Framework

SESAM
THEORY MANUAL
Framework
Wind Fatigue Design
DET NORSKE VERITAS
SESAM
Theory Manual
Framework
Wind Fatigue Design
June 1st, 2001
Valid from program version 2.8
Developed and marketed by
DET NORSKE VERITAS
DNV Software Report No.: 93-7076 / Revision 1, June 1st, 2001
Copyright © 2000 Det Norske Veritas
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FRAMEWORK
Program version 2.8
SESAM
01-JUN-2001
i
TABLE OF CONTENTS
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
PURPOSE
SCOPE
REFERENCES
OVERVIEW AND ASSUMPTIONS
4.1 Overview
4.2 Theoretical Assumptions
4.3 Scope of Analysis
THE SPECTRAL APPROACH
THE NATURE OF THE WIND
WIND SPECTRA
A SPECTRAL FORCING FUNCTION
8.1 Wind Force on a Member
8.2 The Wind Force on a Degree Of Freedom
8.3 Spectral Relationships
VORTEX SHEDDING INDUCED VIBRATIONS
9.1 Determination of Mode Shape and Fundamental Frequency of a Brace
9.2 Brace Oscillation Amplitude Calculation
9.3 Member End Damage Calculation
9.4 Member Centre Damage Calculation
CONDENSATION OF FORCES
STRESS CONCENTRATION FACTORS
11.1 An Overview
11.2 Components of the HSS
11.3 The SCF Schemes
11.4 Non-standard Joints
11.5 Efthymiou Scheme
THE HOT SPOT STRESS TRANSFER FUNCTION FROM POINT FORCING
12.1 The Dynamic Equation
12.2 Results from the Eigenvalue Problem
12.3 The Structure Displacement Vector from a Point Force
12.4 Hot Spot Stresses from a Point Force
HOT SPOT STRESS POWER SPECTRUM
13.1 The Hot Spot Stress Power Spectra
13.2 Integrating the Hot Spot Stress Power Spectra
CALCULATION OF FATIGUE LIFE
14.1 Assumptions
14.2 Damage Evaluation
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APPENDIX 1
APPENDIX 2
APPENDIX 3
APPENDIX 4
APPENDIX 5
APPENDIX 6
APPENDIX 7
APPENDIX 8
APPENDIX 9
Notation Used Within the Text
Power Spectral And Correlation Functions
Algebra Supporting Section 10.0: Condensation Of Forces
SCF Schemes
Lloyd's Register of Shipping Formulae (ISOPE 1991)
Lloyd's 1996 Recommendations for SCFs
Wind Spectra and Coherence Functions
Algebra Supporting Section 14.0: The Damage Integral
Pre-processing of Wind Loads by Wajac
SESAM
FRAMEWORK
Program version 2.8
1.0
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1-1
PURPOSE
The purpose of this manual is to describe the theory underlying the Framework wind fatigue
application.
Wind fatigue of Framework is according to the wind fatigue program Gusto (Reference 3.9),
which has been implemented into Framework for the purpose of making wind fatigue
application available in SESAM.
This manual is a reprint of the theory manual of Gusto (Reference 3.10), except for matters
and references related to external programs, which are not relevant for Framework
applications.
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2-1
SCOPE
This manual describes the calculation methods used in Framework for the determination of
fatigue damage to frame structures subjected to wind loading. Buffeting loads due to gusting
are treated by the power spectral density approach. The damage is a function of the overall
structural response. The effects of vortex shedding induced fatigue due to steady winds are
treated by separate evaluation of individual member responses. Framework calculates the two
effects on the assumption that they are uncoupled. The resultant damages are summed to give
overall fatigue lives of joints and members.
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3-1
REFERENCES
3.1
Power Spectrum of Horizontal Wind Speed.
Van der Hoven, J Met. (14), 1957.
3.2
Dynamic Response of Structures to Wind & Earthquake Loading.
Gould and Abu-Sitta, Pentech Press, 1980.
3.3
The Modern Design of Wind-Sensitive Structures.
CIRIA, 1970.
3.4
Wind Engineering in the Eighties.
CIRIA, 1980.
3.5
Reduction of Stiffness and Mass Matrices.
Guyan, AIAA Journal Vol. 3 no 2, 1965.
3.6
Cumulative Damage in Fatigue.
Miner, Journal of Applied Mechanics. 12(3), 1945.
3.7
Offshore Installations: Guidance on Design and Construction. 4th Edition
Department of Energy, HMSO 1990.
3.8
Dynamic Analysis of Offshore Structures.
Brebbia and Walker, Butterworths 1979.
3.9
Gusto User Manual - Version 6.00.
Brown and Root Document Number 308-7222-MA-13-013-D.
3.10 Gusto Theory Manual - Version 6.00.
Brown and Root Document Number 308-7222-ST-13-046-B.
3.11 Wind Load and Dynamic Response of Marine Structures.
NTNF Research Project Programme for Marine Structures, Report No 4, May 1984.
3.12 Vibration Problems in Engineering. Third edition.
Timoshenko S. P. and Young D. H. D. Van Nostrand and Company, 1955
3.13 Stress Concentration Factors for Simple Tubular Joints
Smedley and Fisher (of Lloyd's Register of Shipping), ISOPE Conference, 1991
3.14 Lloyd's Register of Shipping Recommended Parametric Stress Concentration Factors.
Lloyd's Register of Shipping Document Number OD/TN/95001, January 1996
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3.15 A Criterion for Assessing Wind Induced Crossflow Vortex Vibrations In Wind Sensitive
Structures.
Robinson R. W. and Hamilton J. Health and Safety Executive Offshore Technology
Report OTH 92 379, 1992.
3.16 SESAM, Sestra, Superelement Structural Analysis, User Manual,
May 1999.
3.17 SESAM, Wajac, Wave and Current Loads on Fixed Rigid Frame Structures, User
Manual, May 2001.
3.18 NPD, Regulations and provisions for the petroleum activities, June 1997
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01-JUN-2001
4.0
OVERVIEW AND ASSUMPTIONS
4.1
Overview
4-1
Framework is an application that evaluates wind buffeting and vortex shedding induced
fatigue damage to a structure. It receives input information relating to the eigenmodes of a
structure, coupled with statistical data on the annual wind distribution and associated drag
factors.
The annual wind data are characterized by a set of wind states, considered to represent the
climate for the year. For each of these wind states, the response stress power spectra at a local
“hot spot” within a particular joint are evaluated.
For buffeting calculations the hot spot power spectrum response is divided into the quasistatic response and the dynamic response.
The dynamic response consists of the excited resonant modes. It is partitioned into the
separate resonant modal responses; for each of these an independent damage assessment is
made. This assumes that each response is narrow band and independent of the others, but
sometimes several modes, very close in frequency, are taken as one.
Vortex shedding from brace members may induce oscillations in individual braces. These are
local modes rather than overall structural modes.
The quasi-static part of the power spectra covers the low frequency non-resonant response.
The wind spectrum has a broad peak at low frequencies but is treated as narrow band at its
peak frequency with one third of the stress variance of the low frequency broad band peak
stress spectra. The resultant damage is then multiplied by 10. This approach assumes that the
quasi-static contribution to damage is small, so that a rigorous evaluation is not required.
For each of these dynamic and static partitions the narrow band assumption implies a
Rayleigh distribution for the “hot spot” stress range versus number of cycles. The variance is
given by the integral under the power spectrum. Fatigue damage may then be evaluated by
application of the Palmgren-Miner relationship and reference to a recognised S-N curve (see
Section 14.0).
The overview of this solution method is shown in Figures 4.1 to 4.4.
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OVERVIEW OF SOLUTION METHOD
7
WIND
SPECTRUM
HOT SPOT
STRESS
LOADING
TRANSFER
FUNCTION
10, 11, 12
GEOMETRY
8
WIND
FORCE CROSS
SPECTRA
HOT SPOT
STRESS
SPECTRUM
13
Figure 4.1 Generation of Hot Spot Stress Spectrum with Cross Reference to Principal
Sections of this Manual.
Figure 4.2 Typical Hot Spot Stress Spectrum
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USE OF HOT SPOT STRESS SPECTRUM IN PREDICTION OF FATIGUE LIFE
FOR A GIVEN WINDSTATE (REF. SECTION 13.2)
For each peak shown above
(Close peaks are combined as one)
Treat as narrow-band response with variance equal to the integrand.
Stresses taken as Rayleigh-distributed with the above variance
Apply stress amplitude distribution to SN curve to obtain damage
due to this peak.
Accumulate damage due to each peak
CALCULATION OF TOTAL DAMAGE
For each windstate at specified hot spot
Add damage for that windstate
Figure 4.3 Summary of Calculation Methods
4-3
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DERIVATION OF CROSS-POWER SPECTRUM OF WIND
FORCING (Section 8)
Demonstrate a linear matrix relationship between fluctuating wind
components of a member and member hot spot stresses.
Create the cross-correlation function for wind forcing
Evaluate the cross-power spectral density as the Fourier Transform
of the cross-correlation function
Partition the cross-power density function into a self-power density
function and a spatial function
DERIVATION OF FREQUENCY DOMAIN STRUCTURE TRANSFER
FUNCTION (Sections 10 to 12)
Demonstrate Guyan Reduction Method
(Section 10)
Describe Joint Stress Concentration Factors
(Section 11)
Describe the Hot Spot Stress TransferFunction from Point
Forcing (Section 12)
COMBINATION TO FORM HOT SPOT STRESS SPECTRUM
(Section 13)
The Frequency Domain hot spot stress spectrum is the product
of the forcing cross-spectrum and the structure force to stress
transfer function
Figure 4.4 Expanded Derivation of Hot Spot Stress Spectrum
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Theoretical Assumptions
The major theoretical assumptions made are:
a)
Buffeting damage is dominated by low frequency resonant modes
This implies:
•
separate narrow-band damage evaluation for the resonant modal responses is
conservative
•
the greatest hot spot stresses within a modal response cycle occur at maximum
modal amplitude
•
Guyan reduction may be used to reduce the mass and stiffness matrices without
significant loss of accuracy to the low modes, see Section 8.0. (Note! The
Framework wind fattigue assumes all non-fixed translational degrees of freedoms
(dof) as master dof.)
b)
parametric SCF equations are used to evaluate joint stress concentrations.
c)
the structure is made of welded tubular members.
d)
wind forces are parameterized as linear fluctuating components super-imposed upon
mean wind profiles.
e)
wind gust velocities in the mean wind direction and normal to the mean wind both
horizontally and vertically are statistically independent.
f)
member drag coefficients are invariant under the fluctuating wind component and are
appropriate to the mean wind speed.
g)
vortex shedding induced member oscillations and fatigue are uncoupled from any
buffeting induced vibrations and damage.
4.3
Scope of Analysis
The solution technique used by Framework requires a significant amount of input
information. Eigenvalues, mass normalized eigenvectors, resultant stresses from the
normalized eigendeformations, and forcing functions are generated externally, by Sestra
(Reference 3.16).
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THE SPECTRAL APPROACH
The most appropriate technique for determining wind-induced cyclic stresses is referred to as
the frequency-domain or power-spectral density approach.
A power spectrum describes a time-dependent variable relating the energy distribution over a
range of frequencies. All phase information is averaged out. Analysis methods whereby
output spectra are obtained from input spectra via transfer functions are required for random
processes such as wind or wave loading, where only a statistical description of the
environmental forces can be given. In the spectral analysis method of fatigue due to wind,
the stress spectrum is obtained from the input wind spectrum via the structure stress transfer
function. Because of the nature of the fluctuating wind force, there is, to good accuracy, a
direct linear relationship between the wind speed and force spectra allowing structure stress
spectra to be linearly related to wind speed spectra.
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6-1
THE NATURE OF THE WIND
The energy content of wind as a function of frequency is demonstrated in the form of a power
spectrum averaged over a year, as shown in Figure 6.1, taken from Reference 3.1. Two
dominant peaks occur in this figure. The lower frequency peak is associated with cycles with
a period of the order 2-3 days and is due to the passage of large-scale atmospheric
depressions. The higher frequency peak has significant energy in the range 10 minutes to 1
second. It is this part of the spectrum, which is of interest to structural designers and is
commonly known as the gust spectrum. There is apparently a lack of wind energy with
periods between 10 minutes and two hours. It can clearly be assumed that fatigue, which
depends on changes in wind speed, is not significantly affected by cycles of length greater
than two hours.
For the purpose of fatigue analysis, the wind speed is averaged over a suitable period and the
wind is then represented in that time as having a constant mean value and direction, upon
which fluctuations or gusts are superimposed. Although a period of about ten minutes would
appear to be desirable as an averaging period in order to reflect the influence of the shortterm storms, a period of one hour has traditionally been used. It is for this time that data are
usually available, see, for example, Reference 3.2.
While a speed and direction represent the mean wind in any given hour, the gust components
are statistically described by three parameters: probability distribution, power spectrum and
cross-correlation function.
The probability distribution describes the ratio or percentage of time a certain wind speed is
likely to occur, the power spectra reflect the energy content of the wind as a function of
frequency, and the cross-correlation function indicates the way in which the gusts are
spatially correlated.
The probability distribution can be obtained from measurements; but standard formulae exist
for describing this as a function of mean wind speed, (References 3.3 and 3.11).
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FIGURE 6.1
SPECTRUM OF HORIZONTAL WIND SPEED
(AFTER VAN DER HOVEN, REFERENCE 3.1)
Program version 2.8
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7-1
WIND SPECTRA
Framework requires the definition of a set of hourly mean wind speed vs. height profiles,
their bearings and for what fraction of a year they exist. For each of these, three
parameterized gust spectra are calculated, and a resultant damage assessment made. The total
annual damage may be obtained by adding these damage assessments in proportion to the
fraction of a year in which they are generated.
Obtaining a set of such hourly mean profiles may in itself be a major undertaking when the
measured data are not totally adequate. Engineering judgement may be required in assessing
what approximations are valid. To gain the necessary knowledge base from which to do this,
the reader's attention is drawn to References 3.3 and 3.4.
In general, over a period of one year, wind measurements can be taken to show the number of
hours per year the hourly mean wind is blowing for each speed and direction. These
measurements are taken at 10m above ground or sea level. By applying a power law to
represent the variation of mean wind speed with height relationship based on the drag at the
earth's surface, (Reference 3.3), the requirements are met.
Typically the wind data are such that the easiest set of hourly mean wind speeds to find are over
the eight major compass points and the twelve/fourteen divisions of the Beaufort scale. A
hypothetical table for the percentage of the year occupied by each such wind state is shown in
Table 7.1.
DIRECTION
BEAUFORT
SCALE
1
N
NE
E
SE
S
SW
W
NW
TOTAL
0.24
0.13
0.14
0.14
0.18
0.16
0.19
0.23
1.44
2
0.77
0.64
0.61
0.46
0.77
0.59
0.74
0.68
5.27
3
1.52
1.42
1.27
1.30
2.08
1.82
1.84
1.71
12.97
4
3.14
2.59
1.67
2.22
4.77
3.64
4.01
2.94
24.98
5
2.57
1.95
0.48
1.71
4.26
3.12
2.74
2.06
18.89
6
2.90
1.47
0.39
1.84
4.72
2.85
2.41
1.80
18.37
7
1.58
0.73
0.14
1.42
2.84
1.52
1.36
1.23
10.82
8
0.67
0.20
0.04
0.75
1.57
0.62
0.82
0.58
5.26
9
0.12
0.02
0.00
0.27
0.41
0.16
0.18
0.13
1.28
10
0.02
0.01
0.00
0.13
0.16
0.02
0.06
0.04
0.44
11
0.00
0.00
0.00
0.02
0.05
0.00
0.00
0.02
0.09
12
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
100.0
TABLE 7.1 PROBABILITY DISTRIBUTION OF MEAN WIND SPEED EXPRESSED AS % OF YEAR
FOR WHICH IT OCCURS
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In such a case the bearing of the normal to one of the structure faces does not usually
correspond to one of the compass points. For convenience, the data may be linearly
interpolated to provide wind directions meeting this criterion.
Towers, which exhibit symmetry, may allow opposing wind directions to be treated as one.
Table 7.2 shows data for the 8 directions of Table 7.1 expressed as four composite directions.
The annual mean wind distribution is therefore fully characterized in Table 7.2 by 48 wind
states, that is 12 in each of four directions.
BEAUFORT
N/S
NE/SW
E/W
SE/NW
1
0.42
0.29
0.33
0.37
2
1.54
1.23
1.35
1.14
3
3.60
3.24
3.11
3.01
4
7.91
6.23
5.68
5.16
5
6.83
5.07
3.22
3.77
6
7.62
4.32
2.80
3.64
7
4.42
2.25
1.50
2.65
8
2.24
0.82
0.86
1.33
9
0.53
0.18
0.18
0.40
10
0.18
0.03
0.06
0.17
11
0.05
0.00
0.00
0.04
12
0.00
0.00
0.00
0.00
TABLE 7.2
PROBABILITY DISTRIBUTION OF COMPOSITE WIND STATES EXPRESSED
AS % OF YEAR FOR WHICH IT OCCURS
For each wind state, the wind speed at a height of 10m above the earth's surface is used to
help compute a single-sided gust spectrum S s w( f ) , where f is in cycles/sec. There are five
available spectra for wind fatigue:
•
the HARRIS spectra (Reference 3.3)
•
the DAVENPORT spectra (Reference 3.11)
•
the NPD spectra (Reference 3.18)
•
the PANOFSKY LATERAL spectra (Reference 3.11)
•
the PANOFSKY VERTICAL spectra (Reference 3.11)
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The first three are spectra representing the gusts in the same direction as the mean wind, the
fourth is for lateral (horizontal) gusts across the mean wind and the fifth is for vertical gusts
across the mean wind. Their formulae are presented in Appendix 7.0.
Figure 7.1 shows a plot of the HARRIS spectra, associated with wind cases 1-11 in Tables
7.1 and 7.2.
This spectra (and the others) exhibit a number of important features:
•
the eddy spectra is independent of height; simplifying the problem
•
at typical tower modal frequencies (~ 1Hz) gust energy increases greatly with mean wind
speed. Typically damage per unit time is proportional to the mean velocity raised to a
power between 8 and 12
•
the eddy spectra is linearly dependent on the drag at the earth's surface
s
( f ) in cycles/sec, the double-sided spectra in rads/sec are
From these single-sided spectra S vv
then
Svv (w) =
1 s
Svv ( f )
4π
f = ω / 2π
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FIGURE 7.1
GUST SPECTRA FOR WIND STATES 1-11 IN TABLES 7.1 AND 7.2 (SI UNITS)
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8-1
A SPECTRAL FORCING FUNCTION
In this section, an approximation to the cross-power spectral density function of the buffeting
wind loads is presented in terms of the power spectra for the fluctuating wind. This may then be
used in the derivation of hot spot stress power spectra. The linear relationship between the forcing
spectra and hot spot stress spectra forms the backbone of the spectral approach to wind fatigue
analysis.
In Sections 8.1 and 8.2 a linear relationship between the fluctuating wind components and the
fluctuating forcing at a given node is established. This involves first evaluating the forcing on
the members, and then defining a relationship between member forces and nodal forces.
Because the linear relationship exists and the wind spectra applied in the three directions may
be regarded as statistically independent the cross-power spectral density function of the
forcing from all three spectra is the linear sum of that from the individual spectra.
In Section 8.3 the cross-power spectrum of forcing is found from the wind cross-power
spectral density function of the wind. The wind-cross power spectral density function is then
expressed in terms of the power spectrum of this wind, multiplied by a simple function.
For detailed descriptions of the mathematical definitions of the various spectral functions
above, see Appendix 2.0.
8.1
Wind Force on a Member
The general form for the wind force on a member is given by:
F = 12 ρ Cd DL U n U n
Where ρ , Cd , L, D and U n are the air density, member drag coefficient, member length,
member diameter and vector normal velocity respectively. All structural members are
assumed to be tubular with drag coefficients of 1.2 in the sub-critical regime and varying as
per Figure 8.1 in the critical and super-critical regimes. Although Reynolds' number varies
with wind speed, the assumption is made that the drag coefficient is not time dependent.
When the above form is expanded out and the fluctuating terms considered small compared
to the time mean terms, the wind force may be expressed as
 U ( z ,t )

