development of a reliability-based design …s3.amazonaws.com/zanran_storage/ lighting structural...
TRANSCRIPT
Report No. SRR-91 Final Report
DEVELOPMENT OF A RELIABILITY-BASED DESIGN PROCEDURE FOR HIGH-MAST LIGHTING STRUCTURAL SUPPORTS IN COLORADO John W. van de Lindt Jonathan S. Goode
March 2006 COLORADO DEPARTMENT OF TRANSPORTATION SAFETY AND TRAFFIC ENGINEERING BRANCH AND STAFF BRIDGE BRANCH
The contents of this report reflect the views of the
authors, who are responsible for the facts and
accuracy of the data presented herein. The contents
do not necessarily reflect the official views of the
Colorado Department of Transportation. This
report does not consitute a standard, specification,
or regulation. Any information contained herein
should be used at the users own risk.
i
Technical Report Documentation Page 1. Report No. Structural Research Report (SRR)-91
2. Government Accession No.
3. CDOT Project Manager Richard Osmun
4. Title and Subtitle DEVELOPMENT OF A RELIABILITY-BASED DESIGN PROCEDURE FOR HIGH-MAST LIGHTING STRUCTURAL SUPPORTS IN COLORADO
5. Report Date March 2006
7. Author(s) John W. van de Lindt and Jonathan S. Goode
6. Performing Organization Code Colorado State University
9. Performing Organization Name and Address Colorado State University Department of Civil Engineering Campus Mail Stop 1372 Fort Collins, CO 80523-1372
8. Performing Org Report No. SRR-91
10. Work Unit No. (TRAIS)
12. Sponsoring Agency Name and Address Colorado Department of Transportation – Safety and Traffic Engineering Branch 4201 E. Arkansas Ave. Denver, CO 80222
11. Contract Number:
13. Type of Report & Period Covered Final Report, 2004-2005
15. Supplementary Notes
14. Sponsoring Agency Code File: 81.30
16. Abstract High mast lighting structures are being used to provide illumination for large intersections, particularly for highways located in rural areas. These structures, ranging from approximately 50 – 130 ft (15 – 40 m) in height, are exposed to high wind forces that in turn produce a tremendous number of loading cycles each year. A recent high mast lighting structural support failure in the high plains of Colorado near Denver International Airport provided the impetus for this study. Specifically, a numerical investigation to determine the nature of the complex dynamic response, estimate the fatigue life, and determine the effect of extreme wind gusts known as micro-bursts on this dynamic response as well as the effect on the resulting fatigue life. These high-mast structures are less than 3.28 ft (1 m) in diameter and are quite flexible relative to many civil engineering structures. This flexibility results in large deformations when compared to their diameter, i.e. when combined with the height of these structures. Furthermore, large forces and moments at the base are produced that result in large stresses and stress reversals during multi-mode excitation. Morison’s equation, which provides relative force for slender bodies as a function of flow velocity, was applied within a dynamic finite element framework in order to account for the relative motion between the wind and the motion of the structure. Then, a well-known random vibrations approach was coupled with Miner’s rule to estimate the fatigue life of the structural support. Six different design parameters as well as mean wind velocities were examined and an approximate reliability-based design procedure was developed based on these results. Several examples are presented to illustrate the new approach. However, in order for the approach to be appropriately applied to high-mast lighting structural supports in Colorado it is strongly recommended that a state-wide wind study be undertaken to provide accurate reliability indices for all traffic and safety structures such as signal poles, overhead signs, and high-mast lights. 17. Key Words: high mast lighting, fatigue, structural
reliability, steel design, wind loading
18. Distribution Statement No restrictions. This document is available to the public through: National Technical Information Service 5825 Port Royal Road Springfield, VA 22161
19. Security Classification (report) None
20. Security Classification (Page) None
21. No of Pages 99
22. Price
ii
DEVELOPMENT OF A RELIABILITY-BASED DESIGN
PROCEDURE FOR HIGH-MAST LIGHTING
STRUCTURAL SUPPORTS IN COLORADO
by
John W. van de Lindt, Associate Professor
Jonathan S. Goode, Ph.D. Student
Report No. SRR-91
Sponsored by the
Colorado Department of Transportation
March 2006
Colorado Department of Transportation
Safety and Traffic Engineering Branch
4201 E. Arkansas Ave.
Denver, CO 80222
(303) 757-9506
iii
ACKNOWLEDGEMENTS
This project was funded by the Colorado Department of Transportation (CDOT) – Safety and
Traffic Engineering Branch and Staff Bridge Branch. That support is gratefully acknowledged
by the authors. Dick Osmun, the project manager, provided advice and support during the
project and the authors thank him for his assistance. The second author would also like to thank
the American Institute of Steel Construction (AISC) – Rocky Mountain Region, for providing a
Graduate Fellowship, which provided funding for the latter portion of his participation in this
study.
Opinions expresses in this report are, however, those of the writers and do not necessarily reflect
those of CDOT or AISC.
iv
EXECUTIVE SUMMARY
High mast lighting structures are being used to provide illumination for large intersections,
particularly for highways located in rural areas. These structures, ranging from approximately
50 – 130 ft (15 – 40 m) in height, are exposed to high wind forces that in turn produce a
tremendous number of loading cycles each year. A recent high mast lighting structural support
failure in the high plains of Colorado near Denver International Airport provided the impetus for
this study. Specifically, a numerical investigation to determine the nature of the complex
dynamic response, estimate the fatigue life, and determine the effect of extreme wind gusts
known as micro-bursts on this dynamic response as well as the effect on the resulting fatigue life.
These high-mast structures are less than 3.28 ft (1 m) in diameter and are quite flexible relative
to many civil engineering structures. This flexibility results in large deformations when
compared to their diameter, i.e. when combined with the height of these structures. Furthermore,
large forces and moments at the base are produced that result in large stresses and stress reversals
during multi-mode excitation. Morison’s equation, which provides relative force for slender
bodies as a function of flow velocity, was applied within a dynamic finite element framework in
order to account for the relative motion between the wind and the motion of the structure. Then,
a well-known random vibrations approach was coupled with Miner’s rule to estimate the fatigue
life of the structural support. Six different design parameters as well as mean wind velocities
were examined and an approximate reliability-based design procedure was developed based on
these results. Several examples are presented to illustrate the new approach.
The procedure for the development of the reliability-based design procedure can be broken into
six steps. These steps constitute the six major components of the general analysis procedure
presented in this report. This general analysis procedure is briefly described as:
Step 1: Construction of the finite element model of the HML structural support.
Step 2: Fatigue analysis performed in order to determine the fatigue life of the structure
for a specified wind velocity. Within the fatigue analysis, Steps 3 – 5 must also
be performed in a repetitive fashion.
v
Step 3: Construction of the wind-loading model to determine the loading applied to the
finite element model.
Step 4: Dynamic analysis performed to determine the motion of the system as a function
of time.
Step 5: Resolve non-linearity of the wind-loading model applied to the finite element
model.
Step 6: Reliability analysis performed in order to determine the reliability index for a
specified target fatigue life.
However, in order for the approach to be appropriately applied to high-mast lighting structural
supports in Colorado it is strongly recommended that a state-wide wind study be undertaken to
provide accurate reliability indices for all traffic and safety structures such as signal poles,
overhead signs, and high-mast lights.
vi
TABLE OF CONTENTS
SECTION PAGE
CHAPTER 1 INTRODUCTION
1.1 Background and Motivation ..........................................................................................1
1.2 Scope of Research Project and Objectives.....................................................................3
1.3 Summary of Current Design Guidelines: AASHTO 2001 ............................................4
1.4 Overview of Report........................................................................................................7
CHAPTER 2 ANALYSIS PROCEDURES AND METHODS
2.1 General Analysis Procedure...........................................................................................8
2.2 Finite Element Model ..................................................................................................10
2.3 Wind Load Model ........................................................................................................14
2.4 Dynamic Analysis........................................................................................................21
2.5 Relative Motion ...........................................................................................................23
2.6 Fatigue Analysis...........................................................................................................25
2.7 Reliability Analysis......................................................................................................31
CHAPTER 3 SENSITIVITY AND RELIABILITY ANALYSES
3.1 Sensitivity Analysis Background.................................................................................34
3.2 The Mean Wind Velocity.............................................................................................38
3.3 Sensitivity Analysis .....................................................................................................39
3.4 Reliability Analysis......................................................................................................47
CHAPTER 4 RELIABILITY-BASED DESIGN METHODOLOGY
4.1 Design Methodology Background ...............................................................................67
4.2 Design Charts – Single Variable..................................................................................68
4.3 Design Method – Single Variable................................................................................68
4.4 Design Charts – Multiple Variables.............................................................................74
4.5 Design Method – Multiple Variables...........................................................................74
vii
SECTION PAGE
CHAPTER 5 ILLUSTRATIVE DESIGN EXAMPLES FOR FATIGUE PERFORMANCE
5.1 Example 1 – Single Variable .......................................................................................78
5.2 Example 2 – Multiple Variables ..................................................................................79
CHAPTER 6 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS..........................81
REFERENCES ..............................................................................................................................83
APPENDIX A
A.1 Vortex-Induced-Vibration......................................................................................... A-1
A.2 Selected Mean Wind Velocities................................................................................ A-3
viii
LIST OF FIGURES
FIGURE PAGE
1-1 HML Structure in Colorado .....................................................................................1
1-2 Base-Line Reliability Indices for HML Structural Support.....................................7
2-1 General Analysis Procedure.....................................................................................9
2-2 Finite Element Model Procedure ...........................................................................11
2-3 Boundary Support Conditions Model ....................................................................13
2-4 Wind Velocity Time Series Procedure ..................................................................15
2-5 In-Line Wind Velocity Power Spectrum ...............................................................16
2-6 In-Line Wind Velocity Time Series.......................................................................18
2-7 Wind Loading Procedure .......................................................................................18
2-8 Logarithmic Wind Velocity Profile .......................................................................19
2-9 Drag Coefficient for a Smooth Cylinder................................................................21
2-10 Dynamic Analysis Procedure.................................................................................22
2-11 Relative Motion Procedure ....................................................................................24
2-12 Relative Velocity of Wind and HML Structure.....................................................25
2-13 Fatigue Analysis Procedure ...................................................................................26
2-14 Lognormal PDF .....................................................................................................30
2-15 Lognormal PDF Bins .............................................................................................30
2-16 Reliability Analysis Procedure ..............................................................................31
3-1 Benchmark HML Structural Support in Colorado.................................................35
3-2 Wind Velocity Parent Distribution ........................................................................39
3-3 Fatigue Life Sensitivity – Pole Outside Diameter .................................................40
3-4 Fatigue Life Sensitivity – Pole Wall Thickness.....................................................41
3-5 Fatigue Life Sensitivity – Pole Length ..................................................................42
3-6 Fatigue Life Sensitivity – Luminaire Structure Weight.........................................42
3-7 Fatigue Life Sensitivity – Luminaire Structure Projected Area.............................43
3-8 Fatigue Life Sensitivity – Structure Damping .......................................................44
3-9 Fatigue Life Sensitivity – Wind Velocity COV.....................................................45
ix
FIGURE PAGE
3-10 Fatigue Life Sensitivity – Wind Gust ....................................................................45
3-11 Reliability Analysis – Pole Outside Diameter .......................................................48
3-12 Reliability Analysis – Pole Wall Thickness...........................................................51
3-13 Reliability Analysis – Pole Length ........................................................................54
3-14 Reliability Analysis – Luminaire Structure Weight...............................................57
3-15 Reliability Analysis – Luminaire Structure Projected Area ..................................60
3-16 Reliability Analysis – Structure Damping .............................................................63
3-17 Reliability Analysis – Wind Velocity COV...........................................................65
5-1 Example 1 – Single Variable .................................................................................78
5-2 Example 2 – Multiple Variables ............................................................................80
A-1 Selected Locations in Colorado .......................................................................... A-4
x
LIST OF TABLES
TABLE PAGE
2-1 Common Reliability Indices, β ..............................................................................33
3-1 HML Structural Support Properties – Pole Sections .............................................36
3-2 HML Structural Support Properties – Joint Sections.............................................37
3-3 HML Structural Support Properties – Boundary Support Conditions ...................37
3-4 HML Structural Support Properties – Luminaire Structure Properties .................38
4-1 Benchmark HML Structural Support Properties....................................................69
4-2 Pole Outside Diameter Properties..........................................................................70
4-3 Pole Wall Thickness Properties .............................................................................70
4-4 Pole Section Length Properties..............................................................................71
4-5 Luminaire Structure Projected Area Properties .....................................................71
4-6 Design Method Confirmation Examples – Single Variable ..................................72
4-7 Design Method Confirmation Summary – Single Variable...................................72
4-8 Design Method Confirmation Properties – Single Variable ..................................73
4-9 Design Method Confirmation Percent Changes – Single Variable .......................73
4-10 Design Method Confirmation Examples – Multiple Variables .............................75
4-11 Design Method Confirmation Summary – Multiple Variables..............................76
4-12 Design Method Confirmation Properties – Multiple Variables.............................76
4-13 Design Method Confirmation Percent Changes – Multiple Variables ..................77
A-1 Wind Speed and Vibration Mode Combinations at which Lock-In was
Numerically Identified ........................................................................................ A-3
A-2 Mean Wind Velocities for Selected Locations in Colorado ............................... A-4
1
CHAPTER 1
INTRODUCTION
1.1 Background and Motivation
The Colorado Department of Transportation (CDOT) maintains a large number of high mast
lighting (HML) structures in the state of Colorado. These structures, as shown in Figure 1-1, are
primarily used to illuminate large intersections along major arterials and in rural areas. Quite
often, HML structures have heights ranging from 50 – 130 ft (15 – 40 m) with base diameters of
approximately 1.5 ft (460 mm). The main advantage of using HML structures, as opposed to
standard luminaire structures, is the substantial reduction in the total number of structures
required for a specified area.
Figure 1-1: HML Structure in Colorado
On April 11, 2004, two HML structures located at the intersection of E-470 (a toll road) and
Pena Boulevard near Denver International Airport (DIA) failed. These failures resulted in
several HML structures being retrofitted and many more being constantly monitored in that area.
As a result, a substantial economic investment by the E-470 Authority has been required
following these collapses. As always, the safety of the general public is a primary concern since
many HML structures in Colorado, as well as other states, are relatively close to the roadway.
2
This combination of economic factors and safety concerns has provided the impetus for this
study.
