A nonlinear dynamic analysis of the traffic signal cable system

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A nonlinear dynamic analysis of the traffic signal cable system
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xii, 225 leaves : ill. ; 29 cm.
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Adediran, Adeola Kehinde, 1966-
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Thesis (Ph. D.)--University of Florida, 1997.
Bibliography:
Includes bibliographical references (leaves 221-224).
Statement of Responsibility:
by Adeola Kehinde Adediran.
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Typescript.
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Vita.

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A NONLINEAR DYNAMIC ANALYSIS OF THE
TRAFFIC SIGNAL CABLE SYSTEM













By

ADEOLA KEHINDE ADEDIRAN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1997


























To him who is, and who was and is to come the Almighty.



"For I know the plans I have for you, declares the Lord, "plans to prosper you
and not to harm you, plans to give hope and a future. "

Jeremiah 29:11










ACKNOWLEDGMENTS


The author would like to express profound gratitude to Dr. Marc Hoit and Dr.

Ronald Cook for their unwavering support throughout the length of this doctoral

program. Their affable personalities maintained an invaluable open door policy to

students that has been most beneficial to this author. Special thanks also go to Dr.

Fernando Fagundo, Dr. John Lybas and Dr. James Hobert for their various

contributions to this dissertation and their professional guidance. It has indeed been

an honor to have worked with such a distinguished caliber of lecturers. Other people

that have proven to be a tremendous inspiration and support throughout this

dissertation include Dr. Duane Ellifritt for, among other things, helping to obtain

some needed reference articles; Petros Christou, for his help with programming; Dr.

Glagola for all the friendly advice and Mrs. Clara Osinulu for the idea and

encouragement to go for a doctorate degree.

The author will also like to thank fellow graduate students for the moral

support, the Engineering Minority office and the American Association of University

Women for financial aid at various times in the course of this doctoral program.

Finally and more important, the author would like to express gratitude to God,

the author's mother, Adenihun, sisters, Adeoti and Bolaji and brother, Bayo for all

their loving support.













TABLE OF CONTENTS


ACKNOWLEDGEMENT ........................................ iii

LIST OF FIGURES ............................................. vii

LIST OF TABLES .............................................. ix

ABSTRACT .................................................... xi

CHAPTERS

1. INTRODUCTION .............................................. 1
Problem Definition ......................................... 1
The Structure ........................................ 1
Theory ............................................ 3
The Computer Program .................................. 6
Overview .................................................. 6
Content ............................................. 7
Scope ............................................. 8

2. LITERATURE BACKGROUND ................................ 10
Introduction .............................................. 10
Fundamentals of Cable Analysis ............................... 10
Cable Dynamics .......................................... 18
Closed Form Solutions ................................. 19
Finite Element Method ....................................... 22
Formulation of Element Properites .............................. 23
Beam Properties ................................... 23
Truss Element Properties ................................ 27
Cable Element Properties .............................. 29
Connector Element .................................... 36
Transformation Matrix ................................. 46
Damping Matrix ................ ..................... 50
Dynamic Solution M ethods ................................... 53







Explicit Methods ...................................... 54
Implicit Methods ...................................... 55
Newmark Beta Method ....................................... 56

3. THEORETICAL DEVELOPMENT ............................... 59
Introduction ................................................ 59
Two Node Non-linear Cable Element ................... ...... 61
Equation of M otion .................................... 73
Modified Newmark Beta Method ......................... 77
Modified Newton Raphson ............................. 81
Solution Algorithm ........................................ 86
Solution Flowchart ........................................ 89

4. LOADING ............................................... 91
Introduction .................................... .......... 91
Variable Area W ind Load .................................... 92
Variable Velocity Dynamic Wind Load ......................... 95
Autoregressive Filters ....................................... 103
Test Data ................................................. 107

5. PROGRAM VERIFICATION FOR STATIC LOADS ................ 113
Static Problem Definition .................................... 113
Single Cable Structure Example ....................... 114
Static Test System Configuration .................... ... 116
Loading Sequence of Tests Performed ............... 120
Test Results and Predicted Values .................. 123
Discussion of Results ............................ 130

6. PROGRAM VERIFICATION FOR DYNAMIC LOADS .............. 144
The Structure's Configuration ............... .............. 144
Coordinates of Wind Load Test System ................... 146
Wind Test Set-up ........................................ 147
Description of Wind Load Test ......................... 148
Comparison of Results ................................. 149
Test Data ....................................... 150
Wind Load Modeling .................................. 153

7. CONCLUSION ................ .............................. 156
Summary ................................................ 156







Review of Analysis Method Developed ................... 157
Verification of Analysis Method ......................... 159
Advantages and Disadvantages of Solution Method ............... 159
Recommendations for Further Research ......................... 160


APPENDICES

A. CABLE ELEMENT DERIVATION ............................. 162

B. USER'S MANUAL FOR DYNASS ............................ 179

LIST OF REFERENCES ........................................ 221

BIOGRAPHICAL SKETCH ..................................... 225







LIST OF FIGURES


Figure page
1.1 Traffic Signal System ................... ....... 2
1.2 Static load model ..................... .......... 4

1.3 Mean and time varying components of wind Speed .......... 5
2.1 Cartesian Coordinate... ....... ... ............. 11
2.2 Displacement of a cable segment . ... .. 12
2.3 Shape of cable due to selfwt.. . ..... .... 14
2.4 Notations for Eqn. 2-12 ................. ........ 21
2.5 Element degrees of freedom for a beam element . .. 24
2.6 Degrees of freedom for a truss element . ... 27
2.7 Truss element showing direction cosines . ... 28

2.8 Straight nonlinear truss element used for cables . ... 30
2.9 Deformed length due to displacements . ..... 31
2.10 Joint displacement and corresponding end forces . .... 37
2.11 Relative deformation and associated forces . ..... 38
2.12 Rigid body and natural modes for large displacement beams .. 39

2.13 Rigid body rotations .............. .......... .. 41
2.14 Relative nodal rotations . . . 43
2.15 Euler angles for transformation . . 48

3.1 CableElement. ................ ........ ...... 61

3.2 Newton Raphson Iteration . . .. 82
3.3 Flowchart of solution algorithm . . 90
4.1 Resolved wind load on signals . .. .. 92
4.2 Lift and Drag coefficients .. ....... ...... ..... .. 94








Figure page
4.3 Distribution of sample generated . .. .. 98
4.4 Spectral density of fluctuating part of wind . .. .. 101
4.5 Numerical filteration process .. ..... ..... .. 105
4.6 Wind speed generated using the spectral density function in Eqn. 4-10 .. 106
4.7 Wind load test systems with final cable tensions ..... 108
4.8 Spectral Density (All wind speed ranges) . .. .
109
4.9 Spectral Density (Low wind speeds). . ..... 110
4.10 Spectral Density (Medium wind speeds) . .... 111
4.11 Spectral Density (High wind speeds) . . 112
5.1 Single Cable Problem ..................... .......... 114
5.2 Static Load Test System Dimensions and Node Numbers. ..... 118
5.3 Cable tensions Static Test # 1 . . .. 133
5.4 Cable tensions Static Test # 2 . ..... 135
5.5 Cable tensions Static Test # 3 . . 137
5.6 Dynamic Structural Response at Catenary Node 6 (Test 1) ...... 141
5.7 Dynamic Structural Response at Catenary Node 6 (Test 2) .. .... 142
5.8 Dynamic Structural Response at Catenary Node 6 (Test 3) .. ...... 143
6.1 Wind Load Test System Dimension and Node Numbering .. ..... 145


viii













LIST OF TABLES


page


Table

5-1

5-2

5-3

5-4

5-5

5-6

5-7

5-8

5-9

5-10

5-11

5-12

5-13

5-14

5-15

5-16
5-17

5-18

5-19


6-1 Dimensions and node coordinates for Fig. 6.1 .


Displacement Comparison for Suspended Cable . ... 115

Stress Comparison for Suspended Cable ........ .. 115

Section Properties for Members shown in Fig. 5.2 .. ... 117

Dimensions and node coordinate for Fig. 5.2 .. ..... 119

Loading Sequence for test # 1 . . 120

Loading Sequence for test # 2 . .. 121
Loading Sequence for test # 3 ........... ..... 122

Catenary Cable Results Comparison for Test # 1 . 124

Messenger Cable Results Comparison for Test # 1 .. ... 125

Catenary Cable Results Comparison for Test # 2 .. 126

Messenger Cable Results Comparison for Test # 2 .. ... 127

Catenary Cable Results Comparison for Test # 3 .. .. 128

Messenger Cable Results Comparison for Test # 3 .. ... 129

Test 1 Cable Tensions . . 132

Test 2 Cable Tensions . .. . 134

Test 3 Cable Tensions . . 136
Displacement Comparisons for Test 1 ..... . 138
Displacement Comparisons for Test 2 .... 139

Displacement Comparisons for Test 3 .. . ... 140


. 146







-6-2 Section Properties for Members shown in Fig 6.1 ... .. 148
6-3 Wind Data for Low Wind Speed Test ........ .. .. 150
6-4 Wind Data for MediumWind Speed Test ... .. 151
6-5 Wind Data for High Wind Speed Test ....... .. 152
6-6 Wind Speeds and Resulting Drag Forces used in Dynamic Analysis. 153
6-7 Cable Tensions for Wind Load Test # 1 ....... 154













Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


A NONLINEAR DYNAMIC ANALYSIS OF THE
TRAFFIC SIGNAL CABLE SYSTEM

By

ADEOLA KEHINDE ADEDIRAN

May 1997

Chairperson: Marc I. Hoit
Cochairperson: Ronald A. Cook
Major Department: Civil Engineering Department

Finite element analysis of linear structures is an increasingly popular tool of

choice for analysis by engineers. Only recently has this tool been applied to the

case of nonlinear cable structures with a sufficient degree of veritable accuracy.