1
F (t ) = 2 ρ Cd DLa A ⋅ 
0


0

 2 b c   U ( z ,t ) .U 1 




 + 1 ρ Cd La A  0 1 0   U ( z ,t ) .V 1 
 2
 0 0 1   U ( z ,t ) .W 1





2
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FIGURE 8.1
VARIATION OF DRAG COEFFICIENT FOR A CYLINDRICAL MEMBER
WITH REYNOLDS' NUMBER
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8-3
where A is a transformation matrix, and a, b and c are constants depending on the mean
wind direction and member orientation. The wind velocity vector over a period of time has
U1
 U (z ,t ) 
 


been split into a time mean,  0  , plus fluctuations  V 1  .
 1
 0 


W 
The drag coefficient is dependent on the Reynolds' number as below
In the absence of ice
Re ≤ 2.5x105
C d = 1 .2
2.5x105 < Re ≤ 3.85x105
Cd = 10(10.8599 −1.997199 log10 (Re))
3.85x105 < Re
Cd = 10( 0.2557 log10 (Re) −1.7109)
In the presence of ice the relationship is
8.2
Re ≤ 5.0 x105
Cd = 1.2
Re > 5.0 x105
Cd = 0.7
The Wind Force on a Degree Of Freedom
By summing these member forces and distributing them to the degree of freedom (dof), the
fluctuating force at dof r from wind fluctuations may then be expressed in the form
g r ′ (t ) = 2 ErUU < v( zr , t ) > u′(t )
+ 2 ErVV < v( zr ,t ) > v′(t )
+ 2 ErWW
Equation 8.2.1
< v( zr ,t ) > w′(t )
and the static force at degree of freedom r may be expressed as
g r (t ) = E UU
V ( zr ,t )
r
2
Equation 8.2.2
VV
and EWW
are constant at each dof. They depend upon ρ and Cd and the
where EUU
r , Er
r
values of D, L, A , a, b, and c for each member connected to that degree of freedom. z r is the
height of the dof r.
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Spectral Relationships
The cross-power spectral density function is the Fourier transform of the cross correlationfunction. The cross-power spectral density function S gg (r ,s : ω ) of the forcing between dof r
and dof s is therefore
S gg (r ,s : ω ) =
1 ∞ − izω
g r (t ) g s (t + z ) dτ
∫ e
2π − ∞
Equation 8.3.1
Using Equations 8.2.1 and 8.2.2
g r (t ) g s (t + z ) = g r′ (t ) g ′s (t + z ) + g r (t ) g ′(t ) g s (t + z ) + g r (t ) g s (t + z )
As g ′r (t ) = g ′s (t + z ) = 0 this becomes
g r (t ) g s (t + z ) = g ′r (t ) g ′s (t + z ) + g r (t ) g s (t )
So the power spectra may be divided into two integrals as below
S gg (r ,s : ω ) =
∞
1 ∞ − jzω
1
− jτω
′
′
(
)
(
)
(
)
(
)
+
+
t
t
τ
d
τ
t
t
dτ
g
g
g
g
∫e
∫e
r
s
s
2π − ∞
2π r
−∞
The first integral is simply S gg ′ (r ,s : ω ) , so
S gg (r ,s : ω ) = S g ′g ′ (r ,s : ω ) +
∞
1
g r (t ) g s (t ) ∫ e − jτω dτ
2π
−∞
and noting the Fourier transform identity
δ (ω ) = ∫ e− jτωdτ
where δ (ω ) is the Dirac delta function with the property
δ (ω ) = 1 ω = 0
=0 ω ≠0
the second integral may be re-written
g r (t ) g s (t )
∞
2π
−∞
∫ exp (− jτω ) dτ =
δ (ω )
g r (t ) g s (t )
2π
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01-JUN-2001
8-5
allowing the cross-power spectral density of the forcing to be expressed as
S gg (r ,s : ω ) = S g ′g ′ (r ,s : ω ) +
δ (ω )
g r (t ) g s (t )
2π
Equation 8.3.2
The first and second terms in the above, may be further expanded in terms of wind data. The
first term S g ′g ′ (r ,s : ω ) is defined as
S g ′g ′ (r ,s : ω ) =
1 ∞
∫ exp (− jτω ) g ′r (t ) g ′s (t + τ ) dτ
2π − ∞
and by substitution from Equation 8.2.1
 UU UU U 1 (t ), U 1 (t + τ )
r
s
 Er Es
1 ∞
 UV UV 1
1
V (τ ,r ,t )
∫ exp (− jτω ) + E r E s V r (t ), V s (t + τ )
2π − ∞
 UW UW 1
1
+ E r E s W r (t ), W s (t + τ )
UU
 E UU
r E s S u ′u ′ (r , s : ω ) 


UV
V ( zs, t )  EUV
r E s S v ′v ′ (r , s : ω ) 
 EUW EUW S (r , s : ω ) 
s
w ′w ′
 r

S g ′g ′ (r ,s : ω ) = 4 V (τ ,r ,t )
= 4 V ( zr , t )



 dτ


where S u ′u ′(r , s : ω ) etc. are the cross-power spectral densities of the wind in the directions
along, laterally across and vertically across the mean wind respectively
The variation of the mean wind with height, may be parameterized by the following power
law identity (Reference 3.3)
V (z, t )
α
 z
= V (10, t )  
 10 
Equation 8.3.3
The cross-power spectra of wind S U ′U ′ (r ,s : ω ) may be approximated in terms of S U ′U ′ (ω ) ,
the power spectral density of the wind and a coherence function (References 3.3 and 3.11),
coh(r ,s : ω )
S UU (r ,s : ω ) = coh(r ,s : ω ) S UU (ω )
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8-6
01-JUN-2001
Giving
Program version 2.8
 ErUU ESUU Su 'u ' (ω ) 


2 z   z 
UV
S g ′g ′ (r ,s : ω ) = 4 V (10, t )  s   r  coh (r , s : ω ) + EUV
r E S S v 'v ' (ω ) 
 10   10 
+ UW UW S (ω )
w' w'
 Er ES

α
α
The forms of the coherence function used are presented in Appendix 1.0. The cross-power
spectral density function of the fluctuating forcing is then given by
α
α
2 zs   z 
S g ′g ′ (r , s : ω ) = 4 V (10, t )    r  .
 10   10 
UU
 EUU
r E s S u 'u ' (ω ) 

 UV UV
coh (r ,s : ω ) + E r E s S v 'v ' (ω ) 
 UW UW S (ω )
w' w '

+ E r E s
Equation 8.3.4
Where all the terms on the RHS may be readily evaluated from the structure geometry and
the basic wind data presented in Section 7.0.
As S v 'v ' (ω ) = 0 at ω = 0 , it is clear that S g ′g ′ (r ,s : ω ) = 0 at ω = 0 . Hence Equation 8.3.1
may be re-written as
S gg (r ,s : ω ) = S g ′g ′ (r , s : ω )
(t )
= gr
g s (t )
2π
ω ≠0
ω =0
Now substituting the values of g r (t ) and V ( z ,t ) from the Equations 8.2.2 and 8.3.3.
α
z 
g r (t ) = E r V (10, t )  r 
 10 
giving
S gg (r , s : ω ) = S g ′g ′ (r , s : ω )
=
EVV
r
EVV
s
2
4π
V (10, t )
ω ≠0
α
α
z  z 
 r  s
 10   10 
2
ω =0
Equation 8.3.5
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Program version 2.8
9.0
SESAM
01-JUN-2001
9-1
VORTEX SHEDDING INDUCED VIBRATIONS
In this section the procedures followed to determine the amplitudes of oscillation excited by
vortex shedding in steady winds are described. The determination of the first natural
frequency and its associated mode shape are described, followed by the estimation of the
amplitude of oscillation in a given steady wind. The stress levels at the member ends and
centre then follow from application of the stress concentration factors to the raw member
behaviour.
It is assumed that the vortex shedding effects are only of any significance if they induce
oscillations in the first mode of the brace. This is a reasonable assumption for tubular
structural steel members that are used in typical flare towers. This assumption would not be
valid if applications were to assess the vibration amplitude and stresses associated with long
slender tie rods or guy ropes, where a higher mode may be excited.
The mode and frequency are highly dependent on the conditions of member end fixity. In
general these are not known to any great degree of accuracy, so Framework allows the user to
investigate ranges of fixity. Low end fixity reduces the natural frequency and the member end
damage that occurs. In the extreme a pin-ended member suffers no end damage because the
pure bending deformation induced by vortex shedding produces no end moments, stresses or
damage. High end fixity produces a higher natural frequency and associated with it the
possibility of higher end moments. The amplitude of excitation may, however, be much
smaller because there may be no resonance between the frequency of shedding of vortices
and the natural frequency of the member. For a pin-ended member the damage at the member
centre will exceed that of the member ends, for fixed end members the damage needs to be
checked at the member centre and at both ends.
9.1
Determination of Mode Shape and Fundamental Frequency of a Brace
The brace may be considered as a beam element with end supports. The ends are assumed to
be restrained against lateral translation. The rotational supports may be different at each end
and are allowed to vary between pin-ended (i.e. no rotational resistance) and fully fixed (i.e.
fully restrained against rotation). The basic theory for the solution of this class of problem is
given in Timoshenko and Young (Reference 3.12).
The fundamental equation for the dynamic bending behaviour of a thin beam (i.e. one in
which shear deformations are negligible) is given by
EI
∂ 4w
∂ 2w
m
=
−
∂ x4
∂ t2
where E is the Young's modulus of the material; I is the second moment of area of the beam,
w is the transverse deflexion of the beam, m is its mass per unit length, x is the co-ordinate
along the beam's neutral axis and t is time.
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This fourth-order differential equation has a general solution of the form
w = ( A cos kx + B sin kx + C cosh kx + D sinh kx) cos(ω t + φ )
where A, B, C and D are constants that will be determined from the boundary conditions
applied at the beam ends. ω is the natural frequency of the response (in radians per second),
φ is the phase lag and k is defined by the relationship
k=
4
mω 2
EI
For a brace member in a flare tower the displacement boundary conditions applied at both
ends are that the deflexions, w, are zero. For a beam of length L this gives two equations
A+C = 0
A cos kL + B sin kL + C cosh kL + D sinh kL = 0
In addition to the displacement boundary conditions it is necessary to apply boundary
conditions to the rotations. For simple supports the conditions are that there is no curvature at
either beam end. Equating the second derivative of the displacement to zero gives
− A+C = 0
− A cos kL − B sin kL + C cosh kL + D sinh kL = 0
For fixed ends the corresponding boundary conditions, which are that the slopes are zero at
either beam end, leads to
B+D = 0
− A sin kL + B cos kL + C sinh kL + D cosh kL = 0
For more general support conditions, where there are dissimilar rotational springs at either
end of the beam, the following relationships apply
K 0 {B + D} = EIk{− A + C}
K L {− A sin kL + B cos kL + C sinh kL + D cosh kL} =
EIk{− A cos kL − B sin kL + C cosh kL + D sinh kL}
These equations are derived from the ratios of the end moments to end rotations, which are,
by definition equal to the spring stiffnesses for linear elastic spring supports. The rotational
spring stiffnesses at x = 0 and x = L are given by K0 and KL respectively. The relationships for
simply supported or fully fixed ends may be derived as special cases of the last pair of
equations, with zero or infinite spring stiffnesses substituted as appropriate.
From the set of four equations, i.e. two displacements and two rotational boundary conditions
selected as appropriate, it is possible to solve for any three of the unknowns in terms of the
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Program version 2.8
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01-JUN-2001
9-3
fourth. This solution gives the mode shape, but not its amplitude, and the corresponding
frequency. Strictly speaking the process will give an infinite number of solutions for the
pairs of mode shapes and frequencies, as there is an infinite number of choices of kL that will
satisfy the relationships. As noted above, the fundamental frequency, corresponding to the
first mode is the only one likely to be of any significance for structural applications.
The solution of this set of four equations is carried out iteratively in Framework. The starting
approximation to the value of kL is made using a simplified assumption, which is close to the
value corresponding to the fundamental mode. This ensures that convergence is to the lowest
frequency and not to a spurious higher mode.
The iterative scheme is based upon the Newton-Raphson approach. The determinant of the
matrix of the four simultaneous equations is evaluated together with the determinant of the
derivative of the matrix with respect to the variable kL. The objective is to produce a zero
value for the matrix. If FN is an approximation to the numerical value of determinant of the
four homogeneous equations then
FN +1 = FN {1 −
∂ FN
}
∂ (kL)
is likely to be a better approximation. FN and its derivative are evaluated at the current (trial)
value of kL. This process is repeated until the values of FN and FN+1 differ by less than a
suitably small relative value. Currently this is set to 10-6 in Framework. The iteration limit is
set to 100.
As an example consider the set of four equations for a simply supported beam. In this case
the solution may be derived by inspection. The conditions are that
A=0
B sin kL = 0
C=0
D sinh kL = 0
The only solution for kL that gives deflections that are other than zero at all times is
kL = nπ
D=0
Substitution of integer values of n in the above relation and referring to the equation for k
above gives the relationship for the natural frequency corresponding to mode n. This equation
is valid for all positive integer values of n.
Similar procedures may be followed for other boundary condition sets. The resulting
relationships for kL are more complex and generally involve the use of iterative solution
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9-4
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Program version 2.8
techniques. Reference. 3.12 gives details of many more examples, including free ended
beams and cantilevers.
9.2
Brace Oscillation Amplitude Calculation
The frequency at which vortices are shed from opposite sides of the brace member is
dependent on the Reynolds' number of the fluid flow. If the vortex shedding frequency is
sufficiently removed from the natural frequency of transverse oscillations of the brace there
will not be any resonance and the amplitude of any oscillations will be negligible. If the ratio
of the two frequencies is close to unity, the amplitude of the oscillations will be significant,
that is high stress levels and hence structural fatigue will be caused. The critical velocity of
the flow is defined as that which will cause resonant vortex shedding. Wind velocities in the
range of 60 to 140 per cent of the critical velocity will excite oscillations that cause damage.
Velocities outside this range do not cause appreciable damage and their effects may be
ignored.
Depending on the member end fixities the damage may be higher at the brace ends or at the
member centre.
For each wind attack direction the wind is resolved into components normal and tangential to
each brace. The velocity used is that computed at the member centroid, it being assumed that
any variation of velocity, with height, along the member length will be relatively
unimportant. The normal component of the velocity is used to calculate the Reynolds' number
in conjunction with the outer diameter of the brace. Note that the value used is the member
total diameter, which will include any non-structural cladding or insulation material.
From the Reynolds' number the vortex shedding frequency may be estimated.
This frequency is then compared with the natural frequency of the member itself, again
taking into account the effects of any non-structural mass due to cladding.
The critical velocity is defined as
Vcrit = ω 0 Dinc / St
where ω0 is the natural frequency of the brace member, Dinc is the diameter of the member
including any coating material and St is the Strouhal number.
For each brace member the wind velocities that occur throughout the year are resolved into
normal components. This is done by decomposing the statistical data on wind speeds,
directions and the proportion of the year that such winds occur, into discrete ranges at
constant speeds. The effect of each wind range and its associated velocity is then considered
in isolation. The total damage induced by each wind speed range from each direction is then
summed to give the total structural damage. Note that the effect of wind from opposing
directions will be identical so use of the composite wind data, as described in Section 7.0,
will reduce the volume of data required.
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9-5
If the normal component of a selected range of wind speeds lies in the range
0.6 Vcrit < Vnormal < 1.4 Vcrit
then the contribution to the member end and centre damage is accumulated. If the normal
velocity is outside this range there is no damage contribution. For each wind speed range the
following process is used. Reference 3.15 contains background information.
The stability coefficient is defined as
2
K s = 4πε me /( ρ Dinc
)
where ε is the structural damping ratio, me is the effective mass per unit length of the brace
and ρ is the density of air.
The structural damping ratio is assumed to be defined, as a percentage, by
ε = 0.14 + 0.36e −0.0855 L / Dext
where Dext is the external diameter of the structural member, i.e. excluding any insulation or
cladding material.
The response parameter, SG , is given by
SG = 2π ( St ) 2 K S
The structural lift coefficient, CL , is dependent on the Reynolds' number of the flow.
Re ≤ 2.0x104
2.0 x104 < Re ≤ 4.71x105 / Rtran
Re > 4.71x105 / Rtran
CL = 0.42 − 0.33e( −2.0 x10
4
Re 2 / 1012 )
CL = 0.09 + 0.33e −3200(Re x10
6
/ Rtran )10
CL = 0.15
where Rtran is the transition ratio.
The Van der Pol coefficient for the brace is given by
CVDP
L 2
∫0 w dx
= L 4
∫0 w dx
where w(x) is the mode shape determined from the frequency equation described above.
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9-6
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01-JUN-2001
Program version 2.8
The amplitude of the response at the resonant vortex shedding frequency is then given by
Ar = (3.82 DincCVDPC L ) / (1 + 0.19 SG / C L ) 3.35
At wind velocities away from the critical the non-resonant amplitude for a broad band
response is given by the expression
AB = e ( − (1−VR )
2
5
/ ( 0.02 + 0.1 ( 0.4 − 0.32 e −25000 turb ) ) )
Ar
where the velocity ratio is given by
VR = V / Vcrit
and turb is the turbulence intensity of the flow.
For narrow-band response the amplitude is dependent on the value of the stability factor, KS.
2
K s < 4π
AN = e − 30 (1−VR ) Ar
4π ≤ K S ≤ 12π or 0.725 < VR < 1.0
AN = e − 30 (1−VR )
2
AN = e − 30 (1−VR )
2
12π < K S
( K S / 4π )1.8
1.8
3.0
Ar
Ar
Note that in all cases the amplitude of the response is given as a factor of the resonant
amplitude. The factor defines a bell shaped curve between the lower and upper limits of the
velocity ratio, with maximum amplitude at the resonant frequency. A broad band response
gives a flatter bell shaped curve than the pronounced peak of the narrow-band response
curve.
9.3
Member End Damage Calculation
The member end damage calculation closely mirrors that used in the buffeting damage
calculations. From the calculated forcing frequency, as given by the vortex shedding
characteristics of the brace, and the time per year that the wind blows, the total number of
oscillations of the brace may be determined. The amplitude of the vibrations is determined
using the approach outlined in Section 9.2 above. From the displacement amplitude and the
mode shape the beam section properties are used to calculate the member stresses at the two
ends.
The raw member stresses are then factored by the stress concentration factors (SCFs) to give
the local hot spot stresses. The evaluation of the SCFs is described in Section 11.0. Note that
the stress range, which is twice the stress amplitude, is needed for fatigue damage
calculations. The damage is evaluated using the Miner's law approach in an analogous
manner to the buffeting damage.
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Program version 2.8
9.4
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01-JUN-2001
9-7
Member Centre Damage Calculation
The member centre damage is calculated in a similar manner to the member end damage. The
SCF for the member centre is applied as a blanket value to the entire structure. This value is
supplied from the input data and there are no calculations involved to derive the value. This
user specified SCF should represent the typical value that would be associated with a singlesided girth closure weld. It will depend on the quality control of the welding process, the out
of roundness and the mismatch that are permissible in the fabricated tubular structure.
The approach used is conservative. The damage is evaluated at the section on the brace's
length that has the maximum curvature, and hence bending moment. The member's displaced
shape is examined at 100 equally spaced positions along its length to determine the greatest
curvature. It is unlikely, although possible, that the position of maximum moment would
coincide with a closure weld.
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Program version 2.8
10.0
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01-JUN-2001
10-1
CONDENSATION OF FORCES
The transfer function linking the cross-power spectral density function of fluctuating forcing
to the power spectral density function of the response depends upon knowledge of the
structure's modal frequencies and shapes.
The extraction of a full set of eigenvalues and mode shapes for a structure with a large
number of degrees of freedom is computationally expensive. Consequently, the technique
known as static condensation, or Guyan reduction, is used (Reference 3.5). This reduces the
analysis to a more manageable size by retaining a set of master degrees of freedom, which are
chosen to characterize the structure's kinetic energy. The degrees of freedom are reduced out
are known as the slave freedoms. The algebra of the reduction to the master degrees of
freedom of the forcing vector is given in Appendix 3.0
The Framework wind fatigue keeps all free translational degrres of freedom as master
freedoms and the user can not select the master freedoms.
FRAMEWORK
Program version 2.8
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01-JUN-2001
11.0
STRESS CONCENTRATION FACTORS
11.1
An Overview
11-1
Stress concentrations occur in the welded tubular joints. For fatigue calculations it is
therefore important to take them into account. To evaluate the stress concentrations or “hot
spot stresses” (HSSs), empirically derived stress concentration factors (SCFs) based on joint
geometry are used.
Using these the distribution of the HSSs around the welds is found, under loading produced
by a mean wind state (static loading). When the tower is subject to buffeting, i.e., dynamic
eddy loading, the maximum HSS at a joint, for each mode of response, is assumed to occur in
the same place as for the static loading. This is a reasonable assumption, in that buffeting
fatigue effects on flare towers are normally dominated by the cantilever modes of response.
These strongly resemble the tower's static response.
At each joint, connected members may be either braces or chords. The chord is taken as the
pair of elements of greatest diameter which are collinear. (If there is more than one pair of
collinear elements of the same maximum diameter, the chord is assumed to be the pair with
the greatest thickness.) All other members are assumed to be braces. HSSs are found for each
brace/chord intersection separately, for both chordside and braceside of the weld.
If collinear elements are not found, the joint classification of Framework is tried. A chord
may be identified in Framework without the presence of collinear elements at the joint.
11.2
Components of the HSS
For each brace the axial, in-plane bending (IPB) and out-of-plane bending (OPB) stresses
( σ AX , σ IPB and σ OPB respectively) are all deemed to contribute to both the chordside and
braceside HSS distribution around the weld. Chordside and braceside IPB and OPB
components are approximated by:
HSS IPB (φ ) = SCFIPBσ IPB sin(φ )
HSSOPB (φ ) = SCFOPBσ OPB cos(φ )
where SCF IPB and SCF OPB are the IPB (crown) and OPB (saddle) SCFs respectively and φ
is the angle around the weld, measured from the saddle point (see Figure 11.1). The axial
components are approximated by:
 ( SCFAS + SCFAC ) ( SCFAS − SCFAC )