An investigation conducted by the Advanced Technology for Large Structural Systems (ATLSS)
Engineering Research Center at Lehigh University (Kaufmann, 2005) revealed several
possibilities for the failure of these structures. Conclusions concerning one of the collapsed
structures indicated failure primarily due to the initiation and propagation of fatigue cracks
occurring under high stress cycles. Further investigation of the collapsed structure showed little
evidence of significant corrosion on the crack surface. Reasons for this could be due to the
evidence of high crack propagation rates. Thus, the failure of the structure was likely due to a
short time period event. Indeed, during the hours preceding the failure, there were reports of a
high wind weather event in the area near DIA.
During the investigation, Kaufmann (2005) also conducted chemical and strength property
analyses of the structural steel tubing comprising the HML structural support. The chemical
composition and tensile properties were similar and consistent with Grade 60 steel utilized in
fabricating structural steel tubing. The investigation concluded that the properties of the
structural steel tubing did not directly contribute to the development of the fatigue cracks.
Kaufmann’s investigation also made note of several quality assurance issues that may have
contributed to the failure of these structures. Concerning the collapsed structure, the specified
wall thickness of the structural steel tubing according to design documents was to be 0.25 in
(6.35 mm). However, the actual wall thickness was measured as 0.235 in (5.97 mm), or 6% less
than specified requirements. The sensitivity of fatigue life to wall thickness is significant, as will
be shown in Chapter 3 of the current report. Other structures sent to ATLSS, none of which
collapsed but did show signs of fatigue cracking, had specified wall thicknesses of 0.1875 in
(4.76 mm). Actual measurements indicated wall thicknesses of 0.189 in (4.80 mm), or 0.8%
above specified requirements. One final structure was also tested that did not show any signs of
fatigue cracking. The specified wall thickness was given as 0.25 in (6.35 mm). The actual wall
thickness was determined to be 0.230 in (5.84 mm), or 8% less than specified requirements.
3
The American Society for Testing Materials (ASTM) Standard A595 (ASTM, 2002), Standard
Specification for Steel Tubing, Low-Carbon, Tapered for Structural Use, specifies dimensions
and tolerances related to the HML structural supports. The specification covers three grades of
seam-welded, tapered steel tubes for structural use with diameters ranging from 2.375 – 30 in
(60.325 – 762.0 mm) and wall thicknesses ranging from 0.1046 – 0.375 in (2.657 – 9.525 mm).
Tolerances for wall thickness are given as +10% to –5% of the specified wall thickness exclusive
of the weld area. Based on Kaufmann’s investigation, the collapsed structure, with an actual
wall thickness of 6% less than specified requirements, falls outside the acceptable tolerance
based on the ASTM standard. The structure that showed no signs of fatigue cracking also falls
outside the acceptable range of tolerance. Finally, the structures that did show signs of fatigue
cracking, with actual wall thicknesses 0.8% greater than specified, did fall within the ASTM
tolerance range.
1.2 Scope of Research Project and Objectives
Due to the overwhelming evidence of fatigue damage being the primary cause for the failure of
the aforementioned HML structural supports, the focus of this project is on fatigue damage
induced by cyclic loading. The American Association of State Highway and Transportation
Officials (AASHTO, 2001) define fatigue as “damage resulting in fracture caused by stress
fluctuations”. Due primarily to wind fluctuations, HML structures are subjected to a tremendous
number of loading cycles each year. These wind fluctuations in combination with the height of
these structures result in large forces and moments at their bases. Adding to the large number of
potential loading cycles is the possibility of higher modes of vibration. For slender structures of
this height, a common occurrence can be the presence of higher modes of vibration thus
substantially increasing an already large number of loading cycles. There is also the possibility
of vortex-induced vibration, commonly referred to as vortex shedding, which results in a
phenomenon known as lock-in. A preliminary investigation into HML structure lock-in
frequencies is included in this report for completeness.
This project will develop a semi-prescriptive design procedure for HML structural supports in
Colorado. This design procedure will statistically provide a minimum, or target, structural
4
reliability index, as prescribed by CDOT. The design procedure will consider many factors
affecting HML structural supports. At the front of this effort are wind-engineering principles
concerning fluid/structure interaction that will help to determine the dynamic response of the
structure to a simulated wind event. Using a finite element approach, the resulting forces and
stresses produced at the base of the structure can then be determined. A linear damage
accumulation model will predict the fatigue life of the HML structural support. Probabilistic
methods will be used to determine reliability indices and then transformed into appropriate
design guidelines. Base-line reliability indices for existing structural supports will also be
determined.
1.3 Summary of Current Design Guidelines: AASHTO 2001
AASHTO 2001 Standard Specifications for Structural Supports for Highway Signs, Luminaires
and Traffic Signals is the national standard for the current design guidelines for HML structural
supports. This section considers the current design guidelines, AASHTO 2001, applied to the
benchmark HML structural support outlined in Chapter 3. AASHTO considers two wind-
loading cases for fatigue design for HML structural supports. The intent of the fatigue design is
that the connection detail or structural member, defining a detail category, will have infinite
fatigue life and, as such, stresses produced will remain below a threshold limit. The detail
category used for the HML structural support is questionable. Based on the attachment of the
HML structural support to the base plate with a 0.25 in (6.35 mm) thick backing ring attached to
both the HML structural support and the base plate with seam welds, the detail category is E′ if
the backing ring is not removed accordingly. However, this detail category is very undesirable
in terms of fatigue design. Thus, it is possible that a detail category of E could be achieved by
removing the backing ring after the initial placement of the HML structural support and welding
the pole to the base plate. For completeness, this report will consider both detail categories.
AASHTO defines fatigue importance factors for the two wind-loading cases. Assuming that all
HML structural supports are located on major highways and are critical structures, an importance
factor category of I is assigned. Thus, for the two wind-loading cases, the fatigue importance
factor, IF, is 1.0.
5
Natural Wind Gust
The equivalent static natural wind gust pressure, PNW, is defined as,
(Pa)I250CP
(psf)I5.2CP
FdNW
FdNW
=
= (1-1)
where IF is 1.0 and the drag coefficient, Cd, is determined as a function of a conversion factor, Cv
= 1 for a 3-second 50-year wind gust, the 50-year wind velocities of 90 MPH (144.84 km/hr),
100 MPH (160.93 km/hr), and 110 MPH (177.03 km/hr), and the average diameter of each
section, d. The drag coefficient is determined to be 0.45 for all sections and all wind velocities.
The luminaire structure has a drag coefficient of 1.0 as given by CDOT design specifications
referenced in Chapter 3 of this report. The equivalent static natural wind gust pressure along the
height of the pole is calculated as 2.34 psf (112.5 Pa) for all three wind velocities above and the
resulting pressure on the luminaire structure is calculated as 5.2 psf (250 Pa). Applying this
pressure along the height of the pole, making note of the reduction in outside diameter as height
increases, and on the luminaire structure, the overturning moment at the base of the structure is
calculated to be 345 k-in (38.98 kN-m). The stress due to this overturning moment at the base of
the structure is finally calculated to be 1.81 ksi (12.41 MPa).
For detail category E, AASHTO indicates that the constant-amplitude fatigue threshold is given
as 4.5 ksi (31 MPa). Thus, the stress due to the overturning moment produced by the equivalent
static natural wind gust is lower than the threshold indicated. For detail category E′, AASHTO
indicates that the constant-amplitude fatigue threshold is given as 2.6 ksi (18 MPa). Thus, the
stress due to the overturning moment produced by the equivalent static natural wind gust is also
lower than the threshold indicated.
Vortex-Shedding
The second wind-loading case for fatigue that AASHTO requires HML structural supports to be
checked is vortex shedding. A more complete discussion of this topic is provided in the
appendix of this report. AASHTO indicates that the critical wind velocity, Vc, at which vortex
shedding lock-in can occur is,
6
n
nc S
V df= (1-2)
where fn is the first transverse natural frequency of the structure in cyc/sec, d is the diameter of the
structure in ft (or m), and Sn is the Strouhal number given as 0.18 for circular sections. From a
finite element program written by the authors, fn is determined to be 0.5275 cyc/sec and d is taken
as the average diameter of the structure, 1.5075 ft (0.4595 m). Thus, the critical wind velocity at
which vortex shedding lock-in can occur is calculated to be 4.4178 ft/s (1.3465 m/s) or 3.01
MPH (4.848 km/hr). The vortex shedding model discussed in the appendix for the first
transverse mode confirms this same velocity for the HML.
Using the critical wind velocity, the equivalent static vortex shedding pressure, PVS, can be
calculated as,
(Pa)2ξ
IC0.613VP
(psf)2ξ
IC0.00118VP
Fd2C
VS
Fd2C
VS
=
=
(1-3)
where ξ is the damping ratio given as 0.005 by AASHTO. Using the critical wind velocity as
4.4178 ft/s (1.3465 m/s), the drag coefficient as 1.10 updated for a wind velocity of 3.01 MPH
(4.848 km/hr), and the importance factor as 1.0, the equivalent static vortex shedding pressure is
calculated as 2.53 psf (121.14 Pa). Noting that this pressure is not an appreciable increase over
the pressure caused by a natural wind gust, 2.34 psf (112.5 Pa), it is concluded that the stress due
to the overturning moment at the base of the structure will not exceed the constant-amplitude
fatigue threshold for detail category E of 4.5 ksi (31 MPa) or E′ of 2.6 ksi (18 MPa).
Base-Line Reliability Indices
The benchmark HML structural support considered in this study, further discussed in Chapter 3,
has a reliability index for a given wind velocity and target fatigue life. Figure 1-2 depicts the
base-line reliability indices for this benchmark HML structural support in Colorado for target
7
fatigue life’s of 25 years, 50 years, and 75 years for detail category E, figure (a), and detail
category E′, figure (b). As will be discussed in Chapter 3 of this report, wind velocities in Figure
1-2 remain as mean or average wind velocities over the life of the structure. Also in Chapter 3 of
this report, several characteristics of this benchmark structure will be varied and analyzed.
(a): Detail Category E (b): Detail Category E′
Figure 1-2: Base-Line Reliability Indices for HML Structural Support
1.4 Overview of Report
Chapters 2 – 5 consist of the development of the design procedure as previously discussed.
Chapter 2 illustrates an outline of the general procedure used in modeling the HML structural
support. Each step of the general procedure is further broken into individual procedures that
contribute to the end result. The approach used in this study regarding the mathematical
formulation is also presented to explain the theoretical derivation encompassing each step of the
procedure. Chapter 3 presents a sensitivity analysis of the fatigue life for given wind-loading
events with respect to variations in properties such as structure height, pole diameter, and pole
wall thickness among others. Reliability indices for various cases are also presented. Chapter 4
presents design guidelines as a result of the analyses shown in Chapter 3. Using the results of
Chapter 4, two illustrative design examples for fatigue performance are presented in Chapter 5.
A step-by-step method is given for both examples considered. Chapter 6 summarizes the design
procedure as given in Chapter 4 and illustrated in Chapter 5 of this study. Conclusions based on
this study are offered. Recommendations for future work and consideration are also given.
8
CHAPTER 2
ANALYSIS PROCEDURES AND METHODS
2.1 General Analysis Procedure
The procedure for the development of the reliability-based design procedure can be broken into
six steps. These steps, as outlined in the flowchart shown in Figure 2-1, constitute the six major
components of the general analysis procedure presented in this report. This general analysis
procedure is briefly described as:
Step 1: Construction of the finite element model of the HML structural support.
Step 2: Fatigue analysis performed in order to determine the fatigue life of the structure
for a specified wind velocity. Within the fatigue analysis, Steps 3 – 5 must also
be performed in a repetitive fashion.
Step 3: Construction of the wind-loading model to determine the loading applied to the
finite element model.
Step 4: Dynamic analysis performed to determine the motion of the system as a function
of time.
Step 5: Resolve non-linearity of the wind-loading model applied to the finite element
model.
Step 6: Reliability analysis performed in order to determine the reliability index for a
specified target fatigue life.
9
Construct FiniteElement Model(Section 2.2)
Apply FatigueAnalysis
(Section 2.6)
Construct WindLoading Model(Section 2.3)
Resolve Non-LinearWind Load
(Section 2.5)
Solve DynamicAnalysis of the
System(Section 2.4)Apply Reliability
Analysis(Section 2.7)
Figure 2-1: General Analysis Procedure
As indicated in Figure 2-1, each step corresponds to a section in Chapter 2 of this report.
Sections 2.2 – 2.7 further explain each of these steps. The theoretical details behind the models
and analyses used are discussed. This project only considered in-line wind velocity and the in-
line motion produced. As such, all details presented herein will be for two-dimensional analysis.
Changes to the finite element model, wind loading model, and stress combination would only be
needed for three-dimensional analysis. A discussion on vortex-induced-vibration, transverse
motion produced by an in-line wind velocity, is included in Appendix A of this report.
Section 2.2 discusses the finite element model and the procedure used in its construction. The
types of elements used are further explained for both stiffness and mass matrices. Application of
the boundary support and luminaire structure conditions is also discussed. Finally, the details on
the derivation of the structure damping matrix are presented. Section 2.3 discusses the wind-
loading model and the procedure used in its construction. The in-line wind velocity power
spectrum is presented. The procedure for determining the artificial wind velocity time series is
given. Finally, the details on the logarithmic profile and Morison’s equation are presented.
Section 2.4 details the dynamic analysis procedure as well as the theoretical details for its use.
The Newmark-Beta method is further discussed giving a step-by-step procedure for its use in this
project. Section 2.5 details the relative motion procedure. The calculation of the relative
velocity and application of Morison’s equation to obtain the updated forcing function to be used
in the dynamic analysis are presented. Section 2.6 details the fatigue analysis procedure. The
10
method proposed by Crandall and Mark (1963), using the Palmgren-Miner rule, is derived. The
division of the lognormal probability density function (PDF) for wind velocity used to describe
the loading event is presented. Finally, Section 2.7 details the reliability analysis procedure.
Determination of the probability of failure and reliability index is also presented.
2.2 Finite Element Model
The procedure for constructing the finite element model of the HML structural support is
presented in Figure 2-2. The purpose for using a finite element approach is to determine the
properties of the structure in a discretized manner. These properties are then directly related to
solving the dynamic motion of the system using the equation of motion, which can be expressed
as,
[ ]{ } [ ]{ } [ ]{ } { })(tFxKxCxM =++ (2-1)
where the three matrices, M, C, and K, are the mass, damping, and stiffness matrices,
respectively. The forcing function, F(t), is nonlinear due to the relative motion between the wind
flow and the structure, and is discussed further in Section 2.3. The vectors, x, x , and x ,
represent the position, velocity, and acceleration, respectively, of the nodal points, or discretized
points representing the continuous system, of the structure.