Work in this dissertation provides a finite element approach to the analysis of

cable structures with verification of the approach done for a traffic signal cable

structure. There has been a preponderance of work done on the dynamic analysis

of single cables and cable networks but very little is known about dynamic

analysis of non-grid-like cable elements in conjunction with other elements such as

beams and trusses. The approach of analysis presented in this dissertation

xi







provides a valuable bridge in available theories by creating a program that

dynamically analyzes a cable in context of other types of elements also present

within the structure. Using the principle of virtual work, the cable element

developed in this dissertation is based upon a total Lagrangian formulation. The

cable element models geometric nonlinear capability. Also modeled in the

program developed is a nonlinear loading capability. A realistic wind load

generation was accomplished by the use of auto-regressive numerical filters. With

a complete finite element model sufficiently exhibiting the elastic and kinematic

properties of the true structure, dynamic analysis was accomplished by numerical

integration of the equation of motion. An incremental form of the general equation

of motion is used because of a piecewise linear assumption. The numerical

integration scheme presented in this dissertation is the Newmark's Beta method.

The Newmark's Beta method reduces the equation of motion to a set of nonlinear

algebraic equations. These equations are then solved using a modified Newton-

Raphson. The finite element approach presented in this dissertation is

programmed into a software, DYNASS. Verification of the program DYNASS

and subsequently the solution approach is done by comparing predictions from

DYNASS to test results obtained from recent traffic signal experiments performed

at the Civil Engineering Department, University of Florida.












CHAPTER 1
INTRODUCTION



Problem Definition



The use of cables as load-carrying elements is common for structures such

as bridges, cable reinforced membrane roofs, traffic signal systems, spatial cable

nets and mooring lines. The set of forces seen in each cable application is different

but there is a common trend in the behavior of these structural systems. This

behavior is nonlinear. One source of nonlinearity is the large displacement the

cables undergo. This type of nonlinear behavior is known as geometric

nonlinearity. Several theories have been proposed to handle this problem. One

such theory is presented in this report.


The Structure


The structure investigated here is the traffic signal system. The choice of

this type of cable stayed structure was a result of the recent work done for the

Florida Department of Transportation. The Civil Engineering Department at the







2

University of Florida developed the software program, ATLAS, to perform a static

analysis of a two-point connection traffic signal system as shown in Fig. 1.1. The

restrictions encountered during developing this program spurred the need for

another analysis approach. The typical system considered consists of a catenary

cable (slack cable configuration), a messenger cable (taut cable configuration), the

signal heads and signs (points of dead and wind load application), and the

prestressed concrete poles (as the support system).



Catenary Cable Prestressed concrete pole
Connection hanger {
... ...... -. _< _


Disconnect box A A

Messenger cable


Signal head


Fig. 1.1 Traffic Signal System










The traffic signal system considered includes that shown in Fig. 1.1 but is

not limited to the two-point connection traffic signal system shown. The method

presented in this research is capable of handling virtually any system of cables and

frame elements, for example, the one-point connection traffic signal system and

the suspended box system.



The Theory


The approach presented here uses the theories of structural dynamics to

model the behavior of cables subjected to both dynamic wind loads and static

loads. The static load case can be considered a special case of the general dynamic

model. The static load is modeled as a constant load with duration equal to the full

time of analysis. The structural response often of interest in the static load case is

the damped out response, i.e. the stresses, moments, forces and deflections at die-

down. For this response the initial rate of application of load is inconsequential. It

is, however, good practice to maintain a slow ascent to full load in order to

minimize impact effects of the load and to obtain a dynamic response more

reflective of a stationary load.




















P /
pp/
o /





t> 3 mins
time


Fig. 1.2 Static load model







The wind load is modeled as both static and dynamic. The equivalent static

wind load as determined by AASHTO (American Association of State Highway

Officials, "Standard Specifications for Structural Supports for Highway Signs,

Luminaries and Traffic signals") is applied as a horizontal force (called the drag

force) and a vertical force (called the lift force). An aspect of nonlinear behavior

of traffic signal cable is the nonlinear wind loading. The research work presented









provides a dynamic wind model comprising a steady wind pressure and a variable

wind component. The variable wind component models wind gust effects. A

wind gust is a short duration wind turbulence often characterized by its frequency

content and its duration. Fig. 1.3 shows a typical wind gust representation.






Vw



W t L ,t


Fig. 1.3 Mean and time varying components of wind speed


IM







6

The dynamic theory presented uses geometric and time discretization. The

geometric discretization or discrete element technique employed is the finite

element method. Numerical integration is used in the solution of the equations of

motion in the discrete time domain. Generally a nonlinear response analysis

involves a significant amount of computation as it was in this case. Consequently,

a computer program was written.



The Computer Program


The resulting program is called DYNASS. This program performs a

nonlinear dynamic analysis of the traffic signal cable system. This program is

written in Fortran with some wind generation routines in C. The results of this

program are compared with results from other programs where possible and with

test results from full size traffic signal system tests.





Overview



Each chapter in this report discusses topics or themes built upon in

subsequent chapters. Topics discussed start from the theory proposed to the

program written, to the comparison of the results from the program with test







7

results and finally to the typical applications and conclusion. Figures and formulas

are numbered consecutively throughout each chapter.



Content


Chapter two presents a literature background. This includes an overview of

existing work done on this topic and an explanation of the fundamentals of

structural dynamic theories and cable analysis theories.

Chapter three contains a detailed presentation of the theories and methods

developed in this research. These include the various assumptions made and the

equations derived. This chapter is essentially subdivided into two parts. The first

part is the theoretical development just mentioned, and the second part discusses

the algorithm used in the program, DYNASS.

Chapters four discusses the wind load models developed plus a general

description of the other loads on the traffic signal system. There are two wind

models formulated, the variable area wind load and the variable velocity wind

load. Both models are discussed in some detail in chapter four. Finally, chapter

four presents a brief discussion of the wind load data obtained from dynamic

experiments performed. These experiments were conducted previously for a

Florida Department of Transportation sponsored research.









Chapter five and six present comparisons of the results from DYNASS to

other software, where available, and to test data obtained from experiments done.

Chapter five compares results for static load cases only and chapter six deals with

the comparison of results for dynamic wind load cases. Each chapter is concluded

with a discussion of strengths and limitations of either the theory proposed and/or

the program developed, for each load case.

The final chapter, chapter seven, deals with conclusions observed from the

research and recommendations. Areas needing further research are also identified

and noted.

Also included are Appendices A and B. Appendix A shows a detailed

derivation of the 2-node cable element developed in this research. Appendix B

gives a user's manual of the computer program developed. In appendix B, defaults

assumed in the program are explained and for portions of the program not

developed by the author, references are provided.



Scope


The scope of the research presented is limited by the inherent assumptions

made in the cable analysis. The assumptions made, common to other types of

cable analysis, are as follows:









+ The cables are perfectly flexible and possess no bending stiffness, only

axial stiffness.

* Cable materials are of hookean materials. Ferrous materials are commonly

used for cables.

* Only small strains are involved, i.e. no yielding of cables ever reached.

By the nature of the type of structure investigated (traffic signal systems),

the loads are essentially predefined. For this research, loads examined are

restricted to the dead load from the weight of the signal heads and signs, and wind

loads. Other assumptions discussed in chapter four further define the application

of the theory and subsequently the program.













CHAPTER 2

LITERATURE BACKGROUND



Introduction



Several authors have dealt with the problem of the geometric nonlinearity

of cables and have provided some solutions to the problem. A brief summary of

pertinent results of their work is given in this chapter as they provide a foundation

to the author's work. Also presented is a general theoretical background of cable

analysis and dynamic analysis. The work covered here reflects only the solutions

provided for cable dynamics. It is, however, congruent to consider the static cable

problem as a special case of the cable dynamics problem hence extending these

solutions to cover static cable analysis. Other methods strictly for static solutions

include the force density method [1].



Fundamentals of Cable Analysis



Consider a uniform cable which hangs in static equilibrium in a vertical









plane suspended between two supports. The forces acting on a differential

element of the cable is dependent of the profile assumed by the cable at-rest

position and the cable final deformed position. Let any point on this cable be

described in the 3-D Cartesian coordinates shown in Fig. 2.1.


z


(w)
Y
(v)

x


(u)


Fig. 2.1


Cartesian Coordinate


Let the cable ordinate after deformation be denoted x, y, z as shown in Fig. 2.2.

X=x+u

y=y+v










Z=Z+W

The cable segment shown in Fig. 2.2 below can be described by a vector along the

chord joining both ends of the cable segment.




L= t+ Yi Y + j k
ds ds ds

The terms xj xi Yi and zj z are referred to as the direction cosines for
ds ds ds

the element.










00< -'


I uj ,, i,



Fig. 2.2 Displacement of a cable segment





The relationship between the nodal forces and displacements at any point along

the cable relative to the next is defined as follows,










du
uj = ui + s (2-1)





This says that the displacement at nodej is the sum of the displacement at node i

and the differential strain in the cable multiplied by the arc length of cable.

Similarly for the displacements in the y and z direction, the relationship is

dv
v = v + -7ds (2-2)
d s




d w
w = W + d s (2-3)





The above displacement-strain relationship is referred to as the linear strain -

displacement relationship. The relationship between the forces at node i andj

both along the cable can similarly be summarized as follows,

Tj = Ti + dT (2-4a)


Fxj = Fxi + dFx (2-4b)


Fyj = Fy + dF (2-4c)


Fzj = Fzi + dF (2-4d)


For static analysis the forces acting on a finite length of a cable are










T = Cable tension

Fx, Fy, Fz= equivalent nodal loads due to for example dead weight of cable

and/or cladding or applied nodal load. In the "unloaded

configuration" otherwise referred to as the null state, the applied

load is taken as zero.