+
cos(2φ )σ AX
HSS A (φ ) = 
2
2


where SCFAS and SCFAC are the axial saddle and crown SCFs respectively.
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11-2
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01-JUN-2001
FIGURE 11.1
Program version 2.8
FRAMEWORK
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Program version 2.8
11.3
01-JUN-2001
11-3
The SCF Schemes
The wind fatigue module of Framework has three schemes by which the HSSs may be
evaluated, the “I”, “E” and "O" schemes. They use different SCF formulae, and combine the
component HSSs in slightly different manners. Details of the equations used to derive the
SCFs may be found in Appendix 1.
These schemes classify joints as either:
•
•
•
•
•
T joints
K joints
KT joints
X joints
or non-standard joints (see Section 11.4)
The "I" and "O" schemes will not recognise an X-joint, which is classified as a non-standard
joint for these schemes. Under the "E" scheme, X-joints may be included.
The “I” scheme
•
treats a KT joint as a K joint plus a T joint. The K joint makes up the outer
braces, while the T joint makes up the middle brace
•
sets saddle SCFs = 0 (i.e. SCF OPB = SCF AS = 0 )
•
evaluates the HSS distribution around the weld as
B HSS
C HSS
(φ ) =
(φ ) =
B HSS IPB
C HSS IPB
(φ ) +
(φ ) +
[ B HSS OPB (φ )]
(≡ 0)
[ C HSS OPB (φ )] +
(≡ 0)
+
B HSS A
C HSS A
(φ )
(φ ) + σ *CHORD
where the new B subscript denotes braceside of the weld and the C subscript denotes
chordside. σ *CHORD is the chord stress, with sign. C HSS (φ ) is the locally enhanced stress.
The “O” scheme
•
evaluates the HSS distribution around the weld, both chordside and braceside as
HSS (φ ) = HSS IPB (φ ) + HSS OPB (φ ) + HSS A (φ )
The “E" scheme is described in Appendix 4.0.
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11.4
Program version 2.8
Non-standard Joints
The wind fatigue module of Framework recognizes joints for which
•
there is a chord and more than three braces
•
the gap between braces is either zero or negative (Figure 11.1 shows a positive
gap). This is called an OVERLAPPING joint.
For both of these joint types each brace will be treated as a T joint, and then take the
maximum HSS over all joints.
To consider overlapping joints it is recommended that the facilities available in Framework
basic are applied.
11.5
Efthymiou Scheme
The Efthymiou equations which allows X-joints as well as K, KT, and T are made available
in Framework. This is the "E" scheme. Details are given in Appendix 4.0. Note that there
are some differences in the implementation of the Efthymiou equations and treatment of valid
ranges of geometric parameters of the equations in Framework basic and the wind fatigue
module. In particular overlapping braces of K joints are not handled by the wind fatigue
application implementation.
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12-1
12.0
THE HOT SPOT STRESS TRANSFER FUNCTION FROM POINT FORCING
12.1
The Dynamic Equation
A finite element model of a structure is characterized by its mass and stiffness matrices M
and K . The structural damping may be modelled as an imaginary component of the stiffness
matrix. The forced response of a structure to a force vector G( ω ) at frequency ω, where ω is
in radians/sec is given by
[(1 + jγ )K − ω 2 M ]U exp( jwt ) = G(ω )exp( jwt )
where U is the displacement vector
When Guyan reduction is employed to give reduced mass and stiffness matrices M M and
K M with an equivalent force vector G M applied at the master degrees of freedom to produce
a displacement vector U M at these degrees of freedom; and the exp( jwt ) term is dropped
then
[(1 + jγ ) K M − ω 2 M M ]U M (ω ) = G M (ω )
12.2
Equation 12.1
Results from the Eigenvalue Problem
The homogeneous part of equation 12.1 may be written as
 −1

ω2
M M KM −
 U M (ω ) = 0
I


+
(
1
j
γ
)


Equation 12.2.1
This is the classic eigenvalue problem for the values
λ S = ω 2S / (1 + jγ )
Let φ$ be a matrix of eigenvectors φ S . This set of vectors is not unique because any linear
multiple of an eigenvector is also an eigenvector. If there are coincident eigenvalues the
situation is more complicated. Any linear combination of the eigenvectors corresponding to
the coincident modes is also an eigenvector. To establish a unique matrix φ$ of vectors φ S , φ$
is constrained to follow the relationship
T
φˆ M M φˆ = I
Equation 12.2.2a
FRAMEWORK
SESAM
12-2
01-JUN-2001
Program version 2.8
This is called "normalizing on the (reduced) mass matrix".
Additionally the eigenvectors are mass orthogonal, that is,
φrT M M φ s = 0
for r ≠ s
Equation 12.2.2b
Equation12.2.1 holds for each eigensolution, and may therefore be written as a set of
simultaneous eigensolutions.
K φˆ − diag (λ ) M φˆ = 0
M
S
M
S
T
Premultiplying by φ$ gives
φˆ K M φˆ − diag (λ S )φˆ M M φˆ = 0
T
T
1424
3
S
Term1
substituting into term 1 from Equation 12.2.2 gives
φˆ K Mφˆ = diag (λ S )
T
S
T
Left multiplication by  φˆ 
 
−1
()
−1
followed by right multiplication by φˆ gives
−1
−1
T
K M =  φˆ  diag (λs )φˆ
 
S
Equation 12.2.3a
Equation 12.2.2a may similarly manipulated to give
T
M M =  φˆ 
 
12.3
−1
(I )φˆ
−1
Equation 12.2.3b
The Structure Displacement Vector from a Point Force
Values for M M and K M in terms of eigensolutions (Equation 12.2.3) may be substituted in
Equation 12.1 to give
−1
−1

 φˆT  diag ( )φˆ −1 − 2  φˆT  Iφˆ −1 U = G
(
)
γ
1
j
+
 
λS
ω  
 M

M
 
 


S
gathering the terms in the square brackets together gives
FRAMEWORK
SESAM
Program version 2.8
01-JUN-2001
12-3
 T  −1
−1
2
 φˆ  diag ( (1 + jλ ) λ S − ω )φˆ  U M = G M
S
 

 T


1
 and φ$ gives
Successively left multiplying by  φˆ  , diag 
2
(
)
1
j
+
−
γ
λ
ω
S

 
S 
 ˆT

1
φ G M
U M = φˆ diag 
2
(
)
1
+
j
−
γ
λ
ω
S

S 


1

Q = φˆ diag 
2
(
)
1
+
−
j
γ
λ
ω
S

S 
S = φˆ G M (ω )
T
[
Q rs = φˆ rs / (1 + jγ )λ S − ω 2
]
S s = ∑ φˆ KS G M K
K
where the r and s subscripts refer to the rth row and sth column of their respective matrices.
Hence Equation 12.3.1 may be expressed as
U M = QS

φˆ rs
 ˆ

U M r = ∑ Q rs S S = ∑
2  ∑ φ KS G M K  
S

 S (1 + jγ )λ S − ω  K
and reversing the order of summation


φˆ rsφˆ KS
U M r = ∑ ∑
2 GMk
K 
 S (1 + jγ )λ S − ω 
This gives the displacement at the rth degree of freedom as a summation of the displacements
from a force at each degree of freedom k. Hence if G M (k ) is the displacement at master
freedom r from an equivalent force G M at master freedom s then
k
U M r (k ) = ∑
S
φˆ KSφˆ rsG M k
(1 + jγ )λ S − ω 2
FRAMEWORK
SESAM
12-4
01-JUN-2001
Program version 2.8
which may be extended to include the displacement vector for all master freedoms U M (k )
resulting from an equivalent force G M at master freedom k
k
[
[
]
]
[
]
 ∑ φˆ KSφˆ1S / (1 + jγ ) λ S − ω 2 

 S
 ∑ φˆ φˆ / (1 + jγ ) λ − ω 2 
S

 S KS 2 S


.
U Mk = 
 GMk
.




.
 ˆ ˆ
2 
 ∑ φ KSφ MS / (1 + jγ ) λ S − ω 

S
this may then be re-expressed in terms of the normalized eigenvectors φ$ S
UM = ∑
S
12.4
φˆ KS
⋅ φˆ
(1 + jγ )λ S − ω 2 SG M k
Equation 12.3.2
Hot Spot Stresses from a Point Force
The displacement vector of the structure's master degrees of freedom, U M (k ) , from an
equivalent force at master freedom k may be directly translated into the member stresses of
member N by substituting the member stresses for the normalized eigenvectors into Equation
12.3.2.


φˆ KS
σ ˆ 
N σ (U M (k ) ) =  ∑
2 N φ S  GMk

 S (1 + jγ ) λ S − ω
( )
Equation 12.4.1a
The stresses given by the above relationship are the stresses derived from the eigenvector
displacements factored by the appropriate stimulated amplitudes. To find the HSS at a joint
requires application of the appropriate SCFs to these stresses. This then allows HSS pk (ω ) ,
the hot spot stress at joint p from point forcing at degree of freedom k, to be written in terms
of a function and the value of the point forcing.
HSS pk (ω ) = H pk (ω )G M k (w)
Equation 12.4.1b
FRAMEWORK
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Program version 2.8
01-JUN-2001
13-1
13.0
HOT SPOT STRESS POWER SPECTRUM
13.1
The Hot Spot Stress Power Spectra
Section 12.0 established a transfer function between a forcing point load and the hot spot
stress at a selected joint. In this section the resultant hot spot stress power spectrum at a joint
from the ensemble of forcing point loads will be described.
To derive the power spectra the hot spot stress given by Equation 12.4.1 needs to be reexpressed in the time domain. This is achieved by Fourier transform inversion using the
convolution theorem:
(
)
hss pk (t ) = ∫ h pk (t1) g k t − t dt1
∞
M
−∞
1
Equation 13.1.1
where hss pk (t ) is the transform of HSS pk (ω ) and
M
gk
(t ) is the transform of G M (ω ) .
k
M
gk
(t ) is
the time series of the equivalent forcing at one master dof.
By summing this equation over all the forcings at each master dof, the total hot spot stress
∑ hss pk (t ) at any joint p from the ensemble of equivalent point forces in the time domain
k
may be found. The auto-correlation function of the joint's hot spot stress is then given by
C hss p hss p (τ ) = ∑ ∑ hss pr (t ) hss ps(t + τ )
Equation 13.1.1a
r s
From Equation 13.1.1, and by taking the summations outside the integrals
1
C hss p hss p (τ ) = ∑ ∑ ∫ ∫ h pr (ε ) h ps (ε 1) g r(t − ε ) g s(t + τ − ε 1) dεdε
∞ ∞
M
M
r s −∞ −∞
and noting that the time averaging operator ⋅ only applies to functions of time
C hss p hss p
(τ ) = ∑ ∑
∫ ∫ h pr (ε ) h ps (ε
∞ ∞
r s −∞ −∞
) Mg r(t − ε ) Mg (t + t − ε 1 )
1
the power spectra
S hss p hss p (ω ) =
may be expressed as
1 ∞ − jwτ
C hss p hss p (τ ) dτ
∫e
2π − ∞
s
dεdε 1
FRAMEWORK
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13-2
01-JUN-2001
S hss p hss p (ω ) =
Program version 2.8
∞ ∞
1 ∞ − jwτ 
1
∫e
∑ ∑ ∫ ∫ h pr (ε ) h ps (ε )
2π − ∞
r
s
−∞ −∞

M
gr
(t − ε ) Mg
(t + τ − ε )
1
s

dε dε 1  dτ

By exchanging the order of integration and summation this becomes
∞ ∞
∞
1  1
− jωτ
S hss p hss p (ω ) = ∑ ∑ ∫ ∫ h pr (ε ) h ps (ε )
∫e
2
π
r
s −∞ −∞
 −∞
M
gr
(t − ε ) Mg
(t + τ − ε )

dτ  dε dε 1

1
s
The substitution τ 1 = τ − ε 1 + ε then gives
dτ 1 = dτ
(
e − jωτ = e − jω τ
1+ 1− ε
ε
) = e − jωτ 1 e − jωε 1 e jωε
t +τ + ε1 = t +τ − ε
so that
( )
∞ ∞
∞
1  1
− jωτ 1 − jωε 1 jωε
S hss p hss p (ω ) = ∑ ∑ ∫ ∫ h pr (ε ) h ps ε 
e
e
∫ e
r
s −∞ −∞
 2π − ∞
M
gr
(t − ε ) Mg r (t + τ 1 − ε ) dτ 1  dε dε 1

and exchanging the order of integration(s)
∞
S hss p hss p (ω ) = ΣΣ ∫ e
jωε
−∞
∞
h pr (ε ) ∫ e
− jωε
1
−∞
 ∞ ε − jωτ 1
h ps (ε ) ∫
−∞ 2π
1
M
gr
(t − ε )Mg
(t +τ 1 −ε ) dτ  dεdε
1
s
1