11
Loop throughNumber of Finite
Elements
Determine ElementStiffness Matrix
Determine ElementMass Matrix
End Loop
Apply BoundarySupport Conditions
Apply LuminaireStructure Conditions
Determine StructureDamping Matrix
Assemble StructureStiffness Matrix
Assemble StructureMass Matrix
Figure 2-2: Finite Element Model Procedure
The procedure presented in Figure 2-2 begins with the assembly of K and M. Each element of
the discretized structure has its own stiffness and mass matrix. These matrices are referred to as
the element stiffness and mass matrices. The elements used to construct the finite element model
are six degree-of-freedom beam elements (twelve degree-of-freedom beam elements for three-
dimensional analysis). Each end of the element has 3 degrees-of-freedom: axial deformation,
shear deformation, and bending-moment rotation. The element stiffness matrix can be expressed
as (Paz, 2004),
12
[ ]
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−
−
−−−
−
−
=
3232
22
3232
22
12061206
0000
604602
12061206
0000
602604
LEI
LEI
LEI
LEI
LEA
LEA
LEI
LEI
LEI
LEI
LEI
LEI
LEI
LEI
LEA
LEA
LEI
LEI
LEI
LEI
K e (2-2)
where E is the modulus of elasticity of the material comprising the element, A is the constant
cross-sectional area of the element, I is the constant moment of inertia of the element, and L is
the length of the element. The element mass matrix can be expressed as (Paz, 2004),
[ ]
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−−−−
−
=
156022540130140007002204130354013156022070001400
13032204
420 22
22
LL
LLLLLL
LLLL
M HMLe
ρ (2-3)
where ρHML is the mass density per volume of the material comprising the element. Using a
transformation matrix, T, from local coordinate directions to global coordinate directions, the
stiffness and mass matrices used in Equation (2-1) are formed by,
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]TMTM
TKTK
eT
eT
=
= (2-4)
Finally, all element stiffness and mass matrices are assembled into the structure stiffness and
mass matrices. This is accomplished based on a numbering or indexing scheme assigned to the
degrees-of-freedom for the structure.
13
Boundary support conditions are applied to model the attachment of the HML structural support
to its foundation. These support conditions are represented by linear translational and rotational
springs placed at the base of the HML structural support, as shown in Figure 2-3, and are a
function of the foundation type and soil properties under the foundation. Because these springs
are considered to have no mass, only changes to the structure stiffness matrix are required.
RotationalSpring
TranslationalSpring
x
y
Figure 2-3: Boundary Support Conditions Model
Application of this type of model allows for experimental research to be conducted on actual
foundation connections. For this project, however, the foundation was modeled as a rigid
support. A rigid support is a conservative assumption whereas a foundation that allowed some
movement would actually increase the fatigue life of the structure.
Luminaire structure conditions are applied to model the luminaire structure attached at the top of
the HML structural support. The luminaire is considered to have a specified mass and projected
area for determining wind load. The mass is evenly distributed to translational degrees-of-
freedom to the top point of the structure mass. No contribution to the structure stiffness is
assumed.
The last step as indicated by Figure 2-2 is to determine the structure damping matrix, C. This
matrix is a linear combination of the structure stiffness and mass matrices and is given as
(Chopra, 2001),
[ ] [ ] [ ]KMC βα += (2-5)
14
where the parameters, α and β, are determined as,
⎥⎦
⎤⎢⎣
⎡+
=
⎥⎦
⎤⎢⎣
⎡+
=
21
21
21
2
2
ωωξβ
ωωωωξα
(2-6)
where ω1 and ω2 are the first two circular natural frequencies of vibration for the restrained
structure, boundary support conditions applied, with the luminaire structure. The damping ratio,
ξ, is also used to determine the parameters for Rayleigh damping.
2.3 Wind Load Model
The procedure for applying the wind load model consists of two distinct parts. First, the wind
velocity time series for a specific height must be determined. Included in this part is the use of
an approximation of the in-line wind velocity power spectrum, which is essentially the energy
density as a function of frequency. Second, the wind velocity time series is then extruded along
the height of the structure following ASCE-7 (2005) and thus producing a wind velocity profile
through time. Using this wind velocity profile, the loading due to wind velocity is determined
based on drag force theory, or specifically a relative force equation known as Morison’s
equation.
The procedure for determining the wind velocity time series is presented in Figure 2-4.
15
Loop throughFrequency
Determine FrequencyInterval
Calculate PowerSpectral Density
End Loop
Loop through Time
Assemble WindVelocity Time Series
End Loop
Start
Figure 2-4: Wind Velocity Time Series Procedure
The in-line wind velocity power spectrum, S(z,n), is given by (Simiu, 1996),
3/52* )501(
200),(ff
unznS
+= (2-7)
where n is frequency, u* is wind shear velocity, and f is given by,
( )zunzf = (2-8)
where z is height above ground and u(z) is the wind velocity at height z. By stepping through
frequency values, the in-line wind velocity power spectrum is calculated for a given wind
velocity at a specific, or characteristic, height. Using the international standard for wind velocity
measurements, and consistent with ASCE-7 (2005), the characteristic height is always assumed
to be 33 ft (10 m) for the calculation of the in-line wind velocity power spectrum.
A graphical representation of Equation (2-7) is presented in Figure 2-5. The velocity considered
in this representation is 20 MPH (32.19 km/hr) at a height of 33 ft (10 m). It is important to note
that wind velocities having a low frequency contain most of the energy in the power spectrum. It
16
must also be noted that the power spectrum is dependent on the height above ground, z. As
noted, all power spectrums are generated for a height of 33 ft (10 m) above ground level.
Figure 2-5: In-Line Wind Velocity Power Spectrum
The artificial wind velocity time series can then be generated via the use of the second loop
shown in Figure 2-4. The dynamic analysis must be divided into time increments or time steps.
Thus, for each time step a wind velocity must be generated to compose the wind velocity time
series. This process involves summation of the energy contained within a frequency interval by
generating a large number of sinusoids having the desired amplitudes and frequencies. This is
repeated for all time steps needed for the wind velocity time series.
The process begins by dividing the power spectrum into equal frequency intervals. The
incremental frequency interval and midpoint frequency is determined by,
imid
ii
nnn
nnn
+∆
=
−=∆ +
2
1
(2-9)
where ni, ni+1, and nmid are the frequencies at the lower bound of the frequency interval, the upper
bound of the frequency interval, and the mid-point of the frequency interval, respectively. The
power spectrum located at the midpoint frequency interval can then be calculated using linear
17
interpolation. This approximation is valid only if the frequency interval is small enough such
that the power spectrum at the lower and upper bounds of the frequency interval is
approximately linear. This approximation is given as,
21
1 nnnSSSS
ii
iiimid
∆⎟⎟⎠
⎞⎜⎜⎝
⎛−−
+=+
+ (2-10)
where Si, Si+1, and Smid are the power spectrum values at the lower bound of the frequency
interval, the upper bound of the frequency interval, and the mid-point of the frequency interval,
respectively. For each frequency interval, the results of Equations (2-9) and (2-10) are used to
obtain the wind velocity at a given time step, t. For each time step, the entire power spectrum is
considered. The mean wind velocity, u , is added to the summation to generate a wind velocity
time series with this mean value. A random phase angle, φ , is also applied to the summation.
The wind velocity time series is calculated as,
( ) ( )∑∆
−∆+=nAll
midmid tnnSutu φπ2cos2 (2-11)
Figure 2-6 provides an example of a generated wind velocity time series record of 3-minutes or
180-seconds. This time series, or record, was generated using a reference velocity, mean wind
velocity, of 20 MPH (32.19 km/hr). Notice that the mean wind velocity of this record, indicated
by the horizontal solid line, is given as nearly 20 MPH (32.19 km/hr) or the reference velocity.
The actual mean wind velocity of this record is calculated as 19.8 MPH (31.87 km/hr), which is
within 1% of the reference velocity. Notice the wind gusts (and wind relaxation, i.e. wind speeds
well below the mean value), which are characteristic of a random Gaussian process.
18
Figure 2-6: In-Line Wind Velocity Time Series
The procedure for determining the loading due to the wind velocity time series is presented in
Figure 2-7.
Start Loop
Fit Wind Velocity toLogarithmic Profile
Evaluate LogarithmicProfile at Nodal Points
Determine Wind Loadat Nodal Points
End Loop
Figure 2-7: Wind Loading Procedure
The wind velocity time series generated from the procedure outlined in Figure 2-4 is for a single
point at the characteristic height. Thus, if perfect spatial correlation is assumed along the height
of the structure, each single wind velocity point in the time series can be fit to a profile such that
wind velocities at other points may be easily determined. The profile used in this project is the
19
logarithmic profile. This profile, common in wind engineering studies, is given as (Simiu,
1996),
)ln()(0
*
zz
kuzu = (2-12)
where u* is the wind shear velocity, k is the von Karman constant (≈ 0.4), z is the height above
the ground, z0 is the roughness coefficient, and u(z) is the wind velocity at height z. The wind
shear velocity, u*, is assumed to be constant along the height of the structure. This assumption is
often valid for a height up to approximately 150 ft (45.7 m) above the ground (Simiu, 1996). For
this study, a roughness coefficient, z0, was chosen conforming to an area similar to grassy areas,
z0 equal to 0.787 in (2 cm). An example of the logarithmic profile is provided in Figure 2-8 for a
wind velocity of 20 MPH (32.19 km/hr) at a height of 33 ft (10 m).
Figure 2-8: Logarithmic Wind Velocity Profile
From Equation (2-1), the forcing function, F(t), must be specified at all nodal points. Thus,
using the height of each nodal point, the wind velocity along the height of the structure is
obtained. Morison’s equation, relating fluid flow past an object to determine force, is used to
obtain the wind loading produced by the in-line wind velocity at nodal points as (Morison,
1950),
20
windwinddair uuACF ρ21
= (2-13)
where ρair is the mass density of air, A is the tributary projected area for the nodal point, Cd is the
drag coefficient, uwind is the wind velocity at the nodal point, and F is the force produced by the
wind velocity. In heavier fluids an inertial term is also present, but is neglected here due to the
low mass density of air. The mass density of air, ρair, is determined as a function of altitude and
air temperature. It can be determined by using the following relations,
( )
KkgJ
RsluglbftR
zzz
RTgzPP
RTP
basealtalt
altsealevel
air
−=
°−−
=
+=
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −=
=
05.2871716
exp
ρ
(2-14)
where P is the pressure at the height where the mass density is desired, R is the universal gas
constant, and T is the absolute temperature in degrees Rankin or Kelvin. Other variables are
defined as: Psealevel is the standard pressure at sea level, g is acceleration due to gravity, z is the
height above ground where the mass density is desired, zbasealt is the altitude of the base of the
structure above sea-level, and zalt is the altitude above sea-level where the mass density is
desired.
The drag coefficient, Cd, is determined experimentally and is, in general, a function of the
Reynolds number, projected area of the object normal to flow, and other factors such as surface
roughness. For flow past a smooth cylinder, Cd can be obtained from Figure 2-9. This figure
was a result of experimental results for flow past a smooth cylinder. For this project, a drag
coefficient for the HML structural support was assumed to be 0.45 for all wind velocities. A
drag coefficient of 1.0 was assumed for the luminaire structure at the top. These values are
21
consistent with example design calculations provided by CDOT and a study by the report’s
authors using published values for flow past a smooth cylinder.
Figure 2-9: Drag Coefficient for a Smooth Cylinder (Wilcox, 2000)
2.4 Dynamic Analysis
The procedure for solving the dynamic analysis of the system is presented in Figure 2-10. The
method chosen for this project was the Newmark-Beta method (Newmark, 1959; Paz, 2004).
This method provides a means to numerically integrate the equation of motion, Equation (2-1),
incrementally over a very small time interval, ∆t. It is assumed that the deformations of the
system remain in the linear elastic range for the material of the HML structure, steel, and thus
relatively small. Because of this assumption, a time step of 0.1 seconds was assumed. However,
as will be discussed in Section 2.5, the forcing function due to wind velocity becomes non-linear
as the structure begins to move with some velocity. Therefore, a procedure was also written to
check for unstable solutions. If a specified number of peak displacements continue to grow, an
unstable condition is noted and the dynamic analysis procedure is repeated with a time step equal
to half of the previous time step. This assumes proper convergence and subsequently accurate
results at all wind velocities.
22
Start Loop
Determine EffectiveStiffness and
Equivalent ForceChange
Determine IncrementalPosition and Velocity
Change
DetermineAcceleration
Resolve Non-LinearWind Load
(Section 2.5)
End Loop
Figure 2-10: Dynamic Analysis Procedure
Initial values at time equal to 0 seconds are assumed for position, x, and velocity, x . Initial
acceleration, x , is obtained as,
[ ] { } [ ]{ } { }{ }{ }xKxCtFMx −−= − )(1 (2-15)
where all terms have been previously defined. A static equivalent method is used to obtain the
incremental position and velocity over time step i. The effective stiffness of the structure, Keq, is,
[ ] [ ] [ ] [ ]Ct
Mt
KKN
N
Neq ∆
+∆
+=βγ
β 2
1 (2-16)
where βN and γN are parameters for the Newmark-Beta method. The parameter γN is assumed to
be ½. Values of γN other than ½ may introduce unintended damping effects into the system (Paz,
2004). A range of values for βN is suggested as 1/6 ≤ βN ≤ 1/2 (Newmark, 1959). For this project,
βN was assumed to be ¼, corresponding to the constant acceleration method. The equivalent
force change, Feq, over time step i is,
23
{ } { } [ ] [ ]{ } [ ] [ ] { }iN
N
Ni
N
N
Neq xCtMxCM
tFF ⎥
⎦
⎤⎢⎣
⎡∆⎟⎟⎠
⎞⎜⎜⎝
⎛−−++
∆+∆=
βγ
ββγ
β 21
211 (2-17)
where ∆F is the force change over time step i. The incremental position, ∆x, and velocity
change, x∆ , over time step i are then,
{ } [ ] { }
{ } { } { } { }iN
Ni
N
N
N
N
eqeq
xtxxt
x
FKx
∆⎟⎟⎠
⎞⎜⎜⎝
⎛−+−∆
∆=∆
=∆ −
βγ
βγ
βγ
21
1
(2-18)
The position, xi+1, and velocity, 1+ix , at the end of the time step, i+1, are then calculated as,
{ } { } { }
{ } { } { }xxx
xxx
ii
ii
∆+=
∆+=
+
+
1
1 (2-19)
Using the equation of motion, Equation (2-1), and forcing dynamic equilibrium, the acceleration,
1+ix , at the end of the time step is calculated as,
{ } [ ] { } [ ]{ } [ ]{ }{ }1111
1 +++−
+ −−= iiii xKxCFMx (2-20)
Thus, by stepping through time the position, velocity, and acceleration at each point along the
height of the HML structure can be determined.