The tension in the undeformed slack cable is computed from the cable profile.

The basic shape of a slack cable subjected to its own weight is a catenary profile.

However, if the slope of the cable is everywhere small, the profile adopted by the

cable can be accurately described by a parabola. It is advantageous as a means of

simplification to describe cable shape in two dimensions. Here the x-z plane is

used.








w"



SV2 L"


Ti -1 x L


Fig. 2.3 Shape of cable due to self wt.










According to Ref. [2] the parabolic profile is given by Eqn. 2-5.

4hx
z(x) = xtan(e) (L-x) (2-5)




The tension in the cable at any point x along the span of the cable is as given in

Eqn. 2-6.


WL2 [i 4h+8hxl (2-6)
T(x) w + tan(0)--+
8h L ,2



S = length of the cable along the sag (Fig 2.3).

z (x) = vertical distance of any point along the cable relative the starting

node.

tan (0) = slope of the chord joining both ends of the cable.

h = distance from the chord at mid-span of the cable. This is used to

define cable sag.

w = weight of the cable per horizontal distance.

L = span of the cable.

T (x) = tension in the cable (null state) at any point along the cable.

sag = h


Comparing the above profile with the catenary profile in Ref. [3], the following









equations form the catenary profile equivalent of Eqns 2-5 and 2-6 above. Note

that the catenary shape is more difficult to define and indeed is only defined by an

iterative process of trial and error.

H x
z(x)=-H cosh(E)-cosh(20- -e) (2-7)





tan (0)
E = sinh [ tan( )+ 0 (2-8)



wL
S=2H (2-9)





S WL tan(0)
T (x)=H 1+ sinh sinh 2H --wL wx (2-10)
sinh wL 2H H
2 2H I




H = horizontal component of the tension in the cable. This is invariant
along the cable.


The other variables retain their definitions given on previous page.

There are two unknowns in the above equations, z(x) and H. The value of H is








17

often obtained by approximately setting z(x) to sag x span (L) at x = L/2. The rest

of the configuration is obtained by using the H computed. It is apparent that it is

more convenient to use the parabolic approximation. The error introduced by the

approximation is of the order of 1% 2% for slack cables with small sag. For

example, if the tension at the first third point and the z coordinate were required of

the cable with the following properties:

sag = 10%, w = 0.12 plf, L = 3500 ft, take tan(0) as zero i.e. both cable ends are

on the same level. The results will be the following.

Assuming the cable shape is parabolic -

z (x = /3L) = 311.111 ft. And T (x = /3L) = 529.646 lbf.

Using the catenary equations above the results -

z (x = 1/3L) = 311.557 ft. And T (x = 1/3L) = 536.839 lbf.

An error of just 1.4%. For the remainder of this dissertation, the parabolic (null)

profile will serve as the reference state for the slack catenary cable. The null state

shape is important in the derivation of the stiffness matrix of the cables as well as

in determining the initial tensions in the cable. Any assumption about this initial

configuration could significantly influence analysis results.

Cables, like trusses, are idealized as not having any bending stiffness

(in reality they possess some bending stiffness). This introduces equation of

conditions in the equilibrium analysis of cables making them inherently unstable.










In addition, they can undergo large displacements and cannot resist compression.

These complexities invalidate the more conventional linear theories often based

on the assumption of small displacements and unconditional stability. To solve

the problem of inherent cable instability it is necessary to introduce additional

resisting forces that can keep the cable in equilibrium. The problem of large

displacement is best dealt with by applying the load in small enough increments as

to achieve small enough displacements in each increment. Lastly, the inability of

cables to sustain compression can be taken into account by solving the problem in

steps and keeping track of the stresses in the cable. The cable elements are

rendered inactive when in compression and active when in tension. A very logical

method of solving a cable problem, therefore, is to use the theories of structural

dynamics. It is advantageous to use this technique because it introduces inertia

and damping forces that can be used to stabilize the cable and it provides a time

step solution which allows the application of load in small increments and

supplies a framework for keeping track of cable stresses.



Cable Dynamics



The solutions to the cable dynamics problem can be discussed in two major

categories, closed form solutions otherwise referred to as the "exact" solution, and










the finite element solutions i.e. explicit solutions such as the Euler's method,

Central Difference method and Runge-Kutta method and the more common

implicit methods such as the Wilson Theta method and the Newmark's Beta

method.



Closed Form Solutions


Most closed form solutions, for taut cables and cables with small sag to

span ratios, are based on a linear theory (i.e. small displacement theory). Irvine

and Caughey [4], Uwe Starossek [5] and Veletsos and Darbre [6] have all

presented semi-empirical solutions based on this linear theory. Of importance is

the experimental results of Irvine and Caughey [4]. Their work revealed the

observed dynamic behavior of cables with small sag subjected to free vibration.

When a cable vibrates it does so in two major motions. The transverse horizontal

motion and the in-plane motion. Both motions are observed to be uncoupled. The

transverse motion is essentially a swinging motion as the cable rocks in and out of

its at-rest plane. The in-plane motion consists of two components, the symmetric

and anti-symmetric components. The anti-symmetric components consist of in-

line motion similar to a wave traveling along the length of the cable and in-plane

transverse motions which are unsymmetrical in-plane deflections of the cable.

The symmetrical components are analogous to simple harmonic motions occurring








20

along the cable. For small sag to span ratios (< 1:8) these components of motion

constitute the modal shapes that could then be analyzed. The results of most of

the aforementioned authors establish the mode shapes of cable vibration and their

corresponding frequencies. The limitation of the linear theory is that for catenary

cables with large sag the theory ceases to be accurate The theory also becomes

problematic when the cable is considered as part of a structure and not standing

alone.

The equation of motion from linear theory is given in Eqn. 2-12.

From Uwe Starossek [6],


m g12 (2-11)
H =
s f



H = Horizontal component of the cable tension

m = mass of the cable per unit length

g = acceleration due to gravity

I = cable span

f = sag of the cable

The in-plane equation of motion is for sag 1/20;

d2w d2z d2w dw (2-12)
H + hI = m + c
dx2 dx 3t2 t











dx


ds
X, U

z, w


I z
dz


T+dT


Fig. 2.4 Notations For Eqn. 2-12 for a
cable segment


= damping force per unit length

= displacement in the x, z directions respectively.


dx
hr = 't--
ds


S = dynamic part of the total cable tension. This is invariant along the

cable.



An alternative to the exact solution is the discrete idealization of the

continuum. There are two major techniques of discrete idealization of a structural


c

U, W










continuum, the lumped mass method and the finite element method. The finite

element method is more commonly used because, among other reasons, the

numerical modeling has a clearer physical interpretation. With the finite method,

the dynamic solution for cable vibrations are either explicit solutions or implicit

solutions.



Finite Element Method



Due to the nonlinear behavior of cable systems, direct superposition of

loads and displacements as would be necessary with the typical modal

superposition analysis is not strictly valid. However, according to Leonard [7]

and others, if an incremental load approach is adopted and the load increment is

small enough, then within each step it is reasonable to assume linearity of the

superposed dynamic response. An additional condition for a linearized solution is

that the prestressed cable configuration prior to load application be known and

that the cable system possess a significant stiffness. Using finite element analysis

the traffic signal system is idealized as follows; the strain poles as beam elements,

the cables as nonlinear cable elements and the connectors as large displacement

beam elements. A common practice is the use of steel bracing for the poles.

These braces could be modeled as linear truss elements.












Formulation of Element Properties



The properties of the linear beam elements are given below, their

derivations can be found in most structural analysis books. The typical

geometrically nonlinear truss element often used for cable analysis is also given in

this chapter. The cable element adopted for this dissertation, however, is derived

in chapter 3.



Beam Properties


According to Ref. [8] the stiffness matrix for the 3-D beam element is given in

Eqn. 2-13.























ff'

f6 j f1




i
F y4 4
Zf.

ff






Fig. 2.5 Element degrees of freedom for a beam element.






EAx/1 0 0 0 0 0 -EAx// 0 0 0 0 0
0 12EI3/13 0 0 0 6EI3/12 0 -12E13/13 0 0 0 6E13/12
0 0 12E12//13 0 -6E12/2 0 0 0 -12EI2/13 0 -6EE12/2 0
0 0 0 GJ/1 0 0 0 0 0 -G J/1 0 0
0 0 -6E 1/12 0 4EI2// 0 0 0 6E12/12 0 2EI2/1 0
= 0 6E13/12 0 0 0 4EI3/1 0 -6EI3/12 0 0 0 2EI3//
K= -EAx// 0 0 0 0 0 EAx/l 0 0 0 0 0
0 -12EI3/13 0 0 0 -6E13/12 0 12EI3/13 0 0 0 -6E3//12
0 0 -12EI2/13 0 6E12/12 0 0 0 12E2/13 0 6E12/12 0
0 0 0 -G J/1 0 0 0 0 0 G J// 0 0
0 0 -6EI2/12 0 2E12// 0 0 0 6E12//2 0 4EI2// 0
0 6E13/12 0 0 0 2EI3// 0 -6E3/12 0 0 0 4EI3/l

(2-13)



E = Elastic Modulus (Young's Modulus)

12 = Moment of inertia about the strong axis

13 = Moment of inertia about the weak axis

S = Length of the beam element







The consistent mass matrix for the above 3-D beam element is as follows [9]:


[M] pAL
420 g


140
0
0
0
0
0
70
0
0
0
0
0


0
156
0
0
0
-22L
0
54
0
0
0
13L


0
0
156
0
22L
0
0
0
54
0
-13L
0


0
0
0
140/A

0
0
0
0
0
707

0
0


0 0 70 0
0 -22L 0 54
22L 0 0 0
0 0 0 0
4L2 0 0 0
0 4L2 0 -13L
0 0 140 0
0 -13L 0 156
13L 0 0 0
0 0 0 0
-3L2 0 0 0
0 -3L2 0 22L


0 0 0 0
0 0 0 13L
54 0 -13L 0
0 7A 0 0
13L 0 -3L2 0
0 0 0 -3L2
0 0 0 0
0 0 0 22L
156 0 -22L 0
0 140 0 0
-22L 0 4L2 0
0 0 0 4L2


(2-14)


J = Polar moment of inertia

A = Cross sectional area

p = Density of beam material (lb/ft3)

g = Acceleration due to gravity (32.2 ft/sec2)









Truss Element Properties


The degree of freedom for a typical truss element is shown in Fig. 2.6.