For a stationary random process the time average with respect to t − ε should be the same as a
time average with respect to t, so that
M
gr
(t − ε ) Mg
s
(t +τ
=
M
gr
(t ) Mg
s
(t +τ )
1 ∞ − jωτ 1
∫ e
2π − ∞
M
gr
(t ) Mg
s
(t +τ )
1
−ε
)
1
and by definition
MM
(r , s : ω ) =
S gg
1
dτ 1
Equation 13.1.2
where S Mg Mg (r , s : ω ) is the cross-power spectral density function of the reduced force.
The hot spot stress power spectra may then be written
∞
∞
( )
1
1
S hss p hss p (ω ) = ∑ ∑ ∫ e jωε h (prε ) ∫ e − jωε h ps ε S Mg Mg (r , s : ω ) dεdε
r s −∞
−∞
1
FRAMEWORK
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Program version 2.8
01-JUN-2001
13-3
Noting that S Mg Mg (r , s : ω ) is not a function of ε or ε1 so that
S hss p hss p (ω ) = ∑ ∑ S g
r s
MM
g
∞
∞
−∞
−∞
(r , s : ω ) ∫ e jωε h pr ( ε ) dε
∫e
− jωε 1
( )
1
1
h ps ε dε
The following two identities may now be used
H ps (ω ) =
( )
1 ∞ − jωε 1
1
1
h ps ε dε
∫e
2π − ∞
because H ps (ω ) is the Fourier transform of h ps (t ) and
H *pr
(ω ) =
1 ∞ jωε
∫ e h pr ( ε ) dε
2π − ∞
where H *pr (ω ) is the complex conjugate of H pr (ω ) . This second identity may be easily
verified by expressing the exponent as cosine plus imaginary sine terms. Hence Equation
13.1.2 may be re-written
ω)
= 4π 2 ∑ ∑ H ps (ω ) H *pr (ω ) S Mg Mg (r , s : ω )
S (hss
p hss p
Equation 13.1.3
r s
where S Mg Mg (r , s : ω ) was evaluated in Equation 8.3.5.
13.2
Integrating the Hot Spot Stress Power Spectra
The hot spot stress power spectrum shown in equation 13.1.3 is frequency dependent in the
cross power spectra of the forcing and in the functions H pr (ω ), H *pr (ω ) .
At frequencies far below the modal frequencies the structure behaves in a quasi-static manner
and H pr (ω ) may be approximated by H pr (0) . If corrections are made for the variation in
drag coefficient with wind speed, the integral of the hot spot stress power spectrum may then
be evaluated by simple numerical integration.
For frequencies near the modal frequency, the variation of the forcing cross power spectra
with frequency is small compared to the variation in H pr (ω ) . Hence near a modal frequency
1
λ q2 the integral may be expressed as
FRAMEWORK
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13-4
01-JUN-2001
1
1
λ q2 + ∆2
1
λ q2
∫
Program version 2.8
λ q2 + ∆2
S hss p hss p (ω ) dω = 4π 2 ∑ ∑ S gg (r , s : ω ) ∫
r s
− ∆1
1
λ q2
H ps (ω ) H *pr (ω )dω
Equation 13.2.1
− ∆1
Neglecting the affects of the SCFs, and substituting Equation 12.4.1a we get
1
λq2 + ∆ 2
∫
1
λq2 − ∆1
( )
1
H ps (ω )
H *pr
( )
σ
σ
φˆsm ⋅ p φˆ n
φˆrn ⋅ p φˆm

⋅
(ω ) dω = ∫  ∑ ∑
2
2
1
m n (1 + jγ ) λ m − ω (1 + jγ ) λ n − ω
2

λq − ∆1
λq2 + ∆ 2

 dω


Taking the summation out of the integration
1
λq2 + ∆ 2
∫
1
λq2 − ∆1
( )
( )
σ
σ
H ps (ω ) H *pr (ω ) dω = ∑ ∑ φˆsm ⋅ p φˆm φˆr ⋅ p φˆn
m n
1
λq2 + ∆ 2
∫
1
λq2 − ∆1
dω
[(1 + jγ )λ m − ω 2] [(1 − jγ )λ n − ω 2]
Equation 13.2.2
Now the integral within the summation on the RHS may be re-expressed as
1
1
λ q2 + ∆
1
∫
λ q2 − ∆
dω
[(1 + jγ )λ m − ω ] [(1 − jγ )λ n − ω ]
2
2
λ q2 + ∆
= ∫
1
λ q2 − ∆
(
dω
C 2m − ω 2
)(d 2n − ω 2)
where
1
2
(
)
1
2
(
)
12
2

1+ 1+γ 2

=
Cm

2


1+ 1 + γ 2

=
dn

2

(
)
(
)

2
 λ + j  1+ γ
 m

2


1
2
1

2
 λ 12 − j  1 + γ

 n
2


1

2 −1
1
2
1
 λ2
 m

1

2 −1
1
2
 λ 12
 n

evaluating this integral
1
λ q2 + ∆2

 ω − Cm 
ω − d n 
 log e 
 
 log e 
+
ω
ω
+
1
dω
d
C
m
n





= 2
−
∫
2
2
2
2
2

1
(C m − ω )(d n − ω ) C m − d n  C m
dn
λ q2 − ∆1
 1

 λq2 − ∆1

1
λ q2 + ∆1
and substituting back into Equation 13.2.2 gives
FRAMEWORK
Program version 2.8
SESAM
01-JUN-2001
13-5
1
λq2 + ∆ 2

ω − dn 
 ω − Cm 




log
log
σ

e
e
ˆ


φˆsm ⋅ p φ m φˆrn ⋅ p (φ n ) 
ω + Cm 
ω + d n  


*
−
∫ H ps (ω ) H pr (ω ) dω = ∑ ∑
2
2


1
m n
C
C
dn
m − dn
m
λq2 − ∆1

 1

λq2 −∆1
1
λq2 + ∆ 2
σ
( )
back-substitution into Equation 13.2.1 with SCFs added then gives the integral of the hot spot
stress power spectrum.
FRAMEWORK
Program version 2.8
SESAM
01-JUN-2001
14.0
CALCULATION OF FATIGUE LIFE
14.1
Assumptions
14-1
The following assumptions are made:
14.2
•
The hot spot stress power spectrum is characterized by a quasi-static response
with several separated sharp peaks at the structure resonances. The stress
spectrum is discretized into a finite number of frequency bands covering the submodal and modal peaks
•
The integral under these peaks (or frequency bands) is the variance of the stress
amplitude at the frequency associated with the peaks
•
The stress amplitude within each frequency band has a Rayleigh distribution.
This is true for narrow band processes (Reference 3.8). The sub-modal section is
split into three portions, each of which is treated as having a Rayleigh distribution
•
For each frequency band fatigue is directly related to the number of cycles
experienced in each stress range through the Palmgren-Miner relationship (Reference
3.6)
•
The number of cycles to failure at any stress range (amplitude) may be found
from standard SN curves (Reference 3.7)
Damage Evaluation
The Palmgren-Miner relationship (Reference 3.6), for the damage sustained due to stress
cycles, may be written in terms of the stress amplitude
D ( A) = n c ( A) / N c ( A)
Where D( A) is the cumulative damage at stress amplitude A, n c ( A) is the number of cycles
experienced at the stress amplitude A, and N c ( A) is the number of cycles to cause failure at
stress amplitude A.
The damage evaluated over all stress ranges is then obtained by integrating over all the
possible stress amplitudes, i.e.
∞
( A) dA
D = ∫ nc
− ∞ N c ( A)
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14-2
01-JUN-2001
Program version 2.8
With an SN curve of the following form (θ is to allow for thickness corrections, where
appropriate)
−m
N c ( A)= K 2 [2 A] θ
= K L [2 A]−(m + 2 )θ
m
4
m+2
4
for N c ( A) ≤ 107
A = A0 at N c ( A) = 107
for N c ( A) > 107
the annual damage for any frequency band is found in Appendix 8. It is
 3( m+2 ) −1 − (m + 2 ) 2 m2+ 2  m + 4   m + 4 Ao  
, 2  +
Γ
Γp
2 2 K L θ 4 σ hss
 2   2 2σ hss  

D = N ω ⋅ fn 

m
2 32m K −1θ − m4 2 2 Γ m + 2  1 − Γ  m + 2 , Ao  
σ hss
p
2
2 

 2  
 2 2σ hss  

[
]
[ ]
where
∞
Γ(a ) = ∫ t a −1 e−t dt
−∞
Γ p (a, x ) =
1 x a −1 −t
∫ t e dt
Γ(a ) 0
2
is the variance of the stress amplitude, N w is the number of seconds for
and where σ hss
which the parameterized wind-state forcing the tower is deemed to last within each year; and
fn is the frequency associated with the peak in the hot spot stress power spectrum.
The total annual damage is therefore the sum of the damages over all the frequency bands
and all the wind states. The estimated life is then the reciprocal of the total annual damage.
See Appendix 8 for supporting mathematical background.
FRAMEWORK
Program version 2.8
SESAM
01-JUN-2001
APPENDIX 1
Notation Used Within the Text
Appendix 1 -1
FRAMEWORK
SESAM
Appendix 1 -2
01-JUN-2001
Program version 2.8
SYMBOLS AND SUBSCRIPTS LIST
Symbol/Subscript
(⋅)
Description
time average of (.)
(⋅)1
time varying component of (.)
{ (⋅) }
a set of subscripted items (.)
(ˆ⋅ )
normalised form of (.)
(~⋅ )
first order approximation to (.)
(~~⋅ )
second order approximation to (.)
(⋅)
matrix (⋅)
diag(⋅)s
diagonal matrix with elements (⋅)s
s
(⋅)ps
( p × q ) sub-matrix of matrix (⋅)
(⋅)rs
matrix element of matrix (⋅) at row r column s (integer subscripts)
 (⋅) 
 pq  rs
matrix element of sub-matrix (⋅)ps at row r column s (integer
subscripts)
(⋅)r
vector forming column r of matrix (⋅)
(⋅) f
sub-vector of vector (⋅) with p elements
(⋅)s
component s to vector (⋅) , if (.) is a vector (integer subscripts)
(⋅)
generalised multi-dimensional column vector/matrix (.)
(⋅)
vector in 3-D space
(⋅)T
transposed matrix
FRAMEWORK
Program version 2.8
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01-JUN-2001
(⋅)T
transposed vector
(⋅)−1
inverted matrix
(⋅)xx
N
(⋅ )
Appendix 1 -3
see FUNCTION LIST - SPECTRAL (below) where subscripts
are not integers
identifies a tower member N to which (.) applies
FRAMEWORK
Appendix 1 -4
SESAM
01-JUN-2001
Program version 2.8
MATRIX LIST
The following matrix names are used within the text. Column vectors from these matrices are
not found within the VECTOR LIST, functions which are matrix elements are not described
elsewhere.
Matrix Name
Description
K
Stiffness matrix
M
Mass matrix
KM
MM
Reduced stiffness matrix
Reduced mass matrix
φ
Matrix of eigenvectors
I
Identity matrix
H
Matrix of hot spot stress transfer functions, element (r,s) is the
transfer function between a point force at dof k and joint r
HSS
Matrix of hot spot stresses element (r,s) is the hot spot stress at
joint r from a point force at dof k
A, B & Q
Dummy matrices used to clarify mathematical relationships
FRAMEWORK
Program version 2.8
SESAM
01-JUN-2001
VECTOR LIST
The following vector names are used within the text.
Vector Name
Description
U
Displacement vector for dofs at frequency ω
G
Forcing function vector for dofs at frequency ω
g
Fourier transform of time series for G
UM
Reduced displacement vector for master dofs at frequency
Gm
Reduced force vector for master dofs at frequency
M
g
(t )
Fourier transform of time series
k
V1
Wind velocity
S,T
Dummy vectors used to clarify mathematical relationships
Appendix 1 -5
FRAMEWORK
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Appendix 1 -6
01-JUN-2001
Program version 2.8
FUNCTION LIST - SPECTRAL
Spectral Function
Description
S yy (ω )
Double sided power spectral density function of function y
S yy (ω )
Single sided power spectral density function of function y
S yy (r , s : ω )
Double side cross-power spectral density function of function y
between joint/degree of freedom locations r and s
C yy (τ )
Auto-correlation function of function y
C yy (r , s : τ )
Cross-correlation function of function between joint/degree of
freedom locations r and s
coh (r , s : ω )
Coherence function between degree of freedom r and degree of
freedom s
s
FUNCTION LIST - MEMBER MAPPINGS
These are mappings which related to members rather than degrees of freedom or joints
Member Mapping
N
N
Description
g (t )
a function: the wind load on a member
σ (U )
a mapping between a displacement vector U and the set of
stresses in a member
FRAMEWORK
Program version 2.8
SESAM
01-JUN-2001
Appendix 1 -7
FUNCTION LIST - SPECIAL
Special Functions
δ (ω )
Description
Dirac delta function
= 0 when ω < 0
= 1 when ω = 0
= 0 when ω > 0
FUNCTION LIST - DAMAGE
Damage Functions
Description
N c ( A)
number of cycles to failure at stress amplitude A
n c ( A)
number of cycles at stress amplitude A
Nw
number of seconds annually at wind state w
D ( A)
damage at stress amplitude A
Dw
annual damage at wind state w
σ2hss
integral under a peak within the hot spot stress power spectra; used to
define a Rayleigh distribution
FRAMEWORK
Appendix 1 -8
SESAM
01-JUN-2001
Program version 2.8
CONSTANTS
ρ
air density
Cd
drag coefficient
D
member diameter
L
member length
UU
contribution to wind force at dof r from U U terms, normalised by 2U U
UV
contribution to wind force at dof r from U V terms, normalised by 2U V
Er
UW
contribution to wind force at dof r from U W terms, normalised by 2U W
α
power law exponent for mean wind change with height
λs
eigenvalue s
λs
eigenfrequency
γ
damping
π
Pi
j
square root of -1
a,b,c,C m & d m
constants used to simplify expressions
Er
Er
1
2
1
1
1
INDEPENDENT VARIABLES
z
height
t
time
ω
frequency (radians/sec)
f
frequency (cycles/sec)
A
stress amplitude
τ ,τ 1 ,ε ,ε 1 & t1
dummy variables used in integration
1
1
1
FRAMEWORK
Program version 2.8
SESAM
01-JUN-2001
APPENDIX 2
Power Spectral and Correlation Functions
Appendix 2 -1
FRAMEWORK
SESAM
Appendix 2 -2
A2.1
01-JUN-2001
Program version 2.8
Auto-correlation Function
The auto-correlation function C yy (τ ) is given by
C yy (τ ) = y (t ) y (t + τ )
where
&
y is a function of time
τ is a time lag
It should be noted that C yy (0 ) is the variance of y (t )
A2.2
Power Spectral Density Function
The power spectral density function S yy (ω ) of a time dependent function y (t ) is the
Fourier transform of its auto-correlation function
S yy (ω ) =
=
1 ∞
∫ exp (− jωτ ) C yy (τ )dτ
2π −∞
1 ∞
∫ exp (− jωτ ) yr (t ) y (t + τ ) dτ
2π −∞
The power spectral density function is normally referred to as the power spectra of that
function. The integral of the power spectra with respect to frequency is the variance of
that process:
∞  1 ∞

∫ S yy (ω )dω = ∫ 
∫ exp (− jωτ ) C yy (τ )dτ dω
−∞
− ∞  2π − ∞

∞
and changing the order of integration
=
1 ∞
∞

(
)
τ
∫ C yy  ∫ exp(− jωτ )dω dτ
2π − ∞
− ∞

then using the identity
∞
∫ exp(− jωτ )dω = δ (τ ) = 1 when τ = 0
−∞
= 0 when τ ≠ 0
it follows that
FRAMEWORK
SESAM
Program version 2.8
01-JUN-2001
Appendix 2 -3
−1 ∞
∫ C yy (τ )δ (τ )dτ = C yy (0)
2π − ∞
−∞
where C yy (0 ) is the variance of the process. (See A2.1)
∞
∫ S yy (ω )dω
A2.3
Cross-Correlation Functions
The cross-correlation function C yy (r , s : τ ) is given by
C yy (r , s : τ ) = y (t )y s (t + τ )
where
&
&
A2.4
y is a function of time and space
r & s denote particular locations in space
t is time
Cross-Power Spectral Density Functions
The cross-power spectral density function S yy (r , s : ω ) of a space and time dependent
function y (t ) is the Fourier transform of its cross-correlation function.
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Program version 2.8
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01-JUN-2001
APPENDIX 3
Algebra Supporting Section 10.0: Condensation of Forces
Appendix 3 -1
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Appendix 3 -2
01-JUN-2001
Program version 2.8
This appendix deals with the algebraic manipulation of equations 10.3-10.5 to form equations
10.6.
The forced response of a structure to a force vector G(ω ) where damping is neglected, may
be expressed within a finite element model as;
{K − ω M } U (ω ) exp ( jωτ ) = G (ω ) exp ( jωτ )
2
where
K& M
are the stiffness and mass matrices
U
is the displacement vector
If the displacement vector is partitioned into
(
U T = U TM ,U TS
)
where
T
represents the displacements of the M master degrees of freedom
T
represents the displacements of the S slave degrees of freedom
UM
US
Then equation 9.1 may be re-expressed as
 K MM

 K SM
K MS 
M
 − ω 2  MM
K SS 
 M SM


M MS  

M SS  

U M   G M 

=

 U S   GS 

 

where G M and G S are the forcing vectors on these master and slave degrees of freedom
respectively; K ij and M ij are i × j matrices building up K and M .
The lower set of equations may be written as
)
(
K SM U M + K SSU S − ω 2 M SM U M + M SSU S = G S
The reduction method assumes terms in the lower set of equations are small. By multiplying
−1
these lower terms by K SS an expression for U S may be obtained;
[
−1
−1
]
−1
−1
U S = − K SS K SM + ω 2 K SS M SM U M + ω 2 K SS M SS U M + K SS G s
2
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Program version 2.8
01-JUN-2001
Appendix 3 -3
Neglecting forcing and inertia terms, U S may be approximated to a first order as
−1
~
U S = − K SS K SM U M
3
back substitution into equation 10.2 yields the more accurate approximation containing first
order inertia and forcing terms.
[
{
~
−1
−1
−1
−1
2
~
U S = − K SS K SM + ω K SS M SM − K SS M SS K SS K SM
}]U
−1
M
+ K SSG S
4
The upper set of equations may be written as
(
)
K MMU M + K MSU S − ω 2 M MMU M + M MSU S = G M
which may now be expressed in terms of U M , by using the approximation for the inertia
terms, and the U%% S approximation otherwise, so that
[K
MM
]
~
− ω 2M MM U M + K MSU~ − ω 2M MSU~ S ≈ G M
S
5
Algebraic manipulation, covered in Attachment 3, then leads to