2.5 Relative Motion
The procedure for resolving the non-linear wind load forcing function due to relative motion is
presented in Figure 2-11. The cause for this non-linearity is due to Morison’s equation that is
used to determine the force from the wind velocity in Equation (2-13). As the HML structural
support, herein referred to as the pole, moves, each point on the pole has some velocity, positive
24
or negative depending on the motion. The wind is also moving with some velocity, positive or
negative. The true forcing function must be computed from the relative velocity between the
pole and the wind.
Loop for Convergence
Determine In-LinePole Velocity
at Nodal Points
DetermineRelative Velocity
Update Wind Loadat Nodal Points
SolveDynamic Analysis
of the System(Section 2.4)
Check forConvergence
End Loop
Figure 2-11: Relative Motion Procedure
To check the forcing function, the pole velocity before and after the forcing function is updated
and the new relative velocity must be considered. First, the in-line pole velocity at each nodal
point is extracted. These pole velocities, obtained at the end of the time step being considered,
are combined with the wind velocity at the appropriate time to obtain the relative velocity given
as,
{ } { } { }HMLwindrel uuu −= (2-21)
where urel is the relative velocity between the wind velocity, uwind, and the pole velocity, uHML.
Figure 2-12 illustrates the use of Equation (2-21). Here, positive velocities are assumed to be in
the positive x-direction. The wind velocity is moving with some velocity, uwind, in the positive
direction. The pole is also moving with some velocity, uHML, in the positive direction. Thus, the
net or relative velocity, urel, is the difference between the wind velocity and pole velocity.
25
uwind uHML urel
_ =
x
y
Figure 2-12: Relative Velocity of Wind and HML Structure
Using the relative velocity between the pole and wind, an updated forcing function using a
modified version of Morison’s equation is determined for each nodal point as,
relreldair uuACF ρ21
= (2-22)
where urel is the relative velocity determined from Equation (2-21). Dynamic analysis using the
Newmark-Beta method from Section 2.4 is completed to reevaluate the position, velocity, and
acceleration of each node of the pole at the end of each time step. In-line pole velocities are
again extracted. Comparisons are made between the previously extracted pole velocities and
these new pole velocities at each node. If a set tolerance is exceeded, this procedure is repeated
until convergence is reached. If the tolerance is not exceeded, convergence is achieved and the
relative motion for the particular time step is completed.
2.6 Fatigue Analysis
The procedure for applying the fatigue analysis to determine the fatigue life for a given wind
speed is presented in Figure 2-13. This procedure makes use of the steps also presented in
Sections 2.3 – 2.5 of this report as noted in Figure 2-13. These steps will only be noted with no
further explanation given.
26
The fatigue analysis assumes that the response of the HML structural support is linear elastic, i.e.
no permanent deformations caused by the loading. The derivation to calculate or estimate the
fatigue life of the structure follows the random vibration approach developed by Crandall and
Mark (1963). According to the Palmgren-Miner rule, each stress cycle causes some amount of
damage over some time duration. At some point, the accumulated damage reaches unity,
indicating failure of the system. Divide Wind Velocity
PDF into Bins
Loop throughBins of PDF
Assign Wind Velocityto Bin Value
Construct WindLoading Model(Section 2.3)
SolveDynamic Analysis
of the System(Section 2.4)
End Loop
Extract StressTime History
Resolve Non-LinearWind Load
(Section 2.5)
DetermineExpected Damage
from Bin
DetermineFatigue Life
Figure 2-13: Fatigue Analysis Procedure
First, it is assumed that +0v is the mean up-crossing rate (i.e., the number of mean stress crossings
with a positive slope over one second for the stress time history) such that during a time period
the number of stress cycles is +0v . The fraction of these cycles that would have amplitudes
between some value a and a+da would be p(a)da where p(a) is the probability density function
(PDF) of the peaks of the stress time history. The expected number of peaks, n(a), between these
values is then calculated as,
( ) ( )daaTpvan += 0 (2-23)
27
where T is the time duration of the stress time history. According to the Palmgren-Miner rule, a
single peak a causes an incremental damage of,
( )aNdD 1
= (2-24)
where N(a) is the number of cycles to failure at the stress amplitude a for that material from test
data. For all cycles between a and a+da over time, the expected damage is determined as,
( )( )
( )( )aN
daaTpvaNan +
= 0 (2-25)
To account for all possible values of a and to determine the total expected damage, integration
over the entire possible range from zero to infinity yields,
( )[ ] ( )( )∫
∞+=0
daaNapTvTDE o (2-26)
where ( )[ ]TDE is the total expected damage. Assuming that the stress process is a Gaussian
process, a process that fits a normal distribution, then the peaks follow a special case of the
Weibull distribution known as the Rayleigh distribution (shape parameter = 2.0). The PDF for
the Rayleigh distribution is given as,
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−= 2
2
2 2exp
yy
aaapσσ
(2-27)
where a is the stress amplitudes and 2yσ is the variance of the stress process as a function of
time. Substitution of Equation (2-27) into (2-26) produces,
( )[ ] ∫∞ +
+
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
0 2
21
20
2exp daaa
cTvTDE
y
b
y σσ (2-28)
28
where b and c are fatigue constants related to the material (in the present case steel) of the
structure. These constants, b and c, are determined from,
cNS b = (2-29)
where N is the number of cycles at stress amplitude S. As noted previously, b and c are the
parameters that define the fatigue (S-N) curve for a particular stress category. Integrating the
expected damage from Equation (2-28) yields,
( )[ ] ( ) ⎟⎠⎞
⎜⎝⎛ +Γ=
+
21220 b
cvTDE
b
yσ (2-30)
where σy is the standard deviation of stress process as a function of time and the gamma function
is defined as,
( ) ∫∞ −−=Γ
0
11 dtetx x (2-31)
Finally, setting ( )[ ]ii TDEF = for the ith wind speed (from the PDF of wind velocity) the fatigue
life can be calculated according to the Palmgren-Miner rule as,
∑=
= n
iii
life
PFF
10
1 (2-32)
where P0i is the probability of occurrence of the wind force for the ith wind speed and causing the
associated damage in Equation (2-30).
In order to estimate the fatigue life using Crandall and Mark’s (1963) method, it is necessary to
procure the wind statistical distribution information. Statistics for locations along the Front
Range are not readily available and, therefore, a lognormal distribution was assumed. This
29
corresponds to a reasonable fit for data developed by the National Oceanographic and
Atmospheric Administration (NOAA) for the contiguous United States. The PDF for the
lognormal distribution of wind velocity, u, can be expressed as,
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ −−=
2ln
21exp
21
ζλ
ζπu
uufU (2-33)
where the parameters ζ and λ are defined as,
2
2
22
21ln
1ln
ζµλ
µσζ
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
(2-34)
and the parameters µ and σ are defined as the mean wind velocity and standard deviation of wind
velocity, respectively. The mean wind velocity and standard deviation of wind velocity are
related by the coefficient of variation (COV) as,
µσ
=COV (2-35)
Figure 2-14 illustrates Equation (2-33), the PDF, for a mean velocity of 20 MPH (32.19 km/hr)
and COV of 25%.
30
Figure 2-14: Lognormal PDF
The dynamic response of the structure was determined for each of twenty-five different wind
velocities. This required the lognormal PDF to be divided into twenty-five bins of equal width
and each having a probability of occurrence, P0i, equal to their area. Figure 2-15 illustrates the
bins that would be used for the PDF in Figure 2-14.
Figure 2-15: Lognormal PDF Bins
The area under each bin represents the probability of occurrence of a wind velocity having a
magnitude between the upper and lower bounds of the bin. The probability of occurrence is
paired with the wind velocity occurring at the midpoint of that bin for analysis. As the wind
velocities increase beyond the mean value, the area within each bin typically decreases.
31
However, although their occurrence probability is very small, high wind velocities must be
considered as they may cause extensive damage to the structure, decreasing its fatigue life even
with a low probability of occurrence.
2.7 Reliability Analysis
The procedure used for determining the reliability index, β, for a given target fatigue life is
presented in Figure 2-16. The reliability approach used in this study makes use of a numerically
traditional approach to determine the probability of failure of a complex system. The Monte
Carlo Simulation (MCS) method consists of a repetitive procedure in which a large number
(>10,000) of data points are needed to obtain an accurate assessment of the probability of failure.
Upon completion, the reliability index, β, can also be determined.
Determine COVfor Each Fatigue
Parameter
Monte CarloSimulation Loop
Generate RandomNumbers and
Fit to StatisticalDistribution
EvaluateFatigue Life
EvaluateLimit State Function
End Monte CarloSimulation Loop
Evaluate Probabilityof Failure and
Reliability Index
Figure 2-16: Reliability Analysis Procedure
This procedure utilizes Equations (2-30) and (2-32) from the fatigue analysis procedure and thus
requires all of the values indicated in those equations. It does not require, however, that the
entire fatigue analysis be repeated. For this project, the S-N curve is assumed to possess the
model and time uncertainty in the fatigue estimation. Specifically, the fatigue, or S-N
32
parameters, constants b and c in Equation (2-29), used to describe the S-N curve for steel are
considered as random variables and account for all other randomness.
The parameters used to describe the S-N curve are considered to have a combined COV of 40%
(Assakkaf and Ayyub, 2000). The COV for each S-N parameter is assumed to be equal and can
thus be determined as,
22 COVCOVCOV CBF += (2-36)
where COVF is the combined COV of 40%, COVB and COVC are the COV for each S-N
parameter. This assumption is only valid if statistical independence between the parameters is
assumed. Furthermore, the parameters are assumed to behave as log-normally distributed
random variables for reliability analysis purposes.
The MCS method requires several steps to be performed. The method, specific to this project, is
outlined as and is repeated i times:
Step 1: Generate two uniformly distributed random numbers within the range –1 ≤ x ≤ 1
where x is the random number.
Step 2: Fit uniformly distributed random numbers to standard normal distributed random
numbers where the standard normal distribution has mean of zero and standard
deviation of one.
Step 3: Fit standard normal distributed random numbers to lognormal distributed random
numbers with given mean values and standard deviations determined from the
COV calculated from Equation (2-36).
Step 4: Re-evaluate expected damage for each bin using Equation (2-30).
Step 5: Re-evaluate fatigue life using Equation (2-32).
Step 6: Evaluate limit state function given in Equation (2-37).
target lifeg F F= − (2-37)
33
where g is the limit state function, Flife is the fatigue life calculated for simulation i, and Ftarget is
the target fatigue life as prescribed by CDOT. Thus, we seek the probability that the limit state
function, g, is less than zero, P(g < 0). Counting the number of failures, g < 0, for all simulations
i, the probability of failure, Pf, is calculated as,
simulation
failure
NNPf = (2-38)
where Nfailure is the number of failures and Nsimulation is the total number of simulations or data
points generated. The reliability index, β, can be calculated directly from the probability of
failure as,
( )fP−Φ= − 11β (2-39)
where Φ-1 is the inverse of the standard normal distribution function. Some common values, i.e.
a range typically seen in civil engineering problems, are presented in Table 2-1.
Pf β 0.16 1 0.023 2 0.0013 3 0.000032 4 0.00000029 5
Table 2-1: Common Reliability Indices, β
34
CHAPTER 3
SENSTIVITY AND RELIABILITY ANALYSES
3.1 Sensitivity Analysis Background
The approach used to determine the reliability-based design procedure for HML structural
supports procedure can be used to identify the key factors (design parameters and loading
parameters) that contribute to its fatigue life. To accomplish this, a sensitivity analysis
considering six primary physical and two primary wind velocity factors was performed. The six
primary physical factors focused on the physical characteristics of the HML structural supports,
herein referred to as the pole, and how these factors affect the fatigue life. These included the
outside diameter of the pole, the wall thickness of the pole, the height or length of the pole, the
weight of the luminaire structure, the projected area of the luminaire structure, and the damping
ratio of the structure. The two primary wind velocity factors considered the sensitivity of the
fatigue life to wind velocity factors. First, the assumed coefficient of variation (COV) for the
lognormal distribution describing the wind velocity was considered. Second, the probability of
strong wind gusts associated with periodic weather events in Colorado were considered.
To conduct the sensitivity analysis, a benchmark, or representative, HML structural support was
needed. Structural drawings were provided by CDOT. This submittal was in reference to the I-
25 / Broadway Viaduct Phase II project and prepared by Dynalectric Company. Figure 3-1
illustrates a benchmark HML structural support used in this project. For each section shown in
Figure 3-1, a corresponding table of properties for that given section is given in Tables 3-1 – 3-4.
35
120 ft
Luminaire Structure(Table 3-4)
Section 3(Table 3-1)
Joint Section 2(Table 3-2)
Section 2(Table 3-1)
Joint Section 1(Table 3-2)
Section 1(Table 3-1)
Structural Foundation(Table 3-3)
Figure 3-1: Benchmark HML Structural Support in Colorado
Table 3-1 provides the properties for pole sections 1 – 3 from Figure 3-1. Properties for each
section were directly obtained from the specification submittals obtained from CDOT.
36
Property Section 1 Section 2 Section 3
Total Length 53.49 ft (16.30 m)
52.52 ft (16.01 m)
20.16 ft (6.14 m)
Bottom Outside Diameter
26.00 in (660.40 mm)
19.50 in (495.30 mm)
13.00 in (330.20 mm)
Top Outside Diameter
18.51 in (470.15 mm)
12.15 in (308.61 mm)
10.18 in (258.57 mm)
Taper 0.14 in/ft (11.67 mm/m)
0.14 in/ft (11.67 mm/m)
0.14 in/ft (11.67 mm/m)
Wall Thickness 0.375 in (9.525 mm)
0.250 in (6.350 mm)
0.2391 in (6.0731 mm)
Material Code S22-65 S22-65 S105-55
Yield Strength 65 ksi (448 MPa)
65 ksi (448 MPa)
55 ksi (379 MPa)
Modulus of Elasticity
29,000 ksi (200 GPa)
29,000 ksi (200 GPa)
29,000 ksi (200 GPa)
Mass Density 15.235 slug/ft3
(7850 kg/m3)
15.235 slug/ft3
(7850 kg/m3)
15.235 slug/ft3
(7850 kg/m3)
Poisson’s Ratio 0.3 0.3 0.3
Table 3-1: HML Structural Support Properties – Pole Sections
Table 3-2 provides the properties for joint sections 1 and 2 from Figure 3-1. As indicated in the
specifications, these joint sections are characterized as lap-splices, one section overlapping the
other section. Thus, all section properties are identical to those given in Table 3-1. These joints
do however affect the stiffness of the structure due to the overlapping of the sections as indicated
with the effective wall thickness given in Table 3-2. Properties for each joint section were
directly obtained from the specification submittals obtained from CDOT.