-f

-V




Fig. 2.6 Degrees of freedom for a truss element




The stiffness matrix for the truss is given in Eqn 2-15 [10]. This stiffness

matrix is in the element local coordinate.

1 0 0 -1 0 0
EA 0 0 0 0 0 0

[K] EAx 0 0 0 0 0 0 (2-15)
1 -1 0 0 1 0 0
0 0 0 0 0 0
0 0 0 0 0 0

In the structure's global coordinate, the stiffness matrix is given in Eqn. 2-16.
















t Az


//





Ax


Fig. 2.7 Truss element showing direction cosines


C2
a

cacr
Ca Cr
-C2
-a


-Ca Cy


cnc
r2



-Ca
-cc


ya
SCP


-Cy C
Ya


-C2
1r


-c2
a

-C C8


a


Ca Cy


-C C

-cc


c c
Pia


c2



O Cr


AE
[K] AE
L


-c C


ya




qC2
1'


(2-16)










Ax
L C


Ay
L 1'


Az
L


L = J(Ax)2 + (A y)2 + (Az)2


The mass matrix in global coordinates is also given below [9].


[M] m
6


(2-17)


(2-18)


(2-19)


= mass per unit length for the element

= length of the element


Cable Element Properties


A straight truss element capable of large displacement is commonly used

for cables. The derivation [3] of the stiffness matrix for this element is briefly








30

discussed here so as to be differentiated from that in chapter 3. Fig. 2.8 presents

the straight element nonlinear truss in its local axis.












u _+ su ---ds (---220 I
dsds s
Fig. 2.8 Straight nonlinear truss element used for cables






From Eqn. 2-1, the component of horizontal displacement is,

du du (2-20)
ui + -sds ui cs





vi +-ds vi ds (2-21)




wi + ds w ds (2-22)
as ds



The resultant axial strain along the member due to the above displacements can be

computed by determining the deformed element length Ms. The deformed length











due to displacement in the x direction is simply,


du
ds + -ds
ds


(2-23)


The deformed length due to displacements in the y and z directions are shown in

Fig. 2.9.


ds
asds

[(ds) +( ds)2




ds


awds
ads
as


(ds) +(aw s)


Fig. 2.9 Deformed length due to displacements









The axial strain from the above in the x, y and z directions is given in Eqns 2-24,


2-25 and 2-26.










du
ds +- -dsu
-x s du
xd ds


as dv




ds2 + ds -ds
=- = 1d+ )








Sds ds


Applying the binomial expansion


1+a2 = l+la2 4
2 8


and neglecting the higher order terms Eqns 2-25 and 2-26 become




S2 (+2
Y 2 TS 12 dv
= l+- s -1 =2
Y 2 Os2 s


(2-24)








(2-25)


(2-26)


(2-27)











E = l+-, -1 =-
S= 2+ d21= I 2(2-28)
z 2 (sJ 2 ds



The total axial strain in the element is the summation of the strain

expressions in Eqns 2-26, 2-27 and 2-28. This summation is referred to as the

large deflection strain-displacement equation.

2 2du
E = + -L + l 2 (2-29)
ds 2 ds) 2 ds


If the nodal displacements are known for a finite length L of the nonlinear truss

element, Eqn 2-29 becomes,

lvV._i '2 i W Wi- 2
E = u iv w, L (2-30)
L 2 L 2 L



Using Castigliano's theorem for the force-displacement relationship, the

expression for the strain energy in the element is given in Eqn 2-31.

U = 2JTA = -2 ds (2-31)
2 2


Substituting the expression for strain and expanding the brackets we obtain




U AE +u) du v2+ du (dw2 1 (v4 1(Ov (w2 1 dw 4
+ 2 + s ds +- ++ ds
2 ds ss jds Es 4 ds 2das s +4 J ds
(2-32)








34

Ignoring the fourth order terms and substituting the notation used in Eqn 2-30, we

obtain a new expression for U.

E j i u Ui Vj Vi Uj Ui ] 2i 2 (2-33)
S2 (LL L L sL



Note that,

T AE (u -i (2-34)


Where T is the tension force in the truss element. Solving the integral we get Eqn

2-35.




U = u) + A( i j vi) + "-(Uj -u wi)2 (2-35)



Differentiating the Eqn 2-35 with respect to the nodal displacements gives the

stiffness coefficient associated with that displacement. The expression for

tension, from Eqn 2-34 above is substituted into Eqn 2-35. The assumption is

made that the tension in the cable is approximately a constant.

dU E (u,-_ u) (2-36)
du L (uu-


dU T i ) (2-37)
d vi L










U T Wi wj (2-38)
dwi L


U = AE ( i) (2-39)
duj L (


dU = Ij ,) (2-40)
dvj L 'i


dU T(wj -i) (2-41)
dwj L
.I


In matrix form, the geometric stiffness of a nonlinear truss can be written as given

in Eqn 2-42. The first matrix term below is the conventional elastic stiffness of a

truss element. The second term is referred to as the geometric stiffness term. This

geometric stiffness accounts only for initial cable stress and not for nonlinear

strain displacement relationship.



1 0 0 -1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 -1 0
EA 0 0 0 0 0 0 T0 0 1 0 0 -1
[K] -- + -
L -1 0 0 1 00 L 0 0 0 0 0 0
0 0 0 0 0 0 0 -1 0 0 1 0
0 0 0 0 00 0 0-1 0 0 1


(2-42)












Connector Element


The load on the traffic signal system is applied by the traffic signal heads

and the traffic signs. These are attached to the system via connectors. The

connectors have translational and rotational degrees of freedom. In this they are

identical to the beam element discussed previously. The connectors, however,

undergo large displacements. These displacements affect how the stiffness of the

connector is added to the structure's overall stiffness. The connector stiffness

matrix presented next is a beam element stiffness accounting for large

displacements. The element presented here and adopted for this dissertation was

derived by Chen and Agar [11]. The element stiffness is broken into the elastic

component and the geometric component. The elastic stiffness is as given in the

earlier described beam element, the geometric stiffness is set up in terms of the

natural displacements (deformation) of the beam shown in Fig 2.11. The

derivation of the geometric stiffness is given in detail in Ref. [11]. The derivation

presented in Ref. [11] rely predominantly on work done by Bathe and Bolourchi

[12]. In this dissertation, only the definition of the coefficients of the geometric

stiffness and its application is described. The degrees of freedom for the large

displacement beam element used as connectors is identical to that of regular

beams. The resulting beam end forces corresponding to these degrees of freedom










are shown in Fig. 2.10. To accurately determine the end forces another set of

forces referred to as the natural forces or basic beam forces must be defined. The

natural beam forces are the forces developed in the beam due to relative

deformation of the beam ignoring rigid body motions. These forces are shown in

Fig. 2.11.








M\ Mz
Myj V.
Deformed Position Vyj Mxj

\i ;Mzi J I
Mv'W
M V'i \w



Sui Original Position
Fig. 2.10 Joint displacements and
corresponding end forces














\/ Weak Axis Bending
z 'o

J k P M,
p Mi 0'- Myj

y Mi / Strong Axis Bending



Fig. 2.11 Relative deformations and associated
forces








From Fig. 2.10, the nodal degrees of freedom are,


dIT = f "i VI W. xi yi zi Uj v wj Xj y j zj (2-43)
{ = { 1 I 9.9 u. w. .9 'J' (2-43)


The nodal displacements in Eqn 2-43 consist of the two major components the

rigid body component and the natural displacements (i.e. relative deformation)

component. From Ref. [13] six rigid body modes and six natural displacement

modes completely defines any displacement that may be seen in a large

displacement beam. The identified modes are as given in Fig 2.12.


















Rigid Body Modes


Sequence of modes







H
V dN

x d


z






Y
do, /


Natural Modes


z
Y
Xd


V2 dV






dNd
V2 du


X X.
x dJ / A- dx
Md^y vLd^


Fig. 2.12 Rigid body and natural modes for large
displacement beams








40

From Fig. 2-12 it can be seen that natural mode 1 can be defined by the change in

length of the member i.e. axial deformation. The rest of the modes of deformation

are bending deformations. These can be defined by the end rotations. The natural

displacements (i.e. relative deformations) that are necessary for the computation of

the end forces can therefore, be summarized as given in Eqn 2-44. Notations are

shown in Fig. 2.11.

