eurocode 1 – Basis of design and actions on structures – Part 2-4: Actions on structures –
Wind Actions, ENV 1991-2-4.
7. Standard Specifications for Structural Supports for Highway Signs, Luminaires and Traffic
Signals, American Association of State Highway and Transportation Officials, 2001.
8. ASCE7-2005 – Minimum Design Loads for Buildings and Other Structures, American
Society of Civil Engineering, Reston, VA, 2005.
9. AS170.2, SAA Loading Code, Part 2 – Wind forces, AS1170.2-89, Australian Standards,
1989.
10. AIJ-RLB – Recomandations for Loads on Buildings, Arhitectural Institute of Japan, 1993
(English version 1996).
11. NTE 003/04/00 – Design Standards for Electrical Transmission Lines over 100 kV
(Replaced Standard PE 104).
12. PE 105 – Design Methodology for Electrical Transmission Lines Steel Towers.
13. STAS 10101/20-90 Structural Loads Induced by Wind – Romanian Standardization Institute
– 1990.
14. Yukio Tamura, Ahsan Kareem, Giovanni Solari, Kenny C.S. Kwok, John D. Holmes –
Report by Working Group WCE – Dynamic Response, 2001.
15. Davenport A.G. – The application of statistical concepts to the wind loading of structures,
Proceedings of the Institution of Civil Engineers, London, UK, 1961;19 :449-472.
16. Davenport A.G. – Note on the distribution of the largest value of a random function with
application to gust loading. Proceedings of the Institution of Civil Engineers, London, UK,
1964;24 :187-196.
17. Davenport, A.G. – New Approaches to Design Against Wind Action, Unpublished Text.
Boundary Layer Wind Tunnel Laboratory, University of Western Ontario, London, Ontario
– 1976.
18. Vikery, B.J. – On the reliability of gust loading factors. Proceedings of Technical Meeting
concerning Wind Loads on Buildings and Structures. National Bureau of Standards,
Washington, DC: 1970;93-104.
19. Simiu E. – Equivalent Static Wind Loads for Tall Buildings Design. Journal of Structural
Div., ASCE 1976; 102:719-737.
20. ESDU 76001 – The response of flexible structures to atmospheric turbulence. London, UK,
Engineering Science Data Unit, 1976.
21. ECCS – Recommendations for the calculation of wind effects on buildings and structures.
Brussels: European Convention for Constructional Steelworks, 1978.
22. Solari G. – Equivalent Wind Spectrum Technique: theory and applications. Journal of
Structural Engineering, ASCE 1988; 114: 1303-1323.
23. Kaspersky M. – Extreme wind load distributions for linear and nonlinear design.
Engineering Structures, 1992;14:27-34.
40
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
24. Holmes J.D. – Along-wind response of lattice towers: part 1 – derivation of expressions for
gust response factors. Engineering Structures, 1994;16: 287-292.
25. Holmes J.D. – Along-wind response of lattice towers: part 2 – Effective load distributions.
Engineering Structures, 1996;18: 489-494.
26. Davenport, A.G. – How can we simplify and generalize wind loads. Lournal of Wind
Engineering in Industrial Aerodynamics, 1995; 54-55: 657-669.
27. Tamura, Y., Kawai, H., Uematsu, Y., Marukawa, H., Fuji, K., Taniike, Y., - Wind loads and
wind induced response estimations in the Recommendations for loads on buildings, AIJ
1993. Engineering Structures 1996; 18: 399-411.
28. Solari, G. – Towards a global model for calculating 3D equivalent static wind forces on
structures. Proceedings on 7th US National Wind Engineering Conference, Los Angeles:
1993, 719-728.
29. Piccardo, G., Solari, G. – A refined model for calculating 3D equivalent static wind forces
on structures. Journal of Wind Engineering in Industrial Aerodynamics, 1996; 65: 21-30.
30. Solari, G. – Mathematical Model to Predict 3D Wind Loading on Buildings. Journal of
Engineering Mechanics, ASCE, 1985; 111: 254-276.
31. Piccardo, G., Solari, G. – Closed Form Prediction of 3D Wind-excited response of slender
structures. Journal of Wind Engineering in Industrial Aerodynamics, 1998; 74-76: 697-708.
32. Piccardo, G., Solari, G. – 3D wind-excited response of slender structures: Closed form
solution. Journal of Structural Engineering, ASCE 2000; 126: 936-943.
33. Piccardo, G., Solari, G. – Generalized equivalent spectrum technique. Wind and Structures
Journal, 1998; 1: 161-174.
34. P.A.B. de Sampaio, A.L.G.A. Coutinho – Simulating Vortex Shedding at High Reynolds
Numbers, Proceedings of the 10th International Offshore and Polar Conference, Seattle,
USA, 2000.
35. Yin Zhou, Ahsan Kareem, Ming Gu – Mode Shape Corrections for Wind Load Effects,
Journal of Engineering Mechanics, January, 2002, 15-23.
36. Maria Pia Repetto, Solari, G. – Directional Wind-Induced Fatigue on Slender Vertical
Structures, Journal of Structural Engineering, ASCE, July, 2004, 10032-1040.
37. Dexter, R.J., Ricker, M.J. – Fatigue-Resistant Design of Cantilevered Signal, Sign, and
Light Supports – NCHRP Report 469, University of Minnesota, USA, Published by:
National Academy Press, 2002.
38. Christopher M. Foley, Scott J. Ginal, John L. Peronto, Raymond A. Fournelle – Structural
Analysis of Structures and Luminaire Supports, Marquette University, Wisconsin, USA,
2004.
39. Giosan, I., Brockman, T. - Streamlining Metal Poles, Mechanical Engineering Magazine,
April, 2000, 68-70.
40. Giosan, I. – Building Better Free Standing Structures, Electrical Line Magazine, February,
2006, 63-70.
41. Harris, Cyril, M and Crede, Charles E. – Shock and Vibration Handbook, Section 29,
“Vibration of Structures Induced by Wind,” Part II, 2nd Edition, McGraw-Hill Book Co. –
1976.
42. Vikery, B.J., – “Load Fluctuations in Turbulent Flow,” ASCE, J. Eng. Mech. Div., Vol. 94,
February 1968.
43. Vikery, B.J. and Clark, A. W., – “Lift of Across Wind Response of Tapered Stacks,”
ASCE, J. Eng. Mech. Div., Vol. 98, January 1972.
41
Structural Vortex Shedding Response Estimation Methodology and Finite Element Simulation
44. Wooton, L.R., – “The Oscilation of Large Circular Stacks in Wind”, Proceedings I.C.E.,
August 1969.1969.
45. Simiu, E., Scalon, R.H. – Wind Effects on Structures: Fundamentals and Applications to
Design – 3rd Edition, John Willey & Sons, New York, 1996.
46. ESDU-96030 – Response of Structures to Vortex Shedding. Structures of Circular or
Polygonal Cross Section. Engineering Science Data Unit, London, UK, 1998.
47. Wieringa, J. – Representative roughness parameters for homogeneous terrain. BoundaryLayer Meorology, 63, 323-363, 1993.
48. Lettau, H. – Note of aerodynamic roughness-parameter estimation on the basis of roughnesselement description. Journal of Applied Meteorology, 8, 828-832, 1969.
49. Handbook of Steel Construction – Canadian Institute of Steel Construction, 3rd Edition,
1982.
42