37
Property Joint Section 1 Joint Section 2
Section Type Lap-Splice Lap-Splice
Sections Joined 1 – 2 2 – 3
Effective Wall Thickness
0.625 in (15.875 mm)
0.4891 in (12.4231 mm)
Overlap Length
3.492 ft (1.064 m)
2.683 ft (0.818 m)
Height of Overlap
53.49 ft (16.30 m)
102.59 ft (31.25 m)
Table 3-2: HML Structural Support Properties – Joint Sections
Table 3-3 provides the properties for the structural foundation. Recall from Chapter 2 that this
foundation is modeled as a set of translational and rotational springs. For this project, springs
were assumed to have a stiffness representing a fixed structural boundary condition for the
foundation. A fixed structural boundary condition is a conservative assumption in determining
the fatigue life of the structure.
Property Spring Stiffness
Axial Stiffness
1 x 1030 lb/in (1.75 x 1029 kN/m)
Lateral Stiffness
1 x 1030 lb/in (1.75 x 1029 kN/m)
Overturning Stiffness
1 x 1030 lb*in/rad (1.75 x 1029 kN*m/rad)
Table 3-3: HML Structural Support Properties – Boundary Support Conditions
Table 3-4 provides the properties for the luminaire structure mounted at the top of the HML
structural support. Properties for the luminaire structure were directly obtained from the
specification submittals obtained from CDOT.
38
Property Luminaire Structure
Mounting Height
120 ft (36.576 m)
Weight 666 lbs (302 kg)
Projected Area 11.50 ft2 (1.0684 m2)
Table 3-4: HML Structural Support Properties – Luminaire Structure Properties
3.2 The Mean Wind Velocity
ASCE-7 (2005) provides design wind velocities for consideration in the design process of a
structure. These design wind velocities represent 3-second wind gusts that have an annual
probability of exceedance of 2%, or a mean recurrence interval (MRI) of 50 years. These design
wind velocities are obtained from the consideration of many years of data recorded at stations
such as airports. Extreme value statistics are then used to determine the 3-second wind gusts
associated with the desired MRI.
For this study, however, the fatigue analysis requires the use of the wind velocity parent
distribution for a given area. As given in Chapter 2, the lognormal PDF is assumed as the
statistical distribution describing the loading event, or in this case the wind velocity. It has been
shown that Weibull distribution is typically a better fit for the parent distribution. However, the
Weibull distribution was not considered at this time due to the lack of knowledge concerning the
characteristics of the wind along the Front Range of Colorado. The lognormal parent
distribution is illustrated again in Figure 3-2 for a mean wind velocity of 10 MPH (16.09 km/hr)
and COV of 25% at a height of 33 ft (10 m). Attempts have been made in the past to predict the
design, or extreme, wind velocities, as used in ASCE-7 (2005), from these parent wind
distributions. These attempts have proven to be unsuccessful (Holmes, 2001).
39
Figure 3-2: Wind Velocity Parent Distribution
Thus, the remainder of this study will only consider the mean wind velocity obtained from the
parent wind distribution at 33 ft (10 m). Furthermore, this mean wind velocity is only an
assumed value and not associated with any obtained data. The procedure to be developed will
take this into consideration. The appendix of this report provides some initial consideration of
mean wind velocities within the state of Colorado at 33 ft (10 m). A more detailed statistical
study of the wind along the Front Range of Colorado is needed to accurately determine the
appropriate parent distribution and mean wind velocity. With the completion of this statistical
study, the determination of the design wind velocities along the Front Range of Colorado would
also be capable of being accomplished.
3.3 Sensitivity Analysis
The sensitivity analysis considers eight total factors and their affect on the fatigue life of the
HML structural support for a particular wind-loading event. Among these factors are six
associated with the physical properties or characteristics of the HML structural support or pole.
The other two factors consider the properties of the wind velocity and, in particular, the
statistical distribution used to describe the in-line wind velocity used in the fatigue analysis
computations. Based on Section 1.3, AASHTO 2001, both detail categories E and E′ are used as
the basis for all sensitivity analysis results presented herein.
40
Considering first the six factors related to the physical properties of the HML structural support,
the fatigue life is determined for varying values of these properties. For five of the six factors,
outside diameter, wall thickness, section length, luminaire weight, and luminaire projected area,
variations consist of percentage reductions or increases to determine the variation in calculated
fatigue life. If a 30% reduction is considered, indicated by –30%, then all values associated with
that factor are reduced by 30% along all points of the pole. For the sixth factor, damping ratio,
various values are assumed to determine the variation in calculated fatigue life.
Pole Outside Diameter
Figure 3-3 illustrates the sensitivity of the fatigue life for variations in the outside diameter of the
pole. Large changes in the outside diameter of the pole substantially affect the fatigue life of the
structure. Though a decrease in outside diameter yields less cross-sectional material at the base
of the structure and therefore higher stresses, the loading caused by the wind velocity is also
reduced. An increase in outside diameter yields higher loads due to wind velocity, but there is
also more cross-sectional area to resist the forces caused at the base of the structure.
(a): Detail Category E (b): Detail Category E′
Figure 3-3: Fatigue Life Sensitivity – Pole Outside Diameter
Pole Wall Thickness
Figure 3-4 illustrates the sensitivity of the fatigue life for variations in the wall thickness of the
pole. As can be seen, there is substantial variability in fatigue life for decreases in wall
thickness. This, however, is not the case for increases in wall thickness. This result compares
41
well with the investigation recently conducted by Kaufmann (2005) concerning the exact
measurements as compared to the design specifications. As a result of this analysis, the
sensitivity due to changes in wall thickness must be carefully monitored.
(a): Detail Category E (b): Detail Category E′
Figure 3-4: Fatigue Life Sensitivity – Pole Wall Thickness
Pole Length
Figure 3-5 illustrates the sensitivity of the fatigue life due to changes in the length of each
section comprising the total height of the structure. Recall from Figure 3-1, the pole is
comprised of 3 sections. Sections 1 and 2 overlap to form the lap-splice at Joint Section 1.
Sections 2 and 3 overlap to form Joint Section 2. As expected, reductions in the height of the
structure yield a higher fatigue life. However, dramatic increases do not have an equivalent
effect in regards to reducing the fatigue life drastically different from the original height or no
change.
42
(a): Detail Category E (b): Detail Category E′
Figure 3-5: Fatigue Life Sensitivity – Pole Length
Luminaire Structure Weight
Figure 3-6 illustrates the sensitivity of the fatigue life due to changes in the weight or mass of the
luminaire structure mounted to the top of the pole. Changes in the luminaire structure weight
only effect the applied concentrated nodal mass and the subsequent degrees-of-freedom in the
structure mass matrix. As can be seen, changes to the luminaire structure weight do not effect or
create large variability in the calculated fatigue life of the structure.
(a): Detail Category E (b): Detail Category E′
Figure 3-6: Fatigue Life Sensitivity – Luminaire Structure Weight
Luminaire Structure Projected Area
Figure 3-7 illustrates the sensitivity of the fatigue life due to changes in the projected area of the
luminaire structure mounted to the top of the pole. Changes in the projected area only affect the
43
load created by the wind velocity as determined by Morison’s equation. It is rational to assume
that increases in the area would also correlate well with changes in the luminaire structure
weight. However, to determine if either of these variables affects the fatigue life, they were
considered separately. As can be seen, changes in the projected area of the luminaire structure
do not produce as much variability as changes in the wall thickness of the pole.
(a): Detail Category E (b): Detail Category E′
Figure 3-7: Fatigue Life Sensitivity – Luminaire Structure Projected Area
Structure Damping
Figure 3-8 illustrates the sensitivity of the fatigue life due to changes in the damping ratio, ξ, of
the entire structure. The damping ratio of the structure, however, is somewhat of an unknown
quantity. For a structure of this height and slenderness it would be appropriate to assume a
damping ratio of approximately 1%, which has been used in previous studies on similar
structures. However, this value could vary as the fatigue life certainly does vary for changes in
the damping ratio. For purposes of this project, it will be assumed that the damping ratio is
always 1%. Further experimental verification is needed to ensure this assumption is correct.
Finally, it should be noted that variations in damping ratio values did exhibit a wide deviation in
fatigue life.
44
(a): Detail Category E (b): Detail Category E′
Figure 3-8: Fatigue Life Sensitivity – Structure Damping
Wind Velocity COV
Finally, the last two factors considered are related to the wind velocity. Figure 3-9 illustrates the
sensitivity of the fatigue life due to changes in the assumed coefficient of variation (COV) for the
statistical distribution describing the in-line wind velocity. This distribution is instrumental in
determining the expected damage from each bin of the distribution. Recent studies have
indicated a COV of 25% is appropriate for wind velocity, 35% for wind-loading. A 25% COV
represents a high degree of uncertainty in wind velocities and thus wind-loading. As shown, a
higher COV produces a lower fatigue life for the structure. This unknown quantity cannot be
resolved without extensive statistical studies of wind velocities along the Front Range of
Colorado. Thus, it is assumed that a COV of 25% is appropriate.
45
(a): Detail Category E (b): Detail Category E′
Figure 3-9: Fatigue Life Sensitivity – Wind Velocity COV
Wind Gust
Figure 3-10 illustrates the sensitivity of the fatigue life due to simulated wind gusts. The mean
wind velocity was assumed to be 10 MPH (16.09 km/hr) with a COV of 25%. Wind gusts
varying from 50 MPH (80.47 km/hr) to 100 MPH (160.93 km/hr) were chosen with varying
probabilities of occurrence as indicated in Figure 3-10. It should be noted that these occurrence
rates, even 1%, are extremely high and are used here to examine sensitivity only. As a result of
this analysis, it can be seen that the fatigue life is extremely sensitive to wind gusts, as one might
expect. Further development of this realization is needed.
(a): Detail Category E (b): Detail Category E′
Figure 3-10: Fatigue Life Sensitivity – Wind Gust
46
The sensitivity analysis yielded several factors that directly contribute to the loss of desired
fatigue life. The luminaire structure weight and luminaire structure projected area did not
contribute to large variations in fatigue life. Variations in luminaire structure weight appear to
affect the fatigue life less than variations in luminaire structure projected area. Thus, based on
these two parameters, it would be appropriate to state that the projected area of the luminaire
structure is more important to the resulting fatigue life of the HML structural support.
Sensitivity of the fatigue life to structural damping did have an effect on the fatigue life.
However, this quantity cannot be exactly determined without experimental verifications. Thus, a
damping ratio of 1% is consistent with dynamic investigations of traffic and lighting structures.
Variations in the assumed COV for describing the wind velocity distribution did contribute
significantly to the fatigue life. However, as noted earlier, an extensive statistical study of wind
velocities along the Front Range of Colorado is recommended to determine the appropriate
COV. Thus, a COV of 25% is assumed at this point in time and is approximately consistent with
COV’s in North America.
Sensitivity of fatigue life to wind gusts did contribute to high degrees of variability in fatigue
life. The details behind appropriate gusts and occurrence probabilities associated with those
gusts would need to be determined. This could be accomplished with the same study to
determine the appropriate COV for wind velocity. This analysis is beyond the scope of the
present project but exacerbates the need for a conclusive wind study along the Front Range of
Colorado, where it is known that there are sometimes extreme wind conditions.
The primary factors that contributed to fatigue life that are under the control of the designer are
pole outside diameter, pole wall thickness, and pole length. Due to these observations, the
design method must consider these three factors with respect to wind velocity. The design
method will also include the effects of the luminaire structure attached to the top the HML
structural supports. Only the projected area of the luminaire structure will be considered in this
regard. As shown, the other luminaire structure factor did not display a wide range of fatigue life
values based on variations in its weight.
47
3.4 Reliability Analysis
The reliability analysis considers the results from the sensitivity analysis and their effect on the
reliability of the HML structural support for a particular wind-loading event. From the
sensitivity analysis, all variations in outside diameter, wall thickness, section length, luminaire
weight, and luminaire projected area are considered in the reliability analysis. All variations in
structure damping are considered for the reliability analysis. Fatigue life results are only given
for the assumed damping ratio of 1%. Additionally, all variations in wind velocity COV are
considered for the reliability analysis. Fatigue life results are only given for the assumed wind
velocity COV of 25% and wind gusts are not considered.
All reliability analyses are conducted for 50,000 simulations using the Monte Carlo method. The
limit state function, given in Equation (2-37), is evaluated using a target fatigue life of 50 years.
It should be noted that this target fatigue life was selected somewhat arbitrarily to illustrate the
methods and can be any value in future studies. Failure of the HML structural support is thus
indicated by a calculated fatigue life for a particular wind-loading event as being less than the
target fatigue life or 50 years. The probability of failure is calculated in accordance with
Equation (2-38) where the total number of simulations is 50,000. Finally, the reliability index, β,
is calculated using Equation (2-39) that uses the results from Equation (2-38), the probability of
failure. Using 50,000 simulations, reliability indices of β ≈ 4 are capable of being estimated.
This is a direct result of the minimum probability of failure capable of being obtained using
50,000 simulations.
Pole Outside Diameter
Figures 3-11 (a) – (c) illustrate the fatigue life for 30% reduction, no change, and 30% increase
in pole outside diameter, respectively, for detail category E (designated by E in the figure
subtitle). The solid line illustrates the mean fatigue life for the variation considered. Plus and
minus one standard deviation with respect to the mean fatigue life are given as dashed lines. In
cases where the mean minus one standard deviation becomes negative, these values are not
plotted and hence the appearance of incomplete data sets.
48
(a): Fatigue Life (E) – 30% Reduction (b): Fatigue Life (E) – No Change
(c): Fatigue Life (E) – 30% Increase
Figure 3-11: Reliability Analysis – Pole Outside Diameter
49
Figures 3-11 (d) and (e) illustrate the probability of failure and reliability index for all variations
considered for changes in pole outside diameter, respectively, for detail category E (designated
by E in the figure subtitle). These figures are determined based on a target fatigue life of 50
years. As can be seen, an increase in outside diameter correlates well with a decrease in
probability of failure and an increase in reliability index.