U
0' .

dN} = (2-44)

zj
0'
t





The computation of { d } in Eqn. 2-44 is not intuitively obvious. For illustration

of the process of computing the relative rotations, an assumption is made here that

the connector is initially straight as shown in Fig. 2.13. The relative rotational

angles are the total nodal rotations obtained from analysis, i.e. static or dynamic

analysis minus any member rigid body chord rotations. In Fig. 2.13, it can be

seen that the rigid body rotations about the y, x and z global axes are as follows:

-1 d -1 dx
S= sin Oy = sin d
"dy2 +dz2 dx2 +dz2












0


projection

4 -


ed chord Position



Position


X Z plane projection --

Fig. 2.13 Rigid Body Rotations


= X, X

= y, yi

= Zj Z

= dx2 + dy2 + dz2


d x

d y

dz

I'








42

Using the cartesian coordinate system shown in Fig 2.1, clockwise rotation about

the y axis is considered positive and counter-clockwise rotation about the x axis is

positive. Therefore when dx is positive (,) gets a positive sign but when dy is

positive .x must be assigned a negative sign. The relative rotational deformation

is then computed as shown in Eqn. 2-45 below.

u I -I

yi yi y

YJ (2-45)
6'z. 6 .O -x

'zj zj x
6' 8 -
t xI xJ





Eqn 2-45 above assumes that the nodal rotational displacements (with the nodal

designation i orj) have been resolved to the local coordinate system but the chord

rotations (() are in global reference. Nodal rotations are measured relative to the

original orientation of the beam. Relative rotations are measured relative to the

chord of the beam. T is the deformed length of the beam and I is the original

length of the beam. To check the validity of Eqn. 2-45, consider the chord

rotation and bending deformation of the next connector example. In Fig. 2.14, for

ease of illustration the assumption is made that deformation exist only in the 2 -











dimensional, x z plane.



Chord of deformed beam
y axis
y axis



Undeformed Position '

/


Deformed beam
o' Rigid body chord rotation
8' Relative deformation
8 Nodal rotation



Fig. 2.14 Relative Nodal Rotations











Note that the nodal rotation at node (i) is clockwise hence positive, at node (/) the

nodal rotation is counter-clockwise hence negative. The chord rotation is

positive at both ends. The resultant relative rotation at node (i) is smaller in

magnitude but with the same sign as the nodal rotation, i.e. clockwise. The

resultant relative rotation at node (j) is bigger in magnitude but with the same sign

as the nodal rotation, i.e. counter-clockwise. This conforms with results from









44

applying Eqn. 2-45. The natural end forces of the beam element, as shown in Fig

2-11, can be obtained by multiplying the above relative deformation vector by the


basic beam stiffness, [kNE].


{Pu} [=NE{ d]N (2-46)


Substituting the matrix terms into


AE
A 0 0 0o
L
4EI 2EI
0 Y Y 0
L L
2El 4El
0 LY Y 0
IP = L -- L
\' N 4 4El
0 0 0 ---"
L
2El
0 0 0 -
L
0 0 0 0


Eqn. 2-46, Eqn 2-47 is obtained.


0

0

0
2EI
z
L
4EI
z
L
0


0

0

0

0

0
GJ
L
o










L,


u
0' .
yi
8'

8' .
8'
zj
t


The end force vector computed from Eqn 2-47 can be seen in Fig. 2.11.


P
M.
yL

zN} i (2-48)
M.
zi
M


(2-47)










The coefficients of geometric stiffness are obtained from the basic end forces.

The geometric stiffness matrix for a large displacement beam according to Chen

and Agar [11] is shown in Eqn. 2-49 below.


k] =


70 0 0
a 0
a


0
e

-f
g
i


0
-a
0
-b
- e

-f
0
a


0
0
-a
-c

f
-e
0
0
a


SYMM.


0
k
A
-I
-d

-g
-h
0
-k
-l
d


0
-e

-f
-g
-n
-m
0
e

f
g
i


0

f
-e
h
m
-n
0

-f
e
-h
-J
i


(2-49)

The terms in the matrix above are as follows:


e= Mt/l, f= P/10


= (Mzi+Mzj/6, h=(Myi+Myj/6, i= 2P1/15,


j= Mt (Iz-Iy)
2= y k= My jl,
z y


I= M i/l, m= Mt/2, n= Pl/30,


(2-50)


a=6P/51, b= Myil,


c= Mzi/ 1, d= P(ly + z/Ax,








46
Note that I is the length of the beam and the other variables are as defined in Eqn

2-48. The final expression for the stiffness of the connector element with

reference to the element local axes is,


kT] = [ke]+[kg] (2-51)

The above stiffness in global coordinates can be obtained by applying the standard

coordinate transformation.



Transformation Matrix


The translational transformation matrix can be obtained by using the k node

method. This method is used in the SSTAN program developed at the civil

engineering department, University of Florida. The transformation matrix is a 3x3

matrix. The terms of the transformation matrix [T] is obtained as follows. The

first 1 x 3 row entries represent a unit vector lying along the member local axis,


{dx dy dz


1 = length of the member

dx, dy and dz are similar to Ax, Ay, Az in Fig. 2.7 and similar to those given on

page 40. Another unit vector must be defined along a node in the plane of the

element but not along the member. This node is referred to as the k-node.










dxk dyk dzk,
1k 1k k'



The second 1 x 3 row in the transformation matrix is obtained from the cross

product of the k-node unit vector and the member unit vector. The third 1 x 3 row

is obtained by getting the cross product of the previous member unit vector (in the

first row of the transformation matrix) and the unit vector gotten in the first cross

product (now in the second row of the transformation matrix). The resultant 3 x 3

matrix [T] is used to transform nodal displacements from global to local

coordinate systems.

For the case of the beam with large displacement, in addition to the initial

transformation matrix [T] described above, it is often advantageous to use another

transformation matrix [R] based on Euler angles. This is because this

transformation matrix formulation can better keep track of the twisting of the

element than the former. A combination of both transformation matrices could be

adopted with the first transformation [T] used to transform element properties

from the local coordinate system to the initial member global position. The

second transformation [R] can then be used to transform from the initial global

position to the member's deformed position. The transformation matrix, [R],

used in the context just described is described next. The Euler transformation

matrix [R] is also explained in the context of a numerical time step solution,










which is reflective of its usage in chapter 3.


I +


Fig. 2.15 Euler angles for transformation









I = original length of the beam.

ii = axial relative displacement.

v = relative displacement in the y direction.

w = relative displacement in the z direction.

The Euler angles in Fig. 2.15 can be computed as shown in Eqn. 2-52 and Eqn. 2-

53. Note that u, v and w above are computed using the cumulative nodal

displacements in the case of dynamic time step solution.










-1 v
= tan + (2-52)



-1 W
/ = tan1 (2-53)
(+ 2 +2 (-




The final Euler angle that is needed is the angle of twist, y. The angle of twist can

be obtained from the rotational displacement about the local x axis. It is assumed

that the nodal incremental displacements have been transformed to the element's

initial local coordinate system using just the initial k-node transformation [T].

The angle of twist is cumulated at each iteration using the change in nodal

rotations.

y = t-At + I{cosfcosa(AO .+A x+cos3sina AO .+AO +sinflAO .+AO .)
2 x xI X) ( yi yj zz zy

(2-54)

Y' = the angle of twist at time t

yt +A = the angle of twist at time t + At

A {axis)(node) = change in nodal rotation about the axis {x, y or z) at node (i

orj}.

a and P are as computed from Eqn. 2-52 and Eqn. 2-53.

The final transformation matrix [R] is presented in Eqn. 2-55.







50

cos(p)cos() cos(p)sin(a) sin(p)
[R] = -cosy)sin(a)- sin(y)sin(3)cos(a) cos(y)cos(a)- sin(y)sin(j8)sin(a) sin(y)cos(p)
sin(y)sa)- cs(y)sin(P)cos(a) -sin(y)coa)- cos(y)sin(P)sin(a) cos(y)cos(3)

(2-55)


The final transformation matrix for the large displacement beam is therefore:
[A] = [R][T]

[A] = final transformation.
Applying the transformation matrix, the stiffness of the member in terms of the

global nodal displacements is

A o T A O

[KT]G = A A A(2-56)
0 A O A









Damping Matrix


The last property that needs to be defined is the damping matrix. The

damping matrix for a structural system is easier defined proportional to either the

mass, the stiffness or both. There are two classical damping matrices. The








51

Rayleigh damping and the Caughey damping [14]. The Rayleigh damping matrix

is described below. With Rayleigh damping, the damping matrix is defined as

follows.

[C] = ao[M] + al[K] (2-57)


ao and a, are damping constants depending on the controlling modal frequencies

and the corresponding specified modal damping ratio.

20; M 2
o = i aJ a (2-58)
a0 +i+j = Zi+ j






= modal damping ratio

i, = modal frequency corresponding to mode i.

j = modal frequency corresponding to modej.

Controlling modes are often the higher modes corresponding to the number of

degrees of freedom for the system.

For nonlinear analysis the eigenvalues are never obtained as part of the

solution. There needs to be a way to estimate the natural frequency of the system

prior to a rigorous nonlinear analysis. For the structural system defined, an

estimate of the natural frequency of the system can be taken as either the natural

frequency of the cables on infinitely rigid supports or the natural frequency of the









poles without the cables. Both estimates are given below.

For the catenary cable subjected to self weight, no initial pre-tension and a

specified sag, the natural frequency [3] is


S= n n = 1, 2,3,... (2-59)


n = modes to investigate

g = acceleration due to gravity

f = sag at mid-span of the cable

1 = span of the cable

For cables where the sag is assumed to be zero the above equation ceases to be a

good estimate. Therefore for a messenger cable with initial pre-tension and a near

zero sag the natural frequency can be estimated as

nn irTg (2-60)
rn = l nm-





For the poles, taken as cantilever beams, the natural frequency can be

estimated as follows [9]:

n= x2 m (2-61)
n 2 m







53
Matrix representation of the element properties have now been developed.

The solution techniques that will be used to determine the time history response of

the discretized structure is considered next.



Dynamic Solution Methods



The equation of motion [14] for a typical structure is

[M]{d} + [C]{d} + [K]{d} = {P} (2-62)



[ M ] = assembled mass matrix of the structure.

[C ] = assembled damping matrix of the structure.

[ K ] = assembled stiffness matrix.

{d} = acceleration vector.


{d} = velocity vector.

{d} = displacement vector.

{ P } = applied load vector.