 K M − ω
2

U =
M M  M G M
6a
are respectively the reduced stiffness and mass matrices and G M , is
where K and M
M
M
the equivalent force vector, as below
−1
K M = K MM − K MS K SS K SM
−1
6b
−1
−1
−1
M M = M MM − M MS K SS K SM − K MS K SS M SM + K MS K SS M SS K SS K SM
−1
G M = G M − K MS K SSG S
6c
6d
The spectral method of fatigue analysis requires a dynamic model. With Guyan reduction this
means using the equivalent mass and stiffness matrices M M and K M along with the
equivalent force vector G M . To convert the point force time series of Section 8.0 into
equivalent forces at master degrees of freedom each component in 10.6c is Fourier
transformed into a time series, so that
FRAMEWORK
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Appendix 3 -4
01-JUN-2001
M
Program version 2.8
−1
g = g M − K MS K SS g S
M
where g , g M and g S are the vectors of the transformed components of G M , G M and G S
M
−1
respectively. Multiplying out the matrices K MS and K SS gives the rth component of g , as
( )−∑
M
g
gr = M
r
p =1 q =1
(gM )
(g S )
where
( K MS )rq ( K −SS1 )qp ( g S )q
7
is the rth component of g M .
r
is the sth component of g S .
S
(K MS )
(K −SS1 )
∑
is the rth row and qth column of K MS .
rq
( )
is the qth row and pth column of K SS
qp
−1
As an input requirement, all dof that will attract significant fluctuating wind loading are to be
master dof. The dynamic forcing at slave degrees of freedom for dynamic response is
ignored. e.g. within the context of a symmetric tower, rotational and vertical degrees of
freedom are assumed to attract no fluctuating load.
Hence equation 7 may be simplified as
g r ≈ (g M )r
M
ω ≠0
g r ≅ (g M )r − p∑= r ∑
q
M
over slaves
( K MS ) ( K −SS1 ) ( g S ) ω = 0
rq
qp
q
This allows the power spectra of the forcing at master dof's to be directly evaluated in terms
of the power spectra of the wind, from the equations 8.3.3 and 8.3.4.
MM
S g g (r , s : ω ) = 4 v(10, t )
2
UU
 E UU
r E s S UU (ω ) + 


 zs   zr 
UV
    coh (r , s : ω )  E UV
r E s S UV (ω ) +  ω ≠ 0
 10   10 
 E UW E UW S

s
W ′W ′ (ω ) 
 r
α
α
α
2
   over slaves
= v(10, t  E s  z s  − ∑ ∑ K MS
p
q
  10 
(
α
   over slaves
×  E r  z r  − ∑ ∑ K MS
q
p
  10 
(
α
)sq (K −SS1 )qp E q  z s 


10
  
8
α
)rq (K −SS1 )qp E q  z r 

 ω =0
 10  
This is a large gain in simplicity but normally means that extra masters are required on top of
those required to model the kinetics of the tower.
FRAMEWORK
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Program version 2.8
01-JUN-2001
Appendix 3 -5
Equation 3 is
−1
~
U S = K SS K SMU M
Equation 4 is
[
}]
{
−1
~
−1
−1
−1
−1
2
~
U S = − K SS K SM + ω K SS M SM − K SS M SS K SS K SM U M + K SS G S
and Equation 5 is
[
]
~
~
G M = K MM − ω 2 M MM U M + K MS U~ S − ω 2 M MS U S
Let
−1
A = K SS K SM
and
−1
−1
−1
B = K SS M SM − K SS M SS K SS K SM
Then 3 and 4 may be re-written as
[
]
~
−1
~
2
U S = − AU M & U~ S = − A + ω B U M + K SS G S
Back substitution into Equation 5 gives
[
{[
]
]
−1
G M = K MM − ω 2 M MM U M + K MS − A + ω 2 B U M + K SSG S
} −ω
2
M MS [− AU M ]
and then collecting the ω 2 terms together gives
}{
{
−1
G M = K MMU M − K MS AU M + K MS K SS G S
{
}
+ − ω 2 M MM U M + ω 2 K MS BU M + ω 2 M MS AU M
}
Separating forcing terms to the LHS & taking the & function outside the brackets then gives
[{
} {
}]
−1
G M = K MS K SS G S + K MM − K MS A − ω 2 M MM − K MS B − M MS A U M
Let
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Appendix 3 -6
01-JUN-2001
Program version 2.8
K M = K MM − K MS A
M M = M MM − K MS B − M MS A
−1
G M = G M − K MS K SS G S
to give
[
]
GM = K M −ω 2 M M U M
where back substitution of A and B into A3.1 & A3.2 gives
−1
K M = K MM − K MS K SS K SM
−1
−1
−1
−1
M M = M MM − M MS K SS K SM − K MS K SS M SM + K MS K SS M SS K SS K SM
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Program version 2.8
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01-JUN-2001
APPENDIX 4
Parametric SCF Schemes of the wind fatigue module
Appendix 4 -1
FRAMEWORK
Appendix 4 -2
SESAM
01-JUN-2001
Program version 2.8
This appendix details the two SCF schemes. For each SCF scheme, there are validity ranges
to the geometric parameters. For joints with parameters outside the validity range SCFs are
calculated with the actual parameter and with the parameters set to the broached limit. The
greater SCF is used.
NOMENCLATURE
INDEX
D
T
L
d
t
θ
g
φ
=
=
=
=
=
=
=
=
Chord Diameter
τ
=
t/T
Chord Thickness
β
=
d/D
Chord Length
γ
=
D/2T
Brace diameter
α
=
2L/D
Brace Thickness
ζ
=
g/D
Brace to Chord Inclination
Brace Separation
Angle around the brace/chord intersection (0º saddle, 90º crown)
SCF
=
Stress Concentration Factor - ratio of stress to nominal brace stress
SCFCS ⋅ SCF at the chord saddle
SCFBS ⋅ SCF at the brace saddle
SCFCC ⋅ SCF at the chord crown
SCFBC SCF at the brace crown
FIGURE A4.1
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Program version 2.8
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Appendix 4 -3
THE LLOYD'S SCHEME
Parametric Equations for T Joints
Wordsworth/Smedley equations for all loadcases (Ref. 1)
Chordside SCFs
Axial Load
(
SCF CS = γτβ 6.78 − 6.42 β
0.5
)sin (
1.7 + 0.7 β
3
)θ
SCF CC = K ′C + K O K ′′C
{
K ′C = 0.7 + 1.37 γ
where
KO =
NOTE:
0.5
}(
τ (1 − β ) 2 sin 0.5θ − sin 3 θ
)
τ (β − τ / (2γ ))(α / 2 − β / sin θ )sin θ
(1 − 3 / (2γ ))
This applies for the simply supported condition only and represents the
overall bending in the chord.
K ′′C = 1.05 +
30τ 1.5(1.2 − β ) (cos 4θ + 0.15)
γ
Out-of-Plane Bending
2
SCFCS = γτβ (1.6 − 1.15 β 2) sin (1.35 + β ) θ
In-Plane Bending
SCFCC = 0.75 γ
τ
0.6
0.8
(1.6 β
0.25
− 0.7 β 2) sin (1.5 −1.6 β ) θ
Braceside SCFs
For the above modes of loading the SCF on the braceside of the weld may be
estimated from the following equation:
SCF Braceside(BSorBC ) = 1+ 0.63SCF Chordside(CSorCC )
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Appendix 4 -4
01-JUN-2001
Program version 2.8
Validity Ranges
0.13
12
0.25
30º
8
≤
≤
≤
≤
≤
β
γ
τ
θ
α
≤
≤
≤
≤
≤
1.0
32
1.0
90º
40
for β > 0.98 use β = 0.98 at saddle position
Parametric Equations for K Joints
Kuang Equations for Balanced Axial Load (Ref. 3)
SCF CHORD = 1.506 γ
SCF BRACE = 0.92 γ
0.666
0.157
τ
τ 1.104β (− 0.059 ) (g / D ) 0.067 sin 1.521θ
0.56
β (− 0.441) (g / D ) 0.058exp (1.448 sinθ )
Wordsworth Equations for Unbalanced O.P.B. (Ref. 2)
{
(
5
A
SCF CS = γ τ Aβ A 1.6 − 1.15 β
)}
sin (1.35 + β 5A ) + (0.016 γ β ) (0.45 + g / D ) ( / ) 0.3sin (1.35 + β 5B ) 
θA
θ B
θA θB

B


{1 − 0.1 (
1.0 + 2 g / R )
}
SCF BS = 1 + 0.63 × SCF CS
Kuang Equations for Balanced I.P.B. (Ref. 3)
Bending moment applied to one brace only
SCF CHORD = 1.822 γ
0.38
SCF BRACE = 2.827 τ
0.35
τ 0.94 β
0.06
sin 0.9θ
β (−0.35 ) sin 0.5θ
Validity Range for Axial Load and IPB
0.5
8.333
0.2
0º
≤
≤
≤
≤
β
γ
τ
θ
≤
≤
≤
≤
0.8
33.3
0.8
90º
unless stated otherwise
unless stated otherwise
FRAMEWORK
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Program version 2.8
01-JUN-2001
Appendix 4 -5
Validity Range for OPB
0.13
12
0.25
30º
≤
≤
≤
≤
β
γ
τ
θ
≤
≤
≤
≤
for β > 0.98 use β = 0.98 at saddle position
1.0
32
1.0
90º
Parametric Equations for KT Joints
Kuang Equations for Balanced Axial Load (Ref. 3)
(Outer braces only loaded)
SCF CHORD = 1.83 γ
0.54
SCF BRACE = 6.06 γ
0.1
0.1
SCF BRACE = 138 γ
τ 0.68β
τ
= 4.89 γ
SCF (BRACE
CENTRAL )
τ 1.068β
0.68
0.12
sin θ
0° < θ ≤ 90°
{(g AB + g BC )/ D}0.126 sin 0.5θ
0° < θ ≤ 45°
(− 0.36 )
0.126
β (− 0.36 ) {(g AB + g BC ) / D } sin 2.88θ
0.123 0.672
τ
0.159
β (− 0.396 ) {(g AB + g BC ) / D } sin 2.267 θ
Wordsworth Equations for Unbalanced O.P.B. (Ref. 2)
Central Brace
{
(
5
B
SCF CS = γ τ B β B 1.6 − 1.15 β
)}
 sin (1.35 + β 5B )
θB


+ (θ A /θ B ) 0.3 (0.016 γ β A)
+ (θ B /θ C )0.3 (0.016γ β C )
(0.45 + g AB / D )sin (1.35 + β 5A )
θA
(0.45+ g BC / D )
{1−1.0 (1.0 + ( g AB + g BC )/ R )}
2
45° < θ ≤ 90°
(
5
)
sin 1.35 + β C θ C
}
FRAMEWORK
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Appendix 4 -6
01-JUN-2001
Program version 2.8
Outer brace (for brace A)
{
(
SCF CS = γ τ A β A 1.6 − 1.15 β
5
A
)}
 sin (1.35 + β 2A )
θA


+ (θ A /θ B ) 0.3 (0.016 γ β B )
( 0.45 + g AB / D )sin (1.35+ β 2B )
+ (θ A /θ C ) 0.3 (0.016 γ β C )
( 0.45+ g AC / D )sin (1.35 + β C2 ) 
θ
θB
C

{1−0.1 (1.0+ 2 g AB / R ) } {1−0.1 (1.0+ 2 g AC / R )}
2
2
SCF BS = 1 + 0.63 × SCF CS
Kuang Equations for Balanced I.P.B. (Ref. 3)
Bending moment applied to one brace only
SCF CHORD = 1.822 γ
0.38 0.94
SCF BRACE = 2.827 τ
0.35
τ
β 0.06 sin 0.9θ
β (−0.35 ) sin 0.5θ
Validity Range for Axial Load and IPB
0.3
8.333
0.2
0º
≤
≤
≤
≤
β
γ
τ
θ
≤
≤
≤
≤
0.8
33.3
0.8
90º
unless stated otherwise
unless stated otherwise
Validity Range for OPB
0.13
12
0.25
30º
≤
≤
≤
≤
β
γ
τ
θ
≤
≤
≤
≤
1.0
32
1.0
90º
for β > 0.98 use β = 0.98 at saddle position
FRAMEWORK
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Program version 2.8
01-JUN-2001
THE ORIGINAL INPLANE ONLY SCHEME
Parametric Equations for T Joints
Axial Load SCF CS = 0
Out Plane Bending SCF CS = 0
As “O” Scheme except:
Parametric Equations for K Joints
Chordside SCFs
Axial Load
SCF CS = 0
= 1.1 γ
0.65
τ  sin θ 1 / sin

[θ 1 = max(θ A ,θ B ),
1 
(2ζ
2θ 2 
) 0.05 / β (1.5 0β.25 − β 2 )
θ 2 = min(θ A ,θ B ) ]
Out of Plane Bending
SCF CS = 0
In Plane Bending
SCF S as “O” Scheme T Joint
Appendix 4 -7
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Appendix 4 -8
01-JUN-2001
Program version 2.8
The “E” Scheme (Efthymiou Scheme)
T-Joints
The geometrical parameters are as shown in Figure A4.1. C fix is an end fixity
parameter taken as 0.7. befar = 0. for our geometries. The output SCFs are as follows:
X1
X2
X3
X4
axial SCF at saddle chordside of weld
axial SCF at crown chordside of weld
axial SCF at saddle braceside of weld
axial SCF at crown braceside of weld
Axial SCFs
CC, CS and chordside crown, saddle SCFs
BC, BS are braceside crown, saddle SCFs
f 1 = 1,
(
f 2 = 1 unless α < 12 when
)
)
 f = 1 − 0.83β − 0.56 β 2 − 0.02 γ
1

 f 2 = 1 − 1.43β − 0.97 β 2 − 0.03 γ
(
0.23
e − 0.21γ
−1.16
α
2 .5
e − 0.71γ
−1.38
α
2 .5
0.04
C1 = (C fix − .5) ∗ 2
C 2 = C fix / 2.0
C 3 = C fix / 5.0
for C fix < 0.8 f = f 1 , C fix > 0.8 f = f 2
xstif = 1.0
(
)
2
1.6
t1 = γ ∗ T 1.1 1.11 − 3 (β − 0.52 ) sin (θ )
(1 − β ) f ⋅ xstif
CS = γ τ (2.65 + 5 (β − 0.65 ) ) + τβ (C ⋅ α − 3 )sin (θ )
BC = 1.3 + γτ α (0.187 − 1.25 β (β − 0.96 )sin θ )
BS = 3 + γ (0.12e β + 0.011β − 0.045 )+ βτ (C α − 1.2 )
CC = t 1 + C1 (0.8α − 6 )Tβ
0.2
2
2
2
0.52
1.2
2
0.1
−4
2.7 − 0.01α
1.1
2
3
FRAMEWORK
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Program version 2.8
01-JUN-2001
Appendix 4 -9
Inplane Bending SCFs
CSCFC = Crown SCF chordside
CSCFB = Crown SCF braceside
CSCFC = 1.45βτ
γ (1− 0.68 β ) sin 0.7θ
0.8
CSCFB = 1 + 0.65βτ
(1.09 − 0.77 β )
γ
0.4
sin (0.06γ −1.16) (θ )
Out of Plane Bending SCFs
SSCFB
SSCFC
=
=
saddle SCF braceside
saddle SCF chordside
F 3 is 1, or if α < 12,
(
f 3 = 1 − 55 β 1.8γ
SSCFC = γτβ 1.7 − 1.05 β
SSCFB = τ
− 0.54
γ
− 0.05
3
)sin
1.6
e − 0.49γ
−0.89
α 1.8
θ f3
1.6
(0.99 − 0.47 β + 0.08β ) T
4
10
K-Joints
Geometrical parameters as shown in Figure A4.1..
Axial SCFs
Geometric parameters as in Figure A4.1.. Subscripts a and b refer to braces a and b.
β max = max(β a ,β b ),
β min = min (β a ,β b )
θ max = max(θ a ,θ b ),
θ min = min (θ a ,θ b )
K1 = τ
0.9
a
γ
0.5
(0.67 − β
2
a + 1.16 β a
)sin θ
a
[sin (θ max ) / sin θ min ] 0.3 [β max / β min ] 0.3
[1.64 + .29 β −a 0.38 a tan (8ζ ) ]
(
K 2 = 1 + K1 1.97 − 1.57 β
+ cgapot β
0.25
a
)τ
1.5 0.5 −1.22
τ a sin 1.8
a γ
−0.14
sin 0.7
a
(θ a + θ b )
(0.131 − 0.084 A tan(14ζ + 4.2 β a ) )
(θ a )
FRAMEWORK
SESAM
Appendix 4 -10
01-JUN-2001
Program version 2.8
KC1=K1
KC2=K1
KB1=K2
KB2=K2
KC1 = chordside saddle SCF
KC2 = chordside crown SCF
KB1 = braceside saddle SCF
KB2 = braceside crown SCF
Inplane Bending
ChCR is inplane bending SCF chordside crown
BrCR is inplane bending SCF braceside crown
τ = τ a,
β = β a,
θ =θa
θ max = max(θ a ,θ b )
θ min = min (θ a ,θ b )
β max = max(β a ,β b )
β min = min (β a ,β b )
ζ 1 = −0.3 β max / sin (θ max )
T8 = 1.45 β τ
0.85
γ (1− 0.68 β ) sin 0.7θ
γ (1.09 −.77 β ) sin θ
k 9= T 9 ∗ (0.9 + 0.4 β )
T9 = 1 + 0.65 β τ
0.4
0.06
γ
−1.16
ChCR = T8 unless 3 < ζ 1 , when ChCR = 1.2T8
for ζ 1 > 0 BrCR = T9 else BrCR = K9
FRAMEWORK
SESAM
Program version 2.8
01-JUN-2001
Appendix 4 -11
Out of Plane Bending
β mx = max(β a ,β b )
1
α ≥ 12
−1.06
f4=
α 2.4
1 − 1.07 β 1a.88 e − 0.16 γ
{
(
)sin
(1.7 − 1.05 β )sin
t10 a = γ τ a β a 1.7 − 1.05 β
3
a
t10b = γ τ b β b
x = 1 + ζ ∗ sin θ a / β a
3
b
α ≤ 12
(θ a )
1.6
(θ b )
1.6
1
K14 = S a t10 a 1 − .0.08 (βα ⋅ γ ) 2  e − 0.8 x