(d): Probability of Failure (E) (e): Reliability Index (E)
Figure 3-11 (cont.): Reliability Analysis – Pole Outside Diameter
Figures 3-11 (f) – (j) provide the same results given in (a) – (e) except for detail category E′
(designated by E′ in the figure subtitle).
(f): Fatigue Life (E′) – 30% Reduction (g): Fatigue Life (E′) – No Change
Figure 3-11 (cont.): Reliability Analysis – Pole Outside Diameter
50
(h): Fatigue Life (E′) – 30% Increase
(i): Probability of Failure (E′) (j): Reliability Index (E′)
Figure 3-11 (cont.): Reliability Analysis – Pole Outside Diameter
Pole Wall Thickness
Figures 3-12 (a) – (c) illustrate the fatigue life for 30% reduction, no change, and 30% increase
in pole wall thickness, respectively, for detail category E (designated by E in the figure subtitle).
The solid line illustrates the mean fatigue life for the variation considered. Plus and minus one
standard deviation with respect to the mean fatigue life are given as dashed lines. In cases where
the mean minus one standard deviation becomes negative, these values are not plotted and hence
the appearance of incomplete data sets.
51
(a): Fatigue Life (E) – 30% Reduction (b): Fatigue Life (E) – No Change
(c): Fatigue Life (E) – 30% Increase
Figure 3-12: Reliability Analysis – Pole Wall Thickness
52
Figures 3-12 (d) and (e) illustrate the probability of failure and reliability index for all variations
considered for changes in pole wall thickness, respectively, for detail category E (designated by
E in the figure subtitle). These figures are determined based on a target fatigue life of 50 years.
As can be seen, an increase in wall thickness correlates well with a decrease in probability of
failure and an increase in reliability index.
(d): Probability of Failure (E) (e): Reliability Index (E)
Figure 3-12 (cont.): Reliability Analysis – Pole Wall Thickness
Figures 3-12 (f) – (j) provide the same results given in (a) – (e) except for detail category E′
(designated by E′ in the figure subtitle).
(f): Fatigue Life (E′) – 30% Reduction (g): Fatigue Life (E′) – No Change
Figure 3-12 (cont.): Reliability Analysis – Pole Wall Thickness
53
(h): Fatigue Life (E′) – 30% Increase
(i): Probability of Failure (E′) (j): Reliability Index (E′)
Figure 3-12 (cont.): Reliability Analysis – Pole Wall Thickness
Pole Length
Figures 3-13 (a) – (c) illustrate the fatigue life for 30% reduction, no change, and 30% increase
in pole section length, respectively, for detail category E (designated by E in the figure subtitle).
The solid line illustrates the mean fatigue life for the variation considered. Plus and minus one
standard deviation with respect to the mean fatigue life are given as dashed lines. In cases where
the mean minus one standard deviation becomes negative, these values are not plotted and hence
the appearance of incomplete data sets.
54
(a): Fatigue Life (E) – 30% Reduction (b): Fatigue Life (E) – No Change
(c): Fatigue Life (E) – 30% Increase
Figure 3-13: Reliability Analysis – Pole Length
55
Figures 3-13 (d) and (e) illustrate the probability of failure and reliability index for all variations
considered for changes in pole section length, respectively, for detail category E (designated by
E in the figure subtitle). These figures are determined based on a target fatigue life of 50 years.
As can be seen, a decrease in section length correlates well with a decrease in probability of
failure and an increase in reliability index.
(d): Probability of Failure (E) (e): Reliability Index (E)
Figure 3-13 (cont.): Reliability Analysis – Pole Length
Figures 3-13 (f) – (j) provide the same results given in (a) – (e) except for detail category E′
(designated by E′ in the figure subtitle).
(f): Fatigue Life (E′) – 30% Reduction (g): Fatigue Life (E′) – No Change
Figure 3-13 (cont.): Reliability Analysis – Pole Length
56
(h): Fatigue Life (E′) – 30% Increase
(i): Probability of Failure (E′) (j): Reliability Index (E′)
Figure 3-13 (cont.): Reliability Analysis – Pole Length
Luminaire Structure Weight
Figures 3-14 (a) – (c) illustrate the fatigue life for 30% reduction, no change, and 30% increase
in luminaire structure weight, respectively, for detail category E (designated by E in the figure
subtitle). The solid line illustrates the mean fatigue life for the variation considered. Plus and
minus one standard deviation with respect to the mean fatigue life are given as dashed lines. In
cases where the mean minus one standard deviation becomes negative, these values are not
plotted and hence the appearance of incomplete data sets.
57
(a): Fatigue Life (E) – 30% Reduction (b): Fatigue Life (E) – No Change
(c): Fatigue Life (E) – 30% Increase
Figure 3-14: Reliability Analysis – Luminaire Structure Weight
58
Figures 3-14 (d) and (e) illustrate the probability of failure and reliability index for all variations
considered for changes in luminaire structure weight, respectively, for detail category E
(designated by E in the figure subtitle). These figures are determined based on a target fatigue
life of 50 years. As can be seen, changes in weight do not appear to greatly affect the probability
of failure or reliability index of the structure.
(d): Probability of Failure (E) (e): Reliability Index (E)
Figure 3-14 (cont.): Reliability Analysis – Luminaire Structure Weight
Figures 3-14 (f) – (j) provide the same results given in (a) – (e) except for detail category E′
(designated by E′ in the figure subtitle).
(f): Fatigue Life (E′) – 30% Reduction (g): Fatigue Life (E′) – No Change
Figure 3-14 (cont.): Reliability Analysis – Luminaire Structure Weight
59
(h): Fatigue Life (E′) – 30% Increase
(i): Probability of Failure (E′) (j): Reliability Index (E′)
Figure 3-14 (cont.): Reliability Analysis – Luminaire Structure Weight
Luminaire Structure Projected Area
Figures 3-15 (a) – (c) illustrate the fatigue life for 30% reduction, no change, and 30% increase
in luminaire structure projected area, respectively, for detail category E (designated by E in the
figure subtitle). The solid line illustrates the mean fatigue life for the variation considered. Plus
and minus one standard deviation with respect to the mean fatigue life are given as dashed lines.
In cases where the mean minus one standard deviation becomes negative, these values are not
plotted and hence the appearance of incomplete data sets.
60
(a): Fatigue Life (E) – 30% Reduction (b): Fatigue Life (E) – No Change
(c): Fatigue Life (E) – 30% Increase
Figure 3-15: Reliability Analysis – Luminaire Structure Projected Area
61
Figures 3-15 (d) and (e) illustrate the probability of failure and reliability index for all variations
considered for changes in luminaire structure projected area, respectively, for detail category E
(designated by E in the figure subtitle). These figures are determined based on a target fatigue
life of 50 years. As can be seen, changes in projected area do not appear to greatly affect the
probability of failure or reliability index of the structure.
(d): Probability of Failure (E) (e): Reliability Index (E)
Figure 3-15 (cont.): Reliability Analysis – Luminaire Structure Projected Area
Figures 3-15 (f) – (j) provide the same results given in (a) – (e) except for detail category E′
(designated by E′ in the figure subtitle).
(f): Fatigue Life (E′) – 30% Reduction (g): Fatigue Life (E′) – No Change
Figure 3-15 (cont.): Reliability Analysis – Luminaire Structure Projected Area
62
(h): Fatigue Life (E′) – 30% Increase
(i): Probability of Failure (E′) (j): Reliability Index (E′)
Figure 3-15 (cont.): Reliability Analysis – Luminaire Structure Projected Area
Structure Damping
Figure 3-16 (a) illustrates the fatigue life for an assumed damping ratio of 1% for detail category
E (designated by E in the figure subtitle). The solid line illustrates the mean fatigue life. Plus
and minus one standard deviation with respect to the mean fatigue life are given as dashed lines.
In cases where the mean minus one standard deviation becomes negative, these values are not
plotted and hence the appearance of incomplete data sets.
63
(a): Fatigue Life (E) – 1% Damping
Figure 3-16: Reliability Analysis – Structure Damping
Figures 3-16 (b) and (c) illustrate the probability of failure and reliability index for all variations
considered for changes in structure damping, respectively, for detail category E (designated by E
in the figure subtitle). These figures are determined based on a fatigue life of 50 years. As can
be seen, an increase in structure damping correlates well with a decrease in probability of failure
and an increase in reliability index.
(b): Probability of Failure (E) (c): Reliability Index (E)
Figure 3-16 (cont.): Reliability Analysis – Structure Damping
Figures 3-16 (d) – (f) provide the same results given in (a) – (c) except for detail category E′
(designated by E′ in the figure subtitle).
64
(d): Fatigue Life (E′) – 1% Damping
(e): Probability of Failure (E′) (f): Reliability Index (E′)
Figure 3-16 (cont.): Reliability Analysis – Structure Damping
Wind Velocity COV
Figure 3-17 (a) illustrates the fatigue life for an assumed wind velocity COV of 25% for detail
category E (designated by E in the figure subtitle). The solid line illustrates the mean fatigue
life. Plus and minus one standard deviation with respect to the mean fatigue life are given as
dashed lines. In cases where the mean minus one standard deviation becomes negative, these
values are not plotted and hence the appearance of incomplete data sets.
65
(a): Fatigue Life (E) – 25% COV
Figure 3-17: Reliability Analysis – Wind Velocity COV
Figures 3-17 (b) and (c) illustrate the probability of failure and reliability index for all variations
considered for changes in wind velocity COV, respectively, for detail category E (designated by
E in the figure subtitle). These figures are determined based on a fatigue life of 50 years. As can
be seen, an increase in the COV correlates well with an increase in probability of failure and a
decrease in reliability index.
(b): Probability of Failure (E) (c): Reliability Index (E)
Figure 3-17 (cont.): Reliability Analysis – Wind Velocity COV
Figures 3-17 (d) – (f) provide the same results given in (a) – (c) except for detail category E′
(designated by E′ in the figure subtitle).
66
(d): Fatigue Life (E′) – 25% COV
(e): Probability of Failure (E′) (f): Reliability Index (E′)
Figure 3-17 (cont.): Reliability Analysis – Wind Velocity COV
The results of the reliability analysis yielded interesting results in regard to the selected target
fatigue life of 50 years. As will be seen with the presentation of the design methodology, a target
fatigue life of 50 years at this point appears to be reasonable for mean wind velocities of 15 MPH
(24.14 km/hr) or less for detail category E and 12 MPH (19.31 km/hr) or less for detail category
E′. Due to the variability in wind events along the Front Range of Colorado, these results may be
of concern. However, these results could also be a combination of the two major assumptions
regarding the HML structural support itself and the statistics related to the wind velocity
distribution. As was seen from the sensitivity analysis, structure damping and wind velocity
COV variations produced relatively large variations in fatigue life. These results were
reciprocated in the reliability analysis for both parameters.
67
CHAPTER 4
RELIABILITY-BASED DESIGN METHODOLOGY
4.1 Design Methodology Background
The method consists of several design charts for each of the design parameters considered in the
sensitivity and reliability analyses of Chapter 3. Included among these parameters are variations
in outside diameter, wall thickness, section length or height of the structure, and luminaire
structure projected area. Excluded are two of the more sensitive parameters, structure damping
and wind velocity coefficient of variation (COV). Thus, at this time, a damping ratio for the
structure is assumed to be 1% and the wind velocity COV is assumed to be 25%. Recall,
however, that other studies have concluded that an appropriate COV for wind-loading is 35%,
thus yielding an approximate COV of 25% for wind velocity, if all other variables in Morison’s
equation are considered deterministic, i.e. not random. The luminaire structure weight is not
considered as a parameter in the design charts. As shown in Chapter 3, the luminaire structure
weight was shown to not severely affect the fatigue life or reliability of the HML structural
support. The luminaire projected area was more critical in this aspect. Any luminaire structure
that would be used would have a given weight and projected area. Thus, from these two
parameters it is concluded that the projected area of the luminaire structure is more important.
The first set of design charts, for single parameter variations, were developed in a similar fashion
to the reliability index figures in Section 3.4. However, in this case, the target fatigue life was
varied for 25 years, 50 years, and 75 years. As a result of this, reliability indices were calculated
for each variation of a particular parameter and mean wind velocity for a given target fatigue life.
As noted previously in Chapter 3, 50-year wind velocities, or design wind velocities, were
unable to be produced based on the sole knowledge of the parent wind velocity distribution.
Thus, the use of the single parameter design charts requires the knowledge of the mean, or
average, wind velocity for a particular site in Colorado. The appendix of this report does provide
an initial summary of mean wind velocities for various sites in Colorado. However, the data
used to compile this summary is limited and should not be used exclusively. Use of the single
parameter design charts, based on a given factor or parameter value such as outside diameter,
68
target fatigue life and mean wind velocity produces a reliability index. A summary of the design
method used in conjunction with the single parameter design charts is presented in Section 4.3.
The first set of design charts were developed based on the assumption that only a single
parameter, such as wall thickness, is changing from the benchmark HML structure given in
Figure 3-1 and described in Tables 3-1 through 3-4. As this is likely never the case, a second set
of design charts for multiple parameter variations have been developed for multiple parameter
variations. The multiple parameter design charts do not require the use of the single parameter
design charts. A summary of the design method used in conjunction with the multiple parameter
design charts is presented in Section 4.5.
4.2 Design Charts – Single Variable
The design charts for a single variable are provided in the supplemental reports for detail
categories E and E′. These design charts were determined based on varying only the parameter
for which it is given: pole outside diameter, pole wall thickness, pole length, or luminaire
structure projected area. Thus, it is assumed that, for example, changes in outside diameter and
changes in wall thickness are independent. Or, changing one parameter does not require that
another parameter be changed.
4.3 Design Method – Single Variable
The design charts for a single variable were developed based on the assumption that only a single
parameter, or factor such as outside diameter, is changing from the benchmark HML structure
given in Figure 3-1 and described in Tables 3-1 through 3-4. If, however, it is known that only
one parameter is being varied from the benchmark HML structure, then the design procedure for
determining the single parameter based on a target reliability index, βHML, can be summarized as:
Step 1: Determine detail category at the base of the HML structural support based on
AASHTO specifications: E or E′.
69
Step 2: Select single parameter to be varied: pole outside diameter, pole wall thickness,
pole length, or luminaire structure projected area.
Step 3: Determine the appropriate mean wind velocity, target fatigue life, and target
reliability index, βHML.
Step 4: Using the appropriate design chart based on the single parameter and target
fatigue life, determine the single parameter value based on mean wind velocity
and target reliability index, βHML.
Step 5: Determine all physical properties of HML structural support.
Step 5 requires that the properties of the benchmark HML structural support be modified. Table
4-1 provides the properties for this benchmark structure based on the four design parameters.
Tables 4-2 through 4-4 provide further details for pole outside diameter, pole wall thickness, and
pole section length. Table 4-5 provides further details for luminaire structure projected area. If
the single parameter varied is the pole outside diameter, pole wall thickness, or pole section
length then modification of the parameter must be done for all points along the height of the
pole. For example, if from Step 4, the pole outside diameter was found to be 20.8 in (528.32
mm), which corresponds to a 20% reduction from the original pole outside diameter, for a
particular mean wind velocity, target fatigue life, and target reliability index then the
corresponding reduction in all outside diameters along the height of the pole would be 20%.
Thus, reading from Table 4-2, all other outside diameters for the three sections that comprise the
pole could be easily determined. For cases in which the percent reduction or increase is not
given in the tables, linear interpolation is permitted.
Property Benchmark HML Structural Support
Outside Diameter (Base of Structure)
26.00 in (660.40 mm)
Wall Thickness (Base of Structure)
0.375 in (9.525 mm)
Total Length 120 ft (36.576 m)
Luminaire Structure Projected Area
11.50 ft2 (1.0684 m2)
Table 4-1: Benchmark HML Structural Support Properties
70
Pole Outside Diameter Benchmark Properties
(No Change) -30% -20% -10% +10% +20% +30%
Bottom of Section
26.00 in (660.40 mm)
18.2 in (462.28 mm)
20.8 in (528.32 mm)
23.4 in (594.36 mm)
28.6 in (726.44 mm)
31.2 in (792.48 mm)
33.8 in (858.52 mm)
Sect
ion
1
Top of Section
18.51 in (470.15 mm)
12.957 in (329.11 mm)
14.808 in (376.12 mm)
16.659 in (423.14 mm)
20.361 in (517.17 mm)
22.212 in (564.18 mm)
24.063 in (611.20 mm)
Bottom of Section
19.50 in (495.30 mm)
13.65 in (346.71 mm)
15.6 in (396.24 mm)
17.55 in (445.77 mm)
21.45 in (544.83 mm)
23.4 in (594.36 mm)
25.35 in (643.89 mm)
Sect
ion
2
Top of Section
12.15 in (308.61 mm)
8.505 in (216.03 mm)
9.72 in (246.89 mm)
10.935 in (277.75 mm)
13.365 in (339.47 mm)
14.58 in (370.33 mm)
15.795 in (401.19 mm)
Bottom of Section
13.00 in (330.20 mm)
9.1 in (231.14 mm)
10.4 in (264.16 mm)
11.7 in (297.18 mm)
14.3 in (363.22 mm)
15.6 in (396.24 mm)
16.9 in (429.26 mm)
Sect
ion
3
Top of Section
10.18 in (258.57 mm)
7.126 in (181.00 mm)
8.144 in (206.86 mm)
9.162 in (232.71 mm)
11.198 in (284.43 mm)
12.216 in (310.29 mm)
13.234 in (336.14 mm)
Table 4-2: Pole Outside Diameter Properties
Pole Wall Thickness Benchmark Properties
(No Change) -30% -20% -10% +10% +20% +30%
Section 1 0.375 in (9.525 mm)
0.2625 in (6.6675 mm)
0.3 in (7.62 mm)
0.3375 in (8.5725 mm)
0.4125 in (10.478 mm)
0.45 in (11.43 mm)
0.4875 in (12.383 mm)
Section 2 0.250 in (6.350 mm)
0.175 in (4.445 mm)
0.2 in (5.08 mm)
0.225 in (5.715 mm)
0.275 in (6.985 mm)
0.3 in (7.62 mm)
0.325 in (8.255 mm)
Section 3 0.2391 in (6.0731 mm)
0.16737 in (4.2512 mm)
0.19128 in (4.8585 mm)
0.21519 in (5.4658 mm)
0.26301 in (6.6805 mm)
0.28692 in (7.2878 mm)
0.31083 in (7.8951 mm)
Table 4-3: Pole Wall Thickness Properties
71
Pole Section Length Benchmark Properties
(No Change) -30% -20% -10% +10% +20% +30%
Section 1 53.49 ft (16.30 m)
37.443 ft (11.4126 m)
42.792 ft (13.0430 m)
48.141 ft (14.6734 m)
58.839 ft (17.9341 m)
64.188 ft (19.5645 m)
69.537 ft (21.1949 m)
Section 2 52.52 ft (16.01 m)
36.764 ft (11.2057 m)
42.016 ft (12.8065 m)
47.268 ft (14.4073 m)
57.772 ft (17.6089 m)
63.024 ft (19.2097 m)
68.276 ft (20.8105 m)
Section 3 20.16 ft (6.14 m)
14.112 ft (4.3013 m)
16.128 ft (4.9158 m)
18.144 ft (5.5303 m)
22.176 ft (6.7592 m)
24.192 ft (7.3737 m)
26.208 ft (7.9882 m)
Total Length 120 ft (36.576 m)
84 ft (25.6032 m)
96 ft (29.2608 m)
108 ft (32.9184 m)
132 ft (40.2336 m)
144 ft (43.8912 m)
156 ft (47.5488 m)
Table 4-4: Pole Section Length Properties
Luminaire Structure Projected Area
Benchmark Properties
(No Change) -30% -20% -10% +10% +20% +30%
Projected Area 11.50 ft2 (1.0684 m2)
8.05 ft2 (0.74787 m2)
9.2 ft2 (0.85471 m2)
10.35 ft2 (0.96155 m2)
12.65 ft2 (1.1752 m2)
13.8 ft2 (1.2821 m2)
14.95 ft2 (1.3889 m2)
Table 4-5: Luminaire Structure Projected Area Properties
Verification of the design method for a single variable is provided for four examples as outlined
in Table 4-6 for detail categories E and E′. Table 4-7 provides a summary of the results of these
examples. In Table 4-7, the column marked “Full Simulation” indicates that a full analysis using
the computer simulation program written by the authors was used to determine the reliability
index. The columns contained by the heading “Design Chart Method” makes use of the design
charts provided in the supplemental report for detail categories E and E′. Tables 4-8 and 4-9
provide the resulting properties and percent change from the benchmark HML structural support
from the examples considered in Table 4-6 respectively. The reader is also referred to Chapter 5
for an additional detailed example of the design method for a single variable.
72
Example Number
Outside Diameter at Bottom of Structure
Wall Thickness at Bottom
of Structure
Total Length
Luminaire Structure Projected
Area
Mean Wind Velocity
Target Fatigue
Life
AASHTO Detail
Category
Example 1 Variable No Change No Change No Change 12 MPH (19.31 km/hr) 50 Years E
Example 2 No Change Variable No Change No Change 14 MPH (22.53 km/hr) 25 Years E
Example 3 No Change No Change Variable No Change 14 MPH (22.53 km/hr) 75 Years E′
Example 4 No Change No Change No Change Variable 12 MPH (19.31 km/hr) 75 Years E′
Table 4-6: Design Method Confirmation Examples – Single Variable
Full Simulation Design Chart Method
Example Number
β Unknown Parameter Target β
β Ratio
Example 1 2.998 Outside Diameter 3.0 0.999
Example 2 2.687 Wall Thickness 2.5 1.075
Example 3 2.511 Total Height 2.5 1.004
Example 4 2.851 Projected Area 3.0 0.950
Table 4-7: Design Method Confirmation Summary – Single Variable
73
Example Number
Outside Diameter at Bottom of Structure
Wall Thickness at Bottom of
Structure Total Length
Luminaire Structure
Projected Area
Example 1 20.8 in (528.32 mm)
0.375 in (9.525 mm)
120 ft (36.576 m)
11.5 ft2 (1.0684 m2)
Example 2 26 in (660.4 mm)
0.2625 in (6.668 mm)
120 ft (36.576 m)
11.5 ft2 (1.0684 m2)
Example 3 26 in (660.4 mm)
0.375 in (9.525 mm)
108 ft (32.918 m)
11.5 ft2 (1.0684 m2)
Example 4 26 in (660.4 mm)
0.375 in (9.525 mm)
120 ft (36.576 m)
14.375 ft2 (1.335 m2)
Table 4-8: Design Method Confirmation Properties – Single Variable
Example Number
Outside Diameter at Bottom of Structure
Wall Thickness at Bottom of
Structure Total Length
Luminaire Structure
Projected Area
Example 1 -20% No Change No Change No Change
Example 2 No Change -30% No Change No Change
Example 3 No Change No Change -10% No Change
Example 4 No Change No Change No Change +25%
Table 4-9: Design Method Confirmation Percent Changes – Single Variable
Based on the summary given in Table 4-7, the design charts for a single variable are, on average,
approximately 1% conservative. Example 4, however, does provide a reliability index from the
design charts that is higher than the reliability index calculated from the full computer
simulation. The design chart is thus approximately 5.2% higher than the computer simulation.
This result is not unexpected due to the complexity of the analysis. Furthermore, the results are
still reasonably accurate.
74
4.4 Design Charts – Multiple Variables
The design charts for multiple variables are provided in the supplemental reports for detail
categories E and E′. These design charts were determined based on assumed values for pole
outside diameter, pole length, and luminaire structure projected area. Reasonable values were
assumed for these three variables. Pole wall thickness values were varied for several
combinations of the three assumed variables.
4.5 Design Method – Multiple Variables
The design charts for multiple variables were developed based on assumed values for pole
outside diameter, pole length, and luminaire structure projected area. Reasonable values were
assumed for these three variables. Pole wall thickness values were varied for several
combinations of the three assumed variables. Prior conversations with CDOT personnel
conveyed the desire to select the pole outside diameter and pole wall thickness for a given pole
length and luminaire structure. Thus, the design procedure for determining the parameters based
on a target reliability index, βHML, can be summarized as:
Step 1: Determine detail category at the base of the HML structural support based on
AASHTO specifications: E or E′.
Step 2: Select pole length, or height of the HML structural support, from three choices:
100 ft (30.48 m), 120 ft (36.576 m), or 140 ft (42.672 m).
Step 3: Select luminaire structure projected area from three choices: 7.5 ft2 (0.6968 m2),
9.5 ft2 (0.8826 m2), or 11.5 ft2 (1.0684 m2).
Step 4: Determine the appropriate mean wind velocity, target fatigue life, and target
reliability index, βHML.
Step 5: Select an appropriate pole outside diameter from three choices: 23.4 in (594.36
mm), 26 in (660.4 mm), or 28.6 in (726.44 mm).
Step 6: Using the appropriate design chart based on the pole outside diameter, mean wind
velocity, and target fatigue life, determine the pole wall thickness based on the
pole length, luminaire structure projected area, and target reliability index, βHML.
75
Step 7: Determine all physical properties of HML structural support.
Step 7 requires that the properties of the benchmark HML structural support be modified. As
before, Table 4-1 provides the properties for this typical structure based on the four design
parameters. Tables 4-2 through 4-4 provide further details for pole outside diameter, pole wall
thickness, and pole section length. Table 4-5 provides further details for luminaire structure
projected area. The same procedure as was explained for Step 5 of the design method for a
single variable is applicable to Step 7 for the design method for multiple variables.
Verification of the design method for multiple variables is provided for four examples as
outlined in Table 4-10 for detail categories E and E′. Table 4-7 provides a summary of the
results of these examples. In Table 4-11, the column marked “Full Simulation” indicates that a
full analysis using the computer simulation program written by the authors was used to
determine the reliability index. The columns contained by the heading “Design Chart Method”
makes use of the design charts provided in the supplemental report for detail categories E and E′.
Tables 4-12 and 4-13 provide the resulting properties and percent change from the benchmark
HML structural support from the examples considered in Table 4-10 respectively. The reader is
also referred to Chapter 5 for an additional detailed example of the design method for a single
variable.
Example Number
Outside Diameter at Bottom of Structure
Wall Thickness at Bottom
of Structure
Total Length
Luminaire Structure Projected
Area
Mean Wind Velocity
Target Fatigue
Life
AASHTO Detail
Category
Example 1 Assumed:
28.6 in (726.44 mm)
Variable Assumed:
140 ft (42.672 m)
Assumed: 11.5 ft2
(1.0684 m2)
12 MPH (19.31 km/hr) 75 Years E
Example 2 Assumed:
26 in (660.4 mm)
Variable Assumed:
120 ft (36.576 m)
Assumed: 9.5 ft2
(0.8826 m2)
14 MPH (22.53 km/hr) 50 Years E
Example 3 Assumed:
23.4 in (594.36 mm)
Variable Assumed:
120 ft (36.576 m)
Assumed: 7.5 ft2
(0.6968 m2)
12 MPH (19.31 km/hr) 25 Years E′
Example 4 Assumed:
26 in (660.4 mm)
Variable Assumed:
140 ft (42.672 m)
Assumed: 9.5 ft2
(0.8826 m2)
14 MPH (22.53 km/hr) 50 Years E′
Table 4-10: Design Method Confirmation Examples – Multiple Variables
76
Full Simulation Design Chart Method
Example Number
β Unknown Parameter Target β
β Ratio
Example 1 3.187 Wall Thickness 3.0 1.062
Example 2 2.524 Wall Thickness 2.5 1.010
Example 3 2.763 Wall Thickness 2.75 1.005
Example 4 2.037 Wall Thickness 2.0 1.019
Table 4-11: Design Method Confirmation Summary – Multiple Variables
Example Number
Outside Diameter at Bottom of Structure
Wall Thickness at Bottom of
Structure Total Length
Luminaire Structure
Projected Area
Example 1 28.6 in (726.44 mm)
0.345 in (8.763 mm)
140 ft (42.672 m)
11.5 ft2 (1.0684 m2)
Example 2 26 in (660.4 mm)
0.3 in (7.62 mm)
120 ft (36.576 m)
9.5 ft2 (0.8826 m2)
Example 3 23.4 in (594.36 mm)
0.25 in (6.35 mm)
120 ft (36.576 m)
7.5 ft2 (0.6968 m2)
Example 4 26 in (660.4 mm)
0.445 in (11.303 mm)
140 ft (42.672 m)
9.5 ft2 (0.8826 m2)
Table 4-12: Design Method Confirmation Properties – Multiple Variables
77
Example Number
Outside Diameter at Bottom of Structure
Wall Thickness at Bottom of
Structure Total Length
Luminaire Structure
Projected Area
Example 1 +10% -8% +16.7% No Change
Example 2 No Change -20% No Change -17.4%
Example 3 -10% -33.33% No Change -34.8%
Example 4 No Change +18.67% +16.7% -17.4%
Table 4-13: Design Method Confirmation Percent Changes – Multiple Variables
Based on the summary given in Table 4-11, the design charts for multiple variables are, on
average, approximately 2% conservative. As with the design charts for a single variable, these
results are not unexpected due to the complexity of the analysis. Furthermore, the results are still
reasonably accurate for all four examples.
78
CHAPTER 5
ILLUSTRATIVE DESIGN EXAMPLES FOR FATIGUE PERFORMANCE
5.1 Example 1 – Single Variable
Illustrative example 1 considers an HML structural support design for a mean wind velocity of
12 MPH (19.31 km/hr) with a target fatigue life of 50 years and detail category E. A target
reliability index, βHML, of 3.0 is desired for the HML structural support. Only pole wall
thickness is considered to vary from the benchmark HML structural support used in this study.
All other factors, pole outside diameter, pole length, and luminaire structure projected area, are
considered to be consistent with the benchmark HML structural support. Therefore, there values
are 26 in (660.4 mm), 120 ft (36.576 m), and 11.5 ft2 (1.0684 m2) respectively.
Using the design charts for a single variable from the supplemental report for detail category E
with a mean wind velocity of 12 MPH (19.31 km/hr), a target fatigue life of 50 years, and a
target reliability index of 3.0, the pole wall thickness is determined as shown in Figure 5-1.
Figure 5-1: Example 1 – Single Variable
Based on the results from Figure 5-1, the wall thickness has been reduced 25% to 0.28125 in
(7.144 mm) compared to the benchmark HML structure wall thickness of 0.375 in (9.525 mm) at
the base of the structure.
79
The target reliability index for the structure is 3.0. A full analysis using the computer simulation
program written by the authors determined that the reliability index of the structure used in this
example was 2.972. Recall that the design method proposed for a single variable produces, on
average, designs that are approximately 1% conservative. For this example, the target reliability
index of 3.0 is 0.9% higher than the calculated reliability index of 2.972.
The final step in the design process that must be completed is the adjustment of the physical
properties of the HML structural support, or Step 5 in the design process for a single variable.
The wall thickness of the entire structure must be reduced by 25% at all points. There were no
changes to the outside diameter, length, or luminaire structure projected area. Therefore, their
values remain at 26 in (660.4 mm), 120 ft (36.576 m), and 11.5 ft2 (1.0684 m2) respectively.
5.2 Example 2 – Multiple Variables
Illustrative example 2 considers an HML structural support design for a mean wind velocity of
14 MPH (22.53 km/hr) with a target fatigue life of 50 years and detail category E′. A target
reliability index, βHML, of 2.5 is desired for the HML structural support. Initial requirements for
pole length and luminaire structure projected area are 100 ft (30.48 m) and 9.5 ft2 (0.8826 m2)
respectively. Values for pole outside diameter and pole wall thickness are thus needed for the
desired target reliability index. It is assumed that the pole outside diameter is set at 23.4 in
(594.36 mm). Thus, the pole wall thickness is considered to vary.
Using the design charts for multiple variables from the supplemental report for detail category E′
with a mean wind velocity of 14 MPH (22.53 km/hr), a target fatigue life of 50 years, a pole
outside diameter of 23.4 in (594.36 mm), and a target reliability index of 2.5, the pole wall
thickness is determined as shown in Figure 5-2.
80
Figure 5-2: Example 2 – Multiple Variables
Based on the results from Figure 5-2, the wall thickness has been decreased 15.33% to 0.3175 in
(8.065 mm) compared to the benchmark HML structure wall thickness of 0.375 in (9.525 mm) at
the base of the structure. Furthermore, the pole outside diameter has been reduced 10% to 23.4
in (594.36 mm) compared to the benchmark HML structure outside diameter of 26 in (660.4
mm) at the base of the structure. The pole length has been decreased 16.7% to 100 ft (30.48 m)
compared to the benchmark HML structure length of 120 ft (36.576 m). Finally, the luminaire
structure projected area has been decreased 17.4% to 9.5 ft2 (0.8826 m2) compared to the
benchmark luminaire structure projected area of 11.5 ft2 (1.0684 m2).
The target reliability index for the structure is 2.5. A full analysis using the computer simulation
program written by the authors determined that the reliability index of the structure used in this
example was 2.472. Recall that the design method proposed for multiple variables produces, on
average, designs that are approximately 2% conservative. For this example, the target reliability
index of 2.5 is 1.1% higher than the calculated reliability index of 2.472.
Again, one final step in the design process that must be completed is the adjustment of the
physical properties of the HML structural support, or Step 7 in the design process for multiple
variables. The outside diameter and wall thickness of the entire structure must be decreased by
10% and 15.33%, respectively, at all points.
81
CHAPTER 6
SUMMARY, CONCLUSIONS AND RECOMMENATIONS
In this report the method and results for the development of a reliability-based design procedure
for high-mast lighting structural supports in Colorado were presented. A computational fluid
dynamics (CFD) approach was coupled with finite element theory in order to model the in-line
dynamic motion of high mast lighting structural supports. Common frequencies and wind
velocity combinations associated with vortex shedding were also examined and presented in the
appendix of the report. Six structural and two wind-related parameters were varied and their
sensitivity on fatigue life examined. Based on these sensitivities, design charts were developed
for single and multiple variable variations and are given in the supplemental reports for detail
categories E and E′ defining the AASHTO connection type at the base of the HML structural
support. The design charts for the multiple variable variations were developed based on
conversations with CDOT personnel. Results determined from the design charts compared well
with those obtained from a full computer simulation. The two wind-related parameters in this
study were based on expert judgment, but will require verification beyond the present study’s
scope, prior to implementation of the methodology around the state of Colorado.
The approach presented herein is general and can be applied to virtually any type of high mast
lighting structure that has a reasonable number of design parameters. Tabulated values could
also be generated rather than plots, when it is sought to implement the methodology. It can be
concluded based on the results of the two examples that for the general population of high mast
lighting structural supports in Colorado, the methodology will work well provided that accurate
wind statistics can be developed.
Thus, it is highly recommended that CDOT consider a wind study along Colorado’s Front Range
(and other locations, if desired), thus mapping any inconsistencies with ASCE-7 mapped values,
which are known to be quite coarse. Additionally, this study must also create a link between the
mean wind velocities used in this report and appropriate 50-year wind velocities along
Colorado’s Front Range. Further, current 50-year wind velocities used in the state of Colorado
by CDOT, which range from 90 to 110 MPH (144.84 to 177.03 km/hr) may or may not be
82
justified. The answer to this question is unknown both at the state and national level and is
deserving of the amount of attention that would be required to solve this important problem.
Such a study would have immediate implications on all traffic-related structures, such as signal
poles and arms, whose numbers are far greater than high mast lighting structural supports.
83
REFERENCES
AASHTO (2001). With interims. “Standard Specifications for Structural Supports for Highway Signs, Luminaires and Traffic Signals”, 4th Edition. American Association of State Highway and Transportation Officials, Washington, D.C.
Ang, A.H-S. and W.H. Tang (1975). Probability Concepts in Engineering Planning and Design,
Volume 1. John Wiley and Sons, New York. ASCE 7-05 (2005). “Minimum Design Loads for Buildings and Other Structures”. American
Society of Civil Engineers, Reston, VA. Assakkaf, I., and Ayyub (2000), B.M., “Load and Resistance Factor Design Approach for
Fatigue of Marine Structures”, 8th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability, ASCE, University of Notre Dame, South Bend, IN.
Chopra, A.K. (2001). Dynamics of Structures, 2nd Edition. Prentice Hall, Upper Saddle River,
NJ. Crandall, S.H. and W.D. Mark (1963). Random Vibration of Mechanical Systems. Academic
Press, New York and London. Holmes, J.D. (2001). Wind Loading of Structures. Spon Press, New York. Kaufmann, E.J. (2005). “Investigation of Cracking in High-Mast Light Poles at E-470 and Pena
Boulevard, Denver, Colorado”. ATLSS, Lehigh University, Bethlehem, PA. Kwon, Y.W. and H. Bang (2000). The Finite Element Method Using MATLAB, 2nd Edition. CRC
Press, New York. Miner, M.A. (1945). “Cumulative Damage in Fatigue”. Journal of Applied Mechanics, 12, A159. Morison, J.R., O’Brien, M.P., Johnson, J.W. and S.A. Schaaf (1950). “The Force Exerted by
Surface Waves on Piles”. AIME Petroleum Transactions, 189, 149-157. Newmark, N.M. (1959). “A Method of Computation for Structural Dynamics”. Trans. ASCE,
American Society of Civil Engineers, 109(111). Paz, M. and W. Leigh (2004). Structural Dynamics, 5th Edition. Kluwer Academic Publishers,
Boston. Simiu, E. and R.H. Scanlan (1996). Wind Effects on Structures, 3rd Edition. John Wiley and
Sons, New York. Wilcox, D.C. (2003). Basic Fluid Mechanics, 2nd Edition. DCW Industries, La Cañada, CA.
A-1
APPENDIX A
A.1 Vortex-Induced-Vibration
One major type of loading that slender structures are subjected to during wind loading is a
phenomenon known as vortex shedding. Vortex shedding occurs when vortices are formed and
shed in the wake just behind the cylinder, i.e. the mast of the HML. This section presents the
results of an investigation into vortex-induced-vibration (VIV) of HML structural supports.
Finite Element Model
In order to determine the fundamental period of vibration, Tn, and the harmonics, 1 2, , , nT T T , a
finite element model (FEM) that utilized twelve degree-of-freedom beam elements was
employed. The FEM details were presented in Chapter 2 of this report
Vortex Shedding Model
It has been reasoned that an empirical linear model for VIV provides significant and accurate
information for many engineering structures (Simiu, 1996). In order to derive this approach, we
begin with the forced equation of motion of a cylinder in fluid flow,
( )2
1sin2 LDmx cx kx C K sm
ρ θ+ + = − (A-1)
where m is the mass, c is the damping coefficient, and k is the structure stiffness. The value x is
the displacement and a dot indicates a time derivative, ρ is the mass density of air, D is the
diameter of the cylinder, θ is the phase angle, LC is an experimentally determined lift
coefficient, and
1nDK
Uω
= (A-2)
A-2
where nω is the natural cyclic frequency of the structure, and U is the velocity of the wind flow,
and
UtsD
= (A-3)
where t is time. According to Simiu (1996) solving closed form for the steady state response of
the oscillator described by equations (A-1) and (A-2) yields an amplitude of
( ) ( )
2
2 22 21 1
2
2L
n n
D C m
K c K
ρηω ω
=− +
(A-4)
Notice that all quantities are known except the lift coefficient, CL. However, extensive
experiments have been performed on cylinders in fluids with uniform flow. Again, utilizing the
data provided in Simiu (1996), one can express the amplitude given in equation (A-4) as a
function of K1. In order to solve for CL one can rearrange equation (A-4) as
( ) ( )2 22 21 12
2 2L n nmC K c K
Dη ω ωρ
= − + (A-5)
Then, fitting the appropriate polynomial for the experimental data gives
( ) ( )1
2 20 2 21 12
22
nn i n
iL n n
m a KC K c K
Dω ω
ρ
−
=
⎧ ⎫⎨ ⎬⎩ ⎭= − +∑
(A-6)
where n is the order of the polynomial and i is simply the index in the summation.
For the illustration presented in this appendix it is only desired to determine if lock in occurs at
one or more of the three fundamental transverse frequencies. Because there were slight
variations in the wall thickness of the poles this will be varied as well. Table A-1 presents the
A-3
results of the vortex induced vibration lock-in study described above. An “×” was placed in the
table when lock-in was identified by the numerical model.
First Transverse Mode Second Transverse Mode Third Transverse Mode Wall Thickness
Wind Speed MPH
(km/hr) -30% NC +30% -30% NC +30% -30% NC +30% 5
(8.05) × × × 10
(16.09) 15
(24.14) × × × 20
(32.19) × × × 25
(40.23) × × × 30
(48.28) 35
(56.33) × 40
(64.37) × × × 45
(72.42) × × × 50
(80.47) × × × 55
(88.51) × × × 60
(96.56) × × × 65
(104.61) × × 70
(112.65)
Table A-1: Wind Speed and Vibration Mode Combinations at which Lock-In was Numerically Identified
Interestingly, notice that some type of lock-in occurs at almost every wind speed. However, it is
very important to note that no taper of the HML structural support was considered here and these
results are presented for consideration of VIV sensitivity only.
A.2 Selected Mean Wind Velocities
A preliminary survey of readily available data for locations within Colorado was conducted to
determine mean wind velocities at the characteristic height of 33 ft (10 m). This data was
obtained from the Plains Organization for Wind Energy Resources (POWER) and Energy and
A-4
Environmental Research Center (EERC) at the University of North Dakota. This data is
available online with the Internet address provided under the caption of the table. Data was
recorded hourly. Table A-2 summarizes the data obtained and Figure A-1 provides the locations
given in the table.
Location Record Length Mean Wind Velocity MPH (km/hr)
Mean Standard Deviation
MPH (km/hr)
Mean Turbulence Intensity (COV)
%
Boulder 4 years 8.38 (13.49)
1.57 (2.53) 18.68
Calhan 1 year 13.03 (20.97)
2.39 (3.85) 18.34
Cheyenne Wells 1 year 13.29 (21.39)
2.34 (3.77) 17.59
Genoa 1 year 13.72 (22.08)
2.38 (3.83) 17.31
Gobblers Knob (East) 1 year 13.45
(21.65) 2.25
(3.62) 16.74
Gobblers Knob (West) 1 year 12.90
(20.76) 2.29
(3.69) 17.73
Livermore 1 year 14.93 (24.03)
2.94 (4.73) 19.71
Mesa de Maya 1 year 12.08 (19.44)
2.31 (3.72) 19.09
Pawnee Buttes 1 year 16.21 (26.09)
2.48 (3.99) 15.31
Peetz 1 year 14.34 (23.08)
2.49 (4.01) 17.37
Wauneta 1 year 15.04 (24.20)
2.50 (4.02) 16.62
Table A-2: Mean Wind Velocities for Selected Locations in Colorado
Figure A-1: Selected Locations in Colorado
(Table A-2 and Figure A-1: http://www.undeerc.org/wind/winddb/COwindsites.asp)