For nonlinear analysis direct time integration is used. This means that the solution

to the equation of motion is determined at discrete times (ti i=0,1,2...). The initial








54
conditions at the beginning of the time steps are always known and it is assumed

that some variation in, and self consistent relationship between, the displacements,

velocities and accelerations occur during the time step. The methods of solution

available for dynamic analysis vary depending on the assumptions made for the

variation in displacements, velocities and accelerations within the time step. As

earlier mentioned these methods are classified as either explicit or implicit.



Explicit Methods


A disadvantage of using explicit method is that they are conditionally

stable. The more common explicit solutions are Central Difference method and

Runge-Kunta method. In explicit methods it is assumed that the displacement and

velocity at time t + At are independent of the acceleration at t + At.

Consequently, the displacement at the end of the time step is projected from the

equation of motion at the beginning of the time step. In particular, the elastic

forces and damping forces within the time step are computed using displacements

and velocities at the beginning of the current time step and the beginning of the

previous time step. For details of the available explicit methods, the writer

recommends Ref. [7], [9], [14]. For the cable analysis it is important to apply the

load in small increments, it is, therefore, more convenient to perform an

incremental analysis. Implicit methods of analysis allows us to rewrite the










equation of motion in terms of the change in the displacement, velocity and

acceleration.



Implicit Methods


The incremental steps are measured relative to a reference state at the

beginning of the time step. The equation of motion given in Eqn 2-62 can then be

rewritten.

[M]{Ad(t)} + [C]{Ad (t)} + [K]{Ad(t)}= {AP(t)} (2-63)


The new shape of the cable at any time (t) can be obtained as follows

d(t + At)= d(t)+ Ad(t) (2-64)


The solution of Equation 2-63 above can be obtained by a number of different

implicit methods. The more common methods are the Adam-Stoermer method,

Newmark's Beta method and Wilson Theta method. These methods actually all

have the same form and can be easily inter-changed with a manipulation of the

parameters associated with each method. The Newmark's Beta is used by the

writer and is discussed next.









Newmark Beta Method


Newmark Beta method [14] is based on the following equations written in

terms of incremental parameters.

{Ad} = A {d ()} + 3){ d(t)} + 3{d(t + At)}] (2-65)




{A d} = [{d(t)+ d ( + A t (2-66)



Solve Eqn. 2-65 for {d(t + At)} and Eqn. 2-67 is obtained.


{d(t +)} A = 1 {Ad}- d(t)} 2- ) (2-67)
{d(+A)} At2 P]3 0


From Eqn. 2-67 the change in acceleration within the time step can be expressed

as follows:

Ad} = {d(r+Ar)} {()} = 1 {Ad}- d(t)}- {d()} (2-68)
d = A A 2fAtP


Substituting Eqn. 2-67 into Eqn. 2-66 and the expression for the change in

velocity can be rewritten as,


2 d-I {d() 2 1 t t) (2-69)
1 2 Ad} d(t)} 2 (2-69)
d 2At 2]f Ar {d(P)}









Substituting Eqn. 2-65, 2-66 and 2-68 into 2-63 and rearranging terms we have

the final equations for the Newmark's Beta method.

[K(t)]{Ad} = AF(t)} (2-70)


[(t)] = [M] + [(t)] + [K(t)] (2-71)
SA[ 2 At(2



AP)} = {AP(t)} + [M][ d(r)} + (+) + + [c()] {) + -~ At {d(t)}J


(2-72)

The solution procedure using Newmark's Beta method may then be summarized

as follows:

* From initial conditions, calculate effective stiffness (Eqn. 2-71) and

effective incremental load vector (Eqn. 2-72).

* Solve for incremental displacement (Eqn. 2-70) and cumulative

displacement (Eqn. 2-64).

* Solve for incremental velocity (Eqn. 2-69) and velocity at t + At.

* Impose total equilibrium at the end of the time step to obtain the

acceleration at the end of the time step.

(t + At)} = {P(t + At)}- fd (t + At) (t + At) (2-73)
[M]










fd = damping force vector

fe = elastic force vector.

Repeat the process for the next time step.

The parameter P reflects the variation of the acceleration within the time

step. There are various values that could be used for P. For P = 1/6 we have the

linear acceleration method. For P = 1/4 we have the constant acceleration method.

And finally for P = 1/8, the acceleration is constant, equal to the beginning value

over the first half of the time step and constant, equal to the ending value over the

last half of the time step. When P is taken as zero the Newmark's Beta method

degenerates to the Euler integration scheme. Similarly if P is taken as /2

Newmark's method becomes identical to the central difference integration and to

backward difference when P is unity. The constant acceleration method is

unconditionally stable This means that the solution will not become unbounded

as the integration progresses. With the numerical integration, there are

inaccuracies due to round-off or due to time increment being too large to represent

the forcing function or the structural response. The errors can be redistributed at

each time step by using the Newton-Raphson method.













CHAPTER 3

THEORETICAL DEVELOPMENT



Introduction



In chapter 2, the typical cable element commonly found in literature was

discussed. The formulation of a different 2 node straight cable element adopted

for this dissertation is presented in this chapter. This formulation includes the

equation of motion and solution to the equation for the traffic signal system.



The equations derived in this chapter are based on the Cartesian coordinate

system shown in Fig. 2.1. The deformation parameters u in the x direction, v in the

y direction and w in the z direction are deformations relative to the initial (null)

state. The initial states for the cables are taken as either taut or slack. The taut

cable has an initial pretensioned cable state. The load acting on this cable at the

initial configuration (time t = 0) is the self weight of the cable and the initial

tension. The cable with this null state (condition of zero deformation), in the case

of the traffic signal system, is referred to as the messenger cable. The messenger










cable null state has an initial axial strain but an assumed zero sag. The catenary

cable, in the case of the traffic signal system, has a slack null state. The catenary

cable in this case is presumed to have zero pretension (except for tension from self

weight) and a specified sag. The default sag if none is specified is 5 %. Two

additional reference configurations are of importance. These are the present

configuration at time t and the next configuration at time t + At. The incremental

displacement at the nodes, therefore is the difference between the coordinates of

the nodes at time t and at time t + At. The cumulative displacement at the

beginning of the time step is the difference between the coordinates at time t and

the null state. Similarly the cumulative displacement at the end of the time step is

the difference between the coordinates at time t + At and the null state. The null

configuration is the base reference configuration. Both the messenger and

catenary cable are idealized by the 2-node non-linear cable element derived in the

following section. The main difference between this cable element and that

derived in chapter 2 is the inherent assumption in the strain displacement

relationship.














Two Node Nonlinear Cable Element


The shape function of the element adopted is given in Eqn. 3-2.


6
3 L-4




Fig. 3.1 Cable Element
Fig. 3.1 Cable Element


The element properties for this cable element are derived directly in the

global coordinate system. An explicit derivation of a 2 node cable element from

first principles using some novel manipulations of typical energy expression is

presented next. The derivation presented is based on the work presented by

Henghold and Russell [16] and Haritos and He [17]. Ref. [17] presented a total










Lagrangian formulation of the incremental equation of motion for cable

movement. This writer has adopted this approach with a deviation in that the

cable tension used for the geometric stiffness is cumulated in each time step in an

updated Lagrangian fashion. This is observed to be more convenient in terms of

numerical implementation of the element. Ref. [16], however, uses the method of

perturbation of the virtual work expression for a cable element to derive the

equation of motion of the cable. Though this is very different from the approach

adopted in this dissertation, the writer has chosen to adopt the reference's

approach of defining the element's displacement field (shape function) in such a

way as to derive the element stiffness directly in terms of the deformed global

coordinates as opposed to the more classical approach of using a local coordinate

system with a local to global transformation. The shape function is defined to

relate the coordinates of any generic point along the cable element to the nodal

coordinates.

xl
Y2


Y5
.Z6


In terms of a convenient notation Eqn. 3-1 can be rewritten as











shown in Eqn. 3-2.


{ds = [N]dn} (3-2)



ds = coordinate of an arbitrary point s along the cable

d, = coordinates of the nodal points. n is the degrees of freedom shown

in Fig. 3.1.

N = shape function

The interpolation function for a linear displacement field is given below [18].

s
I- o o
L
0 1-s 0
L
0 0 1
[N]T= s L (3-3)


s
L
s
L






Applying the principle of virtual work, the expression for total work done for a

single element at the time t + At is as follows.


5 T- n -T 0 I -T -T -
SW= JO 3ds qds+ Sd F S d ds- OL A eT a ds =0
j=1


(3-4)










6W = sum of the internal and external virtual work done on the element.

q, = gravity and inertia forces acting along the arc of the cable.

Fj = applied nodal load j is the degrees of freedom. (j = 1...n).

fd = damping force vector.

6d = vector of small virtual variations in coordinates.

Note that, though for convenience the writer has chosen to omit the brackets { }

used as vector notation, the displacement and forces in Eqn. 3-4 and subsequent

equations are vectors. Note also that the straight line overstrike ,-, indicate values

or properties at time t + At. The superscript T, indicate the transpose of the

vector.

The last term in the Eqn. 3-4 is the equivalent internal virtual work due to

stress o at time t + At. This is referred to as the virtual strain energy. The

variation in the coordinate at time t + At is equivalent to the variation in the

difference between the coordinates at time t and time t + At. It is, therefore useful

to express the coordinate at time, t + At, as follows.


{dst+At =dst + {Ads (3-5)



{Ads} = [N]{Adn} (3-6)


= change in nodal displacement vector.


Ad,








65

Ad, = change in displacement vector of any generic point along the cable.

Substituting Eqn. 3-2 into Eqn. 3-4 yields Eqn. 3-7.




W=JoL[N]TS5AdT sds+ [N ]TAdnT F L[N]T AdnTfd ds- L AT ads


(3-7)



o = Kirchhoff stress.

e = Lagrangian strain of deformed cable.

