Framework
Transcription
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 All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. Published by: Det Norske Veritas Veritasveien 1 N-1322 Høvik Norway Telephone: Facsimile: E-mail, sales: E-mail, support: Website: +47 67 57 99 00 +47 67 57 72 72 [email protected] [email protected] www.dnv.com If any person suffers loss or damage which is proved to have been caused by any negligent act or omission of Det Norske Veritas, then Det Norske Veritas shall pay compensation to such person for his proved direct loss or damage. However, the compensation shall not exceed an amount equal to ten times the fee charged for the service in question, provided that the maximum compensation shall never exceed USD 2 millions. In this provision “Det Norske Veritas” shall mean the Foundation Det Norske Veritas as well as all its subsidiaries, directors, officers, employees, agents and any other acting on behalf of Det Norske Veritas. 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 Page 1- 1 2- 1 3- 1 4- 1 4- 1 4- 5 4- 5 5- 1 6- 1 7- 1 8- 1 8- 1 8- 3 8- 4 9- 1 9- 1 9- 4 9- 6 9- 7 10- 1 11- 1 11- 1 11- 1 11- 3 11- 4 11- 4 12- 1 12- 1 12- 1 12- 2 12- 4 13- 1 13- 1 13- 3 14- 1 14- 1 14- 1 FRAMEWORK ii 01-JUN-2001Program version 2.8 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 SESAM 01-JUN-2001 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. FRAMEWORK Program version 2.8 2.0 SESAM 01-JUN-2001 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. FRAMEWORK Program version 2.8 3.0 SESAM 01-JUN-2001 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 FRAMEWORK 3-2 SESAM 01-JUN-2001 Program version 2.8 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 FRAMEWORK Program version 2.8 SESAM 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. FRAMEWORK 4-2 SESAM 01-JUN-2001 Program version 2.8 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 FRAMEWORK Program version 2.8 SESAM 01-JUN-2001 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 FRAMEWORK 4-4 SESAM 01-JUN-2001 Program version 2.8 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 FRAMEWORK Program version 2.8 4.2 SESAM 01-JUN-2001 4-5 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). FRAMEWORK Program version 2.8 5.0 SESAM 01-JUN-2001 5-1 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. FRAMEWORK Program version 2.8 6.0 SESAM 01-JUN-2001 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). FRAMEWORK 6-2 SESAM 01-JUN-2001 FIGURE 6.1 SPECTRUM OF HORIZONTAL WIND SPEED (AFTER VAN DER HOVEN, REFERENCE 3.1) Program version 2.8 FRAMEWORK SESAM Program version 2.8 7.0 03-10-09 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 FRAMEWORK SESAM 7-2 03-10-09 Program version 2.8 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) FRAMEWORK SESAM Program version 2.8 03-10-09 7-3 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 FRAMEWORK 7-4 SESAM 03-10-09 Program version 2.8 FIGURE 7.1 GUST SPECTRA FOR WIND STATES 1-11 IN TABLES 7.1 AND 7.2 (SI UNITS) FRAMEWORK SESAM Program version 2.8 8.0 01-JUN-2001 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 FRAMEWORK 8-2 SESAM 01-JUN-2001 Program version 2.8 FIGURE 8.1 VARIATION OF DRAG COEFFICIENT FOR A CYLINDRICAL MEMBER WITH REYNOLDS' NUMBER FRAMEWORK SESAM Program version 2.8 01-JUN-2001 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. FRAMEWORK SESAM 8-4 8.3 01-JUN-2001 Program version 2.8 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 FRAMEWORK SESAM Program version 2.8 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 ) FRAMEWORK SESAM 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 FRAMEWORK 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. FRAMEWORK 9-2 SESAM 01-JUN-2001 Program version 2.8 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 FRAMEWORK Program version 2.8 SESAM 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 FRAMEWORK 9-4 SESAM 01-JUN-2001 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. FRAMEWORK Program version 2.8 SESAM 01-JUN-2001 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. FRAMEWORK 9-6 SESAM 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. FRAMEWORK Program version 2.8 9.4 SESAM 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. FRAMEWORK Program version 2.8 10.0 SESAM 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 SESAM 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. FRAMEWORK 11-2 SESAM 01-JUN-2001 FIGURE 11.1 Program version 2.8 FRAMEWORK SESAM 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. FRAMEWORK 11-4 SESAM 01-JUN-2001 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. FRAMEWORK Program version 2.8 SESAM 01-JUN-2001 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 SESAM 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 SESAM 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 SESAM 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 SESAM 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 ( ) 12 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) FRAMEWORK SESAM 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 SESAM 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 SESAM 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. FRAMEWORK Program version 2.8 SESAM 01-JUN-2001 APPENDIX 3 Algebra Supporting Section 10.0: Condensation of Forces Appendix 3 -1 FRAMEWORK SESAM 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 FRAMEWORK SESAM 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 SESAM 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 SESAM 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 FRAMEWORK SESAM 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 FRAMEWORK Program version 2.8 SESAM 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 FRAMEWORK SESAM Program version 2.8 01-JUN-2001 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 ) FRAMEWORK SESAM 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 SESAM 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 SESAM 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 SESAM 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 FRAMEWORK SESAM 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 SESAM 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