1
 + . 1 − 0.08 (β ⋅ γ ) 2  e − 0.8 x 
 S b t10b

a




∗ 2.05 β
1
2
mx e
−1.3 x
CHSA = K 14 ∗ f 4
BRSA = τ
− 0.54 − 0.05
γ
a
(0..99 − 0.47 β
CHSA is chordside SCF saddle
BRSA is braceside SCF saddle
a
+ 0.08 β
4
a
)K14
FRAMEWORK
SESAM
Appendix 4 -12
01-JUN-2001
Program version 2.8
KT Joints
a and c are the two diagonal braces
b is the central brace
otherwise nomenclature is as Figure A4.1
ζab = gapab /chord diameter
ζbc = gapbc /chord diameter
for ζab > ζbc
β max = max (β a , β b )
β min = min (β a , β b )
θ max = max (θ a , θ b )
θ min = min (θ a , θ b )
K1 = τ
0.9 0.5
b γ
(0.67 − β
2
b + 1.16 β b
)sinθ
+ (sin (θ max ) / sin (θ min ) ) (β max / β min )
3
(1.64 + 29 β
− 0.38
atan
b
(8ζ ) )
(
K 2 = 1 + K1 1.97 − 1.57 β
0.25
b
−1.22
b
sin
+ Cgapot β 1b.5 γ
0.5
τ
b
0.3
)τ
1.8
− 0.14
sin 0.7
b
(θ a + θ b ) ∗
(0.131 − 0.084 ∗ ATAN / (143 + 4.2 β b )
KC 2 = KC1 = K1
KB 2 = KB1 = K 2
for ζab < ζbc
β max = max(β b ,β c )
β min = min (β b ,β c )
θ max = max(θ b ,θ c )
θ min = min(θ b ,θ c )
K1 = τ
0.9 −0.5
b γ
(0.67 − β
2
b + 1.16 β b
)sin θ
b
(θ b )
FRAMEWORK
SESAM
Program version 2.8
01-JUN-2001
0.3
 sin θ max 
×

 sin θ min 
(
× 1.64 + 0.29 β
(
 β max 


β 
 min 
− 0.38
tan −1
b
K 2 = 1 + K1 1.97 − β
+ Cgapot β 1b.5τ
(
0.3
0.25
b
)τ
−1.22
sin1.8
b
(8ζx ))
−0.14
sin 0.7 θ b
b
(θ b + θ c )
× 0.131 − 0.084 ∗ tan −1 (14 ζ bc + 4.2 β b )
For both cases
KC1
KC2
KB1
KB2
=
=
=
=
chordside saddle SCF
braceside crown SCF
braceside saddle SCF
braceside crown SCF
KT Joint Out of Plane Bending
S a = 1

S b = 1 in our use of the code
S c = 1
β mx = max (β a , β b , β c )
f 4 = 1 unless α ≤ 12when
f 4 = 1 − 1.07 β
1.88
1.06 2.4
α
b exp − 0.16 γ
(
)sin
(1.7 − 1.05 ∗ β )sin
(1.7 − 1.05 ∗ β )sin
t10 a = γ τ a β a 1.7 − 1.05 ∗ β
3
a
1.6
t10b = γ τ b β b
3
b
1.6
3
c
1.6
t10c = γ τ c β c
xab = 1 + ζ ab sin (θ b ) / β b
xbc = 1 + ζ bc sin (θ b ) / β b
powa = (β a / β b )
powc = (β c / β b )
2
2
θa
θb
θc
)
Appendix 4 -13
FRAMEWORK
SESAM
Appendix 4 -14
01-JUN-2001
(
K t10 = t10b 1 − 0.8 ( β a γ
( S (1 − 0.8 (β γ )
0.5
c
b
e − 0.8 xbc
(
+ S a t10 a 1 − 0.8 (β b γ
(
× 2.05 β
1
2
mx e
ab
)
powa
∗
powc
ab
−1.3 x ab
(
(
)
)0.5 e − 0.8 x
+ S c ∗ t10c 1 − 0.08 (β b γ
× 2.05 β
) 0.5 e −0.8 x
Program version 2.8
) 0.5 e − 0.8 x
bc
0.5 −1.3 xba
mx e
Chsad = kt10 ∗ Cfix
BrSAD = τ
−0.54 −0.05
γ
b
( 0.99 − 0.47 ∗ β
b
+ 0.08 β
4
b
)ChSAD
ChSAD is the chordside saddle SCF
BrSAD is the braceside saddle SCF
CCSCF = chordside crown SCF
BCSCF = braceside crown SCF
Inplane Bending SCFs
Reference brace is a diagonal brace. a and c refer to the two diagonal braces.
Diagonal Braces Axial SCF
β min = min (β a , β c )
β max = max(β a , β c ) ,
θ min = min (θ a , θ c )
θ max = max(θ a , θ c ) ,
K1 = τ
0.9 + 0.5
a γ
( 0.67 − β
2
a + 1.16 β a
)sin θ
) (β max / β min )
(1.64 + 0.29 β −a 0.38 ∗ tan -1/ ( 8ζ ) )
∗ ( sin θ max / sin θ min
(
0.3
K 2 = 1 + K1 1.97 − 1.57 β
+ cgapot β a1.5 γ
0.5
τ
0.25
a
)τ
a
0.3
−0.14
0.7
sin θ a
a
−1.22
sin 1.8
a
(θ a + θ c )
× (0.131 − 0.084 atan (14 ( ζ ab + ζ ac + β b ( sin θ b ) ) + 4.2 β a ) )
FRAMEWORK
SESAM
Program version 2.8
01-JUN-2001
Appendix 4 -15
KC1 = chordside saddle SCF
KC2 = chordside crown SCF
KB1 = braceside saddle SCF
KB2 = braceside crown SCF
Diagonal Brace Out of Plane Bending
β mx = max (β a , β b , β c )
f 4 = 1 unless α ≤ 12 when
f 4 = 1 − 1.07 β
1.88 − 0.16γ
a e
1.06
α
2.4
(
) sin
(1.7 − 1.05 β ) sin
(1.7 − 1.05 β ) sin
t10 a = γ τ a β a 1.7 − 1.05 β
3
a
1.6
t10b = γ τ b β b
3
b
1.6
3
c
1.6
t10c = γ τ c β c
θa
θb
θc
x ab = 1 + ζ ab sin (θ a ) β a
x ac = 1 + ( ( ζ ab + ζ ac + β b / sin (θ b ) )∗ sin θ a ) / β a
(
K t10 = t10 a 1 − 0.08 (β b γ
(
∗ 1 − 0.08 (β c γ
(
)
1
2
)
1
2
)
e − 0.8 xab S a
e − 0.8 xab
)
) 0.5 )e − 0.8 xab
0.5
∗ 2.05 β 0mx.5 e −1.3 xab + S c t10c ( 1 − 0.08 (β a γ ) )e − 0.8 xac
0.5
+ S c t10c (1 − 0.08 (β a γ ) ) e − 0.8 xac
+ S b ∗ t10b ∗ 1 − 0.08 (βγ
∗ 2.05 ∗ β
1
2
mx e
−1.3
xac
ChSAD = K t10 ∗ f 4
BRSAD = τ a−0.54 γ − 0.05 0.99 − 0.47 × β a + 0.08 β 4a ChSAD
ChSAD is chordside saddle out of plane bending SCF
BrSAD is braceside saddle out of plane bending SCF
(
)
Diagonal Brace Inplane Bending
CSCF = 1.45 β τ
0.8
γ (1− 0.68 β ) sin 0.7 (θ )
BSCF = 1 + 0.65 β τ
0..4
γ
(1.09 − 0.77 β )
sin ( 0.068γ −1.16) (θ )
CSCF is chordside crown inplane bending SCF
BSCF is braceside crown inplane bending SCF
FRAMEWORK
SESAM
Appendix 4 -16
01-JUN-2001
Program version 2.8
X-Joints SCF
Axial SCFs
X1 = chordside saddle axial SCF
X2 = chordside crown axial SCF
X3 = braceside saddle axial SCF
X4 = braceside crown axial SCF
f 1 = f 2 = 1 unless α < 12 when
(
= 1 − (1.43β − 0.97 β
)
− 0.03 )γ
f 1 = 1 − 0.83β − 0.56 β 2 − 0.02 γ 0.23 e − 0.21γ
f2
2
−1.16
α 2.5
0.04 − 0.71γ −1.38 α 2.5
e
for C fix < 0.8 F = f 1 for C fix > 0.8 F = f 2
(
)
X 1 = 3.87 ∗ γ ∗τ ∗ β 1.1 − β 1.8 sin 1.7 θ . f
X2 =γ
0.2
(
)
)sin
τ 2.65 + 5 ∗ ( β − 0.65 ) 2 − 3τ β sin (θ )
X 3 = 1 + 1.9 γ τ
X 4 = 3+γ
1.2
0.5
(
β 0.9 1.09 − β 1.7
( 0.12 e
−4β
2.5
θf
2
+ 0.011 β − 0.045
)
Inplane Bending
CHCSCF = 1.45 β τ
γ (1− 0.68 β ) sin 0.7 θ
0.8
BRCSCF = 1 + 0.65 β τ
0.4
γ
(1.09 − 0.77 β )
sin ( 0.068 −1.16) (θ )
CHCSCF = chordside crown SCF
BRCSCF = braceside crown SCF
Out of Plane Bending SCF
f 3 = 1 unless α ≤ 12 when
f 3 = 1 − 55 β 1.8 γ
0.16
(
e
− 0.49 γ −0.89 α 1.8
CSA = γτβ 1.56 − 1.34 β
BSA = τ
− 0.54
γ
− 0.05
4
)sin
(θ ) ∗ f 3
( 0.99 − 0.47 β + 0.08 β )∗ CSA
CSA = chordside saddle SCF
BSA = braceside saddle SCF
REFERENCES
1.6
4
FRAMEWORK
Program version 2.8
SESAM
01-JUN-2001
Appendix 4 -17
1.
Wordsworth, A.C., and Smedley, G.P., “Stress Concentration at Unstiffened Tubular
Joints”, Paper 31 of the European Offshore Steels Research Seminar at the Welding
Institute, November 1978.
2.
Wordworth, A.C., “Stress Concentration Factors at K and KT Tubular Joints”, Fatigue
in Offshore Structural Steel, Civil Engineers Conference, February 1981.
3.
Potvin, A.B.,Kuang, J.G.,Leick, R.D., and Kahlick, J.L., “Stress Concentration in
Tubular Joints”, Society of Petroleum Engineers Journal, August 1977.
FRAMEWORK
Program version 2.8
SESAM
01-JUN-2001
Appendix 5 -1
APPENDIX 5
LLOYD'S REGISTER OF SHIPPING FORMULAE (ISOPE 1991)
FRAMEWORK
Appendix 5 -2
SESAM
01-JUN-2001
A5.0
Lloyd's Register of Shipping Formulae (ISOPE 1991)
A5.1
Introduction
Program version 2.8
This appendix gives the SCF parametric equations for unstiffened tubular joints defined by
Lloyd's Register of Shipping in the ISOPE conference paper of 1991.
The equations are derived in terms of functions that may be factored for short chord effects.
The short chord factors are F1 to F3. There are also factors that isolate the localized bending
at saddle positions from the overall beam bending effects. The beam bending terms are B0
and B1. Typically the short chord correction factors are only applicable to the localized
bending terms. Framework automatically takes the appropriate action for the supplied L/D
ratio.
The stiffening effects of unloaded braces in the vicinity of the loaded brace under
consideration are also taken into consideration. These are characterized by the S1 and S2
factors. Similarly the effects of loaded braces in the same area are considered by the
application of influence factors. These are factors IF1 to IF8.
The equations consist of basic relations for T and X joints. These are then amalgamated,
using the appropriate factors for joint geometry and complexity, to build equations for K and
KT joints.
FRAMEWORK
Program version 2.8
SESAM
01-JUN-2001
NOMENCLATURE
FIGURE A.5.1
D
d
T
t
=
=
=
=
θ
=
L
=
g
=
β
=
γ
=
τ
=
α
=
ζ
=
SCFC =
SCFCS =
SCFCC =
SCFB =
SCFBS =
SCFBC =
Outside diameter of can member
Outside diameter of brace member
Thickness of can member
Thickness of brace member
Acute angle between brace and chord
Chord length
Separation between brace toes (i.e. gap) see below
d/D
D/2T
t/T
2L/D
g/D
Maximum SCF on the chordside
SCF at chord saddle
SCF at chord crown
Maximum SCF on the braceside
SCF at brace saddle
SCF at brace crown
FIGURE A.5.2
Appendix 5 -3
FRAMEWORK
Appendix 5 -4
SESAM
01-JUN-2001
Program version 2.8
Lloyd's (ISOPE 1991) T/Y Joint Equations
Axial Load
SCFCS = 1.20T1
(× F1 or F2 for short chord )
SCFCC = 1.2T2 + B0 B1
SCFBS = 1.25T3
(× F1 or F2 for short chord )
SCFBC = 1.23T4
Note: The short chord correction factor F1 is applied for fully fixed ends (C ≤ 0.7). F2 is
applied for pinned ends (C > 0.7). The F factors and the chord stress factors B0 and
B1 are defined in a later section of this appendix.
Out-of-Plane Bending
SCFCS = 1.22T5
(× F3 for short chord )
SCFBS = 1.28T6
(× F3 for short chord )
In-plane Bending
SCFC = 1.15T7
SCFB = 1.18T8
T Joint Factors
NB Apply the modified values of the β parameter when predicting the SCFs at the saddle for
joints that have values of β close to 1.0 and are under axial load or out-of-plane bending.
This affects the equations for T1, T3, T5, and T6.
T1 = τγ 1.2 β (2.12 − 2 β ) sin 2 θ
T2 = τγ 0.2 (3.5 − 2.4 β ) sin 0.3 θ
T3 = 1 + τ 0.6γ 1.3 β (0.76 − 0.7 β ) sin 2.2 θ
T4 = 2.6 β 0.65γ ( 0.3− 0.5 β )
FRAMEWORK
SESAM
Program version 2.8
01-JUN-2001
Appendix 5 -5
T5 = τγβ (1.4 − β 5 ) sin1.7 θ
T6 = 1 + τ 0.6γ 1.3 β (0.27 − 0.2 β 5 ) sin1.7 θ
3
T7 = 1.22τ 0.8 βγ (1− 0.68 β ) sin (1− β ) θ
T8 = 1 + τ 0.2γβ (0.26 − 0.21β ) sin1.5 θ
Lloyd's (ISOPE 1991) X Joint Equations
Balanced Axial Load
SCFCS = 1.22 X 1
(× F1 or F2 for short chord )
SCFCC = 1.33 X 2 + B0 B1 )
SCFBS = 1.19 X 3
(× F1 or F2 for short chord )
SCFBC = 1.13 X 4
Note: The short chord correction factor F1 is applied for fully fixed ends (C ≤ 0.7). F2 is
applied for pinned ends (C > 0.7). The F factors and the chord stress factors B0 and
B1 are defined in a later section of this appendix.
Balanced Out-of-Plane Bending
SCFCS = 1.22 X 5
(× F3 for short chord )
SCFBS = 1.20 X 6
(× F3 for short chord )
Balanced In-Plane Bending
SCFC = 1.23 X 7
SCFB = 1.12 X 8
X Joint Factors
NB Apply the modified values of the β parameter when predicting the SCFs at the saddle for
joints that have values of β close to 1.0 and are under axial load or out-of-plane bending. This
affects equations X1, X3, X5, and X6.
FRAMEWORK
SESAM
Appendix 5 -6
01-JUN-2001
Program version 2.8
X 1 = τβγ 1.3 (1.46 − 1.4 β 2 ) sin 2 θ
X 2 = (0.36 + 1.9τγ 0.5e( − β
1.5 0.5
γ
)
(sin θ + 3 cos 2 θ )
X 3 = 1.0 + 0.6 X 1
X 4 = (1.3 + 0.06τγe( − β
2 0.5
γ
)
(sin θ ) −1
X 5 = τβγ 1.3 (0.63 − 0.6 β 3 ) sin 2 θ
2
X 6 = 1.0 + τβγ 1.5 (0.19 − 0.185β 3 ) sin 7 (1− β ) θ
X 7 = τ 0.8 βγ ( 0.5 β
−0.5
)
(1.0 − 0.32 β 5 ) sin 0.5 θ
X 8 = 1.0 + τ 0.8 βγ (0.32 − 0.25 β ) sin1.5 θ
Lloyd's (ISOPE 1991) K Joint Equations
Note: The short chord correction factor F1 is applied for fully fixed ends (C ≤ 0.7). F2 is
applied for pinned ends (C > 0.7). The F factors, the chord stress factors B0 and B1,
the stiffening factors, S1 and S2 and the influence functions IF1 to IF8 are defined in a
later section of this appendix.
Note: The expression T1A implies that the equation for T1 should be evaluated for the
geometric parameters associated with brace A. Brace A is the brace under
consideration and brace B is the other brace of the K joint.
Single Axial Load (One brace loaded)
SCFCS = 1.18T1 A S1 AB
(× F1 A or F2 A for short chord )
SCFCC = 1.13T2 A S 2 AB + B0 A B1 A
SCFBS = 1.20T3 A S1 AB
(× F1 A or F2 A for short chord )
SCFBC = 1.23T4 A S 2 AB
Single Out-of-Plane Bending (One brace loaded)
FRAMEWORK
Program version 2.8
SESAM
01-JUN-2001
Appendix 5 -7
SCFCS = 1.17T5 A S1 AB
(× F3 A for short chord )
SCFBS = 1.18T6 A S1 AB
(× F3 A for short chord )
Single In-Plane Bending (One brace loaded)
SCFC = 1.15T7 A
SCFB = 1.17T8 A
Balanced Axial Load
SCFCS = 1.22(T1 A S1 AB − T1B S1BA IF1 AB )
(× F1 A or F2 A for short chord )
SCFCC = 1.25(T2 A S 2 AB − T2 B S2 BA IF2 AB ) + B0 A B1 A
SCFBS = 1.12(T3 A S1 AB − T3 B S1BA IF3 AB )
(× F1 A or F2 A for short chord )
SCFBC = 1.26(T4 A S 2 AB − T4 B S2 BA IF4 AB )
Unbalanced Out-of-Plane Bending
SCFCS = 1.14(T5 A S1 AB + T5 B S1BA IF5 AB )
(× F3 A for short chord )
SCFBS = 1.21(T6 A S1 AB + T6 B S1BA IF6 AB )
(× F3 A for short chord )
Balanced In-Plane Bending
SCFC = 1.15(T7 A + T7 B IF7 AB )
SCFB = 1.16(T8 A + T8 B IF8 AB )
Lloyd's (ISOPE 1991) KT Joint Equations
Note: The short chord correction factor F1 is applied for fully fixed ends (C ≤ 0.7). F2 is
applied for pinned ends (C > 0.7). The F factors, the chord stress factors B0 and B1,
the stiffening factors, S1 and S2 and the influence functions IF1 to IF8 are defined in a
later section of this appendix.
FRAMEWORK
Appendix 5 -8
SESAM
01-JUN-2001
Program version 2.8
Note: The expression T1A implies that the equation for T1 should be evaluated for the
geometric parameters associated with brace A. Brace A is the brace under
consideration unless otherwise stated.
Single Axial Load (One brace loaded)
SCFCS = 1.18T1 A S1 AB S1 AC
(× F1 A or F2 A for short chord )
SCFCC = 1.13T2 A S 2 AB + B0 A B1 A
(Outer Brace A)
SCFCC = 1.13T2 B S2 B + B0 B B1B
(Central Brace B)
SCFBS = 1.20T3 A S1 AB S1 AC
(× F1 A or F2 A for short chord )
SCFBC = 1.23T4 A S 2 AB
(Outer Brace A)
SCFBC = 1.23T4 B S 2 B
(Central Brace B)
where S 2 B = Max( S 2 BA , S 2 BC )
Single Out-of-Plane Bending (One brace loaded)
SCFCS = 1.17T5 A S1 AB S1 AC
(× F3 A for short chord )
SCFBS = 1.18T6 A S1 AB S1 AC
(× F3 A for short chord )
Single In-Plane Bending (One brace loaded)
SCFC = 1.15T7 A
SCFB = 1.17T8 A
Balanced Axial Load (Only outer braces A and C loaded)
SCFCS = 1.22(T1B S1 AB S1 AC − T1C S1CB S1CA IF1 AC )
(× F1 A or F2 A for short chord )
SCFCC = 1.25(T2 A S 2 AB − T2C S 2CB IF2 AC ) + B0 A B1 A
SCFBS = 1.12(T3 A S1 AB S1 AC − T3C S1CB S1CA IF3 AC )
SCFBC = 1.26(T4 A S 2 AB − T4C S 2CB IF4 AC )
(× F1 A or F2 A for short chord )
FRAMEWORK
Program version 2.8
SESAM
01-JUN-2001
Appendix 5 -9
The central brace SCFs are evaluated by taking the maxima of the pairs of values that result
from considering brace B and its interaction with the two outer braces (A and C).
SCFCS 1 = 1.22(T1B S1BA S1BC − T1 A S1 AB S1 AC IF1BA )
SCFCS 2 = 1.22(T1B S1BC S1BA − T1C S1CB S1CA IF1BC )
SCFCS = max(SCFCS 1 , SCFCS 2 )
(× F1 B or F2 B for short chord )
SCFCC1 = 1.25(T2 B S 2 B − T2 A S 2 A IF2 BA ) + B0 B B1B
SCFCC 2 = 1.25(T2 B S 2 B − T2C S 2C IF2 BC ) + B0 B B1B
SCFCC = max(SCFCC1 , SCFCC 2 )
SCFBS1 = 1.12(T3 B S1BA S1BC − T3 A S1 AB S1 AC IF3 BA )
SCFBS 2 = 1.12(T3 B S1BC S1BA − T3C S1CB S1CA IF3 BC )
SCFBS = max(SCFBS1 , SCFBS 2 )
(× F1 B or F2 B for short chord )
SCFBC1 = 1.26(T4 B S 2 B − T4 A S 2 A IF4 BA )
SCFBC 2 = 1.26(T4 B S2 B − T4C S 2C IF4 BC )
SCFBC = max(SCFBC1 , SCFBC 2 )
where
S 2 A = max(S2 AB , S2 AC )
S 2 B = max(S2 BA , S2 BC )
S 2C = max(S 2CA , S2CB )
Unbalanced Out-of-Plane Bending (All braces loaded)
SCFCS = 1.14(T5 A S1 AB S1 AC + T5 B S1BA S1BC IF5 AB + T5C S1CB S1CA IF5 AC )
(× F3 A for short chord )
SCFBS = 1.21(T6 A S1 AB S1 AC + T6 B S1BA S1BC IF6 AB + T6C S1CB S1CA IF6 AC )
(× F3 A for short chord )
FRAMEWORK
Appendix 5 -10
SESAM
01-JUN-2001
Program version 2.8
Balanced in-Plane Bending (Only outer braces A and C loaded)
For brace A
SCFC = 1.15(T7 A + T7C IF7 AC )
SCFB = 1.16(T8 A + T8C IF8 AC )
For brace B
SCFCC1 = 1.15(T7 B + T7 A IF7 BA )
SCFCC 2 = 1.15(T7 B + T7C IF7 BC )
SCFCC = max(SCFCC1 , SCFCC 2 )
SCFBC1 = 1.16(T8 B + T8 A IF8 BA )
SCFBC 2 = 1.16(T8 B + T8C IF8 BC )
SCFBC = max(SCFBC1 , SCFBC 2 )
STIFFENING, INFLUENCE AND CHORD STRESS FACTORS
Stiffening Effect of an Additional Brace
The equation for S1AB gives the effect at the saddle of brace A due to the loads on brace B at
the joint. This appears as a reduction in the saddle SCF. Similarly S2AB gives the effect at the
crown of brace A due to loads on brace B at the joint. This increases the crown SCF. Note
that the reduced value of the β parameter should be applied for joints where β approaches 1.0
for the saddle SCF reduction factor, S1AB.
2
S1 AB = {1.0 − 0.4 exp(−30.0 x AB
( β A / β B ) 2 (sin θ A / γ ))}
2
S 2 AB = {1.0 + exp(−2.0 x AB
/(γ 0.5 sin 2 θ B ))}
where x AB = 1.0 + (ζ AB sin θ A / β A )
and
ζ AB = (Gap between weld toes of braces A and B)/(Chord diameter)
IF Factors - Influence Factors for K and KT joints
Equation IFAB gives the influence upon brace A of the applied loading in brace B.
Note that the reduced value of the β parameter should be used for joints where β approaches
1.0 for saddle influence function calculations. This affects equations IF1, IF3, IF5 and IF6.
FRAMEWORK
SESAM
Program version 2.8
01-JUN-2001
Appendix 5 -11
IF1 AB = β Α (2.13 − 2.0 β Α )γ 0.2 sin θ Α (sin θ Α / sin θ Β ) P e( −0.3 x AB )
where
P = 1 if θ Α > θ B and P = 5 if θ Α < θ Β
IF2 AB = {20.0 − 8( β Α + 1.0) 2 }e( −3.0 x AB )
IF3 AB = β Α (2.0 − 1.8β A )γ 0.2 ( β min / β max )(sin θ Α / sin θ Β ) P e( −0.5 x AB )
where P = 2 if θ Α > θ Β and P = 4 if θ Α < θ Β
IF4 AB = (−1.5β Α )e − x AB
IF5 AB = 0.6γ (sin θ Α / sin θ Β )e( −3.0 x AB )
IF6 AB = 0.14 β Αγ 1.5 (sin θ Α / sin θ Β )e( −3.0 x AB )
IF7 AB = 1.5τ A−2.0e( −3.0 x AB )
IF8 AB = {40.0( β Α − 0.75) 2 − 2.5}e( −3.0 x AB )
where x AB = 1.0 + (ζ AB sin θ A / β A )
and
ζ AB = (Gap between weld toes of braces A and B)/(Chord diameter)
Approximation of Chord Stresses for Simple Specimen
B0 =
C τ ( β − τ /( 2γ ))(α / 2 − β / sin θ ) sin θ
(1 − 3 /( 2γ ))
B0 = 0.0
B1 = 1.05 +
For single axial load
For balanced axial load
30.0τ 1.5 (1.2 − β )(cos4 θ + 0.15)
γ
where C is the chord end fixity parameter (0.5 ≤ C ≤ 1.0). For fully fixed chord ends C = 0.5,
for pinned ends C = 1.0.
Short Chord Correction Factors
FRAMEWORK
Appendix 5 -12
SESAM
01-JUN-2001
Program version 2.8
The short chord correction factors account for the reduction of chord ovalization that occurs
due to the presence of chord end restraints close to the joint. Care must be taken in applying
short chord factors in structural analysis. A chord length to diameter (L/D) of 20.0 implies
that short chord effects need not be considered. Accordingly SCFs will be calculated
conservatively; that is they may be higher than the true values, leading to shorter fatigue lives
being predicted.
For joints which satisfy the short chord criteria regarding effective support conditions the
following factors are applicable.
•
For α ≥ 12.0 there are no short chord effects. Note that α = 2(L/D), where (L/D) is the
length to diameter ratio.
•
For α < 12.0 the following expressions are used in the formulae quoted above for SCF
values.
F1 = 1.0 − (0.83β − 0.56 β 2 − 0.02)γ 0.23e( −0.21γ
( −1.16 )
F2 = 1.0 − (1.43β − 0.97 β 2 − 0.03)γ 0.04e( −0.71γ
( −1.38 )
F3 = 1.0 − 0.55β 1.8γ 0.16e( −0.49γ
( −0.89 ) 1.8
α
α 2.5 )
α 2.5 )
)
Validity Range
The equations quoted in this appendix are generally valid for joint parameters within the
following limits
0.13 ≤
10.0 ≤
0.25 ≤
30.0° ≤
4.00 ≤
0.0 ≤
β
γ
τ
θ
α
ζ
≤
≤
≤
≤
1.00
35.0
1.00
90.0°
≤ 1.00
(For K and KT joints)
FRAMEWORK
Program version 2.8
SESAM
01-JUN-2001
APPENDIX 6
LLOYD'S 1996 RECOMMENDATIONS FOR SCFs
Appendix 6 -1
FRAMEWORK
Appendix 6 -2
SESAM
01-JUN-2001
A6.0
Program version 2.8
LLOYD'S 1996 RECOMMENDATIONS FOR SCFs
This attachment outlines the recommendations contained in the Lloyd's Register of Shipping
Recommendations issued in the January 1996 ("Lloyd's Register of Shipping Recommended
Parametric Stress Concentration Factors", document number OD/TN/95001, January 1996).
The equations consist of a mix of those of Efthymiou (see Attachment 4) and those from the
Lloyd's 1991 ISOPE paper (see Attachment 5). Note that for the purposes of clarity the
terminology must be clearly defined. The Lloyd's (1991) equations refer to the ISOPE paper.
These equations are not to be confused with either of the two sets of Lloyd's Recommended
equations, i.e. those of 1988 or 1996.
For K and KT joints, with gaps greater than or equal to zero, Lloyd's Register recommends
the use of the Lloyd's (1991, ISOPE) equations.
For K and KT joints with overlaps (or equivalently, negative gaps) the equations due to
Efthymiou are recommended by Lloyd's Register.
For in-plane bending the Lloyd's Recommendations require the use of the Efthymiou
equations with balanced loads. This contrasts with the Shell recommendations that require
the use of the equivalent equations for unbalanced loads. Accordingly different SCFs will be
generated, for overlapped K or KT joints if the Efthymiou equation set (E) is used instead of
the Lloyd's Recommended equation set (R).
The Lloyd's recommendation for T, Y, and X joints is that the Efthymiou equations are to be
used.
Full details of the Lloyd's (1991, ISOPE) equations are to be found in Attachment 5.
FRAMEWORK
Program version 2.8
SESAM
01-JUN-2001
APPENDIX 7
Wind Spectra and Coherence Functions
Appendix 7 -1
FRAMEWORK
Appendix 7 -2
SESAM
01-JUN-2001
Program version 2.8
POWER SPECTRA AND COHERENCE FUNCTIONS
The Harris Power Cross Spectrum in U′
The Cross-Power Spectra is:
S uu (r , s, f ) := coh(r , s, f ) ⋅ S uu ( f )
where
 Lu ⋅ f 
4.0 ⋅ k ⋅ V ⋅ V 

V 

S uu ( f ) :=
5.0
6.0
2


Lu ⋅ f 

2.0 + 
  ⋅f
V  




 − c ⋅ z (r ) − z (s ) ⋅
coh(r , s, f ) := exp
V

f


and
Lu
f
V
is the turbulence length scale
is the frequency in Hz
is the surface wind speed at 10 m
k
c
is the surface drag coefficient
is a constant controlling the compactness of gust
FRAMEWORK
SESAM
Program version 2.8
01-JUN-2001
Appendix 7 -3
The Davenport Power Spectrum in U′ with Coherence Function
The Cross-Power Spectra is:
S uu (r , s, f ) := coh(r , s, f ) ⋅ S uu ( f )
where:2
 Lu ⋅ f 
4.0 ⋅ k ⋅ V ⋅ V 

V 

S uu ( f ) :=
4.0
3
.0
2

Lu ⋅ f  

1.0 + 
⋅f

V  






z (r ) − z (s ) 
 ⋅ R ⋅
 − c ⋅  2 −
r


coh(r , s, f ) := exp 
 0.5 ⋅ (V (z (r )) + V (z (s )))



f




R :=  x(r )2 − x( s )2  +  y(r )2 − y( s )2  +  z (r )2 − z ( s )2 

 
 

and
Lu
f
V
is the turbulence length scale
k
is the frequency in Hz
c
is the surface wind speed at 10 m
is the surface drag coefficient
is a constant controlling the compactness of gust
FRAMEWORK
Appendix 7 -4
SESAM
01-JUN-2001
Program version 2.8
The Panofsky Lateral (Horizontal) Power Spectrum in V′ with Coherence Function
The Cross-Power Spectra is:
S uu (r , s, f ) := coh(r , s, f ) ⋅ S uu ( f )
S vv ( f ) :=
 Lu ⋅ f 
15.0 ⋅ k ⋅ V ⋅ V 

 V 
5.0

 Lu ⋅ f  3.0
+
⋅
1
.
0
9
.
5

 ⋅ f

 V 



z (r ) − z (s ) 
 ⋅ R ⋅
 − c ⋅  2 −
R


coh(r , s, f ) := exp 
 0.5 ⋅ (V (z (r )) + V (z (s )))



f




R :=  x(r )2 − x( s )2  +  y(r )2 − y( s )2  +  z (r )2 − z ( s )2 

 
 

and
Lu
f
V
is the turbulence length scale
k
is the frequency in Hz
c
is the surface wind speed at 10 m
is the surface drag coefficient
is a constant controlling the compactness of gust
FRAMEWORK
SESAM
Program version 2.8
01-JUN-2001
Appendix 7 -5
The Panofsky Vertical Power Spectrum in W′ with Coherence Function
The Cross-Power Spectra:S uu (r , s, f ) := coh(r , s, f ) ⋅ S uu ( f )
where
S vv ( f ) :=
 Lu ⋅ f 
3.36 ⋅ k ⋅ V ⋅ V 

 V 
5.0

 Lu ⋅ f  3.0
+
⋅
1
.
0
10
.
0

 ⋅ f

 V 



z (r ) − z (s ) 
 ⋅ R ⋅
 − c ⋅  2 −
R


coh(r , s, f ) := exp 
 0.5 ⋅ (V (z (r )) + V (z (s )))



f




R :=  x(r )2 − x( s )2  +  y(r )2 − y( s )2  +  z (r )2 − z ( s )2 

 
 

and
Lu
f
V
is the turbulence length scale
k
is the frequency in Hz
c
is the surface wind speed at 10 m
is the surface drag coefficient
is a constant controlling the compactness of gust
FRAMEWORK
Program version 2.8
SESAM
01-JUN-2001
APPENDIX 8
Algebra Supporting Section 14.0; The Damage Integral
Appendix 8 -1
FRAMEWORK
SESAM
Appendix 8 -2
01-JUN-2001
Program version 2.8
DAMAGE EVALUATION - THEORY
The S-N curve for high amplitude stress cycles (number of cycles < 107) may be expressed as
NU ( A, m ) = K 2 ⋅ (2 ⋅ A)
−m
m
⋅θ 4
where A is the stress amplitude and θ is the thickness factor.
The S-N curve for low amplitude stress cycles (number of cycles > 107) may be expressed as
− (m + 2 )
NL( A, m ) = K L ⋅ (2 ⋅ A)
m+2
⋅θ 4
For a narrow band process, the number of cycles at stress amplitude A is given by
nc( A, σ ) = Time ⋅ freq ⋅
 − A2 

⋅ exp
2
2

σ
 2 ⋅σ 
A
where σ is the variance of the stress amplitude.
The damage integral may then be expressed as

D( A) := Time ⋅ freq ⋅ 

∫
nc( A)
dA +
NL( A, m )
AO
O
∫
∞
AO

nc( A)
dA 
NU ( A, m ) 
where AO is the stress amplitude at 107 cycles, taken from the S-N curve.
Substituting for nc, NL and NU gives DN = D/(Time⋅freq)
DN ( A) :=
∫
AO
O
+
∫
∞
AO
− ( m+ 2 )
4
 − A2 

dA
⋅
exp
2 
σ2
 2 ⋅σ 
−m
 − A2 
A
−1
m
dA
K 2 ⋅ (2 ⋅ A) ⋅θ 4 ⋅ 2 ⋅ exp
2 
σ
 2 ⋅σ 
K L ⋅ (2 ⋅ A)
−1
m+ 2
⋅θ
⋅
A
FRAMEWORK
SESAM
Program version 2.8
01-JUN-2001
Appendix 8 -3
and manipulation gives
DN ( A) := 2 m+2 ⋅ K L ⋅θ
−1
−1
+ 2 ⋅ K 2 ⋅θ
m
− ( m+ 2 )
4
−m
4
⋅
⋅
∫
∫
AO
A m +3
σ2
O
∞
Am+1
AO
σ2
 − A2 
dA
⋅ exp
2 
 2 ⋅σ 
 − A2 
dA
⋅ exp
2 
 2 ⋅σ 
both integrals are of the form
l (a, b, p ) :=
∫
A p+1
b
σ2
a
 − A2 
dA
⋅ exp
2 
 2 ⋅σ 
use the following substitution in l
t ( A) :=
( )
A2
A(t ) := 20.5 ⋅ σ 2
2
2 ⋅σ
0.5
( )
⋅ t 0.5 dA(t ) := 2− 0.5 ⋅ σ 2
0.5
⋅ t − 0.5 ⋅ dt
This gives
l (a, b, p ) :=
∫
( )
20.5⋅ 2 0.5⋅t 0.5
σ


b
2⋅σ
a
 2⋅

2
σ
σ
2


2
p +1
[ ( )
⋅ exp(− t ) ⋅ 2−0.5 ⋅ σ 2
0.5
]
⋅ t −0.5 dt
and manipulation gives
( ) ∫
p
l (a, b, p ) = 2 2 ⋅ σ 2 2 ⋅
P
b
2
2⋅σ
a
 2⋅ 2 
 σ 
t 2 ⋅ exp(− t )dt
p
Hence the damage is given by
AO
m+ 2


m+ 2
−( m + 2 )
m+2
1
−
m+2
2
2
2 ⋅ K L ⋅θ 4 ⋅ 2 2 ⋅ (σ ) 2 ⋅ ⋅σ 2 t 2 ⋅ exp(− t )dt...
0

D( A) := Time ⋅ freq ⋅ 
m
∞
−
m
m
m

 m
−1
2

+ 2 ⋅ K 2 ⋅θ 4 ⋅ 2 2 ⋅ (σ ) 2 ⋅ AO t 2 ⋅ exp(− t )dt
2⋅σ 2


∫
∫
FRAMEWORK
SESAM
Appendix 8 -4
01-JUN-2001
Program version 2.8
and manipulation gives
AO
m+ 2
 3⋅(m+ 2 )

m+ 2
−( m + 2 )
−1
2
2
2 2 ⋅ K L ⋅θ 4 ⋅ (σ ) 2 ⋅ ⋅σ 2 t 2 ⋅ exp(− t )dt...
0

D( A) := Time ⋅ freq ⋅  3⋅m
m
∞
−
m
m


−1
2
+ 2 2 ⋅ K 2 ⋅θ 4 ⋅ (σ ) 2 ⋅ AO t 2 ⋅ exp(− t )dt



2⋅σ 2
∫
∫
Let
G (a, x ) :=
1
⋅
Γ(a )
∫
x
0
t a −1 ⋅ exp(− t )dt
Then
− (m + 2 )
m+2
 3⋅ ( m + 2 )
 m + 4   m + 4 A0  
−1
2 2
2
4
2
⋅ K L ⋅θ
⋅σ
⋅ Γ
,
 ⋅ G
...
2   2 2 ⋅σ 2  


D( A) := Time ⋅ freq ⋅
−m
m
 3⋅ m
m + 2 
 m + 2 A0   
+ 2 2 ⋅ K 2 −1 ⋅ θ 4 ⋅ σ 2 2 ⋅ Γ
,
 ⋅ 1 − G 


 2  
 2 2 ⋅ σ 2   
( )
( )
FRAMEWORK
Program version 2.8
SESAM
01-JUN-2001
Appendix 9
Pre-processing of Wind Loads by Wajac
Appendix 9 -1
FRAMEWORK
SESAM
Appendix 9 -2
01-JUN-2001
Program version 2.8
DAMS
Standard Axis System
Z'
( X T′ , YT′ , ZT′ )
dZ′
dY′
dX′
Member
unit direction
vectors
Y'
( X B′ , YB′ , Z B′ )
W
V
θ
U
Wind direction
X'
Wind velocity
vector
FIGURE A9.1
In the ( X ' , Y ' , Z ' ) coordinate system the wind force vector acting on a member is
F=
1
⋅ ρ ⋅ Cd ⋅ D ⋅ L ⋅ u′n ⋅ un′
2
where ρ
= mass density of air (1.225kg/m3 = 0.002377lbf·sec2/ft4),
Cd
= the member's drag coefficient,
D
L
u′n
= the member's diameter (including ice thickness as appropriate),
= the length of the member within the specified wind profile,
= the magnitude of the wind velocity vector normal to the member, and
un′
= the wind velocity vector normal to the member.
U 
 
With reference to Figure A9.1, let the wind velocity vector u =  V  and the member unit
W 
 
dX ′ 


direction vector m =  dY ′  .
d 
 Z′ 
FRAMEWORK
Program version 2.8
SESAM
01-JUN-2001
Appendix 9 -3
Transforming u into the ( X ' , Y ' , Z ' ) coordinate system gives
 cosθ

u' =  sin θ
 0

− sin θ
0
 U ⋅ cosθ − V ⋅ sin θ 



0  ⋅ u = U ⋅ sin θ + V ⋅ cosθ 


1 
W


cosθ
0
The wind velocity vector normal to the member un′ is then equal to − (u′ × m ) × m .
Magnitude of the normal vector u′n
As the magnitude of m is 1, the magnitude of the normal vector can be found as
u′n
= u′ × m
=
(d Z ′ ⋅ sin θ ⋅ U + d Z ′ ⋅ cosθ ⋅ V − dY ′ ⋅ W )2
+ (d X ′ ⋅ W − d Z ′ ⋅ cosθ ⋅ U + d Z ′ ⋅ sin θ ⋅ V )2
+ (dY ′ ⋅ cosθ ⋅ U − dY ′ ⋅ sin θ ⋅ V − d X ′ ⋅ sin θ ⋅ U − d X ′ ⋅ cosθ ⋅ V )2
2
Expanding, substituting 1 − d 2X ′ − dY2′ for dZ′
and simplifying gives
(dY2′ ⋅ cos2 θ − 2 ⋅ dY ′ ⋅ cosθ ⋅ d X ′ ⋅ sin θ − dY2′ − d X2 ′ ⋅ cos2 θ + 1)⋅U 2
+ 2 ⋅ (d X2 ′ ⋅ sin θ ⋅ cosθ + dY ′ ⋅ d X ′ − dY2′ ⋅ cosθ ⋅ sin θ − 2 ⋅ dY ′ ⋅ cos 2 θ ⋅ d X ′ )⋅ U ⋅ V
u′n =
− 2 ⋅ d Z ′ ⋅ (sin θ ⋅ dY ′ + cosθ ⋅ d X ′ ) ⋅ U ⋅ W
(
)
+ d X2 ′ ⋅ cos 2 θ + 2 ⋅ dY ′ ⋅ sin θ ⋅ d X ′ ⋅ cosθ + 1 − dY2′ ⋅ cos 2 θ − d X2 ′ ⋅ V 2
+ 2 ⋅ d Z ′ ⋅ (sin θ ⋅ d X ′ − cosθ ⋅ dY ′ ) ⋅ V ⋅ W
(
)
+ d X2 ′ + dY2′ ⋅ W 2
= a12 4444
⋅U 2 + b ⋅U
V + c ⋅ U ⋅3
W + d 2 ⋅V 2 + e ⋅V ⋅ W + f 2 ⋅ W 2
2⋅4444
14444244443
dominant terms
for V << U and W << U
(
non-dominant terms
for V << U and W << U
)
where a 2 = dY2′ ⋅ cos 2 θ − 2 ⋅ dY ′ ⋅ cosθ ⋅ d X ′ ⋅ sin θ − dY2′ − d X2 ′ ⋅ cos 2 θ + 1
(
b = 2 ⋅ d X2 ′ ⋅ sin θ ⋅ cosθ + dY ′ ⋅ d X ′ − dY2′ ⋅ cosθ ⋅ sin θ − 2 ⋅ dY ′ ⋅ cos 2 θ ⋅ d X ′
c = −2 ⋅ d Z ′ ⋅ (sin θ ⋅ dY ′ + cosθ ⋅ d X ′ )
(
d 2 = d X2 ′ ⋅ cos 2 θ + 2 ⋅ dY ′ ⋅ sin θ ⋅ d X ′ ⋅ cosθ + 1 − dY2′ ⋅ cos 2 θ − d X2 ′
e = 2 ⋅ d Z ′ ⋅ (sin θ ⋅ d X ′ − cos θ ⋅ dY ′ )
(
f 2 = d X2 ′ + dY2′
)
)
)
FRAMEWORK
SESAM
Appendix 9 -4
01-JUN-2001
Program version 2.8
Define u′n = A ⋅ U + B ⋅V + C ⋅ W .
Equating the left- and right-hand sides from above gives
a 2 ⋅ U 2 + b ⋅ U ⋅ V + c ⋅ U ⋅ W + d 2 ⋅ V 2 + e ⋅ V ⋅ W + f 2 ⋅ W 2 = ( A ⋅ U + B ⋅ V + C ⋅ W )2
= A2 ⋅ U 2 + 2 ⋅ A ⋅ B ⋅ U ⋅ V + 2 ⋅ A ⋅ C ⋅ U ⋅ W
+ B2 ⋅V 2 + 2 ⋅ B ⋅ C ⋅V ⋅W + C 2 ⋅W 2
Equating the dominant terms only gives
u′n ≈ A ⋅ U + B ⋅V + C ⋅ W
where A = a = 1 − (dY ′ ⋅ sin θ + d X ′ ⋅ cosθ )2
(
)(
)
b
d X ′ ⋅ dY ′ ⋅ sin 2 θ − cos 2 θ + d X2 ′ − dY2′ ⋅ sin θ ⋅ cosθ
B=
=
2⋅ A
1 − (dY ′ ⋅ sin θ + d X ′ ⋅ cosθ )2
C=
− d Z ′ ⋅ (sin θ ⋅ dY ′ + cosθ ⋅ d X ′ )
c
=
2⋅ A
1 − (dY ′ ⋅ sin θ + d X ′ ⋅ cosθ )2
The normal vector u′n
u′n = −(u′ × m ) × m
 − d ⋅ W ⋅ d + d 2 ⋅ cosθ ⋅ U − d 2 ⋅ sin θ ⋅ V + d 2 ⋅ cosθ ⋅ U 
X′
Z′
Z′
Y′

 Z ′

 − d 2′ ⋅ sin θ ⋅ V − d ′ ⋅ d ′ ⋅ sin θ ⋅ U − d ′ ⋅ d ′ ⋅ cosθ ⋅ V
Y
X
Y
X

 Y
2


 − d ′ ⋅ d ′ ⋅ cosθ ⋅ U + d X ′ ⋅ dY ′ ⋅ sin θ ⋅ V + d X ′ ⋅ sin θ ⋅ U

=  X Y


2
2
2
  + d X ′ ⋅ cosθ ⋅ V + d Z ′ ⋅ sin θ ⋅ U + d Z ′ ⋅ cosθ ⋅ V − d Z ′ ⋅ W ⋅ dY ′  


 − d ′ ⋅ d ′ ⋅ sin θ ⋅ U − d ′ ⋅ d ′ ⋅ cosθ ⋅ V + d 2′ ⋅ W 
Y
Z
Y


 Y Z



 + d 2 ⋅ W − d ⋅ d ⋅ cosθ ⋅ U + d ⋅ d ⋅ sin θ ⋅ V 
X ′ Z′
X ′ Z′
 X′



2
Substituting 1 − d 2X ′ − dY2′ for dZ′
and simplifying gives
FRAMEWORK
SESAM
Program version 2.8
01-JUN-2001
  − d ⋅ U ⋅ d ⋅ sin θ − d ⋅ V ⋅ d ⋅ cosθ − V ⋅ sin θ
 
X′
Y′
X′
 Y′


2
2

  + U ⋅ cosθ + d X ′ ⋅ V ⋅ sin θ − d X ′ ⋅ U ⋅ cosθ − d X ′ ⋅ d Z ′ ⋅ W  
  − d 2 ⋅ U ⋅ sin θ − d 2 ⋅ V ⋅ cosθ − d ⋅ U ⋅ d ⋅ cosθ

Y′
Y′
X′

′
un =   Y ′


d
V
d
sin
θ
d
d
W
U
sin
θ
V
cos
θ
+
⋅
⋅
⋅
−
⋅
⋅
+
⋅
+
⋅
  Y′
X′
Y′ Z′

 
2  
  − dY ′ ⋅ d Z ′ ⋅ sin θ ⋅ U − dY ′ ⋅ d Z ′ ⋅ cosθ ⋅ V + W ⋅ dY ′  
 
 
2
  + W ⋅ d X ′ − d X ′ ⋅ d Z ′ ⋅ cosθ ⋅ U + d X ′ ⋅ d Z ′ ⋅ sin θ ⋅ V  
Expressing this in matrix form
U 
 
u′n = H ⋅  V 
W 
 
 hUX ′

where H =  hUY ′
h
 UZ ′
and
hVX ′
hVY ′
hVZ ′
hWX ′ 

hWY ′ 
hWZ ′ 
( )
hUY ′ = sin θ ⋅ (1 − dY2′ ) − dY ′ ⋅ d X ′ ⋅ cosθ
hUX ′ = cosθ ⋅ 1 − d X2 ′ − dY ′ ⋅ d X ′ ⋅ sin θ
hUZ ′ = −d Z ′ ⋅ (dY ′ ⋅ sin θ + d X ′ ⋅ cosθ )
(
)
hVX ′ = − sin θ ⋅ 1 − d X2 ′ − dY ′ ⋅ d X ′ ⋅ cosθ
(
)
hVY ′ = cosθ ⋅ 1 − dY2′ + dY ′ ⋅ d X ′ ⋅ sin θ
hVZ ′ = −d Z ′ ⋅ (dY ′ ⋅ cosθ − d X ′ ⋅ sin θ )
hW X ′ = − d X ′ ⋅ d Z ′
hW Y ′ = − dY ′ ⋅ dZ ′
h W Z ′ = d 2X ′ + dY2′
The wind force vector F
From the above the wind force vector F may now be written as
1
F = ⋅ ρ ⋅ Cd ⋅ D ⋅ L ⋅ ( A ⋅ U + B ⋅ V + C ⋅ W ) ⋅ H ⋅ u
2
Appendix 9 -5
FRAMEWORK
SESAM
Appendix 9 -6
01-JUN-2001
Program version 2.8
The wind velocity vector u may be considered to be the sum of both mean velocities and
time-varying velocities, ie
 U   U mean + U ′ 

  
u =  V  =  Vmean + V ′ 

W  W
   mean + W ′ 
where U mean =
U member _ bottom + U member _ top
2
, Vmean = 0 and Wmean = 0 .
The wind force vector may, therefore, be expressed as
U mean + U ′ 


1
V′
F = ⋅ ρ ⋅ Cd ⋅ D ⋅ L ⋅ ( A ⋅ (U mean + U ′) + B ⋅ V ′ + C ⋅ W ′ ) ⋅ H ⋅ 

2


W′






  2 ⋅ A ⋅ U mean ⋅ U ′ + B ⋅ U mean ⋅ V ′ + C ⋅ U mean ⋅ W ′  





 
2
  + A ⋅ U ′2 + B ⋅ U ′ ⋅ V ′ + C ⋅ U ′ ⋅ W ′

⋅
A
U
 
mean   


1
2



= ⋅ ρ ⋅ Cd ⋅ D ⋅ L ⋅ H ⋅
+  A ⋅ U mean ⋅ V ′ + A ⋅ U ′ ⋅ V ′ + B ⋅ V ′ + C ⋅ V ′ ⋅ W ′ 
0
2
 



0
A ⋅ U mean ⋅ W ′ + A ⋅ U ′ ⋅ W ′ + B ⋅ V ′ ⋅ W ′ + C ⋅ W ′2 


 
 14243 

 steady -state 1444444444
424444444444
3
 terms

time − varying terms


(
(
)
)
For a fatigue analysis the steady-state terms may be ignored. Ignoring also the second-order
time-varying terms gives
 2 ⋅ A ⋅ U mean ⋅ U ′ + B ⋅ U mean ⋅ V ′ + C ⋅ U mean ⋅ W ′ 


1
A ⋅ U mean ⋅ V ′
F = ⋅ ρ ⋅ Cd ⋅ D ⋅ L ⋅ H ⋅ 

2


A ⋅ U mean ⋅ W ′


which may be expressed as
 2 B′ C ′   U mean ⋅ U ′ 

 

1
F = ⋅ ρ ⋅ Cd ⋅ D ⋅ L ⋅ H ⋅ A ⋅  0 1 0  ⋅  U mean ⋅ V ′ 
2
 0 0 1  U


  mean ⋅ W ′ 
(
)(
)
B d X ′ ⋅ dY ′ ⋅ sin 2 θ − cos 2 θ + d X2 ′ − dY2′ ⋅ sin θ ⋅ cosθ
where B′ = =
A
1 − (dY ′ ⋅ sin θ + d X ′ ⋅ cosθ )2
FRAMEWORK
SESAM
Program version 2.8
C′ =
01-JUN-2001
Appendix 9 -7
C − d Z ′ ⋅ (sin θ ⋅ dY ′ + cosθ ⋅ d X ′ )
=
A 1 − (dY ′ ⋅ sin θ + d X ′ ⋅ cosθ )2
The Framework application requires the wind force vector to be passed as three separate
vectors representing the force due to, respectively, U mean ⋅ U ′ (the first load condition
generated by Wajac for each wind direction), Vmean ⋅V ′ (the second load condition generated
by Wajac for each wind direction) and Wmean ⋅ W ′ (the third load condition generated by
Wajac for each wind direction), ie
F = FU mean ⋅U ′ + FU mean ⋅V ′ + FU mean ⋅W ′
 2 B′ C ′  U mean ⋅ U ′ 
 


1
0
= ⋅ ρ ⋅ Cd ⋅ D ⋅ L ⋅ H ⋅ A ⋅  0 1 0  ⋅ 

2

0 0 1  
0
 


0
 2 B′ C ′  


 

1
+ ⋅ ρ ⋅ Cd ⋅ D ⋅ L ⋅ H ⋅ A ⋅  0 1 0  ⋅ U mean ⋅ V ′ 
2
0 0 1  

0

 

0
 2 B′ C ′  


 

1
0
+ ⋅ ρ ⋅ Cd ⋅ D ⋅ L ⋅ H ⋅ A ⋅  0 1 0  ⋅ 

2
 0 0 1  U


  mean ⋅ W ′ 
The Framework application also requires U ′ , V ′ and W ′ to be set equal to 0. 5 ⋅U10 where
U10 is the wind velocity U at a reference height of 10m. Substituting 0. 5 ⋅U10 for U ′ , V ′ and
W ′ gives
2

1
⋅ ρ ⋅ Cd ⋅ D ⋅ L ⋅ H ⋅ A ⋅  0
F=
2
0

2

1
+ ⋅ ρ ⋅ Cd ⋅ D ⋅ L ⋅ H ⋅ A ⋅  0
2
0

2

1
+ ⋅ ρ ⋅ Cd ⋅ D ⋅ L ⋅ H ⋅ A ⋅  0
2
0

B′ C ′  U mean ⋅ 0.5 ⋅ U10 
 

1 0 ⋅
0




0 1 
0

0
B′ C ′  


 
1 0  ⋅ U mean ⋅ 0.5 ⋅ U10 

0
0 1  

0
B′ C ′  


 
0
1 0 ⋅



0 1  U mean ⋅ 0.5 ⋅ U10 