A nonlinear strain-displacement relationship is adopted, therefore the strain is as

given in Eqn. 3-8.


s 2,s -1 (3-8)






5s = stretched length of a cable segment at time t + At

aso = unstressed length of a cable segment at time t = 0



In terms of the shape function expression in Eqn. 3-3, Eqn. 3-8 may be rewritten

as,











E 1 dd T dds (3-9)
s 2 ds ds




es = { {}dn}T[N,]T[N"]idn} (3-10)



1
S-0 0
L
0 0
L
0 0
[N']T= L (3-11)
0 0
L
0 0
0--0
L
0 0
L





Note that N' is the differential of the shape function with respect to the generic

distance, s, along the cable. The analysis approach described in chapter 2 is that

of incremental displacement. It is therefore important to define the corresponding

incremental strain. Subtracting the nodal coordinates at time t from the

corresponding coordinates at time t + At, the incremental nodal displacement is

obtained. Similarly, subtracting the expression for strain (Eqn. 3-8) at time, t,

from a corresponding expression at time t + At yields the change in strain.


1 ( s dds 1 ( ds Tds
Ae = ff s-1 It .ds- (3-12)
2 s s s2









= coordinate of an arbitrary point s along the cable at time t + At


Ads = d ds


(3-13)


Solving for din Eqn. 3-13 and substitute into Eqn. 3-12 results in Eqn. 3-14.


A 2 a s ) T Ads j
2 As d


+ dds T(dAds
ds d s )
(3-14)


a+ s J da Js


Substituting the notation used in Eqn. 3-10 into 3-14, Eqn. 3-15 is obtained.



Ae = -I{AdT[N']T[N']Ad +d T[N'T[N']Ad,+AdnT[N'T[N'] d}
(3-15)


The last two terms in Eqn. 3-15 are equal, therefore the above equation can be


simplified further as shown in Eqn. 3-16.

Ae = !{AdnT[N']T[N']Ad}+ dnN'IT[N'] Adn


The virtual work of internal forces as given by the last term of Eqn. 3-4 is given


below.


(3-16)









L -T- (3-17)
8U = AO 6E ads (3


Applying the unit-load method [19], the virtual strain energy expression in Eqn. 3-

17 can be rearranged to be in terms of the stiffness for the cable element.

WU = AEJL5eT-Eds (3-18)


The strain in the cable element at time t + At is equivalent to the strain at time t

plus the change in strain given in Eqn. 3-16. The variation in strain at time t + At

is equivalent to the variation of the change in strain.

6 3T = Ae (3-19)

Substituting Eqn. 3-16 into 3-19 and then into 3-18 the equation given in Eqn. 3-

20 is obtained.

3U=AE jL(- {AdT[N']T[N'] }SAdn + SLAdT[N']T[N'] }Ad + dnT[N']T [N']SAdnr eds
(3-20)

6Adn = variation of incremental nodal displacement vector.

Again, note that the first two terms in Eqn. 3-20 are equivalent.

sU =AE L( AdnT[N'IT[N'] }Adn + dnT[N']T[N']5Adn ) 1ds
(3-21)

From Eqn. 3-21, it can be noted that the term, 6Ad,, is common to all the terms of

Eqn. 3-4, and hence cancels out. Substituting the term for strain,e, Eqn. 3-21

reduces to an expression in terms of stiffness.










e = e + AE (3-22)



{fs} = AEJ {Ad T[N ]T ['] + d T [N ]T [N x
e + (i{Ad T [N']T [N']Ad} + dT [NjT [N']Adn ds

(3-23)


Rearranging and collecting sub-expressions in Eqn. 3-23, Eqn. 3-24 is obtained.

fs = KEAdn + KL Ad + E (3-24)


L T
K = A E J E [N [N ']ds (3-25)
0

KL = AE ({AYT[N'] [N'] + {dT[N'][N]} x {Ad7[N][N']}+ {dnfT[N' [N'}d

(3-26)

L
RE = A E J [N ]T [N '] dn ds (3-27)
0


KE is the referred to as the initial stress matrix and RE is the force vector due initial

stress. Note that the term AEe is equivalent to tension in the cable at the

beginning of the time step. For t = 0, this term is the pretension in the element.

For subsequent time steps the tension term can be computed as follows,

T = T + AEAtAt (3-28)


Note that KL is dependent on unknown displacements. This is the non-linear








70

component of the element stiffness, [K]. This can not be computed at the start of

analysis and must be substituted by an initial approximate stiffness. Let this

approximation be

L dT[N]T[NI xdT[N']T N'Ids (3-29)


The expression in Eqn. 3-29 was achieved by simply setting Ad, in Eqn. 3-26 to

zero. If the shape functions for the 2-node element is substituted into the Eqn. 3-

25, 3-26, 3-27 and 3-29, the following expressions for KE, KL and KA on the next

page are obtained. See Appendix A for complete derivation of the expression.

The following notations are used.

xi ui
Yi Vi
Szi = 1...3 Awi i = 1...3
S 'j = 4...6 dj j = 4...6
yj Vj
zJ. w.



SxjX -i du ui
S= Yj -Yi dv = -v
S- Zi dw w -wi


The expression for KE is as follows:















E t T + AE AE
SE I rL


1 0 0 -1 0 0
0 1 0 0 -1 0
0 0 1 0 0 -1
-1 0 0 1 0 0
0 -1 0 0 1 0
0 0 -1 0 0 1


(3-30)
= initial cable element length.









[KAi AL












[KLi AE
2L3


dx2 dx dy dx dz -d2 -dy dx -dzdx
dxdy dy2 dydz -dyd -dy2 -dydz
dxdz dydz dz2 -dzdx -dydz -dz2
-dx2 -dydx -dzdx dx dxy dxdz
-dy dx -dy2 -dydz dxdy dy2 dydz
-dzdx -dvdz -dz2 dxdz dvdz dz2


(3-31)


(dA+du)(2d +du) (d + u)(2dy + dv) (d + du)(2 + dw) -( + du)(2d + ) -(dr+ d)(2dy+dv) -( +du)(2 +dw)
(y+ dv) (2d + d) (dy + dv) (2dy + dv) (dy + dv) (2c + dw) -(dy + dv)(2 + du) -dy + dv)(2dy + dv) -(dy + dv)(2 + dw)
( + dw)(2dr + ) ( + dw)(2dy + dv) ( + dw) (2 + dw) -( + dw) (2d + d) -(k + dw)(2dy + dv) --( + dw)(2, + dw)
-({+dau)(2d+du) -(ad +u)(2dy+dv) -{dr+ u)(2 + dw) (dc + au)(2d + du) (d + u)(2dy + dv) (d + du)(2 + dw)
-(y+dv)(2d+du) -(dy + dv)(2dy + dv) -(dy + dv)(2t + dw) (dy + v)(2 + (dy + dv) (2y+dv) (dy + dv) (2 +dw)
-(t+dw)(2d+diu) -{ +dw)(2dy+dv) -{, +dw)(2c+c+dw) (c+ dw)(2d + d) ( + dw)(2dy + dv) ( + dw)(2 + dw)
(3-32)










Eqn. 3-33 can be deduced from Eqns. 3-25 and 3-27.

{RE} = [KE{d,} (3-33)


-dx

t-At+AEAE d
RE= L dx (3-34)

dy
dz








Equation of Motion


The above portion calculated the strain energy portion of Eqn. 3-4. It is

now necessary to consider the other terms in Eqn. 3-4. The kinetic energy term is

the first term in Eqn. 3-4. This is in terms of the variable q, defined in Eqn. 3-35

below.

qs = gs + fi (3-35)



q, = distributed load along the element.

gs = gravity load due to self weight.

fi = inertial forces.

Rewriting the first term in Eqn. 3-7 yields Eqn. 3-36.










L L
gs-6K = fJwsN]Thds fms[NT [N]{dn}ds (3-36)
0 0



ws = cable self weight per unit length.

m, = cable mass per unit length.

8K = kinetic energy due to inertia forces.

{ = nodal acceleration.


h, is an index array based on the coordinate system adopted. It is used to

determine how the weight of the element is added to the structure.

The self weight acts in the negative z direction.

h- ={0 0 -1}


2 0 0 -1 0 0 iui
0 2 0 0 -1 0
SK = L 0 0 2 0 0 -1 w i (3-37)
6g -1 0 0 2 0 0 iu
0 -1 0 0 2 0 v
0 0 -1 0 0 2 w


g = acceleration due to gravity.










0
0
ws L -1
s 2 o (3-38)
0
-1










The second term in Eqn. 3-7 and Eqn. 3-4 is the applied nodal load.


f2

FI f (3-39)

f5








This is the computed nodal load due to dead weight of cladding and/or traffic

signal heads and/or wind load. Since the cable element is a two node element, the

nodal distribution of a body load is an even split between the two adjacent nodes.

The third term in Eqn. 3-7 is the damping term. This is obtained as discussed in

chapter 2. The equation of motion for the cable element is therefore given as

shown in Eqn. 3-40.










[MId}+[c]{Jd}+[K]{Ad} = {Fj}+{gs}-RE} (3-40)



The solution to Eqn. 3-40 above can be obtained by a myriad of numerical

integration schemes mentioned in chapter 2. When the cable element is assembled

with the other element types, i.e. beams and connectors The RE term in Eqn. 3-40

above can be replaced by a similar term FE. Where FE is the initial stress vector for

all the elements in the structure.



[M]( [c](J }+ [K]{A dn = {Fj}+{gs,-( E (3-41)


It should be noted that the self weight vector remains the same at the beginning of

the time and at the end. The change in force due to the self weight is zero. When

the wind load is considered the nodal force vector would include the wind force.

At the beginning of each time step the incremental displacement vector is zero and

the velocity and acceleration vector are as obtained from the previous time step

analysis. The initial conditions of the dynamic analysis is obtained from setting

both the displacement and velocity vectors to zero and obtaining the acceleration

from equilibrium of the structure.

{ \ + g, }- { E (3-42)
[M I










The use of the incremental displacement facilitates the use of implicit

numerical integration schemes for solution to Eqn. 3-41. Chapter 2 discusses the

Newmark Beta method. This method in its modified form was adopted by the

writer. The modification adopted is presented in the following section. This

modification is referred to as the alpha modification.



Modified Newmark Beta Method


Due to the inherent problem of "blow up" in the analytically unstable

traffic signal cable system and the fact that the unconditionally stable Newmark

Beta method possesses no intrinsic numerical damping, Wood et al [20] proposed

the use of an alpha factor. This factor introduces a positive artificial damping.

This modified Newmark's method with positive artificial damping of higher

vibrational modes is observed not only to reduce the "blow up" effect but also

reduce the cumulative error which inevitably occurs in a numerical integration

procedure.

Applying the alpha modification to Eqn. 3-40 results in an averaging effect

of the inertia forces at the beginning and at the end of the time step.

(I-a)[MuIj{ }+(a)[M]{ } + [C]{J } +[K]{Adn = (t+A
(3-43)










P} = {F I + Ig -IF (3-44)


Note that if a is zero Eqn. 3-43 reduces to Eqn. 3-40. Apply Newmark's method

based on Eqns. 3-45 and 3-46 to Eqn. 3-44. For convenience the nodal

designation is dropped for Eqn. 3-45 to Eqn 3-50. It should however be

understood that these are nodal displacement, velocity and acceleration.

dt+At = dt+[(l-y)At]dt +(yAt)d;+At (3-45)



dt + At =dt + (A t)t + [(2- )(A t)2 t + [ (A t)2 t + At (3-46)


Solving Eqn. 3-46 for acceleration at time t + At yields Eqn. 3-47.

d -1 -Ad t I (3-47)
t+At At2 ftAt2 t


Substituting Eqn. 3-47 into 3-45 yields Eqn. 3-48.

t+At + = A dt- I d + At Jd (3-48)



Following the steps described in chapter 2 and substituting Eqn. 3-47 and 3-48

into Eqn. 3-43, the final expressions for the effective stiffness and load vectors are

obtained.








79

(,-) -Ad2 d -1d +a[M] t +[CJ Y Adt -L) Id(t +At -

+ [KAdj = P+A}
(3-49)

Note that

{A d,}= {d l, }- {d,} (3-50)


Collecting terms and rearranging Eqn. 3-49 gives Eqn. 3-51.

[i]{Adn = [AP (3-51)


([K = (1 a)[Ml( + ([C + [K] (3-52)
PAt2 pAt





(3-53)



When a is zero the above described Newmark's method reduces to that described

in chapter 2. The values of P and y are as described in chapter 2. If the alpha

modification is to be used, the values of the constants are given below.

When P = 4 and y = /2 for the regular Newmark's method, the constant

acceleration method is obtained. This method is a second order solution with zero

artificial damping and unconditional stability. The intent of the alpha factor is to

insert positive artificial damping while keeping the property of unconditional










stability. For a second order method with positive artificial damping and

unconditional stability the following condition must be satisfied.

l-> > & a<- & a+y>- (3-54)
2 4 2 2


Wood et al [18] in their paper suggest the use of the following as giving the best

results.

a =0.1 = 0.3025 y = 0.6

a = -0.1 =0.5 y = 0.6

a =-0.1 = 0.3025 y = 0.6

The writer suggests the use of a = 0.1 P = 0.3025 y = 0.6 for the traffic

signal cable system as it yields the best artificial damping effect for the system.

When the incremental displacement is obtained from Eqn. 3-51, in order to

maintain dynamic equilibrium, this computed incremental displacement needs to

be refined. The reasons for the need of refinement are as follows;

a) The stiffness matrix at time t + At is dependent on the recently

computed displacements values.

b) The { FE } term in the equation of motion contain values dependent

on the nonlinear cable stiffness and nonlinear connector stiffness.

The Newton-Raphson method is adopted to refine the computed

incremental displacements. Application of the Newton Raphson is discussed

next.










Modified Newton Raphson Method


A graphical representation of the steps involved in the modified Newton

Raphson method is given in Fig. 3.2. There are two additional modifications

made to the regular non-recursive Newton Raphson procedure [9], [14]. One

has to do with the nonlinear loading and the other the nonlinear stiffness of the

cable elements. Note that both the loading and the stiffness are dependent on the

incremental displacements being computed. The writer has taken these into

account in modifying the Newton Raphson procedure adopted.

Recall from Eqn. 3-41 that,

{P} = {F,} {RE} (3-55)





{Fj = Applied nodal loads. These consist of dead weight of cladding

and/or wind loads applied at the nodal points. Discussion of

loading on the traffic signal cable system is presented in chapter 4.

{RE} = Equivalent nodal force vector for member stress at time t. This

accounts for the stresses in the structure at the beginning of the time

step.








82















Force


(K1)se K
ec K,


AR K_
K
A -- I0




-( I
(K2 )secI I
4 Ise

Ad Ad
Ad n Ad2

Ad
S Displacement


Fig. 3.2 Newton Raphson Iteration













The complete procedure to refine Ado shown in Fig. 3.2 on the previous

page is given to illustrate the modified Newton-Raphson method :

1. Initialize data.

K =[i] From Eqn (3-52)
P = PI From Eqn(3-53)
AR = P
o
Ad = K -1AR From Eqn(3-51)




In computing the above initial values, AP is assumed to be zero in Eqn.

3-53. It is assumed that if At is taken small enough the approximation error

introduced by this assumption is negligible.

2. Compute the instantaneous cable stiffness at a displacement Ado and set it

equal to k1.


3. The incremental resisting force:

A 1 = (KI)sec A do (3-56)


Where the secant stiffness (K1)s can be approximated by Eqn. 3-57.


(K 0 + i (3-57)
() sec 2











AR + Ad
.. AF o 21
1 2


(3-58)


4. The residual force at the end of the first iteration is as shown in Eqn. 3-59

AR = P AF (3-59)


5. Adl is computed as in step 1.


(3-60)


Adl = KAl-R1


The sign of the displacement computed in Eqn. 3-60 is negative because

the

residual force is negative.

6. Repeat step 2, compute k2.


7. Repeat step 3, compute new resisting force.


. AF =
2


io(Ado+Ad)+ k2 Ado + Ad,)


(3-61)


8. Repeat step 4, compute the residual force.

A 2 = P AF2


9. Using Eqn. 3-60 above compute Ad2. The whole procedure is continued

until the residual force is smaller than some tolerance value. The true










displacement, therefore, is as given in Eqn. 3-62.

Adn = Ad + Ad + Ad2 (3-62)



There are many convergence checks that could be used. Some authors

suggest a check on subsequent displacement computations; that is

Ad < d i = 0,1,2 ... (3-63)
Ad
n


where S is a prescribed tolerance. The problem faced when convergence is strictly

based on the convergence of the nodal displacement is that of different rates of

convergence for rotational displacements and translational displacements. The

units and range of values for rotations differ greatly compared to translations. As

a result the translational displacements may converge while the rotational

displacements may not simultaneously converge. The writer has chosen a

convergence criterion based on the work done by the residual forces.


T
{A_ R A id= 0,1,2... (3-64)
{AP} Adn





The use of the modified Newmark's method and the above described Newton-

Raphson method in the entire solution procedure is described next.












Solution Algorithm



Based on the theory developed in the previous sections of this chapter the

solution to the equation of motion given in Eqn. 3-40 was obtained as follows.



1. Compute the effective load vector as given in Eqn. 3-53. Note that AP

is given by the right hand side of Eqn. 3-41. For the first time step analysis

AP is taken to be zero. At the end of each time step P,+ t is computed and

the actual change in load, AP, for that time step is obtained. This change in

load is used as the estimate for the next time step analysis. The

acceleration at the beginning of each time step is, however, obtained by

imposing total equilibrium at the end of the previous time step and using

the true load in the equilibrium equation.



2. Compute the effective stiffness matrix using Eqn. 3-52. The initial

effective stiffness matrix at time t = 0, uses [K] = [KE] + [KA],

where [KE] and [KA] are as defined in Eqn. 3-25 and 3-28. At subsequent

times, the effective stiffness matrix uses [K] = [KE] + [KJ. The nonlinear

stiffness [KJ is computed as given in Eqn. 3-26.









3. Solve for change in displacements.

{Ad} = [1 ]t-l { }, (3-65)



4. The Newton Raphson iteration scheme is used to refine the incremental

displacement obtained in Eqn. 3-65. The Newton-Raphson procedure

gives a better estimate of { Ad }. This is done as shown in steps 1 9 on

pages 83 85.

5. Compute the final condition at the end of the time step.

(d} = {d" Ad (3-66)


dt + A-t A t + 2
ntAt = #A + ( )nU t 2p ni
(3-67)

{P}t + At a[M]{d } [C](n }+ A [K]{Adn t
{dnh+At (1-a)[M]


(3-68)


6. Set the incremental displacement of the next time step as follows and repeat

the above steps

Ad n= O0








88

The dynamic time step analysis is concluded when the free-vibration response dies

down. The complete analysis gives the full dynamic response history for the

transient dynamic loads applied and/or the equivalent static solution of the dead

load applied. For the static equivalent solution the "die-down" criteria is as

follows.


[K ]{A d}2 [K]{Ad,2 (3-69)
(3-69)
p- q- p





This means that the strain energy of the structure is averaged over several time

steps and compared to the average of the next several time steps. When the

difference between the two strain energies computed is negligible, die-down is

achieved.

1, p and q are beginning and ending time step cycles for strain energy computation.




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REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E1Z491VED_RXKO2Z INGEST_TIME 2013-10-24T18:22:16Z PACKAGE AA00017648_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES